\documentclass[final,reqno]{amsart} \usepackage{amsaddr} \usepackage{amssymb,amsfonts,amsmath,amsthm} \usepackage{bm} \usepackage{enumerate} \usepackage{physics} \usepackage{bbold} \usepackage{tikz} \usetikzlibrary{cd} \usepackage{accents} \usepackage{mathrsfs} \usepackage{xcolor} \usepackage{hyperref} \hypersetup{ colorlinks=true, allcolors=blue } \usepackage[capitalise]{cleveref} \DeclareMathOperator{\ran}{ran} \let\Span\relax % replaces \undef \DeclareMathOperator{\Span}{Span} \let\div\relax \DeclareMathOperator{\div}{div} \DeclareMathOperator{\supp}{supp} \DeclareRobustCommand\bar[1]{\accentset{\rule{.4em}{.3pt}}{#1}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \crefname{lemma}{Lemma}{Lemmas} \Crefname{lemma}{Lemma}{Lemmas} \newtheorem{corollary}[theorem]{Corollary} \crefname{corollary}{Corollary}{Corollaries} \Crefname{corollary}{Corollary}{Corollaries} \newtheorem{proposition}[theorem]{Proposition} \crefname{proposition}{Proposition}{Propositions} \Crefname{proposition}{Proposition}{Propositions} \crefname{equation}{}{} \theoremstyle{remark} \newtheorem{remark}{Remark} \crefname{remark}{Remark}{Remarks} \Crefname{remark}{Remark}{Remarks} \newcommand{\RR}{\mathbb{R}} \newcommand{\CC}{\mathbb{C}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\NN}{\mathbb{N}} \newcommand{\wconv}{\rightharpoonup} \newcommand{\iden}{\mathbb{1}} \renewcommand{\i}{\mathrm{i}} \newcommand{\dua}[2]{\langle #1, #2 \rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\wt}[1]{\widetilde{#1}} \newcommand{\HC}{\mathcal{H}} \newcommand{\NC}{\mathcal{N}} \newcommand{\SC}{\mathcal{S}} \newcommand{\SymG}{S} \newcommand{\AF}{\bm{\mathfrak{A}}} \newcommand{\BF}{\bm{\mathfrak{B}}} \newcommand{\EF}{\bm{\mathfrak{E}}} \newcommand{\Af}{\mathfrak{A}} \newcommand{\Ef}{\mathfrak{E}} \newcommand{\vjp}{\mathbf{j}^{\mathrm{p}}} \newcommand{\vjph}{\mathbf{j}^{\mathrm{ph}}} \newcommand{\vjc}{\mathbf{j}^{\mathrm{c}}} \newcommand{\vjcmodel}[1]{\mathbf{j}^{\mathrm{c},#1}} \newcommand{\vjcdip}{\mathbf{j}^{\mathrm{c},\dip}} \newcommand{\LL}{\mathrm{PFLL}} \newcommand{\LI}{\mathrm{PFL}} \newcommand{\VR}{\mathrm{VR}} \newcommand{\PF}{\mathrm{PF}} \newcommand{\ph}{\mathrm{ph}} \newcommand{\el}{\mathrm{el}} \newcommand{\rad}{\mathrm{rad}} \newcommand{\dip}{\mathrm{d}} \newcommand{\fin}{\mathrm{fin}} \newcommand{\sol}{\mathrm{sol}} \newcommand{\rdens}{\mathcal{I}} \newcommand{\rcdens}{\mathcal{J}} \newcommand{\Fock}{\mathcal{F}} \newcommand{\Hil}{\mathfrak{H}} \newcommand{\dm}{\mathcal{D}} \let\phi=\varphi \newcommand{\vsigma}{{\bm{\sigma}}} \newcommand{\vzeta}{{\bm{\zeta}}} \newcommand{\vxi}{{\bm{\xi}}} \newcommand{\vgamma}{{\bm{\gamma}}} \newcommand{\vepsilon}{{\bm{\epsilon}}} \newcommand{\vomega}{{\bm{\omega}}} \newcommand{\vnu}{{\bm{\nu}}} \newcommand{\vtheta}{{\bm{\theta}}} \newcommand{\vGamma}{{\bm{\Gamma}}} \renewcommand{\va}{\mathbf{a}} \newcommand{\ve}{\mathbf{e}} \renewcommand{\vu}{\mathbf{u}} \newcommand{\vz}{\mathbf{z}} \newcommand{\vq}{\mathbf{q}} \newcommand{\vw}{\mathbf{w}} \newcommand{\vy}{\mathbf{y}} \newcommand{\vc}{\mathbf{c}} \newcommand{\vk}{\mathbf{k}} \newcommand{\vx}{\mathbf{x}} \newcommand{\lapl}{\bm{\Delta}} \newcommand{\vt}{\mathbf{t}} \newcommand{\vG}{\mathbf{G}} \newcommand{\vv}{\mathbf{v}} \newcommand{\vj}{\mathbf{j}} \newcommand{\vf}{\mathbf{f}} \newcommand{\vg}{\mathbf{g}} \newcommand{\vT}{\mathbf{T}} \newcommand{\vK}{\mathbf{K}} \newcommand{\vM}{{\mathbf{M}}} \newcommand{\vA}{{\mathbf{A}}} \newcommand{\vP}{{\mathbf{P}}} \newcommand{\vX}{{\mathbf{X}}} \newcommand{\vZ}{{\mathbf{Z}}} \newcommand{\vD}{{\mathbf{D}}} \newcommand{\vR}{{\mathbf{R}}} \newcommand{\vY}{{\mathbf{Y}}} \let\vb\relax \newcommand{\vb}{{\mathbf{b}}} \newcommand{\wh}[1]{\widehat{#1}} \renewcommand{\epsilon}{\varepsilon} \title[Density functionals for the Pauli--Fierz Hamiltonian]{Density functionals for the\\ Pauli--Fierz Hamiltonian} \author[M.\,A. Csirik, A. Laestadius, M. Penz, and E.\,I. Tellgren]{Mih\'aly~A.~Csirik$^{1,*}$, Andre~Laestadius$^{1,2,3}$, Markus~Penz$^{3,4,5}$, \and Erik~I.~Tellgren$^{2}$} \address{$^1$Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, G\"oteborg, Sweden} \address{$^2$Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, Norway} \address{$^3$Department of Computer Science, Oslo Metropolitan University, Norway} \address{$^4$Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universit\"at M\"unchen, Germany} \address{$^5$Max Planck Institute for the Structure and Dynamics of Matter and Center for Free-Electron Laser Science, Hamburg, Germany} \thanks{$^*$Corresponding author, email: \href{mailto:csirik@gmail.com}{\texttt{csirik@gmail.com}}} \subjclass[2020]{Primary 81V10; Secondary 81V74, 81V55, 46N50} \keywords{Density-functional theory, Pauli--Fierz Hamiltonian, nonrelativistic quantum electrodynamics, current-density-functional theory, $N$-representability, Lieb functional} \begin{document} \begin{abstract} We develop a density-functional theory for the Pauli--Fierz Hamiltonian of nonrelativistic quantum electrodynamics in the spirit of the convex-analytic formulation of Lieb. We consider finitely many electrons, subject to external scalar and vector potentials, coupled to the full continuum of massless transverse photons with an ultraviolet cutoff. The natural basic variables are the standard electron density and a combined current density, the sum of the paramagnetic current density and a field-induced current density generated by the quantized vector potential. We prove representability results for these density--current pairs and show that the mixed-state constrained-search functional attains its infimum and coincides with the Lieb functional obtained by convex conjugation. Since arbitrarily soft photons purify mixed states at a negligible energy cost without changing their density and combined current density, the pure-state Levy--Lieb functional equals the Lieb functional and is therefore convex, even though its infimum need not be attained. In addition, we establish the zero-coupling limit, which recovers the mixed-state Vignale--Rasolt functional of current-density-functional theory. Finally, we consider the long-wavelength (dipole) limit and show that the limiting functional can be written as a constrained search for the well-known length-gauge dipole Hamiltonian. \end{abstract} \maketitle \clearpage \enlargethispage{\baselineskip} \tableofcontents \clearpage \section{Introduction}\label{sec-intro} Strong coupling of molecules and solids to the quantized electromagnetic field of an optical cavity has been reported to modify chemical and material properties~\cite{ebbesen2016hybrid,Basov2020,garciavidal2021manipulating,Bloch2022}. At the energy scales of chemistry and condensed-matter physics, such effects are described by nonrelativistic quantum electrodynamics (QED) in the form of the Pauli--Fierz Hamiltonian~\cite{pauli1938theorie,spohn2004dynamics}, in which electrons are minimally coupled to the quantized transverse vector potential. To treat realistic systems, density-functional theory (DFT) has been extended to this setting as quantum-electrodynamical DFT (QEDFT)~\cite{tokatly2013time,ruggenthaler2014quantum,Ruggenthaler2017}. See \cite{flick2015kohn,schafer2021making} for approximate functionals and \cite{ruggenthaler2018quantum,ruggenthaler2023understanding} for reviews. While the Pauli--Fierz Hamiltonian itself is well understood mathematically~\cite{hiroshima2002selfadjoint,hasler2008selfadjoint,bach1999spectral,griesemer2001ground,lieb2003existence}, the density-functional side has been analyzed in mathematical detail only for quantum-optical model systems such as the Rabi and Dicke models~\cite{Novokreschenov2023,bakkestuen2025dicke,bakkestuen2025rabi}, in which the system is reduced to two-level systems coupled to finitely many field modes. The mathematical structure of DFT was clarified by Lieb~\cite{Lieb1983}, who placed Levy's constrained search~\cite{levy1979universal} into the framework of convex analysis (see~\cite{lewin2023universal,penz2023structureI} for recent accounts). In the presence of magnetic fields, the density is supplemented by the paramagnetic current density $\vjp$, which couples linearly to the external vector potential $\vA$ in the current-density-functional theory (CDFT) of Vignale and Rasolt~\cite{vignale1987density,vignale1988current}. Absorbing the diamagnetic term $\frac12|\vA|^2$ into the scalar potential makes the ground-state energy jointly concave in the potentials~\cite{tellgren2012choice}, which also leads to a convex formulation of CDFT~\cite{penz2023structureII}. Several cornerstones of DFT are more delicate here: the density--current pair does not in general determine the potentials uniquely~\cite{capelle2002nonuniqueness,laestadius2014hohenberg}. Furthermore, only sufficient conditions for $N$-representability are known, and it is open whether pure- and mixed-state $N$-representability coincide~\cite{lieb2013current,tellgren2014fermion,csirik2024thermodynamic}. Lower semicontinuity of the (mixed-state) Vignale--Rasolt functional was proved only recently~\cite{kvaal2021lower}. In this article, we develop density-functional theory for the Pauli--Fierz Hamiltonian itself, in the spirit of Lieb's analysis. We consider $N$ spinless electrons in $\RR^3$ coupled to the full continuum of massless transverse photons, with a fixed ultraviolet cutoff but no infrared cutoff, in an external scalar potential $v$ and an external classical vector potential $\vA$. The photon mode $\vk$ couples proportionally to $e\wh\chi(\vk)$, where $e\geq0$ is the coupling constant and the form factor $\wh\chi$ is any function satisfying \cref{cutoff-assumption}, such as the hard cutoff $\iden_{|\vk|\leq\Lambda}/\sqrt{|\vk|}$. We indicate these choices by the superscripts $e,\chi$. A new feature compared with CDFT is that a finite-energy state may contain arbitrarily many photons of arbitrarily small energy. This infrared freedom causes the main technical difficulties, but it is also behind some of the most striking results. \emph{The combined current density.} For a finite-energy state $\Psi$, \begin{equation*} \dua{\Psi}{H^{v,\vA}\Psi}=\dua{\Psi}{H^{0,0}\Psi} +\dua{v+\tfrac12|\vA|^2}{\rho_\Psi}+\dua{\vA}{\vjc_\Psi}, \qquad \vjc_\Psi=\vjp_\Psi+\vjph_\Psi, \end{equation*} where $H^{0,0}$ is the Pauli--Fierz Hamiltonian without external potentials \eqref{q00-def}, $\rho_\Psi$ is the electronic density, and $\vjph_\Psi(\vx)=-eN\Re\dua{\Psi}{\AF(\vx)\Psi}_{\vx}$ is the \emph{field-induced current density}, that is, $-e$ times the density-weighted mean of the quantized vector potential $\AF$ at the position $\vx$ of an electron, the partial inner product being taken in all other variables. Thus the external potentials see a state only through $\rho_\Psi$ and the \emph{combined current density} $\vjc_\Psi$, and these are our basic variables. The combined current density obeys a kinetic-energy bound similar to that for $\vjp$ (\cref{prop-current-bounds}), but although $\div\AF=0$, its field-induced part need not be divergence-free for entangled electron--photon states (\cref{prop-div-jph}). Ground states carry the magnetostatic fields of their own physical current density $\vjc+\rho\vA$ (\cref{prop-ground-state-fields}). \emph{Representability.} The photon field purifies the electrons: the pairs $(\rho_\Psi,\vjp_\Psi)$ of pure electron--photon states with finite energy are exactly those of electronic \emph{mixed} states with finite kinetic energy (\cref{thm-photon-purification}). For $e>0$, coherent photon displacements depending on the electron positions produce field-induced currents $-e\rho\va$, where the vector field $\va$ need not be transverse. Hence, every $(\rho,\rho\vv)$ with $\int_{\RR^3}\rho=N$, $\sqrt\rho\in H^1(\RR^3)$ and $\int_{\RR^3}\rho(|\vv|^2+|\grad\vv|^2)<\infty$ is a pure-state $N$-representable density--combined-current pair (\cref{thm-photonic-representability,cor-photonic-combined-class}). \emph{Universal functionals.} The Pauli--Fierz Levy--Lieb (PFLL) functional is the pure-state constrained search \begin{equation*} F_\LL^{e,\chi}(\rho,\vjc) =\inf_{\substack{\|\Psi\|=1\\ \rho_\Psi=\rho\\ \vjc_\Psi=\vjc}} \dua{\Psi}{H^{0,0}\Psi}, \end{equation*} and the Pauli--Fierz Lieb (PFL) functional $F_\LI^{e,\chi}(\rho,\vjc)$ is the convex conjugate of the ground-state energy, reparametrized as in~\cite{tellgren2012choice}. As in Lieb's theory, it is the lower semicontinuous convex envelope of $F_\LL^{e,\chi}(\rho,\vjc)$. Our first main result (\cref{thm-unrestricted-attainment}) is the counterpart of~\cite{Lieb1983,kvaal2021lower}: the mixed-state constrained-search functional is lower semicontinuous, equals $F_\LI^{e,\chi}(\rho,\vjc)$, and attains its infimum at a density matrix of finite mean photon number. Since an energy bound does not control the photon number, the proof removes soft photons and restores the combined current density exactly, at a controlled energy cost. Our second main result (\cref{thm-many-electron-soft-purification}) points in the opposite direction and sets QEDFT apart from DFT or CDFT: soft photons purify any finite-energy mixed state at arbitrarily small energy cost without changing $(\rho,\vjc)$, so that \[ F_\LL^{e,\chi}(\rho,\vjc)=F_\LI^{e,\chi}(\rho,\vjc). \] Hence, unlike the Levy--Lieb functional of electronic DFT, which is not convex~\cite{Lieb1983}, the PFLL functional is convex and lower semicontinuous, and pure- and mixed-state $N$-representability coincide (\cref{cor-many-electron-pure-functional}). The price to pay is that the pure-state infimum is in general not attained. This is true already for one electron and at every coupling (\cref{prop-pure-nonattainment}). Optimizers need not be ground states, and in \cref{sec-optimizer-properties} we discuss which ground-state properties carry over to optimizers. Mixed-state optimizers contain few soft photons: for the hard cutoff, the expected number and energy of photons with frequency below $\kappa$ are $O(\kappa)$ and $O(\kappa^2)$, respectively, so the probability of finding any such photon vanishes as $\kappa\downarrow0$ (\cref{prop-optimizer-soft-spectrum}). \emph{Zero-coupling and dipole limits.} At $e=0$, the photons still purify electronic mixed states, so that $F_\LL^{0,\chi}(\rho,\vjp)=F_\LI^{0,\chi}(\rho,\vjp)=F_\VR(\rho,\vjp)$, the mixed-state Vignale--Rasolt functional (\cref{thm-soft-photon-relaxation}), and $F_\LL^{e,\chi}$ converges to $F_\VR$ pointwise and in the sense of Mosco as $e\downarrow0$ (\cref{thm-zero-coupling-convergence}). The long-wavelength (dipole) approximation, ubiquitous in practical QEDFT~\cite{flick2015kohn,rokaj2018lightmatter,schafer2020relevance}, applies when the system size is small compared to the wavelength of the relevant photon modes, so that the vector potential is effectively uniform across the system. We realize this limit by instead rescaling the system: let $F_{\LI,\varepsilon}$ be the PFL functional of the model in which $\AF(\vx)$ is replaced by $\AF(\varepsilon\vx)$, so that $\varepsilon=1$ is the original model and $\varepsilon$ plays the role of the ratio of system size to wavelength. As $\varepsilon\downarrow0$, these functionals converge in the sense of Mosco to the dipole functional $F_\dip$ (\cref{thm-dipole-limit}), whose current argument $\vjcdip$ is the combined current density formed with the spatially uniform field $\AF(0)$. In the length gauge, $F_\dip$ is the constrained search for the usual dipole Hamiltonian (\cref{cor-dipole-length-gauge}), and \cref{prop-dipole-electronic-comparison} gives $F_\dip\to F_\VR$ as $e\downarrow0$. For fixed $N$ and form factor, these relations are summarized in the diagram \begin{equation*} \begin{tikzcd}[column sep=large,row sep=large] F_\LL^{e,\chi}(\rho,\vjc)=F_\LI^{e,\chi}(\rho,\vjc) \arrow[r,"\vjc=0"] \arrow[d,"\text{dipole, }\varepsilon\downarrow0"] \arrow[dd,bend right=60,"e\downarrow0"'] & F_\LL^{e,\chi}(\rho,0)=F_\LI^{e,\chi}(\rho,0) \arrow[dd,"e\downarrow0"] \\ F_\dip(\rho,\vjcdip) \arrow[d,"e\downarrow0"] & \\ F_\LI^{0,\chi}(\rho,\vjp)=F_\VR(\rho,\vjp) \arrow[r,"\vjp=0"] & F_\LI^{0,\chi}(\rho,0)=F_{\mathrm L}(\rho) \end{tikzcd} \end{equation*} where $F_{\mathrm L}$ is the Lieb functional~\cite{Lieb1983}. The arrow $\varepsilon\downarrow0$ denotes Mosco convergence, and the arrows $e\downarrow0$ denote pointwise convergence, which for the curved arrow also holds in the Mosco sense. \emph{Organization of the paper.} \Cref{sec-prelim,sec-nrqed} introduce the Fock-space setting and the Pauli--Fierz Hamiltonian. The main results are stated in \cref{sec-main-results} and proved in \cref{sec-proofs}. The appendices collect transformation properties of the model (dilations, gauge covariance, Weyl displacements, and the length-gauge transformation, among others) and further technical tools. \section*{Acknowledgements} M.A.Cs., A.L., and M.P. acknowledge funding from the European Research Council through ERC-2021-STG Grant Agreement No.~101041487 (REGAL). A.L. acknowledges support through QSTAR (Centre for Quantum Computing and Applications), funded by the Research Council of Norway (RCN) under project No. 361617. A.L. and E.I.T. also acknowledge support through the Hylleraas Centre for Quantum Molecular Sciences, funded by the RCN under its Centres of Excellence scheme, project No. 262695. \section*{Disclosure of AI use and Lean verification} The authors made substantial use of the large language models Claude Opus 5.5 (Anthropic) and GPT 6 Astra (OpenAI) to edit the text, to generate and refine statements and proofs, to check arguments and find errors, and to write the Lean formalization described below. All output was reviewed and edited by the authors, who take full responsibility for the content. A Lean~4 formalization in the directory \texttt{gpt\_lean}\footnote{\url{https://github.com/xyzw/qedft_pf_pub}} verifies all new results except those on the long-wavelength limit (\cref{sec-dipole-limit}). Standard results not involving the Pauli--Fierz model enter as explicit hypotheses: the canonical commutation relations, Weyl operators, Fock-space factorization and the Wiener--It\^o--Segal isomorphism; normal and partial traces and closed quadratic forms; completed tensor powers; Sobolev representatives, a local Sobolev embedding, the Hardy and fractional Hardy inequalities, Rellich compactness for $H^{1/2}$ and density in $\dot H^{-1/2}$; and $L^p$ duality and the Fenchel--Moreau theorem. \clearpage \section{Preliminary notions}\label{sec-prelim} \subsection{\texorpdfstring{$N$-particle spaces and the bosonic Fock space}{N-particle spaces and the bosonic Fock space}}\label{sec-fock-space} Let $d\in\NN$ be the number of spatial dimensions, $N\in\NN$ the number of particles, and $q\in\NN$, $q\geq1$, the number of internal states (for example, $q=2$ for spin-$1/2$ particles or for the two transverse polarizations of a photon). We denote the fermionic (a) or bosonic (s) $N$-particle spaces by \[ \Hil^N_{a/s}=L_{a/s}^2((\RR^d\times\{1,\ldots,q\})^N), \] where $L^2_a$ (resp. $L^2_s$) is the Hilbert space of totally antisymmetric (resp. symmetric) square-integrable complex-valued functions both equipped with the inner product \begin{multline*} \dua{\phi}{\psi}_{\Hil^N} =\sum_{\sigma_1,\ldots,\sigma_N\in\{1,\ldots,q\}} \int_{\RR^{dN}} \ol{\phi(\vx_1,\sigma_1,\ldots,\vx_N,\sigma_N)}\\ {}\times\psi(\vx_1,\sigma_1,\ldots,\vx_N,\sigma_N) \,\dd\vx_1\ldots\dd\vx_N. \end{multline*} Total antisymmetry (resp. symmetry) is understood with respect to simultaneous permutations of the spatial and spin variables. In what follows, we will often aggregate the spatial variables as $\vX=(\vx_1,\ldots,\vx_N)$, and the spin variables as $\vsigma=(\sigma_1,\ldots,\sigma_N)$ and the spatial-spin variables as $\vZ=(\vz_1,\ldots,\vz_N)$, where $\vz_j=(\vx_j,\sigma_j)$. By abuse of notation, we write $\psi(\vX,\vsigma)=\psi(\vZ)$. Whenever necessary, we distinguish the number of particles $N$ in the aggregate vectors as $\vX^{(N)}$, $\vsigma^{(N)}$ and $\vZ^{(N)}$. We equip $\RR^d\times\{1,\ldots,q\}$ with the product of Lebesgue and counting measure and use the shorthand \[ \int_{\RR^d\times\{1,\ldots,q\}}h(\vz)\,\dd\vz :=\sum_{\sigma=1}^q\int_{\RR^d}h(\vx,\sigma)\,\dd\vx. \] Accordingly, for $\vz=(\vx,\sigma)$ and $\vz'=(\vx',\sigma')$, \[ \delta(\vz-\vz'):=\delta(\vx-\vx')\delta_{\sigma,\sigma'}. \] Electrons will be described by the antisymmetric space $\Hil_a^N$ with fixed $N$, and photons by the bosonic Fock space built from the symmetric spaces. The bosonic Fock space is defined as the infinite direct sum \[ \Fock_s=\Hil^0 \oplus \Hil^1 \oplus \Hil_s^2 \oplus \ldots,\quad\text{with the convention}\quad \Hil^0\equiv \CC, \] consisting of infinite tuples $\Psi=(\psi^0,\psi^1,\psi^2,\ldots)$, where $\psi^N\in \Hil^N_s$, such that \[ \|\Psi\|^2=\sum_{N\ge 0} \|\psi^N\|^2 <\infty. \] This space is separable, since $\Hil^1$ is. Equipped with the corresponding inner product, i.e., \[ \dua{\Phi}{\Psi}_\Fock=\sum_{N\ge 0} \dua{\phi^N}{\psi^N}_{\Hil^N}, \] the space $\Fock_s$ is a Hilbert space. Instead of the tuple notation, $\Psi=\psi^0\oplus\psi^1\oplus\psi^2\oplus\ldots$ is used sometimes. The canonically embedded subspaces $\Hil_s^N\subset\Fock_s$ are called the $N$-particle sectors. For any $N\in\NN$, the truncated Fock spaces are given by \[ \Fock_s^{\le N}=\Hil^0 \oplus \Hil^1 \oplus \Hil_s^2 \oplus \ldots \oplus \Hil_s^N. \] Moreover, the finite-particle vectors form the dense subspace \[ \Fock_s^\fin=\bigcup_{N\ge 1} \Fock_s^{\le N}. \] We denote by $\Omega$ the canonical vacuum vector $\Omega=1\oplus0\oplus0\oplus\ldots$ of $\Fock_s$. It is possible to symmetrically compose two elements $\psi_1\in\Hil_s^{N_1}$ and $\psi_2\in\Hil_s^{N_2}$ and obtain an element in $\Hil_s^{N_1+N_2}$, namely the symmetric tensor product \begin{multline*} \psi_1\vee\psi_2=(N_1!N_2!(N_1+N_2)!)^{-1/2} \sum_{\tau\in\SymG_{N_1+N_2}} \psi_1(\vz_{\tau(1)},\ldots,\vz_{\tau(N_1)})\\ {}\times\psi_2(\vz_{\tau(N_1+1)},\ldots, \vz_{\tau(N_1+N_2)}), \end{multline*} where $\SymG_N$ denotes the symmetric group on $\{1,\ldots,N\}$. Next, we define the creation and annihilation operators \cite[Section~5.2]{bratteli-robinson-2}. For any $f\in\Hil:=\Hil^1$, define the creation operator $a^\dag(f):\Fock_s^\fin\to\Fock_s^\fin$ sectorwise by \[ a^\dag(f)\psi=f\vee\psi \] for $\psi\in\Hil_s^N$, and extend the definition linearly to $\Fock_s^\fin$. Thus $a^\dag(f)\Hil_s^N\subset\Hil_s^{N+1}$. On the same finite-particle space, define the annihilation operator by \[ (a(f)\psi)(\vz_1,\ldots,\vz_{N-1}) =\sqrt{N}\int_{\RR^d\times\{1,\ldots,q\}} \ol{f(\vz)}\psi(\vz,\vz_1,\ldots,\vz_{N-1})\,\dd\vz, \qquad\psi\in\Hil_s^N, \] and extend it linearly to $\Fock_s^\fin$. We set $a(f)\Omega=0$, while $a^\dag(f)\Omega=f$. Moreover, \[ \dua{\Phi}{a^\dag(f)\Psi}_\Fock =\dua{a(f)\Phi}{\Psi}_\Fock, \qquad\Phi,\Psi\in\Fock_s^\fin. \] The canonical commutation relations (CCR) \begin{align*} [a(g),a^\dag(f)]&=\dua{g}{f}_\Hil \iden_{\Fock_s},\\ [a(g),a(f)]&=[a^\dag(g),a^\dag(f)]=0 \end{align*} hold for any $f,g\in\Hil$ on $\Fock_s^\fin$. These operators are closable, and we use the same symbols for their closures, which are mutual Hilbert-space adjoints. The Segal field operator \[ \boxed{\varPhi(f)=a^\dag(f)+a(f)} \] is essentially self-adjoint on $\Fock_s^\fin$, and we denote its closure by the same symbol. The \emph{conjugate} Segal field operator, also called the field-momentum operator, is \begin{equation}\label{conjugate-segal-field-def} \boxed{\varPi(f)=\varPhi(\i f)=\i\bigl(a^\dag(f)-a(f)\bigr).} \end{equation} It is likewise essentially self-adjoint on $\Fock_s^\fin$, and the same symbol denotes its closure. For the vacuum, $\|\varPhi(f)\Omega\|=\|\varPi(f)\Omega\|=\|f\|$. For $f\ne0$, the creation and annihilation operators are unbounded on $\Fock_s$, but their restrictions to truncated Fock spaces are bounded. More precisely, for $N\in\NN$, \begin{align*} a(f)&:\Fock_s^{\le N}\longrightarrow\Fock_s^{\le N-1}, &\|a(f)\Psi\|&\leq\sqrt{N}\,\|f\|\|\Psi\|,\\ a^\dag(f)&:\Fock_s^{\le N}\longrightarrow\Fock_s^{\le N+1}, &\|a^\dag(f)\Psi\|&\leq\sqrt{N+1}\,\|f\|\|\Psi\|. \end{align*} In contrast, the symbols $a(\vz)$ and $a^\dag(\vz)$ are operator-valued distributions, not operators defined at each fixed $\vz$. If $\psi^N\in\Hil_s^N$, then $\vz\mapsto a(\vz)\psi^N$ denotes the $\Hil_s^{N-1}$-valued $L^2$ function defined, for almost every $\vz$, by \[ (a(\vz)\psi^N)(\vz_1,\ldots,\vz_{N-1}) =\sqrt{N}\psi^N(\vz_1,\ldots,\vz_{N-1},\vz). \] In particular, \[ \int_{\RR^d\times\{1,\ldots,q\}} \|a(\vz)\psi^N\|^2\,\dd\vz =N\|\psi^N\|^2. \] Then, the creation operator is defined by the weak adjoint relation \[ \dua{\Phi}{a^\dag(\vz)\Psi}_\Fock :=\dua{a(\vz)\Phi}{\Psi}_\Fock, \qquad \Phi,\Psi\in\Fock_s^\fin. \] Let $\SC_{N+1}$ denote the normalized symmetrization operator, \[ (\SC_{N+1}F)(\vz_1,\ldots,\vz_{N+1}) =\frac{1}{(N+1)!}\sum_{\tau\in\SymG_{N+1}} F(\vz_{\tau(1)},\ldots,\vz_{\tau(N+1)}). \] Then, as an identity of distributions, \[ (a^\dag(\vz)\psi^N)(\vz_1,\ldots,\vz_{N+1}) =\sqrt{N+1}\,\SC_{N+1}\left[ \psi^N(\vz_1,\ldots,\vz_N)\delta(\vz_{N+1}-\vz) \right]. \] The smeared identities \begin{align*} a^\dag(f)&=\int_{\RR^d\times\{1,\ldots,q\}} f(\vz)a^\dag(\vz)\,\dd\vz,\\ a(f)&=\int_{\RR^d\times\{1,\ldots,q\}} \ol{f(\vz)}a(\vz)\,\dd\vz \end{align*} hold as quadratic forms on $\Fock_s^\fin$. In the same sense, without smearing, the point-field CCR are \begin{align*} [a(\vz),a^\dag(\vz')]&=\delta(\vz-\vz')\iden_{\Fock_s},\\ [a^\dag(\vz),a^\dag(\vz')]&=[a(\vz),a(\vz')]=0. \end{align*} Smearing the first identity against $\ol{g(\vz)}f(\vz')$ recovers the first CCR above. \subsection{Partial traces and reduced states} Let $\HC_1$ and $\HC_2$ be separable Hilbert spaces and let $\Gamma\in\mathfrak S_1(\HC_1\otimes\HC_2)$ be trace class. Here, $\mathfrak S_p$ is the Banach space of $p$-Schatten-class operators. The partial trace of $\Gamma$ over the second factor is the unique trace-class operator $\Tr_2\Gamma$ on $\HC_1$ characterized by \begin{equation}\label{partial-trace-duality} \Tr_{\HC_1}\big((\Tr_2\Gamma)B\big) =\Tr_{\HC_1\otimes\HC_2} \big(\Gamma(B\otimes\iden_{\HC_2})\big), \qquad \text{for all}\; B\in\mathcal B(\HC_1). \end{equation} Here $\mathcal B(\HC_1)$ denotes the bounded operators on $\HC_1$. Similarly, $\Tr_1\Gamma$ is characterized by testing against $\iden_{\HC_1}\otimes C$, with $C\in\mathcal B(\HC_2)$. If $(e_j)_{j\geq1}$ is any orthonormal basis of $\HC_2$, then \begin{equation*} \Tr_2\Gamma =\sum_{j\geq1} (\iden_{\HC_1}\otimes\langle e_j|)\,\Gamma\, (\iden_{\HC_1}\otimes|e_j\rangle), \end{equation*} where $\iden_{\HC_1}\otimes|e_j\rangle:\HC_1\to\HC_1\otimes\HC_2$ maps $\phi$ to $\phi\otimes e_j$, and $\iden_{\HC_1}\otimes\langle e_j|$ is its adjoint. The sum converges in trace norm and is independent of the chosen basis. Partial trace preserves positivity and the trace. Thus, if $\Gamma\geq0$ and $\Tr\Gamma=1$, then both partial traces are again density matrices. In the electron--photon system below, $\HC_1=\Hil_\el^N$ and $\HC_2=\Fock_\ph$. We denote density matrices on the full composite space by $\vGamma$, retaining unbolded symbols for reduced density matrices. The states \[ \Gamma_\el=\Tr_\ph\vGamma,\qquad \Gamma_\ph=\Tr_\el\vGamma \] are called the electronic and photon reduced states. When the total state is a wavefunction $\Psi$, the phrase \emph{trace out the photons} means first forming the rank-one density matrix $|\Psi\rangle\langle\Psi|$ and then taking $\Tr_\ph|\Psi\rangle\langle\Psi|$. For a normalized pure state, its Schmidt decomposition has the form \begin{equation}\label{Schmidt-decomposition} \Psi=\sum_{j=1}^{r}\sqrt{p_j}\,\Phi_j\otimes\Xi_j, \qquad p_j>0,\qquad\sum_{j=1}^{r}p_j=1, \end{equation} where $r\in\NN\cup\{\infty\}$ is the Schmidt rank, with $r\leq\min\{\dim\HC_1,\dim\HC_2\}$, and $(\Phi_j)_{j=1}^{r}$ and $(\Xi_j)_{j=1}^{r}$ are orthonormal families in $\HC_1$ and $\HC_2$, respectively. The two reduced states are then \begin{equation}\label{Schmidt-reduced-states} \Tr_2|\Psi\rangle\langle\Psi| =\sum_{j=1}^{r}p_j|\Phi_j\rangle\langle\Phi_j|, \qquad \Tr_1|\Psi\rangle\langle\Psi| =\sum_{j=1}^{r}p_j|\Xi_j\rangle\langle\Xi_j|. \end{equation} Thus they have the same nonzero eigenvalues, and either reduced state is pure exactly when $r=1$, equivalently when $\Psi$ is a product vector. More precisely, a pure marginal is itself a tensor factor of $\Psi$: for a unit vector $\Xi\in\HC_2$, \begin{equation}\label{pure-marginal-product} \Tr_1|\Psi\rangle\langle\Psi|=|\Xi\rangle\langle\Xi| \quad\Longrightarrow\quad \Psi=\Phi\otimes\Xi\ \text{ for a unit vector }\Phi\in\HC_1. \end{equation} Indeed, \cref{Schmidt-reduced-states} then forces $r=1$ and $|\Xi_1\rangle\langle\Xi_1|=|\Xi\rangle\langle\Xi|$, so $\Xi_1=e^{\i\theta}\Xi$ for some $\theta\in\RR$ and $\Psi=\Phi_1\otimes\Xi_1=(e^{\i\theta}\Phi_1)\otimes\Xi$. More generally, a normalized vector $\Psi\in\HC_1\otimes\HC_2$ is called a \emph{purification} of a density matrix $\gamma$ on $\HC_1$ if \[ \Tr_2|\Psi\rangle\langle\Psi|=\gamma. \] Such a purification in the fixed auxiliary space $\HC_2$ exists exactly when $\operatorname{rank}\gamma\leq\dim\HC_2$. Indeed, if $\gamma=\sum_{j=1}^{r}p_j|\Phi_j\rangle\langle\Phi_j|$, one may choose an orthonormal family $(\Xi_j)_{j=1}^{r}$ in $\HC_2$ and use \cref{Schmidt-decomposition}. Since $\Fock_\ph$ is infinite dimensional, every electronic density matrix admits such an abstract Hilbert-space purification. However, the purification might have infinite photon energy. For a positive trace-class $\Gamma$, the defining identity \cref{partial-trace-duality} also applies to positive unbounded observables by monotone approximation. In particular, \begin{equation}\label{partial-trace-positive-observables} \Tr\big((A\otimes\iden_{\HC_2})\Gamma\big) =\Tr\big(A\,\Tr_2\Gamma\big),\qquad A\geq0, \end{equation} where both sides are form traces. In particular, the left-hand side means \[ \Tr\!\left((\sqrt{A}\otimes\iden_{\HC_2})\Gamma (\sqrt{A}\otimes\iden_{\HC_2})\right). \] The analogous identity holds for an observable acting only on $\HC_2$. This is the sense in which finite electronic kinetic energy and finite photon energy pass to the corresponding reduced states. The electronic density and paramagnetic current density used below depend only on $\Gamma_\el$, while the free-field energy depends only on $\Gamma_\ph$. By contrast, the field-induced current density contains a joint electron-position--photon observable and is generally not determined by either reduced state separately. The pair of reduced states retains all expectations of observables acting on only one factor, but does not in general determine the electron--photon correlations needed for the field-induced current density. \subsection{Second quantization of operators}\label{sec-second-quantization} We briefly recall the second quantization of one-particle operators and refer to~\cite[Section~7]{solovej2014many} for details. Let $A$ be a self-adjoint operator on $\Hil$. Its $N$-particle lift $A^{(N)}$ is the closure of $\sum_{j=1}^NA_{\vz_j}$, initially defined on the algebraic symmetric tensor product of $N$ copies of $D(A)$, where $A_{\vz_j}$ acts on the $j$th variable. It is self-adjoint on $\Hil_s^N$, and we set $A^{(0)}=0$. The second quantization of $A$ is the self-adjoint operator \[ \dd\varGamma(A)=\bigoplus_{N\geq0}A^{(N)} \] on $\Fock_s$, with domain consisting of all $\Psi=\bigoplus_N\psi^N$ such that $\psi^N\in D(A^{(N)})$ and $\sum_N\|A^{(N)}\psi^N\|^2<\infty$. If $(f_j)_{j\geq1}\subset D(A)$ is an orthonormal basis of $\Hil$, then \[ \dd\varGamma(A)=\sum_{j,k\geq1}\dua{f_j}{Af_k}\,a^\dag(f_j)a(f_k) \] in the sense of quadratic forms on finite-particle vectors~\cite[Lemma~7.8]{solovej2014many}. In particular, the number operator $\NC=\dd\varGamma(\iden_\Hil)=\sum_{j\geq1}a^\dag(f_j)a(f_j)$ acts as multiplication by $N$ on $\Hil_s^N$. For a tuple $\boldsymbol A=(A_1,\ldots,A_m)$ of strongly commuting self-adjoint operators and $\boldsymbol\alpha\in\RR^m$, we write $\dd\varGamma(\boldsymbol A)=(\dd\varGamma(A_1),\ldots,\dd\varGamma(A_m))$ and $\boldsymbol\alpha\cdot\dd\varGamma(\boldsymbol A) :=\dd\varGamma(\boldsymbol\alpha\cdot\boldsymbol A)$. The multiplicative second quantization of a unitary $U$ on $\Hil$ is the unitary operator $\varGamma(U)=\bigoplus_{N\geq0}U^{\otimes N}|_{\Hil_s^N}$ on $\Fock_s$, with $U^{\otimes0}=1$~\cite[Remark~7.10]{solovej2014many}. It satisfies \begin{gather*} \varGamma(U)a^\#(f)\varGamma(U)^\dag=a^\#(Uf),\qquad \varGamma(U)\dd\varGamma(A)\varGamma(U)^\dag=\dd\varGamma(UAU^\dag),\\ \varGamma(e^{-\i tA})=e^{-\i t\dd\varGamma(A)}, \end{gather*} where $a^\#$ stands for either $a^\dag$ or $a$. When no confusion can arise, we use the same symbol $U$ for $\varGamma(U)$. \section{Nonrelativistic quantum electrodynamics}\label{sec-nrqed} For the remainder of the paper we take $d=3$ and $q=1$ (spinless electrons), and write \[ \Hil_\el^N:=\Hil_a^N,\qquad \Fock_\ph:=\Fock_s(\Hil_\ph),\qquad \HC_N:=\Hil_\el^N\otimes\Fock_\ph, \] where the one-photon space $\Hil_\ph$ is defined in \cref{sec-one-photon-space} below. The restriction to spinless electrons is for notational simplicity only, and all results extend to the spinful case with the appropriate modifications. Writing $\Hil_\ph^{\otimes_sM}$ for the $M$-photon sector of $\Fock_\ph$, with $\Hil_\ph^{\otimes_s0}=\CC$ and $\Hil_\ph^{\otimes_s1}=\Hil_\ph$, every $\Psi\in\HC_N$ has the unique decomposition into $M$-photon components \begin{equation}\label{psifock} \Psi=\bigoplus_{M=0}^\infty\psi^{(M)},\qquad \psi^{(M)}\in\Hil_\el^N\otimes\Hil_\ph^{\otimes_sM},\qquad \sum_{M=0}^\infty\|\psi^{(M)}\|^2<\infty. \end{equation} We often view the elements of $\HC_N$ as Fock-space-valued functions of the electronic positions. The canonical unitary identification $L^2(\RR^{3N})\otimes\Fock_\ph\cong L^2(\RR^{3N};\Fock_\ph)$ maps $\HC_N$ onto the functions $\vX\mapsto\Psi(\vX)\in\Fock_\ph$ that are antisymmetric under permutations of $\vx_1,\ldots,\vx_N$, and \[ \|\Psi\|^2=\int_{\RR^{3N}}\|\Psi(\vX)\|_{\Fock_\ph}^2\dd\vX. \] Derivatives in the electronic positions are weak derivatives of such functions. For $\Psi\in L^2(\RR^{3N};\Fock_\ph)$, $1\leq n\leq N$ and $\mu=1,2,3$, we say that $\partial_{x_n^\mu}\Psi\in L^2(\RR^{3N};\Fock_\ph)$ if there is an element $\partial_{x_n^\mu}\Psi$ of $L^2(\RR^{3N};\Fock_\ph)$ such that \[ \int_{\RR^{3N}}\partial_{x_n^\mu}\Phi(\vX)\,\Psi(\vX)\dd\vX =-\int_{\RR^{3N}}\Phi(\vX)\,\partial_{x_n^\mu}\Psi(\vX)\dd\vX \qquad\text{for every }\Phi\in C_c^\infty(\RR^{3N}), \] where the integrals are Bochner integrals in $\Fock_\ph$. Pairing with a fixed $\Xi\in\Fock_\ph$ shows that this holds if and only if, for every $\Xi\in\Fock_\ph$, the scalar function $\vX\mapsto\dua{\Xi}{\Psi(\vX)}_{\Fock_\ph}$ has weak derivative $\vX\mapsto\dua{\Xi}{\partial_{x_n^\mu}\Psi(\vX)}_{\Fock_\ph}$. In terms of \cref{psifock}, the $M$-photon component of $\partial_{x_n^\mu}\Psi$ is the weak partial derivative of $\psi^{(M)}$, viewed as a function of the electronic and photon variables, with respect to $x_n^\mu$. We write $\grad_{\vx_n}\Psi=(\partial_{x_n^1}\Psi,\partial_{x_n^2}\Psi, \partial_{x_n^3}\Psi)$ and $\|\grad_{\vx_n}\Psi\|^2=\sum_{\mu=1}^3\|\partial_{x_n^\mu}\Psi\|^2$. \subsection{One-photon space}\label{sec-one-photon-space} Photons have two physical polarizations. One may either use the polarization representation \[ \Hil_\ph=L^2(\RR^3\times\{1,2\}) \] or, without choosing a polarization frame, the solenoidal representation \[ \Hil_\sol=\{\vf\in L^2(\RR^3,\CC^3):\vk\cdot\vf(\vk)=0 \text{ for a.a.\ }\vk\}. \] Choose measurable, real, orthonormal polarization vectors $\vepsilon_1,\vepsilon_2$ such that $\vepsilon_\lambda(\vk)\cdot\vk=0$ and use the identity \begin{equation}\label{polarization-projector} \sum_{\lambda=1}^2\vepsilon_\lambda(\vk)\otimes \vepsilon_\lambda(\vk) =\iden-\frac{\vk\otimes\vk}{|\vk|^2}=:P_\sol(\vk) \end{equation} to define the solenoidal projector $P_\sol$. All identities involving $P_\sol$ are understood almost everywhere, and its value at $\vk=0$ may be fixed arbitrarily. Pointwise multiplication by $P_\sol$ is an orthogonal projection on $L^2(\RR^3,\CC^3)$, and \[ \Hil_\sol=\operatorname{Ran}P_\sol \] is therefore closed. Under the inverse Fourier transform it is the space of square-integrable vector fields that are divergence-free in the sense of distributions. More explicitly, the map \begin{equation*} \iota_{\vepsilon}:\Hil_\ph\longrightarrow\Hil_\sol, \qquad \iota_{\vepsilon} f=\sum_{\lambda=1}^2f_\lambda\vepsilon_\lambda, \qquad (\iota_{\vepsilon}^\dag\vf)_\lambda =\vepsilon_\lambda\cdot\vf, \end{equation*} is unitary. If $\vepsilon'$ is another measurable polarization frame, then $\iota_{\vepsilon'}^\dag\iota_{\vepsilon}$ is unitary on $\Hil_\ph$, and its second quantization satisfies \[ \varGamma(\iota_{\vepsilon'}^\dag\iota_{\vepsilon})\, a^\#(\iota_{\vepsilon}^\dag\vf)\, \varGamma(\iota_{\vepsilon'}^\dag\iota_{\vepsilon})^\dag =a^\#(\iota_{\vepsilon'}^\dag\vf), \qquad \vf\in\Hil_\sol. \] Thus the creation and annihilation operators of a solenoidal mode $\vf$, and all operators built from them below, are unitarily equivalent in the two frames. \subsection{The vector potential operator} The photon \emph{dispersion relation}, or simply the \emph{dispersion}, is $\omega(\vk)=|\vk|$. The free-field Hamiltonian is $H_\rad=\dd\varGamma(\omega)$, discussed further in \cref{sec-free-field}. We fix a momentum-space coupling profile, or form factor, $\wh\chi$, incorporating the ultraviolet cutoff, and assume that it satisfies \begin{equation}\label{cutoff-assumption} \boxed{\wh\chi(\vk)=\ol{\wh\chi(-\vk)},\qquad \wh\chi\in L^2(\RR^3),\qquad \frac{\wh\chi}{\sqrt\omega}\in L^2(\RR^3).} \end{equation} Such a form factor is necessary to obtain a well-defined Hamiltonian \cite[Chapter~11]{lieb2010stability}. The $L^2$ condition controls, in particular, the vacuum term in the creation-operator estimate. The weighted condition gives relative bounds with respect to $\sqrt{H_\rad}$ and also imposes an infrared condition at $\vk=0$. The hard ultraviolet cutoff used in the examples is \begin{equation}\label{harduv} \wh\chi_\Lambda(\vk)=\frac{\iden_{|\vk|\leq\Lambda}(\vk)}{\sqrt{|\vk|}}. \end{equation} For $0\leq\Lambda<\infty$ this satisfies \cref{cutoff-assumption}, with the value at $\vk=0$ again irrelevant, and \begin{equation}\label{harduv-norms} \|\wh\chi_\Lambda\|^2=2\pi\Lambda^2, \qquad \left\|\frac{\wh\chi_\Lambda}{\sqrt\omega}\right\|^2 =4\pi\Lambda. \end{equation} The assumptions and representability constructions below apply to a general fixed $\wh\chi$. For $\mu=1,2,3$, define the coupling function \begin{equation}\label{Gx-def} G^\mu_{\vx}(\vk,\lambda) =\frac{\epsilon_\lambda^\mu(\vk)\wh\chi(-\vk) e^{-\i\vk\cdot\vx}}{\sqrt{2}} \end{equation} and, equivalently in the solenoidal representation, \begin{equation}\label{Gx-solenoidal} \vG^\mu_{\vx}(\vk) =\frac{P_\sol(\vk)\ve_\mu\wh\chi(-\vk) e^{-\i\vk\cdot\vx}}{\sqrt{2}}, \qquad \iota_{\vepsilon} G^\mu_{\vx}=\vG^\mu_{\vx}, \end{equation} where $\ve_\mu$ is the $\mu$th coordinate vector. The identity for the solenoidal projector~\eqref{polarization-projector} gives the exact, uniform relations \begin{equation}\label{Gx-norms} \sum_{\mu=1}^3\|G^\mu_{\vx}\|^2 =\|\wh\chi\|^2, \qquad \sum_{\mu=1}^3 \left\|\frac{G^\mu_{\vx}}{\sqrt\omega}\right\|^2 =\left\|\frac{\wh\chi}{\sqrt\omega}\right\|^2. \end{equation} Dominated convergence also shows that $\vx\mapsto G^\mu_{\vx}$ is continuous both in $\Hil_\ph$ and in the norm $f\mapsto\|f/\sqrt\omega\|$. On the finite-particle core $\Fock_\ph^\fin$, define the quantized vector potential, componentwise, by the Segal field \begin{equation}\label{Afield-def} \boxed{\Af^\mu(\vx)=\varPhi(G^\mu_{\vx})=a^\dag(G^\mu_{\vx})+a(G^\mu_{\vx}).} \end{equation} It is symmetric and essentially self-adjoint there, and the same notation will denote its self-adjoint closure. We suppress the dependence of $G^\mu_{\vx}$ and $\Af^\mu(\vx)$ on the fixed form factor $\wh\chi$. A superscript $\chi$ on the representability sets and universal functionals records this choice. The field $\Af^\mu(\vx)$ maps each $M$-photon sector into the direct sum of the $(M-1)$- and $(M+1)$-photon sectors (with the negative sector omitted when $M=0$). In the point-field notation of \cref{sec-fock-space}, with $\vz=(\vk,\lambda)$, in the weak smeared sense on $\Fock_\ph^\fin$, \begin{equation*} a^\dag(f)=\sum_{\lambda=1}^2\int_{\RR^3} f(\vk,\lambda)a^\dag(\vk,\lambda)\dd\vk, \qquad a(f)=\sum_{\lambda=1}^2\int_{\RR^3} \ol{f(\vk,\lambda)}a(\vk,\lambda)\dd\vk. \end{equation*} We may write \cref{Afield-def} as \begin{align} \boxed{ \AF(\vx)=\frac{1}{\sqrt{2}} \sum_{\lambda=1}^2\int_{\RR^3}\vepsilon_\lambda(\vk) \big[\wh\chi(-\vk)e^{-\i\vk\cdot\vx}a^\dag(\vk,\lambda) +\wh\chi(\vk)e^{\i\vk\cdot\vx}a(\vk,\lambda)\big]\dd\vk .} \label{Afield-momentum} \end{align} The integral in \cref{Afield-momentum} is an operator-valued distribution. On the electron--photon space the notation $\Af^\mu(\vx_n)$ means operator-valued multiplication: \begin{equation*} (\Af^\mu(\vx_n)\Psi)(\vX) =\Af^\mu(\vx_n)\Psi(\vX), \qquad \vX=(\vx_1,\ldots,\vx_N), \end{equation*} initially on the natural finite-particle form core. The continuity and uniform bounds above make this multiplication strongly measurable and extend it to the domain used in the Pauli--Fierz form. In the sense of operator-valued distributions we have the Coulomb gauge \begin{equation*} \boxed{\div\AF=0.} \end{equation*} More explicitly, for every $\eta\in C_c^\infty(\RR^3)$, \begin{equation}\label{coulomb-gauge-weak} \sum_{\mu=1}^3\int_{\RR^3} (\partial_\mu\eta)(\vx)\Af^\mu(\vx)\dd\vx=0. \end{equation} This is first an identity on $\Fock_\ph^\fin$ and, by \cref{ann-Hrad-bound,cre-Hrad-bound} below, also a form identity on $D(\sqrt{H_\rad})$. It follows directly from $\vk\cdot\vepsilon_\lambda(\vk)=0$. \subsection{The free field and field bounds}\label{sec-free-field} The free-field Hamiltonian is the second quantization of the dispersion, \begin{equation*} H_\rad=\dd\varGamma(\omega) =\sum_{\lambda=1}^2\int_{\RR^3}|\vk|\, a^\dag(\vk,\lambda)a(\vk,\lambda)\dd\vk. \end{equation*} On the $M$-photon sector, $H_\rad$ is multiplication by $\sum_{m=1}^M|\vk_m|$. It is positive and self-adjoint, \[ \sigma(H_\rad)=[0,\infty),\qquad \ker H_\rad=\Span\{\Omega\}, \] and its restriction to $\Omega^\perp$ has purely absolutely continuous spectrum. The standard creation--annihilation estimates imply, uniformly in $\vx$, \begin{align} \|a(G^\mu_{\vx})\Psi\|&\leq \|G^\mu_{\vx}/\sqrt\omega\|\,\|\sqrt{H_\rad}\Psi\|, \label{ann-Hrad-bound}\\ \|a^\dag(G^\mu_{\vx})\Psi\|&\leq \|G^\mu_{\vx}/\sqrt\omega\|\,\|\sqrt{H_\rad}\Psi\| +\|G^\mu_{\vx}\|\,\|\Psi\|. \label{cre-Hrad-bound} \end{align} The same estimates hold with $G^\mu_{\vx}$ replaced by any $f\in\Hil_\ph$ such that $f/\sqrt\omega\in\Hil_\ph$, for every $\Psi\in D(\sqrt{H_\rad})$. Consequently, for a constant $C_\chi<\infty$ depending only on the form factor, \begin{equation}\label{A-Hrad-bound} \|\AF(\vx)\Psi\|^2\leq C_\chi \big(\|\sqrt{H_\rad}\Psi\|^2+\|\Psi\|^2\big). \end{equation} These standard field estimates and the associated Pauli--Fierz form construction are discussed, for example, in \cite{lieb2003existence}. The transverse electric and magnetic fields are \begin{equation}\label{EB-field-def} \EF_\perp(\vx)=-\i[H_\rad,\AF(\vx)],\qquad \BF(\vx)=\grad\times\AF(\vx), \end{equation} where the commutator and the curl are understood in the sense of operator-valued distributions, that is, after smearing with test fields as in \cref{coulomb-gauge-weak}. The subscript records that $\EF_\perp$ is only the divergence-free part of the electric field. In Coulomb gauge the longitudinal part is the instantaneous Coulomb field of the charges, which manifests itself as the Coulomb interaction in the Hamiltonian~\cite{spohn2004dynamics}. More explicitly, let $\vu\in C_c^\infty(\RR^3;\RR^3)$ and $g_\vu=\sum_{\mu=1}^3\int_{\RR^3}u^\mu(\vx)G^\mu_{\vx}\dd\vx$, so that $\int_{\RR^3}\vu\cdot\AF=a^\dag(g_\vu)+a(g_\vu)$. The Fourier transform of $\vu$ decays rapidly, hence $\omega g_\vu\in\Hil_\ph$, and \[ \int_{\RR^3}\vu\cdot\EF_\perp =-\i\big(a^\dag(\omega g_\vu)-a(\omega g_\vu)\big), \qquad \int_{\RR^3}\vu\cdot\BF =\int_{\RR^3}(\grad\times\vu)\cdot\AF\,\dd\vx \] are self-adjoint Segal fields. For a general form factor the fields need not be defined pointwise. If $\sqrt\omega\,\wh\chi\in L^2(\RR^3)$, the estimates \cref{ann-Hrad-bound,cre-Hrad-bound} define $\EF_\perp(\vx)$ and $\BF(\vx)$ as forms on $D(\sqrt{H_\rad})$ for every $\vx$. Let us summarize the basic properties of these fields. Identities between fields at points are understood after smearing with test fields $\vu,\vv\in C_c^\infty(\RR^3;\RR^3)$, as operator identities on $\Fock_\ph^\fin$. Both fields are transverse, $\div\EF_\perp=\div\BF=0$. In fact, for $\EF_\perp$ the vector $g_{\grad\eta}$ vanishes for every $\eta\in C_c^\infty(\RR^3)$, exactly as in \cref{coulomb-gauge-weak}, and for $\BF$ this follows from $\grad\times\grad\eta=0$. The components of $\AF$ commute, as do those of $\EF_\perp$, and the two fields form a canonical pair: \begin{gather*} [\Af^\mu(\vx),\Af^\nu(\vy)]=[\Ef^\mu_\perp(\vx),\Ef^\nu_\perp(\vy)]=0,\\ [\Af^\mu(\vx),\Ef^\nu_\perp(\vy)] =-\i\int_{\RR^3}\omega(\vk)|\wh\chi(\vk)|^2 P_\sol(\vk)^{\mu\nu}e^{\i\vk\cdot(\vx-\vy)}\dd\vk. \end{gather*} Indeed, $[a^\dag(f)+a(f),a^\dag(h)+a(h)]=2\i\Im\dua{f}{h}$. The scalar products $\dua{g_\vu}{g_\vv}$ and $\dua{\omega g_\vu}{\omega g_\vv}$ are real, since $\vu,\vv$ are real and $|\wh\chi|$, $\omega$ and $P_\sol$ are even in $\vk$, while \[ \Big[\int_{\RR^3}\vu\cdot\AF,\int_{\RR^3}\vv\cdot\EF_\perp\Big] =-2\i\Re\dua{g_\vu}{\omega g_\vv}. \] Commutators involving $\BF$ follow by taking the curl in the corresponding variable. In particular, the components of $\BF$ commute with those of $\AF$ and with each other. For the hard cutoff \cref{harduv}, $\omega|\wh\chi_\Lambda|^2$ is the indicator function of the ball of radius $\Lambda$, so \begin{equation}\label{hard-cutoff-commutator} [\Af^\mu(\vx),\Ef^\nu_\perp(\vy)] =-\i\int_{|\vk|\leq\Lambda}P_\sol(\vk)^{\mu\nu} e^{\i\vk\cdot(\vx-\vy)}\dd\vk. \end{equation} The right-hand side is $-\i(2\pi)^3$ times the transverse delta function with momenta cut off at $\Lambda$. The factor $(2\pi)^3$ reflects the normalization of $\wh\chi$. Since $\EF_\perp$ and $\BF$ carry the form factor, $H_\rad$ is not their field energy. Write ${:}\,\cdot\,{:}$ for normal ordering, which places all creation operators to the left of all annihilation operators. Inserting \cref{Afield-momentum}, the terms with two creation or two annihilation operators cancel between the electric and magnetic parts, and a formal computation gives \[ \frac12\int_{\RR^3}{:}\,|\EF_\perp|^2+|\BF|^2\,{:} =(2\pi)^3\dd\varGamma\big(\omega^2|\wh\chi|^2\big). \] This equals $H_\rad$ exactly when $|\wh\chi|=(2\pi)^{-3/2}\omega^{-1/2}$, that is, without cutoff. \subsection{The Pauli--Fierz Hamiltonian}\label{sec-PF-hamiltonian} We first construct the internal Hamiltonian. Put \begin{equation}\label{PF-reference-form-domain} \mathcal Q_\PF:=\left\{\Psi\in\HC_N: \grad_{\vx_n}\Psi\in L^2(\RR^{3N};\Fock_\ph)^3\ (1\leq n\leq N),\quad \sqrt{H_\rad}\Psi\in\HC_N\right\} \end{equation} and equip it with the reference norm \begin{equation}\label{PF-reference-form-norm} \|\Psi\|_{\mathcal Q_\PF}^2 :=\|\Psi\|^2+\sum_{n=1}^N\|\grad_{\vx_n}\Psi\|^2 +\|\sqrt{H_\rad}\Psi\|^2. \end{equation} Thus $\mathcal Q_\PF$ is the antisymmetric, Fock-space-valued electronic $H^1$ space intersected with the form domain of the radiation energy. The algebraic tensor product \begin{equation}\label{PF-algebraic-core} \mathscr C_\PF= \big(C_c^\infty(\RR^{3N})\cap\Hil_\el^N\big) \otimes_{\mathrm{alg}} \Fock_s^{\mathrm{alg}}\!\left( C_c^\infty\big((\RR^3\setminus\{0\})\times\{1,2\}\big)\right) \end{equation} is dense in $\mathcal Q_\PF$ for $\|\cdot\|_{\mathcal Q_\PF}$. Here $\Fock_s^{\mathrm{alg}}(\mathcal D)$ is the algebraic symmetric Fock space over a space $\mathcal D$ of one-photon functions, that is, the finite linear combinations of the vacuum and of symmetric tensor products of finitely many elements of $\mathcal D$. Fix $e\geq0$\footnote{The photon parity $(-1)^\NC=\varGamma(-\iden)$ satisfies $(-1)^\NC a^\#(f)(-1)^\NC=-a^\#(f)$, so it reverses the sign of $\AF(\vx)$, commutes with $H_\rad$ and acts trivially on the electrons. It therefore maps the Hamiltonians below at coupling $e$ to those at $-e$ and preserves $\rho_\Psi$ and the combined current density $\vjc_\Psi$ of \cref{jc-def}. Hence the representability sets and universal functionals are the same for $e$ and $-e$.} and set \begin{equation*} \boxed{ \vD_n=-\i\grad_{\vx_n}-e\AF(\vx_n).} \end{equation*} The internal Pauli--Fierz Hamiltonian is \begin{equation}\label{q00-def} \boxed{ H^{0,0}=\frac12\sum_{n=1}^N\vD_n^2 +\sum_{1\leq n0$ and $\wh\chi\not\equiv0$. For every $N\in\NN$, \[ \left\{(\rho,\rho\vv): \begin{array}{l} \rho\geq0,\quad\int_{\RR^3}\rho=N,\quad\sqrt\rho\in H^1,\\ \vv\in H^1_{\mathrm{loc}}(\RR^3;\RR^3),\quad \displaystyle\int_{\RR^3}\rho\bigl(|\vv|^2+|\grad\vv|^2\bigr)<\infty \end{array}\right\} \subset\rcdens_N^{e,\chi}. \] The same conclusion holds for velocities $\grad S+\vw$ with $S\in H^1_{\mathrm{loc}}(\RR^3;\RR)$, $\vw\in H^1_{\mathrm{loc}}(\RR^3;\RR^3)$, and \[ \int_{\RR^3}\rho\bigl(|\grad S|^2+|\vw|^2+|\grad\vw|^2\bigr)<\infty. \] \end{corollary} \begin{proof} Apply \cref{thm-photonic-representability} with $\va=-\vv/e$, $S=0$ and $\vb=0$. For the second assertion use $\va=-\vw/e$, the given $S$, and $\vb=0$. Then \cref{jc-def} gives the prescribed combined current density. \end{proof} \subsection{Universal density functionals} By \cref{prop-current-bounds}, every $N$-representable pair $(\rho,\vjc)$ belongs to the Banach space $X\times Y$, where \begin{equation}\label{density-current-spaces} X=L^1(\RR^3)\cap L^3(\RR^3),\qquad Y=L^1(\RR^3,\RR^3)\cap L^{3/2}(\RR^3,\RR^3), \end{equation} with the max norms $\|\rho\|_X=\max\{\|\rho\|_1,\|\rho\|_3\}$ and $\|\vjc\|_Y=\max\{\|\vjc\|_1,\|\vjc\|_{3/2}\}$. Their duals are $X^*=L^\infty(\RR^3)+L^{3/2}(\RR^3)$ and $Y^*=L^\infty(\RR^3,\RR^3)+L^3(\RR^3,\RR^3)$, with dual norms given by $\|v\|_{X^*}=\inf_{v=v_\infty+v_{3/2}} \big(\|v_\infty\|_\infty+\|v_{3/2}\|_{3/2}\big)$ and $\|\vA\|_{Y^*}=\inf_{\vA=\vA_\infty+\vA_3} \big(\|\vA_\infty\|_\infty+\|\vA_3\|_3\big)$, where the infima run over decompositions in the respective summand spaces. The same spaces were considered for CDFT in \cite{MY-CDFTpaper2019}. For $(\rho,\vjc)\in X\times Y$, define the \emph{Pauli--Fierz Levy--Lieb universal functional} \begin{equation}\label{FLL-def} \boxed{ F_\LL^{e,\chi}(\rho,\vjc) :=\inf_{\substack{\Psi\in \mathcal Q_\PF,\,\|\Psi\|=1\\ \rho_\Psi=\rho\\ \vjc_\Psi=\vjc}} \dua{\Psi}{H^{0,0}\Psi}}, \end{equation} with value $+\infty$ if the constraint set is empty. The variational principle implies \begin{equation}\label{density-variational-principle} E(v,\vA)=\inf_{(\rho,\vjc)\in X\times Y} \left[F_\LL^{e,\chi}(\rho,\vjc) +\dua{v+\tfrac12|\vA|^2}{\rho}+\dua{\vA}{\vjc}\right]. \end{equation} The quadratic diamagnetic term $\tfrac12\dua{|\vA|^2}{\rho}$ means that $E(v,\vA)$ need not be jointly concave in $(v,\vA)$. To obtain a jointly concave energy, we follow \cite{tellgren2012choice} and absorb this term into the scalar potential by setting $u=v+\frac{1}{2}|\vA|^2$, and define \begin{equation*} \bar E(u,\vA)=E(u-\frac{1}{2}|\vA|^2,\vA). \end{equation*} For each fixed $N$-representable pair $(\rho,\vjc)$, the expression inside the infimum in \cref{density-variational-principle} is then affine in $(u,\vA)$. Thus $\bar E$ is a pointwise infimum of affine functions and is jointly concave, providing the basis for the following convex-duality construction. The \emph{Pauli--Fierz Lieb universal functional} is then defined as the convex conjugate \begin{equation}\label{FL-def} F_\LI^{e,\chi}(\rho,\vjc) =\sup_{(u,\vA)\in X^*\times Y^*} \big[\bar E(u,\vA)-\dua{u}{\rho}-\dua{\vA}{\vjc}\big]. \end{equation} It is thus convex and lower semicontinuous in the norm topology of $X\times Y$, and biconjugation gives \begin{equation}\label{L-vs-LL} F_\LI^{e,\chi}=\ol{\operatorname{conv}}F_\LL^{e,\chi} \leq F_\LL^{e,\chi}. \end{equation} Here $\ol{\operatorname{conv}}$ denotes the norm-lower-semicontinuous convex envelope. Equivalently, it is the weak-lower-semicontinuous convex envelope. For comparison, recall the class $\dm$ of finite-energy density matrices from \cref{finite-energy-PF-ensembles} and define the mixed-state functional \begin{equation}\label{FLcs-def} \boxed{\wt{F}_\LI^{e,\chi}(\rho,\vjc) =\inf_{\substack{\vGamma\in\dm\\ \rho_\vGamma=\rho\\ \vjc_\vGamma=\vjc}} \Tr(H^{0,0}\vGamma)}. \end{equation} Directly from the variational principle we have \begin{equation}\label{L-cs-relations} F_\LI^{e,\chi}\leq \wt{F}_\LI^{e,\chi} \leq F_\LL^{e,\chi}. \end{equation} As in the case of ordinary DFT and CDFT, equality of the first two functionals requires a separate lower-semicontinuity and compactness argument, supplied by \cref{thm-unrestricted-attainment} below. A closely related result is attainment of the defining infimum of $\wt{F}_\LI^{e,\chi}$, also called expectation-valuedness of the functional. \subsection{Existence and nonexistence of optimizers} \label{sec-unrestricted-attainment} We now establish attainment of the mixed-state constrained search and identify it with the Lieb functional. The main difficulty comes from the possible occurrence of soft photons: an energy bound alone does not control photon number, since a minimizing sequence may contain arbitrarily many photons of arbitrarily small frequency. \begin{theorem}[Existence of a PFL optimizer] \label{thm-unrestricted-attainment} Whenever finite, the mixed-state functional $\wt{F}_\LI^{e,\chi}(\rho,\vjc)$ has an optimizer. Every optimizer $\vGamma$ has finite mean photon number, namely \[ \Tr(\NC\vGamma)\leq C_{N,e,\chi} \big(1+\wt{F}_\LI^{e,\chi}(\rho,\vjc)\big), \] with a constant $C_{N,e,\chi}$ depending only on $N$, $e$ and $\chi$. The functional $\wt{F}_\LI^{e,\chi}(\rho,\vjc)$ is norm lower semicontinuous on $X\times Y$, and \begin{equation}\label{unrestricted-Lieb-equality} \boxed{\wt{F}_\LI^{e,\chi}(\rho,\vjc)=F_\LI^{e,\chi}(\rho,\vjc)} \end{equation} for all $(\rho,\vjc)\in X\times Y$. \end{theorem} The proof removes soft photons and restores the combined current density exactly, at a controlled energy cost. The electronic current shift of Appendix~\ref{app-current-shift} preserves the density, fermionic antisymmetry, and the remaining photon marginal. Applied to a minimizing sequence, it yields a uniform bound on mean photon number. Photon localization and passage to the limit then give a mixed-state minimizer. The same estimates establish norm lower semicontinuity and hence equality with the Lieb functional. However, pure-state attainment fails in general even for $N=1$. \begin{proposition}[Nonexistence of a PFLL optimizer] \label[proposition]{prop-pure-nonattainment} Let $N=1$ and fix the form factor. For every $e\geq0$, there is a density--current pair $(\rho_e,\vj_e)\in\rcdens_1^{e,\chi}$ such that $F_\LL^{e,\chi}(\rho_e,\vj_e)<\infty$ and its infimum is not attained. The pair may depend on both $e$ and the fixed form factor. In particular, this holds for the standard hard ultraviolet cutoff \cref{harduv} at every $e\geq0$ and every finite $\Lambda$. \end{proposition} The density $\rho_e$ is supported in a thin solid torus around a long, narrow ellipse, and the velocity $\vj_e/\rho_e$ has circulation $\pi$ around each loop of the torus, half the circulation quantum. A single orbital cannot carry this velocity, since the phase of a nonvanishing wavefunction winds by a multiple of $2\pi$, but the equal mixture of two orbitals with winding numbers $0$ and $1$ can. Attaching orthogonal soft-photon states to the two orbitals turns this mixture into a pure state with the same density and combined current density, at an energy cost that can be made arbitrarily small but not zero. To exclude a pure minimizer, we compare a lower bound for pure states with an upper bound from mixed states. As the ellipse is narrow, the field has small circulation around the loops, and the two bounds force a pure minimizer to be almost photon-free. Its vacuum component is then bounded away from zero along some loop, so its phase winds by a multiple of $2\pi$, which contradicts the circulation $\pi$. An obvious sufficient condition for pure-state attainment is the existence of a pure ground state with the prescribed density-current pair ($v$-representability). In fact, a pure ground state of $H^{v,\vA}$, with potentials in \cref{potential-spaces}, minimizes the PFLL constrained search at its own density--current pair: the external terms in \cref{energy-decomposition} are the same for all states with that pair, so lowering the internal energy would contradict the ground-state property. In particular, the PFLL infimum at the density--current pair of a ground-state density matrix is attained whenever the corresponding ground state is non-degenerate, since every ground-state density matrix is then the rank-one projector onto that ground state. \subsection{Soft-photon purification} \label{sec-many-electron-soft-purification} Arbitrarily soft photons allow a mixture to be represented by a pure electron--photon state at arbitrarily small energy cost, while the density and combined current density remain unchanged. \begin{theorem}[Soft-photon purification and equality of functionals] \label{thm-many-electron-soft-purification} Fix $N\in\NN$ and $e\geq0$. For every $\vGamma\in\dm$ and every $\delta>0$, there is a normalized $\Psi\in \mathcal Q_\PF$ with finite mean photon number such that \begin{equation}\label{many-electron-pure-approximation} (\rho_\Psi,\vjc_\Psi)=(\rho_\vGamma,\vjc_\vGamma),\qquad \dua{\Psi}{H^{0,0}\Psi}\leq\Tr(H^{0,0}\vGamma)+\delta. \end{equation} Consequently, \begin{equation}\label{many-electron-pure-ensemble-equality} \boxed{\quad F_\LL^{e,\chi}(\rho,\vjc) =F_\LI^{e,\chi}(\rho,\vjc)\quad} \end{equation} for all $(\rho,\vjc)\in X\times Y$. If this value is finite at $(\rho,\vjc)$, the pure-state infimum is attained if and only if the total Pauli--Fierz mixed-state minimization has a rank-one minimizer. \end{theorem} This does not contradict Levy's and Lieb's nonconvexity result for the electronic Levy--Lieb functional \cite{levy1982,Lieb1983}, since a pure electron--photon state can have a mixed electronic marginal and, already at zero coupling, arbitrarily soft photons purify electronic mixed states at arbitrarily small energy cost. \begin{corollary}[Convexity, $N$-representability, and pure minimizing sequences] \label[corollary]{cor-many-electron-pure-functional} For every $N\in\NN$ and $e\geq0$, $F_\LL^{e,\chi}$ is proper, convex, and norm lower semicontinuous on $X\times Y$. Moreover, \begin{equation}\label{unrestricted-pure-mixed-PF-class} \boxed{\rcdens_N^{e,\chi} =\{(\rho_\vGamma,\vjc_\vGamma):\vGamma\in\dm\} =\operatorname{dom}F_\LI^{e,\chi}.} \end{equation} In particular, the pure-state $N$-representability class is convex. Let $(\rho,\vjc)\in X\times Y$ with $F_\LI^{e,\chi}(\rho,\vjc)<\infty$, and let $\vGamma_*\in\dm$ be a minimizer in \cref{FLcs-def}, so that $(\rho_{\vGamma_*},\vjc_{\vGamma_*})=(\rho,\vjc)$. Then there are normalized $\Psi_j\in\mathcal Q_\PF$ with \[ (\rho_{\Psi_j},\vjc_{\Psi_j})=(\rho,\vjc),\qquad \dua{\Psi_j}{H^{0,0}\Psi_j}\longrightarrow F_\LL^{e,\chi}(\rho,\vjc), \] and \begin{equation}\label{pure-minimizing-sequence-number-bound} \sup_j\dua{\Psi_j}{\NC\Psi_j} \leq 1+\Tr(\NC\vGamma_*) \leq 1+C_{N,e,\chi} \big(1+F_\LI^{e,\chi}(\rho,\vjc)\big). \end{equation} \end{corollary} This result extends \cref{prop-spectator-purification} to general form factors. Equality of the infima and the bound \cref{pure-minimizing-sequence-number-bound} do not imply pure-state attainment: the soft photons can still prevent the necessary compactness. The construction uses the infinitely many continuum modes of arbitrarily small positive frequency. In contrast to \cref{prop-spectator-purification}, it requires no region on which the coupling vanishes, so it also strengthens the representability conclusion of that previous result. At zero combined current density, both functionals reduce to searches constrained by the density alone. \begin{proposition}[Vanishing combined current density] \label[proposition]{prop-zero-current-density-functional} For $\sqrt{\rho}\in H^1(\RR^3)$ with $\rho\ge0$ and $\int_{\RR^3}\rho=N$, the following relations hold true: \begin{align} F_\LL^{e,\chi}(\rho,0) &=\inf_{\substack{\Psi\in\mathcal Q_\PF,\ \|\Psi\|=1\\ \rho_\Psi=\rho}} \dua{\Psi}{H^{0,0}\Psi}, \label{zero-current-pure-density-search}\\ F_\LI^{e,\chi}(\rho,0) &=\min_{\substack{\vGamma\in\dm\\\rho_\vGamma=\rho}} \Tr(H^{0,0}\vGamma) \label{zero-current-density-search}\\ &=\sup_{v\in X^*}\big[E(v,0)-\dua{v}{\rho}\big]. \label{zero-current-scalar-dual} \end{align} \end{proposition} Using \cref{thm-many-electron-soft-purification}, the above two zero-current functionals coincide. Note that the current constraint can be dropped from both constrained searches. These functionals still include the quantized field and its coupling to the electrons. At nonzero coupling, their common value generally differs from the purely electronic Lieb functional $F_{\mathrm{L}}(\rho)=F_\VR(\rho,0)$ recalled below in \cref{sec-zero-coupling}. The difference is a nontrivial variational contribution arising from electronic and photonic relaxation and their correlations at fixed density. The vacuum shift $e^2N\|\wh\chi\|^2/2$ gives an upper bound on this difference (\cref{thm-zero-coupling-convergence}). \subsection{Properties of optimizers} \label{sec-optimizer-properties} By \cref{thm-unrestricted-attainment}, the constrained search has an optimizer whenever its value is finite. In this section we describe such optimizers: their fields, their symmetries and dark modes, and how many soft photons they contain. By dark modes we mean photon modes on which $\wh\chi$ vanishes; these do not couple to the electrons. In each proof we compare an optimizer with other states that have the same density and combined current density, so we never assume that the optimizer is a ground state of some $H^{v,\vA}$. \subsubsection{Fields of constrained optimizers} \label{sec-optimizer-mean-fields} We now ask which conclusions of \cref{prop-ground-state-fields} require a ground state, and which follow from constrained minimization alone. Here an optimizer means any $\vGamma\in\dm$ satisfying \[ \rho_\vGamma=\rho,\qquad \vjc_\vGamma=\vjc,\qquad \Tr(H^{0,0}\vGamma)=F_\LI^{e,\chi}(\rho,\vjc)<\infty. \] There need not be supporting external potentials (no $v$-representability required). In their absence, the prescribed current is $\vjc$, whereas a physical current $\vj=\vjc+\rho\vA$ requires a specified external vector potential. All traces below are understood in the form sense, with the fields smeared as in Section~\ref{sec-free-field}. \begin{proposition}[Fields of optimizers] \label[proposition]{prop-optimizer-mean-fields} Let $\vGamma$ be a PFL optimizer at $(\rho,\vjc)$. The following properties hold true. \begin{enumerate}[(i)] \item For real $\vu_1,\ldots,\vu_n\in C_c^\infty(\RR^3;\RR^3)$ and every real $\phi\in\mathcal S(\RR^n)$, \begin{equation}\label{optimizer-field-statistics} \sum_{j=1}^n\Re\Tr\Big(\Big[(\partial_j\phi)\Big(\int_{\RR^3}\vu_1\cdot\AF,\ldots, \int_{\RR^3}\vu_n\cdot\AF\Big)\Big]\Big[\int_{\RR^3}\vu_j\cdot\EF_\perp\Big]\vGamma\Big)=0. \end{equation} \item In particular, the transverse electric field vanishes, $\Tr(\EF_\perp\vGamma)=0$ in the sense of distributions. This means that \begin{equation}\label{optimizer-mean-electric-zero} \Tr\Big(\Big[\int_{\RR^3}\vu\cdot\EF_\perp\Big]\vGamma\Big)=0 \qquad\text{for every }\vu\in C_c^\infty(\RR^3;\RR^3). \end{equation} \item If the prescribed current has the form \[ \vjc=\rho\grad S,\qquad S\in W^{1,1}_{\rm loc}(\RR^3;\RR),\qquad \grad S\in L^3(\RR^3;\RR^3)+L^\infty(\RR^3;\RR^3), \] then every coherent photon amplitude vanishes: $\Tr(a(h)\vGamma)=0$ for all $h\in\Hil_\ph$ with $h/\sqrt\omega\in\Hil_\ph$. In particular, the vector potential vanishes pointwise and the magnetic field vanishes in the sense of distributions: \begin{equation*} \Tr(\AF(\vx)\vGamma)=0\quad(\vx\in\RR^3), \qquad \Tr(\BF\vGamma)=0. \end{equation*} \end{enumerate} \end{proposition} Item (i) extends \cref{optimizer-mean-electric-zero}, its limiting linear case, from the expectation value to all equal-time statistics of the smeared transverse field. The kinetic, Coulomb, and external terms of $H^{v,\vA}$ commute with $\phi(\int\vu_1\cdot\AF,\ldots,\int\vu_n\cdot\AF)$. Formally, the left-hand side of \cref{optimizer-field-statistics} is therefore minus the time derivative at $t=0$ of $\Tr\big(\phi(\int\vu_1\cdot\AF,\ldots,\int\vu_n\cdot\AF)\, e^{-\i tH^{v,\vA}}\vGamma e^{\i tH^{v,\vA}}\big)$: under the full dynamics, every such expectation value is stationary to first order at an optimizer. In a Schr\"odinger representation of the field, in which the smeared fields act as coordinates (Appendix~\ref{app-soft-gaussian} with $\mathfrak h=\Hil_\ph$), \cref{optimizer-field-statistics} says that the probability current in field configuration space is weakly divergence-free. It is thus the field-space counterpart of \cref{prop-ground-state-fields}(i). The field variations in the proof of \cref{prop-optimizer-mean-fields} preserve both constraints on the density and the combined current density. By contrast, an electronic phase changes $\vjc$ by $\rho\grad S$, and a general photon displacement changes it by $-e\rho\vc_f$, with $\vc_f$ as in \cref{weyl-classical-field}. Consequently, the variations proving continuity and the Amp\`ere law for ground states are not generally available at a fixed pair. The next proposition shows that a constrained optimizer can have a combined current density with nonzero divergence at positive coupling. It also shows that even a unique optimizer need not be a ground state for any external potentials in \cref{potential-spaces}. \begin{proposition}[Ground-state conclusions that fail for optimizers] \label[proposition]{prop-optimizer-counterexamples} Let $N=1$, and set \[ \rho(\vx)=\pi^{-3/2}e^{-|\vx|^2},\qquad \vjc(\vx)=\rho(\vx)\vq,\qquad 0\ne\vq\in\RR^3. \] \begin{enumerate}[(i)] \item Let $e>0$, and let $\wh\chi$ satisfy \cref{cutoff-assumption}. The pair $(\rho,\vjc)$ has a PFL optimizer, and every such optimizer satisfies \[ \Tr(\AF\vGamma) =\Tr(\BF\vGamma) =\Tr(\EF_\perp\vGamma)=0, \qquad \div\vjc=-2(\vq\cdot\vx)\rho\ne0. \] The vector-potential identity holds pointwise, and the magnetic and electric field expectation values hold in the sense of distributions. \item Let $e=0$. The unique PFL optimizer at $(\rho,\vjc)$ is \begin{equation}\label{unique-gaussian-optimizer} \begin{gathered} \vGamma_\vq=|\Psi_\vq\rangle\langle\Psi_\vq|, \qquad \Psi_\vq(\vx)=e^{\i\vq\cdot\vx}\pi^{-3/4}e^{-|\vx|^2/2}\Omega, \\ F_\LI^{0,\chi}(\rho,\rho\vq) =\frac34+\frac{|\vq|^2}{2}. \end{gathered} \end{equation} It is not a ground state, or even an eigenstate, of any $H^{v,\vA}$ with potentials in \cref{potential-spaces}. \end{enumerate} \end{proposition} Part (i) shows that continuity and Amp\`ere's law are not consequences of constrained optimality. Indeed, since it involves the physical current $\vjc + \rho \vA$, Amp\`ere's law is most natural in a situation where the external potential $\vA$ is given. By part (ii), uniqueness of a constrained optimizer cannot replace the ground-state assumptions in \cref{prop-ground-state-fields}(iv): the unique constrained optimizer can have nonzero current. As is made clearer in the proof below (in \cref{proof:prop-optimizer-counterexamples}), the restriction on the potentials in \cref{potential-spaces} plays an essential role. For example, in the case $e=0$, the given density-current pair does come from the ground state of an $H^{v,\vA}$ with $v(\vx) = \frac{1}{2} |\vx|^2 \not\in L^{3/2}(\RR^3) + L^{\infty}(\RR^3)$ and $\vA(\vx) = -\vq$. \subsubsection{Symmetry and dark modes} \label{sec-optimizer-symmetry} For $\vR\in{\rm O}(3)$, define in the solenoidal representation of the one-photon space the rotation $(\Upsilon_\vR\vf)(\vk)=\vR\vf(\vR^\top\vk)$, and put \[ (U_\vR\Psi)(\vX) =\Upsilon_\vR\Psi(\vR^\top\vx_1,\ldots,\vR^\top\vx_N) \] on $\HC_N$, as in \cref{prop-orthogonal-covariance}. Let $\Theta$ be the time reversal \cref{time-reversal-def}. For $g=(\vR,s)\in{\rm O}(3)\times\{1,-1\}$, put \[ V_g=\begin{cases} U_\vR, & s=1,\\ U_\vR\Theta, & s=-1. \end{cases} \] \begin{proposition}[Symmetry and dark modes of optimizers] \label[proposition]{prop-optimizer-symmetry} Let $F_\LI^{e,\chi}(\rho,\vjc)<\infty$. \begin{enumerate}[(i)] \item Use the hard cutoff \cref{harduv}, and let $G\subset{\rm O}(3)\times\{1,-1\}$ be a closed subgroup such that \[ \rho(\vR^\top\vx)=\rho(\vx),\qquad s\vR\vjc(\vR^\top\vx)=\vjc(\vx) \qquad((\vR,s)\in G). \] Then $\vGamma\mapsto V_g\vGamma V_g^{-1}$, $g\in G$, maps the set of PFL optimizers at $(\rho,\vjc)$ onto itself, and this set contains a $G$-invariant optimizer. \item Suppose that $\wh\chi=0$ almost everywhere on a measurable set $S\subset\RR^3$. Let $\Hil_S\subset\Hil_\ph$ consist of the one-photon states supported in $S$, and factor $\Fock_\ph\simeq\Fock_s(\Hil_S^\perp)\otimes\Fock_s(\Hil_S)$. Then every PFL optimizer at $(\rho,\vjc)$ has the form $\vGamma'\otimes|\Omega_S\rangle\langle\Omega_S|$, where $\Omega_S$ is the vacuum of the second factor. \end{enumerate} \end{proposition} For instance, if $\rho$ is radial and $\vjc=0$, part (i) with $G={\rm O}(3)\times\{1,-1\}$ gives an optimizer invariant under all rotations, reflections, and time reversal. The group $G=\{(\iden,\pm1)\}$ requires only $\vjc=0$, with no symmetry of $\rho$. Its Haar average is $\frac12(\vGamma+\Theta\vGamma\Theta^{-1})$, so every optimizer $\vGamma$ at $(\rho,0)$ yields a time-reversal-invariant one. If $\rho$ and $\vjc$ are invariant under rotations about an axis, as for a circulating current, there is an optimizer invariant under these rotations even though time reversal is broken. For the hard cutoff, part (ii) applies with $S=\{|\vk|>\Lambda\}$: optimizers carry no photons above the cutoff. \subsubsection{Soft-photon energy distributions and infrared laws} \label{sec-soft-photon-spectrum} We describe photon energy distributions of constrained optimizers and their infrared behavior. Here photon energy and frequency are identified by $\omega(\vk)=|\vk|$, in the units $\hbar=c=1$ used above. Fix a density--current pair $(\rho,\vjc)$ with $F_\LI^{e,\chi}(\rho,\vjc)<\infty$, and let $\vGamma_*$ be any mixed-state optimizer supplied by \cref{thm-unrestricted-attainment}. For a Borel set $B\subset(0,\infty)$, define the photon-number and photon-energy measures \begin{align*} \nu_*(B)&=\Tr\!\left(\dd\varGamma(\iden_{\{|\vk|\in B\}}) \vGamma_*\right), \\ \mu_*(B)&=\Tr\!\left(\dd\varGamma(|\vk|\iden_{\{|\vk|\in B\}}) \vGamma_*\right) =\int_B\varepsilon\,\nu_*(\dd\varepsilon). \end{align*} Both are finite. They describe expected photon counts and expected energy in frequency bands, but neither is a probability measure in general. The total mass of $\mu_*$ is the expected total radiation energy, $\mu_*((0,\infty))=\Tr(H_\rad\vGamma_*)$. By contrast, $\Tr(\iden_B(H_\rad)\vGamma_*)$ is the probability that the \emph{total photon energy} lies in $B$. There is also a density formula. Let $\gamma_*^{(1)}$ be the one-photon reduced density operator, characterized by \[ \Tr(T\gamma_*^{(1)})=\Tr(\dd\varGamma(T)\vGamma_*) \] for bounded one-photon operators $T$. It is positive semidefinite and since its trace gives the mean photon number, it is also trace class. Write $\gamma_*^{(1)}=\sum_j s_j|f_j\rangle\langle f_j|$, with $s_j\geq0$ and $(f_j)$ orthonormal. The momentum-space photon-number density is defined almost everywhere by \[ n_*(\vk,\lambda)=\sum_j s_j|f_j(\vk,\lambda)|^2. \] Polar coordinates give absolutely continuous measures \begin{equation*} \begin{aligned} \nu_*(\dd\varepsilon)&=\mathcal N_*(\varepsilon)\dd\varepsilon, &\mathcal N_*(\varepsilon) &=\varepsilon^2\sum_{\lambda=1}^2 \int_{|\vtheta|=1}n_*(\varepsilon\vtheta,\lambda)\dd\vtheta,\\ \mu_*(\dd\varepsilon)&=\mathcal U_*(\varepsilon)\dd\varepsilon, &\mathcal U_*(\varepsilon)&=\varepsilon\mathcal N_*(\varepsilon). \end{aligned} \end{equation*} Summing over polarizations makes these distributions independent of the polarization frame. \begin{proposition}[Infrared distribution bounds for optimizers] \label[proposition]{prop-optimizer-soft-spectrum} There are $C<\infty$ and $0<\kappa_0\leq1$, depending only on $N,e,\chi$, such that every PFL optimizer satisfies, for $0<\kappa\leq\kappa_0$, \begin{align} E_*(\kappa):=\int_0^\kappa\mathcal U_*(\varepsilon)\dd\varepsilon &\leq C(\kappa^2+b_\kappa^2), \label{optimizer-soft-energy-bound}\\ N_*(\kappa):=\int_0^\kappa\mathcal N_*(\varepsilon)\dd\varepsilon &\leq C(\kappa+a_\kappa^2), \label{optimizer-soft-number-bound} \end{align} where \begin{equation}\label{infrared-kernel-errors} a_\kappa=\left\| \frac{\iden_{\{|\vk|<\kappa\}}\wh\chi}{\sqrt{|\vk|}}\right\|, \qquad b_\kappa=\|\iden_{\{|\vk|<\kappa\}}\wh\chi\|. \end{equation} For the hard cutoff \cref{harduv}, with $\Lambda>0$ and $\kappa\leq\min\{\kappa_0,\Lambda\}$, these reduce to \begin{equation}\label{optimizer-hard-cutoff-spectrum} E_*(\kappa)\leq C\kappa^2,\qquad N_*(\kappa)\leq C\kappa. \end{equation} \end{proposition} By evaluating the constants $a_\kappa$ and $b_\kappa$ explicitly, the hard cutoff gives the frequency-bin estimates \begin{equation*} \frac{2}{\kappa}\int_{\kappa/2}^\kappa \mathcal U_*(\varepsilon)\dd\varepsilon\leq C\kappa, \qquad \frac{2}{\kappa}\int_{\kappa/2}^\kappa \mathcal N_*(\varepsilon)\dd\varepsilon\leq C. \end{equation*} One cannot differentiate \cref{optimizer-hard-cutoff-spectrum} to deduce the corresponding pointwise inequalities, so these must be seen as cumulative bounds, not as an assured soft-photon spectrum or a proved leading asymptotic. In particular, neither a Planck law nor a temperature follows only from these estimates. If $P_{<\kappa}^{\mathrm{vac}}$ is the projection onto the vacuum in the soft Fock factor, tensored with the identity on the other factors, then \[ \Tr((\iden-P_{<\kappa}^{\mathrm{vac}})\vGamma_*) \leq N_*(\kappa)\longrightarrow0. \] Thus the probability of any photon below $\kappa$ tends to zero in an exact optimizer, with rate $O(\kappa)$ for the hard cutoff. \subsection{Zero coupling and soft-photon relaxation} \label{sec-zero-coupling} At $e=0$, the combined current density reduces to the paramagnetic current density and the internal Pauli--Fierz Hamiltonian separates into electronic and photon parts: \begin{align*} \vjc_\Psi&=\vjp_\Psi,\\ \left.H^{0,0}\right|_{e=0} &=H_\el^0\otimes\iden_{\Fock_\ph} +\iden_{\Hil_\el^N}\otimes H_\rad,\\ H_\el^0&:=\frac12\sum_{n=1}^N(-\Delta_{\vx_n}) +\sum_{1\leq n0$, $e_k\to0$, and $(\rho_k,\vj_k)\wconv(\rho,\vj)$ weakly in $X\times Y$, then \[ F_\VR(\rho,\vj) \leq\liminf_{k\to\infty} F_\LL^{e_k,\chi}(\rho_k,\vj_k). \] \item If $F_\VR(\rho,\vj)<\infty$, then the constant sequence of constraints is a recovery sequence. More precisely, \[ F_\LL^{e,\chi}(\rho,\vj) \leq F_\VR(\rho,\vj) +\frac{N}{2}\|\wh\chi\|^2e^2,\qquad e\geq0. \] Moreover, $(\rho,\vj)\in\rcdens_N^{e,\chi}$ for every $e\geq0$. \end{enumerate} Consequently, for every $(\rho,\vj)\in X\times Y$, \[ \lim_{e\downarrow0}F_\LL^{e,\chi}(\rho,\vj) =F_\LL^{0,\chi}(\rho,\vj) =F_\VR(\rho,\vj). \] In particular, the convergence holds in the sequential Mosco sense on $X\times Y$, hence also in the norm-Gamma sense, and every point in the effective domain has a constant recovery sequence. \end{theorem} Part~(i) holds along arbitrary weakly convergent constraints, while part~(ii) uses only the constant recovery sequence, which suffices for Mosco convergence. An upper bound along every norm-convergent sequence $(\rho_k,\vj_k)\to(\rho,\vj)$ would fail in general, since norm convergence in $X\times Y$ does not control $\int_{\RR^3}|\vj_k|^2/\rho_k$. \subsection{The long-wavelength (dipole) limit} \label{sec-dipole-limit} We next keep $N$, $e$, the form factor, and the photon dispersion fixed. For $0\leq\varepsilon\leq1$, introduce a parameter in the spatial dependence of the quantized field: \begin{equation*} \Af^\mu(\varepsilon\vx)=\varPhi(G^\mu_{\varepsilon\vx}),\qquad \vD_{n,\varepsilon}:=-\i\grad_{\vx_n}-e\AF(\varepsilon\vx_n). \end{equation*} The original model corresponds to $\varepsilon=1$. At $\varepsilon=0$, the field is the spatially uniform operator $\AF(0)$, giving the velocity-gauge dipole approximation. A photon dilation shows that $\varepsilon\downarrow0$ is a long-wavelength limit at fixed system size: rescaling $\vx\mapsto\varepsilon\vx$ is equivalent to keeping the electrons fixed while stretching the relevant photon wavelengths by a factor $1/\varepsilon$. For $\varepsilon>0$, the scaling $\varepsilon^{-3/2}\vf(\cdot/\varepsilon)$ is unitary on $\Hil_\sol$. As in \cref{prop-dilation-covariance}, conjugation by its second quantization maps $\AF(\varepsilon\vx)$ to the field at $\vx$ built with $\varepsilon^{-3/2}\wh\chi(\cdot/\varepsilon)$ in place of $\wh\chi$, and maps $H_\rad$ to $\varepsilon^{-1}H_\rad$. The model at $\varepsilon$ is therefore unitarily equivalent to one with photon dispersion $|\vk|/\varepsilon$, that is, with the \emph{speed of light} formally set to $1/\varepsilon$ while the system size is kept fixed. In that picture the frequencies and $\|\wh\chi\|^2$, the vacuum expectation of $|\AF(0)|^2$, are unchanged, while for the hard cutoff the coupled wave numbers satisfy $|\vk|\leq\varepsilon\Lambda$, so all coupled wavelengths diverge as $\varepsilon\downarrow0$, becoming large compared to any fixed system size. Correspondingly, with the fields \cref{EB-field-def}, the electrons in the model at $\varepsilon$ feel the transverse electric field $-\i[H_\rad,\AF(\varepsilon\vx)]=\EF_\perp(\varepsilon\vx)$ at unchanged strength, whereas the magnetic field $\grad_\vx\times\AF(\varepsilon\vx)=\varepsilon\BF(\varepsilon\vx)$ carries a factor $\varepsilon$. Let $H_\varepsilon^{0,0}$ be the internal Hamiltonian obtained from $H^{0,0}$ by replacing $\vD_n$ with $\vD_{n,\varepsilon}$. As in \cref{sec-PF-hamiltonian}, $\dua{\Psi}{H_\varepsilon^{0,0}\Psi}$ denotes its form value for $\Psi\in\mathcal Q_\PF$, and energies of density matrices are written as $\Tr(H_\varepsilon^{0,0}\vGamma)$ in the form sense. The combined current density changes with the field: \begin{equation*} \vjcmodel{\varepsilon}_{\Psi}(\vx) :=N\Re\dua{\Psi}{\vD_{1,\varepsilon}\Psi}_{\vx} =\vjp_\Psi(\vx) -eN\Re\dua{\Psi}{\AF(\varepsilon\vx)\Psi}_{\vx}. \end{equation*} The mixed-state current density is defined by affine extension. The bounds in \cref{prop-current-bounds} hold uniformly in $\varepsilon$ on sets of bounded form energy. Using the same spaces $X\times Y$ and external potentials as in \cref{density-current-spaces}, set \begin{align*} \bar E_\varepsilon(u,\vA) &:=\inf_{\substack{\Psi\in\mathcal Q_\PF\\\|\Psi\|=1}} \big[\dua{\Psi}{H_\varepsilon^{0,0}\Psi}+\dua{u}{\rho_\Psi} +\dua{\vA}{\vjcmodel{\varepsilon}_{\Psi}}\big], \\ F_{\LI,\varepsilon}(\rho,\vjc) &:=\sup_{(u,\vA)\in X^*\times Y^*} \big[\bar E_\varepsilon(u,\vA)-\dua{u}{\rho}-\dua{\vA}{\vjc}\big]. \end{align*} Let $F_{\LL,\varepsilon}$ be the pure-state constrained search for $H_\varepsilon^{0,0}$, with $(\rho_\Psi,\vjcmodel{\varepsilon}_{\Psi})=(\rho,\vjc)$. As in \cref{L-vs-LL}, \begin{equation}\label{dipole-convex-envelope} F_{\LI,\varepsilon} =\ol{\operatorname{conv}}F_{\LL,\varepsilon}. \end{equation} The fixed parameters $N,e,\chi$ are suppressed in this notation. We use the label $\dip$ for the endpoint $\varepsilon=0$ throughout, keeping the coupling $e$ fixed, and write $F_\dip:=\left.F_{\LI,\varepsilon}\right|_{\varepsilon=0}$ and $\vjcdip_\Psi:=\left.\vjcmodel{\varepsilon}_{\Psi}\right|_{\varepsilon=0}$. The external classical vector potential $\vA(\vx)$ is unchanged. \begin{theorem}[Dipole limit of the PFL functional] \label{thm-dipole-limit} Fix $N\in\NN$, $e\geq0$, and the form factor. The functionals $F_{\LI,\varepsilon}$ converge to $F_\dip$ in the sequential Mosco sense on $X\times Y$: \begin{enumerate}[(i)] \item If $\varepsilon_k>0$, $\varepsilon_k\to0$, and $(\rho_k,\vjc_k)\wconv(\rho,\vjc)$ weakly in $X\times Y$, then \begin{equation}\label{dipole-functional-liminf} F_\dip(\rho,\vjc) \leq\liminf_{k\to\infty}F_{\LI,\varepsilon_k}(\rho_k,\vjc_k). \end{equation} \item For every $(\rho,\vjc)\in X\times Y$, there are pairs $(\rho_\varepsilon,\vjc_\varepsilon)\to(\rho,\vjc)$ in the norm of $X\times Y$ such that \begin{equation}\label{dipole-functional-recovery} \limsup_{\varepsilon\downarrow0} F_{\LI,\varepsilon}(\rho_\varepsilon,\vjc_\varepsilon) \leq F_\dip(\rho,\vjc). \end{equation} \end{enumerate} \end{theorem} The dipole functional can be compared directly with the electronic mixed-state functional defined above. In this comparison, the same current density $\vjcdip$ is used as the paramagnetic-current argument of $F_\VR$. \begin{proposition}[Electronic comparison for the dipole functional] \label[proposition]{prop-dipole-electronic-comparison} For every $N\in\NN$, $e\geq0$, and $(\rho,\vjcdip)\in X\times Y$, \begin{equation}\label{dipole-electronic-comparison} F_\VR(\rho,\vjcdip) \leq F_\dip(\rho,\vjcdip) \leq F_\VR(\rho,\vjcdip)+\frac{Ne^2}{2}\|\wh\chi\|^2. \end{equation} Consequently, \begin{equation}\label{dipole-effective-domain} \operatorname{dom}F_\dip=\operatorname{dom}F_\VR. \end{equation} For the standard hard cutoff \cref{harduv}, the upper correction in \cref{dipole-electronic-comparison} is $N\pi e^2\Lambda^2$. \end{proposition} At the dipole endpoint $\varepsilon=0$, with $e$ fixed, the internal Hamiltonian is \begin{equation}\label{dipole-internal-hamiltonian} H_\dip =\frac12\sum_{n=1}^N \big(-\i\grad_{\vx_n}-e\AF(0)\big)^2 +\sum_{1\leq n0\}$, \[ \partial_\mu\sqrt{\rho_\Psi} =\frac{N\Re\dua{\Psi}{\partial_\mu\Psi}_{\vx}}{\sqrt{\rho_\Psi}}. \] One can obtain this formula by first differentiating $(\|\Psi\|_{\vx}^2+\epsilon^2)^{1/2}-\epsilon$ and then letting $\epsilon\downarrow0$. The derivatives are dominated by $\|\partial_\mu\Psi\|_{\vx}$. On $\{\rho_\Psi=0\}$, the weak gradient of $\sqrt{\rho_\Psi}$ vanishes almost everywhere. Since $\Af^\mu(\vx)$ is self-adjoint, the partial inner product $\dua{\Psi}{\Af^\mu(\vx)\Psi}_{\vx}$ is real. Set $\zeta^\mu(\vx):=\dua{\Psi}{D_1^\mu\Psi}_{\vx}$. The identity $\partial_\mu\Psi=\i D_1^\mu\Psi+\i e\Af^\mu(\vx)\Psi$ and the definition of the combined current density then give \[ \Im\zeta^\mu=-\frac{\sqrt{\rho_\Psi}\,\partial_\mu\sqrt{\rho_\Psi}}{N}, \qquad \Re\zeta^\mu=\frac{(\vjc_\Psi)^\mu}{N}. \] These identities express the gradient of $\sqrt{\rho_\Psi}$ and the combined current through the imaginary and real parts of the same partial inner product. Cauchy--Schwarz in the remaining variables gives, on $\{\rho_\Psi>0\}$, \begin{align*} |\grad\sqrt{\rho_\Psi}|^2+ \frac{|\vjc_\Psi|^2}{\rho_\Psi} &=\frac{N^2}{\rho_\Psi}\sum_{\mu=1}^3 \big(|\Im\zeta^\mu|^2+|\Re\zeta^\mu|^2\big)\\ &=\frac{N^2}{\rho_\Psi}\sum_{\mu=1}^3|\zeta^\mu|^2 \leq N\|\vD_1\Psi\|_{\vx}^2. \end{align*} On $\{\rho_\Psi=0\}$ we have $\|\Psi\|_{\vx}=0$, so all three current densities vanish there. With the stated convention for the quotient, both terms on the left vanish almost everywhere on this set, and the inequality holds there as well. Exchanging electronic coordinates sends $\vD_1$ to the corresponding $\vD_n$, and the sign from antisymmetry disappears upon taking squared norms. Consequently, \[ N\|\vD_1\Psi\|^2=\sum_{n=1}^N\|\vD_n\Psi\|^2. \] Integrating the pointwise joint bound proves \cref{joint-density-current-bound}. We now prove the bound \cref{ph-current-bound} on the field-induced current density. Applying conditional Cauchy--Schwarz to the definition of $\vjph_\Psi$ and then integrating gives \[ |\vjph_\Psi(\vx)|^2 \leq e^2N\rho_\Psi(\vx)\|\AF(\vx)\Psi\|_{\vx}^2, \qquad \int_{\RR^3}\frac{|\vjph_\Psi|^2}{\rho_\Psi} \leq e^2N\|\AF(\vx_1)\Psi\|^2. \] For almost every electronic configuration $\vX$, apply \cref{A-Hrad-bound} to the Fock-space vector $\Psi(\vX)$ at the position $\vx_1$. Its constant is independent of that position, so integration in $\vX$ gives \[ \|\AF(\vx_1)\Psi\|^2 \leq C_\chi\big(\|\sqrt{H_\rad}\Psi\|^2+\|\Psi\|^2\big) =C_\chi\big(\|\sqrt{H_\rad}\Psi\|^2+1\big). \] This proves \cref{ph-current-bound}. It remains to prove the integrability of the density and the current densities. The following integrability argument extends the estimates for electronic paramagnetic currents in~\cite{laestadius2014density,MY-CDFTpaper2019} to the combined and field-induced current densities, using the bounds established above. Since $\sqrt{\rho_\Psi}\in H^1(\RR^3)$, the Sobolev embedding gives $\sqrt{\rho_\Psi}\in L^2\cap L^6$, hence $\rho_\Psi\in L^1\cap L^3$. For either $\vj=\vjc_\Psi$ or $\vj=\vjph_\Psi$, the bounds just proved show that $\vj/\sqrt{\rho_\Psi}\in L^2$, with value zero where the density vanishes. Factoring $\vj=\sqrt{\rho_\Psi}\,(\vj/\sqrt{\rho_\Psi})$ and applying H\"older's inequality gives \[ \|\vj\|_{L^1} \leq\|\sqrt{\rho_\Psi}\|_{L^2} \left\|\frac{\vj}{\sqrt{\rho_\Psi}}\right\|_{L^2}, \qquad \|\vj\|_{L^{3/2}} \leq\|\sqrt{\rho_\Psi}\|_{L^6} \left\|\frac{\vj}{\sqrt{\rho_\Psi}}\right\|_{L^2}. \] Finally, $\vjp_\Psi=\vjc_\Psi-\vjph_\Psi$ belongs to the same two spaces. For a density matrix $\vGamma=\sum_r p_r|\Psi_r\rangle\langle\Psi_r|\in\dm$, apply the preceding argument to the auxiliary vector $(\sqrt{p_r}\Psi_r)_r$, taking the Hilbert direct sum over $r$ as part of the space of remaining variables. This vector is normalized and has finite form energy. Its partial density and currents are the ensemble averages, and the operators act componentwise in $r$. Their squared norms are therefore the corresponding form traces, which proves the mixed-state extension. \end{proof} \phantomsection \label{proof:prop-div-jph} \begin{proof}[Proof of \cref{prop-div-jph}] Let $\Psi\in \mathcal Q_\PF$, write $\partial_\mu$ for the derivative with respect to the $\mu$th component of $\vx=\vx_1$, and denote the right-hand side of \cref{div-jph} by \[ f_\Psi(\vx):=-2eN\Re\sum_{\mu=1}^3 \dua{\partial_\mu\Psi}{\Af^\mu(\vx)\Psi}_{\vx}. \] Let $\|\cdot\|_{\mathcal Q_\PF}$ be the reference norm \cref{PF-reference-form-norm}. As in the proof of \cref{prop-current-bounds}, applying \cref{A-Hrad-bound} to $\Phi(\vX)$ for almost every $\vX$ gives \begin{equation}\label{div-jph-field-bound} \|\AF(\vx_1)\Phi\|\leq C_\chi^{1/2}\|\Phi\|_{\mathcal Q_\PF}, \qquad \Phi\in\mathcal Q_\PF. \end{equation} We first show that $f_\Psi$ is integrable. The conditional Cauchy--Schwarz inequality $|\dua{\partial_\mu\Psi}{\Af^\mu(\vx)\Psi}_{\vx}| \leq\|\partial_\mu\Psi\|_{\vx}\|\Af^\mu(\vx)\Psi\|_{\vx}$, followed by Cauchy--Schwarz in $\vx$ and in $\mu$, gives \begin{equation}\label{div-jph-f-bound} \int_{\RR^3}|f_\Psi| \leq2eN\sum_{\mu=1}^3\|\partial_\mu\Psi\|\,\|\Af^\mu(\vx_1)\Psi\| \leq2eN\|\grad_{\vx_1}\Psi\|\,\|\AF(\vx_1)\Psi\|. \end{equation} Both factors are finite by \cref{PF-reference-form-domain} and \cref{div-jph-field-bound}. We now prove \cref{div-jph} in the sense of distributions. Let $v\in C_c^\infty(\RR^3)$ be real. For $\Phi,\Phi'\in\mathcal Q_\PF$ put \[ \begin{aligned} s_v(\Phi,\Phi') :=\sum_{\mu=1}^3\Big( &\dua{\Phi}{(\partial_\mu v)(\vx_1)\Af^\mu(\vx_1)\Phi'} +\dua{\partial_\mu\Phi}{v(\vx_1)\Af^\mu(\vx_1)\Phi'}\\ &+\dua{v(\vx_1)\Af^\mu(\vx_1)\Phi}{\partial_\mu\Phi'}\Big). \end{aligned} \] Since $\dua{\Psi}{\Af^\mu(\vx)\Psi}_{\vx}$ is real, Fubini's theorem and the definitions give \[ \sum_{\mu=1}^3\int_{\RR^3}(\partial_\mu v)(\vjph_\Psi)^\mu +\int_{\RR^3}vf_\Psi =-eN s_v(\Psi,\Psi), \] so \cref{div-jph} holds in the sense of distributions once $s_v(\Psi,\Psi)=0$. The form $s_v$ is sesquilinear, and Cauchy--Schwarz together with \cref{div-jph-field-bound} gives \[ |s_v(\Phi,\Phi')| \leq C_\chi^{1/2}\big(\sqrt3\|\grad v\|_\infty+2\|v\|_\infty\big) \|\Phi\|_{\mathcal Q_\PF}\|\Phi'\|_{\mathcal Q_\PF}. \] By \cref{PF-form-norm-equivalence,PF-canonical-form-domain}, the algebraic core $\mathscr C_\PF$ of \cref{PF-algebraic-core} is dense in $\mathcal Q_\PF$ for $\|\cdot\|_{\mathcal Q_\PF}$. It therefore suffices to show $s_v(\Phi,\Phi')=0$ for the product vectors $\Phi=\phi\otimes\Sigma$ and $\Phi'=\phi'\otimes\Sigma'$ spanning $\mathscr C_\PF$, where $\phi,\phi'\in C_c^\infty(\RR^{3N})\cap\Hil_\el^N$ and $\Sigma,\Sigma'$ are finite-particle vectors in $D(\sqrt{H_\rad})$. Since $\Af^\mu(\vx)$ is symmetric on $D(\sqrt{H_\rad})$, the three terms of $s_v$ combine by the product rule into \[ s_v(\Phi,\Phi') =\sum_{\mu=1}^3\int_{\RR^{3N}} \partial_\mu\big(v(\vx_1)\,\ol{\phi(\vX)}\,\phi'(\vX)\big)\, \dua{\Sigma}{\Af^\mu(\vx_1)\Sigma'}\dd\vX. \] The function $\vx\mapsto\dua{\Sigma}{\Af^\mu(\vx)\Sigma'}$ is bounded and continuous by \cref{A-Hrad-bound} and the continuity of $\vx\mapsto G^\mu_{\vx}$. For fixed $\vx_2,\ldots,\vx_N$, the function $\vx_1\mapsto v(\vx_1)\ol{\phi(\vX)}\phi'(\vX)$ lies in $C_c^\infty(\RR^3)$. By Fubini's theorem we may first integrate in $\vx_1$, and that integral vanishes by the weak Coulomb gauge \cref{coulomb-gauge-weak}, applied to the real and imaginary parts of this test function. Hence $s_v=0$ on $\mathcal Q_\PF$, which proves \cref{div-jph}. Now $\div\vjph_\Psi=f_\Psi$, and \cref{div-jph-f-bound} is \cref{div-jph-L1}. Finally, the right-hand side of \cref{div-jph} contains no derivative of the field and need not vanish. In the one-electron example following \cref{prop-div-jph}, with $\vjph_\Psi$ given by \cref{entangled-example}, it equals $-2e\Re\big(\vg_\xi\cdot\grad(\ol{\phi_0}\phi_1)\big)$, which is nonzero for $e>0$ and generic $\phi_0,\phi_1$. Choosing $\phi_0,\phi_1$ such that $\rho_\Psi=|\phi_0|^2+|\phi_1|^2$ is constant on an open set where this expression does not vanish shows that $\vjph_\Psi/\rho_\Psi$ is not divergence-free either. \end{proof} \subsection{Proof of \texorpdfstring{\cref{prop-ground-state-fields}}{Proposition \getrefnumber{prop-ground-state-fields}}} \label{proof:prop-ground-state-fields} Let $\Psi$ be a ground state of $H^{v,\vA}$. Items (i)--(iii) are first-order conditions of the variational principle \cref{ground-energy-def}: a one-parameter family of trial states through $\Psi$ has energy at least $E(v,\vA)$, with equality at $t=0$. Item (iv) follows from time-reversal symmetry. By \cref{prop-current-bounds}, $\vjc_\Psi\in L^1\cap L^{3/2}$ and $\rho_\Psi\in L^1\cap L^3$, so H\"older's inequality gives $\rho_\Psi\vA\in L^1\cap L^{3/2}$ for $\vA\in L^3+L^\infty$. We first prove (i). For real $\varphi\in C_c^\infty(\RR^3)$ and $t\in\RR$, the gauge unitary $U_{t\varphi}$ of \cref{gauge-unitary} preserves $\mathcal Q_\PF$, and by \cref{H-gauge,energy-decomposition} \[ \dua{U_{t\varphi}\Psi}{H^{v,\vA}U_{t\varphi}\Psi} =E(v,\vA)+t\int_{\RR^3}\grad\varphi\cdot\vj_\Psi +\frac{t^2}{2}\int_{\RR^3}|\grad\varphi|^2\rho_\Psi. \] The minimum at $t=0$ gives $\int\grad\varphi\cdot\vj_\Psi=0$, which is (i). We next derive an identity for the mean photon amplitude, with \[ g_{\vj_\Psi}=\sum_{\mu=1}^3\int_{\RR^3}j_\Psi^\mu(\vx)G^\mu_{\vx}\dd\vx. \] Let $f\in D(\omega)$. By \cref{cor-weyl-PF-shifts}, applied to $\vGamma=|\Psi\rangle\langle\Psi|$, $W(tf)$ preserves $\mathcal Q_\PF$, and the energy of $W(tf)\Psi$ is the quadratic polynomial \cref{weyl-PF-energy-shift} in $t$. Its linear coefficient is \[ 2\Re\dua{\Psi}{a(\omega f)\Psi}-e\int\vc_f\cdot\vj_\Psi =2\Re\dua{\Psi}{a(\omega f)\Psi}-2e\Re\dua{f}{g_{\vj_\Psi}}, \] with $\vc_f$ as in \cref{weyl-classical-field}. This coefficient vanishes, and applying this also to $\i f$ in place of $f$ gives \[ \dua{\Psi}{a(\omega f)\Psi}=e\dua{f}{g_{\vj_\Psi}} \qquad\text{for all }f\in D(\omega). \] We next prove (ii) and (iii). For test fields $\vu,\vv\in C_c^\infty(\RR^3;\RR^3)$, the kernels of $\int\vu\cdot\EF_\perp$ and of $\int(\grad\times\vv)\cdot\BF=\int(\grad\times\grad\times\vv)\cdot\AF$ are $-\i\omega g_\vu$ and $g_{-\Delta\vv}=\omega^2g_\vv$, since $\grad\times\grad\times\vv=-\Delta\vv+\grad\div\vv$ and gradients have zero kernel. The choices $f=g_\vu$ and $f=\omega g_\vv$ in this identity therefore give \begin{gather*} \Big\langle \Psi, \Big(\int\vu\cdot\EF_\perp\Big)\Psi\Big\rangle=-2e\Im\dua{g_\vu}{g_{\vj_\Psi}} =0,\\ \Big\langle \Psi, \Big(\int(\grad\times\vv)\cdot\BF\Big)\Psi \Big\rangle =2e\Re\dua{\omega g_\vv}{g_{\vj_\Psi}}. \end{gather*} The first scalar product is real as in Section~\ref{sec-free-field}, which proves (ii). Since \[ 2\Re\dua{\omega G^\mu_{\vx}}{G^\nu_{\vy}} =\int_{\RR^3}\omega(\vk)|\wh\chi(\vk)|^2P_\sol(\vk)^{\mu\nu} e^{\i\vk\cdot(\vx-\vy)}\dd\vk, \] the second identity is (iii) paired with $\vv$. We finally prove (iv). By \cref{prop-time-reversal}, the time reversal $\Theta$ of \cref{time-reversal-def} reverses $\AF$, $\BF$, and $\vD_n$ and commutes with $H^{v,0}$. Hence $\Theta\Psi$ is a ground state, and by nondegeneracy a multiple of $\Psi$. On the other hand, $\vjc_{\Theta\Psi}=-\vjc_\Psi$ and $\dua{\Theta\Psi}{\AF\Theta\Psi}=-\dua{\Psi}{\AF\Psi}$, and likewise for $\BF$. All three quantities therefore vanish. \subsection{Representability results} \phantomsection \label{proof:thm-photon-purification} \begin{proof}[Proof of \cref{thm-photon-purification}] Let $\Gamma=\sum_{j\geq1}p_j|\Phi_j\rangle\langle\Phi_j|$ be an electronic mixed state in \cref{electronic-mixed-class}, written in its spectral decomposition. For $\epsilon>0$, choose an orthonormal family \[ f_j\in L^2(\{0<|\vk|<\epsilon\}\times\{1,2\}) \] and put $\Xi_j=a^\dag(f_j)\Omega$. All $\Xi_j$ are normalized and lie in the one-photon sector. The vector \begin{equation}\label{one-photon-purification} \Psi_\epsilon=\sum_{j\geq1}\sqrt{p_j}\, \Phi_j\otimes\Xi_j \end{equation} is normalized and $\Tr_\ph|\Psi_\epsilon\rangle\langle\Psi_\epsilon|=\Gamma$. It therefore has density $\rho_\Gamma$ and paramagnetic current density $\vjp_\Gamma$. Moreover, $\AF(\vx)$ changes photon number by exactly one, whence \[ \dua{\Xi_j}{\AF(\vx)\Xi_k}=0 \quad\text{for all }j,k, \] and consequently $\vjph_{\Psi_\epsilon}=0$. It remains to check that $\Psi_\epsilon\in\mathcal Q_\PF$. Orthonormality of the photon vectors $\Xi_j$ eliminates the cross terms in the electronic kinetic energy, so \[ \sum_{n=1}^N\|\grad_{\vx_n}\Psi_\epsilon\|^2 =\sum_j p_j\sum_{n=1}^N\|\grad_{\vx_n}\Phi_j\|^2 =\Tr(-\lapl\Gamma)<\infty. \] Likewise, orthonormality of the electronic vectors $\Phi_j$ eliminates the cross terms in the photon energy, and $H_\rad$ acts as multiplication by $\omega(\vk)=|\vk|$ on the one-photon sector. Hence \begin{align*} \dua{\Psi_\epsilon}{H_\rad\Psi_\epsilon} &=\sum_j p_j\dua{\Xi_j}{H_\rad\Xi_j}\\ &=\sum_j p_j\|\sqrt{\omega}f_j\|^2 \leq\epsilon\sum_j p_j\|f_j\|^2=\epsilon, \end{align*} where the inequality uses the support condition $|\vk|<\epsilon$, and the last equality uses $\|f_j\|=1$ and $\sum_j p_j=1$. Thus $\Psi_\epsilon\in\mathcal Q_\PF$, proving the reverse inclusion in \cref{electronic-projection} as well as the reverse inclusion in \cref{zero-jph-slice}. Conversely, let $\Psi\in \mathcal Q_\PF$ be normalized and define its electronic reduced state by \[ \Gamma:=\Tr_\ph|\Psi\rangle\langle\Psi|. \] Partial trace preserves positivity and the trace, so $\Gamma\geq0$ and $\Tr\Gamma=1$. We must check that this electronic density matrix has finite kinetic energy and reproduces the pair $(\rho_\Psi,\vjp_\Psi)$. The positive-observable partial-trace identity \cref{partial-trace-positive-observables} gives, in the quadratic-form sense, \begin{align*} \Tr(-\lapl\Gamma) &=\sum_{n=1}^N\|\grad_{\vx_n}\Psi\|^2\\ &\leq 2\sum_{n=1}^N\|\vD_n\Psi\|^2 +2e^2\sum_{n=1}^N\|\AF(\vx_n)\Psi\|^2<\infty. \end{align*} The equality follows by applying the definition \cref{partial-trace-duality} of the partial trace first to bounded spectral truncations of $-\lapl$ and then using monotone convergence. The inequality uses $-\i\grad_{\vx_n}\Psi=\vD_n\Psi+e\AF(\vx_n)\Psi$. Both sums are finite by \cref{PF-canonical-form-domain,A-Hrad-bound}. Thus $\Gamma$ satisfies all the state conditions in \cref{electronic-mixed-class}. To see explicitly why the electronic density and paramagnetic current density are preserved, we use a Schmidt decomposition as in \cref{Schmidt-decomposition,Schmidt-reduced-states}, \[ \Psi=\sum_j\sqrt{p_j}\,\Phi_j\otimes\Xi_j, \qquad \Gamma=\sum_j p_j|\Phi_j\rangle\langle\Phi_j|, \qquad \sum_j p_j=1. \] The electronic vectors $\Phi_j$ and the photon vectors $\Xi_j$ form orthonormal families. The finite kinetic trace implies that each $\Phi_j$ with $p_j>0$ has finite kinetic energy and that the Schmidt series also converges in the electronic $H^1$ norm. Inserting this series into \cref{density-def,jp-def}, the photon inner products $\dua{\Xi_j}{\Xi_k}=\delta_{jk}$ remove all cross terms, giving \[ \rho_\Psi=\sum_j p_j\rho_{\Phi_j}=\rho_\Gamma, \qquad \vjp_\Psi=\sum_j p_j\vjp_{\Phi_j}=\vjp_\Gamma. \] The density series converges in $L^1$ by normalization, and the current series by Cauchy--Schwarz and the finite kinetic trace. These identities therefore hold almost everywhere, and prove the forward inclusion in \cref{electronic-projection}. If the original state additionally satisfies $\vjph_\Psi=0$, its triple is $(\rho_\Gamma,\vjp_\Gamma,0)$ with $(\rho_\Gamma,\vjp_\Gamma)\in\rcdens_{N,\mathrm{mix}}^{\el}$. This proves the forward inclusion in \cref{zero-jph-slice}. The reverse inclusion was supplied by the one-photon purification above. \end{proof} \phantomsection \label{proof:prop-spectator-purification} \begin{proof}[Proof of \cref{prop-spectator-purification}] \noindent\textbf{Step 1. Splitting off the uncoupled photon modes.} Replacing $S$ by $S\cap B_r$ for sufficiently large $r$, we may assume that $S$ is bounded. By the symmetry $\wh\chi(\vk)=\ol{\wh\chi(-\vk)}$ in \cref{cutoff-assumption}, $\wh\chi(-\vk)=0$ for almost every $\vk\in S$, so by \cref{Gx-def} every $G^\mu_{\vx}$ vanishes almost everywhere on $S\times\{1,2\}$. The orthogonal decomposition \[ \Hil_\ph=\mathfrak h_R\oplus\mathfrak h_S,\qquad \mathfrak h_R=L^2(S^c\times\{1,2\}),\qquad \mathfrak h_S=L^2(S\times\{1,2\}), \] induces the canonical factorization $\Fock_s(\Hil_\ph)\simeq\Fock_s(\mathfrak h_R)\otimes\Fock_s(\mathfrak h_S)$, and hence \[ \HC_N\simeq\mathcal K_R\otimes\mathcal F_S,\qquad \mathcal K_R:=\Hil_\el^N\otimes\Fock_s(\mathfrak h_R),\qquad \mathcal F_S:=\Fock_s(\mathfrak h_S). \] Under this identification $\AF(\vx)=\AF_R(\vx)\otimes\iden_{\mathcal F_S}$, where $\AF_R$ is the field on $\Fock_s(\mathfrak h_R)$ with the same kernels $G^\mu_{\vx}$, and $H_\rad=H_{\rad,R}\otimes\iden_{\mathcal F_S} +\iden\otimes H_{\rad,S}$, where $H_{\rad,R}$ and $H_{\rad,S}$ are the second quantizations of the restrictions of $\omega$ to $S^c$ and $S$. For real $v\in C_c^\infty(\RR^3)$ and $\vA\in C_c^\infty(\RR^3;\RR^3)$, let $H_R^{v,\vA}$ be the Hamiltonian on $\mathcal K_R$ defined like $H^{v,\vA}$, with $\AF_R$ and $H_{\rad,R}$ in place of $\AF$ and $H_\rad$. Then, as closed forms, \begin{equation}\label{spectator-Hamiltonian-splitting} H^{v,\vA}=H_R^{v,\vA}\otimes\iden_{\mathcal F_S} +\iden_{\mathcal K_R}\otimes H_{\rad,S}, \end{equation} and in particular $Q(H^{0,0})=Q(H_R^{0,0}\otimes\iden_{\mathcal F_S}) \cap Q(\iden_{\mathcal K_R}\otimes H_{\rad,S})$. \noindent\textbf{Step 2. The density and current density depend only on the coupled marginal.} Let $\vGamma\in\dm$, and put $\Gamma_R=\Tr_{\mathcal F_S}\vGamma$ and $\Gamma_S=\Tr_{\mathcal K_R}\vGamma$. Both terms in \cref{spectator-Hamiltonian-splitting} are nonnegative after $N\|v\|_\infty$ is added to the first. The positive-observable identity \cref{partial-trace-positive-observables} therefore gives \begin{equation}\label{spectator-energy-trace-splitting} \Tr(H^{v,\vA}\vGamma) =\Tr(H_R^{v,\vA}\Gamma_R)+\Tr(H_{\rad,S}\Gamma_S), \end{equation} where all terms are finite by \cref{external-PF-form-domain}. Subtracting the case $v=0$, $\vA=0$ removes the last term. Applying \cref{energy-decomposition} to the spectral decomposition of $\vGamma$, we obtain \[ \int_{\RR^3}\big(v+\tfrac12|\vA|^2\big)\rho_\vGamma +\int_{\RR^3}\vA\cdot\vjc_\vGamma =\Tr(H_R^{v,\vA}\Gamma_R)-\Tr(H_R^{0,0}\Gamma_R). \] The right-hand side depends on $\vGamma$ only through $\Gamma_R$. Taking first $\vA=0$ and then general $\vA$ shows that two density matrices in $\dm$ with the same marginal on $\mathcal K_R$ have the same density and the same combined current density. \noindent\textbf{Step 3. Purification by uncoupled one-photon states.} Fix $\vGamma\in\dm$ and write $\Gamma_R=\sum_rp_r|\Phi_r\rangle\langle\Phi_r|$ with $p_r>0$ and orthonormal $\Phi_r$, the index $r$ ranging over a finite or countable set. Since $S$ has positive measure, $\mathfrak h_S$ is infinite dimensional and contains an orthonormal family $(f_r)$ indexed by the same set. Put $\Xi_r=a_S^\dag(f_r)\Omega_S$ and $R_S=\operatorname*{ess\,sup}_{\vk\in S}|\vk|<\infty$. The $\Xi_r$ are orthonormal, so by \cref{Schmidt-decomposition,Schmidt-reduced-states} the unit vector \[ \Psi=\sum_r\sqrt{p_r}\,\Phi_r\otimes\Xi_r \] satisfies \[ \Tr_{\mathcal F_S}|\Psi\rangle\langle\Psi|=\Gamma_R, \qquad \Tr_{\mathcal K_R}|\Psi\rangle\langle\Psi| =\sum_rp_r|\Xi_r\rangle\langle\Xi_r|. \] Applying \cref{partial-trace-positive-observables} to the two terms of \cref{spectator-Hamiltonian-splitting} with $v=0$ and $\vA=0$, and using \cref{spectator-energy-trace-splitting} for $\vGamma$, we get \begin{align*} \|(H_R^{0,0}\otimes\iden_{\mathcal F_S})^{1/2}\Psi\|^2 &=\Tr(H_R^{0,0}\Gamma_R)\leq\Tr(H^{0,0}\vGamma),\\ \|(\iden_{\mathcal K_R}\otimes H_{\rad,S})^{1/2}\Psi\|^2 &=\sum_rp_r\dua{f_r}{\omega f_r}\leq R_S. \end{align*} Thus $\Psi\in \mathcal Q_\PF$, and $|\Psi\rangle\langle\Psi|\in\dm$ has the same marginal $\Gamma_R$ on $\mathcal K_R$ as $\vGamma$. By Step 2, $(\rho_\Psi,\vjc_\Psi)=(\rho_\vGamma,\vjc_\vGamma)$. This proves that the right-hand side of \cref{pure-mixed-PF-class} is contained in the left-hand side. Conversely, if $\Psi\in \mathcal Q_\PF$ is normalized, then $|\Psi\rangle\langle\Psi|\in\dm$, which proves the reverse inclusion. Finally, $\dm$ is convex, its form energy is affine, and the map $\vGamma\mapsto(\rho_\vGamma,\vjc_\vGamma)$ is affine by the weak trace interpretation of \cref{mixed-PF-density-current}. Its image is therefore convex, and the equality \cref{pure-mixed-PF-class} proves the last assertion. \end{proof} \subsection{Proof of \texorpdfstring{\cref{thm-photonic-representability}}{Theorem \getrefnumber{thm-photonic-representability}}} \label{proof:thm-photonic-representability} The idea is to distribute the electrons independently, with the real amplitude $\prod_{n=1}^N\sqrt{\rho(\vx_n)/N}$, and to let the photon state depend on their positions. The electronic amplitude gives density $\rho$ and no paramagnetic current density, but it is symmetric, so antisymmetry is supplied by a real unit vector built from the pair directions $\vx_i-\vx_j$ and placed in a few auxiliary photon modes. This factor contributes neither current density nor field expectation. The field-induced current density comes from a coherent state in a separate band of modes, displaced along the coupling functions at the electron positions with weights $\vu(\vx_n)$. Its conditional field expectation at $\vx$ is $(\vT\vu)(\vx)$ for a positive operator $\vT$, and solving $\vT\vu=\va$ gives $\vjph_\Psi=-e\rho\va$. All photon factors are real, so no paramagnetic current density is created. The paramagnetic current density $\rho(\grad S+\vb)$ is produced by a further factor in the tag: a phase times a coherent state whose Berry connection is $\grad S+\vb$ (\cref{lem-coherent-tag}), made permutation invariant by the lift of \cref{lem-symmetric-tensor-lift}. \noindent\textbf{Step 1. An antisymmetric unit vector.} Let $m=N(N-1)/2$ and let $V=\bigvee^m\RR^3$ be the $m$th symmetric power of $\RR^3$, with its Hilbert tensor norm and dimension $d=\binom{m+2}{2}$. If $N=1$, then $V=\RR$ and we take $\eta=1$. For $N>1$, let \[ \mathcal O_N:=\big\{\vX=(\vx_1,\ldots,\vx_N)\in\RR^{3N}: \vx_i\neq\vx_j\text{ for all }i\neq j\big\} \] be the complement of the collision diagonals, and define $\tau,\eta:\mathcal O_N\to V$ by \[ \tau(\vX)=\bigvee_{i0$ on $\mathcal O_N$, so $\eta$ is well defined. Since $\grad\eta=\|\tau\|_V^{-1}(1-|\eta\rangle\langle\eta|)\grad\tau$, $\tau$ is multilinear in the $\vepsilon_{ij}$, and $|\grad_{\vx_n}\vepsilon_{ij}|\leq C|\vx_i-\vx_j|^{-1}$ for $n\in\{i,j\}$, we obtain \begin{equation}\label{photonic-antisymmetric-tag-bound} \|\eta(\vX)\|_V=1,\qquad \sum_{n=1}^N\|\grad_{\vx_n}\eta(\vX)\|_V^2 \leq C_N\sum_{i0$. Define the translation-invariant matrix kernel $\vK=(K^{\mu\nu})_{\mu,\nu=1}^3$ by \[ K^{\mu\nu}(\vx,\vy) =2\Re\dua{\iden_A\vG^\mu_{\vx}}{\iden_A\vG^\nu_{\vy}},\qquad \vM=\vK(\vx,\vx). \] The function $\iden_A\vG^\mu_{\vx}$ is $J$-real. All its spatial derivatives are uniformly bounded both in the one-photon norm and after multiplication by $\sqrt\omega$, since their momentum supports are bounded. The real matrix $\vM$ is independent of $\vx$, and \begin{equation*} \vc^{\mathsf T}\vM\vc =\int_A|\wh\chi(\vk)|^2|P_\sol(\vk)\vc|^2\dd\vk>0 \qquad(\vc\in\RR^3\setminus\{0\}). \end{equation*} Indeed, $P_\sol(\vk)\vc=0$ only when $\vk$ is parallel to $\vc$, which has measure zero. Thus $\vM$ is positive definite. The kernel $\vK$ is smooth with bounded derivatives and $\vK(\vy,\vx)=\vK(\vx,\vy)^{\mathsf T}$. Write $\Hil_A=\iden_A\Hil_\sol$ and $\Hil_{\RR^3\setminus A}=\iden_{\RR^3\setminus A}\Hil_\sol$. Under the Fock factorization \[ \Fock_s(\Hil_\sol) \simeq\Fock_s(\Hil_A)\otimes\Fock_s(\Hil_{\RR^3\setminus A}), \] choose orthonormal $\xi_1,\ldots,\xi_d\in\Hil_{\RR^3\setminus A}$, supported in the annulus $\{\vk:r/2<|\vk|0$ is the smallest eigenvalue of $\vM$, then $\vT\geq\lambda\iden$. Hence for $\vu=\vT^{-1}\va$, \begin{equation}\label{photonic-field-equation} \|\sqrt\rho\,\vu\|\leq\lambda^{-1}\|\sqrt\rho\,\va\|. \end{equation} Because $\vK$ has bounded derivatives and $\int_{\RR^3}\rho|\vu|\leq\sqrt N\|\sqrt\rho\,\vu\|$, the function $\vK_\rho\vu$ is smooth with bounded derivatives. The solution $\vu$ is determined only $\rho$-almost everywhere. The equation $\vT\vu=\va$ is equivalent to $\vM\vu=\va-\frac{N-1}N\vK_\rho\vu$ $\rho$-almost everywhere, and the right-hand side is defined almost everywhere on $\RR^3$, we fix $\vu$ on all of $\RR^3$ by \[ \vu(\vx)=\vM^{-1}\Bigl(\va(\vx)-\frac{N-1}{N}(\vK_\rho\vu)(\vx)\Bigr). \] Then $\vu$ is in $H^1_{\mathrm{loc}}$ and satisfies \begin{equation}\label{photonic-t-Sobolev-bound} \int_{\RR^3}\rho\bigl(|\vu|^2+|\grad\vu|^2\bigr) \leq C_{N,\chi}\int_{\RR^3}\rho\bigl(|\va|^2+|\grad\va|^2\bigr). \end{equation} Indeed, the zeroth-order part is \cref{photonic-field-equation}. For the gradient, Cauchy--Schwarz with $\int_{\RR^3}\rho=N$ gives \[ \sup_{\vx}|\grad(\vK_\rho\vu)(\vx)| \leq\|\grad\vK\|_\infty\int_{\RR^3}\rho|\vu| \leq\sqrt N\lambda^{-1}\|\grad\vK\|_\infty\|\sqrt\rho\,\va\|, \] and $\grad\vu=\vM^{-1}(\grad\va-\frac{N-1}N\grad\vK_\rho\vu)$. Only the bounded first derivatives of $\vK$ are used here, and no derivative of the density is taken. \noindent\textbf{Step 4. The representing vector and its energy.} Let $W(\vf)$ be the Weyl operator \cref{weyl-definitions} on $\Fock_s(\Hil_A)$, and let $\Omega_A$ be the vacuum of this factor. Let $B(\vX)=\prod_{n=1}^N\sqrt{\rho(\vx_n)/N}$ be the electronic amplitude. It has norm one and $\sum_n\|\grad_{\vx_n}B\|^2=\|\grad\sqrt\rho\|^2$. Define the displacement \[ \vf_{\vX}=\sum_{n=1}^N\sum_{\mu=1}^3 u^\mu(\vx_n)\iden_A\vG^\mu_{\vx_n} \] and the state \begin{equation}\label{photonic-conditional-state} \boxed{\Psi(\vX)=B(\vX)W(\vf_{\vX})\Omega_A\otimes\zeta(\vX).} \end{equation} The displacement $\vf_\vX$ is symmetric in the electronic positions and $J$-real. Thus $\Psi$ is antisymmetric and \[ \|\Psi(\vX)\|^2=B(\vX)^2=\prod_{n=1}^N\frac{\rho(\vx_n)}N. \] Here $\|\Psi(\vX)\|$ is the norm in the photon Fock space. The state $\Psi$ is normalized and has density $\rho$. Since $\vX\mapsto\vf_\vX$ is $J$-real and lies in $H^1_{\rm loc}(\RR^{3N};\Hil_A)$, \cref{prop-weyl-coherent-derivatives}(iv) shows that $W(\vf_\vX)\Omega_A$ is a unit $H^1_{\rm loc}$ vector and gives, for $n=1,\ldots,N$, \begin{equation}\label{photonic-real-coherent-derivatives} \begin{gathered} \dua{W(\vf_\vX)\Omega_A}{\grad_{\vx_n}\bigl(W(\vf_\vX)\Omega_A\bigr)} =0,\\ \|\grad_{\vx_n}\bigl(W(\vf_\vX)\Omega_A\bigr)\|^2 =\|\grad_{\vx_n}\vf_\vX\|^2. \end{gathered} \end{equation} Together with \cref{photonic-tag-zero-current}, these identities make the three terms obtained by differentiating \cref{photonic-conditional-state} pairwise orthogonal. Hence \begin{align*} \sum_n\|\grad_{\vx_n}\Psi\|^2 &=\sum_n\|\grad_{\vx_n}B\|^2 +\int_{\RR^{3N}} B(\vX)^2\sum_n\|\grad_{\vx_n}\zeta\|^2\dd\vX\\ &\quad+\int_{\RR^{3N}} B(\vX)^2\sum_n\|\grad_{\vx_n}\vf_{\vX}\|^2\dd\vX\\ &\leq C_N\|\grad\sqrt\rho\|^2 +C_{N,\chi}\int_{\RR^3}\rho\bigl(|\va|^2+|\grad\va|^2\bigr). \end{align*} To justify the last inequality, first consider the term involving $\grad_{\vx_n}\zeta$. The identity $\|\grad_{\vx_n}\zeta\|=\|\grad_{\vx_n}\eta\|_V$ and \cref{photonic-antisymmetric-tag-bound} reduce it to $C_N\int B^2\sum_{i0$, the field also enters the combined current, and its circulation around a loop could make up the difference. The remaining steps rule this out for a minimizer. Step~2 shows that the field has small circulation around thin loops. Steps~3 and~4 give matching lower and upper energy bounds, from which Step~5 deduces that a pure minimizer is almost photon-free. Step~6 then applies the winding argument of the claim to its vacuum component. \noindent\textbf{Step 2. The flux through the loops.} For $\vt\in(-r_n,r_n)^2$, let \[ g_{n,s,\vt}=\sum_{\mu=1}^3G^\mu_{\vX_n(s,\vt)}\, \partial_sX_n^\mu(s,\vt),\qquad \mathcal F_{n,\vt}=\int_0^{L_n}g_{n,s,\vt}\dd s. \] Thus $\varPhi(g_{n,s,\vt})=\AF(\vX_n)\cdot\partial_s\vX_n$, and $\varPhi(\mathcal F_{n,\vt})$ is the circulation of the field around the loop $s\mapsto\vX_n(s,\vt)$. For $\nu=0,1$, put $M_\nu=\|\omega^{-\nu/2}\wh\chi\|$ and $d_\nu(\ell)=\|\min\{1,\ell\omega\}\,\omega^{-\nu/2}\wh\chi\|$, with the norm of $L^2(\RR^3)$. By \cref{cutoff-assumption}, $M_\nu<\infty$ and $d_\nu(\ell)\downarrow0$ as $\ell\downarrow0$. By \cref{Gx-norms} and $|e^{\i t}-1|\leq2\min\{1,|t|\}$, \begin{equation}\label{thin-loop-kernel-continuity} \sum_{\mu=1}^3\|\omega^{-\nu/2}G^\mu_{\vx}\|^2=M_\nu^2,\qquad \sum_{\mu=1}^3\|\omega^{-\nu/2}(G^\mu_{\vx}-G^\mu_{\vy})\|^2 \leq4d_\nu(|\vx-\vy|)^2. \end{equation} We now choose $a_n$. As $d_\nu(\ell)\downarrow0$, we may take $a_n$ so small that \begin{equation}\label{thin-loop-scales} a_n\leq b_n^3,\qquad d_0(2a_n)+d_1(2a_n)\leq b_n^2. \end{equation} We claim that, with $C$ depending only on the form factor, \begin{equation}\label{thin-loop-flux-kernel} \sup_{\vt\in(-r_n,r_n)^2}\|\omega^{-\nu/2}\mathcal F_{n,\vt}\|\leq Cb_nL_n^2, \qquad\nu=0,1. \end{equation} All norms in this step are $\|\omega^{-\nu/2}\cdot\|$ on $\Hil_\ph$. Parametrize the central ellipse by $\phi\mapsto(a_n\cos\phi,b_n\sin\phi,0)$. Since $x_1$ has total variation $4a_n$, the $\mu=1$ part of $\mathcal F_{n,0}$ is at most $4a_nM_\nu$. In the $\mu=2$ part, $\phi\mapsto\pi-\phi$ pairs points with equal $x_2$, opposite orientation, and distance at most $2a_n$, so \cref{thin-loop-kernel-continuity} bounds it by $4b_nd_\nu(2a_n)$. Off the central ellipse, $\partial_s\vX_n=(1-\kappa_nt_1)\dot{\vgamma}_n$ with $|\kappa_nt_1|\leq\kappa_nr_n$, and $|\vX_n(s,\vt)-\vgamma_n(s)|\leq\sqrt2r_n$. Since $\int_0^{L_n}\kappa_n\dd s=2\pi$, this gives $\|\mathcal F_{n,\vt}-\mathcal F_{n,0}\|\leq2L_nd_\nu(2r_n)+2\pi r_nM_\nu$. By \cref{thin-loop-scales} and $r_n0$ gives \begin{equation}\label{thin-loop-circulation} \Re\dua{u}{D_su}=\frac{\pi}{L_n} \qquad\text{almost everywhere on }\mathcal T_n. \end{equation} As $\|u\|=1$, this gives $\|D_su\|^2=\|(D_s-\pi/L_n)u\|^2+\pi^2/L_n^2$. Keeping only the $\dot{\vgamma}_n$-component of the covariant gradient and using $J_n\leq1+\epsilon_n$, we obtain \begin{equation}\label{thin-loop-energy-split} \begin{aligned} &\dua{\Psi}{H^{0,0}\Psi}-\frac12\int_{\RR^3}|\grad\sqrt{\rho_n}|^2\\ &\qquad\geq\frac{1}{2(1+\epsilon_n)^2}\left[\frac{\pi^2}{L_n^2} +\int_{\mathcal T_n}\rho_n\|(D_s-\pi/L_n)u\|^2\dd\vx\right] +\dua{\Psi}{H_\rad\Psi}. \end{aligned} \end{equation} To bound the last integral, we represent $\Fock_\ph$ by the Wiener--It\^o--Segal isomorphism of Appendix~\ref{app-soft-gaussian} with $\mathfrak h=\Hil_\ph$. Then $\mathcal A_n$ becomes multiplication by a real function $\mathcal A_n(s,\vt;q)$ of the Gaussian point $q$, square integrable in $s$ for almost every $q$, and $\varPhi(\mathcal F_{n,\vt})$ becomes multiplication by its integral over $s$. On $\RR/L\ZZ$, a real potential $\beta\in L^2$ is gauge equivalent to its mean, so Fourier series give $(-\i\partial_s+\beta)^2\geq L^{-2} \operatorname{dist}(\int_0^L\beta,2\pi\ZZ)^2$ as forms on $H^1(\RR/L\ZZ)$. For almost every $\vt$, the map $s\mapsto u(\vX_n(s,\vt))$ lies in $H^1(\RR/L_n\ZZ;\Fock_\ph)$. For such $\vt$ and almost every $q$, apply this with $\beta=-\pi/L_n-e\mathcal A_n(\cdot,\vt;q)$, and use $\operatorname{dist}(\pi+x,2\pi\ZZ)^2\geq\pi^2-2\pi|x|$. Integrating over $q$, and then over $\vt$ by \cref{thin-loop-measure}, gives \[ \int_{\mathcal T_n}\rho_n\|(D_s-\pi/L_n)u\|^2\dd\vx \geq\frac{\pi^2}{L_n^2} -\frac{2\pi e}{L_n^2}\int_{\mathcal T_n}\rho_n \dua{u}{|\varPhi(\mathcal F_{n,\vt})|u}\dd\vx. \] By \cref{ann-Hrad-bound,cre-Hrad-bound}, $\dua{u}{|\varPhi(\mathcal F_{n,\vt})|u}\leq\|\varPhi(\mathcal F_{n,\vt})u\| \leq2\|\mathcal F_{n,\vt}/\sqrt\omega\|\,\|\sqrt{H_\rad}u\| +\|\mathcal F_{n,\vt}\|$, and $\int\rho_n\|\sqrt{H_\rad}u\|\leq\|\sqrt{H_\rad}\Psi\|$. Inserting these bounds and \cref{thin-loop-flux-kernel} into \cref{thin-loop-energy-split} proves \begin{equation}\label{thin-loop-lower-bound} \begin{aligned} &\dua{\Psi}{H^{0,0}\Psi}-\frac12\int_{\RR^3}|\grad\sqrt{\rho_n}|^2\\ &\qquad\geq\frac{\pi^2}{(1+\epsilon_n)^2L_n^2}+\dua{\Psi}{H_\rad\Psi} -\delta_n\big(1+\|\sqrt{H_\rad}\Psi\|\big), \end{aligned} \end{equation} where $\delta_n=2\pi eCb_n$. \noindent\textbf{Step 4. A mixed-state upper bound.} For $K>0$, let $\AF_{0$, there is a finite-rank orthogonal projection $S$ with $\sup_j\Tr((1-S)\Gamma_j)<\varepsilon$. \end{lemma} \begin{proof} Writing $\Gamma-S\Gamma S=(1-S)\Gamma+S\Gamma(1-S)$ and factoring through $\sqrt{\Gamma}$, the Hilbert--Schmidt ideal inequality gives \[ \|\Gamma-S\Gamma S\|_{\mathfrak S_1} \leq2\|\sqrt{\Gamma}\|_{\mathfrak S_2} \|(1-S)\sqrt{\Gamma}\|_{\mathfrak S_2} \leq2\big(\Tr((1-S)\Gamma)\big)^{1/2}. \] To prove precompactness, let $\delta>0$ and choose a finite-rank orthogonal projection $S$ with $\sup_j\Tr((1-S)\Gamma_j)<\delta^2/16$. The bound \cref{density-matrix-compression} then gives the uniform approximation \[ \sup_j\|\Gamma_j-S\Gamma_jS\|_{\mathfrak S_1}<\frac\delta2. \] Each compression is positive and satisfies \[ \Tr(S\Gamma_jS)=\Tr(S\Gamma_j)\leq\Tr\Gamma_j\leq1. \] Thus all the compressions belong to \[ \mathcal K_S :=\{A\in\mathfrak S_1(\mathfrak K): A=SAS,\ A\geq0,\ \Tr A\leq1\}. \] This set is closed in trace norm: the map $A\mapsto SAS$ and the trace are continuous, and positivity is preserved under trace-norm limits. It is bounded because $\|A\|_{\mathfrak S_1}=\Tr A\leq1$ for $A\in\mathcal K_S$. Moreover, the condition $A=SAS$ identifies these operators with operators on the finite-dimensional space $\ran S$. Hence $\mathcal K_S$ is compact in trace norm. Choose a finite $\delta/2$-net $A_1,\ldots,A_m$ for $\mathcal K_S$. For each $j$, some $k\in\{1,\ldots,m\}$ therefore satisfies $\|S\Gamma_jS-A_k\|_{\mathfrak S_1}<\delta/2$, and the triangle inequality gives \[ \|\Gamma_j-A_k\|_{\mathfrak S_1} \leq\|\Gamma_j-S\Gamma_jS\|_{\mathfrak S_1}+\|S\Gamma_jS-A_k\|_{\mathfrak S_1} <\delta. \] The sequence is thus covered by finitely many trace-norm balls of radius $\delta$. Since $\delta>0$ was arbitrary, the set $\{\Gamma_j:j\geq1\}$ is totally bounded. The trace class $\mathfrak S_1(\mathfrak K)$ is complete, so the closure of this set is compact, which proves precompactness in trace norm. \end{proof} For a positive trace-class operator $\vGamma$ on $\HC_N$ with $\Tr\vGamma\leq1$, not necessarily equal to one, define the internal energy $\Tr(H^{0,0}\vGamma)\in[0,\infty]$ by the spectral sum following \cref{finite-energy-PF-ensembles}, and similarly the trace of $\vGamma$ against any nonnegative quadratic form in place of $H^{0,0}$. When the internal energy is finite, define the density $\rho_\vGamma$ and the combined current density $\vjc_\vGamma$ by \cref{mixed-PF-density-current}. These definitions are linear in $\vGamma$ and agree with the earlier ones on $\dm$. \begin{lemma}[Lower semicontinuity and passage to the constraints] \label[lemma]{lem-trace-compactness-and-limits} Let $\vGamma_j$ be positive trace-class operators on $\HC_N$ with $\Tr\vGamma_j\leq1$ and uniformly bounded internal energies, \[ \sup_j\Tr(H^{0,0}\vGamma_j)<\infty, \] and suppose that $\vGamma_j\to\vGamma$ in trace norm. Then $\vGamma$ is positive, $\Tr\vGamma=\lim_j\Tr\vGamma_j$, and \begin{equation}\label{mixed-energy-lsc} \Tr(H^{0,0}\vGamma) \leq\liminf_j\Tr(H^{0,0}\vGamma_j). \end{equation} In particular, $\vGamma$ has finite internal energy, so $\rho_\vGamma$ and $\vjc_\vGamma$ are defined. The same liminf inequality holds for every nonnegative quadratic form on $Q(H^{0,0})$ bounded relative to $H^{0,0}+1$. For bounded scalar $f$ and real $\vzeta\in C_c^\infty(\RR^3,\RR^3)$, \begin{align} \int_{\RR^3} f\rho_{\vGamma_j}&\longrightarrow\int_{\RR^3} f\rho_\vGamma, \label{mixed-density-passage}\\ \int_{\RR^3}\vzeta\cdot\vjc_{\vGamma_j} &\longrightarrow\int_{\RR^3}\vzeta\cdot\vjc_\vGamma. \label{mixed-current-passage} \end{align} \end{lemma} \begin{proof} Positivity and convergence of the traces follow from trace-norm convergence. The Powers--St\o rmer inequality gives \[ \|\sqrt{\vGamma_j}-\sqrt{\vGamma}\|_{\mathfrak S_2}^2 \leq\|\vGamma_j-\vGamma\|_{\mathfrak S_1}\longrightarrow0. \] Equip the form domain $Q(H^{0,0})$ with the energy norm $u\mapsto\dua{u}{(H^{0,0}+1)u}^{1/2}$, making it a Hilbert space continuously embedded in $\HC_N$. Let $\mathfrak S_2(\HC_N,Q(H^{0,0}))$ be the Hilbert--Schmidt operators from $\HC_N$ into this space. Thus an operator $T$ on $\HC_N$ belongs to it when its range lies in the form domain and $(H^{0,0}+1)^{1/2}T$ is Hilbert--Schmidt, and then $\|T\|_{\mathfrak S_2(\HC_N,Q(H^{0,0}))} =\|(H^{0,0}+1)^{1/2}T\|_{\mathfrak S_2}$. Evaluating this norm on an eigenbasis of $\vGamma_j$ gives \[ \|\sqrt{\vGamma_j}\|_{\mathfrak S_2(\HC_N,Q(H^{0,0}))}^2 =\Tr\vGamma_j+\Tr(H^{0,0}\vGamma_j), \] which is uniformly bounded. The continuous embedding into $\mathfrak S_2$ and the strong convergence there identify every weak subsequential limit in this stronger Hilbert space with $\sqrt{\vGamma}$. Thus $\sqrt{\vGamma}$ belongs to that space and $\sqrt{\vGamma_j}\rightharpoonup\sqrt{\vGamma}$ there. Weak lower semicontinuity proves \cref{mixed-energy-lsc}. A nonnegative form bounded relative to $H^{0,0}+1$ is $u\mapsto\|Bu\|^2$ for some operator $B$ bounded from $Q(H^{0,0})$ into a Hilbert space. Composition with $B$ preserves weak Hilbert--Schmidt convergence, so weak lower semicontinuity applies to it as well. For the density, test the trace-norm convergence $\vGamma_j\to\vGamma$ against the multiplication operator $M_f:=\sum_{i=1}^N f(\vx_i)$, acting as the identity on the photon factor. It preserves electronic antisymmetry and satisfies $\|M_f\|_{\mathrm{op}}\leq N\|f\|_\infty$. Hence \begin{align*} \left|\int_{\RR^3}f(\rho_{\vGamma_j}-\rho_\vGamma)\right| &=\left|\Tr\big(M_f(\vGamma_j-\vGamma)\big)\right|\\ &\leq N\|f\|_\infty \|\vGamma_j-\vGamma\|_{\mathfrak S_1}\longrightarrow0, \end{align*} which proves \cref{mixed-density-passage}. For the current density, recall from \cref{sec-PF-hamiltonian} that $\vD_1=-\i\grad_{\vx_1}-e\AF(\vx_1)$ is the covariant momentum of the first electron. Every term in \cref{q00-def} is nonnegative, so $\|\vD_1u\|^2\leq2\dua{u}{H^{0,0}u}$ for $u\in Q(H^{0,0})$. Hence $\vD_1$ is bounded from $Q(H^{0,0})$, with its energy norm, to the ambient vector-valued electron--photon $L^2$ space. We therefore also have $\vD_1\sqrt{\vGamma_j}\rightharpoonup\vD_1\sqrt{\vGamma}$ in the corresponding Hilbert--Schmidt class. As operators into this vector-valued space, the factors $\vzeta(\vx_1)\sqrt{\vGamma_j}$ and $\vD_1\sqrt{\vGamma_j}$ converge strongly and weakly in $\mathfrak S_2$, respectively. The product of the adjoint of the first factor with the second is trace class, and \begin{align*} \int_{\RR^3}\vzeta\cdot\vjc_{\vGamma_j} &=N\Re\Tr\Big[ \big(\vzeta(\vx_1)\sqrt{\vGamma_j}\big)^\dag \vD_1\sqrt{\vGamma_j}\Big]\\ &\longrightarrow N\Re\Tr\Big[ \big(\vzeta(\vx_1)\sqrt{\vGamma}\big)^\dag \vD_1\sqrt{\vGamma}\Big] =\int_{\RR^3}\vzeta\cdot\vjc_\vGamma. \end{align*} This proves \cref{mixed-current-passage}. \end{proof} \begin{remark}[Continuum compactness obstruction] \label{rem-continuum-obstruction} Uniform energy bounds and a fixed electronic density do not ensure trace-norm compactness in the continuum. Choose an orthonormal sequence $\xi_n\in\Hil_\ph$ in a bounded momentum shell and a normalized smooth compactly supported electronic state $\Phi$. Then $\Psi_n=\Phi\otimes a^\dag(\xi_n)\Omega$ has fixed density, one photon, and bounded energy, while $|\Psi_n\rangle\langle\Psi_n|$ has no trace-norm convergent subsequence. Photon loss must therefore be allowed in constructing limits. \Cref{thm-continuum-compactness} implements this using localization and partial trace, and \cref{sec-unrestricted-attainment} supplies its photon-number condition by modifying minimizing sequences at fixed density and combined current density. \end{remark} Photon loss can be measured by localization, which preserves normalization but can turn a pure state into a mixed state. We use geometric localization in Fock space, as developed in \cite[Section~3]{lewin2011geometric}. We first recall the notation of \cref{sec-fock-space,sec-second-quantization}, specialized to the photon Fock space $\Fock_\ph=\Fock_s(\Hil_\ph)$. For $f\in\Hil_\ph$, the Segal field $\varPhi(f)$ is the self-adjoint closure of $a^\dag(f)+a(f)$, initially defined on the finite-particle space $\Fock_\ph^\fin$. It depends real-linearly on $f$, the conjugate field is $\varPi(f)=\varPhi(\i f)$, and the vector potential \cref{Afield-def} is $\Af^\mu(\vx)=\varPhi(G^\mu_{\vx})$. In the vacuum, $\langle\Omega,\varPhi(f)\Omega\rangle=0$ and $\|\varPhi(f)\Omega\|=\|f\|$. For a self-adjoint operator $b$ on $\Hil_\ph$, its second quantization $\dd\varGamma(b)=\bigoplus_{M\geq0}b^{(M)}$ acts on the $M$-photon sector as the closure of $\sum_{j=1}^Mb_j$, where $b_j$ acts on the $j$th photon variable. For nonnegative $b$, the same symbol denotes the corresponding nonnegative quadratic form. For a unitary $T$ on $\Hil_\ph$, or on $\Hil_\ph\oplus\Hil_\ph$ as below, the multiplicative second quantization $\varGamma(T)=\bigoplus_{M\geq0}T^{\otimes M}$, restricted to symmetric tensors, is unitary on the corresponding bosonic Fock space and satisfies $\varGamma(T)\varPhi(f)\varGamma(T)^\dag=\varPhi(Tf)$. Finally, the Weyl operator \cref{weyl-definitions} is $W(f)=\exp(a^\dag(f)-a(f))$, so that $W(\i f)=e^{\i\varPhi(f)}$. Fix the polarization representation $L^2(\RR^3\times\{1,2\})$ and Fourier transform each photon momentum to a position variable $\vy$. Then $H_\rad=\dd\varGamma(\sqrt{-\Delta})$, and $\NC=\dd\varGamma(1)$ denotes photon number. Here $\sqrt{-\Delta}$ is the Fourier multiplier $|\vk|$. The variable $\vy$ is a technical device rather than a physical photon position.\footnote{It is the Fourier dual of $\vk$ in the fixed polarization frame $\vepsilon_\lambda$, which cannot be continuous on spheres $\{|\vk|=r\}$ by the hairy ball theorem, so the localization below depends on this frame. We use the polarization representation because its one-photon space is all of $L^2(\RR^3;\CC^2)$, so multiplication by functions of $\vy$ is available; in the solenoidal representation it would not preserve the divergence-free constraint. Only the photon-number hypothesis \cref{continuum-local-sector-tightness} of \cref{thm-continuum-compactness} refers to the frame; its conclusion involves only the density, the combined current density, and the internal energy.} Fix smooth real functions $\theta_1,\phi_1$ on $\RR^3$ with $0\leq\theta_1,\phi_1\leq1$, $\theta_1^2+\phi_1^2=1$, $\theta_1=1$ on $B_1$, and $\theta_1=0$ outside $B_2$. For $R>0$, set $\theta_R(\vy):=\theta_1(\vy/R)$ and $\phi_R(\vy):=\phi_1(\vy/R)$. Thus $\theta_R^2+\phi_R^2=1$, $\theta_R=1$ on $B_R$, and $\theta_R=0$ outside $B_{2R}$. The one-photon isometry $f\mapsto\theta_Rf\oplus\phi_Rf$ induces a Fock-space isometry $\mathcal U_R$. With the electronic identity suppressed, set \[ \mathcal L_R(\vGamma) :=\Tr_{\mathrm{out}}(\mathcal U_R\vGamma\mathcal U_R^\dag). \] Thus $\mathcal L_R$ retains nearby photons and traces out the second photon factor. \begin{lemma}[Photon localization and field identities] \label[lemma]{lem-photon-localization} For every $R>0$, the map $\mathcal L_R$ is positive and trace preserving, and leaves the electronic reduced state unchanged. For $f\in\Hil_\ph$, its adjoint with respect to the trace pairing satisfies \begin{align} \mathcal L_R^\dag(\varPhi(f))&=\varPhi(\theta_Rf), \label{localized-linear-field}\\ \mathcal L_R^\dag(\varPhi(f)^2)&=\varPhi(\theta_Rf)^2+\|\phi_Rf\|^2\iden. \label{localized-quadratic-field} \end{align} These identities are understood as quadratic-form identities on $D(\varPhi(\theta_Rf))$, with the electronic identity suppressed. \end{lemma} \begin{proof} Since $\mathcal U_R^\dag\mathcal U_R=\iden$, conjugation by $\mathcal U_R$ followed by partial trace is positive and trace preserving. For every bounded electronic operator $A$, \[ \mathcal U_R^\dag(A\otimes\iden\otimes\iden)\mathcal U_R =A\otimes\iden. \] Thus the electronic reduced state is unchanged. By monotone spectral truncation, this also preserves the expectations of nonnegative unbounded electronic observables, including the ordinary kinetic energy and the Coulomb interaction. To prove the field identities, complete the one-photon isometry $j_Ru=(\theta_Ru,\phi_Ru)$ to the unitary \[ T_R=\begin{pmatrix}\theta_R&-\phi_R\\ \phi_R&\theta_R\end{pmatrix} \quad\text{on }\Hil_\ph\oplus\Hil_\ph. \] Unitarity follows from $\theta_R^2+\phi_R^2=1$ and commutation of these multiplication operators. Under the canonical identification $\Fock_s(\Hil_\ph\oplus\Hil_\ph) \simeq\Fock_\ph\otimes\Fock_\ph$, write $V_R=\varGamma(T_R)$ for its second quantization. Then \[ \mathcal U_R\Psi=V_R(\Psi\otimes\Omega),\qquad V_R^\dag(\varPhi(f)\otimes\iden)V_R =\varPhi(\theta_Rf)\otimes\iden -\iden\otimes\varPhi(\phi_Rf). \] For the second identity, \cref{weyl-second-quantization,weyl-factorization} give, for $s\in\RR$, \[ V_R^\dag\big(e^{\i s\varPhi(f)}\otimes\iden\big)V_R =W\big(\i sT_R^\dag(f\oplus0)\big) =e^{\i s\varPhi(\theta_Rf)}\otimes e^{-\i s\varPhi(\phi_Rf)}, \] and the generators of the two sides agree. Since \[ \langle\Omega,\varPhi(\phi_Rf)\Omega\rangle=0,\qquad \|\varPhi(\phi_Rf)\Omega\|^2=\|\phi_Rf\|^2, \] vacuum compression gives \cref{localized-linear-field} and \[ \|(\varPhi(f)\otimes\iden)\mathcal U_R\Psi\|^2 =\|\varPhi(\theta_Rf)\Psi\|^2 +\|\phi_Rf\|^2\|\Psi\|^2. \] This proves \cref{localized-quadratic-field} by polarization. The two fields on the separate factors strongly commute. Their joint spectral calculus, and the finite second moment of the vacuum in the second factor, show that the norm on the left is finite exactly when $\Psi\in D(\varPhi(\theta_Rf))$. Thus these are identities on the stated form domain, without a mean-photon-number assumption. The extra scalar term is the vacuum fluctuation in the discarded modes. \end{proof} \begin{lemma}[Localization of one-photon forms] \label[lemma]{lem-photon-localization-forms} For every $R>0$ and nonnegative self-adjoint one-photon operator $b$, the corresponding nonnegative forms obey \begin{equation}\label{localized-second-quantization} \mathcal L_R^\dag(\dd\varGamma(b)) =\dd\varGamma(\theta_R b\theta_R), \end{equation} where $\theta_R b\theta_R$ denotes the form $u\mapsto\|\sqrt{b}\theta_Ru\|^2$. Moreover, with $C$ depending only on $\theta_1$, \begin{equation}\label{localized-one-photon-bounds} \theta_R^2\leq C R\sqrt{-\Delta},\qquad \theta_R\sqrt{-\Delta}\theta_R\leq C\sqrt{-\Delta} \end{equation} as forms on $H^{1/2}(\RR^3;\CC^2)$, uniformly for $R>0$. \end{lemma} \begin{proof} Write $j_Ru=(\theta_Ru,\phi_Ru)$ for the one-photon localization isometry. For \cref{localized-second-quantization}, observe that $\dd\varGamma(b)\otimes\iden$ corresponds to $\dd\varGamma(b\oplus0)$ and \[ j_R^\dag(b\oplus0)j_R=\theta_R b\theta_R. \] Since $\mathcal U_R$ is induced by $j_R$, it preserves the total photon number and acts on the $m$-photon sector as $j_R^{\otimes m}$, followed by the canonical identification. On this sector, write $B_k$ for the operator $b\oplus0$ acting on the $k$th photon variable, so that $\dd\varGamma(b\oplus0)=\sum_{k=1}^mB_k$ there. Each term factors over the $m$ photon variables, and since $j_R^\dag j_R=\iden$, \[ (j_R^{\otimes m})^\dag B_k\,j_R^{\otimes m} =j_R^\dag j_R\otimes\cdots\otimes j_R^\dag(b\oplus0)j_R \otimes\cdots\otimes j_R^\dag j_R =\iden\otimes\cdots\otimes\theta_Rb\theta_R\otimes\cdots\otimes\iden, \] with the nontrivial factor in the $k$th position. Summing over $k$ gives $\dd\varGamma(\theta_Rb\theta_R)$ on the $m$-photon sector, and summing over $m$ gives the asserted identity. The argument first applies to bounded $b$ on finitely many sectors. Spectral truncation of $b$ and monotone convergence over the sectors give the identity for arbitrary nonnegative $b$, with possibly infinite form values. We next prove \cref{localized-one-photon-bounds}. Work first with $u\in C_c^\infty(\RR^3;\CC^2)$, with the two components corresponding to the photon polarizations. The fractional Hardy inequality, applied to each component, gives \[ \int_{\RR^3}\frac{|u(\vy)|^2}{|\vy|}\dd\vy \leq C_{\mathrm H}\|(-\Delta)^{1/4}u\|^2; \] see \cite[Section~2]{frank2008hardy}. Since $0\leq\theta_R\leq1$ and $\theta_R=0$ outside $B_{2R}$, we have $\theta_R(\vy)^2\leq2R/|\vy|$ almost everywhere. Thus \[ \|\theta_Ru\|^2 \leq2R\int_{\RR^3}\frac{|u(\vy)|^2}{|\vy|}\dd\vy \leq2C_{\mathrm H}R\|(-\Delta)^{1/4}u\|^2, \] which proves the first inequality. For the second inequality, use the integral representation \[ \|(-\Delta)^{1/4}u\|^2 =c_3\iint_{\RR^3\times\RR^3} \frac{|u(\vy)-u(\vz)|^2}{|\vy-\vz|^4}\dd\vy\dd\vz, \qquad c_3>0. \] Expanding the difference as \[ \theta_R(\vy)u(\vy)-\theta_R(\vz)u(\vz) =\theta_R(\vy)\big(u(\vy)-u(\vz)\big) +\big(\theta_R(\vy)-\theta_R(\vz)\big)u(\vz) \] and using $|a+b|^2\leq2|a|^2+2|b|^2$ yields \begin{gather*} \|(-\Delta)^{1/4}(\theta_Ru)\|^2 \leq2\|(-\Delta)^{1/4}u\|^2 +2c_3\int_{\RR^3} K_R(\vz)|u(\vz)|^2\dd\vz,\\ K_R(\vz):=\int_{\RR^3} \frac{|\theta_R(\vy)-\theta_R(\vz)|^2}{|\vy-\vz|^4}\dd\vy. \end{gather*} We claim that $K_R(\vz)\leq C/|\vz|$ for $\vz\neq0$. Indeed, since $\theta_R(\vy)=\theta_1(\vy/R)$, smoothness and boundedness of $\theta_1$ give \[ |\theta_R(\vy)-\theta_R(\vz)| \leq C\min\{|\vy-\vz|/R,1\}. \] Splitting the integral at $|\vy-\vz|=R$ and using spherical coordinates therefore gives, for every $\vz$, \[ K_R(\vz) \leq C\left(R^{-2}\int_0^R\dd r +\int_R^\infty r^{-2}\dd r\right) \leq\frac{C}{R}. \] For $0<|\vz|\leq4R$ this is at most $4C/|\vz|$. If $|\vz|>4R$, then $\theta_R(\vz)=0$, and on the support of $\theta_R$ we have $|\vy-\vz|\geq|\vz|/2$. Hence \[ K_R(\vz) =\int_{B_{2R}}\frac{\theta_R(\vy)^2}{|\vy-\vz|^4}\dd\vy \leq\frac{C R^3}{|\vz|^4} \leq\frac{C}{|\vz|}, \] which proves the claim. Applying the Hardy inequality once more, we obtain \begin{align*} \|(-\Delta)^{1/4}(\theta_Ru)\|^2 &\leq2\|(-\Delta)^{1/4}u\|^2 +C\int_{\RR^3}\frac{|u(\vz)|^2}{|\vz|}\dd\vz\\ &\leq C\|(-\Delta)^{1/4}u\|^2. \end{align*} By density of $C_c^\infty(\RR^3;\CC^2)$ in $H^{1/2}(\RR^3;\CC^2)$ and boundedness of multiplication by $\theta_R$ on $L^2$, these estimates extend to all of $H^{1/2}(\RR^3;\CC^2)$. In particular, $\theta_R$ preserves this form domain. This proves both form inequalities with constants depending only on $\theta_1$. \end{proof} \begin{lemma}[Photon-number and energy bounds] \label[lemma]{lem-photon-localization-energy} There is a constant $C$, depending only on $\theta_1$, such that for every $R>0$ and every positive trace-class operator $\vGamma$, \begin{align} \Tr(\NC\mathcal L_R\vGamma) &=\Tr\big(\dd\varGamma(\theta_R^2)\vGamma\big) \leq\Tr(\NC\vGamma), \label{localized-photon-number}\\ \Tr(\NC\mathcal L_R\vGamma) &\leq C R\Tr(H_\rad\vGamma), \label{localized-number-energy-bound}\\ \Tr(H_\rad\mathcal L_R\vGamma) &\leq C\Tr(H_\rad\vGamma). \label{localized-radiation-bound} \end{align} The nonnegative traces may take the value $+\infty$. In particular, $\mathcal L_R$ maps $\dm$ into $\dm$, and \begin{equation}\label{localized-internal-energy-bound} \Tr(H^{0,0}\mathcal L_R\vGamma) \leq C\big(\Tr(H^{0,0}\vGamma)+\Tr\vGamma\big) \end{equation} for positive $\vGamma$ of finite internal energy, with $C$ independent of $R$ (but also depending on $N,e$, and the form factor). \end{lemma} \begin{proof} Second quantization preserves nonnegative form inequalities by summing them over the photons in each sector. Taking $b=1$ in \cref{localized-second-quantization} and using $0\leq\theta_R^2\leq1$ therefore proves \cref{localized-photon-number}. Taking $b=1,\sqrt{-\Delta}$ and applying \cref{localized-one-photon-bounds} gives \cref{localized-number-energy-bound,localized-radiation-bound}. The trace assertions follow by summing the vector identities and inequalities over a spectral decomposition of $\vGamma$, using monotone convergence. In particular, the localized mean photon number is finite whenever the original radiation energy is finite, even if $\Tr(\NC\vGamma)=\infty$. By \cref{lem-photon-localization}, the ordinary electronic kinetic energy is unchanged, while \cref{localized-radiation-bound} bounds the radiation energy uniformly in $R$. Applying the form-norm equivalence \cref{PF-form-norm-equivalence} before and after localization proves \cref{localized-internal-energy-bound}. Positivity and preservation of the trace then show that $\mathcal L_R(\dm)\subset\dm$. \end{proof} \begin{lemma}[Trace-norm bound for photon localization] \label[lemma]{lem-localization-trace-distance} Let $0\leq\eta\leq1$ be an operator on $\Hil_\ph$, and let $\mathcal U_\eta$ be the Fock-space localization isometry induced by $u\mapsto\eta u\oplus\sqrt{1-\eta^2}\,u$. With the electronic identity suppressed, define \[ \mathcal L_\eta(\tau) :=\Tr_{\mathrm{out}}(\mathcal U_\eta\tau\mathcal U_\eta^\dag). \] Then \begin{align*} \|\tau-\mathcal L_\eta\tau\|_{\mathfrak S_1}&\leq2\sqrt t+t,\\ t:=\Tr\big((1-\varGamma(\eta^2))\tau\big) &\leq\Tr\big(\dd\varGamma(1-\eta^2)\tau\big) \end{align*} for every positive trace-one operator $\tau$ on $\HC_N$. \end{lemma} Here \[ \varGamma(\eta) :=\bigoplus_{m\geq0}\eta^{\otimes m}\big|_{\Hil_\ph^{\otimes_s m}}, \qquad \eta^{\otimes0}=1, \] is the multiplicative second quantization, extending the earlier $\varGamma(U)$ construction from unitaries to contractions. The additive lift $\dd\varGamma(b)$ acts on the $m$-photon sector as $\sum_{j=1}^m b_j$ and vanishes on the vacuum. \begin{proof} Projecting the discarded factor onto its vacuum gives \[ K_0:=(\iden\otimes\langle\Omega|)\mathcal U_\eta =\varGamma(\eta). \] Indeed, on the $m$-photon sector this projection retains precisely the term $\eta^{\otimes m}$. Splitting the partial trace into the discarded vacuum and its orthogonal complement yields \[ \mathcal L_\eta\tau=K_0\tau K_0+\sigma,\qquad \sigma\geq0,\qquad \Tr\sigma=\Tr\big((1-K_0^2)\tau\big)=t, \] where $K_0^2=\varGamma(\eta^2)$. Thus $t\in[0,1]$ is the probability that the discarded factor contains at least one photon. Since $K_0$ is a positive contraction and $(1-K_0)^2\leq1-K_0^2$, Hilbert--Schmidt factorization gives $\|\tau-K_0\tau K_0\|_{\mathfrak S_1}\leq2\sqrt t$. The triangle inequality and $\|\sigma\|_{\mathfrak S_1}=t$ prove the trace-norm bound. Finally, on each photon sector the commuting contractions $(\eta^2)_j$ satisfy $1-\prod_j(\eta^2)_j\leq\sum_j(1-(\eta^2)_j)$. Summing over sectors gives $1-\varGamma(\eta^2)\leq\dd\varGamma(1-\eta^2)$ and hence the bound on $t$. \end{proof} We now combine \cref{lem-trace-compactness-and-limits} with the localization bounds of \cref{lem-photon-localization,lem-photon-localization-forms,lem-photon-localization-energy,lem-localization-trace-distance} to construct a limit state that has the limiting density and current, and whose internal energy is at most the liminf of the internal energies along the sequence. Photons are localized at the scales $R=3^m$, and those that escape to infinity are discarded. \begin{lemma}[Lower semicontinuity with photon loss] \label[lemma]{thm-continuum-compactness} Let $(\vGamma_j)\subset\dm$ have uniformly bounded internal energies and uniformly tight electronic densities: \begin{align*} &\sup_j\Tr(H^{0,0}\vGamma_j)<\infty, \\ &\lim_{R\to\infty}\sup_j\int_{|\vx|>R}\rho_{\vGamma_j}(\vx)\dd\vx=0. \end{align*} Suppose also that the photon-number distributions of the localized states $\mathcal L_{3^m}\vGamma_j$ are asymptotically tight, uniformly in $m$: \begin{equation}\label{continuum-local-sector-tightness} \lim_{M\to\infty}\sup_{m\geq1}\limsup_{j\to\infty} \Tr\big(\iden_{\{\NC>M\}}\mathcal L_{3^m}\vGamma_j\big)=0. \end{equation} If $(\rho_{\vGamma_j},\vjc_{\vGamma_j})\to(\rho,\vjc)$ in the sense of distributions, then there is a $\vGamma\in\dm$ such that \begin{equation}\label{continuum-retained-limit} \rho_\vGamma=\rho,\qquad \vjc_\vGamma=\vjc,\qquad \Tr(H^{0,0}\vGamma) \leq\liminf_{j\to\infty}\Tr(H^{0,0}\vGamma_j). \end{equation} A sufficient condition for \cref{continuum-local-sector-tightness} is \[ \lim_{M\to\infty}\sup_j\Tr(\iden_{\{\NC>M\}}\vGamma_j)=0; \] in particular, a uniform bound on $\Tr(\NC\vGamma_j)$ suffices. \end{lemma} Note that neither a bound on mean photon number nor photon-position tightness is assumed, and the original sequence need not converge in trace norm. The order of limits in \cref{continuum-local-sector-tightness} matters. For each fixed localization scale $3^m$, the limit superior as $j\to\infty$ is taken first, so only the eventual behavior of the localized states at that scale enters. The resulting photon-number tails, one for each $m$, are then required to vanish uniformly in $m$ as $M\to\infty$. Explicitly, for every $\varepsilon>0$ there is an $M$ such that, for every $m\geq1$, some $j_0$ satisfies \[ \Tr\big(\iden_{\{\NC>M\}}\mathcal L_{3^m}\vGamma_j\big)<\varepsilon \qquad\text{for all }j\geq j_0. \] The threshold $j_0$ may depend on $m$, as well as on $\varepsilon$. A uniform bound on the original mean photon number is much stronger. By \cref{localized-photon-number} and $\iden_{\{\NC>M\}}\leq\NC/M$, it gives \[ \Tr\big(\iden_{\{\NC>M\}}\mathcal L_{3^m}\vGamma_j\big) \leq\frac1M\sup_i\Tr(\NC\vGamma_i) \] for all $j$ and $m$ at once, with no threshold. In contrast, \cref{continuum-local-sector-tightness} constrains only the photons retained at each fixed scale, and only for large $j$. This permits the removal of arbitrarily many soft photons. For example, let $\xi$ be a normalized smooth one-photon function supported in $\{0<|\vk|<1\}$, put $\xi_j(\vk,\lambda)=j^3\xi(j^2\vk,\lambda)$, and set $\Xi_j=(j!)^{-1/2}a^\dag(\xi_j)^j\Omega$. Then \[ \dua{\Xi_j}{\NC\Xi_j}=j,\qquad \dua{\Xi_j}{H_\rad\Xi_j}=j^{-1}\dua{\xi}{\omega\xi}. \] Write $\check\xi$ for the inverse Fourier transform of $\xi$ in $\vk$, taken separately for each polarization $\lambda$, so that $\check\xi$ is $\xi$ in the photon position variable $\vy$. Then $\check\xi_j(\vy)=j^{-3}\check\xi(\vy/j^2)$, so for each fixed $R$, \[ \Tr\big(\NC\mathcal L_R(|\Xi_j\rangle\langle\Xi_j|)\big) =j\|\theta_R\check\xi_j\|^2\leq C R^3j^{-5}\longrightarrow0. \] Thus the localized states converge to the vacuum and satisfy \cref{continuum-local-sector-tightness}, although their original photon-number distributions are not uniformly tight. Now fix a normalized $\Phi\in Q(H_\el^0)$ and set $\Psi_j=\Phi\otimes\Xi_j$. We compare $\Psi_j$ with the vacuum product $\Phi\otimes\Omega$. Since $\Xi_j$ is normalized, $\rho_{\Psi_j}=\rho_\Phi$ and $\vjp_{\Psi_j}=\vjp_\Phi$. Since $\Xi_j$ has exactly $j$ photons and $\Af^\mu(\vx)$ changes the photon number by one, $\dua{\Xi_j}{\Af^\mu(\vx)\Xi_j}=0$. Hence $\vjph_{\Psi_j}=0$, and $\Psi_j$ has the same density and combined current density $\vjc_{\Psi_j}=\vjp_\Phi$ as $\Phi\otimes\Omega$. The internal energies differ only through the radiation energy and the field fluctuations. The cross terms in $\|\vD_n\Psi_j\|^2$ vanish because the field expectation in $\Xi_j$ is zero, and the canonical commutation relations give \[ \|\Af^\mu(\vx)\Xi_j\|^2 =\|G^\mu_{\vx}\|^2+2j|\dua{G^\mu_{\vx}}{\xi_j}|^2, \] whose first term is the vacuum value. The Cauchy--Schwarz inequality, \cref{Gx-norms}, and the identity $\|\sqrt\omega\xi_j\|^2=j^{-2}\dua{\xi}{\omega\xi}$ bound the sum over $\mu$ of the second term by $2j^{-1}\|\wh\chi/\sqrt\omega\|^2\dua{\xi}{\omega\xi}$. Integrating against $|\Phi|^2$, summing over the electrons, and using $\int_{\RR^3}\rho_\Phi=N$, we find that the additional internal energy obeys \[ 0\leq\dua{\Psi_j}{H^{0,0}\Psi_j} -\dua{\Phi\otimes\Omega}{H^{0,0}(\Phi\otimes\Omega)} \leq\frac1j\left(1+e^2N\left\|\frac{\wh\chi}{\sqrt\omega}\right\|^2\right) \dua{\xi}{\omega\xi}. \] Thus the states $|\Psi_j\rangle\langle\Psi_j|$ satisfy all hypotheses of \cref{thm-continuum-compactness}, with constant density and current, although $\dua{\Psi_j}{\NC\Psi_j}=j\to\infty$. By the photon-number computation above, their localized states converge in trace norm to $|\Phi\otimes\Omega\rangle\langle\Phi\otimes\Omega|$, so all $j$ photons are lost in the limit. The limit state $\vGamma=|\Phi\otimes\Omega\rangle\langle\Phi\otimes\Omega|$ satisfies \cref{continuum-retained-limit} with equality. \begin{proof}[Proof of \cref{thm-continuum-compactness}] Write $H=H^{0,0}$ and first pass to a subsequence realizing the energy liminf. The form-norm equivalence controls the ordinary electronic kinetic energy and $H_\rad$ uniformly. By \cref{lem-photon-localization,lem-photon-localization-energy}, the localized states have the same electronic reduced state, their radiation energies are uniformly bounded, and \[ \Tr(\NC\mathcal L_R\vGamma_j) \leq C R\Tr(H_\rad\vGamma_j). \] Their full internal energies are uniformly bounded in both $j$ and $R>0$ by \cref{localized-internal-energy-bound}. \noindent\textbf{Step 1. Local trace-norm compactness.} Fix $R>0$. Set \[ -\lapl:=\sum_{n=1}^N(-\Delta_{\vx_n}),\qquad E:=1+\sup_j\Tr(H\vGamma_j)<\infty. \] The form-norm equivalence and the preceding estimates give \[ \sup_j\Tr\big((1-\lapl+H_\rad)\mathcal L_R\vGamma_j\big)\leq CE, \qquad \sup_j\Tr(\NC\mathcal L_R\vGamma_j)\leq CRE. \] All constants in the estimates below are independent of $j,R,M,L,s$. We construct finite-rank orthogonal projections whose complements have uniformly small expectation in $\mathcal L_R\vGamma_j$. First, denote the photon number cutoff by \[ P_M=\iden_{\{\NC\leq M\}}, \] for integers $M\geq0$. The electron--photon interaction terms in the internal Hamiltonian $H=H^{0,0}$ can change photon number, so $P_M$ need not commute with $H$. We deal with these terms later. Since $1-P_M\leq\NC/(M+1)$, \[ \sup_j\Tr\big((1-P_M)\mathcal L_R\vGamma_j\big) \leq\frac1{M+1}\sup_j\Tr(\NC\mathcal L_R\vGamma_j) \leq\frac{CRE}{M+1}. \] Next choose a fixed $\zeta\in C_c^\infty(\RR^3)$ with $0\leq\zeta\leq1$, equal to one on $B_1$ and zero outside $B_2$, and define the symmetric electronic cutoff \[ \zeta_L(\vx_1,\ldots,\vx_N) :=\prod_{n=1}^N\zeta(\vx_n/L),\qquad L\geq1. \] Thus $0\leq\zeta_L\leq1$, $\zeta_L=1$ on $(B_L)^N$, and $\supp\zeta_L\subset(\overline{B_{2L}})^N$: the cutoff localizes all $N$ electrons in the ball $B_{2L}$. Electronic tightness is used through the quantity \[ \epsilon_\el(L):=\sup_j\int_{|\vx|>L}\rho_{\vGamma_j}(\vx)\dd\vx, \] which tends to zero as $L\to\infty$ by the assumed uniform tightness of the electronic densities. Using the elementary fact that \[ (1-\zeta_L)^2\leq\iden_{\{\max_n|\vx_n|>L\}} \leq\sum_{n=1}^N\iden_{\{|\vx_n|>L\}} \] and taking expectations and suprema of both sides yields \[ \sup_j\Tr((1-\zeta_L)^2\mathcal L_R\vGamma_j) \leq\sup_j\int_{|\vx|>L}\rho_{\mathcal L_R\vGamma_j}(\vx)\dd\vx =\epsilon_\el(L). \] Indeed, photon localization leaves the electronic reduced state, and hence the density, unchanged: $\rho_{\mathcal L_R\vGamma_j}=\rho_{\vGamma_j}$ by \cref{lem-photon-localization}. Further, the positive contractions $P_M$ and $\zeta_L$ commute. Thus \[ (1-\zeta_LP_M)^2=(1-P_M)+P_M(1-\zeta_L)^2, \] and hence \begin{align} \sup_j\|(1-\zeta_LP_M)\sqrt{\mathcal L_R\vGamma_j}\|_{\mathfrak S_2}^2 &\leq\sup_j\Tr\big((1-P_M)\mathcal L_R\vGamma_j\big) +\sup_j\Tr\big((1-\zeta_L)^2\mathcal L_R\vGamma_j\big)\nonumber\\ &\leq\frac{CRE}{M+1}+\epsilon_\el(L). \label{cutoff-error} \end{align} The cutoff also preserves the required regularity with an explicit bound. We have \[ \sum_{n=1}^N|\grad_{\vx_n}\zeta_L|^2 \leq\frac1{L^2}\sum_{n=1}^N\big|(\grad\zeta)(\vx_n/L)\big|^2 \leq\frac{N\|\grad\zeta\|_\infty^2}{L^2}. \] Let $\psi\in Q(-\lapl)$ and put $\varphi=P_M\psi$. Since $P_M$ acts only on the photon factor, it commutes with the electronic derivatives, so $\|\grad_{\vx_n}\varphi\|\leq\|\grad_{\vx_n}\psi\|$ and $\|\varphi\|\leq\|\psi\|$. The product rule and $0\leq\zeta_L\leq1$ give \begin{align*} \sum_{n=1}^N\|\grad_{\vx_n}(\zeta_LP_M\psi)\|^2 &\leq2\sum_{n=1}^N\|\grad_{\vx_n}\varphi\|^2 +2\Big\langle \varphi, \sum_{n=1}^N|\grad_{\vx_n}\zeta_L|^2\varphi \Big\rangle\\ &\leq2\sum_{n=1}^N\|\grad_{\vx_n}\psi\|^2+\frac{CN}{L^2}\|\psi\|^2. \end{align*} Applying this to the vectors in a spectral decomposition of $\mathcal L_R\vGamma_j$ and summing, with $\Tr\mathcal L_R\vGamma_j=1$, gives \[ \Tr\big((-\lapl)\zeta_LP_M(\mathcal L_R\vGamma_j)P_M\zeta_L\big) \leq2\Tr\big((-\lapl)\mathcal L_R\vGamma_j\big)+\frac{CN}{L^2}. \] Moreover, $\zeta_LP_M$ commutes with $H_\rad$ and is a contraction, so \begin{align*} \Tr\big(H_\rad\zeta_LP_M(\mathcal L_R\vGamma_j)P_M\zeta_L\big) &\leq\Tr(H_\rad\mathcal L_R\vGamma_j),\\ \Tr\big((1-\lapl+H_\rad)\zeta_LP_M(\mathcal L_R\vGamma_j)P_M\zeta_L\big) &\leq C(E+N/L^2). \end{align*} We now apply Rellich compactness after both cutoffs. In the $m$-photon sector of $\mathcal L_R\vGamma_j$, every retained photon is supported in $B_{2R}$, because its localization factor is $\theta_R$. Consequently, $\zeta_LP_M\sqrt{\mathcal L_R\vGamma_j}$ takes values in the closed subspace \begin{align*} \mathscr H_{L,R,M} :=\bigoplus_{m=0}^M\big\{&\psi^{(m)}\in\Hil_\el^N\otimes L_s^2((\RR^3\times\{1,2\})^m):\psi^{(m)}\equiv 0\text{ a.e.\ outside }\Omega_{L,R,m}\big\}, \end{align*} where \[ \Omega_{L,R,m}:=(B_{2L})^N\times(B_{2R})^m. \] Here the photon variables $(\vy_l,\lambda_l)$ are in the position representation, and the support condition is imposed for every value of the polarization indices $\lambda_l$. On this subspace we restrict the closed form of $1-\lapl+H_\rad$, computing all derivatives after extension by zero to the full configuration space. This gives the form \[ k_{L,R,M}[\psi]:=\sum_{m=0}^Mk_m[\psi^{(m)}],\qquad \psi=\bigoplus_{m=0}^M\psi^{(m)}\in\mathscr H_{L,R,M}, \] whose value in the $m$-photon sector is \[ k_m[\psi^{(m)}]=\|\psi^{(m)}\|^2+\sum_{n=1}^N\|\grad_{\vx_n}\psi^{(m)}\|^2 +\sum_{l=1}^m\big\|(-\Delta_{\vy_l})^{1/4}\psi^{(m)}\big\|^2. \] Its domain $Q(k_{L,R,M})$ consists of those $\psi\in\mathscr H_{L,R,M}$ for which every $k_m[\psi^{(m)}]$ is finite. The form $k_{L,R,M}$ is closed because $\mathscr H_{L,R,M}$ is closed in $\HC_N$. It is densely defined because smooth functions compactly supported in $\Omega_{L,R,m}$, with the prescribed symmetries, are dense in the corresponding $L^2$ space. The form $k_m[\cdot]$ controls the $H^{1/2}$ norm in all $3(N+m)$ variables jointly. Indeed, by Plancherel's theorem, with $\vxi_n$ dual to $\vx_n$ and $\vk_l$ dual to $\vy_l$, the Fourier symbol of $k_m[\cdot]$ is $1+\sum_n|\vxi_n|^2+\sum_l|\vk_l|$. Using $\sum_l|\vk_l|^2\leq\big(\sum_l|\vk_l|\big)^2$, $\sqrt{s+t}\leq\sqrt s+\sqrt t$ for $s,t\geq0$, and $\sqrt{1+s}\leq1+s$, we obtain \begin{align*} \left(1+\sum_{n=1}^N|\vxi_n|^2 +\sum_{l=1}^m|\vk_l|^2\right)^{1/2} &\leq\left(1+\sum_{n=1}^N|\vxi_n|^2\right)^{1/2} +\sum_{l=1}^m|\vk_l|\\ &\leq1+\sum_{n=1}^N|\vxi_n|^2+\sum_{l=1}^m|\vk_l|, \end{align*} and hence $\|u\|_{H^{1/2}(\RR^{3(N+m)})}^2\leq k_m[u]$. Since $\Omega_{L,R,m}$ is a bounded convex domain and hence a Lipschitz domain, the Rellich--Kondrashov theorem for $H^{1/2}$ gives a compact embedding of the $m$-photon component of $Q(k_{L,R,M})$ into $L^2(\Omega_{L,R,m})$ \cite[Theorem~7.1]{dinezza2012hitchhiker}: the Gagliardo seminorm on $\Omega_{L,R,m}$ is bounded by the $H^{1/2}(\RR^{3(N+m)})$ norm \cite[Proposition~3.4]{dinezza2012hitchhiker}, and Lipschitz domains are extension domains \cite[Theorem~5.4]{dinezza2012hitchhiker}. There are only finitely many sectors and finitely many polarization components in each. Therefore $Q(k_{L,R,M})$, normed by $k_{L,R,M}[\cdot]^{1/2}$, embeds compactly into $\mathscr H_{L,R,M}$. Let $K_{L,R,M}\geq1$ be the self-adjoint operator on $\mathscr H_{L,R,M}$ associated with the closed form $k_{L,R,M}[\cdot]$. It has compact resolvent, so for each $s\geq1$ its spectral projection \[ S:=\iden_{[0,s]}(K_{L,R,M}), \] extended by zero on $\mathscr H_{L,R,M}^\perp$, is a finite-rank orthogonal projection on $\HC_N$. The spectral theorem gives \begin{align*} \|(1-S)\zeta_LP_M\sqrt{\mathcal L_R\vGamma_j}\|_{\mathfrak S_2}^2 &=\Tr\big((1-S)\zeta_LP_M(\mathcal L_R\vGamma_j)P_M\zeta_L\big)\\ &\leq s^{-1}\Tr\big(K_{L,R,M}\zeta_LP_M(\mathcal L_R\vGamma_j)P_M\zeta_L\big) \leq\frac{C(E+N/L^2)}{s}. \end{align*} Combining this with the preceding cutoff error \cref{cutoff-error}, we obtain \begin{align*} \Tr\big((1-S)\mathcal L_R\vGamma_j\big) &=\|(1-S)\sqrt{\mathcal L_R\vGamma_j}\|_{\mathfrak S_2}^2\\ &\leq2\|(1-\zeta_LP_M)\sqrt{\mathcal L_R\vGamma_j}\|_{\mathfrak S_2}^2 +2\|(1-S)\zeta_LP_M\sqrt{\mathcal L_R\vGamma_j}\|_{\mathfrak S_2}^2\\ &\leq\frac{2CRE}{M+1}+2\epsilon_\el(L) +\frac{C(E+N/L^2)}{s}. \end{align*} For fixed $R$, first choose $M$, then $L$, and finally $s$ sufficiently large to make this bound arbitrarily small, uniformly in $j$. The states $\mathcal L_R\vGamma_j$ are positive with unit trace, so the compactness part of \cref{lem-trace-compression} shows that the sequence $(\mathcal L_R\vGamma_j)_j$ is precompact in trace norm. \noindent\textbf{Step 2. A trace-norm limit of the retained states.} A diagonal extraction gives a subsequence, still denoted $(\vGamma_j)$, such that \[ \mathcal L_{3^m}\vGamma_j\longrightarrow\vGamma^{(m)} \quad\text{in trace norm as }j\to\infty,\text{ for every }m\geq1. \] Localization maps compose according to multiplication of their one-photon contractions. For $l>m$, the function $\theta_{3^m}$ vanishes outside $B_{2\cdot3^m}$, while $\theta_{3^l}=1$ on $B_{3^l}\supset B_{3^{m+1}}\supset B_{2\cdot3^m}$. Hence $\theta_{3^m}\theta_{3^l}=\theta_{3^m}$, and therefore \[ \mathcal L_{3^m}\vGamma^{(l)}=\vGamma^{(m)}. \] Since photon-sector projections are bounded, the hypothesis implies \[ \delta(M):=\sup_m\Tr(\iden_{\{\NC>M\}}\vGamma^{(m)}) \longrightarrow0. \] We next show that $(\vGamma^{(m)})$ is Cauchy in trace norm, using \cref{lem-localization-trace-distance} with $\tau=\vGamma^{(l)}$ and $\eta=\theta_{3^m}$ for $l>m$. For this, the probability that passing from $\vGamma^{(l)}$ to $\vGamma^{(m)}$ discards at least one photon must be small. The bound on this probability in \cref{lem-localization-trace-distance} involves the mean photon number, which need not be finite here. We therefore track instead the expectation of the bounded function $n\mapsto2^{-n}$ of the photon number, \[ b_m:=\Tr(2^{-\NC}\vGamma^{(m)})\in[0,1], \] which is finite for every state. We show below that discarding photons can only increase this expectation, so $(b_m)$ is nonincreasing and converges. Its decrease $b_m-b_l$, together with the photon-number tightness $\delta(M)$, then controls the probability of discarding a photon. For $l>m$, the localization isometry $\mathcal U_{3^m}$ underlying $\mathcal L_{3^m}$ takes values in $\Hil_\el^N\otimes\Fock_\ph^{\mathrm{in}}\otimes\Fock_\ph^{\mathrm{out}}$, where the two Fock factors contain the retained and discarded photons, respectively. In photon position, the retained photons carry the factor $\theta_{3^m}$ and are supported in the ball $B_{2\cdot3^m}$, while the discarded photons carry the factor $\phi_{3^m}$, which vanishes on $B_{3^m}$. Define their number operators by \[ \NC_{\mathrm{in}} :=\iden_{\Hil_\el^N}\otimes\NC\otimes\iden_{\Fock_\ph^{\mathrm{out}}}, \qquad \NC_{\mathrm{out}} :=\iden_{\Hil_\el^N}\otimes\iden_{\Fock_\ph^{\mathrm{in}}}\otimes\NC. \] Thus on the sector with $p$ retained and $q$ discarded photons they act as multiplication by $p$ and $q$, respectively, and strongly commute. Recall that $\mathcal U_{3^m}=\varGamma(j_{3^m})$ is the second quantization of the one-photon isometry $j_{3^m}f=(\theta_{3^m}f,\phi_{3^m}f)$, followed by the identification $\Fock_s(\Hil_\ph\oplus\Hil_\ph) \simeq\Fock_\ph^{\mathrm{in}}\otimes\Fock_\ph^{\mathrm{out}}$, with the electronic identity suppressed, and that $\mathcal L_{3^m}(\vGamma) =\Tr_{\mathrm{out}}(\mathcal U_{3^m}\vGamma\mathcal U_{3^m}^\dag)$. In the joint state $\mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag$, their sum has the number distribution of $\vGamma^{(l)}$, and the first has that of $\vGamma^{(m)}$. Indeed, $\mathcal U_{3^m}$ maps the $n$-photon sector into the sum of the sectors with $p+q=n$, so $\mathcal U_{3^m}^\dag f(\NC_{\mathrm{in}}+\NC_{\mathrm{out}}) \mathcal U_{3^m}=f(\NC)$ for every bounded function $f$ on $\{0,1,2,\ldots\}$. Moreover, $f(\NC_{\mathrm{in}})$ acts trivially on the discarded factor, and the partial trace over that factor is $\mathcal L_{3^m}\vGamma^{(l)}=\vGamma^{(m)}$. Hence \begin{align*} \Tr\!\left(f(\NC_{\mathrm{in}}+\NC_{\mathrm{out}})\, \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right) &=\Tr\big(f(\NC)\vGamma^{(l)}\big),\\ \Tr\!\left(f(\NC_{\mathrm{in}})\, \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right) &=\Tr\big(f(\NC)\vGamma^{(m)}\big). \end{align*} Take $f(n)=2^{-n}$. The operators $\NC_{\mathrm{in}}$ and $\NC_{\mathrm{out}}$ strongly commute, so $2^{-(\NC_{\mathrm{in}}+\NC_{\mathrm{out}})} =2^{-\NC_{\mathrm{in}}}2^{-\NC_{\mathrm{out}}}$, and the two identities give \begin{align*} b_l&=\Tr\!\left(2^{-\NC_{\mathrm{in}}}2^{-\NC_{\mathrm{out}}} \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right),\\ b_m&=\Tr\!\left(2^{-\NC_{\mathrm{in}}} \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right). \end{align*} Subtracting, we obtain \[ b_m-b_l =\Tr\!\left(2^{-\NC_{\mathrm{in}}} (1-2^{-\NC_{\mathrm{out}}}) \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right)\geq0, \] where the inequality holds because $2^{-p}(1-2^{-q})\geq0$ for all integers $p,q\geq0$. In particular, the sequence $(b_m)$ is nonincreasing. Let \[ t_{m,l}:=\Tr\!\left(\iden_{\{\NC_{\mathrm{out}}\geq1\}} \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right) \] denote the probability that at least one photon is discarded. Splitting according to $\NC_{\mathrm{in}}>M$ and $\NC_{\mathrm{in}}\leq M$ gives \begin{align*} t_{m,l} &\leq\Tr\!\left(\iden_{\{\NC_{\mathrm{in}}>M\}} \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right)\\ &\quad+\Tr\!\left(\iden_{\{\NC_{\mathrm{in}}\leq M\}} \iden_{\{\NC_{\mathrm{out}}\geq1\}} \mathcal U_{3^m}\vGamma^{(l)}\mathcal U_{3^m}^\dag\right)\\ &\leq\delta(M)+2^{M+1}(b_m-b_l). \end{align*} For the first term, we take $f=\iden_{(M,\infty)}$ in the second trace identity above. This shows that the first term equals $\Tr(\iden_{\{\NC>M\}}\vGamma^{(m)})$, which is at most $\delta(M)$. For the second term, the operator $2^{-\NC_{\mathrm{in}}}(1-2^{-\NC_{\mathrm{out}}})$ is at least $2^{-M-1}$ on the joint spectral subspace $\NC_{\mathrm{in}}\leq M$, $\NC_{\mathrm{out}}\geq1$, so the formula for $b_m-b_l$ bounds this term by $2^{M+1}(b_m-b_l)$. Finally, $t_{m,l}$ is the quantity $t$ of \cref{lem-localization-trace-distance} for $\tau=\vGamma^{(l)}$ and $\eta=\theta_{3^m}$, and $\mathcal L_{3^m}\vGamma^{(l)}=\vGamma^{(m)}$, so \[ \|\vGamma^{(l)}-\vGamma^{(m)}\|_{\mathfrak S_1} \leq2\sqrt{t_{m,l}}+t_{m,l}. \] The nonincreasing sequence $b_m$ has a limit. First choose $M$ so that $\delta(M)$ is small, then $m$ so that $b_m-b_l$ is small uniformly for $l>m$. Hence $\vGamma^{(m)}$ is Cauchy in trace norm and converges to a density matrix $\vGamma$ on the original space $\HC_N$, possibly with infinite mean photon number. The rough energy bound \cref{localized-internal-energy-bound} gives \[ \sup_{j,m}\Tr(H\mathcal L_{3^m}\vGamma_j)\leq CE. \] Applying \cref{lem-trace-compactness-and-limits} first to $\mathcal L_{3^m}\vGamma_j\to\vGamma^{(m)}$ and then to $\vGamma^{(m)}\to\vGamma$ yields \[ \sup_m\Tr(H\vGamma^{(m)})\leq CE,\qquad \Tr(H\vGamma)\leq CE<\infty, \] and hence $\vGamma\in\dm$. We next prove the sharp energy inequality. \noindent\textbf{Step 3. The energy inequality.} The argument involves an electronic cutoff at scale $L$, the photon localization scale $R=3^m$, and a truncation of $H_\rad$ by second quantizations of one-photon operators of rank $K$, all introduced below. We pass to the limits in a fixed order: first $j\to\infty$, then $R=3^m\to\infty$ with $L$ and $K$ fixed, then $K\to\infty$, and finally $L\to\infty$. The order matters because the bound on the localized truncated radiation energy below carries an error $\delta_K(R)$ that is established only for fixed $K$, and tends to zero as $R\to\infty$ but not uniformly in $K$. To control the interaction error, we first choose a symmetric electronic cutoff $\eta_L$, equal to one when $\max_i|\vx_i|\leq L$ and zero when $\max_i|\vx_i|\geq2L$, together with a smooth complementary cutoff $\wt\eta_L$. For example, using the electronic cutoff $\zeta_L$ constructed above, take \[ \eta_L=\sin\!\left(\frac\pi2\zeta_L\right),\qquad \wt\eta_L=\cos\!\left(\frac\pi2\zeta_L\right). \] Then $\eta_L^2+\wt\eta_L^2=1$ and \[ \sum_{i=1}^N\left(|\grad_{\vx_i}\eta_L|^2 +|\grad_{\vx_i}\wt\eta_L|^2\right) =\frac{\pi^2}{4}\sum_{i=1}^N|\grad_{\vx_i}\zeta_L|^2 \leq\frac{C}{L^2}. \] Both cutoffs preserve $Q(H)$: their bounded gradients preserve the electronic $H^1$ regularity, and they commute with $\sqrt{H_\rad}$. For either cutoff $f\in\{\eta_L,\wt\eta_L\}$, the covariant derivative obeys \[ \vD_i(f\Psi)=f\vD_i\Psi-\i(\grad_{\vx_i}f)\Psi, \qquad \vD_i=-\i\grad_{\vx_i}-e\AF(\vx_i), \] because $f$ commutes with the field operator $\AF(\vx_i)$. Since $\eta_L$ and $\wt\eta_L$ are real, expanding gives \begin{align*} &\|\vD_i(\eta_L\Psi)\|^2+\|\vD_i(\wt\eta_L\Psi)\|^2\\ &\quad=\dua{\vD_i\Psi}{(\eta_L^2+\wt\eta_L^2)\vD_i\Psi} +\|(\grad_{\vx_i}\eta_L)\Psi\|^2 +\|(\grad_{\vx_i}\wt\eta_L)\Psi\|^2\\ &\qquad+2\Re\dua{\vD_i\Psi}{-\i\big(\eta_L\grad_{\vx_i}\eta_L +\wt\eta_L\grad_{\vx_i}\wt\eta_L\big)\Psi}\\ &\quad=\|\vD_i\Psi\|^2+\|(\grad_{\vx_i}\eta_L)\Psi\|^2 +\|(\grad_{\vx_i}\wt\eta_L)\Psi\|^2. \end{align*} In the last step we used $\eta_L^2+\wt\eta_L^2=1$, and the cross terms cancel because \[ \eta_L\grad_{\vx_i}\eta_L +\wt\eta_L\grad_{\vx_i}\wt\eta_L =\tfrac12\grad_{\vx_i}(\eta_L^2+\wt\eta_L^2)=0. \] The Coulomb interaction is an electronic multiplication operator, and $H_\rad$ acts only on the photon factor. Their localized forms therefore sum exactly to their original forms, using $\eta_L^2+\wt\eta_L^2=1$. With the factor $1/2$ in the kinetic energy, the resulting covariant IMS identity is \begin{align*} &\dua{\eta_L\Psi}{H\eta_L\Psi} +\dua{\wt\eta_L\Psi}{H\wt\eta_L\Psi}\\ &\quad=\dua{\Psi}{H\Psi} +\frac12\sum_{i=1}^N\left( \|(\grad_{\vx_i}\eta_L)\Psi\|^2 +\|(\grad_{\vx_i}\wt\eta_L)\Psi\|^2\right)\\ &\quad\leq\dua{\Psi}{H\Psi}+\frac{C}{L^2}\|\Psi\|^2. \end{align*} The computation first holds on the form core and extends to $Q(H)$ by continuity. Applying it to the vectors in a spectral decomposition of $\vGamma_j$ and summing, with $\Tr\vGamma_j=1$, gives \[ \Tr(H\eta_L\vGamma_j\eta_L) +\Tr(H\wt\eta_L\vGamma_j\wt\eta_L) \leq\Tr(H\vGamma_j)+\frac{C}{L^2}. \] Since $H\geq0$, the second term on the left is nonnegative and can be dropped. Thus \[ \Tr(H\eta_L\vGamma_j\eta_L)\leq\Tr(H\vGamma_j)+C/L^2. \] Let $\vGamma$ be a positive trace-class operator with $\Tr(H\vGamma)<\infty$. Every term in \cref{q00-def} is nonnegative, so $\Tr(H_\rad\vGamma)\leq\Tr(H\vGamma)<\infty$, and we may define \[ \mathcal E(\vGamma):=\Tr(H\vGamma)-\Tr(H_\rad\vGamma), \] the internal energy with the free-photon contribution removed. It still contains the electron--photon coupling through the covariant kinetic energy, as well as the Coulomb repulsion. The field identities \cref{localized-linear-field,localized-quadratic-field}, the annihilation--creation bounds \cref{ann-Hrad-bound,cre-Hrad-bound} relative to $\sqrt{H_\rad}$, and Cauchy--Schwarz give \[ \mathcal E\big(\mathcal L_R(\eta_L\vGamma_j\eta_L)\big) \leq\mathcal E(\eta_L\vGamma_j\eta_L) +C\epsilon_L(R)+C\epsilon_L(R)^2, \] where \[ \epsilon_L(R):=\max_\mu\sup_{|\vx|\leq2L} \big(\|(1-\theta_R)G^\mu_{\vx}\| +\|(-\Delta)^{-1/4}(1-\theta_R)G^\mu_{\vx}\| +\|\phi_RG^\mu_{\vx}\|\big). \] Here the negative Sobolev norm is $\|f\|_{\dot H^{-1/2}}=\|(-\Delta)^{-1/4}f\|$ in photon position representation. The Coulomb term is unchanged and the quadratic-field identity includes the vacuum fluctuations. The constants are uniform in $j$ for fixed $L$. We now show that $\epsilon_L(R)\to0$ as $R\to\infty$ for fixed $L$. Since $0\leq1-\theta_R,\phi_R\leq1$ and both vanish on $B_R$, dominated convergence gives $(1-\theta_R)f\to0$ and $\phi_Rf\to0$ in $\Hil_\ph$ for every $f\in\Hil_\ph$. Smooth multiplication by $\theta_R$ is uniformly bounded on $\dot H^{1/2}$, and therefore by duality on $\dot H^{-1/2}$. Density of compactly supported smooth functions shows that $\theta_R\to1$ strongly in both spaces. The map $\vx\mapsto G^\mu_{\vx}$ is continuous into $\Hil_\ph$ and into $\dot H^{-1/2}$, as noted after \cref{Gx-norms}. Hence the form factors $G^\mu_{\vx}$ with $|\vx|\leq2L$ and $\mu=1,2,3$ form a compact family in both norms. The operators $1-\theta_R$ on both spaces and $\phi_R$ on $\Hil_\ph$ are uniformly bounded in $R$ and converge strongly to zero, and strong convergence of uniformly bounded operators is uniform on compact sets. The supremum over $|\vx|\leq2L$ in the definition of $\epsilon_L(R)$ is therefore harmless, and \[ \lim_{R\to\infty}\epsilon_L(R)=0. \] The rough bound \cref{localized-radiation-bound} only controls the localized radiation energy up to a constant factor. To obtain it with coefficient one, we truncate $H_\rad$ from below by second quantizations of finite-rank one-photon operators: choose an orthonormal basis $(e_k)$ whose finite span is a core for $(-\Delta)^{1/4}$, and set \begin{gather*} f_k=(-\Delta)^{1/4}e_k,\qquad t_K=\sum_{k=1}^K|f_k\rangle\langle f_k|\leq\sqrt{-\Delta},\\ T_K=\dd\varGamma(t_K)=\sum_{k=1}^Ka^\dag(f_k)a(f_k). \end{gather*} Then $T_K\uparrow H_\rad$ in the form sense. \noindent\textbf{Claim.} For fixed $K$ there are numbers $\delta_K(R)\to0$ as $R\to\infty$ such that \[ \theta_Rt_K\theta_R\leq\big(1+\delta_K(R)\big)\sqrt{-\Delta} \] in the sense of quadratic forms. \noindent\textit{Proof of the claim.} Since $\|(-\Delta)^{-1/4}f_k\|=\|e_k\|=1$, each $f_k$ lies in $\dot H^{-1/2}$, and the strong convergence $\theta_R\to1$ on $\dot H^{-1/2}$ established above gives \[ e_{k,R}:=(-\Delta)^{-1/4}\theta_Rf_k\longrightarrow e_k \quad\text{in }\Hil_\ph,\qquad k=1,\ldots,K. \] Let $\Pi_K$ be the orthogonal projection onto the span of $e_1,\ldots,e_K$, and put \[ \delta_K(R):=\Big\|\sum_{k=1}^K|e_{k,R}\rangle\langle e_{k,R}|-\Pi_K\Big\| \leq\sum_{k=1}^K\big(\|e_{k,R}\|+1\big)\|e_{k,R}-e_k\| \longrightarrow0. \] Let $\psi$ be in the form domain of $\sqrt{-\Delta}$. Since $\theta_R$ is real, moving $(-\Delta)^{\pm1/4}$ across the inner product gives \begin{align*} \dua{\psi}{\theta_Rt_K\theta_R\psi} &=\sum_{k=1}^K|\dua{\theta_Rf_k}{\psi}|^2 =\sum_{k=1}^K\big|\dua{e_{k,R}}{(-\Delta)^{1/4}\psi}\big|^2\\ &=\Big\langle(-\Delta)^{1/4}\psi, \sum_{k=1}^K|e_{k,R}\rangle\langle e_{k,R}|\,(-\Delta)^{1/4}\psi\Big\rangle\\ &\leq\big(\|\Pi_K\|+\delta_K(R)\big)\|(-\Delta)^{1/4}\psi\|^2 =\big(1+\delta_K(R)\big)\dua{\psi}{\sqrt{-\Delta}\,\psi}. \end{align*} This proves the claim. The convergence $\delta_K(R)\to0$ is not uniform in $K$. Since second quantization preserves form inequalities, \cref{localized-second-quantization} and the claim give \[ \Tr\big(T_K\mathcal L_R(\eta_L\vGamma_j\eta_L)\big) =\Tr\big(\dd\varGamma(\theta_Rt_K\theta_R)\eta_L\vGamma_j\eta_L\big) \leq\big(1+\delta_K(R)\big)\Tr(H_\rad\eta_L\vGamma_j\eta_L). \] Combining the last two estimates before taking any limits yields \begin{equation}\label{localized-combined-energy-bound} \mathcal E\big(\mathcal L_R(\eta_L\vGamma_j\eta_L)\big) +\Tr\big(T_K\mathcal L_R(\eta_L\vGamma_j\eta_L)\big) \leq\Tr(H\vGamma_j)+C/L^2+o_R(1). \end{equation} Here $o_R(1)$ denotes an error that tends to zero as $R\to\infty$, uniformly in $j$, for fixed $L$ and $K$. It collects the term $C\epsilon_L(R)+C\epsilon_L(R)^2$ from the first estimate and the term $\delta_K(R)\Tr(H_\rad\eta_L\vGamma_j\eta_L)\leq\delta_K(R)(E+C/L^2)$ from the second, and both are bounded independently of $j$. We justify the two limit passages for fixed $L,K$ in detail. Since photon localization commutes with the electronic multiplier $\eta_L$, the positive, possibly subnormalized operators \[ \sigma_{j,m}:=\mathcal L_{3^m}(\eta_L\vGamma_j\eta_L) =\eta_L(\mathcal L_{3^m}\vGamma_j)\eta_L,\qquad \sigma_m:=\eta_L\vGamma^{(m)}\eta_L,\qquad \sigma:=\eta_L\vGamma\eta_L \] satisfy $\sigma_{j,m}\to\sigma_m$ in trace norm as $j\to\infty$ for each $m$, and $\sigma_m\to\sigma$ in trace norm as $m\to\infty$. Indeed, multiplication on both sides by $\eta_L$ is a contraction in trace norm. The rough energy bound \cref{localized-internal-energy-bound} for $\mathcal L_{3^m}\vGamma_j$, followed by the covariant IMS estimate, gives \[ \sup_{j,m}\big(\Tr\sigma_{j,m}+\Tr(H\sigma_{j,m})\big) \leq C_L<\infty. \] By \cref{lem-trace-compactness-and-limits}, the same bound holds for $\sigma_m$, uniformly in $m$, and for $\sigma$. Write $W=\sum_{nM\}}\otimes\iden)\mathcal U_R \leq\iden_{\{\NC>M\}}. \] Taking the trace against $\vGamma_j$ and then the indicated limits gives \[ \sup_{m\geq1}\limsup_{j\to\infty} \Tr\big(\iden_{\{\NC>M\}}\mathcal L_{3^m}\vGamma_j\big) \leq\sup_j\Tr\big(\iden_{\{\NC>M\}}\vGamma_j\big). \] Uniform tails on the right therefore imply \cref{continuum-local-sector-tightness}. If the mean photon numbers are uniformly bounded, the spectral inequality $\iden_{\{\NC>M\}}\leq\NC/(M+1)$ for integers $M\geq0$ gives \[ \sup_j\Tr\big(\iden_{\{\NC>M\}}\vGamma_j\big) \leq\frac{\sup_j\Tr(\NC\vGamma_j)}{M+1} \longrightarrow0, \] which proves the final assertion. \end{proof} \subsection{Soft photon removal lemma} In this section, we prove \cref{lem-exact-soft-removal}, which is a central technical ingredient in the proofs of mixed-state attainment, soft-photon purification, and infrared bounds on optimizers. First, we split the photon space into soft- and hard-photon parts. For $\kappa>0$, let \[ \Hil_{\geq\kappa}=\iden_{\{\omega\geq\kappa\}}\Hil_\ph,\qquad \Hil_{<\kappa}=\iden_{\{\omega<\kappa\}}\Hil_\ph \] be the hard and soft one-photon spaces, so that $\Hil_\ph=\Hil_{\geq\kappa}\oplus\Hil_{<\kappa}$, and let $\Fock_{\geq\kappa}=\Fock_s(\Hil_{\geq\kappa})$ and $\Fock_{<\kappa}=\Fock_s(\Hil_{<\kappa})$. The exponential law for Fock spaces gives the factorization \[ \HC_N\simeq\Hil_\el^N\otimes\Fock_{\geq\kappa} \otimes\Fock_{<\kappa}. \] Accordingly, $\Af^\mu(\vx)=\Af^\mu_{\geq\kappa}(\vx)+\Af^\mu_{<\kappa}(\vx)$, where $\Af^\mu_{\geq\kappa}(\vx)=\varPhi(\iden_{\{\omega\geq\kappa\}}G^\mu_{\vx})$ acts on $\Fock_{\geq\kappa}$ and $\Af^\mu_{<\kappa}(\vx)=\varPhi(\iden_{\{\omega<\kappa\}}G^\mu_{\vx})$ acts on $\Fock_{<\kappa}$. Similarly, $H_\rad=H_{\rad,\geq\kappa}+H_{\rad,<\kappa}$ with \[ H_{\rad,\geq\kappa}=\dd\varGamma(\omega\iden_{\{\omega\geq\kappa\}}),\qquad H_{\rad,<\kappa}=\dd\varGamma(\omega\iden_{\{\omega<\kappa\}}). \] The \emph{hard model} is the internal Pauli--Fierz Hamiltonian $H^{0,0}_{\geq\kappa}$ on $\Hil_\el^N\otimes\Fock_{\geq\kappa}$ with field $\AF_{\geq\kappa}$ and radiation energy $H_{\rad,\geq\kappa}$. The restricted form factor still satisfies \cref{cutoff-assumption}, so the form construction above applies to it. The notation $\Tr_{\el,<\kappa}\vGamma$ denotes the partial trace over the electronic and soft-photon factors, leaving the reduced state on $\Fock_{\geq\kappa}$. Similarly, $\Tr_{<\kappa}$ denotes the partial trace over $\Fock_{<\kappa}$ alone, leaving an operator on $\Hil_\el^N\otimes\Fock_{\geq\kappa}$. We write $\Omega_{<\kappa}$ for the normalized vacuum in $\Fock_{<\kappa}$, so that $|\Omega_{<\kappa}\rangle\langle\Omega_{<\kappa}|$ is the state with no photons in the modes $\omega<\kappa$. We are now in a position to state and prove the main result of this section. \begin{lemma}[Exact soft-photon removal] \label[lemma]{lem-exact-soft-removal} Fix $N\in\NN$. For every $\vGamma\in\dm$ and $0<\kappa\leq1$, there is a $\vGamma_\kappa\in\dm$ with $\rho_{\vGamma_\kappa}=\rho_\vGamma$ and $\vjc_{\vGamma_\kappa}=\vjc_\vGamma$, and with photon marginal \begin{equation}\label{exact-soft-removal-marginal} \Tr_\el\vGamma_\kappa =\big(\Tr_{\el,<\kappa}\vGamma\big) \otimes|\Omega_{<\kappa}\rangle\langle\Omega_{<\kappa}|. \end{equation} Put $E_< =\Tr(H_{\rad,<\kappa}\vGamma)$. There exists a constant $C$, independent of $\vGamma$ and $\kappa$ (but allowed to depend on $N$), such that \begin{equation}\label{exact-soft-removal-cost} \begin{split} \Tr(H^{0,0}\vGamma_\kappa) &\leq\Tr(H^{0,0}\vGamma)-E_<\\ &+C\Big(\kappa^2+e^2b_\kappa^2 +e\kappa\sqrt{a_\kappa^2E_<+b_\kappa^2}+e^2\kappa^2\big(a_\kappa^2E_<+b_\kappa^2\big)\Big). \end{split} \end{equation} In particular, $\Tr(\NC\vGamma_\kappa)\leq\kappa^{-1}\Tr(H_\rad\vGamma)$. \end{lemma} The quantities $a_\kappa,b_\kappa$ defined in \cref{infrared-kernel-errors} both tend to zero as $\kappa\downarrow0$. \phantomsection \label{proof:lem-exact-soft-removal} \begin{proof} Replacing the soft photons by the vacuum removes their radiation energy $E_<$, but it also removes the contribution of the soft field to the combined current density, so by itself it would violate the current constraint. We compensate for this loss by converting the field-induced current density carried by the soft photons into an equal paramagnetic current density. The components of the soft field commute, so we can condition on their values $u$. For each $u$, the soft field becomes a classical vector potential $\va_u$ that varies on the scale $\kappa^{-1}$. After the electrons are localized into clusters of diameter $O(R)$, the field $\va_u$ is nearly constant on each cluster. Its constant part is removed exactly by a suitable gauge transformation in the electronic variables. The small remainder is removed by the current shift of \cref{lem-fermionic-current-repair}, at a kinetic cost controlled by $\kappa R$ and the infrared quantities $a_\kappa,b_\kappa$. \noindent\textbf{Step 1. Electronic localization.} Fix a cluster scale $1\leq R\leq\kappa^{-1}$, to be chosen in Step~6. Choose a locally finite smooth partition $(\eta_a)_{a\in\NN}$ on $\RR^3$ with $\sum_a\eta_a^2=1$, $\supp\eta_a\subset B(c_a,c_0R)$, and $\sum_a|\grad\eta_a|^2\leq C/R^2$. For each multi-index $\alpha=(\alpha_1,\ldots,\alpha_N)\in\NN^N$ with $\alpha_1\leq\cdots\leq\alpha_N$, let $\mathscr S_N\alpha$ be the set of its distinct rearrangements $(\alpha_{\pi(1)},\ldots,\alpha_{\pi(N)})$, $\pi\in\mathscr S_N$, and put \[ \zeta_\alpha(\vX)= \left(\sum_{(a_1,\ldots,a_N)\in\mathscr S_N\alpha} \prod_{i=1}^N\eta_{a_i}(\vx_i)^2\right)^{1/2}. \] These are real symmetric $W^{1,\infty}$ multipliers. Since $\NN^N$ is the disjoint union of the sets $\mathscr S_N\alpha$, expanding $\prod_{i=1}^N\sum_a\eta_a(\vx_i)^2=1$ gives $\sum_\alpha\zeta_\alpha^2=1$. Moreover, $\zeta_\alpha$ is the Euclidean norm of the vector $\big(\prod_i\eta_{a_i}(\vx_i)\big)_{(a_1,\ldots,a_N)\in\mathscr S_N\alpha}$, and the gradient of a norm is bounded by the norm of the gradient. Hence \[ \sum_\alpha\sum_{i=1}^N|\grad_{\vx_i}\zeta_\alpha|^2 \leq\sum_{i=1}^N\sum_{(a_1,\ldots,a_N)\in\NN^N} \Big|\grad_{\vx_i}\prod_{j=1}^N\eta_{a_j}(\vx_j)\Big|^2 =\sum_{i=1}^N\sum_a|\grad\eta_a(\vx_i)|^2 \leq\frac{C_N}{R^2}. \] Set \[ \vGamma_\alpha=\zeta_\alpha\vGamma\zeta_\alpha,\qquad E_\alpha=\Tr(H_{\rad,<\kappa}\vGamma_\alpha). \] All these operators act within the antisymmetric space. Real multiplication and the covariant IMS identity give \begin{align} \sum_\alpha(\rho_{\vGamma_\alpha},\vjc_{\vGamma_\alpha}) &=(\rho_\vGamma,\vjc_\vGamma),\qquad \sum_\alpha\Tr\vGamma_\alpha=1,\qquad \sum_\alpha E_\alpha=E_<,\notag\\ \sum_\alpha\Tr(H^{0,0}\vGamma_\alpha)&\leq\Tr(H^{0,0}\vGamma)+C/R^2. \label{current-preserving-electronic-IMS} \end{align} Here densities, current densities, and form traces are extended linearly to positive trace-class operators of arbitrary finite trace. Whenever current densities are used, the operator is assumed to have finite energy in the relevant model. \noindent\textbf{Step 2. Conditioning on the soft field.} We use the Gaussian representation and the unnormalized conditional operators constructed in Appendix~\ref{app-soft-gaussian}, with the soft space $\mathfrak h=\Hil_{<\kappa}$ and $\mathcal K=\Hil_\el^N\otimes\Fock_{\geq\kappa}$. Under $\iden\otimes\mathcal G$, the soft field $\AF_{<\kappa}(\vx)$ becomes multiplication by the real random field $\va_u(\vx)$ with components \[ a_u^\mu(\vx)=\sum_{n\geq1} \dua{\xi_n}{\iden_{\{\omega<\kappa\}}G^\mu_{\vx}}\,u_n, \] where $(\xi_n)_{n\geq1}$ is a $J$-real orthonormal basis of $\Hil_{<\kappa}$ and $u=(u_n)_{n\geq1}\in\mathcal Q=\RR^{\NN}$ carries independent standard Gaussian coordinates. Let $\Theta_\alpha(u)$ be the family $\Theta(u)$ of \cref{lem-soft-disintegration} for $\vGamma_\alpha$. These are unnormalized conditional operators for the electrons and hard photons at the soft-field value $u$. They satisfy \[ \int_{\mathcal Q}\Tr\Theta_\alpha(u)\dd\mu(u) =\Tr\vGamma_\alpha\leq1, \] but their individual traces can exceed one. More precisely, for every bounded operator $B$ on $\Hil_\el^N\otimes\Fock_{\geq\kappa}$ and every bounded measurable function $f$ on $\mathcal Q$, \begin{equation}\label{conditioned-disintegration} \Tr\big(\vGamma_\alpha\,(B\otimes\mathcal G^{-1}M_f\mathcal G)\big) =\int_{\mathcal Q}f(u)\Tr\big(B\Theta_\alpha(u)\big)\dd\mu(u), \end{equation} where $M_f$ is multiplication by $f$ on $L^2(\mathcal Q,\mu)$. Explicitly, if $(\Psi_j)_{j\geq1}$ is an orthonormal basis of $\HC_N$, then \[ \Theta_\alpha(u)=\sum_{j\geq1} \big|\big((\iden\otimes\mathcal G)\sqrt{\vGamma_\alpha}\Psi_j\big)(u)\big\rangle \big\langle\big((\iden\otimes\mathcal G)\sqrt{\vGamma_\alpha}\Psi_j\big)(u)\big|, \] where $(\iden\otimes\mathcal G)\sqrt{\vGamma_\alpha}\Psi_j$ is regarded as an element of $L^2(\mathcal Q,\mu;\Hil_\el^N\otimes\Fock_{\geq\kappa})$. The series converges in trace norm for $\mu$-almost every $u$, and by \cref{lem-soft-disintegration}(iii) the result does not depend on the choice of the basis, up to a $\mu$-null set. The electronic coordinates remain in the union of support balls selected by $\zeta_\alpha$, without assigning fixed particle labels. The unitary $\iden\otimes\mathcal G$ acts only on the soft factor. It therefore commutes with electronic differentiation and with hard-photon operators, and it maps $\AF_{<\kappa}(\vx_i)$ to multiplication by $\va_u(\vx_i)$. Writing \[ \vD_{i,\geq\kappa}=-\i\grad_{\vx_i}-e\AF_{\geq\kappa}(\vx_i),\qquad \vD_i=\vD_{i,\geq\kappa}-e\AF_{<\kappa}(\vx_i), \] we obtain, for $\mu$-almost every $u$, \[ \big((\iden\otimes\mathcal G)\vD_i\Psi\big)(u) =\big(\vD_{i,\geq\kappa}-e\va_u(\vx_i)\big) \big((\iden\otimes\mathcal G)\Psi\big)(u), \] first for $\Psi$ in the finite-particle core and then, by the field bounds, for every $\Psi$ in the form domain. Apply this to $\Psi=\sqrt{\vGamma_\alpha}\Psi_j$, where $(\Psi_j)_{j\geq1}$ is the orthonormal basis used in the series representation of $\Theta_\alpha$ above. For $\Theta=\sum_j|v_j\rangle\langle v_j|$ one has $\sum_j\|Xv_j\|^2=\|X\sqrt{\Theta}\|_{\mathfrak S_2}^2$, so Tonelli's theorem shows that the kinetic energy of $\vGamma_\alpha$ is exactly \begin{equation}\label{conditioned-soft-kinetic} \begin{aligned} \frac12\sum_{i=1}^N\|\vD_i\sqrt{\vGamma_\alpha}\|_{\mathfrak S_2}^2 &=\frac12\sum_{i=1}^N\sum_{j\geq1}\|\vD_i\sqrt{\vGamma_\alpha}\Psi_j\|^2\\ &=\frac12\int_{\mathcal Q}\sum_{i=1}^N \|(\vD_{i,\geq\kappa}-e\va_u(\vx_i))\sqrt{\Theta_\alpha(u)}\|_{\mathfrak S_2}^2 \dd\mu(u). \end{aligned} \end{equation} We next derive the corresponding formula for the current density. For a positive trace-class operator $\Theta$ on $\Hil_\el^N\otimes\Fock_{\geq\kappa}$ with $\Tr(H^{0,0}_{\geq\kappa}\Theta)<\infty$, let $\vjc_{\Theta,\geq\kappa}$ denote the combined current density of $\Theta$ in the hard model, defined by \cref{jc-def} with $\AF$ replaced by $\AF_{\geq\kappa}$. Fix $\vzeta\in C_c^\infty(\RR^3,\RR^3)$, let $\Psi$ be in the form domain, and write $\psi_u=((\iden\otimes\mathcal G)\Psi)(u)$ for its fibers. Integrating \cref{jc-def} against $\vzeta$ turns the partial inner product into a full one, and $-\i\grad_{\vx_1}-e\AF(\vx_1)=\vD_{1,\geq\kappa}-e\AF_{<\kappa}(\vx_1)$. Hence \begin{align*} \int_{\RR^3}\vzeta(\vx)\cdot\vjc_\Psi(\vx)\dd\vx &=N\Re\dua{\Psi}{\vzeta(\vx_1)\cdot \big(\vD_{1,\geq\kappa}-e\AF_{<\kappa}(\vx_1)\big)\Psi}\\ &=N\Re\int_{\mathcal Q}\dua{\psi_u}{\vzeta(\vx_1)\cdot \big(\vD_{1,\geq\kappa}-e\va_u(\vx_1)\big)\psi_u}\dd\mu(u)\\ &=\int_{\mathcal Q}\int_{\RR^3}\vzeta(\vx)\cdot \big(\vjc_{\psi_u,\geq\kappa}(\vx) -e\va_u(\vx)\rho_{\psi_u}(\vx)\big)\dd\vx\,\dd\mu(u). \end{align*} The second line uses the unitarity of $\iden\otimes\mathcal G$, which commutes with $\vzeta(\vx_1)$, and the fiberwise identity for $\vD_1$ above. In the third line, the hard part is the hard-model current density of $\psi_u$ tested against $\vzeta$. For the soft part, $\va_u(\vx_1)$ is real, so $N\Re\dua{\psi_u}{\vzeta(\vx_1)\cdot\va_u(\vx_1)\psi_u} =\int_{\RR^3}\vzeta\cdot\va_u\,\rho_{\psi_u}$ by \cref{density-def}. Densities and current densities are linear in the state. Apply the last identity to $\Psi=\sqrt{\vGamma_\alpha}\Psi_j$ and sum over $j$. By \cref{A-Hrad-bound}, applied to $\AF_{\geq\kappa}$ and $\AF_{<\kappa}$, the resulting integrand $\vjc_{\Theta_\alpha(u),\geq\kappa}(\vx) -e\rho_{\Theta_\alpha(u)}(\vx)\va_u(\vx)$ is integrable over $\RR^3\times\mathcal Q$, which justifies exchanging the sum with the integrals. Since $\vzeta$ is arbitrary, Fubini's theorem and the fundamental lemma of the calculus of variations give \begin{equation}\label{conditioned-soft-current} \vjc_{\vGamma_\alpha}(\vx) =\int_{\mathcal Q}\big(\vjc_{\Theta_\alpha(u),\geq\kappa}(\vx) -e\rho_{\Theta_\alpha(u)}(\vx)\va_u(\vx)\big)\dd\mu(u) \qquad\text{for a.e. }\vx\in\RR^3. \end{equation} Since $\iden_{\{\omega<\kappa\}}G^\mu_{\vx}$ has norms at most those of $G^\mu_{\vx}$, the estimates \cref{ann-Hrad-bound,cre-Hrad-bound} and hence \cref{A-Hrad-bound} also hold for the removed field $\AF_{<\kappa}(\vx)$. As $\vD_{i,\geq\kappa}=\vD_i+e\AF_{<\kappa}(\vx_i)$ and $\Tr(H^{0,0}\vGamma_\alpha)<\infty$, this gives \[ \sum_{i=1}^N\|\vD_{i,\geq\kappa}\sqrt{\vGamma_\alpha}\|_{\mathfrak S_2}^2 \leq2\sum_{i=1}^N\|\vD_i\sqrt{\vGamma_\alpha}\|_{\mathfrak S_2}^2 +2e^2NC_\chi\big(\Tr(H_\rad\vGamma_\alpha)+\Tr\vGamma_\alpha\big)<\infty. \] The computation leading to \cref{conditioned-soft-kinetic}, with $\vD_i$ replaced by $\vD_{i,\geq\kappa}$, and the fact that $H_{\rad,\geq\kappa}$ acts only on $\Fock_{\geq\kappa}$ therefore give \[ \int_{\mathcal Q}\Big(\sum_{i=1}^N\|\vD_{i,\geq\kappa}\sqrt{\Theta_\alpha(u)}\|_{\mathfrak S_2}^2 +\Tr(H_{\rad,\geq\kappa}\Theta_\alpha(u))\Big)\dd\mu(u)<\infty. \] The Coulomb interaction $W=\sum_{n0$, the density matrix $\Theta_\alpha(u)/\Tr\Theta_\alpha(u)$ has finite hard-model energy. The operations applied to $\Theta_\alpha(u)$ in Step 4, namely multiplication by a phase of modulus one and the current shift of \cref{lem-fermionic-current-repair}, leave $\Tr(W\Theta_\alpha(u))$ unchanged. Hence only the kinetic and radiation energies change in the remaining steps. \noindent\textbf{Step 3. Local bounds on the measured field.} Fix a multi-index $\beta$ and $\vx\in\RR^3$. The differentiated soft field $\partial_{\vx}^\beta\Af^\mu_{<\kappa}(\vx)$ is the Segal field with kernel $\partial_{\vx}^\beta(\iden_{\{\omega<\kappa\}}G^\mu_{\vx}) =(-\i\vk)^\beta\iden_{\{\omega<\kappa\}}G^\mu_{\vx}$. Since $|\vk|<\kappa$ on the soft region, the projector identity behind \cref{Gx-norms} gives \[ \sum_{\mu=1}^3\Big\|\frac{(-\i\vk)^\beta\iden_{\{\omega<\kappa\}} G^\mu_{\vx}}{\sqrt\omega}\Big\|^2 \leq\kappa^{2|\beta|}a_\kappa^2,\qquad \sum_{\mu=1}^3\big\|(-\i\vk)^\beta\iden_{\{\omega<\kappa\}} G^\mu_{\vx}\big\|^2 \leq\kappa^{2|\beta|}b_\kappa^2, \] with $a_\kappa,b_\kappa$ as in \cref{infrared-kernel-errors}. These kernels are soft, that is, they vanish for $\omega\geq\kappa$, so \cref{ann-Hrad-bound,cre-Hrad-bound} hold with $H_{\rad,<\kappa}$ in place of $H_\rad$ and give \[ \sum_{\mu=1}^3\|\partial_{\vx}^\beta\Af^\mu_{<\kappa}(\vx)\Psi\|^2 \leq C\kappa^{2|\beta|} \big(a_\kappa^2\|\sqrt{H_{\rad,<\kappa}}\Psi\|^2+b_\kappa^2\|\Psi\|^2\big). \] Apply this to $\Psi=\sqrt{\vGamma_\alpha}\Psi_j$ and sum over $j$. Then use \cref{conditioned-disintegration} with $B=\iden$ and the unbounded function $f(u)=|\partial^\beta\va_u(\vx)|^2$, to which the identity extends by monotone convergence. As $E_\alpha=\Tr(H_{\rad,<\kappa}\vGamma_\alpha)$, we obtain \begin{equation}\label{conditioned-soft-field-moments} \int_{\mathcal Q} \Tr\Theta_\alpha(u)|\partial^\beta\va_u(\vx)|^2\dd\mu(u) \leq C_\beta\kappa^{2|\beta|}\big(a_\kappa^2E_\alpha+b_\kappa^2\Tr\vGamma_\alpha\big), \end{equation} uniformly in $\vx$ and $\alpha$. The fields have smooth versions locally: this follows from these Gaussian derivative moments and local Sobolev embedding, and remains true under the measures $\Tr\Theta_\alpha(u)\dd\mu(u)$. For a fixed $\alpha$, every electron in $\vGamma_\alpha$ lies in one of the balls $B(c_{\alpha_i},c_0R)$. Consider the graph whose vertices are the distinct centers among $c_{\alpha_1},\ldots,c_{\alpha_N}$, two centers being joined by an edge if their distance is at most $4c_0R$. We call the vertex sets of its connected components the \emph{clusters} of $\alpha$ and index them by $s$. A cluster has at most $N$ centers, so its diameter is at most $4(N-1)c_0R\leq C_NR$, while centers in different clusters are more than $4c_0R$ apart. The measured soft field varies on the scale $\kappa^{-1}\geq R$, so in Step 4 it will be replaced on each cluster by a constant. Choose one center of each cluster as its anchor $c_{\alpha,s}$. The construction is illustrated in \cref{fig:soft-clusters}. Let $\theta_a=1$ on a neighborhood of $B(c_a,c_0R)$, with $\supp\theta_a\subset B(c_a,3c_0R/2)$ and $|\grad\theta_a|\leq C/R$, and define the cluster cutoff \[ \theta_{\alpha,s}=1-\prod_{a\text{ in cluster }s}(1-\theta_a). \] Then \begin{gather*} \theta_{\alpha,s}=1\ \text{ on }\bigcup_{a\text{ in cluster }s}B(c_a,c_0R), \qquad \supp\theta_{\alpha,s}\subset \bigcup_{a\text{ in cluster }s}B(c_a,3c_0R/2),\\ |\grad\theta_{\alpha,s}|\leq\frac{C_N}{R}. \end{gather*} Since centers in different clusters are more than $4c_0R$ apart, the supports of $\theta_{\alpha,s}$ for different $s$ are disjoint. Let $\mathcal B_{\alpha,s}$ be a ball centered at $c_{\alpha,s}$ of radius $C_NR$ containing $\supp\theta_{\alpha,s}$. All constants are independent of the separation between clusters. Put $Z_\alpha(u)=\max_s\sup_{\mathcal B_{\alpha,s}}|\grad\va_u|$. For a ball $B\subset\RR^3$ of radius $r$ and $f\in H^2(B)$, the Sobolev inequality \[ \sup_B|f|^2\leq C\sum_{|\beta|\leq2}r^{2|\beta|-3} \int_B|\partial^\beta f|^2\dd\vx \] holds with $C$ independent of $r$. Apply it with $r=C_NR$ to the components of $\grad\va_u$ on each $\mathcal B_{\alpha,s}$, multiply by $\Tr\Theta_\alpha(u)$, and integrate over $u$. By \cref{conditioned-soft-field-moments} for multi-indices of order $|\beta|+1$, and since $|\mathcal B_{\alpha,s}|\leq CR^3$, this yields \begin{align*} \int_{\mathcal Q}\Tr\Theta_\alpha(u)Z_\alpha(u)^2\dd\mu(u) &\leq C\sum_s\sum_{|\beta|\leq2}R^{2|\beta|-3} \int_{\mathcal B_{\alpha,s}}\int_{\mathcal Q}\Tr\Theta_\alpha(u) |\partial^\beta\grad\va_u|^2\dd\mu(u)\dd\vx\\ &\leq C\kappa^2\big(a_\kappa^2E_\alpha+b_\kappa^2\Tr\vGamma_\alpha\big), \end{align*} because $\kappa R\leq1$ and there are at most $N$ clusters (the constant may depend on $N$). No global supremum of the random field is used. \begin{figure}[!htbp] \centering \begin{tikzpicture}[ x=.82cm, y=.82cm, line cap=round, line join=round ] \fill[black!7] (1.4,2.2) circle (.75) (3.0,2.9) circle (.75) (3.2,1.2) circle (.75) (9.2,2.4) circle (.75) (10.6,1.6) circle (.75) (6.4,4.7) circle (.75); \foreach \c in {(1.4,2.2),(3.0,2.9),(3.2,1.2),(9.2,2.4),(10.6,1.6),(6.4,4.7)}{ \filldraw[fill=black!18,draw=black!55,line width=.4pt] \c circle (.5); } \draw[black!80,dashed,line width=.6pt] (3.0,2.9) circle (2.55); \draw[black!80,dashed,line width=.6pt] (9.2,2.4) circle (2.4); \draw[black!80,dashed,line width=.6pt] (6.4,4.7) circle (1.0); \draw[black,line width=.9pt] (1.4,2.2) -- (3.0,2.9) -- (3.2,1.2); \draw[black,line width=.9pt] (9.2,2.4) -- (10.6,1.6); \foreach \c in {(1.4,2.2),(3.0,2.9),(3.2,1.2),(9.2,2.4),(10.6,1.6),(6.4,4.7)}{ \fill[black] \c circle (1.5pt); } \foreach \c in {(3.0,2.9),(9.2,2.4),(6.4,4.7)}{ \draw[black,line width=.8pt] \c circle (3.6pt); } \foreach \x/\y in {1.2/2.38,1.62/2.04,3.32/2.72,2.98/1.0, 9.02/2.72,10.82/1.42,6.68/4.46}{ \filldraw[fill=white,draw=black,line width=.6pt] (\x,\y) circle (.13); \node[font=\bfseries\tiny,inner sep=0pt] at (\x,\y) {$-$}; } \end{tikzpicture} \caption{The clusters of a multi-index $\alpha$ ($N=7$, $\alpha_1=\alpha_2$). Centers at distance at most $4c_0R$ are joined. Anchors are ringed and the balls $\mathcal B_{\alpha,s}$ are dashed.} \label{fig:soft-clusters} \end{figure} \noindent\textbf{Step 4. Current shift on each conditional state.} For almost every $u$, define the one-body phase and residual velocity \[ \begin{split} \varphi_{\alpha,u}(\vx) &=-e\sum_s\theta_{\alpha,s}(\vx)\va_u(c_{\alpha,s})\cdot(\vx-c_{\alpha,s}),\\ \vv_{\alpha,u}(\vx) &=-e\sum_s\theta_{\alpha,s}(\vx) \big(\va_u(\vx)-\va_u(c_{\alpha,s})\big). \end{split} \] In the electronic variables, let $U_{\alpha,u}$ be multiplication by the permutation-symmetric phase \[ \exp\Big(\i\sum_{i=1}^N\varphi_{\alpha,u}(\vx_i)\Big). \] First replace $\Theta_\alpha(u)$ by $U_{\alpha,u}\Theta_\alpha(u)U_{\alpha,u}^\dag$. This leaves the density and the photon marginal unchanged and adds $\rho_{\Theta_\alpha(u)}\grad\varphi_{\alpha,u}$ to the current density. Where $\theta_{\alpha,s}=1$, the velocity $\grad\varphi_{\alpha,u}=-e\va_u(c_{\alpha,s})$ is constant, and $\grad\varphi_{\alpha,u}+\vv_{\alpha,u}=-e\va_u$. Next apply \cref{lem-fermionic-current-repair} with velocity $\vv_{\alpha,u}$ to the hard model $H^{0,0}_{\geq\kappa}$ instead of $H^{0,0}$. Its proof applies verbatim, since the multipliers act only on the electronic variables. If $\Tr\Theta_\alpha(u)=0$, set $\widetilde\Theta_\alpha(u)=0$. Positivity gives $\Theta_\alpha(u)=0$ in this case. If $\Tr\Theta_\alpha(u)>0$, apply it to the normalized state $U_{\alpha,u}\Theta_\alpha(u)U_{\alpha,u}^\dag/\Tr\Theta_\alpha(u)$, multiply the result by $\Tr\Theta_\alpha(u)$, and call the resulting operator $\widetilde\Theta_\alpha(u)$. Its current density is $\vjc_{\Theta_\alpha(u),\geq\kappa}-e\rho_{\Theta_\alpha(u)}\va_u$, the integrand in \cref{conditioned-soft-current}, while \[ \|\vv_{\alpha,u}\|_\infty\leq CeR Z_\alpha(u),\qquad \|\grad\vv_{\alpha,u}\|_\infty\leq CeZ_\alpha(u). \] The wavenumber chosen in the proof of \cref{lem-fermionic-current-repair} depends measurably on $\vv_{\alpha,u}$. When the velocity vanishes, use the identity channel. The tensor lift is fixed independently of the velocity, so this defines a measurable family of channels. The kinetic remainder on an unnormalized conditional operator is bounded by \[ C\Tr\Theta_\alpha(u)(\|\vv_{\alpha,u}\|_\infty^2+\|\grad\vv_{\alpha,u}\|_\infty). \] We now compare the hard-model kinetic energy of the output with the integrand of \cref{conditioned-soft-kinetic}, that is, we estimate \[ \delta_\alpha(u) =\frac12\sum_{i=1}^N \big\|\vD_{i,\geq\kappa}\sqrt{\widetilde\Theta_\alpha(u)}\big\|_{\mathfrak S_2}^2 -\frac12\sum_{i=1}^N \big\|(\vD_{i,\geq\kappa}-e\va_u(\vx_i))\sqrt{\Theta_\alpha(u)}\big\|_{\mathfrak S_2}^2. \] The phase is exact: since $\vD_{i,\geq\kappa}U_{\alpha,u} =U_{\alpha,u}(\vD_{i,\geq\kappa}+\grad\varphi_{\alpha,u}(\vx_i))$, it replaces $\vD_{i,\geq\kappa}$ by $\vD_{i,\geq\kappa}+\grad\varphi_{\alpha,u}(\vx_i)$. By \cref{fermionic-repair-cost}, the current shift then adds \[ \int_{\RR^3}\vjc_{U_{\alpha,u}\Theta_\alpha(u)U_{\alpha,u}^\dag,\geq\kappa} \cdot\vv_{\alpha,u}, \] the hard-model current density of $U_{\alpha,u}\Theta_\alpha(u)U_{\alpha,u}^\dag$ paired with $\vv_{\alpha,u}$, plus the kinetic remainder. For the square completion, abbreviate $\Theta=\Theta_\alpha(u)$, $\vg=\grad\varphi_{\alpha,u}$, $\vv=\vv_{\alpha,u}$, and $\vD_i^{\mathrm h}=\vD_{i,\geq\kappa}$. The phase-transformed current is $\vjc_{\Theta,\geq\kappa}+\rho_\Theta\vg$, so \[ \begin{aligned} &\frac12\sum_{i=1}^N \big\|(\vD_i^{\mathrm h}+\vg(\vx_i))\sqrt\Theta\big\|_{\mathfrak S_2}^2 +\int_{\RR^3}(\vjc_{\Theta,\geq\kappa}+\rho_\Theta\vg)\cdot\vv\\ &\quad=\frac12\sum_{i=1}^N \big\|(\vD_i^{\mathrm h}+\vg(\vx_i)+\vv(\vx_i))\sqrt\Theta\big\|_{\mathfrak S_2}^2 -\frac12\int_{\RR^3}\rho_\Theta|\vv|^2\\ &\quad=\frac12\sum_{i=1}^N \big\|(\vD_i^{\mathrm h}-e\va_u(\vx_i))\sqrt\Theta\big\|_{\mathfrak S_2}^2 -\frac12\int_{\RR^3}\rho_\Theta|\vv|^2. \end{aligned} \] The last equality uses $\vg+\vv=-e\va_u$ on the electronic support. Dropping this nonpositive term and using the bounds on $\vv_{\alpha,u}$, \[ \delta_\alpha(u) \leq C\Tr\Theta_\alpha(u)\big(\|\vv_{\alpha,u}\|_\infty^2 +\|\grad\vv_{\alpha,u}\|_\infty\big) \leq C\Tr\Theta_\alpha(u)\big(e^2R^2Z_\alpha(u)^2+eZ_\alpha(u)\big). \] Integrate over $u$. The first term is controlled by the Step 3 bound on the integral of $\Tr\Theta_\alpha(u)Z_\alpha(u)^2$. For the second, the Cauchy--Schwarz inequality, together with $\int_{\mathcal Q}\Tr\Theta_\alpha\dd\mu=\Tr\vGamma_\alpha$, gives the same bound under a square root. Hence \[ \int_{\mathcal Q}\delta_\alpha(u)\dd\mu(u) \leq Ce^2\kappa^2R^2\big(a_\kappa^2E_\alpha+b_\kappa^2\Tr\vGamma_\alpha\big) +Ce\kappa\sqrt{\big(a_\kappa^2E_\alpha+b_\kappa^2\Tr\vGamma_\alpha\big) \Tr\vGamma_\alpha}. \] Thus constant measured fields on each cluster are compensated without a kinetic remainder depending on their magnitude. \noindent\textbf{Step 5. Assembly.} Using $\HC_N\simeq\Hil_\el^N\otimes\Fock_{\geq\kappa}\otimes\Fock_{<\kappa}$, define \[ \vGamma_{\kappa,R} =\sum_\alpha\int_{\mathcal Q}\widetilde\Theta_\alpha(u)\otimes |\Omega_{<\kappa}\rangle\langle\Omega_{<\kappa}|\dd\mu(u). \] The map $u\mapsto\widetilde\Theta_\alpha(u)$ is measurable by Step 4, and the integral converges in trace norm because the conditional operations preserve traces, so that $\Tr\widetilde\Theta_\alpha(u)=\Tr\Theta_\alpha(u)$. They also preserve photon marginals. By \cref{conditioned-disintegration} with $f=1$, $\int_{\mathcal Q}\Theta_\alpha\dd\mu=\Tr_{<\kappa}\vGamma_\alpha$, and $\vGamma_{\kappa,R}$ is therefore a positive trace-class operator with \[ \Tr\vGamma_{\kappa,R} =\sum_\alpha\Tr\big(\Tr_{<\kappa}\vGamma_\alpha\big) =\sum_\alpha\Tr\vGamma_\alpha=1, \] and its photon marginal is \[ \sum_\alpha\Tr_\el\big(\Tr_{<\kappa}\vGamma_\alpha\big)\otimes |\Omega_{<\kappa}\rangle\langle\Omega_{<\kappa}| =\big(\Tr_{\el,<\kappa}\vGamma\big)\otimes |\Omega_{<\kappa}\rangle\langle\Omega_{<\kappa}|, \] since $\sum_\alpha\zeta_\alpha^2=1$. This is \cref{exact-soft-removal-marginal}. Preparing the soft vacuum adds only its covariant-kinetic contribution $e^2N b_\kappa^2/2$, and the soft radiation energy $E_<$ is removed. For the constraints, let $\Theta$ be a positive trace-class operator on $\Hil_\el^N\otimes\Fock_{\geq\kappa}$ with finite hard-model energy, $\Tr(H^{0,0}_{\geq\kappa}\Theta)<\infty$. Since $\dua{\Omega_{<\kappa}}{\Af^\mu_{<\kappa}(\vx)\Omega_{<\kappa}}=0$, the definitions \cref{density-def,jc-def} give \[ \rho_{\Theta\otimes|\Omega_{<\kappa}\rangle\langle\Omega_{<\kappa}|} =\rho_\Theta,\qquad \vjc_{\Theta\otimes|\Omega_{<\kappa}\rangle\langle\Omega_{<\kappa}|} =\vjc_{\Theta,\geq\kappa}. \] By linearity in the state, Step 4, \cref{conditioned-soft-current}, and \cref{current-preserving-electronic-IMS}, for almost every $\vx$, \begin{align*} \vjc_{\vGamma_{\kappa,R}}(\vx) &=\sum_\alpha\int_{\mathcal Q} \vjc_{\widetilde\Theta_\alpha(u),\geq\kappa}(\vx)\dd\mu(u)\\ &=\sum_\alpha\int_{\mathcal Q} \big(\vjc_{\Theta_\alpha(u),\geq\kappa}(\vx) -e\rho_{\Theta_\alpha(u)}(\vx)\va_u(\vx)\big)\dd\mu(u)\\ &=\sum_\alpha\vjc_{\vGamma_\alpha}(\vx) =\vjc_\vGamma(\vx). \end{align*} The phase and the current shift leave the density unchanged, and the density depends only on the electronic reduced state. Hence \cref{conditioned-disintegration} with $f=1$ and \cref{current-preserving-electronic-IMS} give \[ \rho_{\vGamma_{\kappa,R}} =\sum_\alpha\int_{\mathcal Q}\rho_{\Theta_\alpha(u)}\dd\mu(u) =\sum_\alpha\rho_{\vGamma_\alpha} =\rho_\vGamma. \] Since $\sum_\alpha(a_\kappa^2E_\alpha+b_\kappa^2\Tr\vGamma_\alpha) =a_\kappa^2E_<+b_\kappa^2$ and, by the Cauchy--Schwarz inequality, \[ \sum_\alpha\sqrt{\big(a_\kappa^2E_\alpha+b_\kappa^2\Tr\vGamma_\alpha\big) \Tr\vGamma_\alpha} \leq\sqrt{a_\kappa^2E_<+b_\kappa^2}, \] we obtain \begin{equation}\label{exact-soft-removal-cost-R} \begin{split} \Tr(H^{0,0}\vGamma_{\kappa,R}) &\leq\Tr(H^{0,0}\vGamma)-E_< +C\Big(R^{-2}+e^2b_\kappa^2 +e^2\kappa^2R^2\big(a_\kappa^2E_<+b_\kappa^2\big)\\ &\qquad+e\kappa\sqrt{a_\kappa^2E_<+b_\kappa^2}\Big), \end{split} \end{equation} with $C$ independent of $\vGamma$, $\kappa$ and $R$. It also proves finite form energy of the output. Only the output is asserted to have finite full form energy, so no soft-radiation regularity of an intermediate measured state is needed. The marginal identity proves the photon-number bound. \noindent\textbf{Step 6. Choice of the cluster scale.} Only the terms $R^{-2}$ and $e^2\kappa^2R^2(a_\kappa^2E_<+b_\kappa^2)$ in \cref{exact-soft-removal-cost-R} depend on $R$. Put $q=e\kappa\sqrt{a_\kappa^2E_<+b_\kappa^2}\geq0$. If $\kappa^2\leq q\leq1$, then $q>0$, and we choose $R=q^{-1/2}\in[1,\kappa^{-1}]$. The two terms then balance and both equal $q$. If $q<\kappa^2$, including $q=0$, take $R=\kappa^{-1}$. Then $e^2(a_\kappa^2E_<+b_\kappa^2)<\kappa^2$, so the two terms add up to less than $2\kappa^2$. If $q>1$, take $R=1$. Then the two terms add up to less than $2q^2=2e^2\kappa^2(a_\kappa^2E_<+b_\kappa^2)$. In each case, $\vGamma_\kappa=\vGamma_{\kappa,R}$ satisfies \cref{exact-soft-removal-cost} after changing $C$. The constant $C$ in \cref{exact-soft-removal-cost-R} does not depend on $\vGamma$, so $R$ may depend on $E_<$. \end{proof} \subsection{The proof of \texorpdfstring{\cref{thm-unrestricted-attainment}}{Theorem \getrefnumber{thm-unrestricted-attainment}}} \label{proof:thm-unrestricted-attainment} We remove the soft photons of near-minimizers using \cref{lem-exact-soft-removal}, which keeps $\rho$ and $\vjc$ fixed at a small energy cost and bounds the mean photon number. \Cref{thm-continuum-compactness} then gives a minimizer and lower semicontinuity, and convex conjugation identifies the functional with $F_\LI^{e,\chi}$. \noindent\textbf{Step 1. Soft energy of near-minimizers.} Apply \cref{lem-exact-soft-removal}, with $H_{\rad,<\kappa}$ as defined above. The resulting state $\vGamma_\kappa$ replaces photons with $\omega<\kappa$ by the vacuum and adjusts the electronic current density so that its density and combined current density agree exactly with those of $\vGamma$. The estimate \cref{exact-soft-removal-cost}, together with $\sqrt{s+t}\leq\sqrt s+\sqrt t$ and $\kappa\leq1$, gives \begin{align*} &\Tr(H^{0,0}\vGamma_\kappa) -\Tr(H^{0,0}\vGamma)+\Tr(H_{\rad,<\kappa}\vGamma)\\ &\quad\leq C\Big[\kappa^2+e^2b_\kappa^2+ e^2\kappa^2a_\kappa^2\Tr(H_{\rad,<\kappa}\vGamma) +e\kappa a_\kappa \sqrt{\Tr(H_{\rad,<\kappa}\vGamma)} +e\kappa b_\kappa\Big]. \end{align*} Since $\kappa a_\kappa\to0$, choose $\kappa_0\leq1$, depending only on $N,e$ and $\chi$, so the term proportional to $\kappa^2a_\kappa^2\Tr(H_{\rad,<\kappa}\vGamma)$ can be absorbed by the term $\Tr(H_{\rad,<\kappa}\vGamma)$ on the left-hand side for $0<\kappa\leq\kappa_0$. Young's inequality absorbs the square-root term as well, giving \begin{equation}\label{unrestricted-soft-energy-gain} \Tr(H^{0,0}\vGamma_\kappa) \leq\Tr(H^{0,0}\vGamma) -\tfrac12\Tr(H_{\rad,<\kappa}\vGamma) +C_{N,e,\chi}(\kappa^2+b_\kappa^2). \end{equation} For an exact feasible $\vGamma$, set $d_\vGamma=\Tr(H^{0,0}\vGamma) -\wt{F}_\LI^{e,\chi}(\rho,\vjc)$. The competitor $\vGamma_\kappa$ has exactly the same constraints. Consequently \begin{equation}\label{unrestricted-uniform-soft-energy} \Tr(H_{\rad,<\kappa}\vGamma)\leq 2d_\vGamma+C_{N,e,\chi}(\kappa^2+b_\kappa^2), \qquad 0<\kappa\leq\kappa_0. \end{equation} \noindent\textbf{Step 2. A uniformly number-bounded minimizing sequence.} For $0<\kappa<\kappa_0$, the pointwise inequality \[ \iden_{\{\omega\geq\kappa\}} \leq\frac{\omega}{\kappa_0} +\omega\int_\kappa^{\kappa_0} \frac{\iden_{\{\omega0,\quad\sum_rp_r=1. \] Here $(\Phi_r)$ is orthonormal in $\Hil_\el^N\otimes\Fock_{\geq\kappa}$, and each $\Phi_r$ is antisymmetric in the electronic coordinates. Let $H_{\geq\kappa}^{0,0}$ be the hard model introduced before \cref{lem-exact-soft-removal}, and put $\vD_{n,\geq\kappa}=-\i\grad_{\vx_n} -e\AF_{\geq\kappa}(\vx_n)$. The soft-vacuum expectation and \cref{Gx-norms} give the exact identity \begin{equation}\label{many-electron-soft-vacuum-energy} \Tr(H^{0,0}\vGamma_\kappa) =\sum_rp_r\dua{\Phi_r}{H_{\geq\kappa}^{0,0}\Phi_r} +\frac{Ne^2}{2}b_\kappa^2. \end{equation} In particular, every $\Phi_r$ is in the hard-model form domain and the displayed sum of nonnegative form energies is finite. The soft one-photon space $\Hil_{<\kappa}$ is infinite dimensional. Choose an orthonormal family $(f_r)$ in it, write $\eta_r=a^\dag(f_r)\Omega_{<\kappa}$, and consider \begin{equation}\label{many-electron-soft-tag-state} \Psi_\kappa=\sum_r\sqrt{p_r}\,\Phi_r\otimes\eta_r. \end{equation} The vectors $\Phi_r\otimes\eta_r$ are orthonormal, so the series \cref{many-electron-soft-tag-state} converges in norm to a unit vector. We show that it also converges in the form norm of $H^{0,0}$, so that $\Psi_\kappa$ lies in its form domain $\mathcal Q_\PF$. Each term of the series has exactly one soft photon. Identify $\Hil_{<\kappa}$ with the one-photon sector of $\Fock_{<\kappa}$, and let $\Psi\in\Hil_\el^N\otimes\Fock_{\geq\kappa}\otimes\Hil_{<\kappa}$. For almost every $\vX=(\vx_1,\dots,\vx_N)$, the value $\Psi(\vX)$ lies in $\Fock_{\geq\kappa}\otimes\Hil_{<\kappa}$, and $\Af^\mu_{<\kappa}(\vx_n)$ acts on it as $a^\dag(g)+a(g)$ on the soft factor, with $g=\iden_{\{\omega<\kappa\}}G^\mu_{\vx_n}$. The vector $a^\dag(g)\Psi(\vX)$ has two soft photons and $a(g)\Psi(\vX)$ has none, so these two vectors are orthogonal. Together with the CCR $\|a^\dag(g)\Psi(\vX)\|^2=\|g\|^2\|\Psi(\vX)\|^2+\|a(g)\Psi(\vX)\|^2$ and $\|a(g)\Psi(\vX)\|\leq\|g\|\|\Psi(\vX)\|$, this gives \[ \|[a^\dag(g)+a(g)]\Psi(\vX)\|^2 =\|g\|^2\|\Psi(\vX)\|^2+2\|a(g)\Psi(\vX)\|^2 \leq3\|g\|^2\|\Psi(\vX)\|^2. \] By \cref{Gx-norms} restricted to $\{\omega<\kappa\}$, the sum of $\|g\|^2$ over $\mu$ is $b_\kappa^2$. Summing over $\mu$ and $n$ and integrating over $\vX$ therefore gives \begin{equation}\label{many-electron-soft-tag-variance} \sum_{n=1}^N\|\AF_{<\kappa}(\vx_n)\Psi\|^2 \leq3Nb_\kappa^2\|\Psi\|^2. \end{equation} Consider a finite partial sum $\Psi_I=\sum_{r\in I}\sqrt{p_r}\,\Phi_r\otimes\eta_r$. Both $\Psi_I$ and $\vD_{n,\geq\kappa}\Psi_I$ have exactly one soft photon, while $\AF_{<\kappa}(\vx_n)\Psi_I$ has zero or two. Since $\vD_n=\vD_{n,\geq\kappa}-e\AF_{<\kappa}(\vx_n)$, it follows that \[ \dua{\vD_{n,\geq\kappa}\Psi_I}{\AF_{<\kappa}(\vx_n)\Psi_I}=0, \qquad \|\vD_n\Psi_I\|^2 =\|\vD_{n,\geq\kappa}\Psi_I\|^2+e^2\|\AF_{<\kappa}(\vx_n)\Psi_I\|^2. \] Since the tags $\eta_r$ are orthonormal, tracing out the soft photons gives \begin{equation}\label{many-electron-soft-tag-hard-state} \Tr_{<\kappa}|\Psi_I\rangle\langle\Psi_I| =\sum_{r\in I}p_r|\Phi_r\rangle\langle\Phi_r|. \end{equation} The hard model $H^{0,0}_{\geq\kappa}$ acts only on $\Hil_\el^N\otimes\Fock_{\geq\kappa}$, so \[ \dua{\Psi_I}{H^{0,0}_{\geq\kappa}\Psi_I} =\sum_{r\in I}p_r\dua{\Phi_r}{H^{0,0}_{\geq\kappa}\Phi_r}. \] By the identity for $\|\vD_n\Psi_I\|^2$, the rest of $\dua{\Psi_I}{H^{0,0}\Psi_I}$ is $\frac{e^2}{2}\sum_n\|\AF_{<\kappa}(\vx_n)\Psi_I\|^2$, which \cref{many-electron-soft-tag-variance} bounds, plus the soft radiation energy. Since $\omega<\kappa$ on $\Hil_{<\kappa}$, the latter satisfies $\dua{\Psi_I}{H_{\rad,<\kappa}\Psi_I}\leq\kappa\|\Psi_I\|^2 =\kappa\sum_{r\in I}p_r$. Hence \begin{equation}\label{many-electron-soft-tag-partial-energy} \dua{\Psi_I}{H^{0,0}\Psi_I} \leq\sum_{r\in I}p_r \dua{\Phi_r}{H_{\geq\kappa}^{0,0}\Phi_r} +\left(\kappa+\frac{3Ne^2}{2}b_\kappa^2\right) \sum_{r\in I}p_r. \end{equation} Applying this bound to tails, together with \cref{many-electron-soft-vacuum-energy}, proves convergence of \cref{many-electron-soft-tag-state} in the full form norm. Hence $\Psi_\kappa\in \mathcal Q_\PF$ is normalized and retains electronic antisymmetry. Letting $I$ increase to all indices in \cref{many-electron-soft-tag-hard-state}, by continuity of the partial trace in trace norm, gives \[ \Tr_{<\kappa}|\Psi_\kappa\rangle\langle\Psi_\kappa| =\gamma_\kappa =\Tr_{<\kappa}\vGamma_\kappa. \] So $\Psi_\kappa$ and $\vGamma_\kappa$ have the same reduced state on electrons and hard photons. We next prove that $\Psi_\kappa$ and $\vGamma_\kappa$ have the same density and combined current density. Put $\Psi_r^0=\Phi_r\otimes\Omega_{<\kappa}$, so that $\vGamma_\kappa=\sum_rp_r|\Psi_r^0\rangle\langle\Psi_r^0|$, and let $\vGamma_I=\sum_{r\in I}p_r|\Psi_r^0\rangle\langle\Psi_r^0|$. Let $v\in C_c^\infty(\RR^3)$ and $\vA\in C_c^\infty(\RR^3;\RR^3)$. Integrating \cref{density-def,jc-def} against $v$ and $\vA$ gives, for every $\Psi\in\mathcal Q_\PF$, \begin{equation}\label{many-electron-soft-tag-tested} \int_{\RR^3}v\rho_\Psi=N\dua{\Psi}{v(\vx_1)\Psi}, \qquad \int_{\RR^3}\vA\cdot\vjc_\Psi =N\Re\dua{\Psi}{\vA(\vx_1)\cdot\vD_1\Psi}. \end{equation} Write $\vD_1=\vD_{1,\geq\kappa}-e\AF_{<\kappa}(\vx_1)$. The operators $v(\vx_1)$ and $\vA(\vx_1)\cdot\vD_{1,\geq\kappa}$ act only on $\Hil_\el^N\otimes\Fock_{\geq\kappa}$. Since the tags are orthonormal and $\Omega_{<\kappa}$ is normalized, \begin{align*} \dua{\Psi_I}{v(\vx_1)\Psi_I} &=\sum_{r\in I}p_r\dua{\Psi_r^0}{v(\vx_1)\Psi_r^0},\\ \dua{\Psi_I}{\vA(\vx_1)\cdot\vD_{1,\geq\kappa}\Psi_I} &=\sum_{r\in I}p_r \dua{\Psi_r^0}{\vA(\vx_1)\cdot\vD_{1,\geq\kappa}\Psi_r^0}. \end{align*} The soft field contributes to neither state. The vector $\AF_{<\kappa}(\vx_1)\Psi_I$ has zero or two soft photons while $\Psi_I$ has one, and $\AF_{<\kappa}(\vx_1)\Psi_r^0$ has one soft photon while $\Psi_r^0$ has none. Hence \[ \dua{\Psi_I}{\vA(\vx_1)\cdot\AF_{<\kappa}(\vx_1)\Psi_I} =\dua{\Psi_r^0}{\vA(\vx_1)\cdot\AF_{<\kappa}(\vx_1)\Psi_r^0} =0. \] By \cref{many-electron-soft-tag-tested} and linearity in the state, the last three displays give \begin{equation}\label{many-electron-soft-tag-partial-constraints} \int_{\RR^3}v\rho_{\Psi_I}=\int_{\RR^3}v\rho_{\vGamma_I}, \qquad \int_{\RR^3}\vA\cdot\vjc_{\Psi_I} =\int_{\RR^3}\vA\cdot\vjc_{\vGamma_I}. \end{equation} Now let $I$ increase to all indices. Then $|\Psi_I\rangle\langle\Psi_I|\to|\Psi_\kappa\rangle\langle\Psi_\kappa|$ and $\vGamma_I\to\vGamma_\kappa$ in trace norm. All these operators have trace at most one, and their energies are bounded uniformly in $I$, by \cref{many-electron-soft-tag-partial-energy} and by $\Tr(H^{0,0}\vGamma_I)\leq\Tr(H^{0,0}\vGamma_\kappa)$. Hence \cref{mixed-density-passage,mixed-current-passage} of \cref{lem-trace-compactness-and-limits} pass both sides of \cref{many-electron-soft-tag-partial-constraints} to the limit. Since $v$ and $\vA$ are arbitrary, $\Psi_\kappa$ and $\vGamma_\kappa$ have the same density and combined current density. Together with \cref{lem-exact-soft-removal}, this gives \begin{equation}\label{many-electron-soft-tag-constraints} (\rho_{\Psi_\kappa},\vjc_{\Psi_\kappa}) =(\rho_{\vGamma_\kappa},\vjc_{\vGamma_\kappa}) =(\rho_\vGamma,\vjc_\vGamma). \end{equation} Passing to the limit in \cref{many-electron-soft-tag-partial-energy} and subtracting the vacuum contribution in \cref{many-electron-soft-vacuum-energy} gives \begin{equation}\label{many-electron-soft-tag-cost} \dua{\Psi_\kappa}{H^{0,0}\Psi_\kappa} \leq\Tr(H^{0,0}\vGamma_\kappa) +\kappa+Ne^2b_\kappa^2. \end{equation} The last two terms tend to zero as $\kappa\downarrow0$, since $b_\kappa\to0$ for every form factor. Finally, let $\NC_{\geq\kappa}=\dd\varGamma(\iden_{\{\omega\geq\kappa\}})$. The soft tags have exactly one photon, so $\dua{\Psi_\kappa}{\NC\Psi_\kappa}=1+\Tr(\NC_{\geq\kappa}\gamma_\kappa)$. Tracing \cref{exact-soft-removal-marginal} over $\Fock_{<\kappa}$ gives $\Tr_{\el,<\kappa}\vGamma_\kappa=\Tr_{\el,<\kappa}\vGamma$, so $\Tr(\NC_{\geq\kappa}\gamma_\kappa)=\Tr(\NC_{\geq\kappa}\vGamma)$. Thus \begin{equation}\label{many-electron-soft-tag-number} \dua{\Psi_\kappa}{\NC\Psi_\kappa} =1+\Tr(\NC_{\geq\kappa}\vGamma) \leq1+\kappa^{-1}\Tr(H_\rad\vGamma)<\infty. \end{equation} Together, \cref{many-electron-soft-removal-limit,many-electron-soft-tag-constraints,many-electron-soft-tag-cost} prove \cref{many-electron-pure-approximation} by choosing $\kappa$ sufficiently small. Fix $(\rho,\vjc)\in X\times Y$, and let $\vGamma\in\dm$ satisfy $(\rho_\vGamma,\vjc_\vGamma)=(\rho,\vjc)$. The pure states in \cref{many-electron-pure-approximation} are admissible in \cref{FLL-def}, so $F_\LL^{e,\chi}(\rho,\vjc)\leq\Tr(H^{0,0}\vGamma)+\delta$ for every $\delta>0$. Taking the infimum over $\vGamma$ and using \cref{L-cs-relations} proves $F_\LL^{e,\chi}(\rho,\vjc)=\wt F_\LI^{e,\chi}(\rho,\vjc)$. If no such $\vGamma$ exists, then no admissible pure state exists either, since $|\Psi\rangle\langle\Psi|\in\dm$ for every normalized $\Psi\in\mathcal Q_\PF$. Then $F_\LL^{e,\chi}(\rho,\vjc)=\wt F_\LI^{e,\chi}(\rho,\vjc)=+\infty$. The equality with $F_\LI^{e,\chi}(\rho,\vjc)$ follows from \cref{thm-unrestricted-attainment}. If $F_\LL^{e,\chi}(\rho,\vjc)<\infty$, equality of the two searches identifies a pure minimizer at $(\rho,\vjc)$ with a rank-one minimizer of the total mixed-state problem, in both directions. \end{proof} \phantomsection \label{proof:cor-many-electron-pure-functional} \begin{proof}[Proof of \cref{cor-many-electron-pure-functional}] Properness, convexity, and norm lower semicontinuity follow from \cref{many-electron-pure-ensemble-equality} and the corresponding properties of $F_\LI^{e,\chi}(\rho,\vjc)$. Every pure state defines a rank-one density matrix, and \cref{many-electron-pure-approximation} provides, for every mixed state, a finite-energy pure state with the same density and combined current density. This proves the first equality in \cref{unrestricted-pure-mixed-PF-class}. Since $\Tr(H^{0,0}\vGamma)<\infty$ for every $\vGamma\in\dm$, the middle set in \cref{unrestricted-pure-mixed-PF-class} is the domain of $\wt{F}_\LI^{e,\chi}$, and \cref{unrestricted-Lieb-equality} identifies $\wt{F}_\LI^{e,\chi}$ with $F_\LI^{e,\chi}$. This proves the second equality. The pure-state class is therefore the domain of the convex functional $F_\LI^{e,\chi}$, hence convex. Fix any mixed-state minimizer $\vGamma_*$ at $(\rho,\vjc)$. By \cref{thm-unrestricted-attainment}, it has finite mean photon number. For $0<\kappa\leq1$, let $\Psi_\kappa$ be the pure state \cref{many-electron-soft-tag-state} built from $\vGamma_*$ in the proof of \cref{thm-many-electron-soft-purification}. That is, first remove the soft photons from $\vGamma_*$ by \cref{lem-exact-soft-removal}, then attach orthonormal one-photon soft tags to the eigenvectors of the result. Choose a sequence $\kappa_j\downarrow0$ and put $\Psi_j=\Psi_{\kappa_j}$. By \cref{many-electron-soft-tag-constraints}, $(\rho_{\Psi_j},\vjc_{\Psi_j}) =(\rho_{\vGamma_*},\vjc_{\vGamma_*})=(\rho,\vjc)$ for every $j$. By \cref{many-electron-soft-removal-limit,many-electron-soft-tag-cost} and $b_{\kappa_j}\to0$, \[ \limsup_{j\to\infty}\dua{\Psi_j}{H^{0,0}\Psi_j} \leq\Tr(H^{0,0}\vGamma_*) =F_\LL^{e,\chi}(\rho,\vjc), \] where the equality holds because $\vGamma_*$ is a minimizer, by \cref{unrestricted-Lieb-equality,many-electron-pure-ensemble-equality}. Each $\Psi_j$ is admissible in \cref{FLL-def}, so $\dua{\Psi_j}{H^{0,0}\Psi_j}\geq F_\LL^{e,\chi}(\rho,\vjc)$, and hence $\dua{\Psi_j}{H^{0,0}\Psi_j}\to F_\LL^{e,\chi}(\rho,\vjc)$. Since $\NC_{\geq\kappa_j}\leq\NC$, \cref{many-electron-soft-tag-number} and the photon-number bound in \cref{thm-unrestricted-attainment} give \cref{pure-minimizing-sequence-number-bound}. \end{proof} \phantomsection \label{proof:prop-zero-current-density-functional} \begin{proof}[Proof of \cref{prop-zero-current-density-functional}] Let $\Theta$ be the time reversal \cref{time-reversal-def}, with the properties in \cref{prop-time-reversal}, and let $\rdens_N:=\{\rho\geq0:\ \int_{\RR^3}\rho=N,\ \sqrt\rho\in H^1(\RR^3)\}$. We first show that, for either $F\in\{F_\LL,F_\LI\}$, every $\rho\in\rdens_N$, and every $\vjc\in Y$, \begin{equation}\label{zero-current-minimum} F^{e,\chi}(\rho,\vjc)=F^{e,\chi}(\rho,-\vjc),\qquad F^{e,\chi}(\rho,0)\leq F^{e,\chi}(\rho,\vjc). \end{equation} Evenness of both functionals in the current density is \cref{F-time-reversal}. For the PFL functional, convexity then gives \[ F_\LI^{e,\chi}(\rho,0) \leq\frac12\big[F_\LI^{e,\chi}(\rho,\vjc) +F_\LI^{e,\chi}(\rho,-\vjc)\big] =F_\LI^{e,\chi}(\rho,\vjc). \] This is \cref{zero-current-minimum} for $F_\LI$. By \cref{time-reversal-laws}, $\bar E(u,\vA)=\bar E(u,-\vA)$, and concavity of $\bar E$ gives $\bar E(u,\vA)\leq\bar E(u,0)=E(u,0)$, proving \cref{zero-current-scalar-dual} directly from \cref{FL-def}. For the mixed-state search, replace any $\vGamma$ by $\tfrac12(\vGamma+\Theta\vGamma\Theta^{-1})$. This preserves its density and energy and sets its combined current density to zero. \Cref{thm-unrestricted-attainment} then gives \cref{zero-current-density-search}, including the mixed-state minimum. The minimizing density matrix can be chosen time-reversal invariant. The density bound makes $\rho\in\rdens_N$ necessary for finiteness. Conversely, every $\rho\in\rdens_N$ has a finite-kinetic-energy electronic representative \cite{Lieb1983}. Vacuum embedding and the same averaging then give a finite-energy density matrix at $(\rho,0)$. For the PFLL functional, to prove minimality at zero current density, take any normalized $\Psi\in\mathcal Q_\PF$ with pair $(\rho,\vjc)$ and form \[ \vGamma_\Psi=\frac12\bigl( |\Psi\rangle\langle\Psi| +|\Theta\Psi\rangle\langle\Theta\Psi|\bigr). \] This density matrix has pair $(\rho,0)$ and energy $\dua{\Psi}{H^{0,0}\Psi}$. For every $\delta>0$, the approximation statement \cref{many-electron-pure-approximation} supplies a normalized pure state $\Phi_\delta$ with the exact pair $(\rho,0)$ and \[ \dua{\Phi_\delta}{H^{0,0}\Phi_\delta} \leq\dua{\Psi}{H^{0,0}\Psi}+\delta. \] Letting $\delta\downarrow0$ and taking the infimum over $\Psi$ proves \cref{zero-current-minimum} for $F_\LL$, without using its convexity or its equality with $F_\LI$. Taking the infimum over all pure states of density $\rho$ gives \cref{zero-current-pure-density-search}. The density bound and the same construction applied to a vacuum-embedded electronic representative give the finite domain $\rdens_N$. Finally, apply \cref{many-electron-pure-approximation} to a PFL optimizer at $(\rho,0)$. Letting the energy error tend to zero gives $F_\LL^{e,\chi}(\rho,0)\leq F_\LI^{e,\chi}(\rho,0)$. The reverse inequality follows from \cref{L-cs-relations}. Only the constructive approximation is used here, not the general equality of the two functionals. \end{proof} \subsection{Properties of optimizers} \phantomsection \label{proof:prop-optimizer-mean-fields} \begin{proof}[Proof of \cref{prop-optimizer-mean-fields}] For $h\in\Hil_\ph$ with $h/\sqrt\omega\in\Hil_\ph$, the annihilation estimate and Cauchy--Schwarz in a spectral decomposition of $\vGamma$ give \begin{equation}\label{optimizer-annihilation-expectation-bound} |\Tr(a(h)\vGamma)| \leq\|h/\sqrt\omega\|\, \big(\Tr(H_\rad\vGamma)\big)^{1/2}. \end{equation} Thus all linear traces below are well defined, even without finite mean photon number. For (i), put $g_j=\sum_\mu\int u_j^\mu(\vx)G^\mu_{\vx}\dd\vx$ and $\vec\AF=(\int\vu_1\cdot\AF,\ldots,\int\vu_n\cdot\AF)$, so that $\int\vu_j\cdot\AF=\varPhi(g_j)$ and, as in Section~\ref{sec-free-field}, $g_j\in D(\omega)$. By the symmetry of the form factor, all scalar products among the kernels $g_j$, $\omega g_j$, and $G^\mu_{\vx}$ are real. By \cref{prop-weyl-algebra}(iii), the components of $\vec\AF$ therefore strongly commute with each other and with every $\Af^\mu(\vx)$, so \[ U_t=e^{\i t\phi(\vec\AF)},\qquad t\in\RR, \] is a photon unitary commuting with every $\Af^\mu(\vx)$. It commutes with each $\vD_n$ and leaves the density, the paramagnetic current density, and the conditional field expectations unchanged, hence also $\vjc$. For $\xi\in\RR^n$, $e^{\i\xi\cdot\vec\AF}=W(\i\sum_j\xi_jg_j)$, and the pointwise shift \cref{weyl-weighted-point-shift} gives \[ a(\vk,\lambda)e^{\i\xi\cdot\vec\AF}\eta =e^{\i\xi\cdot\vec\AF}\Big(a(\vk,\lambda)\eta +\i\sum_{j=1}^n\xi_jg_j(\vk,\lambda)\eta\Big), \qquad\eta\in D(\sqrt{H_\rad}). \] Since $e^{\i t\phi}-1\in\mathcal S(\RR^n)$, integrating this identity against its Fourier transform, and using closedness of the annihilation map, gives \[ a(\vk,\lambda)U_t\eta =U_t\Big(a(\vk,\lambda)\eta +\i t\sum_{j=1}^ng_j(\vk,\lambda)(\partial_j\phi)(\vec\AF)\eta\Big). \] As $g_j\in D(\omega)$, $U_t$ preserves $D(\sqrt{H_\rad})$, hence $\mathcal Q_\PF$, and only the radiation energy changes. Recall that $\int\vu_j\cdot\EF_\perp=-\i\big(a^\dag(\omega g_j)-a(\omega g_j)\big)$. Squaring and integrating against $\omega$ gives the expansion below. In its linear term we used that the commutator \[ [a(\omega g_j),(\partial_j\phi)(\vec\AF)] =\sum_{l=1}^n\dua{\omega g_j}{g_l}(\partial_l\partial_j\phi)(\vec\AF) \] is self-adjoint: \[ \begin{aligned} \Tr(H^{0,0}U_t\vGamma U_t^\dag) &=\Tr(H^{0,0}\vGamma) -t\sum_{j=1}^n\Re\Tr\Big((\partial_j\phi)(\vec\AF) \int\vu_j\cdot\EF_\perp\,\vGamma\Big)\\ &\quad+t^2\sum_{j,l=1}^n\dua{\sqrt\omega g_j}{\sqrt\omega g_l} \Tr\big((\partial_j\phi)(\vec\AF)(\partial_l\phi)(\vec\AF) \vGamma\big). \end{aligned} \] Optimality at $t=0$ proves \cref{optimizer-field-statistics}. For (ii), apply (i) with $n=1$, $\vu_1=\vu$, and $\phi(q)=m\psi(q/m)$, where $\psi\in\mathcal S(\RR)$ is real with $\psi'(0)=1$. Then $\phi'(\int\vu\cdot\AF)=\psi'(m^{-1}\int\vu\cdot\AF)$ is bounded uniformly in $m$ and converges strongly to $\iden$ as $m\to\infty$. Since $\vGamma$ has finite radiation energy, dominated convergence over a spectral decomposition of $\vGamma$ gives $\Re\Tr\big(\int\vu\cdot\EF_\perp\,\vGamma\big)=0$. This trace is real, because $\int\vu\cdot\EF_\perp$ is self-adjoint, which proves \cref{optimizer-mean-electric-zero}. For (iii), suppose first that $\vjc=0$. By \cref{zero-current-density-search}, $\vGamma$ also minimizes the internal energy among all density matrices with density $\rho$. For any $f\in D(\omega)$, \cref{cor-weyl-PF-shifts} shows that $W(tf)\vGamma W(tf)^\dag$ has density $\rho$, and that its internal energy is the quadratic polynomial \cref{weyl-PF-energy-shift} in $t$, with linear coefficient $2\Re\Tr(a(\omega f)\vGamma)-e\int\vc_f\cdot\vjc$. This coefficient vanishes. Since $\vjc=0$, using both $f$ and $\i f$ gives $\Tr(a(\omega f)\vGamma)=0$. For any $h$ as in \cref{optimizer-annihilation-expectation-bound}, set \[ h_m=\iden_{\{m^{-1}\leq\omega\leq m\}}h, \qquad f_m=h_m/\omega\in D(\omega). \] Then $\|(h_m-h)/\sqrt\omega\|\to0$, so \cref{optimizer-annihilation-expectation-bound} shows that $\Tr(a(h)\vGamma)=0$. Taking $h=G^\mu_{\vx}$ proves the pointwise assertion for $\AF$, and taking $h=g_{\grad\times\vu}$ proves it for $\BF$. Finally, if $\vjc=\rho\grad S$, the gauge covariance identities \cref{current-gauge,energy-gauge,F-gauge}, applied to spectral decompositions, show that $U_{-S}\vGamma U_{-S}^\dag$ is an optimizer at $(\rho,0)$. The electronic unitary $U_{-S}$ leaves the photon reduced state unchanged. All coherent photon amplitudes therefore agree with those of this zero-current optimizer. \end{proof} \phantomsection \label{proof:prop-optimizer-counterexamples} \begin{proof}[Proof of \cref{prop-optimizer-counterexamples}] (i) The vacuum product $\sqrt\rho\otimes\Omega$ has finite energy and pair $(\rho,0)$. By \cref{thm-unrestricted-attainment} an optimizer $\vGamma_0$ at this pair exists. Gauge covariance with $S(\vx)=\vq\cdot\vx$ maps it to an optimizer at $(\rho,\rho\vq)$. \Cref{prop-optimizer-mean-fields} gives the claimed fields for every optimizer there, and direct differentiation gives the divergence. (ii) Write $\phi=\sqrt\rho$. For any finite-energy density matrix $\vGamma=\sum_r p_r|\Psi_r\rangle\langle\Psi_r|$ with density $\rho$, integration by parts gives \[ \Tr(H^{0,0}\vGamma) =\frac34+\frac12\sum_{r,\mu}p_r \|(\partial_\mu+x^\mu)\Psi_r\|^2 +\Tr(H_\rad\vGamma). \] Indeed, $\int|\vx|^2\rho=3/2$. The identity follows first on smooth vectors and then by approximation in $H^1\cap D(|\vx|)$. The state $\phi\otimes\Omega$ attains $3/4$ at $(\rho,0)$. Equality forces each occupied vector to be $\phi\otimes\xi_r$, and $\ker H_\rad=\Span\{\Omega\}$ then forces the density matrix to be $|\phi\otimes\Omega\rangle\langle\phi\otimes\Omega|$. Gauge covariance is a bijection of the optimizer sets and proves \cref{unique-gaussian-optimizer} and its uniqueness. Suppose that $\Psi_\vq$ were an eigenvector of some admissible $H^{v,\vA}$ with eigenvalue $E$. Removing its electronic phase makes $\phi\otimes\Omega$ an eigenvector with vector potential $\vA'=\vA+\vq\in L^3+L^\infty$. At $e=0$ the photon vacuum decouples. Taking the real part of the weak electronic eigenvalue equation, tested against real compactly supported functions, gives \[ -\frac12\Delta\phi+\left(v+\frac12|\vA'|^2\right)\phi=E\phi. \] Since $\phi>0$, this implies \[ v+\frac12|\vA'|^2 =E+\frac{\Delta\phi}{2\phi} =E+\frac{|\vx|^2-3}{2}. \] The left-hand side belongs to $L^{3/2}+L^\infty$, whereas the quadratically growing right-hand side does not. This is a contradiction. Finally, time reversal maps the constraint $(\rho,\rho\vq)$ to $(\rho,-\rho\vq)$, so uniqueness at $(\rho,\rho\vq)$ does not imply time-reversal invariance. \end{proof} \phantomsection \label{proof:prop-optimizer-symmetry} \begin{proof}[Proof of \cref{prop-optimizer-symmetry}] (i) By \cref{prop-time-reversal}, $\Theta$ commutes with every $U_\vR$, and $\Theta^2=\iden$. Thus $g\mapsto V_g$ is a strongly continuous representation of $G$ by unitary and antiunitary operators. By \cref{H-rotation,rho-rotation,jc-rotation,time-reversal-laws}, each $V_g$ preserves $\mathcal Q_\PF$ and the energy, and maps a density matrix with pair $(\rho',\vj')$ to one with pair $(\rho'\circ\vR^\top,s\vR\vj'\circ\vR^\top)$. For $g\in G$ the pair $(\rho,\vjc)$ is fixed, so $V_g$ maps optimizers to optimizers, with inverse given by $V_{g^{-1}}$. Let $\vGamma$ be an optimizer, and let $\dd g$ be the normalized Haar measure on $G$. Since $V_g$ commutes with $H^{0,0}$, the map \[ g\mapsto\sqrt{H^{0,0}+1}\,V_g\vGamma V_g^{-1}\sqrt{H^{0,0}+1} =V_g\sqrt{H^{0,0}+1}\,\vGamma\sqrt{H^{0,0}+1}\,V_g^{-1} \] is continuous in trace norm. Hence \[ \vGamma_G=\int_G V_g\vGamma V_g^{-1}\dd g \] is a density matrix with $\sqrt{H^{0,0}+1}\,\vGamma_G\sqrt{H^{0,0}+1}$ trace class. As in the proof of \cref{cor-weyl-PF-shifts}, the energy and the pairings of the density and of the combined current density with bounded test functions are continuous linear functionals of $\sqrt{H^{0,0}+1}\,\vGamma\sqrt{H^{0,0}+1}$ in trace norm. They are constant along the orbit, so they pass to the average, and $\vGamma_G$ is an optimizer at $(\rho,\vjc)$. Invariance of Haar measure makes it $G$-invariant. (ii) Every $G^\mu_{\vx}$ lies in $\Hil_S^\perp$. By \cref{weyl-factorization}, the field acts only on the first factor, and $H_\rad$ is the sum of the radiation energies of the two factors. Replacing an optimizer $\vGamma$ by the partial trace of $\vGamma$ over the second factor, tensored with $|\Omega_S\rangle\langle\Omega_S|$, therefore preserves the density, the combined current density, and every term of the energy except the radiation energy of the second factor, which it removes. Optimality forces that term to vanish. Since $\omega>0$ almost everywhere, the radiation energy of the second factor has kernel spanned by $\Omega_S$, so every eigenvector of $\vGamma$ with positive eigenvalue has $\Omega_S$ as its second factor. \end{proof} \begin{proof}[Proof of \cref{prop-optimizer-soft-spectrum}] \phantomsection \label{proof:prop-optimizer-soft-spectrum} The competitor in \cref{unrestricted-soft-energy-gain} has exactly the same density and combined current density as $\vGamma_*$. Optimality therefore gives \[ 0\leq\Tr(H^{0,0}(\vGamma_{*,\kappa}-\vGamma_*)) \leq-\tfrac12E_*(\kappa)+C(\kappa^2+b_\kappa^2), \] which proves \cref{optimizer-soft-energy-bound} after changing $C$. No supporting external potential is needed. Tonelli's theorem and $\mathcal U_*(\varepsilon)=\varepsilon\mathcal N_*(\varepsilon)$ give the exact identity \begin{equation*} N_*(\kappa)=\frac{E_*(\kappa)}{\kappa} +\int_0^\kappa\frac{E_*(t)}{t^2}\dd t. \end{equation*} The form factor contribution obeys a second Tonelli identity, \[ \frac{b_\kappa^2}{\kappa} +\int_0^\kappa\frac{b_t^2}{t^2}\dd t =\int_{\omega<\kappa}\frac{|\wh\chi(\vk)|^2}{\omega(\vk)}\dd\vk =a_\kappa^2. \] Together these prove \cref{optimizer-soft-number-bound}. Finally, for the hard cutoff and $\kappa\leq\Lambda$, $b_\kappa^2=2\pi\kappa^2$ and $a_\kappa^2=4\pi\kappa$. \end{proof} \subsection{Zero coupling} \label{proof:thm-soft-photon-relaxation} \begin{proof}[Proof of \cref{thm-soft-photon-relaxation}] Throughout the proof, $H^{0,0}$ is evaluated at $e=0$. The equality $F_\LL^{0,\chi}=F_\LI^{0,\chi}$ follows from \cref{thm-many-electron-soft-purification}. It remains to prove $F_\LL^{0,\chi}=F_\VR$ and the attainment criterion. \noindent\textbf{Step 1. The lower bound $F_\LL^{0,\chi}\geq F_\VR$.} Let $\Psi\in \mathcal Q_\PF$ be normalized and suppose that $(\rho_\Psi,\vjp_\Psi)=(\rho,\vjp)$. Its electronic and photon reduced states are \[ \Gamma_\el :=\Tr_\ph|\Psi\rangle\langle\Psi|,\qquad \Gamma_\ph :=\Tr_\el|\Psi\rangle\langle\Psi|. \] The partial-trace identity for positive observables \cref{partial-trace-positive-observables}, first for bounded spectral truncations and then by monotone convergence, gives \[ \dua{\Psi}{H^{0,0}\Psi} =\Tr(H_\el^0\Gamma_\el) +\Tr(H_\rad\Gamma_\ph). \] In particular, $\Gamma_\el$ has finite electronic energy. It has the same density as $\Psi$: for $v\in C_c^\infty(\RR^3)$ and the bounded operator $V=\sum_{n=1}^Nv(\vx_n)$ on $\Hil_\el^N$, \[ \int_{\RR^3}v\rho_\Psi =\dua{\Psi}{(V\otimes\iden_{\Fock_\ph})\Psi} =\Tr(V\Gamma_\el) =\int_{\RR^3}v\rho_{\Gamma_\el}. \] It also has the same paramagnetic current density. Indeed, insert a Schmidt decomposition $\Psi=\sum_j\sqrt{p_j}\,\Phi_j\otimes\Xi_j$, for which $\Gamma_\el=\sum_jp_j|\Phi_j\rangle\langle\Phi_j|$ by \cref{Schmidt-reduced-states}, into \cref{jp-def}. The gradient acts only on the electronic factor, and $\dua{\Xi_j}{\Xi_k}=\delta_{jk}$ removes all cross terms, so \[ \vjp_\Psi=\sum_jp_j\vjp_{\Phi_j}=\vjp_{\Gamma_\el}. \] The convergence of these series is checked in the proof of \cref{thm-photon-purification}, which holds for every $e\geq0$. Consequently, $(\rho_{\Gamma_\el},\vjp_{\Gamma_\el})=(\rho,\vjp)$, and \[ \dua{\Psi}{H^{0,0}\Psi} \geq\Tr(H_\el^0\Gamma_\el) \geq F_\VR(\rho,\vjp). \] Taking the infimum over admissible $\Psi$ gives $F_\LL^{0,\chi}(\rho,\vjp)\geq F_\VR(\rho,\vjp)$. \noindent\textbf{Step 2. The upper bound $F_\LL^{0,\chi}\leq F_\VR$.} We purify an arbitrary admissible electronic density matrix with soft photons. Each of its eigenvectors is paired with its own one-photon state, the photon states being orthonormal and supported at momenta below $\epsilon$. The resulting pure state has this density matrix as its electronic reduced state, and hence the prescribed density and current density. Its energy exceeds the electronic energy only by the photon energy, which is at most $\epsilon$ and can therefore be made arbitrarily small. This is the one-photon purification \cref{one-photon-purification} used in the proof of \cref{thm-photon-purification} to prove the reverse inclusions in \cref{electronic-projection,zero-jph-slice}. There the purification only had to realize the prescribed pair with vanishing field-induced current density. Here, at $e=0$, the energy form factorizes, so we can also compute the energy of the purified state exactly. More precisely, let $\Gamma$ be any admissible electronic density matrix of finite energy. Writing only its nonzero eigenvalues, its spectral decomposition is \[ \Gamma=\sum_{j=1}^r p_j|\Phi_j\rangle\langle\Phi_j|,\qquad p_j>0,\qquad \sum_{j=1}^r p_j=1,\qquad r\in\NN\cup\{\infty\}. \] The vectors $\Phi_j$ are orthonormal, belong to $Q(H_\el^0)$, and satisfy \[ \sum_{j=1}^r p_j\dua{\Phi_j}{H_\el^0\Phi_j} =\Tr(H_\el^0\Gamma)<\infty. \] Fix $\epsilon>0$. The one-photon space over the region $\{0<|\vk|<\epsilon\}$ is infinite dimensional, so it contains an orthonormal family $\{f_j^\epsilon\}_{j=1}^r$. Put \[ \Xi_j^\epsilon=a^\dag(f_j^\epsilon)\Omega. \] Then the $\Xi_j^\epsilon$ are orthonormal one-photon vectors and \[ \dua{\Xi_j^\epsilon}{H_\rad\Xi_j^\epsilon} =\dua{f_j^\epsilon}{|\vk|f_j^\epsilon}\leq\epsilon. \] When $r=\infty$, consider first the finite sums \[ \Psi_{\epsilon,M} =\sum_{j=1}^M\sqrt{p_j}\, \Phi_j\otimes\Xi_j^\epsilon. \] For $L>M$, orthogonality in both tensor factors and the form factorization at $e=0$ give \[ \begin{split} &\|\Psi_{\epsilon,L}-\Psi_{\epsilon,M}\|^2 +\dua{\Psi_{\epsilon,L}-\Psi_{\epsilon,M}}{ H^{0,0}(\Psi_{\epsilon,L}-\Psi_{\epsilon,M})}\\ &\quad=\sum_{M0$, there is a finite-rank orthogonal projection $S$ such that \[ \|(1-S)\chi_R\|_{Q(H_\el^0)\to\Hil_\el^N}<\delta. \] Now split \[ (1-S)\sqrt{\Gamma_k} =(1-S)(1-\chi_R)\sqrt{\Gamma_k} +(1-S)\chi_R\sqrt{\Gamma_k}. \] Since $\|1-S\|\leq1$, the Hilbert--Schmidt norm of the first term is at most $\|(1-\chi_R)\sqrt{\Gamma_k}\|_{\mathfrak S_2}$. For the second term, each $\sqrt{\Gamma_k}\psi_j$ lies in $Q(H_\el^0)$ by \cref{electronic-form-HS-bound}, so the operator-norm bound on $(1-S)\chi_R$ gives \[ \|(1-S)\chi_R\sqrt{\Gamma_k}\|_{\mathfrak S_2}^2 =\sum_j\|(1-S)\chi_R\sqrt{\Gamma_k}\psi_j\|^2 \leq\delta^2\sum_j\|\sqrt{\Gamma_k}\psi_j\|_{Q(H_\el^0)}^2 \leq C\delta^2. \] Together, \[ \sup_k\|(1-S)\sqrt{\Gamma_k}\|_{\mathfrak S_2} \leq\sup_k\|(1-\chi_R)\sqrt{\Gamma_k}\|_{\mathfrak S_2} +C^{1/2}\delta. \] First choosing $R$ large and then $\delta$ small makes the right-hand side arbitrarily small. Since $1-S$ is an orthogonal projection, \[ \Tr((1-S)\Gamma_k)=\|(1-S)\sqrt{\Gamma_k}\|_{\mathfrak S_2}^2, \] so for every $\varepsilon>0$ there is a single finite-rank orthogonal projection $S$ with $\sup_k\Tr((1-S)\Gamma_k)<\varepsilon$. This is the hypothesis of \cref{lem-trace-compression}. By the compression estimate \cref{density-matrix-compression}, every $\Gamma_k$ lies within $2\sqrt{\varepsilon}$ in trace norm of its compression $S\Gamma_kS$. These compressions are positive and have trace at most one, and each $A=S\Gamma_kS$ satisfies $A=SAS$, that is, it vanishes on $(\ran S)^\perp$ and maps into $\ran S$. They are therefore operators on the fixed finite-dimensional space $\ran S$, extended by zero, and so lie in a compact set. Since $\varepsilon$ is arbitrary, $(\Gamma_k)$ is precompact in trace norm. After passing to a subsequence, \[ \Gamma_k\longrightarrow\Gamma\quad\text{in }\mathfrak S_1(\Hil_\el^N), \qquad \Gamma\geq0,\qquad\Tr\Gamma=1. \] \noindent\textbf{Step 4. The lower bound (i): passage to the limit.} We pass both the full electronic form and the constraints to this limit. The Powers--St\o rmer inequality gives $\sqrt{\Gamma_k}\to\sqrt{\Gamma}$ in $\mathfrak S_2(\Hil_\el^N)$. By \cref{electronic-form-HS-bound}, $\sqrt{\Gamma_k}$, regarded as a Hilbert--Schmidt operator with values in the form domain, has a weakly convergent subsequence. The continuous embedding $Q(H_\el^0)\hookrightarrow\Hil_\el^N$ and the strong $\mathfrak S_2$ convergence identify its weak limit with $\sqrt{\Gamma}$. Weak lower semicontinuity therefore yields \[ \Tr(H_\el^0\Gamma) \leq\liminf_{k\to\infty}\Tr(H_\el^0\Gamma_k). \] This applies to the full electronic form, including the Coulomb many-body interaction. Moreover, $-\i\grad_{\vx_1}$ is a bounded map from $Q(H_\el^0)$ to $(\Hil_\el^N)^3$. Hence \[ (-\i\grad_{\vx_1})\sqrt{\Gamma_k} \rightharpoonup(-\i\grad_{\vx_1})\sqrt{\Gamma} \quad\text{in the corresponding Hilbert--Schmidt space}. \] Trace-norm convergence passes the density against every bounded multiplication operator. For every real $\vA\in C_c^\infty(\RR^3,\RR^3)$, strong convergence of $\vA(\vx_1)\sqrt{\Gamma_k}$ and the preceding weak convergence give \[ \begin{split} \int_{\RR^3}\vA\cdot\vjp_{\Gamma_k} &=N\Re\big(\vA(\vx_1)\sqrt{\Gamma_k}, (-\i\grad_{\vx_1})\sqrt{\Gamma_k}\big)_{\mathfrak S_2}\\ &\longrightarrow N\Re\big(\vA(\vx_1)\sqrt{\Gamma}, (-\i\grad_{\vx_1})\sqrt{\Gamma}\big)_{\mathfrak S_2} =\int_{\RR^3}\vA\cdot\vjp_\Gamma. \end{split} \] Since $L^\infty\subset X^*$ and $C_c^\infty(\RR^3,\RR^3)\subset Y^*$, the weak convergences $\rho_k\wconv\rho$ in $X$ and $\vjp_{\Gamma_k}\wconv\vj$ in $Y$ also hold against these test functions. Since $\rho_k=\rho_{\Gamma_k}$, uniqueness of limits therefore gives \[ (\rho_\Gamma,\vjp_\Gamma)=(\rho,\vj). \] Using the approximate minimality, the form comparison, and the nonnegativity of the photon energy, we conclude that \[ \begin{split} \liminf_{k\to\infty} F_\LL^{e_k,\chi}(\rho_k,\vj_k) &\geq\liminf_{k\to\infty}\left( \dua{\Psi_k}{H_{e_k,\chi}^{0,0}\Psi_k}-\frac1k\right)\\ &=\liminf_{k\to\infty} \dua{\Psi_k}{H_{0,\chi}^{0,0}\Psi_k}\\ &\geq\Tr(H_\el^0\Gamma) \geq F_\VR(\rho,\vj). \end{split} \] This proves (i). \noindent\textbf{Step 5. The recovery bound (ii).} Suppose $F_\VR(\rho,\vj)<\infty$ and fix $\delta>0$. Choose an admissible electronic density matrix $\Gamma_\delta$ with \[ \Tr(H_\el^0\Gamma_\delta) \leq F_\VR(\rho,\vj)+\delta. \] Apply the construction in \cref{one-photon-purification} as detailed in Step 2 of the proof of \cref{thm-soft-photon-relaxation}, choosing all photon tags in $\{0<|\vk|<\delta\}$. The resulting normalized vector $\Psi_\delta$ has electronic marginal $\Gamma_\delta$ and photon energy at most $\delta$. It lies entirely in the one-photon sector: for an infinite-rank $\Gamma_\delta$, this follows because every partial sum has that property and the one-photon sector is closed. Since $\AF(\vx)$ maps this sector into the orthogonal sum of the vacuum and two-photon sectors, $\dua{\Psi_\delta}{\AF(\vx)\Psi_\delta}_{\vx}=0$ almost everywhere. Thus \[ \rho_{\Psi_\delta}=\rho,\qquad \vjp_{\Psi_\delta}=\vj,\qquad \vjph_{\Psi_\delta}=0,\qquad \dua{\Psi_\delta}{H_\rad\Psi_\delta}\leq\delta. \] It therefore represents the same combined-current pair $(\rho,\vj)$ at every coupling and belongs to $Q(H_{e,\chi}^{0,0})$ for every $e\geq0$ by the common form-domain identity. The electronic derivative preserves photon number, whereas $\AF(\vx_n)$ changes it by one, and hence \[ \Re\dua{(-\i\grad_{\vx_n})\Psi_\delta}{ \AF(\vx_n)\Psi_\delta}=0. \] For almost every $\vX$, the photon factor is $\Psi_\delta(\vX)=a^\dag(\eta)\Omega$ for a one-photon vector $\eta$ supported in $\{\omega<\delta\}$, with $\|\eta\|=\|\Psi_\delta(\vX)\|$. The canonical commutation relations give \[ \|\Af^\mu(\vx)a^\dag(\eta)\Omega\|^2 =\|G^\mu_{\vx}\|^2\|\eta\|^2+2|\dua{G^\mu_{\vx}}{\eta}|^2 \leq\big(\|G^\mu_{\vx}\|^2 +2\|\iden_{\{\omega<\delta\}}G^\mu_{\vx}\|^2\big)\|\eta\|^2. \] Summing over $\mu$ by \cref{Gx-norms}, applied also to $\iden_{\{\omega<\delta\}}\wh\chi$, and integrating over the electronic variables gives \[ \|\AF(\vx_n)\Psi_\delta\|^2 \leq\|\wh\chi\|^2+2\|\iden_{\{\omega<\delta\}}\wh\chi\|^2. \] Expanding the positive-coupling form now gives \[ \begin{split} F_\LL^{e,\chi}(\rho,\vj) &\leq \dua{\Psi_\delta}{H_{e,\chi}^{0,0}\Psi_\delta}\\ &=\Tr(H_\el^0\Gamma_\delta) +\dua{\Psi_\delta}{H_\rad\Psi_\delta} +\frac{e^2}{2}\sum_{n=1}^N \|\AF(\vx_n)\Psi_\delta\|^2\\ &\leq F_\VR(\rho,\vj)+2\delta +\frac{e^2N}{2}\big(\|\wh\chi\|^2 +2\|\iden_{\{\omega<\delta\}}\wh\chi\|^2\big). \end{split} \] Since $\wh\chi\in L^2(\RR^3)$, letting $\delta\downarrow0$ at fixed $e$ gives explicitly \[ F_\LL^{e,\chi}(\rho,\vj) \leq F_\VR(\rho,\vj)+\frac{N}{2}\|\wh\chi\|^2e^2, \] which is the quantitative estimate in (ii). The same construction proves $(\rho,\vj)\in\rcdens_N^{e,\chi}$ for every $e\geq0$. \noindent\textbf{Step 6. The zero-coupling limit.} Applying (i) to the constant constraints and combining it with (ii) gives the asserted pointwise limit on the effective domain. If $F_\VR(\rho,\vj)=+\infty$, (i) applied to the constant sequence shows that the limit is $+\infty$. Finally, since $\vjc=\vjp$ at $e=0$, \cref{electronic-projection} gives $\rcdens_N^{0,\chi}=\rcdens_{N,\mathrm{mix}}^{\el}$, and \cref{soft-photon-equality} identifies the limit with $F_\LL^{0,\chi}$. \end{proof} \subsection{The long-wavelength (dipole) limit} \phantomsection \label{proof:thm-dipole-limit} \begin{proof}[Proof of \cref{thm-dipole-limit}] For a finite-rank positive operator $\vGamma$ whose range lies in $\mathcal Q_\PF$, the density $\rho_\vGamma$ and the current densities $\vjcmodel{\varepsilon}_{\vGamma}$ and $\vjcdip_\vGamma$ are defined as for density matrices, also when $\Tr\vGamma<1$. All constants below may depend on $N,e$, and the fixed form factor, but not on $\varepsilon\in[0,1]$. \noindent\textbf{Step 1. Uniform bounds and fixed-state convergence.} The field bounds \cref{Gx-norms,A-Hrad-bound} are uniform in the spatial argument. The proof of \cref{PF-form-norm-equivalence} therefore gives \begin{equation}\label{dipole-proof-uniform-form-bound} c\|\Psi\|_{\mathcal Q_\PF}^2 \leq \dua{\Psi}{H_\varepsilon^{0,0}\Psi}+\|\Psi\|^2 \leq C\|\Psi\|_{\mathcal Q_\PF}^2. \end{equation} In particular, the form-domain identification is uniform in $\varepsilon$. Define the modulus \begin{equation}\label{dipole-kernel-modulus} d(r)=\sum_{\mu=1}^3\sup_{|\vy|\leq r} \left(\|G^\mu_{\vy}-G^\mu_{0}\| +\left\|\frac{G^\mu_{\vy}-G^\mu_{0}}{\sqrt\omega}\right\|\right). \end{equation} The continuity following \cref{Gx-norms} implies $d(r)\to0$ as $r\downarrow0$. If $\Psi$ is supported where $|\vx_n|\leq2L$ for all $n$, then \begin{equation}\label{dipole-local-field-error} \sum_{n=1}^N \|(\AF(\varepsilon\vx_n)-\AF(0))\Psi\|^2 \leq C d(2\varepsilon L)^2 \big(\|\sqrt{H_\rad}\Psi\|^2+\|\Psi\|^2\big). \end{equation} Indeed, for almost every $\vX$ in the support of $\Psi$, we have $\Psi(\vX)\in D(\sqrt{H_\rad})$ and $|\varepsilon\vx_n|\leq2\varepsilon L$ for every $n$. Fix such $\vX$, an index $n$, and a component $\mu$, and put $g=G^\mu_{\varepsilon\vx_n}-G^\mu_{0}$. Since the Segal field is real-linear in its argument, \[ \big(\Af^\mu(\varepsilon\vx_n)-\Af^\mu(0)\big)\Psi(\vX) =\big(a^\dag(g)+a(g)\big)\Psi(\vX), \] and the definition \cref{dipole-kernel-modulus} gives $\|g\|+\|g/\sqrt\omega\|\leq d(2\varepsilon L)$. The bounds \cref{ann-Hrad-bound,cre-Hrad-bound}, applied with $g$ in place of $G^\mu_{\vx}$, therefore yield \[ \begin{split} \big\|\big(\Af^\mu(\varepsilon\vx_n)-\Af^\mu(0)\big)\Psi(\vX)\big\| &\leq2\|g/\sqrt\omega\|\,\|\sqrt{H_\rad}\Psi(\vX)\|+\|g\|\,\|\Psi(\vX)\|\\ &\leq2d(2\varepsilon L) \big(\|\sqrt{H_\rad}\Psi(\vX)\|+\|\Psi(\vX)\|\big). \end{split} \] Integrating over $\vX$, and using $\|\sqrt{H_\rad}\Psi\|^2=\int\|\sqrt{H_\rad}\Psi(\vX)\|^2\dd\vX$ because $H_\rad$ acts only on the photon variables, gives \cref{dipole-local-field-error} with $C=24N$. Expanding the differences of the squared covariant-derivative norms, and using \cref{dipole-proof-uniform-form-bound}, gives \begin{equation}\label{dipole-local-form-error} |\dua{\Psi}{H_\varepsilon^{0,0}\Psi}-\dua{\Psi}{H_\dip \Psi}| \leq C\big(d(2\varepsilon L)+d(2\varepsilon L)^2\big) \big(\dua{\Psi}{H_\varepsilon^{0,0}\Psi}+\|\Psi\|^2\big). \end{equation} The same estimates hold after summing over any finite-rank positive operator with this electronic support. For a fixed $\Psi\in\mathcal Q_\PF$, the field difference instead converges without a support assumption: \begin{equation}\label{dipole-fixed-state-form-convergence} \|(\AF(\varepsilon\vx_n)-\AF(0))\Psi\|\longrightarrow0, \qquad \dua{\Psi}{H_\varepsilon^{0,0}\Psi}\longrightarrow \dua{\Psi}{H_\dip \Psi}. \end{equation} Indeed, for almost every electronic configuration $\vX$, $\Psi(\vX)\in D(\sqrt{H_\rad})$. The estimates \cref{ann-Hrad-bound,cre-Hrad-bound} and the continuity of the kernels give pointwise convergence on these fibers, dominated in squared norm by $C(\|\sqrt{H_\rad}\Psi(\vX)\|^2+\|\Psi(\vX)\|^2)$. Integration and then expansion of the kinetic form prove \cref{dipole-fixed-state-form-convergence}. The current convergence is in the full space $Y$. For a normalized pure state, put \[ b_{\varepsilon,\Psi}(\vx) =N\|(\AF(\varepsilon\vx)-\AF(0))\Psi\|_{\vx}^2, \] where the partial norm $\|\cdot\|_{\vx}$ is summed over the three field components. Pointwise Cauchy--Schwarz and H\"older's inequality give \begin{align} |\vjcmodel{\varepsilon}_{\Psi}-\vjcdip_{\Psi}| &\leq e\sqrt{\rho_\Psi}\sqrt{b_{\varepsilon,\Psi}}, \notag\\ \|\vjcmodel{\varepsilon}_{\Psi}-\vjcdip_{\Psi}\|_Y &\leq e\big(\sqrt N+\|\sqrt{\rho_\Psi}\|_6\big) \|\sqrt{b_{\varepsilon,\Psi}}\|. \label{dipole-current-difference-bound} \end{align} For a density matrix, define $b_{\varepsilon,\vGamma}$ by linear extension. Cauchy--Schwarz over the state labels proves the same estimates with $\Psi$ replaced by $\vGamma$. The Hilbert-space-valued Sobolev inequality bounds $\|\sqrt{\rho_\vGamma}\|_6$ by the square root of the ordinary electronic kinetic energy. Consequently \begin{equation}\label{dipole-fixed-ensemble-convergence} \vjcmodel{\varepsilon}_{\vGamma}\longrightarrow \vjcdip_\vGamma\quad\text{in }Y, \qquad \Tr(H_\varepsilon^{0,0}\vGamma)\longrightarrow \Tr(H_\dip \vGamma) \end{equation} for every fixed finite-rank density matrix of finite form energy. Moreover, by \cref{dipole-local-field-error}, the current difference in \cref{dipole-current-difference-bound} tends to zero uniformly on electronically supported density matrices of bounded energy whenever $\varepsilon L\to0$. We will also use the uniform bound \begin{equation}\label{dipole-energy-density-current-bounds} \|\rho_\vGamma\|_3+ \|\vjcmodel{\varepsilon}_{\vGamma}\|_{3/2} +\|\vjcmodel{\varepsilon}_{\vGamma}\|_1 \leq C\big(1+\Tr(H_\varepsilon^{0,0}\vGamma)\big) \end{equation} for every density matrix $\vGamma$ with $\Tr(H_\varepsilon^{0,0}\vGamma)<\infty$. To see this, regard the weighted state vectors of a density matrix as a single vector in a Hilbert-space direct sum. Its electronic derivative is controlled by \cref{dipole-proof-uniform-form-bound}, and its covariant derivative by $\Tr(H_\varepsilon^{0,0}\vGamma)$. Sobolev and H\"older then give \cref{dipole-energy-density-current-bounds}, exactly as for pure states in \cref{prop-current-bounds}. \noindent\textbf{Step 2. Finite-rank approximation of the convex relaxation.} By \cref{dipole-convex-envelope}, the epigraph of $F_{\LI,\varepsilon}$ is the norm-closed convex hull of the epigraph of $F_{\LL,\varepsilon}$. Thus, whenever $F_{\LI,\varepsilon}(\rho,\vjc)<\infty$ and $\delta>0$, there is a finite-rank density matrix $\vGamma$, made from states in $\mathcal Q_\PF$, such that \begin{equation}\label{dipole-epigraph-approximation} \|\rho_\vGamma-\rho\|_X +\|\vjcmodel{\varepsilon}_{\vGamma}-\vjc\|_Y<\delta, \qquad \Tr(H_\varepsilon^{0,0}\vGamma) 0$. By \cref{dipole-epigraph-approximation}, choose finite-rank density matrices $\vGamma_k$ such that \begin{equation}\label{dipole-liminf-ensemble-choice} \|\rho_{\vGamma_k}-\rho_k\|_X +\|\vjcmodel{\varepsilon_k}_{\vGamma_k}-\vjc_k\|_Y \leq k^{-1}, \qquad \Tr(H_{\varepsilon_k}^{0,0}\vGamma_k) \leq F_{\LI,\varepsilon_k}(\rho_k,\vjc_k)+k^{-1}. \end{equation} Their energies are uniformly bounded. Since $L^\infty\subset X^*$, $\rho_{\vGamma_k}$ converges weakly in $L^1$ to $\rho$. Positivity and the test function $1$ show $\rho\geq0$ and $\int_{\RR^3}\rho=N$. Testing against the indicator of the complement of a fixed ball gives \[ \int_{|\vx|>R}\rho_{\vGamma_k}\longrightarrow \int_{|\vx|>R}\rho \] for every fixed $R>0$. Now let $\eta>0$. Since $\rho\in L^1$, there is $R_0$ with $\int_{|\vx|>R_0}\rho<\eta$, and the convergence just displayed, with $R=R_0$, gives an index $K$ such that $\int_{|\vx|>R_0}\rho_{\vGamma_k}<\eta$ for all $k\geq K$. Each of the finitely many densities $\rho_{\vGamma_1},\dots,\rho_{\vGamma_{K-1}}$ is integrable, so there is $R_1\geq R_0$ such that $\int_{|\vx|>R_1}\rho_{\vGamma_k}<\eta$ also for $kR}\rho_{\vGamma_k}\leq\eta$ for every $R\geq R_1$. As $\eta>0$ was arbitrary, this proves \begin{equation}\label{dipole-liminf-density-tightness} \lim_{R\to\infty}\sup_k\int_{|\vx|>R}\rho_{\vGamma_k}=0. \end{equation} Choose a smooth real $h:[0,\infty)\to[0,\pi/2]$, equal to zero on $[0,1]$ and to $\pi/2$ on $[2,\infty)$. With $|\vX|^2=\sum_{n=1}^N|\vx_n|^2$, define, for $\varepsilon>0$, \[ \theta_\varepsilon(\vX)=\cos h(\sqrt\varepsilon\,|\vX|),\qquad \wt\theta_\varepsilon(\vX)=\sin h(\sqrt\varepsilon\,|\vX|). \] Both multipliers are smooth and permutation invariant, and $\theta_\varepsilon$ equals $1$ where $|\vX|\leq\varepsilon^{-1/2}$ and vanishes where $|\vX|\geq2\varepsilon^{-1/2}$. They satisfy \[ \theta_\varepsilon^2+\wt\theta_\varepsilon^2=1,\qquad \sum_{n=1}^N\big(|\grad_{\vx_n}\theta_\varepsilon|^2+ |\grad_{\vx_n}\wt\theta_\varepsilon|^2\big)\leq C\varepsilon. \] Since the multipliers commute with the field and the interaction, the covariant IMS localization formula gives, for every $\Psi\in\mathcal Q_\PF$, \begin{equation}\label{dipole-IMS-identity} \begin{split} &\dua{\theta_\varepsilon\Psi}{H_\varepsilon^{0,0}\theta_\varepsilon\Psi} +\dua{\wt\theta_\varepsilon\Psi}{H_\varepsilon^{0,0}\wt\theta_\varepsilon\Psi}\\ &\qquad=\dua{\Psi}{H_\varepsilon^{0,0}\Psi} +\frac12\int_{\RR^{3N}} \sum_{n=1}^N\big(|\grad_{\vx_n}\theta_\varepsilon|^2+ |\grad_{\vx_n}\wt\theta_\varepsilon|^2\big) \|\Psi(\vX)\|^2\dd\vX. \end{split} \end{equation} The identity follows first on a form core and then by form-norm approximation. We now show that the $\vGamma_k$ may be assumed to be supported where every $|\vx_n|\leq2\varepsilon_k^{-1/2}$. By \cref{dipole-liminf-ensemble-choice}, \begin{equation}\label{dipole-liminf-weak-properties} (\rho_{\vGamma_k},\vjcmodel{\varepsilon_k}_{\vGamma_k}) \rightharpoonup(\rho,\vjc)\quad\text{in }X\times Y, \qquad \Tr(H_{\varepsilon_k}^{0,0}\vGamma_k) \leq F_{\LI,\varepsilon_k}(\rho_k,\vjc_k)+o(1), \end{equation} and we check that both properties survive localization by $\theta_{\varepsilon_k}$ followed by normalization. If $|\vX|>\varepsilon_k^{-1/2}$, then $|\vx_n|>(N\varepsilon_k)^{-1/2}$ for some $n$. Hence \cref{dipole-liminf-density-tightness} gives \begin{equation*} 0\leq\Tr(\wt\theta_{\varepsilon_k}^2\vGamma_k) \leq\int_{|\vx|>(N\varepsilon_k)^{-1/2}}\rho_{\vGamma_k}(\vx)\dd\vx \longrightarrow0. \end{equation*} In particular, $\Tr(\theta_{\varepsilon_k}^2\vGamma_k) =1-\Tr(\wt\theta_{\varepsilon_k}^2\vGamma_k)$ is positive for all large $k$, and we discard the remaining indices. Positivity of the complementary localized form in \cref{dipole-IMS-identity} yields \[ \frac{\Tr(H_{\varepsilon_k}^{0,0}\theta_{\varepsilon_k}\vGamma_k\theta_{\varepsilon_k})} {\Tr(\theta_{\varepsilon_k}^2\vGamma_k)} \leq\frac{\Tr(H_{\varepsilon_k}^{0,0}\vGamma_k)+C\varepsilon_k} {\Tr(\theta_{\varepsilon_k}^2\vGamma_k)} =\Tr(H_{\varepsilon_k}^{0,0}\vGamma_k)+o(1), \] since the energies are uniformly bounded. Combined with the energy bound in \cref{dipole-liminf-weak-properties}, this shows that the normalized localized density matrices satisfy that energy bound as well. For the constraints, the density difference $\rho_{\vGamma_k}-\rho_{\theta_{\varepsilon_k}\vGamma_k\theta_{\varepsilon_k}}$ is nonnegative and has mass $N\Tr(\wt\theta_{\varepsilon_k}^2\vGamma_k)$. For the current density, differentiation of the real multiplier $\theta_{\varepsilon_k}$ contributes a purely imaginary term, which disappears from its defining real part. Cauchy--Schwarz, including the state labels, thus gives \begin{equation*} \begin{split} &\|\vjcmodel{\varepsilon_k}_{\vGamma_k} -\vjcmodel{\varepsilon_k}_{\theta_{\varepsilon_k}\vGamma_k\theta_{\varepsilon_k}}\|_1\\ &\qquad\leq\big(N\Tr(\wt\theta_{\varepsilon_k}^2\vGamma_k)\big)^{1/2} \left(\sum_{n=1}^N \Tr(\vD_{n,\varepsilon_k}^\dag\vD_{n,\varepsilon_k}\vGamma_k) \right)^{1/2}\longrightarrow0. \end{split} \end{equation*} The trace in this display denotes the squared-derivative form trace. Division by $\Tr(\theta_{\varepsilon_k}^2\vGamma_k)$ changes the density by another $N\Tr(\wt\theta_{\varepsilon_k}^2\vGamma_k)$ in $L^1$ and the current density by $o(1)$ in $L^1$, by the energy bounds. The original and localized densities are uniformly bounded in $L^3$, and the corresponding current densities in $L^{3/2}$, by \cref{dipole-energy-density-current-bounds}. If $f_k\to0$ in $L^1$ and $(f_k)$ is bounded in $L^p$, $10$, \begin{equation*} E_{0$, the dark subspace with $0<\omega0$ and orthonormal $\Psi_j$, have density $\rho$. Since $|\vR|^2\leq N\sum_n|\vx_n|^2$, \[ \sum_j\lambda_j\dua{\Psi_j}{|\vR|^2\Psi_j} \leq N\int_{\RR^3}|\vx|^2\rho<\infty. \] At each electronic configuration take $h=e\sum_\mu R^\mu G^\mu_{0}$; then $\|\sqrt\omega h\|^2=e^2\vR^\top\vM\vR\leq e^2\|\vM\||\vR|^2$. The bound \cref{weyl-radiation-bound}, applied with $\pm\i h$ and integrated, gives, for every $\Psi$ with $\dua{\Psi}{|\vR|^2\Psi}<\infty$, \[ \|\sqrt{H_\rad}U_{\mathrm d}^{\pm1}\Psi\|^2 \leq2\|\sqrt{H_\rad}\Psi\|^2 +2e^2\|\vM\|\dua{\Psi}{|\vR|^2\Psi}. \] Applying this to $\Psi_j$ and to $U_{\mathrm d}\Psi_j$ shows that $U_{\mathrm d}\Psi_j\in D(\sqrt{H_\rad})$ if and only if $\Psi_j\in D(\sqrt{H_\rad})$. By \cref{prop-dipole-length-gauge}, applied also with $-e$ in place of $e$, $U_{\mathrm d}\Psi_j$ and $\Psi_j$ have finite electronic kinetic energy together, so $U_{\mathrm d}\Psi_j\in\mathcal Q_\PF$ if and only if $\Psi_j\in\mathcal Q_\PF$. Summing the displayed bound over $j$ and using \cref{dipole-length-trace-identity} and the form-norm equivalence of Step~1 then shows that $U_{\mathrm d}\vGamma U_{\mathrm d}^\dag\in\dm$ if and only if $\vGamma\in\dm$. Thus the admissible sets in \cref{dipole-length-functional} and \cref{dipole-length-hamiltonian-functional} coincide. For each $j$, the Weyl shift \cref{dipole-length-Weyl-shift} with $\eta=\Psi_j(\vX)$, integrated over $\vX$, gives \[ \|\sqrt{H_\rad}U_{\mathrm d}\Psi_j\|^2 =\dua{\Psi_j}{\big[H_\rad-e\vR\cdot\EF_\perp(0) +e^2\vR^\top\vM\vR\big]\Psi_j}, \] where $e\sum_\mu R^\mu\varPi(\omega G^\mu_{0})=-e\vR\cdot\EF_\perp(0)$ by \cref{dipole-electric-field-origin}. The bound on $a_\omega$ following \cref{weyl-weighted-field} gives $|\varPi(\omega h)[\eta,\eta]| \leq2\|\sqrt\omega h\|\,\|\sqrt{H_\rad}\eta\|\,\|\eta\|$, so by the Cauchy--Schwarz inequality the weighted sums over $j$ of the three terms converge absolutely. Summing over $j$ turns \cref{dipole-length-functional} into \cref{dipole-length-hamiltonian-functional}, and the attainment statement carries over. Restricting to rank-one $\vGamma$ and using $F_{\LL,\dip}=F_\dip$ from \cref{dipole-search-equality} gives the pure-state version. \end{proof} \appendix \section{Transformation properties}\label{app-transformation-properties} We work in the solenoidal one-photon representation and, unless stated otherwise, specialize to the hard cutoff \cref{harduv}. Second quantization is understood whenever a one-photon unitary is applied to a Fock-space state. The displayed $e,\Lambda$ labels on $\AF_\Lambda$, $H_{e,\Lambda}^{0,0}$, $(\vjc_\Psi)^{e,\Lambda}$, and $F^{e,\Lambda}$ record parameter dependence that was suppressed earlier. Here $\Lambda$ abbreviates the form-factor label $\chi_\Lambda$, so that, for instance, $F^{e,\Lambda}=F^{e,\chi_\Lambda}$. An unsuperscripted $F$ carries the parameters shown in the surrounding formula. All state identities below apply to normalized vectors in the indicated form domains, and Hamiltonian identities are identities of self-adjoint operators established through their closed quadratic forms, unless explicitly stated only as form identities. \subsection{Isometries} The field momentum is the second quantization \[ \vP_\ph=\dd\varGamma(\vk) =\sum_{\lambda=1}^2\int_{\RR^3}\vk\, a^\dag(\vk,\lambda)a(\vk,\lambda)\dd\vk. \] For $\vb\in\RR^3$, define the photon translation \[ (t_\vb\vf)(\vk)=e^{-\i\vb\cdot\vk}\vf(\vk),\qquad T_\ph(\vb)=\varGamma(t_\vb)=e^{-\i\vb\cdot\vP_\ph}, \] and the electron translation \[ (T_\el(\vb)\Phi)(\vx_1,\ldots,\vx_N) =\Phi(\vx_1-\vb,\ldots,\vx_N-\vb). \] The composite system is translated by \[ T(\vb)=T_\el(\vb)\otimes T_\ph(\vb). \] \begin{proposition} \label[proposition]{prop-translation-covariance} $T(\vb)$ is a unitary preserving $\mathcal Q_\PF$, and \begin{equation}\label{A-translation} T_\ph(\vb)^\dag\AF_\Lambda(\vx)T_\ph(\vb) =\AF_\Lambda(\vx-\vb). \end{equation} The internal Hamiltonian, density, and combined current density satisfy \begin{equation}\label{translation-laws} \begin{gathered} T(\vb)^\dag H^{0,0}T(\vb)=H^{0,0},\\ \rho_{T(\vb)\Psi}(\vx)=\rho_\Psi(\vx-\vb),\qquad \vjc_{T(\vb)\Psi}(\vx)=\vjc_\Psi(\vx-\vb). \end{gathered} \end{equation} Consequently, for every $(\rho,\vjc)\in X\times Y$ and $F\in\{F_\LL,F_\LI\}$, \begin{equation}\label{F-translation} F(\rho(\,\cdot-\vb),\vjc(\,\cdot-\vb))=F(\rho,\vjc). \end{equation} \end{proposition} \begin{proof} The solenoidal coupling functions in \cref{Gx-solenoidal} obey $t_\vb^\dag\vG^\mu_{\vx;\Lambda} =\vG^\mu_{\vx-\vb;\Lambda}$. Covariance of creation and annihilation operators under second quantization gives \cref{A-translation}. Moreover, $t_\vb$ commutes with $\omega$, so $T_\ph(\vb)$ commutes with $H_\rad$. Electronic translation preserves weak derivatives and all interparticle distances. Thus $T(\vb)$ preserves $\mathcal Q_\PF$, and the electronic and photon translations cancel in the field term: \[ T(\vb)^\dag\AF_\Lambda(\vx_n)T(\vb)=\AF_\Lambda(\vx_n), \qquad T(\vb)^\dag\vD_nT(\vb)=\vD_n. \] These identities first hold on test vectors and extend by the field bound \cref{A-Hrad-bound} and form-norm approximation. The kinetic, Coulomb, and radiation forms are therefore invariant. Changing the electronic integration variables in \cref{density-def,jc-def} gives the remaining identities in \cref{translation-laws}. The inverse $T(-\vb)$ makes this an energy-preserving bijection between the corresponding constraint sets, proving \cref{F-translation} for $F_\LL$, including empty constraint sets. Translation is a linear homeomorphism of $X\times Y$, so taking the norm-lower-semicontinuous convex envelope in \cref{L-vs-LL} proves the identity for $F_\LI$. \end{proof} For $\vR\in{\rm O}(3)$, define on $\Hil_\sol$ \begin{equation}\label{rotation-one-photon} (\Upsilon_\vR\vf)(\vk)=\vR\vf(\vR^\top\vk), \end{equation} and on $\HC_N$ define \[ (U_\vR\Psi)(\vx_1,\ldots,\vx_N) =\Upsilon_\vR\Psi(\vR^\top\vx_1,\ldots,\vR^\top\vx_N). \] \begin{proposition} \label[proposition]{prop-orthogonal-covariance} These maps are unitary, and $U_\vR$ preserves $\mathcal Q_\PF$. The creation, annihilation, and field operators transform according to \begin{align} \Upsilon_\vR a^\#(\vf)\Upsilon_\vR^\dag &=a^\#(\Upsilon_\vR\vf), \label{rotation-creation}\\ \Upsilon_\vR^\dag\AF_\Lambda(\vx)\Upsilon_\vR &=\vR\AF_\Lambda(\vR^\top\vx). \label{A-rotation} \end{align} The corresponding state identities are \begin{align} U_\vR^\dag H^{0,0}U_\vR&=H^{0,0}, \label{H-rotation}\\ \rho_{U_\vR\Psi}(\vx)&=\rho_\Psi(\vR^\top\vx), \label{rho-rotation}\\ \vjc_{U_\vR\Psi}(\vx)&=\vR\vjc_\Psi(\vR^\top\vx). \label{jc-rotation} \end{align} Consequently, for every $(\rho,\vjc)\in X\times Y$ and $F\in\{F_\LL,F_\LI\}$, \begin{equation}\label{F-rotation} F(\rho\circ\vR^\top,\vR\vjc\circ\vR^\top)=F(\rho,\vjc). \end{equation} \end{proposition} \begin{proof} Orthogonal changes of variables preserve Lebesgue measure. For solenoidal $\vf$, they also preserve transversality, since \[ \vk\cdot\vR\vf(\vR^\top\vk) =(\vR^\top\vk)\cdot\vf(\vR^\top\vk)=0. \] Thus $\Upsilon_\vR$ is unitary on $\Hil_\sol$ with inverse $\Upsilon_{\vR^\top}$, and \cref{rotation-creation} is the second-quantization identity. By \cref{polarization-projector}, \[ P_\sol(\vR\vk)=\vR P_\sol(\vk)\vR^\top. \] Using radiality of the hard cutoff in \cref{Gx-solenoidal}, we obtain \[ \Upsilon_\vR^\dag\vG^\mu_{\vx;\Lambda} =\sum_{\nu=1}^3R^{\mu\nu} \vG^\nu_{\vR^\top\vx;\Lambda}, \] which proves \cref{A-rotation}. Radiality of $\omega$ also gives $\Upsilon_\vR^\dag H_\rad\Upsilon_\vR=H_\rad$. The electronic chain rule and \cref{A-rotation} yield $U_\vR^\dag\vD_nU_\vR=\vR\vD_n$. Orthogonality preserves the squared covariant-derivative norms and the Coulomb interaction. These facts prove form-domain preservation and \cref{H-rotation}, first on test vectors and then by form approximation. Changing variables in the partial inner products gives \cref{rho-rotation,jc-rotation}. Finally, $U_{\vR^\top}$ is the inverse map on constraint sets, so \cref{F-rotation} holds for $F_\LL$. The induced map on $X\times Y$ is a linear homeomorphism, so \cref{L-vs-LL} proves the result for $F_\LI$. \end{proof} \subsection{Dilations} \begin{proposition}[Dilation identities] \label[proposition]{prop-dilation-covariance} For $\lambda>0$, define on $\Hil_\sol$ \begin{equation*} (\Upsilon_\lambda\vf)(\vk)=\lambda^{3/2}\vf(\lambda\vk). \end{equation*} This is unitary, and \begin{align} \Upsilon_\lambda a^\#(\vf)\Upsilon_\lambda^\dag &=a^\#(\Upsilon_\lambda\vf), \label{a-scaling}\\ \Upsilon_\lambda H_\rad\Upsilon_\lambda^\dag &=\lambda H_\rad, \label{Hrad-scaling}\\ \Upsilon_\lambda\AF_\Lambda(\vx)\Upsilon_\lambda^\dag &=\lambda\AF_{\Lambda/\lambda}(\lambda\vx). \label{A-scaling} \end{align} Let \[ (S_\lambda\Phi)(\vX)=\lambda^{3N/2}\Phi(\lambda\vX), \qquad V_\lambda=S_\lambda\otimes\Upsilon_\lambda^\dag, \qquad \Psi_\lambda=V_\lambda\Psi. \] The map $V_\lambda$ is unitary on $\HC_N$ and maps $Q(H_{e,\Lambda/\lambda}^{0,0})$ bijectively onto $Q(H_{e,\Lambda}^{0,0})$. The density and current density satisfy \begin{align} \rho_{\Psi_\lambda}(\vx) &=\lambda^3\rho_\Psi(\lambda\vx), \label{rho-scaling}\\ (\vjc_{\Psi_\lambda})^{e,\Lambda}(\vx) &=\lambda^4(\vjc_\Psi)^{e,\Lambda/\lambda}(\lambda\vx). \label{jc-scaling} \end{align} The internal Hamiltonian satisfies the operator identity \begin{equation}\label{energy-scaling} \begin{split} V_\lambda^\dag H_{e,\Lambda}^{0,0}V_\lambda &=\frac{\lambda^2}{2}\sum_{n=1}^N \big(-\i\grad_{\vx_n}-e\AF_{\Lambda/\lambda}(\vx_n)\big)^2\\ &\quad+\lambda\sum_{1\leq n0$, the form on the right is closed on this domain, with a norm equivalent to the reference form norm. Polarization and uniqueness in the representation theorem for closed forms therefore give the asserted equality of self-adjoint operators. \end{proof} The covariant kinetic term scales quadratically. The Coulomb and photon terms scale linearly. \subsection{Time reversal} \label{app-time-reversal} Time reversal needs only \cref{cutoff-assumption}, not the hard cutoff. In the solenoidal representation, recall the conjugation $(J\vf)(\vk)=\ol{\vf(-\vk)}$. This is an antiunitary involution commuting with $\omega$ and with the rotations \cref{rotation-one-photon}, and \cref{cutoff-assumption} gives $J\vG^\mu_{\vx}=\vG^\mu_{\vx}$. Let $\mathcal C_J$ be the induced antiunitary involution of $\Fock_\ph$, characterized by $\mathcal C_Ja^\#(\vf)\mathcal C_J=a^\#(J\vf)$, and define the time reversal \begin{equation}\label{time-reversal-def} (\Theta\Psi)(\vX)=(-1)^\NC\mathcal C_J\big(\Psi(\vX)\big), \end{equation} where $(-1)^\NC=\varGamma(-\iden)=e^{\i\pi\NC}$ multiplies the $m$-photon component by $(-1)^m$. Equivalently, $\Theta=C\otimes\varGamma(-J)$, where $C$ is complex conjugation of electronic wave functions. \begin{proposition}[Time reversal] \label[proposition]{prop-time-reversal} $\Theta$ is an antiunitary involution of $\HC_N$ that preserves $\mathcal Q_\PF$ and commutes with every $U_\vR$ of \cref{prop-orthogonal-covariance}. The field and the covariant derivatives are odd and the radiation energy is even: \begin{equation}\label{A-time-reversal} \Theta\AF(\vx)\Theta^{-1}=-\AF(\vx),\qquad \Theta\vD_n\Theta^{-1}=-\vD_n,\qquad \Theta H_\rad\Theta^{-1}=H_\rad. \end{equation} Consequently, $\EF_\perp$ is even and $\BF$ is odd, and for external potentials in \cref{potential-spaces}, \begin{equation}\label{H-time-reversal} \Theta H^{v,\vA}\Theta^{-1}=H^{v,-\vA}. \end{equation} The corresponding state identities are \begin{equation}\label{time-reversal-laws} \rho_{\Theta\Psi}=\rho_\Psi,\qquad \vjc_{\Theta\Psi}=-\vjc_\Psi,\qquad \dua{\Theta\Psi}{H^{0,0}\Theta\Psi}=\dua{\Psi}{H^{0,0}\Psi}. \end{equation} Consequently, for every $(\rho,\vjc)\in X\times Y$ and $F\in\{F_\LL,F_\LI\}$, \begin{equation}\label{F-time-reversal} F(\rho,-\vjc)=F(\rho,\vjc). \end{equation} \end{proposition} \begin{proof} Second quantization of $J\vG^\mu_{\vx}=\vG^\mu_{\vx}$ and $J\omega=\omega J$ gives $\mathcal C_J\AF(\vx)\mathcal C_J=\AF(\vx)$ and $\mathcal C_JH_\rad\mathcal C_J=H_\rad$, while $(-1)^\NC a^\#(\vf)(-1)^\NC=-a^\#(\vf)$. Applied pointwise in $\vX$, the antilinear map $\mathcal C_J$ also conjugates the electronic dependence, so $-\i\grad_{\vx_n}$ changes sign. This proves \cref{A-time-reversal} on test vectors. The field bound \cref{A-Hrad-bound} and form-norm approximation extend it, and show that $\Theta$ preserves $\mathcal Q_\PF$. Since $\mathcal C_J^2=\iden$ and $(-1)^\NC$ commutes with $\mathcal C_J$, $\Theta^2=\iden$. The pointwise action preserves fermionic antisymmetry. Because $J$ commutes with the one-photon rotations, $\mathcal C_J$ and $(-1)^\NC$ commute with their second quantizations, so $\Theta U_\vR=U_\vR\Theta$. As $\Theta$ is antilinear, $\EF_\perp=-\i[H_\rad,\AF]$ is even, and $\BF=\grad\times\AF$ is odd. The real multiplication operators $v$, $\vA(\vx_n)$, and the Coulomb interaction commute with $\Theta$, which gives \cref{H-time-reversal} as an identity of closed forms. Pointwise antiunitarity gives $\rho_{\Theta\Psi}=\rho_\Psi$ and, by \cref{A-time-reversal}, \[ \vjc_{\Theta\Psi}(\vx) =N\Re\dua{\Theta\Psi}{\vD_1\Theta\Psi}_\vx =-N\Re\ol{\dua{\Psi}{\vD_1\Psi}_\vx} =-\vjc_\Psi(\vx). \] The energy identity is \cref{H-time-reversal} with $v=0$ and $\vA=0$. Finally, $\Theta$ is an energy-preserving bijection between the pure constraint sets at $(\rho,\vjc)$ and $(\rho,-\vjc)$, which proves \cref{F-time-reversal} for $F_\LL$. The map $(\rho,\vjc)\mapsto(\rho,-\vjc)$ is a linear homeomorphism of $X\times Y$, so \cref{L-vs-LL} gives the identity for $F_\LI$. \end{proof} \subsection{Gauge covariance} \begin{proposition}[Gauge covariance] \label[proposition]{prop-gauge-covariance} Let $g\in W^{1,1}_{\rm loc}(\RR^3,\RR)$ satisfy \[ \grad g\in L^3(\RR^3,\RR^3)+L^\infty(\RR^3,\RR^3), \] and define \begin{equation}\label{gauge-unitary} (U_g\Psi)(\vX)=e^{\i\sum_{n=1}^Ng(\vx_n)}\Psi(\vX). \end{equation} Then $U_g$ is unitary and preserves $\mathcal Q_\PF$. For external potentials in \cref{potential-spaces}, \begin{align} H^{v,\vA-\grad g}&=U_gH^{v,\vA}U_g^\dag, \label{H-gauge}\\ \rho_{U_g\Psi}&=\rho_\Psi, \qquad \vjc_{U_g\Psi}=\vjc_\Psi+\rho_\Psi\grad g. \label{current-gauge} \end{align} In particular, the physical current density $\vjc_\Psi+\rho_\Psi\vA$ is invariant under the simultaneous transformation $(\Psi,\vA)\mapsto(U_g\Psi,\vA-\grad g)$. The internal energy satisfies \begin{equation}\label{energy-gauge} \dua{U_g\Psi}{H^{0,0}U_g\Psi} =\dua{\Psi}{H^{0,0}\Psi}+\int_{\RR^3}\grad g\cdot\vjc_\Psi +\frac12\int_{\RR^3}\rho_\Psi|\grad g|^2. \end{equation} For every $(\rho,\vjc)\in X\times Y$ and $F\in\{F_\LL,F_\LI\}$, \begin{equation}\label{F-gauge} F(\rho,\vjc+\rho\grad g) =F(\rho,\vjc)+\int_{\RR^3}\grad g\cdot\vjc +\frac12\int_{\RR^3}\rho|\grad g|^2. \end{equation} \end{proposition} \begin{proof} The Sobolev chain and product rules give \[ \vD_nU_g\Psi=U_g\big(\vD_n+(\grad g)(\vx_n)\big)\Psi. \] By the vector-valued Sobolev inequality, multiplication by $(\grad g)(\vx_n)$ is bounded from $\mathcal Q_\PF$ to $\HC_N$, so this identity is valid for every $\Psi\in\mathcal Q_\PF$. The phase commutes with $H_\rad$, and its inverse is $U_{-g}$, so it preserves the form domain in both directions. Adding $\vA(\vx_n)-(\grad g)(\vx_n)$ to the transformed derivative proves \cref{H-gauge} as a closed-form identity. Pointwise unitarity and the same derivative formula give \cref{current-gauge} and the asserted physical-current invariance. Expanding the squared covariant-derivative norms, the cross term is \[ \sum_{n=1}^N\Re\dua{\vD_n\Psi}{(\grad g)(\vx_n)\Psi} =\int_{\RR^3}\grad g\cdot\vjc_\Psi, \] and the quadratic term is $\tfrac12\int_{\RR^3}\rho_\Psi|\grad g|^2$. The Coulomb and radiation terms are unchanged, proving \cref{energy-gauge}. The bijection $U_g$ between the two constraint sets therefore proves \cref{F-gauge} for $F_\LL$. For $F_\LI$ the argument is the same, applied to density matrices. By \cref{current-gauge,energy-gauge}, applied to spectral decompositions, $\vGamma\mapsto U_g\vGamma U_g^\dag$ is a bijection of $\dm$ onto itself that maps the constraint set of \cref{FLcs-def} at $(\rho,\vjc)$ onto the one at $(\rho,\vjc+\rho\grad g)$ and shifts the energy by the last two terms in \cref{F-gauge}. This proves \cref{F-gauge} for $\wt F_\LI$, and hence for $F_\LI$ by \cref{unrestricted-Lieb-equality}. \end{proof} \subsection{Weyl displacement operators} \label{app-weyl-displacements} We collect the displacement identities used throughout the paper. The algebraic statements hold on the bosonic Fock space over any complex one-particle space $\Hil$. Define the exponential vector and the Weyl operator by \begin{equation}\label{weyl-definitions} \boxed{\mathcal E(f):=\bigoplus_{n=0}^\infty\frac{f^{\otimes n}}{\sqrt{n!}}, \qquad W(f):=\exp\bigl(-\i\varPhi(\i f)\bigr) =\exp\bigl(a^\dag(f)-a(f)\bigr).} \end{equation} The exponential is defined through the self-adjoint Segal field $\varPhi(\i f)$. This convention gives $W(\i f)=\exp(\i\varPhi(f))$. \begin{proposition}[Weyl relations and field shifts] \label[proposition]{prop-weyl-algebra} For $f,g,h\in\Hil$, the following properties hold true. \begin{enumerate}[(i)] \item The Weyl operators act on exponential vectors and compose according to \begin{align} W(f)\mathcal E(g) &=e^{-\|f\|^2/2-\dua{f}{g}}\mathcal E(f+g), \label{weyl-exponential-action}\\ W(f)^\dag=W(-f),\qquad W(f)W(g) &=e^{-\i\Im\dua{f}{g}}W(f+g). \label{weyl-relations} \end{align} \item The map $f\mapsto W(f)$ is strongly continuous. On the linear span of exponential vectors, $W(f)$ has the normally ordered expression \[ W(f)=e^{-\|f\|^2/2}e^{a^\dag(f)}e^{-a(f)}. \] Here the two normally ordered exponentials act by their convergent power series on exponential vectors. \item The closed creation and annihilation operators and the Segal fields satisfy \begin{align} W(f)^\dag a(h)W(f) &=a(h)+\dua{h}{f}\iden, \label{weyl-annihilation-shift}\\ W(f)^\dag a^\dag(h)W(f) &=a^\dag(h)+\dua{f}{h}\iden, \label{weyl-creation-shift}\\ W(f)^\dag\varPhi(h)W(f) &=\varPhi(h)+2\Re\dua{h}{f}\iden, \label{weyl-field-shift}\\ W(f)^\dag\varPi(h)W(f) &=\varPi(h)+2\Im\dua{h}{f}\iden. \label{weyl-conjugate-field-shift} \end{align} Each identity includes invariance of the domain of the operator being conjugated. If $\Im\dua{h}{g}=0$, then $\varPhi(h)$ and $\varPhi(g)$ strongly commute. \item For any unitary $T$ on $\Hil$, \begin{equation}\label{weyl-second-quantization} \varGamma(T)W(f)\varGamma(T)^\dag=W(Tf). \end{equation} \item Under the canonical Fock-space factorization over $\Hil=\Hil_1\oplus\Hil_2$, \begin{equation}\label{weyl-factorization} W(f_1\oplus f_2)=W(f_1)\otimes W(f_2). \end{equation} \end{enumerate} \end{proposition} \begin{proof} (i) The defining series gives $\dua{\mathcal E(g)}{\mathcal E(h)}=e^{\dua{g}{h}}$, and exponential vectors are eigenvectors of the annihilation operators: \begin{equation}\label{exponential-annihilation-eigenvector} a(f)\mathcal E(g)=\dua{f}{g}\mathcal E(g). \end{equation} Exponential vectors have dense linear span. Indeed, if $\Psi=(\psi^{(n)})_{n\geq0}$ is orthogonal to every $\mathcal E(tg)$, $t\in\RR$, then the power series $\sum_nt^n\dua{g^{\otimes n}}{\psi^{(n)}}/\sqrt{n!}$ vanishes identically, so $\dua{g^{\otimes n}}{\psi^{(n)}}=0$ for all $g$ and $n$. The polarization identity \[ \mathrm{Sym}(g_1\otimes\cdots\otimes g_n) =\frac{1}{2^nn!}\sum_{\epsilon\in\{\pm1\}^n}\epsilon_1\cdots\epsilon_n (\epsilon_1g_1+\cdots+\epsilon_ng_n)^{\otimes n}, \] where $\mathrm{Sym}$ averages over permutations of the factors, then gives $\dua{\mathrm{Sym}(g_1\otimes\cdots\otimes g_n)}{\psi^{(n)}}=0$. These symmetrizations span a dense subspace of the $n$-particle sector, so $\psi^{(n)}=0$. The right-hand side of \cref{weyl-exponential-action} preserves inner products of exponential vectors. Its inverse is the same formula with $-f$, so it defines a unitary. Composing the formulas gives \cref{weyl-relations}, and norm continuity on exponential vectors, followed by density, gives strong continuity in $f$. In particular, $t\mapsto W(tf)$ is a strongly continuous one-parameter unitary group. Differentiating at $t=0$ gives $a^\dag(f)-a(f)=-\i\varPhi(\i f)$ as its generator. The exponential span is a core for this Segal field, so this unitary is precisely \cref{weyl-definitions}. (ii) Strong continuity was shown in (i). The normally ordered formula follows from \cref{exponential-annihilation-eigenvector} and $e^{a^\dag(f)}\mathcal E(g)=\mathcal E(f+g)$. (iii) The eigenvector identity \cref{exponential-annihilation-eigenvector} proves \cref{weyl-annihilation-shift} on the exponential span. This span is an operator core for $a(h)$ and $a^\dag(h)$: finite symmetric tensors are obtained by differentiating exponential vectors in their graph norms, and finite-particle truncation gives a core for the closed operators. Since $W(f)$ and $W(-f)$ map the exponential span onto itself, closure gives \cref{weyl-annihilation-shift} with equality of domains. Taking adjoints gives \cref{weyl-creation-shift}. Applying \cref{weyl-relations} to $W(\i t h)=e^{\i t\varPhi(h)}$ and using uniqueness of the self-adjoint generator proves \cref{weyl-field-shift}. Replacing $h$ by $\i h$ gives \cref{weyl-conjugate-field-shift}. When $\Im\dua{h}{g}=0$, the Weyl relations make the two unitary groups $e^{\i t\varPhi(h)}$ and $e^{\i s\varPhi(g)}$ commute for all $s,t\in\RR$, which is strong commutation of their generators. (iv) The identity $\varGamma(T)\mathcal E(g)=\mathcal E(Tg)$ and \cref{weyl-exponential-action} prove \cref{weyl-second-quantization} on exponential vectors, and density extends it. (v) The canonical factorization sends $\mathcal E(g_1\oplus g_2)$ to $\mathcal E(g_1)\otimes\mathcal E(g_2)$, and applying \cref{weyl-exponential-action} proves \cref{weyl-factorization}. \end{proof} The radiation form is expressed through the annihilation operators. The sectorwise annihilation formulas define a closed map \[ (\mathcal T\eta)(\vk,\lambda) :=\sqrt{\omega(\vk)}\,a(\vk,\lambda)\eta,\qquad \|\mathcal T\eta\|=\|\sqrt{H_\rad}\eta\|, \] from $D(\sqrt{H_\rad})$ to $L^2(\RR^3\times\{1,2\};\Fock_\ph)$. It lets us extend the annihilation operator to kernels that are not square integrable. This is needed below for $g=\omega f$ with only $f\in D(\sqrt\omega)$, in which case $g$ need not lie in $\Hil_\ph$ and $a(g)$ is not defined. For a measurable kernel $g$ with $g/\sqrt\omega\in\Hil_\ph$ and $\eta,\zeta\in D(\sqrt{H_\rad})$, define \begin{align} a_\omega(g)\eta &:=\sum_{\lambda=1}^2\int_{\RR^3} \overline{\frac{g(\vk,\lambda)}{\sqrt{\omega(\vk)}}}\, (\mathcal T\eta)(\vk,\lambda)\,\dd\vk, \label{weyl-weighted-annihilation}\\ \varPhi_\omega(g)[\eta,\zeta] &:=\dua{a_\omega(g)\eta}{\zeta} +\dua{\eta}{a_\omega(g)\zeta}. \label{weyl-weighted-field} \end{align} The integrand pairs two square-integrable factors, so the Cauchy--Schwarz inequality gives \[ \|a_\omega(g)\eta\| \leq\|g/\sqrt\omega\|\,\|\sqrt{H_\rad}\eta\|. \] Thus $a_\omega(g)$ is a bounded operator from the radiation form domain to $\Fock_\ph$, and $\varPhi_\omega(g)$ is a continuous symmetric form on that domain. If also $g\in\Hil_\ph$, then $a_\omega(g)=a(g)$ on $D(\sqrt{H_\rad})$, and $\varPhi_\omega(g)$ is the form of $\varPhi(g)$ there. \begin{proposition}[Radiation energy and photon number under displacement] \label[proposition]{prop-weyl-radiation} On $\Fock_\ph$, let $f\in\Hil_\ph$. The following hold. \begin{enumerate}[(i)] \item If $f\in D(\sqrt\omega)$, then $W(f)$ maps $D(\sqrt{H_\rad})$ bijectively onto itself, and \begin{equation}\label{weyl-radiation-bound} \|\sqrt{H_\rad}W(f)\eta\| \leq\|\sqrt{H_\rad}\eta\| +\|\sqrt\omega f\|\,\|\eta\|. \end{equation} \item If $f\in D(\sqrt\omega)$, the radiation form shifts exactly: for all $\eta,\zeta\in D(\sqrt{H_\rad})$, \begin{equation}\label{weyl-radiation-form-shift} \dua{W(f)\eta}{H_\rad W(f)\zeta} =\dua{\eta}{H_\rad\zeta} +\varPhi_\omega(\omega f)[\eta,\zeta] +\|\sqrt\omega f\|^2\dua{\eta}{\zeta}, \end{equation} where $\dua{\eta}{H_\rad\zeta} =\dua{\sqrt{H_\rad}\eta}{\sqrt{H_\rad}\zeta}$ is the radiation form. \item If $f\in D(\omega)$, then $W(f)$ preserves $D(H_\rad)$, and the operator identity on this domain is \begin{equation}\label{weyl-radiation-operator-shift} W(f)^\dag H_\rad W(f) =H_\rad+\varPhi(\omega f)+\|\sqrt\omega f\|^2\iden. \end{equation} \item The photon number satisfies \begin{equation}\label{weyl-number-bound} W(f)D(\sqrt\NC)=D(\sqrt\NC),\qquad \|\sqrt\NC W(f)\eta\| \leq\|\sqrt\NC\eta\|+\|f\|\,\|\eta\|. \end{equation} \item The normalized coherent vector $W(f)\Omega$ satisfies \begin{equation}\label{weyl-coherent-energy} \dua{W(f)\Omega}{\NC W(f)\Omega}=\|f\|^2,\qquad \dua{W(f)\Omega}{H_\rad W(f)\Omega}=\|\sqrt\omega f\|^2. \end{equation} The second equality allows the value $+\infty$. Thus $f\in D(\sqrt\omega)$ is also necessary for preservation of the radiation form domain. \end{enumerate} \end{proposition} \begin{proof} We use the closed map $\mathcal T$ defined before the proposition. For $h\in D(\sqrt\omega)$, \cref{weyl-exponential-action} gives, first on $\eta=\mathcal E(h)$, \begin{equation}\label{weyl-weighted-point-shift} \mathcal T W(f)\eta =(\iden\otimes W(f)) \bigl(\mathcal T\eta+\sqrt\omega f\otimes\eta\bigr). \end{equation} The span of these exponential vectors is a radiation form core. Indeed, finite symmetric tensors with factors in $D(\sqrt\omega)$ form such a core by particle truncation and one-particle spectral truncation. They are in turn limits, in the form norm, of finite differences of exponential vectors, obtained by polarizing and differentiating their defining series. The triangle inequality in \cref{weyl-weighted-point-shift} proves \cref{weyl-radiation-bound} on this core. For a form-convergent sequence, the right-hand side of \cref{weyl-weighted-point-shift} converges and $W(f)\eta_n$ converges in Fock space. Closedness of $\mathcal T$ extends the identity and the bound to the full form domain. Applying the same argument to $-f$ proves invariance in both directions. Taking the inner product of the two sides of \cref{weyl-weighted-point-shift}, for $\eta$ and $\zeta$, and using \cref{weyl-weighted-annihilation} with $g=\omega f$, so that $g/\sqrt\omega=\sqrt\omega f$, gives \cref{weyl-radiation-form-shift}. For $f\in D(\omega)$ and $\zeta\in D(H_\rad)$, the right-hand side of \cref{weyl-radiation-form-shift} is represented by the vector $(H_\rad+\varPhi(\omega f)+\|\sqrt\omega f\|^2)\zeta$. The representation theorem for closed forms therefore gives $W(f)\zeta\in D(H_\rad)$ and \cref{weyl-radiation-operator-shift}. Repeating this with $-f$ proves equality of the operator domains. The same proof with weight $1$ in place of $\omega$ proves \cref{weyl-number-bound}. Applying the identities to $\Omega$ proves \cref{weyl-coherent-energy} when the right-hand side is finite. In general the coherent annihilation formula $a(\vk,\lambda)W(f)\Omega=f(\vk,\lambda)W(f)\Omega$ gives the energy equality as a nonnegative integral, including the value $+\infty$. \end{proof} \begin{corollary}[Displacement of density, current, and energy] \label[corollary]{cor-weyl-PF-shifts} Assume \cref{cutoff-assumption}, let $f\in D(\sqrt\omega)$, and set \begin{equation}\label{weyl-classical-field} c_f^\mu(\vx)=2\Re\dua{G^\mu_{\vx}}{f}. \end{equation} The field $\vc_f$ is real, bounded, continuous, and divergence-free. The photon unitary $W(tf)$ preserves $\mathcal Q_\PF$ for every $t\in\RR$. For $\vGamma\in\dm$, let \[ \boxed{\vGamma_t=W(tf)\vGamma W(tf)^\dag,} \] with the electronic identity suppressed. Then the following hold. \begin{enumerate}[(i)] \item $\vGamma_t\in\dm$, its electronic reduced state is unchanged, and \begin{equation}\label{weyl-density-current-shift} \rho_{\vGamma_t}=\rho_\vGamma,\qquad \vjp_{\vGamma_t}=\vjp_\vGamma,\qquad \vjc_{\vGamma_t}=\vjc_\vGamma-et\rho_\vGamma\vc_f. \end{equation} \item For admissible external potentials, the exact energy change is \begin{equation}\label{weyl-PF-energy-shift} \begin{split} \Tr(H^{v,\vA}\vGamma_t) &=\Tr(H^{v,\vA}\vGamma)\\ &\quad+t\left[\Tr\bigl(\varPhi_\omega(\omega f)\vGamma\bigr) -e\int\vc_f\cdot(\vjc_\vGamma+\rho_\vGamma\vA)\right]\\ &\quad+t^2\left[\|\sqrt\omega f\|^2 +\frac{e^2}{2}\int\rho_\vGamma|\vc_f|^2\right]. \end{split} \end{equation} Here the trace of the weighted field is the sum of its forms over any spectral decomposition of $\vGamma$. It is finite, with \begin{equation}\label{weyl-field-trace-bound} \left|\Tr\bigl(\varPhi_\omega(\omega f)\vGamma\bigr)\right| \leq2\|\sqrt\omega f\| \bigl(\Tr(H_\rad\vGamma)\bigr)^{1/2}. \end{equation} For $f\in D(\omega)$, this trace equals $2\Re\Tr(a(\omega f)\vGamma)$. \end{enumerate} \end{corollary} \begin{proof} The continuity and uniform bounds of $G^\mu_{\vx}$ give $\|\vc_f\|_\infty\leq2\|\wh\chi\|\|f\|$ and continuity. The weak Coulomb-gauge identity gives $\div\vc_f=0$. Since $W(tf)$ acts independently of the electronic positions, it commutes with electronic weak derivatives. \Cref{prop-weyl-radiation} then gives preservation of $\mathcal Q_\PF$. Partial trace over photons is unchanged by a photon unitary, so the density and paramagnetic current density are unchanged. By \cref{weyl-field-shift}, \[ W(tf)^\dag\AF(\vx)W(tf)=\AF(\vx)+t\vc_f(\vx). \] The definitions of $\vjph$ and $\vjc$ now give \cref{weyl-density-current-shift}. Expanding the kinetic form replaces $\vD_n+\vA$ by $\vD_n+\vA-et\vc_f$. Its linear term is $-et\int\vc_f\cdot(\vjc_\vGamma+\rho_\vGamma\vA)$ and its quadratic term is $\tfrac12e^2t^2\int\rho_\vGamma|\vc_f|^2$. The scalar and Coulomb forms are unchanged. Adding \cref{weyl-radiation-form-shift} proves \cref{weyl-PF-energy-shift}, first for pure states. The weighted annihilation bound and Cauchy--Schwarz in a spectral decomposition prove \cref{weyl-field-trace-bound} and justify summing the identities for density matrices. For independence of the spectral decomposition, put $K=(1+H_\rad)^{1/2}$ and let $B_f$ be the bounded operator with \[ \dua{\eta}{B_f\zeta} =\varPhi_\omega(\omega f)[K^{-1}\eta,K^{-1}\zeta]. \] The form trace equals $\Tr(B_fK\vGamma K)$, where $K\vGamma K$ is the positive trace-class operator defined by the finite radiation energy. Hence it does not depend on the chosen decomposition. All remaining terms are integrable since $\vc_f$ is bounded and $\vjc_\vGamma,\rho_\vGamma\vA\in L^1$. \end{proof} In particular, for real $\vu\in C_c^\infty(\RR^3;\RR^3)$ and $g_\vu=\sum_\mu\int u^\mu(\vx)G^\mu_{\vx}\dd\vx$, symmetry of the form factor gives $\dua{G^\mu_{\vx}}{g_\vu}\in\RR$. Consequently, \[ \vc_{\i g_\vu}=0,\qquad \varPhi(\i\omega g_\vu)=-\int\vu\cdot\EF_\perp. \] These are the displacements preserving both constraints in the proof of \cref{prop-optimizer-mean-fields}. General displacements instead give the current change in \cref{weyl-density-current-shift}. \begin{proposition}[Coherent vectors and position-dependent displacements] \label[proposition]{prop-weyl-coherent-derivatives} Let $\mathcal O\subset\RR^m$ be open. The following hold. \begin{enumerate}[(i)] \item For $f,g,h\in\Hil$, the normalized coherent vectors satisfy \begin{gather} W(f)\Omega=e^{-\|f\|^2/2}\mathcal E(f),\qquad a(h)W(f)\Omega=\dua{h}{f}W(f)\Omega, \label{weyl-coherent-vector}\\ \dua{W(f)\Omega}{W(g)\Omega} =\exp\Bigl(-\frac12\|f-g\|^2+\i\Im\dua{f}{g}\Bigr). \label{weyl-coherent-overlap} \end{gather} \item As a map of real Hilbert spaces, $f\mapsto W(f)\Omega$ is smooth. Its derivative at $f$ in the direction $h$ is \begin{equation}\label{weyl-coherent-derivative} \frac{\dd}{\dd s}W(f+sh)\Omega\Big|_{s=0} =\bigl(a^\dag(h)-\Re\dua{f}{h}\bigr)W(f)\Omega, \end{equation} and this vector satisfies \begin{equation}\label{weyl-coherent-metric} \begin{split} \dua{W(f)\Omega}{\bigl(a^\dag(h)-\Re\dua{f}{h}\bigr)W(f)\Omega} &=\i\Im\dua{f}{h},\\ \big\|\bigl(a^\dag(h)-\Re\dua{f}{h}\bigr)W(f)\Omega\big\|^2 &=\|h\|^2+\bigl(\Im\dua{f}{h}\bigr)^2. \end{split} \end{equation} \item Let $f\in H^1_{\rm loc}(\mathcal O;\Hil)$ and $S\in H^1_{\rm loc}(\mathcal O;\RR)$. Then $e^{\i S}W(f)\Omega\in W^{1,1}_{\rm loc}$ and, almost everywhere, \begin{equation}\label{weyl-coherent-phase-derivatives} \begin{split} \Im\dua{e^{\i S}W(f)\Omega}{\partial_j\bigl(e^{\i S}W(f)\Omega\bigr)} &=\partial_jS+\Im\dua{f}{\partial_jf},\\ \big\|\partial_j\bigl(e^{\i S}W(f)\Omega\bigr)\big\|^2 &=\|\partial_jf\|^2+\bigl|\partial_jS+\Im\dua{f}{\partial_jf}\bigr|^2. \end{split} \end{equation} In particular, $e^{\i S}W(f)\Omega\in H^1_{\rm loc}$ if $\grad S+\Im\dua{f}{\grad f}\in L^2_{\rm loc}$. \item If $J$ is a conjugation on $\Hil$, the map $f\mapsto W(f)\Omega$ is globally Lipschitz on $\Hil_J:=\{f:Jf=f\}$: \begin{equation}\label{weyl-real-coherent-Lipschitz} \|W(f)\Omega-W(g)\Omega\|^2 =2\bigl(1-e^{-\|f-g\|^2/2}\bigr)\leq\|f-g\|^2 \qquad(f,g\in\Hil_J). \end{equation} Thus for $f\in H^1_{\rm loc}(\mathcal O;\Hil_J)$, $W(f)\Omega\in H^1_{\rm loc}$ and \[ \dua{W(f)\Omega}{\partial_j\bigl(W(f)\Omega\bigr)}=0,\qquad \|\partial_j\bigl(W(f)\Omega\bigr)\|^2=\|\partial_j f\|^2. \] \end{enumerate} \end{proposition} \begin{proof} \Cref{weyl-exponential-action} applied to the vacuum gives \cref{weyl-coherent-vector}, and the exponential-vector inner product gives \cref{weyl-coherent-overlap}. The norm-convergent exponential series and its derivatives converge uniformly on bounded sets of $f$, so differentiating gives \cref{weyl-coherent-derivative}. The eigenvector identity \cref{exponential-annihilation-eigenvector} and the CCR yield \[ \dua{W(f)\Omega}{a^\dag(h)W(f)\Omega}=\dua{f}{h},\qquad \|a^\dag(h)W(f)\Omega\|^2=\|h\|^2+|\dua{h}{f}|^2. \] Substitution proves \cref{weyl-coherent-metric}. Differentiating the scalar phase then proves \cref{weyl-coherent-phase-derivatives} for smooth $f,S$. For the stated Sobolev regularity, first compose the map $g\mapsto W(g)\Omega$ with smooth radial truncations of $f$. On bounded sets its derivative is bounded, and \cref{weyl-coherent-metric} gives the global estimate \[ \big\|\bigl(a^\dag(h)-\Re\dua{f}{h}\bigr)W(f)\Omega\big\| \leq(1+\|f\|)\|h\|. \] Since $(1+\|f\|)\|\grad f\|$ is locally integrable, the truncated chain rules pass to the limit in $W^{1,1}_{\rm loc}$. The product rule with $e^{\i S}$ proves the displayed identities almost everywhere. Their right-hand sides give square-integrable derivatives when $\grad S+\Im\dua{f}{\grad f}\in L^2_{\rm loc}$. Finally, scalar products of $J$-real vectors are real, so \cref{weyl-coherent-overlap} gives \cref{weyl-real-coherent-Lipschitz}. In this case $\Im\dua{f}{\partial_jf}=0$, and the last assertions follow from the same derivative identities with $S=0$. \end{proof} These formulas imply \cref{photonic-real-coherent-derivatives} and apply to the tags in \cref{lem-coherent-tag}. The distinction between $W^{1,1}_{\rm loc}$ and $H^1_{\rm loc}$ matters for complex displacements: $f\in H^1_{\rm loc}$ alone does not control the square of $\Im\dua{f}{\grad f}$. \subsection{Length-gauge transformation} \label{app-dipole-length-gauge} In this subsection the form factor need only satisfy \cref{cutoff-assumption}. The additional ultraviolet moment needed to expand the transformed radiation form will be stated separately. Put $\vR=\sum_{n=1}^N\vx_n$, and let $H_\dip$ be the dipole Hamiltonian in \cref{dipole-internal-hamiltonian}. Note first that by \cref{Gx-def,polarization-projector}, \[ \dua{G^\mu_{0}}{G^\nu_{0}} =\frac12\int_{\RR^3}|\wh\chi(\vk)|^2 \left(\delta^{\mu\nu}-\frac{k^\mu k^\nu}{|\vk|^2}\right) \,\dd\vk \in\RR. \] Since $\Af^\mu(0)=\varPhi(G^\mu_{0})$, \cref{prop-weyl-algebra}(iii) shows that the components of $\AF(0)$ strongly commute. At each electronic configuration $\vX$, the operator $e\vR\cdot\AF(0)$ is the Segal field $\varPhi\bigl(e\sum_{\mu=1}^3R^\mu G^\mu_{0}\bigr)$. We define the multiplication operator \begin{equation}\label{dipole-length-unitary} U_{\mathrm d}:=\exp(\i e\vR\cdot\AF(0)),\qquad\text{that is,}\qquad U_{\mathrm d}(\vX)=W\bigl(\i e\textstyle\sum_{\mu=1}^3R^\mu G^\mu_{0}\bigr). \end{equation} \begin{proposition}[Length-gauge transformation] \label[proposition]{prop-dipole-length-gauge} Let $\mathcal Q_{\mathrm d}:=U_{\mathrm d}^\dag\mathcal Q_\PF$ be the transported form domain. The following hold. \begin{enumerate}[(i)] \item $U_{\mathrm d}$ is unitary on $\HC_N$. \item Every $\Psi\in\mathcal Q_{\mathrm d}$ has finite ordinary electronic kinetic energy, and on $\mathcal Q_{\mathrm d}$ \begin{equation}\label{dipole-length-derivative} U_{\mathrm d}^\dag\vD_{n,\dip}U_{\mathrm d}=-\i\grad_{\vx_n}, \qquad 1\leq n\leq N. \end{equation} \item For $\Psi\in\mathcal Q_{\mathrm d}$, the density and current density satisfy \begin{equation}\label{dipole-length-current} \rho_\Psi=\rho_{U_{\mathrm d}\Psi},\qquad \vjp_\Psi=\vjcdip_{U_{\mathrm d}\Psi}. \end{equation} \item If $\sqrt\omega\,\wh\chi\in L^2(\RR^3)$, then $\mathscr C_\PF\subset\mathcal Q_{\mathrm d}$, and on this algebraic test space the following identity holds in the quadratic-form sense: \begin{equation}\label{dipole-length-form} U_{\mathrm d}^\dag H_\dip U_{\mathrm d} =H_\el^0+H_\rad +e\sum_{\mu=1}^3R^\mu\varPi(\omega G^\mu_{0}) +e^2\vR^\top\vM\vR, \end{equation} where $\vM$ is the matrix \cref{dipole-polarization-matrix}. For $f/\sqrt\omega\in\Hil_\ph$, $\varPi(f)$ denotes the form $\varPhi_\omega(\i f)$ of \cref{weyl-weighted-field} on $D(\sqrt{H_\rad})$. Since $\varPi(f)=\varPhi(\i f)$, this extends the conjugate Segal field \cref{conjugate-segal-field-def}. An interpretation as a Segal operator requires $f\in\Hil_\ph$. \end{enumerate} \end{proposition} \begin{proof} By \cref{dipole-length-unitary}, $U_{\mathrm d}$ acts at each electronic configuration by a Weyl operator, which is unitary and, by \cref{prop-weyl-algebra}(ii), strongly continuous in $\vX$. Its dependence on $\vX$ is permutation invariant, and hence it preserves fermionic antisymmetry. For $\Psi\in\mathscr C_\PF$, differentiation of this multiplier gives \begin{equation}\label{dipole-unitary-gradient-proof} -\i\grad_{\vx_n}U_{\mathrm d}^\dag\Psi =U_{\mathrm d}^\dag\vD_{n,\dip}\Psi, \qquad 1\leq n\leq N. \end{equation} Here the photon factors in the core belong to the domains of all three field components, and their strong commutation justifies differentiation of the corresponding unitary groups. For general $\Psi\in\mathcal Q_\PF$, approximate by $\mathscr C_\PF$ in the common form norm. The field bound \cref{A-Hrad-bound} makes $\vD_{n,\dip}:\mathcal Q_\PF\to\HC_N^3$ continuous. Unitarity and closedness of the electronic weak derivative then extend \cref{dipole-unitary-gradient-proof} to the limit. This proves both the finite kinetic energy of every vector in $\mathcal Q_{\mathrm d}$ and \cref{dipole-length-derivative}, but not that $\mathcal Q_{\mathrm d}\subset\mathcal Q_\PF$. Pointwise unitarity preserves the electronic configuration density, and \cref{dipole-unitary-gradient-proof} preserves the local pairings that define the current density. Integrating over the remaining electronic variables gives \cref{dipole-length-current}. It remains to prove (iv), so assume $\sqrt\omega\,\wh\chi\in L^2(\RR^3)$. Then $G^\mu_{0}\in D(\sqrt\omega)$ by \cref{Gx-def}, and the linear term in \cref{dipole-length-form} is defined, since $\omega G^\mu_{0}/\sqrt\omega=\sqrt\omega\,G^\mu_{0}\in\Hil_\ph$. For $h\in D(\sqrt\omega)$, \cref{prop-weyl-radiation}(i) with $f=\i h$ shows that $W(\i h)$ maps $D(\sqrt{H_\rad})$ bijectively onto itself, and \cref{prop-weyl-radiation}(ii) gives, for $\eta\in D(\sqrt{H_\rad})$, \begin{equation}\label{dipole-length-Weyl-shift} \|\sqrt{H_\rad}W(\i h)\eta\|^2 =\|\sqrt{H_\rad}\eta\|^2 +\varPi(\omega h)[\eta,\eta] +\|\sqrt\omega h\|^2\|\eta\|^2. \end{equation} At each electronic configuration take $h=e\sum_\mu R^\mu G^\mu_{0}$, so that $U_{\mathrm d}(\vX)=W(\i h)$, $\varPi(\omega h)=e\sum_\mu R^\mu\varPi(\omega G^\mu_{0})$, and $\|\sqrt\omega h\|^2=e^2\vR^\top\vM\vR$. For $\Phi\in\mathscr C_\PF$ the electronic support is compact, so \cref{weyl-radiation-bound} gives finite radiation energy of $U_{\mathrm d}\Phi$. Differentiation of the unitary multiplier and its commutation with $\AF(0)$ give finite electronic kinetic energy as well. Thus $U_{\mathrm d}\Phi\in\mathcal Q_\PF$, that is, $\Phi\in\mathcal Q_{\mathrm d}$. By \cref{dipole-length-derivative} and invariance of the Coulomb multiplication form, the kinetic and Coulomb parts of $\dua{\Phi}{U_{\mathrm d}^\dag H_\dip U_{\mathrm d}\Phi}$ equal $\dua{\Phi}{H_\el^0\Phi}$. Integrating \cref{dipole-length-Weyl-shift} over $\vX$ gives the radiation part and proves \cref{dipole-length-form}. \end{proof} The integral form of the matrix \cref{dipole-polarization-matrix} follows from \cref{Gx-def,polarization-projector}. For the hard cutoff \cref{harduv}, the radial weight in this integral is the indicator of the ball of radius $\Lambda$. Since $\int_{\mathbb S^2}(\delta^{\mu\nu}-\theta^\mu\theta^\nu) \,\dd\theta=(8\pi/3)\delta^{\mu\nu}$ and $\int_0^\Lambda r^2\,\dd r=\Lambda^3/3$, it follows that \begin{equation*} \vM=\frac{4\pi\Lambda^3}{9}\iden. \end{equation*} The last term in \cref{dipole-length-form} is the dipole self-polarization energy and is part of the unitary transformation. Its role in stability is discussed in \cite{rokaj2018lightmatter}. The expanded formula is asserted on the stated test space. On the full transported domain $\mathcal Q_{\mathrm d}$, the radiation energy is $\|\sqrt{H_\rad}U_{\mathrm d}\Psi\|^2$, which is nonnegative and requires neither the separate terms in the expansion to be finite nor an electronic position moment. Even though $\AF(0)$ is spatially uniform, its conditional expectation in an entangled state can depend on $\vx$. Thus the field-induced current density need not equal $\rho(\vx)$ times the unconditional field expectation. Moreover, this limit keeps the photon continuum: the span of the three $G^\mu_{0}$ is generally not invariant under $\omega$. The coupled one-photon space is the closed span of $f(\omega)G^\mu_{0}$, with $f$ bounded Borel. A finite-mode model requires a further approximation. Likewise, fixing photon displacement expectations as additional density variables would define a different constrained-search functional. \section{Symmetric tensor lifts}\label{app-symmetric-tensor-lift} In this appendix we construct the permutation-invariant multipliers used in Appendix~\ref{app-current-shift} and in the proof of \cref{thm-photonic-representability}. We use the following notation. For a separable complex Hilbert space $\mathcal V$, the permutation group $\mathscr S_N$ acts on configurations $\vX=(\vx_1,\ldots,\vx_N)\in(\RR^3)^N$ and on $\mathcal V^{\otimes N}$ by \[ (\pi\vX)^i=\vx^{\pi^{-1}(i)},\qquad R_\pi(z_1\otimes\cdots\otimes z_N) =z_{\pi^{-1}(1)}\otimes\cdots\otimes z_{\pi^{-1}(N)}, \] and $\Pi_N=\frac1{N!}\sum_{\pi\in \mathscr S_N}R_\pi$ is the projection onto the fully symmetric tensors. Let $\mathcal D_N\subset(\RR^3)^N$ be the open set of configurations with pairwise distinct points. For $t:\RR^3\to\mathcal V$ put \[ T(\vX)=\bigotimes_{i=1}^Nt(\vx_i), \] so that $T(\pi\vX)=R_\pi T(\vX)$. For a map $F$ on $\mathcal D_N$, $\dd F=\sum_{i,\mu}\partial_{x_i^\mu}F\,\dd x_i^\mu$ denotes its differential in $\vX$. The product $T$ has the connection and derivatives we need, but it is not permutation invariant. The next lemma provides isometries $U_N(\vX)$ that absorb the permutations, so that $U_NT$ is invariant. The flatness condition $U_N^\dag\dd U_N=0$ ensures that composition with $U_N$ leaves the connection of $T$ unchanged. The construction glues local maps with a partition of unity whose gradients blow up when all electrons come together. There $T$ is almost fully symmetric, which is why the fully symmetric part is kept fixed. The resulting derivative bound is uniform when $t$ is Lipschitz, even when electron coordinates coincide. For general $t$ it is controlled by the inverse squared pair distances, which are integrable against finite-energy states by Hardy's inequality. \begin{lemma}[Flat symmetric isometries] \label[lemma]{lem-flat-symmetric-isometries} For every $N\in\NN$ and every separable complex Hilbert space $\mathcal V$, there are a separable complex Hilbert space $\mathcal K_N$, finite-dimensional if $\mathcal V$ is, a smooth map $U_N$ from $\mathcal D_N$ to the isometries $\mathcal V^{\otimes N}\to\mathcal K_N$, and a constant $b_N$ depending only on $N$ with the following properties. For $\vX\in\mathcal D_N$ and $\pi\in \mathscr S_N$, \begin{equation}\label{symmetric-tensor-lift-flat} U_N(\pi\vX)R_\pi=U_N(\vX),\qquad U_N^\dag\dd U_N=0,\qquad U_N(\vX)\Pi_N\text{ is independent of }\vX, \end{equation} and for every map $t:\RR^3\to\mathcal V$ with $\|t(\vx)\|=1$, \begin{equation}\label{symmetric-tensor-lift-extra-metric} \sum_{i=1}^N\sum_{\mu=1}^3 \|(\partial_{x_i^\mu}U_N(\vX))T(\vX)\|^2 \leq b_N\min\Big\{\sum_{i0$ on $\mathcal D_N$, and $s$ is smooth there with values in the compact centered unit sphere \[ \mathbb S_N=\Big\{\vY\in(\RR^3)^N:\ \sum_i\vy_i=0,\ \sum_i|\vy_i|^2=1\Big\}. \] Moreover $r(\pi\vX)=r(\vX)$ and $s(\pi\vX)=\pi s(\vX)$. The stabilizer of a point $\vY\in\mathbb S_N$ is the product of the permutation groups of the blocks of indices $i$ with equal $\vy_i$. Every block has fewer than $N$ elements, because $\mathbb S_N$ excludes total coincidence. Choose a ball $O$ in $\mathbb S_N$ centered at $\vY$, so small that $\pi O\cap O=\emptyset$ for every $\pi$ outside the stabilizer. Since permutations act isometrically, $O$ is invariant under the stabilizer, and any two translates $\pi O$, $\pi'O$ are either equal or disjoint. By compactness, finitely many such balls $O_a$, with blocks $I_{a,j}$, cover $\mathbb S_N$. Take smooth bump functions supported in the $O_a$ whose positivity sets still cover $\mathbb S_N$, average each over the stabilizer of $O_a$, extend it to the translates $\pi O_a$ by invariance, and normalize by the square root of the sum of their squares. This gives smooth permutation-invariant functions $\chi_a$ on $\mathbb S_N$, supported compactly inside $\bigcup_\pi\pi O_a$, with $\sum_a\chi_a^2=1$. We write $\chi_a$ also for $\chi_a\circ s$. Then $\sum_a\chi_a\dd\chi_a=0$, and since the derivatives of $s$ are $O(r^{-1})$, \[ \sum_{a,i,\mu}|\partial_{x_i^\mu}\chi_a(\vX)|^2 \leq A_Nr(\vX)^{-2}, \] where $A_N<\infty$ depends only on the chosen atlas, hence only on $N$. \noindent\textbf{Step 2. Local maps.} Fix a chart $a$. For $\vX\in\mathcal D_N$ with $s(\vX)\in O_a$ put \[ V_a(\vX)=\bigotimes_jU_{|I_{a,j}|}(\vX_{I_{a,j}}),\qquad \vX_I=(\vx_i)_{i\in I}, \] where $\mathcal V^{\otimes N}$ is identified with $\bigotimes_j\mathcal V^{\otimes I_{a,j}}$ by a fixed reordering of the tensor factors. By the inductive hypothesis, $V_a$ is a smooth isometry into the separable space $\mathcal K^{(a)}=\bigotimes_j\mathcal K_{|I_{a,j}|}$, it satisfies $V_a(\tau\vX)R_\tau=V_a(\vX)$ for $\tau$ in the stabilizer of $O_a$, and $V_a^\dag\dd V_a=0$ by the product rule. Moreover, $V_a\Pi_N$ is constant: a fully symmetric tensor is symmetric within each block, so $\Pi_N=(\bigotimes_j\Pi_{|I_{a,j}|})\Pi_N$, and each $U_{|I_{a,j}|}\Pi_{|I_{a,j}|}$ is constant. Extend $V_a$ to the translates by \[ \widetilde V_a(\vX)=V_a(\pi^{-1}\vX)R_{\pi^{-1}},\qquad s(\vX)\in\pi O_a. \] If $\pi O_a=\pi'O_a$, then $\pi^{-1}\pi'$ lies in the stabilizer of $O_a$, and the invariance of $V_a$ under the stabilizer shows that $\widetilde V_a$ does not depend on the choice of $\pi$. By construction, $\widetilde V_a(\pi\vX)R_\pi=\widetilde V_a(\vX)$. The map $\widetilde V_a$ inherits $\widetilde V_a^\dag\dd\widetilde V_a=0$, and $\widetilde V_a\Pi_N$ is constant because $R_{\pi^{-1}}\Pi_N=\Pi_N$. \noindent\textbf{Step 3. Definition of $U_N$.} Let $\mathcal K_N=\operatorname{Ran}\Pi_N\oplus\bigoplus_a\mathcal K^{(a)}$, an orthogonal direct sum, and define \[ U_N(\vX)u=\Pi_Nu\ \oplus\ \bigoplus_a\chi_a(\vX)\widetilde V_a(\vX)(\iden-\Pi_N)u, \] where each term is extended by zero outside the support of $\chi_a$. Since $\sum_a\chi_a^2=1$ and each $\widetilde V_a$ is an isometry, $\|U_N(\vX)u\|^2=\|\Pi_Nu\|^2+\|(\iden-\Pi_N)u\|^2=\|u\|^2$. The first identity in \cref{symmetric-tensor-lift-flat} follows from the invariance of $\chi_a$ and $\widetilde V_a$ and from $R_\pi \Pi_N=\Pi_NR_\pi=\Pi_N$. The third holds because $U_N(\vX)\Pi_Nu=\Pi_Nu\oplus0$. For the second, \[ U_N^\dag\dd U_N =(\iden-\Pi_N)\Big(\sum_a\chi_a\dd\chi_a\, \widetilde V_a^\dag\widetilde V_a +\sum_a\chi_a^2\,\widetilde V_a^\dag\dd\widetilde V_a\Big) (\iden-\Pi_N)=0. \] \noindent\textbf{Step 4. The derivative bound.} Let $t$ be as in \cref{symmetric-tensor-lift-extra-metric}, and let $m(\vX)$ denote the minimum on its right-hand side, without the factor $b_N$. Since $T$ is a unit vector, $\|(\iden-\Pi_N)T\|\leq1$. If $t$ is Lipschitz, then, since $t(\bar\vx)^{\otimes N}\in\operatorname{Ran}\Pi_N$ and all $t(\vx_i)$ are unit vectors, telescoping the tensor product gives \[ \|(\iden-\Pi_N)T\| \leq\|T-t(\bar\vx)^{\otimes N}\| \leq\sum_i\|t(\vx_i)-t(\bar\vx)\| \leq\sqrt N\,r(\vX)\|\grad t\|_\infty. \] Moreover, $r^2=N^{-1}\sum_{i3\|\vv\|_\infty$, let $K_{\mu,\sigma}$ be electronic multiplication by \[ K_{\mu,\sigma}(\vx) =\frac1{\sqrt6} \sqrt{1+\frac{3\sigma v^\mu(\vx)}q}\, e^{\i\sigma qx^\mu}, \qquad \mu=1,2,3,\quad\sigma\in\{-1,1\}, \] and set \[ \vGamma_{\vv,q} =\sum_{\mu,\sigma}K_{\mu,\sigma}\vGamma K_{\mu,\sigma}^\dag, \qquad \vGamma_{\vv,q}(\vx,\vx') =\sum_{\mu,\sigma} K_{\mu,\sigma}(\vx) \, \vGamma(\vx,\vx') \, \ol{K_{\mu,\sigma}(\vx')} \] These multipliers are bounded with bounded weak derivatives and hence act boundedly on $\mathcal Q_\PF$. In particular, $\vGamma_{\vv,q}$ has finite form energy. Directly from their definition, \begin{align*} \sum_{\mu,\sigma}|K_{\mu,\sigma}(\vx)|^2 &=\frac16\sum_{\mu,\sigma} \left(1+\frac{3\sigma v^\mu(\vx)}{q}\right)=1,\\ \sum_{\mu,\sigma}\ol{K_{\mu,\sigma}(\vx)} \grad K_{\mu,\sigma}(\vx) &=\frac12\grad\sum_{\mu,\sigma}|K_{\mu,\sigma}(\vx)|^2 +\i q\sum_{\mu,\sigma}\sigma|K_{\mu,\sigma}(\vx)|^2\ve_\mu =\i\vv(\vx), \end{align*} where $\ve_\mu$ is the $\mu$th coordinate unit vector. The above scalar relation means that $\sum_{\mu,\sigma}K_{\mu,\sigma}^\dag K_{\mu,\sigma}=\iden$ as an operator. Each summand defining $\vGamma_{\vv,q}$ is positive and trace class, and cyclicity of the trace gives \[ \Tr\vGamma_{\vv,q} =\Tr\!\left(\vGamma\sum_{\mu,\sigma} K_{\mu,\sigma}^\dag K_{\mu,\sigma}\right) =\Tr\vGamma=1. \] Consequently, $\vGamma_{\vv,q}\in\dm$. For the density, write a spectral decomposition $\vGamma=\sum_rp_r|\Psi_r\rangle\langle\Psi_r|$ with $p_r>0$. We use the density and current formulas also for unnormalized vectors, with their usual quadratic scaling. Since $N=1$ and each multiplier is scalar in the photon variables, $\rho_{K_{\mu,\sigma}\Psi_r} =|K_{\mu,\sigma}|^2\rho_{\Psi_r}$. Hence, almost everywhere, \[ \rho_{\vGamma_{\vv,q}} =\sum_rp_r\sum_{\mu,\sigma} |K_{\mu,\sigma}|^2\rho_{\Psi_r} =\sum_rp_r\rho_{\Psi_r}=\rho_\vGamma. \] To check the photon reduced state, let $B\in\mathcal B(\Fock_\ph)$. Each $K_{\mu,\sigma}$ acts only on the electronic factor and therefore commutes with $\iden_\el\otimes B$. By the definition \cref{partial-trace-duality} of the partial trace and cyclicity of the trace, \begin{align*} \Tr\!\left((\Tr_\el\vGamma_{\vv,q})B\right) &=\sum_{\mu,\sigma}\Tr\!\left( \vGamma K_{\mu,\sigma}^\dag(\iden_\el\otimes B)K_{\mu,\sigma}\right)\\ &=\Tr\!\left(\vGamma(\iden_\el\otimes B) \sum_{\mu,\sigma}K_{\mu,\sigma}^\dag K_{\mu,\sigma}\right)\\ &=\Tr\!\left((\Tr_\el\vGamma)B\right). \end{align*} Since this holds for every bounded $B$, it proves $\Tr_\el\vGamma_{\vv,q}=\Tr_\el\vGamma$. For the current density, fix one multiplier $K=K_{\mu,\sigma}$. The field operator commutes with electronic scalar multiplication, so the weak product rule gives \[ \vD_1(K\Psi)=K\vD_1\Psi-\i(\grad K)\Psi. \] Using $\vjc_\Psi=\Re\dua{\Psi}{\vD_1\Psi}_{\vx}$ for $N=1$, we obtain \[ \vjc_{K\Psi} =\Re\dua{K\Psi}{K\vD_1\Psi-\i(\grad K)\Psi}_{\vx} =|K|^2\vjc_\Psi+\Im(\ol K\grad K)\rho_\Psi. \] Summing over the six multipliers therefore yields \[ \sum_{\mu,\sigma}\vjc_{K_{\mu,\sigma}\Psi} =\vjc_\Psi +\rho_\Psi\Im\!\left( \sum_{\mu,\sigma}\ol K_{\mu,\sigma}\grad K_{\mu,\sigma}\right) =\vjc_\Psi+\rho_\Psi\vv. \] Since the multipliers commute with $\AF(\vx)$, the field-induced current density is unchanged and the shift is entirely paramagnetic. Apply this to each $\Psi_r$ and sum with weights $p_r$ to conclude \[ \vjc_{\vGamma_{\vv,q}} =\sum_rp_r\big(\vjc_{\Psi_r}+\rho_{\Psi_r}\vv\big) =\vjc_\vGamma+\rho_\vGamma\vv. \] The density series converges in $L^1$ by positivity and normalization, and the current series by Cauchy--Schwarz and the finite mean form energy. Thus the identities hold also for an infinite-rank $\vGamma$. Expanding the squared norm in the same product rule gives \begin{align} \Tr(H^{0,0}\vGamma_{\vv,q})-\Tr(H^{0,0}\vGamma) &=\int_{\RR^3}\vjc_\vGamma\cdot\vv+\frac{q^2}{2} \notag\\ &\quad+\frac{3}{8q^2}\sum_{\mu=1}^3 \int_{\RR^3}\rho_\vGamma \frac{|\grad v^\mu|^2}{1-9(v^\mu)^2/q^2}. \label{one-electron-repair-exact-energy} \end{align} The radiation energy is unchanged. These identities extend by form approximation, or termwise in a spectral decomposition of $\vGamma$. Choose \[ q^2=\max\left\{36\|\vv\|_\infty^2, \left(\int_{\RR^3}\rho_\vGamma|\grad\vv|^2\right)^{1/2}\right\}, \] so that $q\geq6\|\vv\|_\infty>3\|\vv\|_\infty$ since $\vv\neq0$. The denominators in \cref{one-electron-repair-exact-energy} are at least $3/4$, and the last two terms are at most \[ \frac{q^2}{2} +\frac{1}{2q^2}\int_{\RR^3}\rho_\vGamma|\grad\vv|^2 \leq q^2 \leq36\|\vv\|_\infty^2 +\left(\int_{\RR^3}\rho_\vGamma|\grad\vv|^2\right)^{1/2}. \] Set $\vGamma_{\vv}=\vGamma_{\vv,q}$ with this choice of $q$. Photon localization acts only on the photon factor, so equality of photon marginals also preserves the localized photon-number distributions. \end{proof} To lift this construction to the antisymmetric $N$-electron space, we replace tensor products of the one-electron multipliers by symmetric multipliers with the same connection. These are constructed in Appendix~\ref{app-symmetric-tensor-lift}, and only \cref{lem-symmetric-tensor-lift} is used below. The lift applies to any unit-vector field with connection $\i\vv$, not only to the six multipliers above. Part (ii) of the next lemma uses this to remove the boundedness assumptions on $\vv$. \begin{lemma}[Exact current shift for arbitrary particle number] \label[lemma]{lem-fermionic-current-repair} Let $N\in\NN$ and $\vGamma\in\dm$. \begin{enumerate}[(i)] \item If $\vv\in W^{1,\infty}(\RR^3,\RR^3)$ and $\vv\neq0$, there is a finite electronic multiplication mixture $\vGamma_{\vv}\in\dm$ satisfying \cref{one-electron-repaired-constraints} and \begin{equation}\label{fermionic-repair-cost} \Tr(H^{0,0}\vGamma_{\vv}) \leq\Tr(H^{0,0}\vGamma)+\int_{\RR^3}\vjc_\vGamma\cdot\vv +C_N\big(\|\vv\|_\infty^2+\|\grad\vv\|_\infty\big). \end{equation} \item Let $\mathcal V$ be a separable complex Hilbert space, let $t\in H^1_{\mathrm{loc}}(\RR^3;\mathcal V)$ satisfy $\|t\|=1$ and $\int_{\RR^3}\rho_\vGamma\|\grad t\|^2<\infty$, and put $\vv=\Im\dua{t}{\grad t}$. Then there is a countable electronic multiplication mixture $\vGamma_{\vv}\in\dm$ satisfying \cref{one-electron-repaired-constraints} and \begin{equation}\label{fermionic-repair-cost-general} \Tr(H^{0,0}\vGamma_{\vv}) \leq(1+C_N)\Tr(H^{0,0}\vGamma)+\int_{\RR^3}\vjc_\vGamma\cdot\vv +\frac12\int_{\RR^3}\rho_\vGamma\|\grad t\|^2. \end{equation} \end{enumerate} Here $C_N$ depends only on $N$. In both cases, $\vGamma_{\vv}$ and $\vGamma$ have the same Coulomb energy. \end{lemma} \phantomsection \label{proof:lem-fermionic-current-repair} \begin{proof} \emph{(i)} Let $K_{\mu,\sigma}$ be the six multipliers from the proof of \cref{lem-one-electron-current-repair}, with $q^2=\max\{36\|\vv\|_\infty^2,\|\grad\vv\|_\infty\}$, and collect them in the unit vector $t(\vx)=(K_{\mu,\sigma}(\vx))_{\mu,\sigma}\in\CC^6$. By the identities established there, $\dua{t}{\grad t}=\i\vv$ and \[ |\grad t|^2 =q^2+\frac{3}{4q^2}\sum_{\mu=1}^3 \frac{|\grad v^\mu|^2}{1-9(v^\mu)^2/q^2} \leq q^2+\frac{\|\grad\vv\|_\infty^2}{q^2} \leq36\|\vv\|_\infty^2+2\|\grad\vv\|_\infty. \] Apply \cref{lem-symmetric-tensor-lift} with $\mathcal V=\CC^6$ to $t$, fix an orthonormal basis $(e_\alpha)_{\alpha=1}^{\dim\mathcal K_N}$ of $\mathcal K_N$, and let $K_\alpha=\langle e_\alpha,K\rangle$. These are finitely many permutation-invariant scalar multipliers with \begin{gather*} \sum_\alpha|K_\alpha|^2=1,\qquad \sum_\alpha\ol K_\alpha\grad_{\vx_i}K_\alpha=\i\vv(\vx_i),\\ \sum_{i,\alpha}|\grad_{\vx_i}K_\alpha|^2 \leq C_N\big(\|\vv\|_\infty^2+\|\grad\vv\|_\infty\big). \end{gather*} Define $\vGamma_{\vv} =\sum_\alpha K_\alpha\vGamma K_\alpha^\dag$. Symmetry and bounded weak derivatives preserve the fermionic form domain. Since the $K_\alpha$ act by multiplication in $\vX$ and $\sum_\alpha|K_\alpha|^2=1$, the first identity preserves the density, the Coulomb energy, and the photon marginal. The second identity gives the current shift, and the third bounds the kinetic remainder after expanding each covariant derivative. The radiation energy is unchanged. \emph{(ii)} Apply \cref{lem-symmetric-tensor-lift} to $t$, fix an orthonormal basis $(e_\alpha)$ of $\mathcal K_N$, and let $K_\alpha=\langle e_\alpha,K\rangle$. These are countably many permutation-invariant scalar multipliers with \begin{gather*} \sum_\alpha|K_\alpha|^2=1,\qquad \sum_\alpha\ol K_\alpha\grad_{\vx_i}K_\alpha=\i\vv(\vx_i),\\ \sum_{i,\alpha}|\grad_{\vx_i}K_\alpha|^2 \leq\sum_i\|\grad t(\vx_i)\|^2 +b_N\sum_{i0$, and put \[ h(\vx)=\frac1{\sqrt2}\sum_{k=1}^3 \Big(\lambda w_k(\vx)+\frac{\i x_k}{\lambda}\Big)\ve_k,\qquad t(\vx)=e^{\i(S(\vx)+\frac12\vx\cdot\vw(\vx))}\, W(h(\vx))\Omega_{\mathcal V}, \] where $(\ve_k)$ is the standard basis of $\CC^3$. Then $t\in H^1_{\mathrm{loc}}(\RR^3;\mathcal V)$, $\|t\|=1$, and almost everywhere \begin{equation}\label{coherent-tag-identities} \Im\dua{t}{\grad t}=\grad S+\vw,\qquad \|\grad t\|^2=|\grad S+\vw|^2 +\frac12\big(\lambda^2|\grad\vw|^2+3\lambda^{-2}\big). \end{equation} \end{lemma} \phantomsection \label{proof:lem-coherent-tag} \begin{proof} The phase $S+\frac12\vx\cdot\vw$ lies in $H^1_{\mathrm{loc}}$, and $h\in H^1_{\mathrm{loc}}(\RR^3;\CC^3)$. Since $\sum_kx_k\partial_jw_k=\partial_j(\vx\cdot\vw)-w_j$, \begin{gather*} \Im\dua{h}{\partial_jh} =\frac12\sum_k\big(w_k\delta_{jk}-x_k\partial_jw_k\big) =w_j-\frac12\partial_j(\vx\cdot\vw),\\ \sum_j|\partial_jh|^2 =\frac12\big(\lambda^2|\grad\vw|^2+3\lambda^{-2}\big). \end{gather*} Hence $\partial_j(S+\frac12\vx\cdot\vw)+\Im\dua{h}{\partial_jh}=\partial_jS+w_j$, and \cref{prop-weyl-coherent-derivatives}(iii) gives \cref{coherent-tag-identities}. Unitarity of $W(h)$ gives $\|t\|=1$. The right-hand side of the second identity in \cref{coherent-tag-identities} is locally integrable, so $t\in H^1_{\mathrm{loc}}$ by the same proposition. \end{proof} With this tag, the last term in \cref{fermionic-repair-cost-general} is $\frac12\int\rho_\vGamma\|\grad t\|^2$, which involves only the weighted norms $\int\rho_\vGamma(|\grad S+\vw|^2+|\grad\vw|^2)$ and $N$. For instance, the rigid rotation $\vw=\vomega\times\vx$ is allowed for every $\vGamma$ with $\int\rho_\vGamma|\vx|^2<\infty$. \section{Gaussian representation of the field} \label{app-soft-gaussian} In this appendix we represent the field in a subspace of the photon modes by multiplication operators and condition states on its values. In the solenoidal representation, recall the conjugation $(J\vf)(\vk)=\ol{\vf(-\vk)}$. The symmetry $\wh\chi(\vk)=\ol{\wh\chi(-\vk)}$ in \cref{cutoff-assumption} gives $J\vG^\mu_{\vx}=\vG^\mu_{\vx}$. Fix an infinite-dimensional closed subspace $\mathfrak h\subset\Hil_\ph$ with $J\mathfrak h\subset\mathfrak h$, and let $P_{\mathfrak h}$ be the orthogonal projection onto it. Since $J$ is an antiunitary involution, also $J\mathfrak h^\perp\subset\mathfrak h^\perp$, so $P_{\mathfrak h}$ commutes with $J$ and every $P_{\mathfrak h}G^\mu_{\vx}$ is $J$-real. The exponential law for Fock spaces gives $\Fock_\ph\simeq\Fock_s(\mathfrak h^\perp)\otimes\Fock_s(\mathfrak h)$. Put $\mathcal K=\Hil_\el^N\otimes\Fock_s(\mathfrak h^\perp)$, so that $\HC_N\simeq\mathcal K\otimes\Fock_s(\mathfrak h)$, and let $\AF_{\mathfrak h}(\vx)$ have components $\Af^\mu_{\mathfrak h}(\vx)=\varPhi(P_{\mathfrak h}G^\mu_{\vx})$. We use two cases. For the whole field, $\mathfrak h=\Hil_\ph$ and $\mathcal K=\Hil_\el^N$. For the soft field, fix $\kappa>0$ and use the notation introduced before \cref{lem-exact-soft-removal}: then $\mathfrak h=\Hil_{<\kappa}$, $P_{\mathfrak h}=\iden_{\{\omega<\kappa\}}$, and $\mathcal K=\Hil_\el^N\otimes\Fock_{\geq\kappa}$. This subspace is $J$-invariant because $\omega(-\vk)=\omega(\vk)$. Scalar products of $J$-real vectors are real, so all components of $\AF_{\mathfrak h}$ commute strongly by \cref{prop-weyl-algebra}(iii). Let $(\xi_n)_{n\geq1}$ be a $J$-real orthonormal basis of $\mathfrak h$, equip $\mathcal Q=\RR^{\NN}$ with its product $\sigma$-algebra and the product measure \[ \mu=\bigotimes_{n\geq1}\gamma,\qquad \dd\gamma(t)=(2\pi)^{-1/2}e^{-t^2/2}\dd t, \] and write $u=(u_n)_{n\geq1}\in\mathcal Q$. Mapping the occupation basis associated with $(\xi_n)$ to products of normalized Hermite polynomials in the variables $u_n$ gives a unitary $\mathcal G:\Fock_s(\mathfrak h)\to L^2(\mathcal Q,\mu)$, the Wiener--It\^o--Segal isomorphism \cite{segal1956tensor,simon1974pphi2,janson1997gaussian}. For its use with the Pauli--Fierz model, see \cite{lorinczi2011feynmankac}. Under $\mathcal G$, the field $\varPhi(g)$ with $g\in\mathfrak h$ and $Jg=g$ becomes multiplication by the real Gaussian variable $\sum_n\dua{\xi_n}{g}u_n$ of variance $\|g\|^2$. In particular, $\varPhi(\xi_n)$ becomes multiplication by $u_n$. Hence $\AF_{\mathfrak h}(\vx)$ becomes multiplication by the real random field $\va_u(\vx)$ with components \[ a_u^\mu(\vx)=\sum_{n\geq1} \dua{\xi_n}{P_{\mathfrak h}G^\mu_{\vx}}\,u_n, \] the series converging in $L^2(\mathcal Q,\mu)$. Now let $\vGamma$ be a positive trace-class operator on $\HC_N$. We write $\mathcal G$ also for $\iden\otimes\mathcal G$, which maps $\HC_N$ unitarily onto $L^2(\mathcal Q,\mu;\mathcal K)$. If $(\Psi_j)_{j\geq1}$ is any orthonormal basis of $\HC_N$, put \[ \Theta(u)=\sum_{j\geq1} |(\mathcal G\sqrt{\vGamma}\Psi_j)(u)\rangle \langle(\mathcal G\sqrt{\vGamma}\Psi_j)(u)|. \] \begin{lemma}[Disintegration over the field] \label[lemma]{lem-soft-disintegration} Let $\vGamma$ and $\Theta$ be as above. \begin{enumerate}[(i)] \item For $\mu$-almost every $u$, the series defining $\Theta(u)$ converges in trace norm to a positive trace-class operator on $\mathcal K$. \item The family $(\Theta(u))_{u\in\mathcal Q}$ disintegrates $\vGamma$ over the values of $\AF_{\mathfrak h}$: for every bounded operator $B$ on $\mathcal K$ and every bounded measurable function $f$ on $\mathcal Q$, \begin{equation}\label{soft-disintegration} \Tr\big(\vGamma\,(B\otimes\mathcal G^{-1}M_f\mathcal G)\big) =\int_{\mathcal Q}f(u)\Tr\big(B\Theta(u)\big)\dd\mu(u), \end{equation} where $M_f$ is multiplication by $f$ on $L^2(\mathcal Q,\mu)$. In particular, $\int_{\mathcal Q}\Theta(u)\dd\mu(u)$ is the partial trace of $\vGamma$ over $\Fock_s(\mathfrak h)$. \item Every family of trace-class operators on $\mathcal K$ satisfying \cref{soft-disintegration} for all such $B$ and $f$ agrees with $\Theta(u)$ for $\mu$-almost every $u$. In particular, $\Theta$ does not depend on the choice of the basis $(\Psi_j)_{j\geq1}$, up to a $\mu$-null set. \end{enumerate} \end{lemma} \begin{proof} (i) Each term of the series is positive and of rank one, and its trace is the squared norm of its vector. By Tonelli's theorem and unitarity of $\mathcal G$, \[ \int_{\mathcal Q}\sum_j\|(\mathcal G\sqrt{\vGamma}\Psi_j)(u)\|^2\dd\mu(u) =\sum_j\|\sqrt{\vGamma}\Psi_j\|^2=\Tr\vGamma<\infty. \] Hence the traces are summable for $\mu$-almost every $u$. (ii) Expanding the trace in the basis $(\Psi_j)$, the left-hand side of \cref{soft-disintegration} equals \begin{align*} &\sum_j\dua{\sqrt{\vGamma}\Psi_j} {(B\otimes\mathcal G^{-1}M_f\mathcal G)\sqrt{\vGamma}\Psi_j}\\ &\qquad=\sum_j\int_{\mathcal Q}f(u) \dua{(\mathcal G\sqrt{\vGamma}\Psi_j)(u)} {B(\mathcal G\sqrt{\vGamma}\Psi_j)(u)}\dd\mu(u). \end{align*} The integrands are bounded by $\|B\|\|f\|_\infty\|(\mathcal G\sqrt{\vGamma}\Psi_j)(u)\|^2$, so the bound in (i) allows the sum and the integral to be exchanged, which gives \cref{soft-disintegration}. For $f=1$, the definition \cref{partial-trace-duality} of the partial trace yields the last identity, the integral converging in trace norm by the same bound. (iii) Let $(\Theta'(u))_{u\in\mathcal Q}$ be another family of trace-class operators satisfying \cref{soft-disintegration}. Its left-hand side does not involve the family, so for every bounded $B$, \[ \int_{\mathcal Q}f(u)\Tr\big(B(\Theta(u)-\Theta'(u))\big)\dd\mu(u)=0 \] for all bounded measurable $f$. Hence $\Tr(B(\Theta(u)-\Theta'(u)))=0$ outside a $\mu$-null set depending on $B$. Since $\mathcal K$ is separable, there is a countable family of finite-rank operators that is norm dense in the compact operators on $\mathcal K$. 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