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Gluing defects for projected coarse graining in finite spin-foam models

Primarily human-written textHuman understanding: all partsmath.MP — Mathematical Physicsmath.CT — Category Theorymath.FA — Functional Analysis

Contributed by Zhenyu Sun ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

Orthogonal truncation can discard states that contribute to a later boundary amplitude. For a self-adjoint contraction TT on a finite-dimensional Hilbert space and an isometric inclusion ss of retained states, we prove that the two-step residual Δ=s∗T2s−(s∗Ts)2\Delta=s^*T^2s-(s^*Ts)^2 controls every finite placement of intermediate projections in every Schatten norm. For blocks of lengths n1,…,nkn_1,\ldots,n_k, the error is at most ∑i<jninj∥Δ∥\sum_{i<j}n_in_j\|\Delta\|, and the constant is optimal even for positive contractions. The residual vanishes exactly when the retained subspace is invariant. Two-dimensional finite-group gauge sums realize an idempotent projected transfer operator with nonzero residual, as well as positive-semidefinite transfer operators with identical one-step retained data and different residuals. We derive their gluing and subdivision identities with explicit boundary measures and BF normalization. We also bound the accumulation of local errors under tensor contraction and the distance to compatible linear amplitude data. Colimits extend compatible amplitude families, and quotients by invariant subspaces preserve the comparison maps. A measurement map that is injective on the retained ansatz gives a unique normalized least-squares fit. Exact rational calculations reproduce the finite-group examples and the blocking comparisons.

Provenance statement

ChatGPT was used for proofreading and language polishing.

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Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentOct 22, 2026