math.MP — Mathematical Physics

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

Contributed by Zhenyu Sun

Orthogonal truncation can discard states that contribute to a later boundary amplitude. For a self-adjoint contraction $T$ on a finite-dimensional Hilbert space and an isometric inclusion $s$ of retained states, we prove that the two-step residual $\Delta=s^*T^2s-(s^*Ts)^2$ controls every finite placement of intermediate projections in every Schatten norm. For blocks of lengths $n_1,\ldots,n_k$, the error is at most $\sum_{i<j}n_in_j\|\Delta\|$, 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.