Remark. The scope of the finite gauge construction [fdip-0087]
Remark. The scope of the finite gauge construction [fdip-0087]
The positive convolution theorem and projected finite-time trace law concern a specified pure-gauge model on a finite spatial graph. Their assumptions are the strictly positive, summable character coefficients of Definition [fdip-007Z], not reflection positivity alone. The trace-square theorem applies to the stated algebra of bounded gauge-invariant spatial observables.
Lüscher's construction includes \(SU(N)\) gauge fields and Wilson fermions. Proposition 1 proves self-adjointness, boundedness, gauge invariance and strict positivity of its transfer operator. Its factorization and argument using tensor characters motivate the pure-gauge proof here. The fermion factors require the stated hopping assumptions, including \(0<K_{\mathrm {hop}}<1/6\); \(K_{\mathrm {hop}}\) is that paper's hopping parameter, not the spatial gauge group \(K\) used here. See [luscher1977transfer, pp. 286–288, Proposition 1].
Proposition 2 in [luscher1977transfer, p. 289] identifies time-ordered expectations under its specified gauge fixing and boundary conditions: periodic gauge fields and anticyclic fermions in time. The qualification on periodic temporal gauge appears in footnote 3 on p. 285. The direct projected integral of Theorem [fdip-0083] retains temporal averaging, rather than using that gauge-fixing identification as a finite-periodic premise.
The spectral Hamiltonian has its stated maximal domain; boundedness of the transfer operator does not make its logarithm bounded. No unique-vacuum or continuum reconstruction claim is required. In particular, the joint Gaussian matter law and lattice fermion operators need their own transfer and positivity hypotheses.