Construction of Euclidean (QED)2 via lattice gauge theory

Let ν=det_ren(1+K _g ) be the renormalized Matthews-Salam determinant of (QED)₂, where \(K_g = ieA_{g,} S = \left( {\sum {\gamma _\mu \partial } _\mu + m} \right)^{ - 1} \) is euclidean fermion propagator of one of the following boundary conditions: (1) free, (2) periodic at ?Λ, Λ=?L/2;L/2]², (3) anti-periodic at ?Λ, and \(A_g (x) = (\sum \gamma _\mu A_\mu (x))g(x)\) . Hereg(x)=1 ifxεΛ₀=?r/2,r/2]² с Λ and 0 otherwise. Then we show

1. νεL ^p (dμ(A)), p>0. Further we prove a new determinant inequality which holds for the QED, QCD-type models containing fermions. This enables us to prove:
2. Z(Λ₀)=∫νdμ(A)≦expc|Λ₀|]. Similar volume dependence is shown for the Schwinger functions.