Continuum limit of QED 2 on a lattice

Abstract

A path integral is defined for the vacuum expectation values of Euclidean QED 2 on a periodic lattice. Wilson's expression is used for the coupling between fermion and gauge fields. The action for the gauge field by itself is assumed to be a quadratic in place of Wilson's periodic action. The integral over the fermion field is carried out explicitly to obtain a Matthews-Salam formula for vacuum expectation values. For a combination of gauge and fermion fields G on a lattice with spacing proportional to N −1, N ϵ Z +, the Matthews-Salam formula for the vacuum expectation 〈 G〉 N has the form ( G ) N=∫ d nu; W N ( G ,f), where dμ is an N-independent measure on a random electromagnetic field ƒ and W N(G, ƒ) is an N-dependent function of ƒ determined by G . For a class of G we prove that as N → ∞, W N(C, ƒ) has a limit W(G, ƒ) except possibly for a set of ƒ of measure zero. In subsequent articles it will be shown that ∫ d nu; W N ( G ,f) exists and lim N→∞∫ d nu; W N ( G ,f).