Strict deformation quantization of abelian lattice gauge fields

Abstract

This paper shows how to construct classical and quantum field C*-algebras modeling a U (1) n -gauge theory in any dimension using a novel approach to lattice gauge theory, while simultaneously constructing a strict deformation quantization between the respective field algebras. The construction starts with quantization maps defined on operator systems (instead of C*-algebras) associated to the lattices, in a way that quantization commutes with all lattice refinements, therefore giving rise to a quantization map on the continuum (meaning ultraviolet and infrared) limit. Although working with operator systems at the finite level, in the continuum limit we obtain genuine C*-algebras. We also prove that the C*-algebras (classical and quantum) are invariant under time evolutions related to the electric part of abelian Yang-Mills. Our classical and quantum systems at the finite level are essentially the ones of [14], which admit completely general dynamics, and we briefly discuss ways to extend this powerful result to the continuum limit. We also briefly discuss reduction, and how the current set-up should be generalized to the non-abelian case. Operator systems and limit C*-algebrasLattices Let us first define what we mean by 'a lattice'. For simplicity, we take our time-slice to be R D in this paper, although any metric space would work. Throughout this article, a lattice is a finite subset l ⊆ R D × R D such that, using the lexicographical ordering of R D , we have x < y for all (x, y) ∈ l, and, we have tx+(1−t)y = sz+(1−s)w for all (x, y), (z, w) ∈ l and all 0 < t, s < 1. The elements e = (x, y) of a lattice l