Quantization and the resolvent algebra

Abstract

We introduce a novel commutative C*-algebra CR(X) of functions on a symplectic vector space (X,σ) admitting a complex structure, along with a strict deformation quantization that maps a dense subalgebra of CR(X) to the resolvent algebra introduced by Buchholz and Grundling [2]. The associated quantization map is a field-theoretical Weyl quantization compatible with the work of Binz, Honegger and Rieckers [1]. We also define a Berezin-type quantization map on all of CR(X), which continuously and bijectively maps it onto the resolvent algebra.The commutative resolvent algebra CR(X), generally defined on a real inner product space X, intimately depends on the finite dimensional subspaces of X. We thoroughly analyze the structure of this algebra in the finite dimensional case by giving a characterization of its elements and by computing its Gelfand spectrum.