Quantization and the resolvent algebra

Abstract

We introduce a novel commutative C*-algebra C R (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 C R (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 C R (X), which continuously and bijectively maps it onto the resolvent algebra.The commutative resolvent algebra C R (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. * This is an author generated post-print of the following article,

Strict deformation quantization of abelian lattice gauge fields

“…) generated by the commutative Weyl C*-algebra W((g * ) l , 0) from [3] and the commutative resolvent algebra C R ((g * ) l ) from [13]. The reason to work with the unital C*-algebra A l 0 is that A l 0 and its Weyl quantization are conserved under fully general dynamics in the sense of [14].…”

“…This subset generates the C*-algebra C(G l ) ⊗ W((g * ) l , 0), which can be seen as a classical Weyl C*-algebra on the torus [3,13,14] that lies inside A l 0 = A l 0 . The image of W l 0 under the above quantization map generates the crossed product C*-algebra…”

“…The direction of this leap, then, might be indicated by strict deformation quantization [11], for it gives a set of axioms that a quantization map between a classical and a quantum C*-algebra should satisfy. These axioms are stringent, and examples are mostly found in finite-dimensional spaces [10,11,12,15,16], with a few exceptions that usually rely on finite-dimensional approximations [3,13]. To quantize a gauge theory, one is therefore advised to first quantize a finite-dimensional regularization, and this is where lattice gauge theory comes in.…”

“…The current paper will add to this program by constructing promising new field algebras for quantum abelian lattice gauge theories in arbitrary dimension, using an approach guided by C*-algebraic quantization. It follows up on [14], where the field algebra corresponding to a quantum abelian gauge theory on a fixed lattice is defined as the closure of the image under a Weyl quantization map of a classical algebra that is the analogue of the commutative resolvent algebra [2,13] when replacing the configuration space R nk by T nk . Here T n is interpreted as the abelian gauge group and k as the number of edges.…”

Extensions of Bundles of C*-algebras

“…The primary example of this is the formulation of the classical → 0 limit of quantum theories in the framework of strict deformation quantization. The general theory is presented in Rieffel (1989Rieffel ( , 1993 and Landsman (1998aLandsman ( , 2007Landsman ( , 2017, and many examples are investigated in Landsman (1993aLandsman ( ,b, 1998b, Binz et al (2004), Honegger and Rieckers (2005), Honegger et al (2008), Bieliavsky and Gayral (2015), and van Nuland (2019). Under modest conditions, a strict deformation quantization determines a continuous bundle of C*-algebras, with so-called equivalent quantizations determining the same continuous bundle.…”