Classical and Quantised Resolvent Algebras for the Cylinder

Buchholz and Grundling (Commun Math Phys 272:699–750, 2007) introduced a \(\hbox {C}^*\)-algebra called the resolvent algebra as a canonical quantisation of a symplectic vector space and demonstrated that this algebra has several desirable features. We define an analogue of their resolvent algebra on the cotangent bundle \(T^*\mathbb {T}^n\) of an n-torus by first generalising the classical analogue of the resolvent algebra defined by the first author of this paper in earlier work (van Nuland in J Funct Anal 277:2815–2838, 2019) and subsequently applying Weyl quantisation. We prove that this quantisation is almost strict in the sense of Rieffel and show that our resolvent algebra shares many features with the original resolvent algebra. We demonstrate that both our classical and quantised algebras are closed under the time evolutions corresponding to large classes of potentials. Finally, we discuss their relevance to lattice gauge theory.