Classical and Quantised Resolvent Algebras for the Cylinder.

Area of Science:

• Mathematical Physics
• Quantum Algebra
• Geometric Quantization

Background:

• The resolvent algebra, introduced by Buchholz and Grundling, is a canonical quantization of symplectic vector spaces with desirable properties.
• Earlier work by van Nuland defined a classical analogue of the resolvent algebra on the cotangent bundle of an n-torus.

Purpose of the Study:

• To define and analyze a quantum resolvent algebra on the cotangent bundle of an n-torus.
• To generalize the classical resolvent algebra and apply Weyl quantization.
• To investigate the properties and applications of this new algebraic structure.

Main Methods:

• Generalization of the classical resolvent algebra on the cotangent bundle of an n-torus.
• Application of Weyl quantization to the generalized classical algebra.
• Analysis of the resulting quantum algebra's properties, including its closure under time evolution.

Main Results:

• A novel resolvent algebra is constructed on the cotangent bundle of an n-torus via Weyl quantization.
• The constructed quantization is proven to be almost strict in the sense of Rieffel.
• The new resolvent algebra exhibits many shared features with the original resolvent algebra.
• Both classical and quantized algebras are shown to be closed under time evolutions for broad classes of potentials.

Conclusions:

• The developed resolvent algebra on the n-torus cotangent bundle is a significant extension of previous work.
• The algebra possesses properties analogous to the original resolvent algebra, validating the quantization approach.
• The study highlights the potential relevance of these algebras in the field of lattice gauge theory.