Generation of Nonlocal Spaces from Moyal Brackets and Penrose Lattices by way of example

TL;DR

This paper explores how Moyal brackets can generate nonlocal spaces through noncommutative lattices, connecting quantum mechanics, Lie algebras, and complex lattice structures like Penrose lattices.

Contribution

It establishes a novel link between Moyal quantization, noncommutative lattices, and nonlocal spaces, with detailed analysis of specific lattice cases.

Findings

Moyal brackets generate SU(N) Lie algebras.

Representations correspond to Bratteli diagrams on Hilbert spaces.

Nonlocal spaces are generated via quantization through Bratteli diagrams.

Abstract

In this paper we consider a connection between the Moyal formuration of quantum mechanics and noncommutative lattices on nonlocal spaces. The Moyal bracket generates SU(N) Lie algebras,and representations of these algebras correspond to Bratteli diagrams on Hilbert spaces of C*-algebras, and also representations of noncommutative lattices on nonlocal spaces correspond to the same diagrams. We will show that the quantization thus generates nonlocal spaces by the intermediation of Bratteli diagrams.The above connection are investigated in details in two definite cases, one is one-dimensional lattice QCD SU(N) fields on the unit circle, the other is Penrose lattice SU-type Lie algebra fields on the Penrose lattice.