A Poisson Algebra on the Hida Test Functions and a Quantization using the Cuntz Algebra

TL;DR

This paper introduces a novel quantization method for classical systems using Cuntz algebra generators, providing an alternative to the Jordan-Schwinger map with properties like Lie bracket conservation.

Contribution

It develops a new quantization scheme based on Cuntz algebra generators, differing from traditional methods by maintaining Lie algebra properties.

Findings

Defines a Poisson algebra on Hida test functions.

Constructs a quantization scheme using Cuntz algebra generators.

Ensures properties similar to Van Hove prequantization.

Abstract

In this note we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan-Schwinger (J.-S.) map which has been known and used for a long time by physicists. The difference, comparing to J.-S. map, is that we use generators of Cuntz algebra $\mathcal{O}_{\infty}$ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a "building blocks" instead of creation-annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization i.e. exact conservation of Lie bracket and linearity.