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

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.

References in corpus (3)

• Quantization Methods: A Guide for Physicists and Analysts
• Quantum Probability, Renormalization and Infinite-Dimensional *-Lie Algebras
• Recursive boson system in the Cuntz algebra ${\mathcal{O}}_{\infty }$