Quantization via Star Products

TL;DR

This paper explores a star product-based quantization method that describes quantum theories purely through function spaces, avoiding Hilbert space reliance, and emphasizes the importance of associativity in solution validity, demonstrated with harmonic oscillators.

Contribution

It introduces a star product quantization scheme that operates entirely within function space, providing an alternative to Hilbert space formulations and highlighting the role of associativity.

Findings

The scheme successfully describes quantum systems without Hilbert spaces.

Associativity ensures exclusion of unwanted solutions in the stargen-value equation.

Explicit demonstration with D-dimensional harmonic oscillator confirms the approach.

Abstract

We study quantization via star products. We investigate a quantization scheme in which a quantum theory is described entirely in terms of the function space without reference to a Hilbert space, unlike the formulation employing the Wigner functions. The associative law plays an essential role in excluding the unwanted solutions to the stargen-value equation. This is demonstrated explicitly with the $D$-dimensional harmonic oscillator.