Equivalence of Weyl Vacuum and Normal Ordered Vacuum in the Moyal Quantization

TL;DR

This paper demonstrates the equivalence of Weyl vacuum and normal ordered vacuum in Moyal quantization of the harmonic oscillator, using a differential equation approach to establish their relationship.

Contribution

It introduces a method to prove the equivalence of two different vacuum definitions in Moyal quantization through differential equations.

Findings

Weyl vacuum and normal ordered vacuum are equivalent in Moyal quantization.

A differential equation characterizes the normal ordered vacuum.

The equivalence is established mathematically.

Abstract

We study the features of the vacuum of the harmonic oscillator in the Moyal quantization. Two vacua are defined, one with the normal ordering and the other with the Weyl ordering. Their equivalence is shown by using a differential equation satisfied by the normal ordered vacuum.