Preferred Quantization Rules: Born-Jordan vs. Weyl: the Pseudo-Differential Point of View

TL;DR

This paper compares Born-Jordan and Weyl quantization methods, analyzing their mathematical properties and relationships within pseudo-differential calculus, highlighting the Born-Jordan rule as a historically older alternative gaining renewed interest.

Contribution

It provides a detailed analysis of Born-Jordan quantization, relating it to Cohen class and pseudo-differential calculus, and explores its properties including symplectic covariance and Weyl symbol representation.

Findings

Born-Jordan quantization is an average over τ of τ-operators.

Born-Jordan operators exhibit symplectic covariance.

The Weyl symbol of Born-Jordan operators is derived.

Abstract

There has recently been evidence for replacing the usual Weyl quantization procedure by the older and much less known Born-Jordan rule. In this paper we discuss this quantization procedure in detail and relate it to recent results of Boggiato, De Donno, and Oliaro on the Cohen class. We begin with a discussion of some properties of Shubin's $\tau$-pseudo-differential calculus, which allows us to show that the Born-Jordan quantization of a symbol $a$ is the average for $\tau\in\lbrack0,1]$ of the $\tau$-operators with symbol $a$. We study the properties of the Born-Jordan operators, including their symplectic covariance, and give their Weyl symbol.