Quantum canonical transformations in Weyl-Wigner-Groenewold-Moyal formalism

TL;DR

This paper investigates the representation of quantum canonical transformations within the Weyl-Wigner-Groenewold-Moyal formalism, demonstrating methods to construct basic transformations in exponential forms and exploring their applications to phase space functions.

Contribution

It introduces a novel approach to construct all three basic quantum canonical transformations in the ordinary exponential form within the star-product formalism.

Findings

Gauge and point transformations are straightforward in star-exponential form.

Interchange transformation requires an ordinary exponential form.

The approach aids in finding generating functions for canonical transformations.

Abstract

A conjecture in quantum mechanics states that any quantum canonical transformation can decompose into a sequence of three basic canonical transformations; gauge, point and interchange of coordinates and momenta. It is shown that if one attempts to construct the three basic transformations in star-product form, while gauge and point transformations are immediate in star-exponential form, interchange has no correspondent, but it is possible in an ordinary exponential form. As an alternative approach, it is shown that all three basic transformations can be constructed in the ordinary exponential form and that in some cases this approach provides more useful tools than the star-exponential form in finding the generating function for given canonical transformation or vice versa. It is also shown that transforms of c-number phase space functions under linear-nonlinear canonical transformations and intertwining method can be treated within this argument.