Quantum Deformed Canonical Transformations, W_{\infty}-algebras and Unitary Transformations

TL;DR

This paper explores the algebraic structure of quantum canonical transformations using symbol calculus, constructing a $W_{} imes W_{}$ algebra of pseudo-differential operators that connects quantum and classical mechanics.

Contribution

It introduces a novel algebraic framework for quantum canonical transformations via pseudo-differential operators and relates it to the $W_{}$-algebra and GNS construction.

Findings

Constructed a $W_{} imes W_{}$ algebra of pseudo-differential operators.

Showed the algebra contracts to classical limits and relates to phase-space functions.

Connected the GNS representation to the star-product formalism.

Abstract

We investigate the algebraic properties of the quantum counterpart of the classical canonical transformations using the symbol-calculus approach to quantum mechanics. In this framework we construct a set of pseudo-differential operators which act on the symbols of operators, i.e., on functions defined over phase-space. They act as operatorial left- and right- multiplication and form a $W_{\infty}\times W_{\infty}$- algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel'fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation-space (a doubled Hilbert space) is closely related to the algebra of functions over phase-space equipped with the star-product of the symbol-calculus.