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Formal solutions of stargenvalue equations

Nuno Costa Dias, João Nuno Prata

Published Oct 5, 2001• 18 citations
OpenAlex

Abstract

The two most important equations of Wigner quantum mechanics are the Moyal and the stargenvalue equations. The latter equation is the Weyl-Wigner transform of the eigenvalue equation of standard operator quantum mechanics and, in the context of the Wigner formulation, is of identical importance. The aim of this letter is to present the formal solution of a general stargenvalue equation in arbitrary dimension, both for continuous and discrete spectra. The properties of the formal solution will be studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. These results provide further insight into the mathematical structure of phase space quantum mechanics and are especially relevant for the construction of a complete formal solution of Wigner quantum mechanics in the Heisenberg picture. 1

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The two most important equations of Wigner quantum mechanics are the Moyal and the stargenvalue equations. The latter equation is the Weyl-Wigner transform of the eigenvalue equation of standard operator quantum mechanics and, in the context of the Wigner formulation, is of identical importance. The aim of this letter is to present the formal solution of a general stargenvalue equation in arbitrary dimension, both for continuous and discrete spectra. The properties of the formal solution will be studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. These results provide further insight into the mathematical structure of phase space quantum mechanics and are especially relevant for the construction of a complete formal solution of Wigner quantum mechanics in the Heisenberg picture. 1
This chapter presents Wigner’s approach to quantum mechanics, based on the Wigner function in phase space. It explains Wigner–Weyl quantization, which makes it possible to associate functions on phase space to wave functions and operators, and it develops the technology to do quantum mechanics in this formalism. This includes the star product, Moyal evolution, and star-eigenvalue equations. It also develops semiclassical methods in this formulation, and it has a section on Berry’s semiclassical formula for the Wigner function in one-dimensional systems.
The Weyl-Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations is still missing. In this paper we present a covariant generalization of the Weyl order prescription and of the Weyl-Wigner map and use them to derive Vey quantum mechanics directly from the standard operator formulation. The procedure displays some interesting features: it yields all the key ingredients and provides a more straightforward interpretation of the Vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the Vey idiom. These features are illustrated through a simple example. 1
The Weyl–Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations, is still missing. In this article we present a covariant generalization of the Weyl order prescription and of the Weyl–Wigner map and use them to derive Vey quantum mechanics directly from the standard operator formulation. The procedure displays some interesting features: it yields all the key ingredients and provides a more straightforward interpretation of the Vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the Vey idiom. These features are illustrated through a simple example.
This is a text on quantum mechanics formulated simultaneously in terms of position and momentum, i.e. in phase space. It is written at an introductory level, drawing on the remarkable history of the subject for inspiration and motivation. Wigner functions density -- matrices in a special Weyl representation -- and star products are the cornerstones of the formalism. The resulting framework is a rich source of physical intuition. It has been used to describe transport in quantum optics, structure and dynamics in nuclear physics, chaos, and decoherence in quantum computing. It is also of importance in signal processing and the mathematics of algebraic deformation. A remarkable aspect of its internal logic, pioneered by Groenewold and Moyal, has only emerged in the last quarter-century: it furnishes a third, alternative way to formulate and understand quantum mechanics, independent of the conventional Hilbert space or path integral approaches to the subject. In this logically complete and self-standing formulation, one need not choose sides between coordinate or momentum space variables. It works in full phase-space, accommodating the uncertainty principle; and it offers unique insights into the classical limit of quantum theory. The observables in this formulation are c-number functions in phase space instead of operators, with the same interpretation as their classical counterparts, only composed together in novel algebraic ways using star products. This treatise provides an introductory overview and supplementary material suitable for an advanced undergraduate or a beginning graduate course in quantum mechanics. Readership: Advanced undergraduates and graduates studying quantum mechanics.