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Covariant Weyl quantization, symbolic calculus, and the product formula

Kamil Serkan Gunturk

Published Aug 16, 2006• 4 citations
OpenAlex

Abstract

A covariant Wigner-Weyl quantization formalism on the manifold that uses\npseudo-differential operators is proposed. The asymptotic product formula that leads\nto the symbol calculus in the presence of gauge and gravitational fields is presented.\nThe new definition is used to get covariant differential operators from momentum\npolynomial symbols. A covariant Wigner function is defined and shown to give\ngauge-invariant results for the Landau problem. An example of the covariant Wigner\nfunction on the 2-sphere is also included.

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A covariant Wigner-Weyl quantization formalism on the manifold that uses\npseudo-differential operators is proposed. The asymptotic product formula that leads\nto the symbol calculus in the presence of gauge and gravitational fields is presented.\nThe new definition is used to get covariant differential operators from momentum\npolynomial symbols. A covariant Wigner function is defined and shown to give\ngauge-invariant results for the Landau problem. An example of the covariant Wigner\nfunction on the 2-sphere is also included.
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.
We demonstrate that Weyl's pioneering idea (1918) to intertwine metric and magnetic fields into a single joint connection can be naturally realized, on the phase space level, by the gauge-invariant quantization of the cotangent bundle with magnetic symplectic form. Quantization, for systems over a noncompact Riemannian configuration manifold, may be achieved by the introduction of a magneto-metric analog of the Stratonovich quantizer — a family of invertible, selfadjoint operators representing quantum δ functions. Based on the quantizer, we construct a generalized Wigner transform that maps Hilbert–Schmidt operators into L2 phase-space functions. The algebraic properties of the quantizer allow one to extract a family of symplectic reflections, which are then used to (i) derive a simple, explicit, and geometrically invariant formula for the noncommutative product of functions on phase space, and (ii) construct a magneto-metric connection on phase space. The classical limit of this product is given by the usual multiplication of functions (zeroth-order term), the magnetic Poisson bracket (first-order term), and by the magneto-metric connection (second-order term).PACS Nos.: 02.40.-k, 11.10.Nx
The Weyl-Wigner-Moyal formalism is developed for spin by means of a correspondence between spherical harmonics and spherical-harmonic tensor operators. The exact asymptotic relation among the P , Q , and Weyl symbols is found, and the analogue of the Moyal expansion is developed for the Weyl symbol of the product of two operators in terms of the symbols for the individual operators. It is shown that in the classical limit, the Weyl symbol for a commutator equals i times the Poisson bracket of the corresponding Weyl symbols.
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 quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
We show that the cross Wigner function can be written in the form [Formula: see text] where [Formula: see text] is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl–Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics.