Weyl–Wigner transform
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| Weyl–Wigner transform | |
|---|---|
| Name | Weyl–Wigner transform |
| Field | Mathematical physics |
| Introduced | 1927–1932 |
| Key people | Hermann Weyl, Eugene Wigner |
Weyl–Wigner transform The Weyl–Wigner transform is an integral transform linking operators on a Hilbert space to functions on phase space, providing a quasi-probability representation that connects quantum mechanics with classical mechanics. It serves as a cornerstone in deformation quantization, signal analysis, and time-frequency methods, and underpins formulations used in semiclassical analysis and quantum optics. The transform and its associated distributions bridge operator theory with symplectic geometry and harmonic analysis.
Introduction
The transform maps operators associated with states or observables in a Hilbert space used in Hermann Weyl's quantization program and Eugene Wigner's quasi-probability formalism to phase-space functions closely related to classical observables. It appears alongside constructions by John von Neumann in algebraic formulations, and complements methods developed in the contexts of Paul Dirac's theory, Max Born's statistical interpretation, and work by Harish-Chandra and Andrey Kolmogorov in harmonic analysis. The representation is central in discussions involving the Stone–von Neumann theorem, the Moyal bracket, and links to approaches by Norbert Wiener and Jean-Baptiste Joseph Fourier in transform theory.
Mathematical Definition and Properties
The transform assigns to a trace-class operator on L^2(R^n) a function on R^{2n} via an oscillatory integral involving the Weyl quantization kernel introduced by Hermann Weyl and formalized using methods related to Ludwig Schlesinger and I. M. Gelfand's distribution theory. Core properties include real-valuedness for self-adjoint operators noted in analyses by Eugene Wigner and positivity properties studied in the context of the Hudson theorem and work by Rudolf Feynman. It intertwines conjugation and symplectic covariance connected to results by André Weil and the Metaplectic group studied by Boris Fedosov and Lionel Schwartz. The transform respects trace and inner-product identities analogous to the Plancherel theorem and engages with the Heisenberg group representations examined by George Mackey and Israel Gelfand.
Relation to Quantum Mechanics and Phase Space
In quantum mechanics the transform produces the Wigner quasiprobability distribution used to represent pure and mixed states in phase space, linking to expectation values of observables constructed via Weyl ordering advocated by Hermann Weyl and operationally interpreted in works by Paul Dirac and Eugene Wigner. The mapping makes explicit the deformation from Poisson brackets to the Moyal bracket developed by José Enrique Moyal and connects with semiclassical asymptotics studied by Marcel de Gosson and Victor Guillemin. It interfaces with experimental frameworks in quantum optics influenced by Roy J. Glauber and with tomographic reconstructions used in Leonard Mandel's and Elekes] ? investigations; furthermore, it is instrumental in phase-space formulations employed in quantum tomography experiments by groups at institutions such as Bell Labs and CERN.
Examples and Applications
Applications span quantum optics, signal processing, and condensed matter physics where the transform is used to analyze coherence and interference patterns in experiments by Roy J. Glauber and to model semiclassical limits in studies by Michael Berry and Mark Srednicki. In signal analysis the Wigner–Ville distribution, developed in connection with Jean Baptiste Joseph Fourier's ideas and formalized by Jean Morlet and Dennis Gabor, provides time-frequency representations used by engineers at Bell Labs and in radar research initially pursued by Herman Goldstine and Norbert Wiener. In quantum chemistry the transform underlies phase-space approaches used in computational methods influenced by Walter Kohn's density functional theory and practical algorithms similar to those from Martin Golubitsky's numerical analysis lineage. The transform also appears in studies of quantum chaos examined by Michael Berry and Olivier Bohigas.
Generalizations and Alternatives
Generalizations include the Cohen class of time-frequency distributions influenced by Leon Cohen and the Husimi Q function introduced by Kôdi Husimi. Alternatives and extensions appear in deformation quantization developed by Flato and Bayen and in the Berezin transform studied by Felix Berezin and Grigori Perelman's mathematical milieu. Phase-space representations adapted to spin systems use Stratonovich–Weyl correspondences attributed to Robert Stratonovich and further elaborated in contexts related to Alexander Kirillov's orbit method and works by Nigel Hitchin and Victor Guillemin on symplectic geometry. Connections to pseudodifferential operators trace to research by Lars Hörmander and Joseph Kohn.
Historical Development and Key Contributors
Origins trace to Hermann Weyl's 1927 proposal for ordering prescriptions and Eugene Wigner's 1932 introduction of a quasiprobability distribution; development continued through contributions by John von Neumann, José Enrique Moyal, and André Weil. Further formalization and applications involved Ludwig Faddeev, I. M. Gelfand, László Erdős, and researchers in the Soviet mathematical physics tradition such as Arkady Migdal and Lev Landau; contemporary expansions and rigorous treatments were advanced by Lars Hörmander, Marcel de Gosson, and Victor Guillemin. Experimental and applied implementations engaged figures in quantum optics and signal processing including Roy J. Glauber, Jean Morlet, Norbert Wiener, and groups at Bell Labs and CERN.