| phase-space formulation of quantum mechanics | |
|---|---|
| Name | Phase-space formulation of quantum mechanics |
| Scientists | Eugene Wigner, Hermann Weyl, José Moyal, Marvin Jay Keas |
| Introduced | 1927–1949 |
| Institutions | Princeton University, University of Cambridge, University of Melbourne |
| Field | Quantum mechanics |
phase-space formulation of quantum mechanics
The phase-space formulation of quantum mechanics is an alternative formulation in which quantum states and observables are represented by functions on classical phase space rather than vectors and operators in a Hilbert space. It provides a bridge between classical mechanics and quantum theory by encoding quantum phenomena as modifications of classical phase-space structures, enabling semiclassical analysis and novel computational techniques in quantum optics and quantum information.
The phase-space approach originated from efforts to relate quantum mechanics to classical statistical mechanics. Early contributions include the Weyl transform by Hermann Weyl (1927) linking operators to phase-space functions and the Wigner quasi-probability distribution introduced by Eugene Wigner (1932). Further formalization came with work by Groenewold (1946) on operator ordering and by José Moyal (1949), who derived the algebraic structure now known as the Moyal bracket. The formulation was developed in concert with research at institutions such as Princeton University and University of Cambridge, and it influenced later studies in quantum chaos and semiclassical methods.
Phase-space quantum mechanics maps operators on a Hilbert space to functions on phase space using transforms like the Weyl transform and its inverse, the Weyl–Wigner correspondence. Central mathematical objects include the phase-space coordinate pair (q,p), the Weyl symbol of operators, and kernel functions determining operator ordering (e.g., Weyl, normal, and antinormal orderings). The formalism makes heavy use of Fourier transform techniques, distributions in the sense of Schwartz distributions, and deformation quantization methods as developed in deformation quantization theory. Connections to functional analysis, operator theory, and representation theory of the Heisenberg group are foundational.
The Wigner function is a real-valued quasi-probability distribution on phase space representing quantum states; it can take negative values, indicating nonclassicality. Closely related are other quasi-probability distributions such as the Glauber–Sudarshan P representation and the Husimi Q function. These distributions correspond to different operator orderings via the Cahill–Glauber s-parameterized distribution family. The Wigner function reproduces correct marginal position and momentum probability densities and is widely used in quantum optics (e.g., in describing states of the electromagnetic field) and in experimental tomography protocols such as quantum state tomography and homodyne detection.
In phase-space language, observables correspond to functions (symbols) with a noncommutative associative product, the star product (or Moyal product). The star product encodes operator multiplication and depends on Planck's constant ħ; its formal series expansion yields bidifferential operators expressing quantum corrections. This perspective is central to deformation quantization where classical Poisson algebras are deformed into quantum algebras. Notable mathematical contributors include Weyl, Kramers (ordering issues), and Groenewold. The star product formalism also links to the Kontsevich formality theorem in mathematical physics.
Time evolution in phase-space is governed by the Moyal bracket, a deformation of the classical Poisson bracket that generates the quantum Liouville equation for the Wigner function. The Moyal bracket reduces to the Poisson bracket in the semiclassical limit ħ → 0. For quadratic Hamiltonians the phase-space evolution reproduces classical trajectories exactly, while for anharmonic systems quantum corrections manifest through higher-order ħ terms. The formalism interfaces with propagator methods (e.g., Van Vleck determinant techniques), path integral approaches such as the Feynman path integral, and numerical schemes like the split-operator method for solving the Moyal equation.
Phase-space methods are extensively used in quantum optics to analyze coherence, squeezing, and nonclassical states in systems studied at laboratories like Bell Labs and in experiments using laser sources and cavity quantum electrodynamics. In quantum chaos, Wigner functions provide diagnostics of eigenstate localization and scars, linking to studies at institutions such as Los Alamos National Laboratory and research by Michael Berry on semiclassical eigenfunction statistics. In quantum information, phase-space representations facilitate continuous-variable quantum computing, Gaussian state analysis, and entanglement criteria (e.g., via negativity of the Wigner function), connecting to platforms like optical quantum computing and proposals by Samuel L. Braunstein and Pieter Kok.
The phase-space formulation makes the quantum–classical correspondence explicit: expansion of the star product and the Moyal bracket in powers of ħ yields classical dynamics plus systematic quantum corrections used in semiclassical approximations such as the WKB method and the Van Vleck–Gutzwiller propagator. This aids in deriving classical transport equations from quantum kinetics (e.g., the quantum Boltzmann equation) and in analyzing decoherence and the emergence of classicality in open quantum systems treated with master equations (Lindblad form). Semiclassical phase-space techniques are widely applied in molecular dynamics, mesoscopic physics, and chemical reaction theory, linking to computational tools developed in computational physics and chemical physics.
Category:Quantum mechanics Category:Mathematical physics Category:Quantum optics