Wigner distribution function

Wigner distribution function
Lifeinthetrees Antony Stewart · CC0 · source
Name	Wigner distribution function
Field	Quantum mechanics; Signal processing
Introduced	1932
Inventor	Eugene Wigner

Contents

Wigner distribution function The Wigner distribution function is a quasi-probability distribution introduced by Eugene Wigner that represents quantum states in phase space. It connects concepts from Eugene Wigner, Paul Dirac, Max Born, Werner Heisenberg, and Erwin Schrödinger to provide a bridge between operator formalisms and classical phase-space descriptions. The function has influenced developments at institutions such as the Institute for Advanced Study, Princeton University, University of Göttingen, University of Vienna, and has been applied in contexts involving John von Neumann, Richard Feynman, and Max Planck.

Definition and basic properties

The Wigner distribution function is defined for a state associated with operators introduced by Eugene Wigner in 1932 and possesses properties discussed by John von Neumann, Paul Dirac, Hermann Weyl, Weyl–Wigner transform, and Hermann Weyl's contemporaries. It is a real-valued, quasi-probability distribution that can assume negative values, linking to analyses by Wolfgang Pauli, Niels Bohr, Louis de Broglie, Lev Landau, and Lev V. Keldysh. The marginals of the function reproduce probability densities related to measurement operators studied by Werner Heisenberg and Max Born, and transformation properties under unitary dynamics relate to work by Richard Feynman, Paul Dirac, John Bell, and David Bohm. Symmetry and covariance under groups examined by Eugene Wigner and Emmy Noether are central to its behavior.

Mathematically the Wigner distribution function is expressed with integrals and operators familiar to John von Neumann, Paul Dirac, Hermann Weyl, Norbert Wiener, and Andrey Kolmogorov. For a pure state given by a wavefunction in the position representation (linked historically to Erwin Schrödinger and Max Born), the function uses a Fourier transform introduced in contexts by Joseph Fourier and furthered by Jean Baptiste Joseph Fourier's successors such as Henri Poincaré and Sofia Kovalevskaya. The Weyl correspondence and the Moyal bracket (named after José Enrique Moyal and connected to Moyal's equation) relate the Wigner function to operator commutators treated by Werner Heisenberg and Paul Dirac. Positivity issues and nonclassicality criteria echo analyses by John Bell, Alain Aspect, Anton Zeilinger, and Samuel Braunstein.

In quantum mechanics the Wigner distribution function provides a phase-space representation that resonates with the matrix mechanics of Werner Heisenberg and the wave mechanics of Erwin Schrödinger. It mediates between the Weyl–Wigner transform and operator formulations used by John von Neumann and connects to semiclassical methods developed by Marcel Brillouin, Michael Berry, Vladimir Arnold, and Martin Gutzwiller. The negative regions of the Wigner function have been associated with quantum contextuality and nonlocality addressed in studies by John Bell, Niels Bohr, Alain Aspect, and Anton Zeilinger, while its time evolution follows equations analogous to the Liouville equation investigated by Joseph Liouville and extended by José Enrique Moyal and Lev Landau.

Applications of the Wigner distribution function span quantum optics as developed by Roy J. Glauber, Loudon Rodney, Leonard Mandel, and E. C. G. Sudarshan, to quantum information theory influenced by Peter Shor, Charles Bennett, Gilles Brassard, and Anton Zeilinger. In signal processing the Wigner–Ville distribution links to work by Jean Ville and Claude Shannon, and engineering applications reference research from Norbert Wiener, Ralph Hartley, Harry Nyquist, and H. W. Bode. Specific examples include the harmonic oscillator treated by Paul Dirac, coherent states related to Erwin Schrödinger and Roy J. Glauber, squeezed states explored by Daniel Walls and H. P. Yuen, and decoherence analyses tied to Wojciech Zurek, Joos Zeh, and Max Tegmark. Experimental reconstructions of Wigner functions have been performed in platforms associated with Serge Haroche, David Wineland, Anton Zeilinger, A. Aspect, and Immanuel Bloch.

Generalizations of the Wigner distribution function include the Cohen class and time-frequency distributions developed by Leon Cohen, modifications inspired by the Weyl correspondence studied by Hermann Weyl and J. E. Moyal, and phase-space methods extended by Marcel B. R.}} and Martin Gutzwiller. Multidimensional and spinor generalizations draw on formalisms by P. A. M. Dirac, Paul Dirac, Lev Landau, Vladimir Fock, and Eugene Wigner's successors such as Yakir Aharonov and Michael Berry. Noncommutative and deformation-quantization approaches relate to work by Flato, Bayen, Antoni Zygmund, and Mikhail Gromov, while computational implementations and tomography techniques reference contributions by Klauder, Sudarshan, Vogel, and Risken. Cross-disciplinary extensions reach into fields connected with Claude Shannon, Norbert Wiener, John Tukey, and Alan Turing.