Wigner functions of permutation operators are obtained and serve as a basis for a phase-space formulation of quantum 1/2 -spin identical fermions. We prove a version of the Pauli principle for this type of particle.
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.
This is a review paper on the formulation of nonrelativistic Quantum Mechanics on Phase Space. In an overlook to the traditional approach, we present the basic concepts including the Weyl mapping, the Wigner function, the Moyal propagator and the Spectral projections. We discuss the advantages and the disadvantages of this formulation. We apply it to the study of quadratic Hamiltonians. The Stratonovich-Weyl correspondence is presented as the natural framework in which the theory can be extended to include spin, relativity and constraints. The theory of identical particles is also considered.