Wigner Quasiprobability Distribution - Evolution Equation For Wigner Function

Evolution Equation For Wigner Function

The Wigner transformation is a general invertible transformation of an operator on a Hilbert space to a function g(x,p) on phase space, and is given by

Hermitean operators map to real functions. The inverse of this transformation, so from phase space to Hilbert space, is called the Weyl transformation,

(not to be confused with another definition of the Weyl transformation).

The Wigner function P(x,p) discussed here is thus seen to be the Wigner transform of the density matrix operator ρ. Thus, the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of g(x, p) with the Wigner function.

The Wigner transform of the von Neumann evolution equation of the density matrix in the Schrödinger picture is Moyal's evolution equation for the Wigner function,

where H(x,p) is Hamiltonian and { {•, •} } is the Moyal bracket. In the classical limit ħ → 0, the Moyal bracket reduces to the Poisson bracket, while this evolution equation reduces to the Liouville equation of classical statistical mechanics.

Formally, in terms of quantum characteristics, the solution of this evolution equation reads,

where and are solutions of so-called quantum Hamilton's equations, subject to initial conditions and, and where -product composition is understood for all argument functions.

Since -composition is thoroughly nonlocal (the "quantum probability fluid" diffuses, as observed by Moyal), vestiges of local trajectories are normally barely discernible in the evolution of the Wigner distribution function. In the integral representation of -products, successive operations by them have been adapted to a phase-space path-integral, to solve this evolution equation for the Wigner function, in principle.