The Phase-Space Structure of the Klein-Gordon Field

Abstract

The formalism based on the equal-time Wigner function of the two-point correlation function for a quantized Klein-Gordon field is presented. The notion of the gauge-invariant Wigner transform is introduced and equations for the corresponding phase-space calculus are formulated. The equations of motion governing the Wigner function of the Klein-Gordon field are derived. It is shown that they lead to a relativistic transport equation with electric and magnetic forces and quantum corrections. The governing equations are much simpler than in the fermionic case which has been treated earlier. In addition the newly developed formalism is applied towards the description of spontaneous symmetry breakdown.

Wigner function and kinetic phenomena for chiral plasma in a strong magnetic field

“…The alternative approach is to use the equaltime Wigner function [43] which breaks explicitly the Lorentz covariance because the Fourier transform with respect to the relative time coordinate is not performed. However, such an equal-time Wigner function poses a mathematically well-defined initial-value problem and its interpretation as a quasiprobability distribution function in the phase space is physically transparent [43,44,61,62]. [Note that the Wigner function is a quasiprobability distribution function because it can take negative values].…”

Pair creation in transport equations using the equal-time Wigner function

“…Applying phase-space calculus [2], the equation of motion for P can be written as a differential equation in phase space,…”

“…The usage of the equal-time Wigner transform of the two-point correlation function of a quantum field theory has been proposed to investigate nonperturbatively the time evolution of the vacuum state [1,2]. This approach is based on the Fourier transform of the two-point correlation function split in the space coordinates but evaluated at equal times.…”

Relativistic kinetic equations for electromagnetic, scalar, and pseudoscalar interactions

“…One of the recent progresses in transport theory [1,2] is the establishment of transport equations for spinor [3] and scalar [4] equal-time Wigner functions with abelian gauge interaction. The main advantage of the equal-time formulation lies in the fact that the initial value of the equal-time Wigner function can be directly obtained from the corresponding initial field operators, since there is only one time scale in the equal-time formulation.…”