Gauge-independent Wigner functions: General formulation

Abstract

We introduce a gauge-invariant Wigner operator (GIWO) and a gauge-independent Wigner function (GIWF) that allow for both quantized and classical electromagnetic fields. If only classical fields are present, a Weyl transform analogous to the one associated with ordinary Wigner functions can be defined for any operator; in the case of quantized fields this is at least possible for Weyl-ordered functions of the position and kinetic momentum operators. We show how the evolution of the operators having a Weyl transform can be followed with the aid of the GIWF defined as the Heisenberg-picture expectation value of the GIWO, and derive for the GIWO the Heisenberg equation of motion which only involves the physical electric and magnetic fields. Some aspects of the case of entirely classical fields are discussed in more detail. (i) The GIWF in conjunction with the postulate that physical observables can be measured without referring to the gauge permits a quantum-mechanical treatment of a full experimental run without the problem of relating the measured values to the gauge-dependent density operator. (ii) A closed equation of motion for the GIWF is obtained. (iii) In the dipole approximation quantum features of the dynamics are lost. (iv) Quantum corrections to dynamics are associated with recoil effects.