Gauge-invariant quantum kinetic equations for electrons in classical electromagnetic fields

Abstract

Gauge-invariant quantum kinetic equations for interacting electrons are deduced using the Keldysh diagrammatic technique. The Dyson equations are transformed using a special type of the Wigner representation that produces gauge-invariant Green functions. As a result, they depend on the variables having a meaning of position and kinetic momentum. The Wigner representation used makes it necessary to modify the diagrammatic technique in such a way that it will be able to account for the momentum-energy exchange between the system and the electromagnetic field. The formalism obtained makes it possible to carry out many-particle calculations for non-linear systems in arbitrary electromagnetic fields. Some particular simple cases are considered. A special discussion is given regarding the meaning of the detailed balance in such a formulation.

A Wigner Quasi-distribution Function for Charged Particles in Classical Electromagnetic Fields

“…We mention, here, a few general things about the collision terms that emerge from their explicit expressions derived from equation (38) of Levanda and Fleurov (1994). The collision force is written in the form…”

“…, thenD(X, y|x 1 ) =D(X, −y|x 1 ) and all the processes in the system are detailed balanced as discussed in Section 4 of Levanda and Fleurov (1994). In particular creation and annihilation of particles,D(X, y) y=0 = 0, and the collision force, ∇ yD (X, y) y→0 = 0, vanish.…”

“…The phase loop function was introduced in our previous paper (Levanda and Fleurov, 1994). Here we consider its shape for systems in a static and homogeneous field.…”

“…Nonetheless, the transformation (1) has been used many times in the past for systems in electromagnetic fields. This may lead to certain difficulties 1 in calculations and in interpretation of the results (see, e.g., Mahan, 1987;Mahan, 1990;Rammer, 1986, and the discussion in Levanda and Fleurov, 1994) since terms explicitly depending on the potential may appear, e.g., in kinetic equations. Fleurov and Kozlov (1978) (see also Al'tshuler, 1978) applied a transformation of the form ρ W (P, X) = d 4 ye − ī h y µ (Pµ− e c Aµ(X))ρ (y, X),…”

“…Since then a number of papers were published on quantum kinetic equations (Mahan, 1987;Rammer and Smith, 1986;Tugushev and Fleurov, 1983;Reizer and Sergeev, 1987;Khan et al, 1987;Reggiani et al, 1987;Davies and Wilkins, 1988;Bertoncini et al, 1989;Ferry and Grubin, 1995) and references therein. A gauge invariant formulation of the quantum kinetic equations for charged particles in an arbitrary classical electromagnetic field was derived in our paper (Levanda and Fleurov, 1994). Here, Section III, briefly revisits this derivation and represents the resulting quantum HamiltonJacobi and Boltzmann equations in a more compact form by using the 'slashed' derivatives.…”