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arXiv:1806.06443

Gauge-independent Husimi functions of charged quantum particles in the electro-magnetic field

Ya. A. Korennoy

Published Jun 17, 2018• 1 citation
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Abstract

Gauge-independent Husimi function ($Q-$function) of states of charged quantum particles in the electro-magnetic field is introduced using the gauge-independent Stratonovich-Wigner function, the corresponding dequantizer and quantizer operators transforming the density matrix of state to the Husimi function and vice versa are found explicitly, and the evolution equation for such function is derived. Also own gauge-independent non-Stratonovich Wigner function is suggested and its Husimi function is obtained. Dequantizers and quantizers for these Wigner and Husimi functions are given.

DOI
10.48550/arxiv.1806.06443

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Gauge-independent Husimi function (Q−Q-Q−function) of states of charged quantum particles in the electro-magnetic field is introduced using the gauge-independent Stratonovich-Wigner function, the corresponding dequantizer and quantizer operators transforming the density matrix of state to the Husimi function and vice versa are found explicitly, and the evolution equation for such function is derived. Also own gauge-independent non-Stratonovich Wigner function is suggested and its Husimi function is obtained. Dequantizers and quantizers for these Wigner and Husimi functions are given.
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.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in the presence of an electromagnetic field in phase space, which is spanned by position and kinetic momentum. This approach is independent of the chosen potentials, as it depends only on the electric and magnetic field variables. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field based solely on differential operators that simplify existing formulations. A gauge-dependent equation is derived first using Moyal's equation. A transformation of the Wigner function, introduced by Stratonovich, is then used to make the equation gauge-invariant, which gives us a strong formulation of the problem. This equation can be transformed into its weak form, which proves that both formulations are equivalent. An analysis of the different properties of the gauge-dependent and gauge-invariant formulations is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak formulations.
Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in phase space, which is spanned by position and kinetic momentum. This approach uses the electromagnetic field variables instead of the electrodynamic potentials. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field, which simplifies existing formulations. First, a gauge-dependent equation is derived using Moyal’s formulation. A transformation of the Wigner function introduced by Stratonovich yields the strong form of the gauge-invariant equation. Expressing the pseudo-differential operators by integral operators gives the weak form of the gauge-invariant equation. An analysis of the different properties of the gauge-invariant equation is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak forms.
We investigate the effects of quantized radiation reaction fields on the motion of a charged particle using the gauge-independent Wigner operator (GIWO) and gauge-independent Wigner function (GIWF) introduced earlier [Phys. Rev. A 33, 2913 (1986)]. To complement the equation of motion of the GIWO, the Heisenberg equations of motion of the quantized electromagnetic fields are solved within the Markov approximation. After considering the operator orderings and orders of magnitude of the radiation reaction terms, we eliminate the quantum fields from the evolution equation of the GIWO, and obtain for the GIWF a closed equation containing relaxation terms. As an example of the formalism we derive a Fokker-Planck equation (FPE) for the GIWF of a particle in a constant magnetic field. To the order \ensuremath{\Elzxh} 0^{0}0 the classical radiation damping ensues, and the first quantum correction proportional to \ensuremath{\Elzxh} emerges as diffusion. The diffusion operator turns out to be indefinite and the FPE consequently defies our attempts at a complete analysis, but we demonstrate that at least the coherent states constructed from the Landau levels exhibit a manifestly physical time evolution under the FPE. We point out that the GIWF calculated with quantized electromagnetic fields is divergent even if the fields are in the vacuum state, and suggest that the GIWF should be associated with the particle state by ignoring the quantized fields altogether.
In this work, we review two formalisms of coherent states for the case of a particle in a magnetic field. We focus our revision on both pioneering (Feldman and Kahn 1970 Phys. Rev. B 1 4584) and recent (Kowalski and Rembieliński 2005 J. Phys. A: Math. Gen. 38 8247 ) formulations of coherent states for this problem. We introduce a general application, the Husimi function, which takes into account collective and environmental effects, and which can be derived in a variety of ways. However, we calculate this function as the expected value of the density operator in a basis of coherent states.
On the basis of the Hamiltonian form of the Klein–Gordon equation for a charged scalar particle field introduced by Feshbach and Villars, the gauge-invariant 2 × 2 Wigner matrix has been constructed whose diagonal elements describe positive and negative charge densities and whose off-diagonal elements correspond to cross-densities in phase space. The system of coupled transport equations has been derived in the case of interaction with an arbitrary external electromagnetic field. A gauge-independent generalization of the free particle representation due to Feshbach and Villars is given, and on the basis of it both the nonrelativistic and the classical limits of the relativistic quantum Boltzmann–Vlasov equation are discussed. In the nonrelativistic limit ( p / mc → 0) the set of equations of motion decouple to two independent quantum transport equations describing the dynamics of oppositely charged positon and negaton densities. In the classical limit (ℏ → 0) two relativistic Boltzmann–Vlasov equations result for the diagonal positon and negaton densities. Even though the Planck constant ℏ is absent from the latter equations, the real part of the positon–negaton cross-density does not vanish.