Remarks on "Relativistic kinetic equations for electromagnetic, scalar, and pseudoscalar interactions"

We derive the kinetic equations for both the covariant and equal-time Wigner functions of Dirac particles with electromagnetic, scalar and pseudoscalar interactions. We emphasize the constraint equations for the spinor components in the equal-time formulation. PACS: 03.65.Bz, 05.60.+w, 52.60.+h. Typeset using REVTEX 1 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. Therefore some quantum problems such as pair production [5,6] in strong electric fields have so far only be solved in the equal-time formulation [3,7]. The advantages of the covariant formulation, on the other hand, are explicit Lorentz invariance and the feature that the kinetic equations naturally split into a transport equation of Vlasov-Boltzmann type and a generalized mass-shell condition [1,2] which makes explicit the off-shell effects generated by the collisions in the system. In Ref. [8] we discussed the relationship between the covariant and equal-time kinetic equations. By taking the energy average of the covariant equations we obtained both transport and constraint equations for the equal-time Wigner functions in scalar and spinor electrodynamics. Recently, Shin and Rafelski [9] discussed a system which, in addition to the electromagnetic vector interaction, included also scalar and pseudoscalar (strong) interactions. For the system defined by the Lagrangian L = ψ̄(iγ∂μ − eγ Aμ −m+ gσσ + igππγ5)ψ − 1 4 FμνF μν + 1 2 (∂σ∂μσ + ∂ π∂μπ)− UM (σ, π) , (1) they derived a generalized group of transport equations for the spinor components of the equal-time Wigner function in the mean field approximation. Here Aμ, σ and π are electromagnetic, scalar and pseudoscalar fields, respectively, gσ and gπ are the scalar and pseudoscalar coupling constants, and UM(σ, π) is the mesonic self-interaction potential. In this short paper, we first study the full covariant kinetic equations for such a system and then derive from them transport and constraint equations also for the equal-time Wigner function, using our recently introduced energy averaging method. We thereby supplement the equal-time transport equations derived in Ref. [9] by a set of non-trivial constraint equa2 tions. The importance of the latter is emphasized both for the semiclassical limit and for the general quantum dynamics. In the following we restrict ourselves to the kinetic equations for the spinor Wigner function in the mean field approximation, by replacing the field operators Aμ, σ and π by their mean values. Such a Hartree approximation can be justified for sufficiently strong fields and has so far been used in most applications of quantum transport theory. Since the meson potential UM does not explicitly appear in the kinetic equations for the spinor Wigner function (it only enters the selfconsistency equations for the scalar and pseudoscalar mean fields), the discussions below holds universally for all potentials UM(σ, π). The covariant spinor Wigner function W4(x, p) is the ensemble average of the Wigner operator Ŵ4(x, p) which is the 4-dimensional Wigner transformation of the covariant density operator Φ̂4(x, y): W4(x, p) = ∫