Comment on “ Relativistic kinetic equations for electromagnetic , scalar and pseudoscalar interactions ”

It is found that the extra quantum constraints to the spinor components of the equal-time Wigner function given in a recent paper by Zhuang and Heinz should vanish identically. We point out here the origin of the error and give an interpretation of the result. However, the principal idea of obtaining a complete equal-time transport theory by energy averaging the covariant theory remains valid. The classical transport equation for the spin density is also found to be incorrect. We give here the correct form of that equation and discuss briefly its structure. PACS: 05.60.+w, 03.65.Bz, 52.60.+h In recent papers [1, 2] two of us (P.Z. and U.H.) investigated the equal-time transport theory for a system with electromagnetic, scalar and pseudoscalar interactions by taking the energy average of the corresponding covariant theory. It was shown that the spinor components of the equal-time Wigner function, which are the zeroth-order energy moments of the corresponding components of the covariant Wigner function, are coupled to the first-order moments and satisfy the generalized BGR equations [3]. When confirming these conclusions by an independent check of the calculations in [2] we found, however, that the extra quantum constraints (ZH21) on the equal-time components should vanish identically. Furthermore we found a related error in the classical transport equation (ZH18) for the spin density. (We refer to specific equations from Ref. [2] by adding ZH in front of the equation number.) We here point out the origin of the error and give the correct derivation. In [2] Eqs. (ZH21) were derived by eliminating the first-order energy moments from the constraint equations (ZH10) by combining them with the BGR transport equations (ZH9) and with the first-order energy moments of those covariant equations (ZH7) whose zerothorder moments gave rise to the BGR equations. The mathematical mistake which leads to Eqs. (ZH21) is that the second term on the r.h.s. of ∂ p (pW ) = p∂ n pW + n∂ n−1 p W (1) was inadvertently dropped for n ≥ 2. In this formula W stands for the equal-time Wigner function or any of its spinor components, and ∂ p is the nth-order momentum derivative which appears in the electromagnetic, scalar and pseudoscalar field operators E,B, σe, σo, πe and πo defined in [2]. This meant that, for instance, the second term was omitted from σepW = pσeW − h̄(∇σo)W/2, (2) and related formulae. The effects of these terms in the field operators are to cancel exactly the terms in Eqs. (ZH21) involving the operators M,L,Fσo ,Fσe and Fσ,Fπo and Fπe and Fπ. Therefore no extra constraints on the equal-time Wigner function arise from equations (ZH10). It is possible to give a deeper interpretation of this result: In [2] the two groups of fundamental equal-time kinetic equations are the BGR transport equations (ZH9) and the 1 constraint equations (ZH10). Eqs. (ZH9) determine the evolution of the zeroth-order moments, while Eqs. (ZH10) give explicit expressions for the first-order moments in terms of the zeroth-order ones. In principle, another group of equations which connects zerothand firstorder energy moments can be derived from the first-order energy moment of the covariant version of the BGR equations. The above calculation shows that this additional set of equations contains no independent information; Eqs. (ZH9) and (ZH10) are the only independent equations controlling the behavior of the zerothand first-order energy moments. Furthermore, quite generally it is impossible to extract any extra relationships among the zeroth-order moments from the constraints (ZH10), except in the classical limit. In this limit, the covariant components satisfy the mass-shell constraints p = E−p = (m), and their energy dependence degenerates to two delta-functions at E = ±Ep = √ p + (m), with m being the constituent quark mass (see Eq. (5) below). In this case (and only in this case) all higher energy moments are algebraically related to the zeroth order moment, and in particular W 1 (x,p) = ∫ dp0p0W (x, p) = ±EpW (x,p) . (3) This extra relationship between the classical limits of the first and zeroth order energy moments turns the constraint equations (ZH10) into a set of essential constraints on the classical transport equations (ZH9), which allow one to reduce the number of independent distribution functions by a factor of two. However, this works only in the classical limit, and no such constraints can be derived in the general quantum case. When redoing the calculations a related error was also discovered in the classical transport equation (ZH18) for the spin density g0 This should read