TPR Equal-Time Hierarchies for Quantum Transport Theory Pengfei Zhuang Gesellschaft fur Schwerionenforschung, Theory Group, P.O.Box , D-64

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1 TPR-96-5 Equal-Time Hierarchies for Quantum Transort Theory Pengfei Zhuang Gesellschaft fur Schwerionenforschung, Theory Grou, P.O.Box 055, D-640 Darmstadt, Germany Ulrich Heinz Institut fur Theoretische Physik, Universitat Regensburg, D-9040 Regensburg, Germany (October 4, 996) Abstract We investigate in the equal-time formalism the derivation and truncation of innite hierarchies of equations of motion for the energy moments of the covariant Wigner function. From these hierarchies we then extract kinetic equations for the hysical distribution functions which are related to loworder energy moments, and show how to determine the higher order moments in terms of these lowest order ones. We aly the general formalism to scalar and sinor QED with classical background elds and comare with the results derived from the three-dimensional Wigner transformation method. PACS: 0.65.Bz, w, h. Tyeset using REVTEX

2 I. INTRODUCTION Transort theory [] based on the Wigner oerator is extensively used to describe the formation and evolution of highly excited nuclear matter roduced in relativistic heavy ion collisions. The Wigner oerator can be dened in 4-dimensional [{6] or -dimensional [7,8] momentum sace, which we denote by ^W(x; ) and ^W (x; ), resectively. Corresondingly, there are two formulations for the hase-sace structure of any eld. Either of these two formulations has its advantages and disadvantages. Besides its manifest Lorentz covariance (which is very useful from a technical oint of view), another characteristic feature of the 4-dimensional formulation for QCD [{5] and QED [6] is that the quadratic kinetic equation can be slit u naturally into a transort and a constraint equation. The comlementarity of these two ingredients is essential for a hysical understanding of quantum kinetic theory [9]. In the classical limit, these two equations reduce to the Vlasov and mass-shell equations, resectively. The main advantage of the -dimensional aroach [7,8] is that it is easier to set u as an initial value roblem: one can directly comute the initial value of the Wigner oerator from the corresonding eld oerators at the same time. In the covariant frame this is not ossible since the covariant Wigner oerator is dened as a 4-dimensional Wigner transform of the density matrix and thus includes an integration over time. Hence in this aroach the initial condition for the Wigner oerator at very early times must be constructed henomenologically. Some true quantum roblems like air roduction [0] in a strong external eld have thus so far been solved only in the -dimensional (or equal-time) formulation [7,]. One way [7] to obtain equal-time kinetic equations which arallels the rocedure in the covariant formulation is to Wigner transform the equation of motion for the equal-time density oerator ^(x; y; t). For sinor QED this rocedure results in the BGR equations [7] for the equal-time Wigner functions. In Ref. [] we suggested a dierent derivation which is based on taking the energy average of the covariant kinetic equations in the 4-dimensional formulation. It exloits the fact that the equal-time Wigner function is the energy average

3 (i.e. zeroth order energy moment) of the covariant one. With this method we showed for sinor QED that the direct energy average of the covariant kinetic equations leads, in addition to the BGR transort equations for the sinor comonents of the equal-time Wigner function, also to a second grou of constraint equations which coule the equal-time Wigner function to the rst order energy moment of the covariant one. In the classical (h! 0) limit, these additional equations rovide essential constraints on the equal-time Wigner function and allow to reduce the number of indeendent distribution functions by a factor of two []. In the general quantum case, the additional equations determine the time evolution of the energy distribution function which in general can not be exressed in terms of the equal-time Wigner function. In this sense the BGR equations do not rovide a comlete set of equal-time kinetic equations. As we will discuss in this aer, this incomleteness has a more general asect. As just mentioned, the equal-time Wigner oerator is related to the covariant one by [] ^W (x; ) = Z de ^W(x; ;E); (.) where we wrote =(E;), E indeendent of.as such it is only the lowest member of an innite hierarchy of energy moments of the covariant Wigner oerator: ^W j (x; ) = Z de E j ^W(x; ;E); j =0;;; ; (.) with ^W0 (x; ) ^W (x; ). Therefore, to set u a comlete equal-time transort theory which contains the same amount of information as the covariant theory one needs dynamical equations for all the energy moments. Any covariant kinetic equation will thus corresond to an innite hierarchy of couled kinetic equations for its energy moments, i.e. for the equal-time Wigner oerators ^Wj (x; ). This innite hierarchy only exists for genuine quantum roblems where the energy can exhibit quantum uctuations. In the classical limit, the covariant Wigner oerator satises the mass-shell constraint = E = m, and the energy deendence of the covariant Wigner function thus degenerates to two delta-functions at E = E = m +. The

4 equal-time Wigner oerator ^W (x; ) then slits into a ositive and a negative frequency comonent, ^W (x; ) = ^W+ (x; )+ ^W (x; ) ; (h! 0) (.) and all energy moments can be exressed algebraically in terms of these as ^W j (x; ) = ^W+ j (x; )+ ^Wj (x; ) = E j ^W + (x; )+( E ) j ^W (x; ) ; j =0;;;::: (h! 0): (.4) The solution of the equal-time kinetic equations for ^W (x; ) thus also determines the dynamics of all higher energy moments. Thus, in the classical limit, a simle zeroth order energy average of the covariant kinetic theory yields a comlete equal-time kinetic theory. In the general quantum case, the higher order energy moments ^W j (x; ), j, contain genuine additional information and can no longer be exressed algebraically through the equal-time Wigner oerator ^W (x; ). This means that in rincile in the equal-time formulation we are stuck with the roblem of solving an innite hierarchy of couled equations. Actually, there are two such hierarchies, one resulting from the covariant transort equation (\transort hierarchy"), the other arising from the generalized mass-shell constraint (\constraint hierarchy"). In ractice this raises the roblem of truncating the hierarchy in ahysically sensible way. Since only the low-order energy moments of the covariant Wigner function have anintuitive hysical interretation, it turns out that hysics itself suggests an aroriate truncation scheme. We will show that the hierarchies of moment equations are structured in such awaythat the rst few low-order moments form a nite and closed subgrou of equations which can be solved as an initial value roblem, and that (surrisingly) all the higher order moments can be derived from these low-order moments recursively using only the constraint hierarchy, i.e. without solving any additional equations of motion. The equations from the transort hierarchy for the higher order moments are redundant. We rst discuss on a general basis, starting from the covariant aroach, the derivation and truncation of equal-time hierarchies of kinetic equations. For illustration we then 4

5 consider in full generality the case of a transort theory for scalar elds with arbitrary scalar otentials. For this case everything can be worked out exlicitly to arbitrary order of the moments. We give the subgrou of equations which fully characterize the rst few low-order moments, rove the indeendence and redundance of the transort equations for the higher order moments outside this subgrou, and obtain from the constraint hierarchy exlicit exressions for all the higher order moments in terms of the solutions of the loworder subgrou. We then aly the general formalism to scalar and sinor QED. Here the equations have a more comlicated structure, and we restrict our attention to the closed subgrou of equations for the lowest order moments, discussing the redundance of the transort equations for the higher order moments and their recursive determination through the constraint hierarchy only for the rst moments outside the closed subgrou of equations for the low-order moments. We will comare our results with the kinetic equations for the equal-time distribution functions obtained reviously in Refs. [7,8,]. Our nal result will be a comlete set of kinetic equations which can be imlemented numerically as an initial value roblem. II. GENERAL FORMALISM The 4-dimensional Wigner transform of the equation of motion for the covariant density oerator leads to a comlex, Lorentz covariant kinetic equation for the Wigner oerator. It coules the one-body Wigner oerator to two-body correlations [], which in turn satisfy an equation which coules them to three-body terms, and so on. After taking an ensemble average this generates the so-called BBGKY hierarchy [] for the n-body Wigner functions. A oular way to get a closed kinetic equation for the one-body Wigner function (i.e. the ensemble average of the one-body Wigner oerator) is to truncate the BBGKY hierarchy at the one-body level, by factorizing the two-body Wigner functions in the Hartree aroximation. So far most alications of quantum transort theory have emloyed this aroximation, and in the following we will also restrict ourselves to it. For us the mean 5

6 eld aroximation rovides a crucial simlication, and at resent it is not obvious to us how to generalize our results in order to include correlations and collision terms. For a scalar eld in mean eld aroximation the comlex equation for the self-adjoint scalar Wigner function can be searated into two indeendent real equations []: ^G(x; ) W(x; ) =0; ^F (x; ) W(x; ) =0: (.a) (.b) The rst equation corresonds to a generalized Vlasov equation; after erforming the energy average it generates a hierarchy of transort equations for the energy moments W j (x; ) (\transort hierarchy"). The second equation is a generalized mass-shell constraint; it generates a hierarchy of non-dynamic constraint equations (\constraint hierarchy"). Equations with the structure given in (.) will be the starting oint for our discussion of scalar eld theories in Secs. III and IV A. Factors of in the dynamical oerators ^G(x; ) and ^F (x; ) arise from the Wigner transformation of the artial in the Klein-Gordon equation. Since the latter contains at most second order time derivatives, at most two owers of 0 occur. In fact, ^G(x; ) is linear in 0 while ^F (x; ) is quadratic in 0. This will be imortant below (see Sec. II B). For sinor elds the covariant Wigner function is a 4 4 matrix in sinor sace which for a hysical interretation must be decomosed into its 6 sinor comonents W s. In this way the comlex kinetic equation for the Wigner function matrix is slit into indeendent real equations for the self-adjoint sinor comonents [6,]. These equations can be further divided into two subgrous according to their anticiated structure after erforming the energy average: 6X s 0 = 6X s 0 = ^G ss0 (x; ) W s0 (x; ) =0; (.a) ^F ss0 (x; ) W s0 (x; ) =0; (s=;;:::;6): (.b) The rst subgrou leads to equations containing only rst order time derivatives and thus generates 6 hierarchies of transort equations for the energy moments of the sinor com- 6

7 onents (\transort hierarchies"). The other subgrou which involves both rst and second order time derivatives leads to a set of 6 hierarchies of constraint equations (\constraint hierarchies") for the equal-time moments of the sinor comonents. For the lowest energy moments, the sinor comonents W s (x; ) of the equal-time Wigner function, the details of this rocedure were worked out in Ref. [], and we will use these results in Sec. IV C. Since the original Dirac equation is linear in the time derivative, the dynamical oerators ss ^G ss0 (x; ) and ^F 0 (x; ) contain at most single owers of 0. In fact, the oerators ^Gss 0 (x; ) are indeendent of 0. A. Hierarchy of energy moments In this subsection we will concentrate for simlicity on a single covariant kinetic equation of the generic form ^G(x; ) W(x; ) =0; (.) where ^G(x; ) contains at most two owers of 0 and of the sace-time derivative but an arbitrary number of derivatives with resect to the momentum sace coordinates (see Aendix C). We will return to the full set of equations (.) res. (.) in the following Sections. We begin by decomosing the energy deendence of the Wigner function into a basis of orthogonal olynomials h j (E): W(x; ;E)= X j=0 w j (x; ) h j (E) : (.4) The exansion coecients w j (x; ) are dened in the equal-time hase sace. Using the orthonormality relation Z d(e) h i (E) h j (E) = ij ; (.5) where d(e) isthearoriate integration measure associated with the chosen set of olynomials h j, the equal-time comonents w j (x; ) can be related to energy moments of the covariant Wigner function constructed with the basis functions h j (E): 7

8 Z w j (x; ) = d(e) h j (E) W(x; ;E) : (.6) If the system has nite total energy the covariant sinor comonents must vanish in the limit E!. Wewill assume that they vanish at innite energy faster than any ower of E such that for any combination of integers i; j; m; n 0, we have @E (h i(e)e m W(x; ;E) =0: (.7) j With exonential accuracy we may therefore restrict the energy integration to a nite interval E. Introducing the scaled energy! = E= we can thus use as our set of basis functions the Legendre olynomials s n+ h n (!) = P n (!) (.8) with the trivial measure d(!) = d! on the interval [ ; ]. As discussed above, the dynamical oerator ^G(x; ) in (.) in general contains owers of E u to second order and an innite number of energy In terms of the new dimensionless energy variable we may thus write ^G(x; ;!)= MX X m=0 n=0! ^G mn (x; )! m ; with M. Substituting this double exansion into equation (.), multilying by h i (!) from the left and integrating over energy! we obtain X m;n Z ^G mn (x; ) d! h i nw(x; ;!)=0: (.0) Inserting the exansion (.4) of the covariant Wigner function W(x; ;!), this can be written as X j=0 ^H ij (x; ) w j (x; ) =0; i=0;;;::: ; (.) where ^H ij (x; ) = MX X C mn ij m=0 n=0 8 ^G mn (x; ) (.)

9 with It is easy to see from (.) that Z C mn ij = d! h i (!)! n! h j(!) : (.) C mn ij =0 for n>j and i>j+m n: (.4) For j i m + n the coecients are in general nonzero. Therefore, the sum over j in Eq. (.) extends over the range j max[0;i M]. For each value of i, Eq. (.) thus contains an innite number of terms. In this form Eqs. (.) are thus not ractically useful. However, one can use the surface condition (.7) to rewrite Eqs. (.) in such a way that each equation contains only a nite number of terms. Returning to Eq. (.0) and integrating by arts, we can relace the integrand by h i (!)! n!w(x; ;!)= (.5) nx l=0 ( l! h i (!)! n l! W(x; ;!) +( ) n! h i (!)! m W(x; ;!): The contribution to the integral in (.0) from each term in the sum is fully cancelled by the surface condition (.7), and only the last term in (.5) survives. Inserting it into (.0) and using again the exansion (.4) we nd instead of Eqs. (.-.) the following set of equations: with and the coecients c mn ij = i+m X ^g ij (x; ) w j (x; ) =0; i=0;;;:::; (.6) j=0 ^g ij (x; ) = MX i+m X m=0 n=0 c mn ij ^G mn (x; ) (.7) q Z (i + )(j +) d! P j ) n P i (!)! m : (.8) The latter can be determined recursively from c 00 ij = ij (which results from the orthogonality relation (.5)) by using the recursion relations for the Legendre olynomials, see Aendix A. Since the nonvanishing coecients c mn ij are now restricted to the domain 9

10 n i + m and j i + m n; (.9) the sum over j in (.6) runs now only over the nite range 0 j i + M. The rst inequality in (.9) was already used in (.7) to limit the sum over n. The P i (!) are olynomials in! of order i, and thus the equal-time comonents w j (x; ) occurring in Eqs. (.6) are linear combinations of the energy moments W k (x; ) of order k i (see Eq. (.)). For each value of i, Eq. (.6) thus rovides a relation among the rst i + M + energy moments of the covariant Wigner function ^W (x; ;!) (including the zeroth order moment W (x; ) = w 0 (x; )). As i is allowed to run over all ositive integers, Eqs. (.6) form an innite hierarchy of relations among the energy moments of the covariant Wigner function. Each covariant equation of the tye (.) generates its own such hierarchy. Only the full set of these innite hierarchies of moment equations constitutes a comlete equal-time kinetic descrition of the system under study. B. Truncating the hierarchy In order to discuss ossible truncation schemes we must return to the comlete set of covariant kinetic equations. Let us concentrate here on the scalar case, Eqs. (.), and write down the two resulting hierarchies of moment equations as i+m X ^g ij (x; )w j (x; ) =0; j=0 (.0a) i+m+ X j=0 ^f ij (x; )w j (x; ) =0; (i=0;;;:::): (.0b) In writing down the uer limits of the sums we already used that ^F (x; ) in (.a) contains one ower of 0 more than ^G(x; ) in (.a). For the scalar eld case one has M =. For the sinor case one obtains from (.) a similar set of equations with M =0. Let us now try to truncate these hierarchies for the moments w j at some order j max. The equations from the \transort hierarchy" (.0a) with hierarchy index i I t involve all moments w j with 0 j I t + M, i.e. the rst I t + M +moments (including the lowest 0

11 moment with index 0). Similarly, the equations from the \constraint hierarchy" (.0b) with hierarchy index i I c involve the rst I c + M +moments 0 j I c + M +. For a closed set of equations both hierarchies must be truncated at the same order j max, i.e. we must have I t + M = I c + M +=j max : (.) Truncating in this waywe are left with I t + equations from the transort hierarchy and I c + equations from the constraint hierarchy. In order solve them the number of equations must at least equal the number of moments. However, if there are more equations than moments, the system may be overdetermined, and therefore we would like to require equality of the number of equations and moments: I t ++I c +=j max +: (.) The two conditions (.) and (.) have a unique solution: I t = I c +=M; j max =M; (.) which yields M + transort and M constraint equations for the rst M + energy moments. Smaller values of j max don't yield enough equations, and larger values lead to an (at least suercially) overdetermined system of equations. For sinor elds (M = 0) the truncated set involves only one transort and no constraint equation; for each sinor comonent it gives a single kinetic equation (the BGR equation [7]) for its lowest energy moment, the equal-time Wigner function W s (x; ). For scalar elds (M = ) the truncated set contains two transort equations and one constraint for the three lowest order moments w 0 ; w ; w. If we gobeyond this minimal closed subset of moments and equations, we get two equations for every additional moment, one from the transort hierarchy and one from the constraint hierarchy. As we will show exlicitly in the next Section, in the constraint hierarchy (.0b) the highest moment always comes with a constant coecient. As we increase

12 the hierarchy index i in Eqs. (.0), at each ste the newly occurring moment can thus be exlicitly exressed in terms of the already known lower order moments using the corresonding constraint equation from (.0b). As we will discuss, these higher order constraint equations contain imortant hysics. But in addition, at each ste there is also a dynamical equation of motion for the new moment from the transort hierarchy (.0a). How can the two equations be consistent? The answer is that this transort equation is not an indeendent new equation, but (with some algebraic eort) can be exressed as a combination of the lower order equations which have already been used. Our roof of this fact uses exlicitly the structure of the dynamical oerators ^Gmn and ^Fmn. It involves cumbersome algebra, and only for scalar elds with only scalar otential or mean eld interactions we have been able to nd a general roof. For scalar and sinor QED the roof is still incomlete, and we will only demonstrate the rst ste for the M + nd moment. A comletion of the roof resumably requires a so far missing deeer insight into the general dynamic structure of the moment equations and their relation to the underlying covariant theory. III. SCALAR FIELD THEORY In this Section we will give an exlicit and comlete discussion of the moment hierarchy for the simlest case of a scalar eld theory in Hartree aroximation. We exemlify the truncation of the hierarchy and the recursive comutation of the higher order moments beyond minimal truncation. The discussion in the following Section for the ractically more relevant case of QED will be technically more involved and, unfortunately, also less comlete. A. Covariant kinetic equations Consider the Klein-Gordon equation with a scalar otential x + m 0 + U(x) ^(x) =0: (.)

13 The covariant Wigner function is the four-dimensional Wigner transform of the covariant density matrix (x; y) = h^(x; y)i: W(x; ) = Z d 4 ye iy (x; y) = Z D d 4 ye iy ^ x + y ^+ x E y : (.) To derive the kinetic equations for the scalar Wigner function, we calculate the second-order y of the covariant density oerator, and then emloy the Klein-Gordon equation (.) and its adjoint. After taking the ensemble average and erforming the Wigner transform we obtain two comlex kinetic equations: x + m 0 + U x + m 0 + U x i@ x W(x; ) =0; (.a) x + + i@x W(x; ) =0: (.b) Since the scalar Wigner function is real, adding and subtracting these two comlex equations yields two real equations of the tye (.). After reinstating h the corresonding oerators ^G and ^F are given by ^G(x; ) =h@ x +Im ^M (x; ) ; (.4a) ^F (x; ) = + h x+re ^M (x; ) ; (.4b) where the mass oerator ^M is dened as ^M (x; ) =m 0+^ e (x; )+i^ o (x; ) ;! U(x) ; h4 ^ e (x; ) = cos ^ o (x; ) = sin h4! U(x) : (.5a) (.5b) (.5c) Here the triangle oerator 4 is dened as 4 where the coordinate x acts only on the scalar otential U(x) acts only on the Wigner function. B. Semiclassical exansion In the general quantum situation the articles have no denite mass due to quantum uctuations around their classical mass shell and collision eects in the medium. In the

14 situation here with only an external otential this is illustrated by the mass oerator ^M. Only in the classical limit h! 0itreduces to the quasiarticle mass Re ^M 0 (x; ) =m (x)=m 0+U(x); (.6a) Im ^M 0 (x; ) =0: (.6b) In this case the constraint equation reduces to the on-shell condition m (x) W 0 (x; ) =0 (.7) for the classical covariant Wigner function W 0. The classical transort equation arises from the general transort equation at rst order in h. The rst order contribution to the mass oerator is Re ^M (x; ) =0; Im ^M (x; ) =hm(x)(@ x m(x))@ ; (.8) and we obtain the covariant Vlasov x + m(x)(@ x m(x))@ W0 (x; ) =0; (.9) with a Vlasov force term induced by x-deendent eective mass term. For scalar elds there is no rst order quantum correction to the oerator F in (.4), and from the zeroth order term we obtain the mass-shell condition for the rst order Wigner function: m (x) W (x; ) =0: (.0) This discussion holds universally for arbitrary otentials. If, for instance, U(x) is generated by the scalar eld ^(x) itself in Hartree aroximation, Z U(x) = C+h^(x)^ + (x)i= C+ d 4 W(x; ) ; (.) () 4 with a mass arameter C and a couling strength, this model rovides a useful tool for a dynamical descrition of sontaneous symmetry breaking [8]. 4

15 C. The -dimensional dynamical oerators We now erform the energy average of the covariant transort and constraint equations (.) and construct the hierarchy (.0) of moment equations. The rst ste is the double exansion of the tye (.9) for the covariant dynamical oerators ^G(x; ) and ^F (x; ): 8 h@ t for m =;n=0 Here ^G mn (x; ;t)= ^F mn (x; ;t)= : 8 : hr x ^ o for m = n =0 n! i n! ih n (@ n t ^ o ) for m =0;n6=0even n ih (@ n t ^ e ) for m =0;n odd 0 else h for m =;n=0 x+ +m 0+^ e for m = n =0 n! ih i n! n (@ n t ^ e ) for m =0;n6=0even n ih (@ n t ^ o ) for m =0;n odd 0 else. h ^ e (x; ;t) = cos ^ o (x; ;t) = sin U(x;t); r! xr h r! xr U(x;t) (.a) (.b) (.a) (.b) are the three-dimensional analogues of the covariant oerators ^ e and ^ o in Eq. (.5). Again, the satial gradients act only on U(x;t), while the momentum gradients act on the equal-time Wigner functions (i.e. on the energy moments w j (x; )). The three-dimensional dynamical oerators ^Gmn (x; ) and ^Fmn (x; ) must now be combined with the coecients c mn ij to obtain the dynamical oerators ^g ij (x; ) and ^f ij (x; ) which are needed in the transort and constraint hierarchies. This is done in Aendix B. 5

16 D. Minimal truncation The resulting transort hierarchy is truncated at I t = M =, the constraint hierarchy at I c = M =0. This yields the following equations for w 0 (x; ), w (x; ), and w (x; ): ^g 00 w 0 +^g 0 w =0; ^g 0 w 0 +^g w +^g w =0; ^f 00 w 0 + ^f 0 w + ^f 0 w =0: (.4a) (.4b) (.4c) The dynamical oerators ^g ij and ^f ij are given in Aendix B and Eqs. (.). Reexressing w j in terms of the energy moments W j from Eq. (.), w 0 (x; ) = W(x;;t); (.5a) w (x; ) = s W (x;;t); w (x; ) = (.5b) s 5 W (x; ;t) W(x;;t) ; (.5c) Eqs. (.4) can be rewritten t W (x; t W (x; ;t)= h W (x; ;t)= 4 r x r x h^ o(x;;t) h^ o(x;;t) t r x + +m 0+^ e (x;;t) W (x;;t)+ (@ t^ e (x;;t)) W (x; ;t);! W(x;;t): (.6a) (.6b) (.6c) Please note that all owers of the cuto cancel in the nal exressions as they should. The two transort equations (.6a,b) do not decoule, not even in the classical limit h! 0. To achieve decouling one must return to the covariant equations in Sec. III A and study their semiclassical limit as given in Sec. IIIB before erforming the energy average. Then the mass-shell condition (.7) can be used to rewrite all higher order energy moments in terms of the zeroth order moment as exlained in the Introduction, Eq. (.4). With this information the constraint (.6c), in the limit h! 0, becomes trivial, 6

17 W (x; ;t)=e (x;t)w(x;;t); E (x;t)= +m (x;t); (.7) while the two transort equations (.6a) and (.6b) become identical and can be written in the form of a Vlasov equation for the charge density (see Sec. t W (x; ;t)+ E r x r x E r! W (x;;t)=0: (.8) The reason why the information contained in Eq. (.4) cannot be easily recovered directly from the -dimensional transort and constraint equations is that in their derivation, through Eq. (.5), we made heavy use of artial integration with resect to the energy. In the classical limit this has the unfortunate eect of sreading the information contained in the on-shell condition over the whole innite hierarchy of -dimensional constraint equations. Although we have always talked about Eq. (.6c) as a \constraint equation" it is clear that, as far as solving the minimal subset (.6) of equal-time kinetic equations is concerned, this terminology is only adequate in the classical limit h! 0. In general quantum situations it is a second order artial dierential equation for the lowest order moment W (x; ;t) which must be solved as an initial value roblem together with the rst order artial dierential equations (.6a) and (.6b). E. Higher order moment equations The next higher moment w is determined by the third equation in the transort hierarchy and the second equation in the constraint hierarchy, ^g 0 w 0 +^g w +^g w +^g w =0; ^f 0 w 0 + ^f w + ^f w + ^f w =0: (.9a) (.9b) With the dynamical oerators from Aendix B and Eqs. (.) and w (x; ) = s 7 5 W (x; ;t) W (x;;t) (.0) we obtain (using Eq. (.6a)) 7

18 @ t W (x; ;t)= h 4 h W (x; ;t)= 4 r x h^ t^ o (x;;t) t r x + +m 0+^ e (x;;t) W (x;;t)+(@ t^ e (x;;t)) W (x; ;t)! W (x;;t) (.a) h (@ t^ o (x;;t)) W (x; ;t): (.b) By substituting (.9b) into (.9a) and taking into account the following commutators: [ ^G0 ; ^F00 ]= ^G0 ^F0 ; (.a) [ ^G00 ; ^F00 ]= ^F0 ^G0 ^G0 ^F0 ; (.b) [ h ; ^ e=o ]= e=o@ + h e=o ; (.c) [r x ; ^ e=o ]=r x^ e=o + h r x^ o=e r x ; (.d) [ ; ^ e=o ]= hr x^ o=e + h 4 r x ^ e=o ; (.e) the third transort equation (.9a) can be rewritten in terms of the rst transort equation (.4a) as ^f 00 (^g 00 w 0 +^g 0 w )=0: (.) This imlies that the transort equation (.9a) for w is redundant. The third-order moment W is comletely determined in terms of the solutions of the minimal subgrou (.6) by the constraint equation (.b). It arises from Eq. (.9b) by noting that ^f is a constant, ^f = C = 5 7 (see Aendix B), and solving for w : w = C ^f0 w 0 + ^f w + ^f w : (.4) Note that, in contrast to (.6c), Eq. (.b) does not require solving a artial dierential equation because everything on the r.h.s. is known from the solution of Eqs. (.6). The above rocedure can be extended to all higher order moment equations, by reeatedly using the commutators listed in (.). In general one nds that a transort equation Xi+ j=0 ^g ij w j =0 (.5) 8

19 with i can be re-exressed in terms of the rst (i Xi j=0 0 ^f i j+ X k=0 ) transort equations as ^g jk w k A =0: (.6) Thus, excet for the rst two, all transort equations are redundant. The higher order moments w i with i>can be comuted from their constraint equations with the constant C i given by w i = Xi C i j=0 ^f i ;j w j ; (.7) C i ^f i ;i = i(i ) (i ) q (i )(i +) : (.8) IV. APPLICATION TO QED A. Scalar QED In this Section we discuss the alication of the general formalism develoed in Sec. II to QED. Since some of the equations will be rather lengthy wewill economize on the notation by droing all factors of h. The latter are correctly given in Refs. [,7] to which we refer in case of need. We begin with the case of scalar QED with external electromagnetic elds. In Ref. [] we discussed the semiclassical transort equations for this theory by energy averaging the semiclassical limit of the covariant transort equations. In this subsection we will derive the general equal-time quantum transort equations by erforming the energy average without any aroximations. In the following subsection the result will be comared with the corresonding equations derived by directly Wigner-transforming the equations of motion for the equal-time density matrix [8]. In scalar QED the scalar eld obeys the Klein-Gordon equation (@ + iea (x))(@ + iea (x)) + m (x) =0; (4.) 9

20 Following an analogous rocedure as in Sec. IIIA (see [] for details) one derives two covariant kinetic equations of tye (.) for the covariant Wigner function Z " Z = W(x; ) = d 4 ye *^ iy x + y ex ie ds A(x + sy)y ^ + x = y + : (4.) The corresonding covariant dynamical oerators ^G and ^F are given by ^G(x; ) =^ (x; ) ^D (x; ) ; (4.a) ^F (x; ) = 4^D (x; ) ^D (x; ) ^ (x; )^ (x; )+m ; (4.b) F A ^ (x; ) = ^D (x; ) =@ ie e Z A is the electromagnetic eld tensor. dssf (x is@ ; (4.c) ds F (x is@ : (4.d) The structure of the equal-time transort theory for scalar QED is very similar to that for a scalar otential U(x) which we considered in the revious section. The dierence resides solely in the exressions for the dynamical oerators ^G mn and ^F mn. In Aendix C we rovide the double exansions (.9) for the basic oerators ^ and ^D as well as for ^G and ^F. In terms of the covariant Wigner function the charge current j and the energymomentum tensor T are exressed as j (x; ) =e W(x; ) ; T (x; ) = W(x; ) : (4.4a) (4.4b) After erforming the energy average these equations translate into relations between the rst three moments w 0 ;w ;w and the equal-time hase-sace distributions for the scalar density W (x; ;t), the charge density (x; ;t), and the energy density (x; ;t): w 0 (x; ;t)= W(x;;t) ; s w (x; ;t)= e (x;;t) ; s w (x; ;t)= 5 (x; ;t) W(x;;t) 0 (4.5a) (4.5b) : (4.5c)

21 The charge current density j(x; ;t) and the momentum density P (x; ;t) can be exressed in terms of W and as j(x; ;t)=ew(x;;t); P(x;;t)= e (x;;t): (4.6) The subgrou (.4) determining the moments w 0, w and w can thus be equivalently rewritten as transort and constraint equations for W, and : ^d t +(^d t^ 0 +^^d)w =0; e (4.7a) ^d t + e ( ^d t^ 0 + ^^d + ^D) +(^0^D+^^I ^dt^a+ ^G^d)W =0; (4.7b) = 4 ( ^d t ^d ) (^ 0 ^ )+m + ^A W ^ 0 e ; (4.7c) The exressions of ^g ij and ^f ij were obtained via the relations given in Aendix B from the equal-time oerators (C) in Aendix C. We have also used the commutators [ ^d t ; ^ 0 ]= ^D; (4.8a) [ ^d t ; ^A] =^E: (4.8b) Due to the line integrals over s in the oerators ^ and ^D which guarantee the gauge invariance of the formalism [], the equal-time oerators ^Gmn and ^Fmn for QED are much more comlicated than Eqs. (.) for the case of a scalar otential. This is the origin of the more comlicated structure of the minimal subgrou (4.7) of -dimensional kinetic equations. Please note the sequence of the oerators in Eqs. (4.7): in articular the generalized time derivatives ^d t act on everything following them. Eqs. (4.7) are thus much more intricately couled than the corresonding equations (.6) for the ure scalar case. It is instructive to integrate Eqs. (4.7) over to obtain equations of motion for the corresonding sace-time t (x;t)+rj(x;t)=0; t (x;t)+rp(x;t) E(x;t)j(x;t)=0; (4.9b) (x;t)= Z! d h () 4 (@ t rx)+ +m W(x;;t): (4.9c)

22 The rst two equations exress the conservation of electric charge and of energymomentum while the last equation gives the energy density in terms of the scalar equaltime hase-sace density W (x; ; t) including quantum corrections. In Eq. (4.9b) P (x; t) = R (d =() )P (x; ;t)isthe momentum density in coordinate sace, and the last term describes the conversion of eld energy into mechanical energy by the work done by the electric eld on moving charges. Let us now consider the next two equations in the hierarchy, the transort and constraint equations (.9) for the third-order moment w. Again the constraint equation can be directly solved for w in terms of the solutions w 0 ;w ;w of Eqs. (4.5), (4.7) (see Eq. (.4)). Our task is to show that the transort equation for w is redundant. To this end we substitute the constraint (.9b) into the transort equation (.9a) and use the scalar QED analogue of the commutators (.), namely (4.8) and [ ^d t ; ^B] =^F; (4.0a) [ ^d t ; ^] = I+r x^ 0 ; (4.0b) [^I; ^] = r x ^G; (4.0c) [^d; ^ 0 ]=r x^ 0 ; [ ^d t ; ^d] = er x r Z [^; ^ 0 ]= ie Z ds E(x + isr ;t); ds s E(x + isr ;t)+er x Z ds s E(x + isr ;t)r ; (4.0d) (4.0e) (4.0f) as well as the identity Z to rewrite (.9a) in the form ds is + r xr s r x r E(x + isr ;t)=0; (4.) 4 ^f 00 (^g 00 w 0 +^g 0 w )+ ^f 0 (^g 0 w 0 +^g w +^g w )=0: (4.) Note that in the derivation of the commutators (4.0) we used Maxwell's ~ F =0; (4.)

23 for the dual eld strength tensor ~ F = F. The identity (4.) (which has no analogue in a theory without gauge invariance) is roven by exanding the electric eld E(x + isr ;t) = e isrxr E(x;t) and integrating term by term. Eq. (4.) exresses the transort equation for w in terms of the transort equations for w and w. This roves that it is redundant. >From the comarison of (4.) with (.) we observe that in the case of scalar QED the third transort equation is exressed in terms of the rst and second transort equations, while only the rst one occurs in the case of scalar otentials. This dierence results from the line integrals which occur in the gauge theory. In a transort theory with gauge invariance, the kinetic momentum is not but ^. The energy average of the zeroth comonent of the second term in (4.c) yields the equal-time oerator ^ 0 given in (C), and ^ 0 = ^f 0 in turn generates the coecient in front of the second bracket in (4.). In a transort theory without gauge freedom all coecients ^f i;i+ vanish due to the absence of linear terms in in the covariant constraint equation. We have not had the atience to carry the above considerations to higher orders in the energy moments. The corresonding calculations become extremely messy. Based on the exerience with ure scalar theory we exect all higher order transort equations to be redundant, but we have failed to discover a simle calculational technique which ermits us to rove this in an elegant way. B. Comarison with the Feshbach-Villars aroach In Ref. [8] a set of equal-time transort equations for scalar QED was derived by directly Wigner transforming the equations of motion for the (non-covariant) equal-time density matrix. Since the Klein-Gordon equation (4.) contains a second-order time derivative, its direct translation into -dimensional hase sace does not lead to a sensible transort equation which should have only rst-order time derivatives. Therefore the -dimensional aroach exloits the Feshbach-Villars reresentation [4] of the eld equations of motion

24 which contains only rst-order time derivatives. The rice to ay (in addition to the loss of manifest Lorentz covariance) is the introduction of an auxiliary eld which results in a rather comlicated matrix structure of the scalar Wigner function. For the energy averaging method advocated here there is no such roblem. In this case second-order time derivatives aear only in the constraint hierarchy, and the transort hierarchy contains only rst-order time derivatives. Exressing the equal-time density oerator (x; y; t) of the scalar eld in terms of the Feshbach-Villars sinors, erforming a -dimensional Wigner transformation and using the Klein-Gordon equation in Feshbach-Villars form, the authors of [8] obtained the following couled equations for the 4 sinor comonents f i (x; ;t) (i = 0;;;) of the Wigner function: m ^d t f 0 = ^^d(f + f ) ; (4.4a) m ^d t f = ( 4 ^d ^ )(f + f )+m f ; (4.4b) m ^d t f =( 4^d ^ )f +^^df 0 m f ; (4.4c) m ^d t f = ( 4 ^d ^ )f ^^df 0 : (4.4d) In Ref. [8] these equal-time hase-sace distributions were related to the hysical hasesace densities ; j;, and P as follows [8,5]: = ef 0 ; = m f + f + mf ; (4.5a) (4.5b) j = e m f + f ; (4.5c) P = f 0 : (4.5d) Unlike the sinor decomosition for sinor QED [6,7] where each comonent of the Wigner function has a denite hysical meaning, there is no obvious hysical interretation for the comonent f in scalar QED. The comarison of the charge currents j in Eqs. (4.5c) and (4.6) leads directly to the identication 4

25 W = f + f : (4.6) m Using the relations (4.5) and (4.6) one nds from a suitable linear combination of the kinetic equations (4.4) the following equations of motion for the hysical hase-sace distributions and, ^d t + ^^dw =0; e ^d t + e ^^d = 4 ^d t ^d 4 ^ ^dt + ^d t A W =0; (4.7a) (4.7b) ^d + (^ + )+m W; (4.7c) as well as the following relation between the unhysical comonent f and the scalar density: f = ^d t W : (4.8) Comaring Eqs. (4.7) from the equal-time Feshbach-Villars aroach with the corresonding Eqs. (4.7) from the covariant aroach using our energy-averaging rocedure, one realizes that they have avery dierent structure. The dierence is even more obvious when looking at the corresonding equations for the sace-time densities which are obtained by integrating Eqs. (4.7) over t t (x;t)+rp(x;t) (x) = Z () t 4 E(x;t)j(x;t) r x h rx t + + m W (x; ; t): Z d W (x; ;t)=0; () (4.9b) (4.9a) (4.9c) The rst equation describes charge conservation and agrees with our Eq. (4.9a). When comaring Eqs. (4.9b) and (4.9b) one sees that the energy-momentum conservation law derived from the Feshbach-Villars aroach features a strange additional term of unknown hysical origin. Similarly, when comared to Eq. (4.9c), Eq. (4.9c) contains a strange factor in front of the Lalace oerator which breaks Lorentz invariance. We believe that these discreancies indicate a technical roblem with the Feshbach-Villars aroach of 5

26 Ref. [8]. Whether the roblem resides in the derivation of the transort equations (4.4) or in the ostulated relations (4.5) between the Feshbach-Villars sinor comonents f i and the hysical densities could not be claried. Only in the case of satially constant electromagnetic elds (which to our knowledge is the only situation to which these equations have so far been alied [8,]) the two sets of equations (4.9) and (4.9) coincide. If we further neglect the magnetic eld B, we derive two decouled ordinary dierential equations in time for W and : d t =0; d t+4( +m )d t +4eE W =0: (4.0) These equations are identical in both aroaches. The rst equation exresses charge conservation in a homogeneous electric eld, and the second equation is the well-known equation of motion for the charge current j [,] which, according to the relation j = e n, +m governs the time evolution of the article density n by air roduction in the electric eld. This latter quantity thus comes out the same in both aroaches. C. Sinor QED The case of sinor QED has been discussed in the context of the energy averaging method in Refs. [,7,8]. The discussion resented in those aers has, however, focussed entirely on the equal-time kinetic equations for the lowest energy moment of the covariant Wigner function. Here we will reformulate the roblem in terms of the equal-time hierarchy of energy moments as introduced in Sec. II and also discuss the equations of motion for the higher order moments. Let us briey review the relevant technical stes. We begin by erforming a sinor decomosition [6] of the covariant Wigner function, searating in a second ste exlicitly the temoral and satial arts [7] of the covariant sinor comonents: W (x; ) = F(x; )+i 5 P (x; )+ V (x; )+ 5 A (x; )+ 4 S (x; ) (4.a) 6

27 = h 0 F 0 (x; )+ 5 0 F (x; )+i 5 F (x; )+F (x; ) 4 5 G 0 (x; ) G (x; )+i 5 G (x; )+ 5 0 G (x; ) i : (4.b) Inserting the decomosition (4.a) into the covariant (VGE) equation of motion [6] for the Wigner function W (x; ), (^ (x; )+ i ^D (x; )) m W (x; ) =0; (4.) with ^ (x; ) and ^D (x; ) from Eqs. (4.c,d), and searating real and imaginary arts one arrives at two grous of couled covariant kinetic equations of tye (.), see Eqs. (74,75) of Ref. []. Performing the energy average then leads to two grous of equal-time kinetic hierarchies (with 6 such hierarchies of equations in each grou) for the energy moments of the 6 covariant sinor comonents. Since for sinor QED M = 0, minimal truncation of these 6 hierarchies according to Sec. II B results in one transort equation from each of the 6 transort hierarchies and no constraint equations. The minimal subgrou of equal-time kinetic equations thus consists only of the 6 transort equations for the 6 zeroth-order energy moments f i (x; ;t) and g i (x; ;t) of the covariant sinor comonents F i (x; ) and G i (x; ) (i =0;;;): ^d t f 0 + ^dg =0; ^d t f + ^dg 0 +mf =0; (4.a) (4.b) ^d t f +^g mf =0; (4.c) ^d t f ^g =0; (4.d) ^d t g 0 + ^df ^ g =0; (4.e) ^d t g + ^df 0 ^ g 0 +mg =0; (4.f) ^d t g + ^d g +^f mg =0; (4.g) ^d t g ^d g ^f =0: (4.h) These 6 equations are identical with the BGR equations derived by Bialynicki-Birula, Gornicki and Rafelski [7] by Wigner transforming the equations of motion for the equal- 7

28 time density matrix. They determine the dynamics of the zeroth-order energy moments. The three-dimensional dynamical oerators occurring in these equations are identical with the ones arising in scalar QED and are given in Aendix C. The rst-order moments f i (x; ) and g i (x; ) satisfy 6 transort equations derived [] from the rst energy moment of the covariant transort equations (.a), ^dt f 0 + ^dg + ^Df0 + Ig =0; (4.4a) ^dt f + ^dg 0 + mf + ^Df + Ig 0 =0; ^dt f + ^g ^dt f mf + ^Df +^Gg =0; (4.4b) (4.4c) ^g + ^Df ^Gg =0; (4.4d) ^dt g 0 + ^df + ^ g + ^Dg0 + ^If ^G g =0; (4.4e) ^dt g + ^df 0 ^dt g + ^d g + ^f ^dt g ^d g ^ g 0 + mg + ^Dg + ^If0 ^G g0 =0; (4.4f) mg + ^Dg + ^I g +^Gf =0; (4.4g) ^f + ^Dg ^I g ^Gf =0; (4.4h) and 6 constraint equations derived [] from the zeroth energy moment of the covariant constraint equation (.b): f 0 = ^g ^ 0 f 0 + mf ; (4.5a) f = ^g 0 ^ 0 f ; (4.5b) f = h ^dg ^ 0 f ; (4.5c) f = h ^dg ^ 0 f + mf 0 ; (4.5d) g 0 = h ^d g + ^f ^ 0 g 0 + mg ; (4.5e) g = h ^d g 0 + ^f 0 ^ 0 g ; (4.5f) g = h ^df + ^ g ^ 0 g ; (4.5g) g = h ^df ^ g ^ 0 g + mg 0 : (4.5h) 8

29 A discussion similar to that in scalar QED reveals [8] that the transort equations (4.4) are not indeendent of the BGR equations (4.) and the contraint equations (4.5). For instance, using Eqs. (4.5), the transort equation for f 0 can be exressed as an oerator combination of the transort equations for f 0 and g : ^ 0 ^dt f 0 + ^dg ^ ^dt g + ^df 0 ^ g 0 +mg =0: (4.6) Therefore, the rst-order moments are fully determined in terms of the solutions of the BGR equations (4.) for the zeroth-order moments by solving the constraint equations (4.5). Again, we have not been able to nd a simle roof that the same is generally true for all higher order energy moments, and we stoed here. We do, however, believe that such a roof must exist [9], and that therefore all higher order energy moments can be directly comuted from the solutions of the BGR transort equations by solving the constraint hierarchy. It was shown in Ref. [] that in the classical limit the simle algebraic relation (.4) changes the structure of the constraints (4.5) for the rst-order moments and turns them into additional constraints for the zeroth-order moments (i.e. the equal-time Wigner functions). These extra constraints reduce the number of indeendent zeroth-order moments from 6 in the quantum case to 4 in the classical limit. They are thus extremely imortant. As a result the BGR equations reduce to two decouled Vlasov-tye transort equations for the charge and sin distribution functions. In the general quantum case there are no such extra constraints on the equal-time Wigner functions [8]. One must solve all 6 couled transort equations (4.), but these solutions then fully determine also all higher order moments. These higher order moments have imortant hysical meaning: the rst-order moment off 0 (x; ), for instance, describes the energy distribution in hase-sace. With the hel of the contraint (4.5a) it is given by (x; ;t)=tr Z de E 0W(x; ) = s f 0(x;;t) = mf (x; ;t) ^ 0 f 0 (x;;t)+^g (x;;t) : (4.7) 9

30 V. CONCLUSIONS We have resented a universal method for the construction of equal-time quantum transort theories from the covariant quantum eld equations of motion. It is based on energy averaging the covariant kinetic equations for the covariant Wigner oerator (which is the Wigner transform of the covariant, \two-time" density matrix) and its energy moments. This rocedure yields a hierarchy of couled transort and constraint equations for the energy moments of the covariant Wigner function, the so-called equal-time Wigner functions. We showed how, in the mean-eld aroximation, this hierarchy can be truncated at a rather low level, requiring the solution of only a small number of equal-time transort equations, and how the higher order energy moments (higher order equal-time Wigner functions) can be constructed from these solutions via constraints. The major advantage of the equal-time formulation of (quantum) transort theory is that the resulting transort equations can be solved as initial value roblems, with boundary values for the equal-time Wigner functions at t = which can be calculated from the elds at t =. This is not the case for the covariant transort equations and the covariant Wigner function. The resent aer thus rovides an essential ste in the direction of ractical comutations of the dynamics of relativistic quantum eld systems out of thermal equilibrium in the language of transort theory, i.e. in a hase-sace oriented aroach. The method resented here imroves uon revious aroaches by being much more systematic: we did not just focus on the lowest energy moments (which contain only a small fraction of the information contained in the covariant Wigner function), but we discussed and showed how to solve the comlete hierarchy of moment equations. We had already before demonstrated for sinor QED that the non-covariant three-dimensional aroach (which starts directly from the equal-time density matrix) yields an incomlete set of equal-time transort equations. In this aer we also discussed the case of scalar QED and again discovered serious roblems with the direct three-dimensional aroach based on the Feshbach-Villars formulation. We conclude that the only safe way of deriving a correct and 0

31 comlete set of equal-time quantum transort equations is by starting from the covariant formulation and taking energy moments of the covariant kinetic equations. The method can be generalized in a straightforward way to other tyes of interactions [6{8], including non-abelian gauge interactions [9]. The structure of the hierarchy of equal-time quantum kinetic equations deends on the structure of the covariant eld equations from which one starts. For scalar or vector theories with second order time derivatives one has to solve a couled set of three equations for the three lowest energy moments, two resulting from the equal-time transort hierarchy and one stemming from the constraint hierarchy. For sinor theories with only rst order time derivatives in the eld equations one ends u with only one equal-time transort equation for the lowest energy moment of each sinor comonent of the Wigner function. All higher order energy moments can be determined from the solutions of these equations by solving constraints. Imortant further simlications occur in the classical limit h! 0: then all higher order energy moments can be exressed algebraically in terms of the zeroth energy moment, and the number of equations is drastically reduced. For scalar theories one obtains just one Vlasov-tye equation for the on-shell charge distribution function. For sinor theories one obtains two decouled Vlasov-tye equations (one scalar and one vector equation) for the on-shell charge and sin density distributions in hase-sace. Again the only systematic way of deriving the constraints leading to these simlications is by energy averaging the (classical limit of the) covariant transort equations. All results in this aer were derived in the mean eld aroximation, i.e. in the collisionless limit. It is generally known that including collision terms in the covariant transort equations leads to the aearance of non-localities in time (\memory eects") in the equaltime transort equations [0]. It is not inconceivable that these memory eects lead to serious comlications for the truncation of the equal-time transort hierarchy. This is certainly an interesting and dicult roblem for future studies.