Non-equilibrium perturbation theory
for spin-tex2html_wrap_inline851 fields

Ian D. Lawrie
Department of Physics and Astronomy, University of Leeds

The problem

In this talk, I describe a partial resummation of perturbation theory that I think is useful when one wishes to follow the evolution in time of a system that is not close to equilibrium. In the simplest case of a scalar field theory, ordinary perturbation theory uses bare time-ordered propagators of the form
together with anti-time-ordered and Wightman functions of a similar kind. In this expression (which applies to a spatially homogeneous system), the mode functions tex2html_wrap_inline853 for particle modes of momentum k are solutions of the equation
where the differential operator tex2html_wrap_inline857 is that associated with the quadratic part of the action tex2html_wrap_inline859. The numbers tex2html_wrap_inline861 are the occupation numbers of these modes that characterize an initial state set up, say, at t=0. These occupation numbers are problematic in the following sense. If there are interactions that cause scattering, one would expect the state of the system at time t to be characterized by quasiparticle excitations of finite width, say tex2html_wrap_inline867, whose occupation numbers tex2html_wrap_inline869 evolve with a relaxation time of the order of tex2html_wrap_inline871. Thus, propagators of the form (1) provide a useful description only for times tex2html_wrap_inline873.

The purpose of the resummation I will describe, then, is to obtain a modified perturbative scheme in which the lowest-order propagators take at least approximate account of the evolution of the non-equilibrium state, so as to be useful over extended periods of time. Broadly, the aim is to describe this state in terms of its own quasiparticle excitations rather than single-particle excitations of the vacuum. This means finding an optimal means of splitting the action I into an unperturbed part tex2html_wrap_inline877 that (loosely speaking) describes free quasiparticles, and an interaction tex2html_wrap_inline879 that accounts for their interactions. Since only Gaussian path integrals can be computed exactly, tex2html_wrap_inline877 should be quadratic in the fields. I will therefore write
where tex2html_wrap_inline877 is the part of the original action I that is quadratic in the fields and tex2html_wrap_inline887 is the rest of it. A partial resummation of the perturbation series for the full 2-point functions is effected by asking the counterterm tex2html_wrap_inline889 that appears in tex2html_wrap_inline879 to cancel some part of the loop contributions to the self energy. The central issue to be addressed is how to construct a suitable differential operator tex2html_wrap_inline889 or, equivalently, a lowest-order action tex2html_wrap_inline877 and the propagators associated with it. I will first review how this can be done for a scalar field [1] and then describe the extent to which I have been able to do something similar for a spin-tex2html_wrap_inline851 field.

Solution for scalar fields

Here, I consider explicitly the usual tex2html_wrap_inline899 theory, defined by the Lagrangian density
but the strategy I describe should apply equally well when the scalar field tex2html_wrap_inline901 is embedded in some larger theory. I consider only a spatially homogeneous system, but allow for a time-dependent mass m(t) which drives this system away from equilibrium. For example, the theory of a scalar field in a Robertson-Walker universe can be cast in the form of a Minkowski-space theory with time-dependent mass. To be concrete, suppose that an initial state of thermal equilibrium with inverse temperature tex2html_wrap_inline905 is set up at time t=0. Then the standard closed-time-path formalism yields a path integral weighted by the action
where the path-integration variables tex2html_wrap_inline909, tex2html_wrap_inline911 and tex2html_wrap_inline913 live on a closed contour in the complex time plane. The Euclidean action tex2html_wrap_inline915 (which uses m(0)) represents the initial density matrix. In this theory, there is a tex2html_wrap_inline919 matrix of 2-point functions tex2html_wrap_inline921, with a, b = 1, 2, 3, but I will need to think explicitly only about the real-time functions, with a, b = 1, 2. For the real-time part of the action, I want to construct a lowest-order version tex2html_wrap_inline927, where, after a spatial Fourier transform, the differential operator tex2html_wrap_inline929 is
Subject to several constraints (about which I will be more explicit when I discuss spinor fields in detail), the most general choice for tex2html_wrap_inline929 is
where tex2html_wrap_inline933, tex2html_wrap_inline935 and tex2html_wrap_inline867 are real functions yet to be determined. Of course, the counterterm tex2html_wrap_inline889 can be read off from (8) and (9).

The tex2html_wrap_inline941 matrix of propagators tex2html_wrap_inline943 is the solution (subject to suitable boundary conditions) of
Suppressing the spatial momentum k, which is omnipresent, this solution can be written in terms of a single complex function h(t,t') as
where tex2html_wrap_inline949 and tex2html_wrap_inline951. The function h is
We see that one of the undetermined functions, tex2html_wrap_inline867, can be interpreted as a quasiparticle width. The quasiparticle energy tex2html_wrap_inline957 is a solution of
Finally, the function tex2html_wrap_inline959, which I hope to interpret in terms of time-dependent occupation numbers, is a solution of

To give substance to the scheme I have described so far, a prescription is needed for determining the three functions tex2html_wrap_inline933, tex2html_wrap_inline935 and tex2html_wrap_inline867 introduced in (9). To this end, define the tex2html_wrap_inline941 self energy matrix tex2html_wrap_inline969 by
This self energy has contributions from the counterterm tex2html_wrap_inline889 and from loop diagrams:
The general strategy is to optimize tex2html_wrap_inline943 as an approximation to the full two-point functions tex2html_wrap_inline975 by arranging for tex2html_wrap_inline889 to cancel some part of tex2html_wrap_inline979. Clearly, since tex2html_wrap_inline979 is non-local in time, only a partial cancellation can be achieved. Various prescriptions might be possible; perhaps the most obvious is the following. Express tex2html_wrap_inline969 in terms of the average time tex2html_wrap_inline985 and the difference (t-t') and Fourier transform on (t-t'). The components of tex2html_wrap_inline991 contain at most one time derivative, so the self energy can be decomposed into contributions that are even and odd in the frequency:
Generalized gap equations to be solved for tex2html_wrap_inline933, tex2html_wrap_inline935 and tex2html_wrap_inline867 can now be obtained by requiring
which amounts to an on-shell renormalization prescription.

These gap equations provide exact implicit definitions of tex2html_wrap_inline933, tex2html_wrap_inline935 and tex2html_wrap_inline867, but they cannot, of course, be exactly solved. If the perturbative expansions for tex2html_wrap_inline1005 and tex2html_wrap_inline1007 are truncated at some finite order, one obtains concrete expressions for them in terms of the propagators tex2html_wrap_inline943. These truncated gap equations, together with equation (13) for the quasiparticle energy and (14) for the function tex2html_wrap_inline959 form a closed system that one might try to solve numerically. It is to some extent illuminating to establish a connection with kinetic theory through some further approximations.Suppose that the gap equations are truncated at two-loop order - the lowest order that yields a nonzero quasiparticle width tex2html_wrap_inline867. Then, assuming sufficiently weak coupling and sufficiently slow time evolution, propagators inside the loop diagrams can be approximated by taking tex2html_wrap_inline1015 and the limit tex2html_wrap_inline1017. Then, with quasiparticle occupation numbers tex2html_wrap_inline869 defined by
a time-derivative expansion of (14), yields the Boltzmann-like equation


Partial solution for spin-tex2html_wrap_inline1021 fields

I will now describe an attempt to apply a similar analysis to the case of spinor fields. This is somewhat more complicated. For reasons that will become apparent, I have achieved so far only a restricted version of the solution and even this has features for which I have no clear interpretation. For purposes of illustration, I will have in mind a simple theory of the form
but again the strategy should apply to more general theories containing spinors. If the tex2html_wrap_inline901 particles are heavy enough to decay into two tex2html_wrap_inline1025 particles, then the one-loop self energy for tex2html_wrap_inline1025 has an imaginary part which gives the tex2html_wrap_inline1025 a finite thermal width. As before, I plan to construct a lowest-order action tex2html_wrap_inline1031 such that the propagator matrix S, which is a solution of
incorporates this thermal width and evolving quasiparticle occupation numbers. I continue to deal with a spatially homogeneous system and will henceforth suppress all momentum arguments. The first step towards constructing a suitable differential operator tex2html_wrap_inline929 is to obtain some general properties of the full two-point functions tex2html_wrap_inline1037 and to insist that these be shared by the lowest-order propagators S(t,t'). The form of tex2html_wrap_inline929 is then constrained by the requirement that (23) admit a solution having these properties.

The full two-point functions are
where tex2html_wrap_inline1043 and tex2html_wrap_inline905 are spinor indices. The Wightman function tex2html_wrap_inline1047 can be written as
To simplify matters, I will assume that the chemical potential for tex2html_wrap_inline1025 particles is zero. Then, using the hermiticty of the density operator, it is simple to show that
where, for any Dirac matrix tex2html_wrap_inline1051, I define tex2html_wrap_inline1053. It would be helpful if, as in the scalar theory, the second Wightman function tex2html_wrap_inline1055 could be expressed in terms of the same matrix tex2html_wrap_inline1057. To this end, I will construct tex2html_wrap_inline877 so as to be CP invariant. This does not imply that the whole theory need be CP invariant - only that CP-violating parts of the full two-point functions will not be resummed. For the purpose of establishing the structure of tex2html_wrap_inline877, however, I suppose that the full two-point functions tex2html_wrap_inline1063 do belong to a CP-invariant theory. In that case, it turns out that
The CP conjugate of a Dirac matrix tex2html_wrap_inline1051 is tex2html_wrap_inline1067, where C is the charge conjugation matrix and tex2html_wrap_inline1071 denotes the transpose.

After a spatial Fourier transform, the full CP-invariant two-point functions are now given by
and I will require the unperturbed propagators S(t,t') associated with tex2html_wrap_inline877 to have the same structure:
The fact that the Wightman functions have unique values at t=t' implies that
and the canonical anticommutation relation tex2html_wrap_inline1079 implies that
I now look for a differential operator tex2html_wrap_inline1081 such that the equation (23) will admit a solution of the form (29). The tex2html_wrap_inline1083 arises from the derivative of tex2html_wrap_inline1085 and, if I insist only on the boundary conditions (30) and (31), then tex2html_wrap_inline1087 and tex2html_wrap_inline1089 can have only a single time derivative, with coefficient tex2html_wrap_inline1091, while tex2html_wrap_inline1093 and tex2html_wrap_inline1095 can have no time derivatives at all. (In the scalar case, similar considerations restrict the diagonal elements of tex2html_wrap_inline929 to have at most two time derivatives and the off diagonal elements to have at most one. In principle, one might allow arbitrary numbers of time derivatives, with enough boundary conditions to eliminate the consequent derivatives of tex2html_wrap_inline1083, but I have not found such a scheme tractable.) Finally, causality requires that

With these restrictions, the most general form of tex2html_wrap_inline929 is
where the as yet undetermined matrices tex2html_wrap_inline1103 and tex2html_wrap_inline1105 satisfy

The general procedure would now be to expand the tex2html_wrap_inline1107 in a complete basis of Dirac matrices
the functions tex2html_wrap_inline1109 being the analogues of tex2html_wrap_inline933, tex2html_wrap_inline935 and tex2html_wrap_inline867 that appeared in the scalar theory. However, I have not yet found the energy to solve this general problem. I will therefore consider a restricted version using a minimal subset of the full Dirac basis, closed under multiplication, that contains the matrices 1, tex2html_wrap_inline1119 and tex2html_wrap_inline1121 appearing in the free theory. (``1'' of course denotes the unit matrix.) A convenient basis is
Taking account of the restrictions (34), I get
There are thus 7 as yet undetermined real functions of t and momentum k (which I here reinstate for emphasis), namely tex2html_wrap_inline1129, tex2html_wrap_inline1131, tex2html_wrap_inline1133 and the real and imaginary parts of tex2html_wrap_inline1135 and tex2html_wrap_inline1137.

With tex2html_wrap_inline1081 given by (33) and S(t,t') by (29), the equation (23) for the propagators reduces to the pair
Because the basis (36) is closed under multiplication, (and because the coefficient of tex2html_wrap_inline1119 in (39) is tex2html_wrap_inline1145) the solution for H(t,t') can be written as
Evidently, the function tex2html_wrap_inline1129 can be identified as a thermal quasiparticle width. The homogeneous equation (39), involving the first time argument of H(t,t') becomes
For orientation, I remark that, if tex2html_wrap_inline877 were taken as simply the quadratic part of (22), we would have tex2html_wrap_inline1155 and tex2html_wrap_inline1157. The hermitian matrix
has real eigenvalues tex2html_wrap_inline1159, where
is the quasiparticle energy. If tex2html_wrap_inline1161 are the corresponding eigenvectors, then the two solutions
are positive- and negative-energy solutions at the initial time t=0. These two solutions are orthogonal under the inner product tex2html_wrap_inline1165 (which is preserved by the time evolution) and are related by
When tex2html_wrap_inline1131 and tex2html_wrap_inline1135 are slowly varying, a useful adiabatic approximation is
Evidently, the solution of (39) is
where the coefficients tex2html_wrap_inline1171 and tex2html_wrap_inline1173 are to be found by solving (40) and applying suitable boundary conditions.

To solve (40), form the linear combinations
Then(40) becomes
where tex2html_wrap_inline1175 is the matrix
The solution of (50) is, of course
The solution of (51) is not of this form, but since tex2html_wrap_inline1177 and tex2html_wrap_inline1179 are linearly independent vectors, it can be written as
The residual t' dependence of tex2html_wrap_inline1183 will shortly find an interpretation in terms of time-dependent quasiparticle occupation numbers. These functions are solutions of
with tex2html_wrap_inline1185, and

These equations can, of course, be formally integrated, but in the form given they will become kinetic equations. The solutions (53) and (54) represent, of course, the same functions as (49). By comparing these two representations, we can determine the coefficients tex2html_wrap_inline1171, tex2html_wrap_inline1173 and tex2html_wrap_inline1191 up to constants, which are themselves determined by boundary conditions. In this way, the propagators (29) are given through the matrix H(t,t') (equation (41) in terms of the function
together with B(t,t'), C(t,t') and D(t,t'), for which similar expressions can be written down. The function N(t'), which multiplies products of one positive-frequency mode and one negative-frequency mode, can clearly be indentified as a quasiparticle occupation number. It is a solution of
which, with a reasonable prescription for determining tex2html_wrap_inline1203, etc. will become a Boltzmann equation. The function tex2html_wrap_inline1205 is a solution of
This looks like a second kinetic equation. However, tex2html_wrap_inline1205 appears as the coefficient of products of two positive-frequency or two negative-frequency mode functions, which have no counterpart in the free-particle propagator, and I have been able to find no simple interpretation for it. In the case of a system that starts at t=0 from a state of thermal equilibrium, continuity conditions on the complete time path yield initial conditions for (59) and (60), which are
Finally, a prescription for determining the seven functions tex2html_wrap_inline1129, etc. can be given, which is analogous to that described for the scalar field in section 2. For the specific model (22), and with similar weak-coupling and adiabatic approximations, the one-loop approximation to the kinetic equation (59) is
This does indeed have the form of a Boltzmann equation where, in the approximation described above, the occupation-number changing processes are the decay of a scalar particle (momentum p=k+k', energy tex2html_wrap_inline1215, occupation number tex2html_wrap_inline1217) into a pair of spin-tex2html_wrap_inline1021 particles (momenta k and k', energies tex2html_wrap_inline957 and tex2html_wrap_inline1227, occupation numbers tex2html_wrap_inline959 and tex2html_wrap_inline1231) and the inverse process of pair annihilation.


I. D. Lawrie, Phys. Rev. D40, 3330 (1989); J. Phys. A25, 6493 (1992)

Laurence Yaffe
Tue Nov 23 19:05:56 PST 1999