Classical Limit of Time-Dependent Quantum Field Theory-a Schwinger-Dyson Approach

We rewrite the Martin-Siggia-Rose (MSR) formalism for the statistical dynamics of classical fields in a covariant second order form appropriate for the statistical dynamics of relativistic field theory. This second order formalism is related to a rotation of Schwinger's closed time path (CTP) formalism for quantum dynamics, with the main difference being that certain vertices are absent in the classical theory. These vertices are higher order in an $\hbar$ expansion. The structure of the second order formulation of the Schwinger Dyson (S-D) equations is identical to that of the rotated CTP formalism apart from initial conditions on the Green's functions and the absence of these vertices. We then discuss self-consistent truncation schemes based on keeping certain graphs in the two-particle irreducible effective action made up of bare vertices and exact Green's functions.


I. INTRODUCTION
Recently there has been a lot of interest in connecting hot relativistic quantum field theory which is important for cosmology and heavy ion physics with classical field theory. Although there have been several insightful studies in perturbation theory [1], a systematic analysis has not been given for the Schwinger Dyson equations. One exception is the work of Wetterich [6], but that work only concerns itself with equal time correlation functions. Since formalisms already exist for obtaining Schwinger Dyson (S-D) equations for both the quantum [2] and classical problems [3], we thought it timely to review the MSR formalism, and cast it in a form so that it could be directly compared with the now more familiar CTP formalism [4]. We show here that the second order formulation of the S-D Equations in the MSR formalism is identical to that of the quantum theory (CTP formalism) apart from initial conditions on the Green's functions and the absence of those vertices which are higher order in .

II. REVIEW OF FIRST ORDER IN TIME MSR FORMALISM
In the paper of Martin-Siggia-Rose (MSR) [3], an operator formalism was developed which allows one to find the generating functional for both the correlation and response functions for first order classical field equations of the type:ẋ (r, t) = A[x(r, t)] , where A[x(r, t)] is a local polynomial in the classical field x(r, t). In the work of MSR, A[x(r, t)] could contain dissipative terms as well as prescribed noise terms. The formalism presented in MSR is first order in time derivatives and not apparently covariant. What we will show is that there is a covariant subset of the MSR equations in terms of which all the MSR Green's functions can be recovered. This subset can be derived from a second order Lagrangian formulation which can be related to the → 0 limit of the CTP formalism of Schwinger and Keldysh. For the statistical classical field evolutions of N interacting classical fields φ a , a = 1, 2 . . . N then x(r, t) is the 2N component field consisting of φ a and the canonical momentum π a =φ a .
If for example we restrict ourselves to cubic interactions, then the vector A is of the form where i = 1 . . . 2N . In the MSR formalism one introduces the operator such that the commutation rule is true. Defining an operator Hamiltonian H the equations of motion can be written in the compact forṁ For Eq. (4) to be true at all times one needs thatx satisfies for consistency. Thereforex(r, t) is a functional of x(r, 0) andx(r, 0). The formal solution to these equations is given by (in what follows we suppress the spatial coordinate r) where The meaning of the expectation value x(t)x(t ′ ) is as follows. Given an initial probability function P [x(0)] then Whereverx(0) appears, it is replaced bŷ and it acts on everything to the right of itself. This definition of the extended averaging procedure has three important properties. (1) The average of a product of x's agrees with the conventional definition. (2)The time dependence of x(t)x(t ′ ) is consistent with the above equations of motion.
(3) The expectation value of a product of x andx which has anx to the left vanishes if P [x(0)] goes to zero fast enough at large |x|. The last property is crucial for the tridiagonal form of the Green's functions and follows from the fact that Thus in particular in the absence of external sources The meaning of the hatted operators is understood in terms of the response of the system to an external source. If one changes the Hamiltonian by one then finds [3] that the response of any observable A is given by: where T corresponds to the usual time ordered product operation. Since x(t) and x(t ′ ) always commute, it is clear that the generating functional for both the fluctuation and response functions is given by In particular the one particle functions are The two particle Green's function matrix is tridiagonal One next doubles the space as in the CTP formalism by introducing the field Φ(r, t) Then the commutators of Φ satisfy in the ( In this larger space the equations of motion arė where the "non-Hermitian" operator H has the form and the equations of motion in the presence of an external source J is where The generating functional W [η,η] of Eq. 12 is now written in compact form by writing The expectation value of the equation of motion in the presence of an external source leads to the equation All the higher n-point functions are obtained by functional differentiation. The complete S-D equations are derived in the original MSR paper [3], and a path integral representation was derived by Jouvet and Phythian [5]. These coupled Green's functions are in a 2N ×2N matrix space and obey first order in time equations rather than in the usual N ×N space obeying second order equations. To see how to obtain the second order formalism we will simplify the discussion by considering a cubic plus quartic anharmonic oscillator.

III. SECOND ORDER MSR FORMALISM
For the anharmonic oscillator with equation of motion x + m 2 x + gx 2 + λx 3 = 0 , the above discussion leads to This yields the first order equations The first order equations for the Green's functions are obtained by taking the expectation value of these equations with sources added and then using functional differentiation. The expectation value is over an initial distribution function ρ[x 0 , p 0 ]. A simple example is is to choose our ensemble of initial conditions at t = 0 from a thermal distribution of the free Hamiltonian, i.e.
One can show that out of the twelve nonzero Green's Functions in the first order formalism only two are independent. These can be chosen as G xx , and G xp .
The harmonic oscillator values of these Green's functions are easily determined from the operator solutions and the initial density matrix. For the thermal initial conditions described above we obtain The response function is In general, in the absence of sources The independent second order equations (adding sources) are Herep (0) is treated as an operator -δ δp when one averages over the initial probability function in phase space.
These equations are derivable from the Lagrangian The vertices of the classical theory are A formal path integral formalism can be generated for the generating functional where the expectation value is over the initial density matrix. We want to compare L M SR with the Lagrangian for the CTP formalism. The result of the CTP formalism [2] is that the action is the difference of two terms, one for each branch of the closed time path contour. Explicitly for the anharmonic oscillator Introducing the change of variables we obtain in this new basis where This formal manipulation can be justified by first obtaining the SD equations in the (+, −) basis and making a rotation by π/4 as discussed in Ref. [1]. We notice that there are five extra vertices in the quantum case not present in the MSR Lagrangian. Note that arises from the fact that the classicalp is the derivative operator -d/dp which for quantum mechanics becomes i d/dp. It may be noted that apart from these extra vertices in the quantum case, dependence also enters in the initial conditions on the Green's functions. This dependence is explicit in the commutator contributions and also occers in the dependence on the initial distribution which is now a Bose-Einstein distribution rather than a Maxwell-Boltzmann one. As an illustration consider the quantum version of Eq. (26) for the Harmonic oscillator with an initial Bose distribution. Then we obtain From this we find where and So we see that enters in many ways in the initial conditions for the Green's functions, and also in the structure of the S-D equations where certain vertices are proportional to 2 . The way one derives the S-D equations from the action is identical for both classical and quantum mechanics. Thus we obtain the same structure classically, but there are fewer vertex functions.
We now derive the S-D equations for a generic cubic self-interacting field theory whether classical or quantum. For N interacting scalar fields, we introduce the column vector Φ α composed of Φ 1 i = φ i and Φ 2 i =π i where i = 1, 2, · · · N . We also need the metric g αβ which is just σ 1 αβ as far as connecting the Φ 1 and Φ 2 sectors. Then we can write generically for cubic interactions: where The generating functional is formally Defining the "classical" field Φ α and the connected 2 point function G via one has that in the presence of sources To obtain the S-D equation for the inverse two point function we first use the connection between the connected 3-point function and the 1-PI vertex function [8] obtained by first Legendre transforming from J to Φ and using the chain rule where the 1-PI three point function is defined by The usual S-D is obtained by multiplying Eq. 40 on the right by G −1 to obtain and We also have However Σ is a proper self energy and can be related [7] to the effective action Γ 2 [G] for the 2-PI irreducible graphs of the theory via Σ(12) αβ = 2 δΓ 2 [G] δG αβ (1,2) , so that it is just a function of the bare vertices and the full Green's functions. Using δΣ(12) αβ δΦ ρ (3) = d5d6 δΣ(12) αβ δG σλ (5,6) δG σλ (5,6) we obtain that (57)H(67; 23) ηλ;βν . (49) The scattering Kernel H is defined by δΣ αβ (12) δG σλ (5,6) = H(1256) αβσλ .
Self consistent approximations [7,8] are determined by keeping a certain class of graphs in Γ 2 [G], the sum of all 2-PI graphs made from bare vertices and full propagators. For cubic interactions, the Bare Vertex Approximation (BVA) is obtained by keeping the graph d1d2γ ijk G il (12)G jm (12))G kn (12)γ lmn , which then leads to the self energy being the one loop diagram, and the scattering Kernel being single particle exchange. By using the variational definitions of Σ and H one is guaranteed an internally consistent approximation. As an example of the difference between the quantum and classical S-D equations, let us restrict ourselves to a cubic anharmonic oscillator and make the bare vertex approximation. The S-D equation for the correlation function becomes which can be identified with Kraichnan's Direct Interaction Approximation [9]. Expanding we can write this in terms of classical and quantum contributions. We have in symbolic form The classical contributions to the self-energy in the BVA are Σ cl F (12) = 1 2 γ 2 pxx G 2 xx (12) , Σ cl R (12) = γ 2 pxx G xp (12)G xx (12) , Σ cl A (12) = γ 2 pxx Gp x (12)G xx (12) , and the quantum contribution is which has formal order of 2 since γppp is of order 2 .
We have recently shown that the classical BVA gives excellent agreement with Monte Carlo simulations in 1 + 1 dimensional φ 4 field theory [10].

IV. CONCLUSIONS
We have shown how the classical S-D equations can be derived from the quantum S-D equations by comparing the CTP formalism with the MSR formalism. Our hope is that this result will allow researchers to make clearer the connection between quantum field thory and classical field theory at high temperatures. We are in the process of comparing the BVA for the quantum and classical cases at high and low temperature to understand for what range of temperatures the classical approximation is valid.