# Self-Consistent Approximations to

Non-Equilibrium Many-Body Theory

###### Abstract

Within the non-equilibrium Green’s function technique on the real time contour, the -functional method of Baym is reviewed and generalized to arbitrary non-equilibrium many-particle systems. The scheme may be closed at any desired order in the number of loops or vertices of the generating functional. It defines effective theories, which provide a closed set of coupled classical field and Dyson equations, which are self-consistent, conserving and thermodynamically consistent. The approach permits to include unstable particles and therefore unifies the description of resonances with all other particles, which obtain a mass width by collisions, decays or creation processes in dense matter. The inclusion of classical fields enables the treatment of soft modes and phase instabilities. The method can be taken as a starting point for adequate and consistent quantum improvements of the in-medium rates in transport theories.

Gesellschaft für Schwerionenforschung mbH, Planckstr. 1,
64291 Darmstadt, Germany

Kurchatov Institute, Kurchatov sq. 1, Moscow 123182,
Russia

Moscow Institute for Physics and Engineering,
Kashirskoe sh. 31, Moscow 115409, Russia

## 1 Introduction

Non-equilibrium Green’s function technique, developed by Schwinger, Kadanoff, Baym and Keldysh [1, 2, 3, 4], is the appropriate concept to study the space–time evolution of many-particle quantum systems. This formalism finds now applications in various fields, such as quantum chromodynamics [5], nuclear physics [6, 7, 8, 9, 10, 11, 12], astrophysics [10, 13, 14], cosmology [15], spin systems [16], lasers [17], physics of plasma [18, 19], physics of liquid He [20], critical phenomena, quenched random systems and disordered systems [21], normal metals and super-conductors [13, 22, 23], semiconductors [24], tunneling and secondary emission [25], etc.

For actual calculations certain approximation steps are necessary. In many cases perturbative approaches are insufficient, as for physical systems with strong couplings, e.g. like those treated in nuclear physics, in physics of liquids He and He, or high-temperature super-conductivity, etc. In such cases, one has to re-sum certain sub-series of diagrams in order to obtain a reasonable approximation scheme. In contrast to perturbation theory for such re-summations one frequently encounters the complication that the resulting equations of motion, even though self-consistent, may no longer comply with the conservation laws, e.g., of charge, energy and momentum. This problem has first been considered in two pioneering papers by Baym and Kadanoff [26, 27] discussing the response to an external perturbation of quantum systems in thermodynamic equilibrium. Baym, in particular, showed [27] that any approximation, in order to be conserving, must be so-called -derivable. Thereby, he exploited the properties of an auxiliary functional, the -functional, introduced by Luttinger and Ward [28] a year earlier for the formulation of the thermodynamic potential (see also [29]). Thereby the -functional is determined in terms of full, i.e. re-summed, Green’s functions and free vertices. The scaling parameter of the vertices can be considered as an expansion parameter of a given approximation level. In the non-equilibrium formalism the problem of conserving approximations could be even more severe than in the case of the systems response to an external perturbation close to thermal equilibrium, since the system may exercise a rather violent evolution. Apart from transport models, mostly based on the quasi-particle approximation like Landau’s Fermi liquid theory, there were only few attempts to discuss the issue of conserving approximations in the context of the non-equilibrium field theory (see, e.g., [2, 6, 9]), which mainly considered Hatree-Fock and T-matrix approximations. However, the general problem of constructing conserving approximations in the non-equilibrium case and, in particular, beyond the quasi-particle limit has not explicitly been addressed yet.

Alongside, the question of thermodynamic consistency is vital. If, as a result of a non-equilibrium evolution, a system arrives at an equilibrium state, the non-equilibrium Green’s functions should properly describe thermodynamic quantities and potentials, such that thermodynamic relations between them are preserved. This problem is also relevant to the thermodynamic Green’s function technique, as already considered by Baym [27]. Baym demonstrated that any -derivable approximation is at the same time thermodynamically consistent.

In this paper we re-address the above problems and extend the concept i) to the genuine non-equilibrium case formulated on the closed real-time contour, ii) to relativistic field-theory Lagrangians in principle of arbitrary type (not just two-body fermion interactions) and iii) to the inclusion of classical fields, i.e. non-vanishing expectation values of the field operators. The generalized scheme permits to construct self-consistent, approximate, coupled dynamical equations of motion for the classical fields and Green’s functions of the system on the closed real-time contour. This set of equations is conserving and thermodynamically consistent. Thereby, the inclusion of classical fields permits to account for the phase-transition phenomena or to describe the coherent dynamics of soft modes, much in the spirit of hard-thermal-loop re-summations [30, 31, 12]. Avoiding the quasi-particle limit enables us to appropriately consider the finite mass-width of all constituents in the dense matter environment. The latter aspect unifies the description of resonances, which have already a mass-width in vacuum, with all particles, which acquire a dynamical width during the collision processes in the dense matter. The proper account of the finite width of the particles is also vital for a non-singular treatment of the soft-mode problem [12], where space-time coherence effects, like the Landau–Pomeranchuk–Migdal effect, defer the use of zero width quasi-particles and require non-perturbative re-summations. In this paper we confine the presentation to the derivation of the closed set of self-consistent couple Kadanoff-Baym and classical field equations. This constitutes the basis for various further steps towards classical-type transport schemes through the gradient approximation, which will be presented in a forthcoming paper[38].

For the sake of clarity, we restrict the presentation to systems of relativistic scalar bosons. This allows us to formulate the basic ideas in a simple and transparent form. The resulting relations can directly be generalized to multi-component systems of relativistic bosons and fermions. In sect. 2 we introduce the general equations of motion and the expressions for the conserved quantities on the operator level. The equations of motion of the corresponding expectation values are formulated within the real-time closed contour formalism (sect. 3). Thereby, it is advantage to formulate the concepts in terms of generating functionals on the non-equilibrium contour, where the special functional plays a central role (sect. 4). This generating functional takes the same status in the space of Green’s functions (two-point functions) and classical fields (one-point functions), as the original Lagrangian for the field operators. Subsequently, we formulate the diagrammatic representation for (sect. 5). We show that any approximation, where all classical field sources and self-energies are -derivable in the sense of a variational principle, has the following properties: (i) it is conserving, (ii) it provides conserved current and energy-momentum tensor, which are identical to the corresponding Noether quantities (sect. 6), and (iii) it is at the same time thermodynamically consistent (sect. 7). In the summary, we formulate the main results and briefly discuss extensions and applications of the derived formalism. The list of diagrammatic rules is deferred to the Appendix A, while Appendix B contains some helpful equilibrium relations.

## 2 Energy-Momentum Tensor and Conserved Currents

We consider a system of relativistic scalar bosons, specified by the free Klein-Gordon Lagrangians

(2.1) |

where and are bosonic field operators. The convention of units is such that . The interaction Lagrangians (for neutral bosons) and

The variational principle of stationary action leads to the Euler–Lagrange equations of motion for the field operators

(2.2) |

and similarly for the corresponding adjoint equation. Thereby, the operator is a local source current of the field , while is the differential operator of the free evolution with the free propagator as resolvent.

The standard canonical energy-momentum tensor [32] has some undesired features, as it is non-symmetric in the Lorentz indices, for example. Alternatively, using the Euler–Lagrange equations of motion and the definition of the source current (2.2), one can show that the following form also defines a conserving energy momentum tensor

For notational simplicity, expression (2) and similar expressions below, which appear symmetric in and , are written in such a way that they directly apply to complex fields. The symbol implies that for real fields the corresponding expressions are obtained from those for complex fields by multiplying by the factor upon equating

For a multi-component system, the latter defines the sum of the potential energy densities of any field with the currents induced by the other fields in the system.

In terms of the differential operator , the alternative energy–momentum tensor of eq. (2) can be written in a charge symmetric form as

(2.5) | |||||

This form of the energy–momentum tensor is equal to the metric one, which results from variation of the action over the metric tensor rather than over the fields. All terms in eq. (2.5) are evidently symmetric in .

For specific interactions, eq. (2) provides simple relations between and . If all vertices of have the same number of field operators attached, one simply deduces

(2.6) |

For instance, for the -theory, where , the interaction energy is half of the potential energy.

If the Lagrangian is invariant under some global transformation of charged fields (with the charge ), e.g.,

(2.7) |

there exists a Noether current defined as [32]

(2.8) |

which is conserved, i.e. Formally, it is derived by applying the local transformation of the form (2.7) and using the stationary condition of the action around physical solutions.

From the Euler–Lagrange equations of motion (2.2), one obtains

(2.9) |

which vanishes, since for the symmetry (2.7) the interaction Lagrangian has to consist of terms, where and appear pairwise. In terms of the operator, the current can again be written in a charge symmetric form as

(2.10) |

which naturally vanishes for the neutral particles ().

One may also define the tensor , which is associated with the Lorentz invariance of the Lagrangian and provides the angular momentum conservation. However, we do not treat this tensor in this paper, since it is of no common use in kinetics.

## 3 Real-Time Contours

In the non-equilibrium case, one assumes that the system has been prepared at some initial time described in terms of a given density operator , where the form a complete set of eigenstates of . All observables can be expressed through -point Wightman functions of Heisenberg operators

(3.11) | |||||

Note the fixed operator ordering for Wightman functions.

The non-equilibrium theory can entirely be formulated on a special contour—the closed real-time contour (see figure 1) with the time argument running from to along the time-ordered branch and back to along the anti-time-ordered branch. Contour-ordered multi-point functions are defined as expectation values of contour ordered products of operators

(3.12) |

where denotes the special time-ordering operator, which orders the operators according to a time parameter running along the time contour . The l.h.s. of eq. (3.12) is written in the Heisenberg representation, whereas the r.h.s. is given in the interaction () representation. Here and below, the subscript ”I” indicates the interaction picture. Functions with points can be expressed in terms of products of other multi-point functions contour integrated over internal coordinates. Ultimately, one likes to express the multi-point functions of interest in terms of one- and two-point functions (Wick’s linked cluster expansion), i.e. in terms of classical fields, classical source-currents, propagators and self-energies. Note that at this level the contour is not a contour in the complex plane, as the figure may suggest, but rather it runs along real time arguments. It is through the placement of external points on the contour that the contour ordering obtains its particular sense.

In certain calculations, e.g., in those that apply the Fourier and Wigner transformations, it is necessary to decompose the full contour into its two branches—the time-ordered and anti-time-ordered branches. One then has to distinguish between the physical space-time coordinates and the corresponding contour coordinates which for a given take two values and () on the time ordered and anti-time ordered branches, respectively (see figure 1). Closed real-time contour integrations can then be decomposed as

(3.13) |

where only the time limits are explicitly given. Thus, the anti-time-ordered branch acquires an extra minus sign if integrated over physical times. For any two-point function , the contour values are defined as

(3.14) |

on the different branches of the contour. The contour -function is defined as

(3.15) |

where, is the -Pauli matrix. For any multi-point function, the external point , which has the largest physical time, can be placed on either branch of the contour without changing the value, since the contour-time evolution from to provides unity. Therefore, one-point functions have the same value on both sides of the contour.

Due to the change of operator ordering, genuine multi-point functions are
discontinuous in general, when two contour coordinates become identical. In
particular, two-point functions like become^{1}^{1}1Quite commonly, like in
refs. [2, 6], the notation 3.16). We prefer the more flexible labeling of contour points. is used for two-point functions instead of
(

(3.16) |

where and are the usual time and anti-time ordering operators. Since there are altogether only two possible orderings of the two operators, in fact given by the Wightman functions and , which are both continuous, not all four components of are independent. From eq. (3.16) follow relations between non-equilibrium and usual retarded and advanced functions

(3.17) | |||||

where is the step function of the time difference.

Discontinuities of a two-point function may cause problems for differentiations, in particular, since they often occur simultaneously in products of two or more two-point functions. The proper procedure is, first, with the help of eq. (3) to represent the discontinuous parts in and by the continuous and times -functions, then to combine all discontinuities, e.g. with respect to , into a single term proportional to , and finally to apply the differentiations. One can easily check that in the following particularly relevant cases

(3.18) | |||

(3.19) | |||

(3.20) |

all discontinuities exactly cancel. Thereby, these values are independent of the placement of and on the contour, i.e. the values are only a function of the physical coordinate .

Boson fields may take non-vanishing expectation values of the field operators , called classical fields. The corresponding equations of motion are provided by the ensemble averaging of the operator equations of motion (2.2)

(3.21) |

Here , while is the freely evolving classical field which starts from at time . Thereby, is the free contour Green’s function

(3.22) |

The reader can easily verify that the contour form (3.21) is equivalent to the standard retarded form of the classical field equations due to eq. (3) and the fact that and are one-point functions, which have identical values on both sides of the contour. The free propagator is resolvent of equation

(3.23) |

on the contour, where is the contour -function (3.15). Subtracting the classical fields via

(3.24) |

we define the full propagator in terms of quantum-fluctuating parts of the fields

(3.25) |

Here and below, the sub-label indicates that uncorrelated parts are subtracted. In terms of diagrams it implies, that the corresponding expectation values are given by sums of entirely connected diagrams.

Averaging the operator equations of motion (2.2) multiplied by and subtracting classical-field parts, one obtains the equation of motion for the propagators as

(3.26) |

where the contour -function appears due to the contour ordering of the operators in .

Eq. (3.26) is still exact and accounts for the full set of initial
correlations contained in . In order to proceed, one may
suggest that the typical interaction time for the change of
the correlation functions is significantly shorter than the typical relaxation
time , which determines the system evolution, when one
neglects those initial correlations. Then, describing the system at times
, one can neglect the initial correlations which
are supposed to be dying at the time in accordance
with the Bogolyubov’s principle of the weakening of initial
correlations.^{2}^{2}2Actually, considering a dilute gas limit, Bogolyubov
suggested the weakening of all the correlations, whereas we use a weaker
assumption on the weakening of only short-range ()
correlations, cf. [33]. As a result one can apply the standard
Wick decomposition dropping higher order correlations for the driving terms on
the r.h.s. of both equations of motion (3.21) and
(3.26). Then both driving terms can be expressed solely as functionals
of the classical fields and one-particle propagators rather than on higher
order correlations. In particular the driving term of (3.26) can then
be expressed through the proper self-energy

(3.27) | |||||

The second line of eq. (3.27) results from the fact that the expectation value is connected and, therefore, in the first line, has to be contracted with one of the operators occurring in