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Cambridge University Press eBooks

Closed-Time-Path Green’s Functions

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter considers the product of operators in the Heisenberg representation and express it in terms of the contour time-ordering operator. Since the relative order in which the operators enter the quantum average matters, this order has to be specified in detail. This procedure leads to considering the single- and two-particle Green’s functions, where the product contains, respectively, two and four field operators, which are at the core of the diagrammatic many-particle theory to be developed in what follows.

DOI
10.1017/9781009411509.008

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This chapter considers the product of operators in the Heisenberg representation and express it in terms of the contour time-ordering operator. Since the relative order in which the operators enter the quantum average matters, this order has to be specified in detail. This procedure leads to considering the single- and two-particle Green’s functions, where the product contains, respectively, two and four field operators, which are at the core of the diagrammatic many-particle theory to be developed in what follows.
The contour-ordered Green's function considered in the previous chapter was ideal for discussing general closed time path properties such as the perturbative diagrammatic structure for non-equilibrium states. However, the contour-ordered Green's function lacks physical transparency and does not appeal to intuition. We need a different approach, which brings quantities back to real time. To accomplish this we introduce a representation where forward and return parts of the closed time path are ordered by numbers, specifying the position of a contour time by two indices, i = 1, 2. Next is the diagrammatic perturbation theory in the real-time technique then formulated in a fashion where the aspects of non-equilibrium states emerge in the physically most appealing way. In particular, we shall construct the representation where spectral properties and quantum statistics show up on a different footing in the diagrams. Lastly, we consider the connection to the imaginary-time treatment of non-equilibrium states, and establish its equivalence to the real-time approach propounded in this chapter. Real-time matrix representation To let our physical intuition come into play; we need to get from contour times back to real times. This is achieved by labeling the forward and return contours of the closed time path, depicted in Figure 4.5, by numbers, specifying the position of a contour time by an index. The forward contour we therefore label c 1 and the return contour c 2 , i.e. a contour time variable gets tagged by the label 1 or 2 specifying its belonging to forward or return contour, respectively.
In Chapter 9 we applied the second quantization to the Hartree–Fock method. In this chapter we provide the basic aspects of the thermal Green function method, which is convenient for systematic perturbative expansion beyond the mean-field approximation. Several problems in quantum statistical mechanics can be elegantly and compactly described in terms of the so-called Matsubara Green functions (Matsubara (1955); Abrikosov et al. (1963)). Hence, if on one hand, the density matrix description, often used in the previous chapters, has a more direct physical interpretation, on the other hand the Green function method is far more powerful and takes advantage of a handy diagrammatic perturbation expansion. In the following chapters, we rely on the Green function for a more complete treatment of some many-body problems. To be self contained we introduce here this technical chapter with a short presentation of the by now standard Green function method. For a comprehensive treatment we refer the reader to the many books available and in particular to Abrikosov et al. (1963). The Matsubara Green function The main aim of quantum statistical mechanics is the evaluation of the partition function (see Eqs. (6.31) and (6.32)), e.g. in the grand canonical ensemble In the case of quantum gases one can evaluate Ƶ exactly as we did in Chapter 7. In general, the evaluation of the partition function is a difficult task and it is useful to use perturbation theory to obtain approximate solutions. To set up a perturbation theory, one begins by splitting the Hamiltonian as H = H 0 + H i, where H 0 represents the “easy” part, which can be solved exactly. In the second step, one could naturally think of expanding the exponential of the interacting part H i in order to write the partition function as a perturbative series. However, such a procedure requires extreme care due to the non-commutative nature of operators in quantum mechanics. The way of solving this problem is via the “ time ” ordering of operators in each term of the perturbative expansion. It can be shown, as we shall see in the present chapter, that the quantum statistical average of any time-ordered sequence of operators with respect to the unperturbed Hamiltonian H 0 can be expressed in terms of elementary quantum statistical averages of two field operators. Such elementary averages are called Green functions and in this section we introduce the basic definitions concerning them.
This chapter considers a general form of the Wick’s theorem, which leads to a perturbation expansion of the (contour) single- and two-particle Green’s functions, which are expressed in terms of the contour time-ordering operator. The strategy for proving the Wick’s theorem is similar to that adopted within the Matsubara formalism for the Green’s functions at finite temperature and relies on the Gibbs form of the statistical operator in the interaction picture. An extension to superfluid Bose and Fermi systems is also considered.
Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].