The structural properties of all diagrammatic expansions developed so far can be analyzed systematically with the help of functional equations. 5.1 Generating Functionals In Chapter 3 we have seen that the correlation functions obtained from the functional derivatives of Z[j] via relation (2.14), and the generating functional itself, contain many disconnected parts. Ultimately, however, we shall be interested only in the connected parts of Z[j]. Remember that a meaningful description of a very large thermodynamic system can only be given in terms of the free energy which is directly proportional to the total volume. In the limit of an infinite volume, also called thermodynamic limit, one has then a well-defined free energy density. The partition function, on the other hand, has no proper infinite-volume limit. We can observe this property directly in the diagrammatic expansion of Z[j]. Each component of a disconnected diagram is integrated over the entire space, thus contributing a volume factor. The expansion of Z[j] therefore diverges at an infinite volume. In thermodynamics, we form the free energy from the logarithm of the partition function, which carries only a single overall volume factor and contains only connected diagrams. Therefore we expect the logarithm of Z[j] to provide us with the desired generating functional W[j]: W[j] = log Z[j]. (5.1) In this chapter we shall see that the functional derivatives of W[j] produce, indeed, precisely the connected parts of the Feynman diagrams in each correlation function. Consider the connected correlation functions G (n) c (x1,...,xn) defined by the functional derivatives G (n) c (x1,...,xn) = δ δ
The structural properties of all diagrammatic expansions developed so far can be analyzed systematically with the help of functional equations. 5.1 Generating Functionals In Chapter 3 we have seen that the correlation functions obtained from the functional derivatives of Z[j] via relation (2.14), and the generating functional itself, contain many disconnected parts. Ultimately, however, we shall be interested only in the connected parts of Z[j]. Remember that a meaningful description of a very large thermodynamic system can only be given in terms of the free energy which is directly proportional to the total volume. In the limit of an infinite volume, also called thermodynamic limit, one has then a well-defined free energy density. The partition function, on the other hand, has no proper infinite-volume limit. We can observe this property directly in the diagrammatic expansion of Z[j]. Each component of a disconnected diagram is integrated over the entire space, thus contributing a volume factor. The expansion of Z[j] therefore diverges at an infinite volume. In thermodynamics, we form the free energy from the logarithm of the partition function, which carries only a single overall volume factor and contains only connected diagrams. Therefore we expect the logarithm of Z[j] to provide us with the desired generating functional W[j]: W[j] = log Z[j]. (5.1) In this chapter we shall see that the functional derivatives of W[j] produce, indeed, precisely the connected parts of the Feynman diagrams in each correlation function. Consider the connected correlation functions G (n) c (x1,...,xn) defined by the functional derivatives G (n) c (x1,...,xn) = δ δ
In this chapter, we introduce and explain one of the fundamental tools in the study of dynamic critical phenomena, namely dynamic perturbation theory. Inevitably, large parts of Chapter 4 need to be rather technical. Complementary to the straightforward iterative method for the solution of non-linear Langevin equations presented in Section 3.2.3, we now describe the more elegant and efficient field-theoretic techniques. Yet both for the derivation of general properties of the perturbation series and for the sake of practical calculations, the response functional and the following elaborations on Feynman diagrams, cumulants, and vertex functions prove indispensable. As is the case with every efficient formalism, once one has become acquainted with the Janssen–De Dominicis response functional for the construction of dynamic field theory, and therefrom the perturbation expansion in terms of vertex functions, it serves to save a considerable amount of rather tedious work. For the purely relaxational O ( n )-symmetric models A and B, we derive the fluctuation-dissipation theorem within this formalism, and then systematically construct the perturbation series and its diagrammatic representation in terms of Feynman graphs. In order to reduce the required efforts to a minimum, the generating functionals first for the cumulants, and then, motivated by Dyson's equation for the propagator, for the one-particle irreducible vertex functions are examined. As an example, the relevant vertex functions for models A and B are explicitly evaluated to two-loop order. Furthermore, alternative formulations of the perturbation expansion are discussed, and the Feynman rules are given in both the frequency and the time domain. The results of this chapter provide the foundation for the renormalization group treatment in the subsequent Chapter 5.
This chapter defines more general statistical systems in an arbitrary number of space dimensions. For convenience, it uses the ferromagnetic language — even though, as a consequence of universality, the results that are derived in this work apply to much more general statistical systems. In addition to complete and connected correlation functions (whose decay properties at large distance — called cluster properties — are recalled), it introduces vertex functions, which are related to the thermodynamic potential. The free energy and thermodynamic potential, like connected correlation functions and vertex functions, are related by a Legendre transformation of which a few properties are discussed. Exercises are provided at the end of the chapter.
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. According to Feynman, as described in Chapter 4, instead of formulating quantum theory in terms of operators, the canonical formulation, for calculational purposes quantum dynamics can conveniently be formulated in terms of a few simple stenographic rules, the Feynman rules for propagators and interaction vertices. In Chapters 4 and 5, we showed how to arrive at the Feynman rules of diagrammatic perturbation theory for non-equilibrium states starting from the Hamiltonian defining the theory. The feature of non-equilibrium states, originally carried by the dynamical indices, could be expressed in terms of two simple universal vertex rules for the RAK-components of the matrix Green's functions. We are thus well acquainted with diagrammatics even for the description of non-equilibrium situations. However, for the situations studied using the quantum kinetic equations in Chapters 7 and 8, only the Dyson equation was needed, i.e. the self-energy, the 2-state one-particle irreducible amputated Green's function. No need for higher-order vertex functions was required, and the full flourishing diagrammatics was not put into action. In this chapter we shall proceed the other way around. We shall show that the diagrammatics of a physical theory, including the description of non-equilibrium states, can be obtained by simply stating quantum dynamics, the superposition principle, as the two exclusive options for a particle: to interact or not to interact ! From this simple Shakespearean approach we shall construct the Feynman diagrammatics of non-equilibrium dynamics.
The Hamiltonian for the many-electron system in finite volume [0, L]d is given by H = H0 + Hint where H0 = 1Ld ∑ eka + kσakσ (6.1) ek = k 2 2m − µ, and Hint = 1L3d ∑ ∑ V (k− p)a+kσa+q−k,τaq−p,τapσ (6.2) As usual, the interacting part may be represented by the following diagram: Since there is conservation of momentum, there are three independent momenta. One may consider three natural limiting cases with two independent momenta: forward exchange BCS In chapter 8 where we prove rigorous bounds on Feynman diagrams we show that the most singular contributions to the perturbation series come from the the third diagram in the above figure. To retain only this term, the q = 0 term of Hint given by (6.2), is the basic approximation of BCS theory. Furthermore V (k−p) is substituted by its value on the Fermi surface, V (k − p) ≈ V (kˆ − pˆ) where kˆ = kF k‖k‖ , kF = √ 2mµ, and the momenta k,p are restricted to values close to the Fermi surface, |ek|, |ep| ≤ ωD, where the cutoff ωD is referred to as the Debye frequency. Thus the BCS approximation reads Hint ≈ HBCS (6.3) where HBCS = 1L3d ∑ ∑ V (kˆ− pˆ) a+kσa+−k,τa−p,τapσ (6.4) We first make a comment on the volume factors. In (6.4) we retained only the q = 0 term of (6.2), but we did not cancel a volume factor. It is not obvious that (6.4) is still proportional to the volume. However, in the case of the forward approximation which is defined by putting k = p in (6.2) without canceling a volume factor, Hint ≈ Hforw := 1L3d ∑ ∑ V (0)a+kσa + p,τap,τakσ (6.5) there is an easy argument which shows that (6.5) is still proportional to the volume (for constant density). Namely, on a fixed N -particle space FN the interacting part Hint is a multiplication operator given by N∑ V (xi − xj) (6.6) Let δy(x) = δ(x − y). Then ϕ(x1, · · · ,xN ) := δy1 ∧ · · · ∧ δyN (x1, · · · ,xN ) is an eigenfunction of Hint with eigenvalue E = 12 N∑ V (yi − yj) = 12Ld N∑ ∑ e−i(yi−yj)qV (q) (6.7) which is, for V (x) ∈ L1, proportional to N or to the volume for constant density. One finds that ϕ is also an eigenvector of the forward term, Hforwϕ = Eforwϕ where Eforw is obtained from (6.7) by putting q = 0 without canceling a volume factor, V (q = 0) (6.8) Although the model defined by the Hamiltonian HBCS in (6.4) is still quartic in the annihilation and creation operators, it can be solved explicitly, without making a quadratic mean field approximation as it is usually done. The important point is that the approximation ‘putting q = 0 without canceling a volume factor’ has the effect that in the bosonic functional integral representation the volume factors enter the formulae in such a way that in the thermodynamic limit the integration variables are forced to take values at the global minimum of the effective potential. That is, fluctuations around the minimum configuration are suppressed for Ld →∞.
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.