Introduction. 1: Evolution of States of Quantum Systems of Finitely Many Particles. 1. Principal Concepts of Quantum Mechanics. 2. Evolution of States of Quantum Systems with Arbitrarily Many Particles. 3. Evolution of States in the Heisenberg Representation and in the Interaction Representation. Mathematical Supplement I. References. 2: Evolution of States of Infinite Quantum Systems. 4. Bogolyubov Equations for Statistical Operators. 5. Solution of the Bogolyubov Equations. 6. Gibbs Distributions. Mathematical Supplement II. Mathematical Supplement III. References. 3: Thermodynamic Limit. 7. Thermodynamic Limit for Statistical Operators. 8. Statistical Operators in the Case of Quantum Statistics. 9. Bogolyubov's Principle of Weakening of Correlations. Mathematical Supplement IV. References. 4: Mathematical Problems in the Theory of Superconductivity. 10. Frohlich Model. 11. Bogolyubov's Compensation Principle for 'Dangerous' Diagrams. Compensation Equations. 12. Bardeen--Cooper--Schrieffer (BCS) Hamiltonian. 13. Microscopic Theory of Superfluidity. Mathematical Supplement V. References. 6: Green's Functions. 14. Green's Functions. Equations for Green's Functions. 15. Investigation of the Equations for Green's Functions in the Theory of Superconductivity and Superfluidity. 16. Green's Functions in the Thermodynamic Limit. References. 6: Exactly Solvable Models. 17. Description of the Hamiltonians of Model Systems. 18. Functional Spaces of Translation--Invariant Function. 19. ModelHamiltonians in the Spaces of Translation Invariant Functions. 20. Model BCS Hamiltonian in the Space hT. Equivalence of General and Model Hamiltonians in the Space of Pairs. 21. Equations for Green's Functions and their Solutions. Mathematical Supplement VI. References. 7: Quasiaverages. Theorem on Singularities of Green's Functions of 1/q2-Type. 22. Quasiaverages. 23. Green's Functions and their Spectral Representations. 24. Theorem on Singularities of Green's Functions of 1/q2-Type. References. Subject Index.
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 →∞.
A general system composed of many weakly interacting bosons and/or fermions is considered. The object is to develop a procedure for renormalizing the single-particle creation operators, so as to remove the interactions to successively higher orders of perturbation. The basic idea is that creation operators ${{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$ must satisfy the Hamiltonian commutator equations $[\mathcal{H},{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}]={\ensuremath{\omega}}_{i}{{\ensuremath{\theta}}_{i}}^{\ifmmode\dagger\else\textdagger\fi{}}$, where
H is the Hamiltonian and ${\ensuremath{\omega}}_{i}$ are the particle energies. For the many-weakly-interacting-particle system, the zeroth-order boson and fermion creation operators satisfy these equations to zeroth order. It is shown that if the creation operators are renormalized so as to satisfy the Hamiltonian commutator equations to order
m, and also to satisfy the appropriate boson commutators and fermion anticommutators to order
m, then the problem is solved to order
m. In particular, the vectors formed by operating on the ground state with the renormalized creation operators, according to the usual boson and fermion occupation-number representation, are eigenvectors of
H and are orthonormal, all to order
m. A procedure is otained for finding the (
m+1)-order contributions to the particle creation energies, in terms of the
m-order operators. Explicit first-order calculations of these general results are provided for a system of bosons and a system of fermions, and these first-order results are shown to include similar results of Rayleigh-Schr\"odinger perturbation theory, the random-phase approximation, and the method of thermodynamic Green's functions. The problem of anharmonic lattice dynamics is studied in detail, and a method of undetermined coefficients is used to renormalize the phonon creation operators to first order. The phonon energies are calculated to second order, and this calculation shows that the interactions between renormalized phonons cannot be removed in second order. Statistical averages of the phonon energies give the energy shifts and lifetimes which have been calculated previously by various propagator techniques. In addition, the renormalized energies are used to calculate the temperature-dependent part of the Helmholtz free energy correct to second order. As a final example, electron-phonon interactions in a normal metal are studied. The electron and phonon creation operators are renormalized to first order, the particle energies are calculated to second order, and statistical averages of the particle energies recover the usual thermodynamic Green's-function results for energy shifts and lifetimes. These examples show the simplicity by which the renormalization procedure obtains a great amount of detailed information about the single-particle nature of a many-particle system.
Introduction Mathematically, given a linear differential operator L x , one encounters the solution to the inhomogeneous differential equation Here ρ ( x ) is a given source function. For a given boundary condition, we assume a solution to exist. The solution can be reduced to a simpler problem. Let G ( x, y ) is the Green's function. This is a function of x with y a parameter. Take G ( x, y ) to satisfy the same boundary conditions as φ ( x ). Then since An example of L x is, of course, the Schrödinger operator We take ρ ( x ) = V ( x ) ψ ( x , t ), ψ ( x , t ) being the wave function and V ( x ) the potential operator. We are interested in Green's functions taken over from the techniques of quantum field theory (Schweber, 1961; Lifshitz and Petaevskii, 1981). We will concern ourselves particularly with one- and two-time Green's functions, since our principal interest is to show a connection to the calculations of linear response theory (Chapter 15) as well as to quantum kinetic equations. Then we wish to compare the methods with those described in Chapter 4. In this we will follow the work of L. P. Kadanoff and G. Baym (1962) and of L. V. Keldysh (1965) and also Zubarev (1974). We will not discuss equilibrium statistical mechanics utilizing Green's function techniques for many-body problems. The literature is exhaustive (see Abrikosov et al. , 1963; Fetter and Walecka, 1971). A good general introduction is the book by G. D. Mahan (2000). One- and two-time quantum Green's functions and their properties Let us introduce the creation operator ψ † ( r, t ) and annihilation operator ψ ( r, t ) of the second quantization formalism (see Schweber, 1961). They have the equal time commutation rules for Bose and Fermi particles: The Hamiltonian operator for the particles is and the number density of particles at r t is the operator Note that r = ( r 1 … r N ) for (1, 2, 3, …, N ) and V (| r − r ′|) is the pair potential depending, as in Chapter 4, on the scalar distance between the particles.