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Fortschritte der Physik

Collective Quantum Fields

Hagen M. Kleinert

Published Jan 1, 1978• 182 citations
Publisher

Abstract

Abstract A general quantum field theory of collective excitations is presented for many‐body systems. Lagrangians involving fundamental particles are transformed to collective fields exactly via Feynman path integrals. Graphically, this amounts to a complete resummation of the perturbation series without the danger of double counting. Plasma and pairing effects provide special examples corresponding to mutually complementary transformations. Bose and Fermi systems can be treated on equal footing. The methods are illustrated by giving the exact collective Lagrangian for the BCS model and approximate ones for type II superconductors and 3He.

DOI
10.1002/prop.19780261102

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Abstract A general quantum field theory of collective excitations is presented for many‐body systems. Lagrangians involving fundamental particles are transformed to collective fields exactly via Feynman path integrals. Graphically, this amounts to a complete resummation of the perturbation series without the danger of double counting. Plasma and pairing effects provide special examples corresponding to mutually complementary transformations. Bose and Fermi systems can be treated on equal footing. The methods are illustrated by giving the exact collective Lagrangian for the BCS model and approximate ones for type II superconductors and 3He.
A systematic method of handling a system composed of a number of particles and an intermediary Bose field has been developed. The main idea consists in linearizing the interaction by taking certain average values so that it may be amalgamated to the free Hamiltonian. Only the remaining fluctuational interaction is treated by perturbation theory. The effect of the averaged interaction appears in the form of two generalized Hartree fields associated with the particle and the Bose quantum, respectively. The self-consistent equations for these Hartree fields are derived from an analysis of the SSS matrix, keeping close analogy to the renormalization procedure in quantum electrodynamics. The formalism includes such approximations as the Bohm-Pines, Hartree-Fock, and Thomas-Fermi theories as special cases. General criteria about the validity and plausibility of the picture involved in the present formulation are discussed from various points of view.
This book describes the theory and selected applications of one of the most important mathematical tools used in the theoretical investigation of collective excitations in statistical physics, such as occur in superfluidity, superconductivity, plasma dynamics, superradiation, and in phase transitions. The author, who is a distinguished physicist and leading researcher in this area, begins with an introduction to functional integral techniques in equilibrium statistical thermodynamics, and discusses the expression of partition functions and Green functions in terms of functional integrals. Subsequent sections deal with the application of functional integrals in superfluid Bose systems, systems with Coulomb interaction, and superfluid Fermi systems. The final section considers the application of the concept of Bose-condensation of auxiliary fields to the theory of crystals, heavy atoms and also to the theory of model Hamiltonians (BCS and Dicke models).
The coupled cluster [or exp(S)] formulation of quantum many-body theory is flrst extensively reviewed. It is then applied to the general problem of pairing correlations within a many-fermion system. The method is illustrated by restricting ourselves to the exact calculations that can be performed within the various generalized ladder approximations. It is shown how the formalism provides an efficient and unified framework in which to describe all aspects of pairing at the same level of approximation. Possible extensions to deal with composite clusters of more than two particles and to incorporate the effects of the collective excitations, are sketched
Functional integration is one of the most powerful methods of modern theoretical physics. The functional integration approach to systems with an infinite number of degrees of freedom turns out to be very suitable for the introduction and formulation of the diagram perturbation theory in quantum field theory and statistical physics. This approach is simpler than that using an operator method. The application of functional integrals in statistical physics allows one to derive numerous interesting results more quickly than when using other methods. The theories of phase transitions of the second kind, superfluidity, superconductivity, lasers, plasma, the Kondo effect, the Ising model – a list of problems which is far from complete, for which the application of the functional integration method appears to be very fruitful. In some of the problems, it allows us to give a rigorous proof of results obtained by other methods. If there is a possibility of an exact solution, the functional integration method gives a simple way to obtain it. In problems far from being exactly solvable (for example, the general theory of phase transitions), the application of functional integrals helps to build up a qualitative picture of the phenomenon and to develop the approximative methods of calculations. Functional integrals are especially useful for the description of collective excitations, such as plasma oscillations in the system of particles with Coulomb interaction, quantum vortices and long-wave phonons in superfluidity and superconductivity, collective modes in 3 He-type systems and in 3 He− 4 He mixtures.
These lecture notes deal with the problem of collective coordinates in many-body systems, which are treated as gauge systems in (0+1) dimensions. The resulting classical Dirac brackets are discussed, as well as the structure of the quantal space of wavefunctions. Emphasis is made on the application of the BRST formalism. Several systems displaying an approximate breakdown of symmetries are treated. Many-body physicists may find attractive both a rigorous formulation of the problem of collective coordinates and being introduced to the BRST formalism, which has become a fundamental tool in gauge theories. Field-theorists may find illuminating the applications to simpler mechanical examples. The notes are self-contained. In particular, they do not require a previous knowledge of either the BRST formalism or of collective transformations.
We develop a squeezed-field path-integral representation for BCS superconductors utilizing a generalized completeness relation of squeezed-fermionic coherent states. We derive a Grassmann path integral of fermionic quasiparticles that explicitly includes the collective degrees of freedom of the order-parameter dynamics governed by the classical Anderson pseudospin model. Based on this method, we analyze the spectral function of the single-particle excitations, and show that the squeezed-field path integral for the BCS Hamiltonian describes a bosonic sideband branch that corresponds to the Higgs amplitude mode of BCS superconductors, in addition to reproducing the quasiparticle and quasihole excitation branches described by the BCS mean-field approximation.