Functional Integrals and Collective Excitations

Abstract

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).

Thermodynamics of a trapped Bose-condensed gas

“…This approximation, which is discussed in depth in Ref. [12], corresponds to the so called Popov approximation [27], and is expected to be appropriate both at high temperatures, where n T ≫ m T , and at low temperatures where the two densities are of the same order, but both negligibly small for the very dilute systems we are considering here. Using decomposition (12) and the mean-field prescriptions (14), and (15) the grand-canonical…”

Renormalization-group approach to the infrared behavior of a zero-temperature Bose system

“…Nevertheless, since the results are valid exactly for 1 < d < 3, as said already, the predicted IR behavior approaches the correct result for d → 1. 9 The usual Green's functions can be readily obtained from Eqs. (4.18), (4.19) and (4.27):…”

Gapless Hartree-Fock-Bogoliubov approximation for Bose gases

“…However, a glance at the original works by Popov [22][23][24][25] shows that he has never suggested this trick. What he actually considered was a narrow region of temperatures T in the vicinity of the condensation temperature T c .…”