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InTech eBooks

Path-Integral Description of Cooper Pairing

Jacques Tempere, P A Devreese Jeroen

Published Oct 17, 2012• 13 citations
Publisher

Abstract

Before we start applying path integration to treat Cooper pairing and superfluidity, it is a good idea to quickly review the concepts behind path integration. There are many textbooks providing plentiful details, such as Feynman’s seminal text [1] and Kleinert’s comprehensive compendium [2], and other works listed in the bibliography [3–5]. We will assume that the reader is already familiar with the basics of path-integral theory, so if the following paragraphs are not merely reminders to you, it is probably better to first consult these textbooks.

DOI
10.5772/48458

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Before we start applying path integration to treat Cooper pairing and superfluidity, it is a good idea to quickly review the concepts behind path integration. There are many textbooks providing plentiful details, such as Feynman’s seminal text [1] and Kleinert’s comprehensive compendium [2], and other works listed in the bibliography [3–5]. We will assume that the reader is already familiar with the basics of path-integral theory, so if the following paragraphs are not merely reminders to you, it is probably better to first consult these textbooks.
This is the second, significantly expanded edition of the comprehensive textbook of 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular of the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's famous formula to include singular 1/r- and 1/r2-potentials. The second is a simple quantum equivalence principle governing the transformation of euclidean path integrals to spaces with curvature and torsion.The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation expansion. In contrast to ordinary perturbation expansions, divergencies are absent. Instead, there is a uniform convergence from weak to strong couplings, opening a way to precise approximate evaluations of analytically unsolvable path integrals.Tunneling processes are treated in detail. The results are used to determine the lifetime of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A new variational treatment extends the range of validity of previous tunneling theories from large to small barriers. A corresponding extension of large-order perturbation theory now also applies to small orders.Special attention is devoted to path integrals with topological restrictions. These are relevant to the understanding of the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern-Simons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.
The following notes introduce Quantum Mechanics at an advanced level addressing students of Physics, Mathematics, Chemistry and Electrical Engineering. The aim is to put mathematical concepts and techniques like the path integral, algebraic techniques, Lie algebras and representation theory at the readers disposal. For this purpose we attempt to motivate the various physical and mathematical concepts as well as provide detailed derivations and complete sample calculations. We have made every effort to include in the derivations all assumptions and all mathematical steps implied, avoiding omission of supposedly ‘trivial’ information. Much of the author’s writing effort went into a web of cross references accompanying the mathematical derivations such that the intelligent and diligent reader should be able to follow the text with relative ease, in particular, also when mathematically difficult material is presented. In fact, the author’s driving force has been his desire to pave the reader’s way into territories unchartered previously in most introductory textbooks, since few practitioners feel obliged to ease access to their field. Also the author embraced enthusiastically the potential of the TEX typesetting language to enhance the presentation of equations as to make the logical pattern behind the mathematics as transparent as possible. Any suggestion to improve the text in the respects mentioned are most welcome. It is obvious, that even though these notes attempt to serve the reader as much as was possible for the author, the main effort to follow the text and to master