Functional Integral Techniques in Condensed Matter Physics

Abstract

Path — integral formulation of quantum mechanics has been proposed long ago by Feynman (Feynman and Hibbs, 1965) and was then widely applied in quantum physics. Nowadays an unified theory based on the use of the functional integrals was firmly established and developed for studying all quantum systems, including many — body systems in statistical physics and condensed matter theory (Sakita, 1985; Wiegel, 1986; de Wit, 1981) as well as those of interacting quantized fields (Fadeev, 1976; Itzykson and Zuber, 1980; Lee, 1976; Ramond, 1981). This review is an introduction to the functional integral techniques in statistical physics and condensed matter theory. It is written for the readers who had no experience to work with the functional integrals. Therefore together with the elementary proof of the basic formulae of the functional integral techniques we explain the meaning of the functional integrals and show how can they be expressed in terms of the conventional integrals. We suppose that the readers had some basic understandings of the traditional methods of the quantum field theory in statistical physics (Abrikosov et al., 1965). The basic formulae of the functional integral techniques are derived in Sec. I for the bosonic systems and in Sec. II for the fermionic ones. They were applied widely in studying many problems of quantum theory of condensed matters. Some examples of these applications are presented in the concluding Sec. III.