Path integrals of spin-J systems in the holomorphic representation

A. Foussats, A. Greco, Alejandro Muramatsu

Jean Zinn‐Justin

This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum statistical physics (in the so-called second quantization formalism). Path integrals then take the form of integrals over trajectories in phase space in a complex parametrization. A parallel formalism, based on integration over anti-commuting type or Grassmann variables, makes it possible to handle fermions in a way quite analogous to bosons. The corresponding path integral representation of the statistical operator is then derived. The holomorphic formalism is specially well adapted to a study of the harmonic oscillator and, more generally, of perturbed harmonic oscillators. As an illustration, the formalism is applied to the Bose-Einstein condensation. This chapter also considers complex integrals and Wick's theorem, kernel of operators, general Hamiltonians, partition function, and the quantum Bose gas.

Horacio Grinberg

M G Cruz

The formalism of functional integration in the holomorphic representation is illustrated by calculating the partition function for the small polaron Hamiltonian in the atomic limit. As an introduction, the standard construction of the functional integral representation of the partition function for a coupled fermion-boson system is reviewed.

Jean Zinn‐Justin

Abstract Chapter 4 has introduced the functional integral representation of the quantum statistical operators and thus, formally, evolution in imaginary or Euclidean time. By contrast, to calculate the evolution operator and the scattering S-matrix elements, quantities relevant to particle physics, it is necessary to make a continuation from imaginary to real time. However, the representation of the S-matrix follows from additional considerations. To illustrate the power of the formalism, we show how to recover the perturbative expansion of the scattering amplitude, some semi-classical approximations, and the eikonal approximation. When the asymptotic states at large time are eigenstates of the harmonic oscillator, instead of free particles, the holomorphic formalism becomes useful. A simple generalization of the path integral of Chapter 4 leads to the corresponding path integral representation of the S-matrix. In the case of the Bose gas, the evolution operator is then given by a holomorphic field integral. A parallel formalism leads to an analogous representation for the evolution operator of a system of non-relativistic fermions.

Reinhold Egger, Manfried Milch, U. Weiss