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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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Path And Functional Integrals In Quantum Statistical Physics

Path And Functional Integrals In Quantum Statistical Physics

Quantum Field Theory and Critical Phenomena


Oxford University Press

Hamiltonians in quantum mechanics can be expressed in terms of creation and annihilation operators, instead of the more usual position and momentum operators, a method well adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism these operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables which in the classical limit correspond to a complex parametrization of phase space. To this formalism corresponds a path integral representation of the statistical operator (the density matrix at thermal equilibrium) where paths belong to complex spaces instead of the more usual position-momentum phase space. Its construction provides a useful introduction to the construction of the path integral for fermion degrees of freedom. Both formalisms can then be generalized to a quantum gas of Bose or Fermi particles in the grand canonical formulation. A functional integral representation of quantum partition functions can be derived, a topic discussed in the second part of the chapter. Finally, these formalisms allow the construction of the functional integral representation of the scattering S-matrix in quantum field theory shown in Chapters 6 and 8.

Keywords:   quantum mechanics, holomorphic formalism, functional integrals, Bose particles, Fermi particles

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