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Il Nuovo Cimento

Derivation of the functional integral formalism for fermi systems from the canonical formalism

W. K. Burton, A. H. de Borde

Published Jun 1, 1957• 2 citations
Publisher
DOI
10.1007/bf02856040

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Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum 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 that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
We present a formalism of the path integral centroid molecular dynamics (CMD) extended to Bose and Fermi statistics as a semiclassical approach to explore the dynamics of quantum many-body systems. The validity of the method is examined in relation to the time correlation functions. The presently proposed scheme, refined from our previous derivation [Chem. Phys. Lett. 307, 187 (1999)], is aimed at the calculations of not the exact quantum-mechanical dynamics but the semiclassical dynamics under certain approximations. The formalism is based on the projection operator with which the Bose/Fermi system is mapped onto a particular type of pseudo-Boltzmann system. In the pseudo-Boltzmann system the correlation due to the Bose/Fermi statistics is introduced via an extra pseudopotential called the permutation potential and its relevant operator. Using the present semiclassical formalism, the time correlation function of centroid position, which is evaluated from the CMD trajectories in the pseudo-Boltzmann system, is an approximation to the Kubo canonical correlation function of position operator of the exact quantum-statistical system composed of bosons or fermions. There is no such apparent relation between the momentum operator and the corresponding momentum centroid.