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Chapman & Hall/CRC eBooks

Chapter 4

Gaussian Integration, Grassmann Integrals

Published Aug 30, 2004• 0 citations
Publisher

Abstract

In this chapter we introduce anticommuting variables and derive the Grassmann integral representations for the partition function and the correlation functions. Grassmann integrals are a suitable algebraic tool for the rearrangement of fermionic perturbation series. We demonstrate this in the first section by considering a model with a quadratic perturbation which is explicitely solvable. In that case the perturbation series can be resumed completely. By doing this in two ways, first a direct computation without anticommuting variables and then a calculation using Grassmann integrals, we hope to be able to convince the reader of the utility of that formalism.

DOI
10.1201/9781482286052-3

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In this chapter we introduce anticommuting variables and derive the Grassmann integral representations for the partition function and the correlation functions. Grassmann integrals are a suitable algebraic tool for the rearrangement of fermionic perturbation series. We demonstrate this in the first section by considering a model with a quadratic perturbation which is explicitely solvable. In that case the perturbation series can be resumed completely. By doing this in two ways, first a direct computation without anticommuting variables and then a calculation using Grassmann integrals, we hope to be able to convince the reader of the utility of that formalism.
The methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric wave functions of bosons. In the case of fermion systems, however, one faces the problem that fermion wave functions, or fermion correlation functions (or Green functions) are antisymmetric with respect to the exchange of a fermion pair. Thus, the construction of generating functions requires the introduction of an antisymmetric or Grassmann algebra of ‘classical functions’. It is then possible to generalize to Grassmann algebras the notions of derivatives and integrals, yielding quite parallel formalisms for bosons and fermions, in particular, to define a path integral for fermion systems, analogous to the holomorphic path integral for bosons. This chapter discusses differentiation and integration in Grassmann algebras, Gaussian integrals and perturbative expansion, partition function, and quantum Fermi gas.
In the path integral representation of quantum amplitudes ( chapter 16 ) the fields are regarded as classical functions. Matrix elements of time-ordered products of bosonic operators could be satisfactorily represented (see the discussion following ( 16.79 )). But something new is needed to represent, for example, the time-ordered product of two fermionic operators: there must be a sign difference between the two orderings, since the fermionic operators anticommute . Thus it seems that to represent amplitudes involving fermionic operators by path integrals we must think in terms of ‘classical’ anticommuting variables.