Some Aspects of Functional Integrals and Many Body Theory

Abstract

Conventionally the quantum many body problem has been formulated in terms of a Hamiltonian language. In second quantization this involves the use of field operators which obey commutation (for bosons) or anticommutation (for fermions) relations for equal times;

$$ \psi \left( {\underline x ,t} \right)\,\,{\psi ^ + }\left( {\underline x ',t} \right) - {\psi ^ + }\left( {\underline x ',t} \right)\,\,\psi \left( {\underline x ,t} \right)\, = \delta \left( {\underline x - \underline x '} \right) $$

(1a)

$$ \psi \left( {\underline x ,t} \right)\,\,{\psi ^ + }\left( {\underline x ',t} \right) + {\psi ^ + }\left( {\underline {x'} ,t} \right)\,\,\psi \left( {\underline x ,t} \right)\, = \delta \left( {\underline x - \underline x '} \right) $$

(1b)

Here, and below, we use x to denote symbolically all the relevant coordinates other than time; i.e. space, spin, etc. The 6-function is to be interpreted as Kronecker or Dirac according to whether the corresponding coordinate element is discrete or continuous.