ABSTRACT
In this chapter we derive the perturbation series for the partition function. Since it is given by a trace of an exponential of an operator H0 +λV , we first expand the exponential with respect to λ. The result is a power series with operator valued coefficients. Then we compute the trace of the n’th order coefficient. This can be done with Wick’s theorem which lies at the bottom of any diagrammatic expansion. It states that the expectation value, with respect to exp[−βH0], of a product of 2n annihilation and creation operators is given by a sum of terms where each term is a product of expectation values of only two annihilation and creation operators. Depending on whether the model is fermionic or bosonic or complex or scalar, this sum is given by a determinant, a pfaffian, a permanent or simply the sum over all pairings which is the bosonic analog of a pfaffian. For the many-electron system, one obtains an n × n determinant. The final result is summarized in Theorem 3.2.4 below.
