Partition Function (mathematics) - Definition

Definition

Given a set of random variables taking on values, and some sort of potential function or Hamiltonian, the partition function is defined as

The function H is understood to be a real-valued function on the space of states, while is a real-valued free parameter (conventionally, the inverse temperature). The sum over the is understood to be a sum over all possible values that each of the random variables may take. Thus, the sum is to be replaced by an integral when the are continuous, rather than discrete. Thus, one writes

for the case of continuously-varying .

When H is an observable, such as a finite-dimensional matrix or an infinite-dimensional Hilbert space operator or element of a C-star algebra, it is common to express the summation as a trace, so that

When H is infinite-dimensional, then, for the above notation to be valid, the argument must be trace class, that is, of a form such that the summation exists and is bounded.

The number of variables need not be countable, in which case the sums are to be replaced by functional integrals. Although there are many notations for functional integrals, a common one would be

Such is the case for the partition function in quantum field theory.

A common, useful modification to the partition function is to introduce auxiliary functions. This allows, for example, the partition function to be used as a generating function for correlation functions. This is discussed in greater detail below.