Correlation Functions
By introducing artificial auxiliary functions into the partition function, it can then be used to obtain the expectation value of the random variables. Thus, for example, by writing
one then has
as the expectation value of . In the path integral formulation of quantum field theory, these auxiliary functions are commonly referred to as source fields.
Multiple differentiations lead to the correlation functions of the random variables. Thus the correlation function between variables and is given by:
For the case where H can be written as a quadratic form involving a differential operator, that is, as
then the correlation function can be understood to be the Green's function for the differential operator (and generally giving rise to Fredholm theory). In the quantum field theory setting, such functions are referred to as propagators; higher order correlators are called n-point functions; working with them defines the effective action of a theory.
Read more about this topic: Partition Function (mathematics)
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