Correlation function

Correlation functions used in astronomy, financial analysis, econometrics, and statistical mechanics differ only in the particular stochastic processes they are applied to.

For possibly distinct random variables X(s) and Y(t) at different points s and t of some space, the correlation function is where

Examples of important spacetime symmetries are — Higher order correlation functions are often defined.

A typical correlation function of order n is (the angle brackets represent the expectation value) If the random vector has only one component variable, then the indices

The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics.

Any probability distribution which obeys a condition on correlation functions called reflection positivity leads to a local quantum field theory after Wick rotation to Minkowski spacetime (see Osterwalder-Schrader axioms).