Correlation function

Correlation functions contain information about the distribution of points or events, or things across some space/time.

A very simple example of a correlation function is the following: Given the existence of a point at a position X in some space, what is the probability of there being another point at a second position Y.

For stochastic processes, including those that arise in statistical mechanics and Euclidean quantum field theory, a correlation function is the correlation between random variables at two different points in space or time. If one considers the correlation function between random variables at the same point but at two different times then one refers to this as the autocorrelation function. If there are multiple random variables in the problem then correlation functions of the same random variable are also sometimes called autocorrelation. The autocorrelation can be intuitively understood as an indicator of how the random variable at a given point changes with time. Correlation functions of different random variables are sometimes called cross correlations. Cross correlations are a useful indicator of the dependencies among different random variables as a function of time.

Correlation functions used in astronomy, financial analysis, quantum field theory and statistical mechanics differ only in the particular stochastic processes they are applied to with the caveat that we are dealing with "quantum distributions" in QFT. (For details, see Correlation function (quantum field theory).)

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


 * $$C(s,t) = \operatorname{corr}( X(s), X(t) ).$$

In this definition, it has been assumed that the stochastic variable is scalar-valued. If it is not, then one can define more complicated correlation functions. For example, if one has a vector Xi(s), then one can define the matrix of correlation functions


 * $$C_{ij}(s,s') = \operatorname{corr}( X_i(s), X_j(s') )$$

or a scalar, which is the trace of this matrix. If the probability distribution has any target space symmetries, i.e. symmetries in the space of the stochastic variable (also called internal symmetries), then the correlation matrix will have induced symmetries. If there are symmetries of the space (or time) in which the random variables exist (also called spacetime symmetries) then the correlation matrix will have special properties. Examples of important spacetime symmetries are &mdash; n is
 * translational symmetry yields C(s,s ' ) = C(s &minus; s ' ) where s and s ' are to be interpreted as vectors giving coordinates of the points
 * rotational symmetry in addition to the above gives C(s, s ' ) = C(|s &minus; s ' |) where |x| denotes the norm of the vector x (for actual rotations this is the Euclidean or 2-norm).


 * $$C_{i_1i_2\cdots i_n}(s_1,s_2,\cdots,s_n) = \langle X_{i_1}(s_1) X_{i_2}(s_2) \cdots X_{i_n}(s_n)\rangle.$$

If the random variable has only one component, then the indices $$i_j$$ are redundant. If there are symmetries, then the correlation function can be broken up into irreducible representations of the symmetries &mdash; both internal and spacetime.

The case of correlations of a single random variable can be thought of as a special case of autocorrelation of a stochastic process on a space which contains a single point.

Properties of probability distributions
With these definitions, the study of correlation functions is equivalent to the study of probability distributions. Probability distributions defined on a finite number of points can always be normalized, but when these are defined over continuous spaces, then extra care is called for. The study of such distributions started with the study of random walks and led to the notion of the Ito calculus.

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 lead to a local quantum field theory after Wick rotation to Minkowski spacetime. The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory.