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For Stochastic Process es, including those that arise in Statistical Mechanics and Euclidean Quantum Field Theory , a correlation function is the Correlation between Random Variable s 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. Correlation functions of different random variables are sometimes called '''cross correlations'''. 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. DEFINITION For random variables ''X''(''s'') and ''X''(''t'') at different points ''s'' and ''t'' of some space, the correlation function is : 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 ''X''''i''(''s''), then one can define the matrix of correlation functions : 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 —
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