Cross-correlation matrix: Difference between revisions

Revision as of 14:16, 27 July 2013

Revision as of 20:41, 19 August 2013

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==Properties of probability distributions==

With these definitions, the study of correlation functions is equivalentsimilar to the study of [[probability distributions]]. Many stochastic processes can be completely characterized by their correlation functions; the most notable example is the class of [[Gaussian processes]].

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 walk]]s and led to the notion of the [[Itō calculus]].

The Feynman [[path integral formulation|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.