Cross-correlation matrix: Difference between revisions

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==Properties of probability distributions==

With these definitions, the study of correlation functions is equivalent to the study of[[ probability distributions]].{{Clarify|date=May 2010}} 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 [[Ito 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.