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. A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In quantum field theory (QFT the forces between particles are mediated by other particles In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way 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. Astronomers describe the distribution of galaxies in the universe by means of a Correlation function. Financial analysis refers to an assessment of the viability stability and profitability of a Business, sub-business or Project. In quantum field theory (QFT the forces between particles are mediated by other particles Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics (For details, see Correlation function (quantum field theory). In Quantum field theory, correlation functions generalize the concept of Correlation functions in statistics )

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 X_i(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. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable 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. Spacetime symmetries refers to aspects of Spacetime that can be described as exhibiting some form of Symmetry. Examples of important spacetime symmetries are —

translational symmetry yields C(s,s') = C(s − 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 − s'|) where |x| denotes the norm of the vector x (for actual rotations this is the Euclidean or 2-norm).

n is

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

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 — both internal and spacetime. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

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. A random walk, sometimes denoted RW, is a Mathematical formalization of a trajectory that consists of taking successive Random steps Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to Stochastic processes such as Brownian motion ( Wiener process)

The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics. This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of 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. In Quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted In quantum field theory (QFT the forces between particles are mediated by other particles In Physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a problem in Minkowski space from a solution to a related problem In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory. In quantum field theory (QFT the forces between particles are mediated by other particles

Definition

Properties of probability distributions

See also