|
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. Correlation functions of different random variables are sometimes called cross correlations. Jump to: navigation, search In the mathematics of probability, a stochastic process is a random function. ...
Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
Quantum field theory (QFT) is the application of quantum mechanics to fields. ...
Jump to: navigation, search In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...
Jump to: navigation, search A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...
Correlation functions used in astronomy, financial analysis, quantum field theory and statistical mechanics differ only in the particular stochastic processes they are applied to. Astronomers describe the distribution of galaxies in the universe by means of a correlation function. ...
Financial analysis is the analysis of the accounts and the economic prospects of a firm. ...
Quantum field theory (QFT) is the application of quantum mechanics to fields. ...
Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
Definition
Consider a probability density functional P[X(s)] for stochastic variables X(s) at different points s of some space, then the correlation function is In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ...
Generally, functional refers to something with and able to fulfill its purpose or function. ...
where the statistical averages are taken with respect to the measure specified by the probability density function. Measure can mean: To perform a measurement. ...
In this definition, it has been assumed that the stochastic variable is a scalar. 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
or a scalar, which is the trace of this matrix. If the probability density P[X(s)] has any target space symmetries, ie, 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 — - 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).
Higher order correlation functions are often defined. A typical correlation function of order n is If the random variable has only one component, then the indices ij 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 mathematics, the term irreducible is used in several ways. ...
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. Jump to: navigation, search In mathematics and physics, a random walk is a formalization of the intuitive idea of taking successive steps, each in a random direction. ...
Itô calculus, named after Kiyoshi Itô, treats mathematical operations on stochastic processes. ...
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. In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. ...
Statistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ...
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. ...
Quantum field theory (QFT) is the application of quantum mechanics to fields. ...
In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ...
In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...
Jump to: navigation, search Figure 1. ...
Quantum field theory (QFT) is the application of quantum mechanics to fields. ...
See also |
Feeds |
Contact