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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. These functions are called the Schwinger functions and they are analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfies a property known as reflection positivity. Quantum field theory (QFT) is the application of quantum mechanics to fields. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
Domain has several meanings: some kind of territory, such as (for example) a demesne or a realm In New Zealand a Town Domain is typically a public sport area administered by a Domain Board. ...
Pick any arbitrary coordinate τ and pick a test function fN with N points as its arguments. Assume fN has its support in the "time-ordered" subset of N points with 0<τ1 < ... < τN. Choose one such fN for each positive N, with the f's being zero for all N larger than some integer M. Given a point x, let be the reflected point about the τ=0 hyperplane. Then, This page deals with mathematical distributions. ...
The word support has several specialized meanings: In mathematics, see support (mathematics). ...
where * represents complex conjugation. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...
The Osterwalder-Schrader theorem states that Schwinger functions which satisfies these properties can be analytically continued into a quantum field theory. Quantum field theory (QFT) is the application of quantum mechanics to fields. ...
Euclidean path integrals satisfy reflection positivity formally. Pick any polynomial functional F of the field φ which doesn't depend upon the value of φ(x) for those points x whose τ coordinates are nonpositive. This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ...
In mathematics, the term functional is applied to certain functions. ...
Then,
- .
Since the action S is real and can be split into S+ which only depends on φ on the positive half-space and S- which only depends upon φ on the negative half-space and if S also happens to be invariant under reflections, then the previous quantity has to be nonnegative.
See also
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