If CP is combined with time reversal ( T-symmetry ), the resulting CPT-symmetry can be shown using only the Wightman axioms to be universally obeyed.

He is one of the founders of the axiomatic approach to quantum field theory, and originated the set of Wightman axioms.

In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory.

Arthur Wightman formulated the axioms in the early 1950s but they were first published only in 1964, after Haag-Ruelle scattering theory affirmed their significance.

One of the Millennium Problems is to realize the Wightman axioms in the case of Yang-Mills fields.

One basic idea of the Wightman axioms is that there is a Hilbert space upon which the Poincaré group acts unitarily.

For that, the Wightman axioms have position dependent operators called quantum fields which form covariant representations of the Poincaré group.

To get around this, the Wightman axioms introduce the idea of smearing over a test function to tame the UV divergences which arise even in a free field theory.

The Wightman axioms restrict the causal structure of the theory by imposing either commutativity or anticommutativity between spacelike separated fields.

Arthur Wightman showed that the vacuum expectation value distributions, satisfying certain set of properties which follow from the axioms, are sufficient to reconstruct the field theory — Wightman reconstruction theorem, including the existence of a vacuum state ; he did not find the condition on the vacuum expectation values guaranteeing the uniqueness of the vacuum ; this condition, the cluster property, was found later by Res Jost, Klaus Hepp, David Ruelle and Othmar Steinmann.

Unlike local quantum field theory, the Wightman axioms restrict the causal structure of the theory explicitly by imposing either commutativity or anticommutativity between spacelike separated fields, instead of deriving the causal structure as a theorem.

If one considers a generalization of the Wightman axioms to dimensions other than 4, this ( anti ) commutativity postulate rules out anyons and braid statistics in lower dimensions.

The Wightman postulate of a unique vacuum state doesn't necessarily make the Wightman axioms inappropriate for the case of spontaneous symmetry breaking because we can always restrict ourselves to a superselection sector.

The cyclicity of the vacuum demanded by the Wightman axioms means that they describe only the superselection sector of the vacuum ; again, that is not a great loss of generality.

The Wightman axioms can be rephrased in terms of a state called a Wightman functional on a Borchers algebra equal to the tensor algebra of a space of test functions.

One can generalize the Wightman axioms to dimensions other than 4.

Currently, there is no proof that the Wightman axioms can be satisfied for interacting theories in dimension 4.

There is a million dollar prize for a proof that the Wightman axioms can be satisfied for gauge theories, with the additional requirement of a mass gap.

This theorem is the key tool for the constructions of interacting theories in dimension 2 and 3 which satisfy the Wightman axioms.

* Wightman axioms

# REDIRECT Wightman axioms

The Wightman framework does not cover effective field theories because there is no limit as to how small the support of a test function can be.

The title was adapted by Raymond F. Streater and Arthur S. Wightman for their ( serious ) textbook on axiomatic quantum field theory, < cite > PCT, Spin and Statistics, and All That </ cite >.

* Arthur Wightman and Lars Gårding: Fields as operator-valued distributions in relativistic quantum theory.

Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, also modern quantum field theory can be considered close to an axiomatic description.

It was originally introduced by Wick, Wightman, and Wigner to impose additional restrictions to quantum theory beyond those of selection rules.

The traditional basis of constructive quantum field theory is the set of Wightman axioms.

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.

It is used in quantum field theory to construct the analytic continuation of Wightman functions.

* Arthur Wightman: Quantum field theory in terms of vacuum expectation values.

* Arthur Wightman: What is the point of so-called " axiomatic field theory "?.

In 1636, John Wightman gave £ 50 for the poor of Hinckley and a field in Earl Shilton was also let, earning £ 3 5s per year.

* Arthur Wightman: The theory of quantized fields in the 50s, in Brown, Dresden, Hoddeson ( eds.

We have a language where is a constant symbol and is a unary function and the following axioms:

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers.

From these two axioms, it follows that for every g in G, the function which maps x in X to g · x is a bijective map from X to X ( its inverse being the function which maps x to g < sup >− 1 </ sup >· x ).

In complete analogy, one can define a right group action of G on X as a function X × G → X by the two axioms:

The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or The signature ( a formal language's non-logical symbols ) for the axioms includes a constant symbol 0 and a unary function symbol S.

For bipartite pure states, the von Neumann entropy of reduced states is the unique measure of entanglement in the sense that it is the only function on the family of states that satisfies certain axioms required of an entanglement measure.

The function N is called a neighbourhood topology if the axioms below are satisfied ; and then X with N is called a topological space.

* Module: an Abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms.

A Kleene algebra is a set A together with two binary operations +: A × A → A and ·: A × A → A and one function *: A → A, written as a + b, ab and a * respectively, so that the following axioms are satisfied.

He chose the axioms ( see Peano axioms ), in the language of a single unary function symbol S ( short for " successor "), for the set of natural numbers to be:

In this view, theories function as axioms: predicted observations are derived from the theories much like theorems are derived in Euclidean geometry.

In the expected utility theory of von Neumann and Morgenstern, four axioms together imply that individuals act in situations of risk as if they maximize the expected value of a utility function.

The successor function is used in the Peano axioms which define the natural numbers.

An orthocomplementation on a bounded lattice is a function that maps each element a to an " orthocomplement " a < sup >⊥</ sup > in such a way that the following axioms are satisfied:

Savage proved that, if you adhere to axioms of rationality, if you believe an uncertain event has possible outcomes each with a utility to you of then your choices can be explained as arising from a function in which you believe that there is a subjective probability of each outcome is, and your subjective expected utility is the expected value of the utility,

In 1955, Patrick Suppes and Muriel Winet solved the issue of the representability of preferences by a cardinal utility function, and derived the set of axioms and primitive characteristics required for this utility index to work.

A characteristic approach proceeds from a set of reasonable axioms of social choice to a social welfare function ( or constitution ).