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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.

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## Some Related Sentences

quantum and field

A unified interpretation

**of**antiparticles is now available**in**__quantum____field__**theory****,**which solves both these problems**.**
This is an example

**of**renormalization**in**__quantum____field__**theory**—**the**__field__**theory**being necessary because**the**number**of**particles changes from one**to**two and back again**.****In**

__quantum__

__field__

**theory**

**,**this process is allowed only as an intermediate

__quantum__state for times short enough that

**the**violation

**of**energy conservation

**can**

**be**accommodated by

**the**uncertainty principle

**.**

This section draws upon

**the**ideas**,**language and notation**of**canonical quantization**of**a__quantum____field__**theory****.**
Practical applications are made impossible due

**to****the**no-cloning theorem**,**and**the**fact that__quantum____field__theories preserve causality**,**so that__quantum__correlations cannot**be**used**to**transfer information**.**
The analogy was completed when Hawking

**,****in**1974**,**showed that__quantum____field__**theory**predicts that black holes should radiate like a black body**with**a temperature proportional**to****the**surface gravity**of****the**black hole**.**
The physical model behind cosmic inflation is extremely simple

**,**however it has not yet been confirmed by particle physics**,**and there are difficult problems reconciling inflation and__quantum____field__**theory****.**
After World War II

**,**several ideas from__quantum____field__**theory**were applied**to**condensed matter problems**.**
These ideas were unified by Kenneth Wilson

**in**1972**,**under**the**formalism**of****the**renormalization group**in****the**context**of**__quantum____field__**theory****.**
Theoretical models have also been developed

**to**study**the**physics**of**phase transitions**,**such as**the**Landau-Ginzburg**theory****,**Critical exponents and**the**use**of**mathematical techniques**of**__quantum____field__**theory**and**the**renormalization group**.**
Goldstone's theorem

**in**__quantum____field__**theory**states that**in**a system**with**broken continuous symmetry**,**there may exist excitations**with**arbitrarily low energy**,**called**the**Goldstone bosons**.****In**1929

**,**Heisenberg gave a series

**of**invited lectures at

**the**University

**of**Chicago explaining

**the**new

__field__

**of**

__quantum__mechanics

**.**

The problem

**of**thinking**in**terms**of**classical measurements**of**a__quantum__system becomes particularly acute**in****the**__field__**of**__quantum__cosmology**,**where**the**__quantum__system is**the**universe**.**
The general concept

**of**a chemical reaction has been extended**to**non-chemical reactions between entities smaller than atoms**,**including nuclear reactions**,**radioactive decays**,**and reactions between elementary particles as described by__quantum____field__**theory****.****In**

__quantum__

__field__

**theory**

**,**

**the**Casimir effect and

**the**Casimir – Polder force are physical forces arising from a quantized

__field__

**.**

quantum and theory

He discovered that

**the**so-called Weil representation**,**previously introduced**in**__quantum__mechanics by Irving Segal and Shale**,**gave a contemporary framework for understanding**the**classical__theory__**of**quadratic forms**.**
A 2008

__quantum__physics experiment performed**in**Geneva**,**Switzerland has determined that**in**any hypothetical nonlocal hidden-variables__theory__**the**speed**of****the**__quantum__non-local connection would have**to****be**at least 10**,**000 times**the**speed**of**light**.****In**computational complexity

__theory__

**,**BQP ( bounded error

__quantum__polynomial time ) is

**the**class

**of**decision problems solvable by a

__quantum__computer

**in**polynomial time

**,**

**with**an error probability

**of**at most 1 / 3 for all instances

**.**

Bootstrapping is using very general consistency criteria

**to**determine**the**form**of**a__quantum____theory__from some assumptions on**the**spectrum**of**particles**.**
Albert Einstein

**,****in**1922**,**said regarding contemporary theories**of**superconductivity that “**with**our far-reaching ignorance**of****the**__quantum__mechanics**of**composite systems we are very far from being able**to**compose a__theory__out**of**these vague ideas ”

quantum and Wightman

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 >.
He is one

**of****the**founders**of****the**axiomatic approach**to**__quantum__**field****theory****,**and originated**the****set****of**__Wightman__axioms**.**
* Arthur

__Wightman__and Lars Gårding: Fields as operator-valued**distributions****in**relativistic__quantum__**theory****.****In**physics

**the**

__Wightman__axioms are an attempt at a mathematically rigorous formulation

**of**

__quantum__

**field**

**theory**

**.**

For that

**,****the**__Wightman__axioms have position dependent operators called__quantum__fields which form covariant representations**of****the**Poincaré group**.**
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**.**
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**.**

quantum and distributions

For example

**,**large voids**in****the**__distributions__**of**galaxies are believed by many cosmologists**to**have originated**in**__quantum__fluctuations during**the**big bang**.**
For example

**,**Maxwell's equations**of**electromagnetism are linear**in****the**electric and magnetic fields**,**and charge and current__distributions__( i**.**e**.****the**sum**of**two solutions is also a solution ); another example is Schrödinger's equation**of**__quantum__mechanics which is linear**in****the**wavefunction**.**
systematic correspondence does

**,****in**fact**,**exist between**the**__quantum__commutator and a deformation**of****the**Poisson bracket**,****the**Moyal bracket**,**and**,****in**general**,**__quantum__operators and classical observables and__distributions__**in**phase**space****.**
For example

**,****the**potential profile across**the**__quantum__well structure**of**a diode laser**can****be**mapped and analyzed ; such a profile could indicate**the**electron and hole__distributions__where light is generated and could lead**to**improved laser designs**.****In**

__quantum__information

**theory**

**,**

**the**notion

**of**entropy is extended from probability

__distributions__

**to**

__quantum__states

**,**or density matrices

**.**

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