Clebsch–Gordan coefficients: Difference between revisions

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In [[physics]], the '''cluster decomposition property''' is related to [[Principle of locality|locality]] in [[quantum field theory]]. In a quantum field theory having this property, the [[vacuum expectation value]] of a product of many [[operator (mathematics)|operator]]s - each of them being either in region A or in region B where A and B are very separated - asymptotically equals the product of the expectation value of the product of the operators in A, times a similar factor from the region B. Consequently, sufficiently separated regions behave independently.


If A<sub>1</sub>, ..., A<sub>n</sub> are n operators each localized in a bounded region and <math>U(a)</math> represents the [[unitary operator]] [[active transformation|actively translating]] the Hilbert space by the vector a, then if we pick some subset of the n operators to translate,
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:<math>\lim_{|a|\rightarrow \infty}\langle\Omega|A'_1\cdots A'_n|\Omega\rangle-\langle\Omega|\prod_{\mbox{unshifted i}}A_i|\Omega\rangle \langle\Omega|\prod_{\mbox{shifted i}}U(a)A_i U^{-1}(a)|\Omega\rangle=0</math>
 
where <math>\vert \Omega \rangle</math> is the [[vacuum state]], and
 
:<math>A'_i=\left\{\begin{matrix}A_i &\mbox{  if it is one of the unshifted operators}\\U(a)A_iU^{-1}(a) &\mbox{  if it is one of the shifted operators}\end{matrix}\right .</math>
 
provided a is a [[spacelike]] vector.
 
Expressed in terms of the [[connected correlation function]]s, it means if some of the arguments of the connected correlation function are shifted by large spacelike separations, the function goes to zero.
 
This property only holds if the vacuum is a [[pure state]]. If the vacuum is [[degenerate energy levels|degenerate]] and we have a [[Mixed state (physics)|mixed state]], the cluster decomposition property fails.
 
If the theory has a [[mass gap]] <math>m>0</math>, then there is a value a<sub>0</sub> beyond which the connected correlation function is absolutely bounded by <math> C e^{-m\vert a \vert }</math> where <math>C</math> is some coefficient and <math>\vert a \vert </math> is the length of the vector <math>a</math> for <math>\vert a \vert > a_0</math>.
 
{{DEFAULTSORT:Cluster Decomposition Theorem}}
[[Category:Quantum field theory]]
[[Category:Theorems in quantum physics]]
 
 
{{Quantum-stub}}

Latest revision as of 19:37, 14 November 2014

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