Cluster decomposition theorem

In physics, the cluster decomposition property is related to locality in quantum field theory. In a quantum field theory having this property, the vacuum expectation value of a product of many operators – 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. Functional average of a number of field operator is called correlation function or correlator. So the spacelike asymptotic behaviour of truncated correlators consisting of field clusters determines how the strength of the correlations between the field degrees of freedom in these clusters changes as the distance between the clusters grows, and this behaviour is characterised by the cluster decomposition theorem.[1]

If ${\displaystyle A_{1},\dots ,A_{n}}$ are ${\displaystyle n}$ operators each localized in a bounded region and ${\displaystyle U(a)}$ represents the unitary operator actively translating the Hilbert space by the vector ${\displaystyle a}$, then if we pick some subset of the n operators to translate,

${\displaystyle \lim _{|a|\rightarrow \infty }\langle \Omega |A'_{1}\cdots A'_{n}|\Omega \rangle -\langle \Omega |\prod _{\mbox{unshifted i}}A_{i}|\Omega \rangle \langle \Omega |\prod _{\mbox{shifted i}}U(a)A_{i}U^{-1}(a)|\Omega \rangle =0}$

where ${\displaystyle \vert \Omega \rangle }$ is the vacuum state, and

${\displaystyle A'_{i}=\left\{{\begin{array}{cl}A_{i}&{\mbox{if it is one of the unshifted operators}}\\U(a)A_{i}U^{-1}(a)&{\mbox{if it is one of the shifted operators}}\end{array}}\right.}$

provided ${\displaystyle a}$ is a spacelike vector.

Expressed in terms of the connected correlation functions, 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 and we have a mixed state, the cluster decomposition property fails.

If the theory has a mass gap ${\displaystyle m>0}$, then there is a value ${\displaystyle a_{0}}$ beyond which the connected correlation function is absolutely bounded by ${\displaystyle Ce^{-m\vert a\vert }}$ where ${\displaystyle C}$ is some coefficient and ${\displaystyle \vert a\vert }$ is the length of the vector ${\displaystyle a}$ for ${\displaystyle \vert a\vert >a_{0}}$.

References

1. Lowdon, Peter (January–March 2017). "Confinement and the cluster decomposition property in QCD". Nuclear and Particle Physics Proceedings. 282–284: 168. Bibcode:2017NPPP..282..168L. doi:10.1016/j.nuclphysbps.2016.12.032.