Cluster decomposition theorem

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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 are operators each localized in a bounded region and represents the unitary operator actively translating the Hilbert space by the vector , then if we pick some subset of the n operators to translate,

where is the vacuum state, and

provided 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 , then there is a value beyond which the connected correlation function is absolutely bounded by where is some coefficient and is the length of the vector for .

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.


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