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.

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,

...
Wikipedia