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Locality condition in quantum field theory
In physics, the cluster decomposition property states that experiments carried out far from each other cannot influence each other. Usually applied to quantum field theory, it requires that vacuum expectation values of operators localized in bounded regions factorize whenever these regions becomes sufficiently distant from each other. First formulated by Eyvind Wichmann and James H. Crichton in 1963 in the context of the S matrix, it was conjectured by Steven Weinberg that in the low energy limit the cluster decomposition property, together with Lorentz invariance and quantum mechanics, inevitably lead to quantum field theory. String theory satisfies all three of the conditions and so provides a counter example against this being true at all energy scales.

If a theory is constructed from creation and annihilation operators, then the cluster decomposition property automatically holds. This can be seen by expanding out the S matrix as a sum of Feynman diagrams which allows for the identification of connected S matrix elements with connected Feynman diagrams. Vertices arise whenever creation and annihilation operators commute past each other leaving behind a single momentum delta function. In any connected diagram with V vertices, I internal lines and L loops, I L of the delta functions go into fixing internal momenta, leaving V (I L) delta functions unfixed. A form of Euler's formula states that any graph with C disjoint connected components satisfies C = V I+L. Since the connected S matrix elements correspond to C=1 diagrams, these only have a single delta function and thus the cluster decomposition property, as formulated above in momentum space in terms of delta functions, holds. Microcausality, the locality condition requiring commutation relations of local operators to vanish for spacelike separations, is a sufficient condition for the S matrix to satisfy cluster decomposition. In this sense cluster decomposition serves a similar purpose for the S matrix as microcausality does for fields, preventing causal influence from propagating between regions that are distantly separated. However, cluster decomposition is weaker than having no superluminal causation since it can be formulated for classical theories as well. One key requirement for cluster decomposition is that it requires a unique vacuum state, with it failing if the vacuum state is a mixed state. The rate at which the correlation functions factorize depends on the spectrum of the theory, where if it has mass gap of mass then there is an exponential falloff while if there are massless particles present then it can be as slow as .