requires that the S-matrix only has a single delta function. Cluster decomposition can also be formulated in terms of correlation functions, where for any two operators and localized to some region, the vacuum expectation values factorize as the two operators become distantly separated This formulation allows for the property to be applied to theories that lack an S-matrix such as conformal field theories. It is in terms of these Wightman functions that the property is usually formulated in axiomatic quantum field theory. In some formulations, such as Euclidean constructive field theory, it is explicitly introduced as an axiom. Properties 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 . References Quantum field theory Axiomatic quantum field theory Theorems in quantum mechanics