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,[1] 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 the initial and final states consist of two clusters, with
The physical interpretation of this is that any two spatially well separated experiments
[3] This condition is fundamental to the ability to doing physics without having to know the state of the entire universe.
, which at the perturbative level are equivalent to connected Feynman diagrams, the cluster decomposition property can be restated as demanding that connected S-matrix elements must vanish whenever some of its clusters of particles are far apart from each other.
on a subset of particles will effectively change the momentum space S-matrix as By translational invariance, a translation of all particles cannot change the S-matrix, therefore
This means that in momentum space the property 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
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.
[5] In some formulations, such as Euclidean constructive field theory, it is explicitly introduced as an axiom.
[6] 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.
[7] However, cluster decomposition is weaker than having no superluminal causation since it can be formulated for classical theories as well.