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 H. 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. Formulation The S-matrix describes the amplitude for a process with an initial state evolving into a final state . If the initial and final states consist of two clusters, with and close to each other but far from the pair and , then the cluster decomposition property requires the S-matrix to factorize as the distance between the two clusters increases. The physical interpretation of this is that any two spatially well separated experiments and cannot influence each other. This condition is fundamental to the ability to doing physics without having to know the state of the entire universe. By expanding the S-matrix into a sum of a product of connected S-matrix elements , 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. This position space formulation can also be reformulated in terms of the momentum space S-matrix . Since its Fourier transformation gives the position space connected S-matrix, this only depends on position through the exponential terms. Therefore, performing a uniform translation in a direction 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 must be proportional to a momentum conserving delta function to ensure that the translation exponential factor vanishes. If there is an additional delta function of only a subset of momenta corresponding to some cluster of particles, then this cluster can be moved arbitrarily far through a translation without changing the S-matrix, which would violate cluster decomposition. This means that in momentum space the property