Cluster decomposition

Formulation

The S-matrix ${\displaystyle S_{\beta \alpha }}$ describes the scattering amplitude of a process with an initial state ${\displaystyle \alpha }$ evolving into a final state ${\displaystyle \beta }$. If the initial and final states consist of two clusters, with ${\displaystyle {\alpha }_1}$ and ${\displaystyle {\beta }_1}$ close to each other but far from the pair ${\displaystyle {\alpha }_2}$ and ${\displaystyle {\beta }_2}$, then the cluster decomposition property requires that the S-matrix factorizes

${\displaystyle S_{\beta \alpha }\to S_{{\beta }_1{\alpha }_1}{S}_{{\beta }_2{\alpha }_2}}$

as the distance between the two clusters increases. The physical interpretation of this is that any two spatially well separated experiments ${\displaystyle {\alpha }_1\to {\beta }_1}$ and ${\displaystyle {\alpha }_2\to {\beta }_2}$ cannot influence each other. This condition is fundamental to the ability of doing physics without having to know the state of the entire universe. By expanding the S-matrix in terms of connected S-matrix elements ${\displaystyle S_{\beta \alpha }^c}$, 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 ${\displaystyle {\tilde{S}}_{\beta \alpha }^c}$. Since its Fourier transformation gives the position space connected S-matrix, this only depends on position through exponential terms. Therefore, performing a uniform translation in a direction ${\displaystyle \mathit{a}}$ on a subset of particles will effectively change the momentum space S-matrix as

${\displaystyle {\tilde{S}}_{\beta \alpha }^c\stackrel{{\mathit{x}}_i\to {\mathit{x}}_i+\mathit{a}}{\to }{e}^{i\mathit{a}\cdot (\sum _i{\mathit{p}}_i)}{\tilde{S}}_{\beta \alpha }^c.}$

By translational invariance, a translation of all particles cannot change the S-matrix, therefore ${\displaystyle {\tilde{S}}_{\beta \alpha }}$ must be proportional to a momentum conserving delta function ${\displaystyle \delta (\mathrm{\Sigma }\mathit{p})}$ 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 cluster decomposition 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 ${\displaystyle {\mathcal{O}}_1(x)}$ and ${\displaystyle {\mathcal{O}}_2(x)}$ localized to some region, the vacuum expectation values factorize as the two operators become distantly separated

${\displaystyle \underset{|\mathit{x}|\to \mathrm{\infty }}{lim}⟨{\mathcal{O}}_1(\mathit{x}){\mathcal{O}}_2(0)⟩\to ⟨{\mathcal{O}}_1⟩⟨{\mathcal{O}}_2⟩.}$

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.