Editing Cluster decomposition (section)

== Formulation ==

The ''S''-matrix <math>S_{\beta \alpha}</math> describes the [[scattering amplitude|amplitude]] for a process with an initial state <math>\alpha</math> evolving into a final state <math>\beta</math>. If the initial and final states consist of two clusters, with <math>\alpha_1</math> and <math>\beta_1</math> close to each other but far from the pair <math>\alpha_2</math> and <math>\beta_2</math>, then the cluster decomposition property requires the ''S''-matrix to factorize

:<math>
S_{\beta \alpha} \rightarrow S_{\beta_1 \alpha_1}S_{\beta_2\alpha_2}
</math>

as the distance between the two clusters increases. The physical interpretation of this is that any two spatially well separated experiments <math>\alpha_1 \rightarrow \beta_1</math> and <math>\alpha_2 \rightarrow \beta_2</math> cannot influence each other.<ref>{{cite book|first=M. D.|last=Schwartz|title=Quantum Field Theory and the Standard Model|publisher=Cambridge University Press|date=2014|chapter=7|pages=96–97|isbn=9781107034730}}</ref> This condition is fundamental to the ability to doing physics without having to know the [[quantum state|state]] of the entire [[universe]]. By expanding the ''S''-matrix into a sum of a product of connected ''S''-matrix elements <math>S_{\beta \alpha}^c</math>, which at the perturbative level are equivalent to [[Feynman diagram#Connected diagrams: linked-cluster theorem|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 [[position and momentum spaces|momentum space]] ''S''-matrix <math>\tilde S^c_{\beta \alpha}</math>.<ref>{{cite book|first=S.|last=Weinberg|author1-link=Steven Weinberg|title=The Quantum Theory of Fields: Foundations|publisher=Cambridge University Press|date=1995|chapter=4|volume=1|pages=177–188|isbn=9780521670531}}</ref> Since its [[Fourier transform]]ation gives the position space connected ''S''-matrix, this only depends on position through the exponential terms. Therefore, performing a uniform [[Translation operator (quantum mechanics)|translation]] in a direction <math>\boldsymbol a</math> on a subset of particles will effectively change the momentum space ''S''-matrix as

:<math>
\tilde S_{\beta \alpha}^c \xrightarrow{\boldsymbol x_i \rightarrow \boldsymbol x_i+\boldsymbol a} e^{i\boldsymbol a\cdot (\sum_i \boldsymbol p_i)} \tilde S_{\beta \alpha}^c.
</math>

By [[translational symmetry|translational invariance]], a translation of all particles cannot change the ''S''-matrix, therefore <math>\tilde S_{\beta \alpha}</math> must be proportional to a momentum conserving [[delta function]] <math>\delta (\Sigma \boldsymbol p)</math> 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 requires that the ''S''-matrix only has a single delta function.

Cluster decomposition can also be formulated in terms of [[correlation function (quantum field theory)|correlation functions]], where for any two operators <math>\mathcal O_1(x)</math> and <math>\mathcal O_2(x)</math> localized to some region, the vacuum expectation values factorize as the two operators become distantly separated

:<math>
\lim_{|\boldsymbol x|\rightarrow \infty}\langle \mathcal O_1(\boldsymbol x)\mathcal O_2(0)\rangle \rightarrow \langle \mathcal O_1\rangle \langle \mathcal O_2 \rangle.
</math>

This formulation allows for the property to be applied to theories that lack an ''S''-matrix such as [[conformal field theory|conformal field theories]]. It is in terms of these [[Wightman axioms|Wightman functions]] that the property is usually formulated in [[axiomatic quantum field theory]].<ref>{{cite book|last1=Bogolubov|first1=N.N.|author1-link=Nikolay Bogolyubov|last2=Logunov|first2=A.A.|author2-link=Anatoly Logunov|last3=Todorov|first3=I.T.|author3-link=|translator-last1=Fulling|translator-first1=S.A.|translator-link1=Stephen A. Fulling|translator-last2=Popova|translator-first2=L.G.|date=1975|title=Introduction to Axiomatic Quantum Field Theory|url=|doi=|location=|edition=1|publisher=Benjamin|chapter=|pages=272–282|isbn=9780805309829}}</ref> In some formulations, such as Euclidean [[Constructive quantum field theory|constructive field theory]], it is explicitly introduced as an [[axiom]].<ref>{{cite book|last=Iagolnitzer|first=D.|author-link=|date=1993|title=Scattering in Quantum Field Theories The Axiomatic and Constructive Approaches|url=|doi=|location=|publisher=Princeton University Press|chapter=3|pages=155–156|isbn=9780691633282}}</ref>