Editing Cluster decomposition (section)

== 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. [[Vertex (graph theory)|Vertices]] arise whenever creation and annihilation operators commute past each other leaving behind a single momentum delta function. In any [[connectivity (graph theory)|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 characteristic|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 [[principle of locality|locality]] condition requiring commutation relations of local operators to vanish for [[spacetime#spacetime interval|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 [[field (physics)|fields]], preventing [[causality (physics)|causal]] influence from propagating between regions that are distantly separated.<ref>{{cite book|last=Brown|first=L.S.|author-link=Lowell S. Brown|date=1992|title=Quantum Field Theory|url=|doi=10.1017/CBO9780511622649|location=Cambridge|publisher=Cambridge University Press|chapter=6|pages=311–313|isbn=978-0521469463}}</ref> However, cluster decomposition is weaker than having no [[faster-than-light|superluminal causation]] since it can be formulated for classical theories as well.<ref>{{cite journal|last1=Bain|first1=J.|author-link1=|date=1998|title=Weinberg on Qft: Demonstrative Induction and Underdetermination|url=http://www.jstor.org/stable/20118095|journal=Synthese|volume=117|issue=1|pages=7–8|doi=10.1023/A:1005025424031|jstor=20118095 |pmid=|arxiv=|s2cid=9049200|access-date=|url-access=subscription}}</ref>

One key requirement for cluster decomposition is that it requires a unique [[vacuum state]], with it failing if the vacuum state is a [[quantum state|mixed state]].<ref>{{cite book|first=S.|last=Weinberg|title=The Quantum Theory of Fields: Modern Applications|publisher=Cambridge University Press|date=1995|chapter=19|volume=2|page=167|isbn=9780521670548}}</ref> The rate at which the correlation functions factorize depends on the spectrum of the theory, where if it has [[mass gap]] of mass <math>m</math> then there is an exponential falloff <math>\langle \phi(x) \phi(0)\rangle \sim e^{-m|x|}</math> while if there are [[massless particle]]s present then it can be as slow as <math>1/|x|^2</math>.<ref>{{cite book|last1=Streater|first1=R.F.|author1-link=Ray Streater|last2=Wightman|first2=A.S.|author2-link=Arthur Wightman|orig-date=1964|publication-date=2000|title=PCT, Spin and Statistics, and All That|url=|doi=|location=Princeton|publisher=Princeton University Press|chapter=3|page=113|isbn=978-0691070629}}</ref>