Editing Cluster decomposition

{{Short description|Locality condition in quantum field theory}}
In [[physics]], the '''cluster decomposition property''' states that [[experiment]]s carried out far from each other cannot influence each other. Usually applied to [[quantum field theory]], it requires that [[vacuum expectation value]]s of [[operator (physics)|operators]] localized in [[bounded set|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|''S''-matrix]],<ref>{{cite journal|last1=Wichmann|first1=E.H.|author-link1=|last2=Crichton|first2=J.H.|author-link2=|date=1963|title=Cluster Decomposition Properties of the S Matrix|url=https://link.aps.org/doi/10.1103/PhysRev.132.2788|journal=Phys. Rev.|volume=132|issue=6|pages=2788–2799|doi=10.1103/PhysRev.132.2788|publisher=American Physical Society|pmid=|arxiv=|bibcode=1963PhRv..132.2788W |s2cid=|access-date=|url-access=subscription}}</ref> it was conjectured by [[Steven Weinberg]] that in the [[effective field theory|low energy limit]] the cluster decomposition property, together with [[Lorentz covariance|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.<ref>{{cite conference|last1=Weinberg|first1=S.|author-link1=|date=1996|title=What is quantum field theory, and what did we think it is?|conference=Conference on Historical Examination and Philosophical Reflections on the Foundations of Quantum Field Theory|url=|journal=|volume=|issue=|pages=241–251|doi=|pmid=|arxiv=hep-th/9702027|s2cid=|access-date=}}</ref>

== 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>

== 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>

==References==
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{{DEFAULTSORT:Cluster decomposition}}
[[Category:Quantum field theory]]
[[Category:Axiomatic quantum field theory]]
[[Category:Theorems in quantum mechanics]]