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Bilingsley (talk | contribs) Split the discussion for 2d and 4d in two different sections. Added the term "A-theorem". |
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*At fixed points of the [[RG flow]], which are specified by a set of fixed-point couplings <math>g^*_i</math>, the function <math>C(g^*_i,\mu)=C_*</math> is a constant, independent of energy scale.
The theorem formalizes the notion that theories at high energies have more degrees of freedom than theories at low energies and that information is lost as we flow from the former to the latter.
[[Alexander Zamolodchikov]] proved in 1986 that two-dimensional quantum field theory always has such a ''C''-function. Moreover, at fixed points of the RG flow, which correspond to [[conformal field theory|conformal field theories]], Zamolodchikov's ''C''-function is equal to the [[central charge]] of the corresponding conformal field theory,<ref>[[Alexander Zamolodchikov|Zamolodchikov, A. B.]] (1986). [http://www.jetpletters.ac.ru/ps/1413/article_21504.pdf "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory], ''JETP Lett'' '''43''', pp 730–732.</ref> and roughly counts the degrees of freedom of the system.▼
==Two-dimensional case==
Until recently, it had not been possible to prove an analog ''C''-theorem in higher-dimensional quantum field theory. However, in 2011, Zohar Komargodski and Adam Schwimmer of the [[Weizmann Institute of Science]] proposed a proof for the physically more important four-dimensional case, which has gained acceptance.<ref>{{cite doi| 10.1038/nature.2011.9352|noedit}}</ref><ref name="komargodski">{{cite doi|10.1007/JHEP12(2011)099|noedit}}</ref> (Still, simultaneous monotonic and cyclic ([[limit cycle]]) or even chaotic RG flows are compatible with such flow functions when multivalued in the couplings, as evinced in specific systems.<ref>{{cite doi|10.1103/PhysRevLett.108.131601|noedit}}</ref>) RG flows of theories in 4 dimensions and the question of whether scale invariance implies conformal invariance, is a field of active research and not all questions are settled (circa 2013).▼
▲[[Alexander Zamolodchikov]] proved in 1986 that two-dimensional quantum field theory always has such a ''C''-function. Moreover, at fixed points of the RG flow, which correspond to [[conformal field theory|conformal field theories]], Zamolodchikov's ''C''-function is equal to the [[central charge]] of the corresponding conformal field theory,<ref>[[Alexander Zamolodchikov|Zamolodchikov, A. B.]] (1986). [http://www.jetpletters.ac.ru/ps/1413/article_21504.pdf "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory], ''JETP Lett'' '''43''', pp 730–732.</ref>
==Four-dimensional case - ''A''-theorem==
Until recently, it had not been possible to prove an analog ''C''-theorem in higher-dimensional quantum field theory. It is known that at fixed points of the RG flow, if such function exists, it will no more be equal to the central charge c, but rather to a different quantity a. For this reason the analog of the ''C''-theorem in four dimensions is called the '''''A''-theorem'''.
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==See also==
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