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| In [[theoretical physics]], specifically [[quantum field theory]], Zamolodchikov's '''C-theorem''' states that there exists a positive real function, <math>C(g^{}_i,\mu)</math>, depending on the [[coupling constant]]s of the quantum field theory considered, <math>g^{}_i</math>, and on the energy scale, <math>\mu^{}_{}</math>, which has the following properties:
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| *<math>C(g^{}_i,\mu)</math> decreases monotonically under the [[renormalization group]] (RG) flow.
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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.
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| [[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.
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| 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).
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| ==See also==
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| *[[Conformal field theory]]
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| ==References==
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| {{reflist}}
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| [[Category:Conformal field theory]]
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| [[Category:Renormalization group]]
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| [[Category:Quantum field theory]]
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| [[Category:Theoretical physics]]
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| [[Category:Mathematical physics]]
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