C-theorem: Difference between revisions

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{{short description|Theorem in quantum field theory}}

{{DISPLAYTITLE:''C''-theorem}}

In [[theoretical physics]], specifically [[quantum field theory]], the '''''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:

*<math>C(g^{}_i,\mu)</math> decreases monotonically under the [[renormalization group]] (RG) flow.

*<math>C(g^{}_i,\mu)</math> decreases monotonically under the [[renormalization group]] (RG) flow.

*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 (physics and chemistry)|degrees of freedom]] than theories at low energies and that information is lost as we flow from the former to the latter.

==Two-dimensional case==

[[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 [[two-dimensional conformal field theory|conformal field theories]], Zamolodchikov's ''C''-function is equal to the [[central charge]] of the corresponding conformal field theory,<ref>[[Alexander{{cite journal Zamolodchikov| last1 = Zamolodchikov, | first1 = A. B.]] (| author-link = Alexander Zamolodchikov | year = 1986). | [http://www.jetpletters.ac.ru/ps/1413/article_21504.pdftitle = "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory], ''| url = http://www.jetpletters.ru/ps/1413/article_21504.pdf | journal = JETP Lett'' '''| volume = 43''', pp| pages = 730–732 |bibcode = 1986JETPL..43..730Z }}</ref> which lends the name ''C'' to the theorem.

==Four-dimensional case -: ''A''-theorem==

[[John Cardy]] in 1988 considered the possibility to generalise ''C''-theorem to higher-dimensional quantum field theory. He conjectured that in four spacetime dimensions, the quantity behaving monotonically under renormalization group flows, and thus playing the role analogous to the central charge {{mvar|c}} in two dimensions, is a certain anomaly coefficient which came to be denoted as {{mvar|a}}.<ref>{{cite journal | last1 = Cardy | first1 = John | year = 1988 | title = Is there a c-theorem in four dimensions? | journal = Physics Letters B | volume = 215 | issue = 4| pages = 749–752 | doi=10.1016/0370-2693(88)90054-8|bibcode =1988PhLB..215..749C }}</ref>

For this reason, the analog of the ''C''-theorem in four dimensions is called the '''''A''-theorem'''.

In perturbation theory, that is for renormalization flows which do not deviate much from free theories, the ''A''-theorem in four dimensions was proved by [[Hugh Osborn]] using the local renormalization group equation.<ref>{{cite journal | last1 = Osborn | first1 = Hugh | year = 1989 | title = Derivation of a Four-Dimensional c Theorem | journal = Physics Letters B | volume = 222 | issue = 1| pages = 97 | doi=10.1016/0370-2693(89)90729-6|bibcode =1989PhLB..222...97O }}</ref><ref>{{cite journal | last1 = Ian | first1 = Jack| last2 = Osborn | first2 = Hugh | year = 1990 | title = Analogs for the c Theorem for Four-Dimensional Renormalizable Field Theories | url = https://cds.cern.ch/record/205908| journal = Nuclear Physics B | volume = 343 | issue = 3| pages = 647–688 | doi=10.1016/0550-3213(90)90584-Z|bibcode =1990NuPhB.343..647J }}</ref> However, the problem of finding a proof valid beyond perturbation theory remained open for many years.

==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 {{mvar|c}}, but rather to a different quantity {{mvar|a}}.<ref>Nakayama, Y. (2015). "Scale invariance vs conformal invariance", ''Physics Reports'' '''569''' 1-93. </ref> For this reason, the analog of the ''C''-theorem in four dimensions is called the '''''A''-theorem'''.

In 2011, [[Zohar Komargodski]] and Adam Schwimmer of the [[Weizmann Institute of Science]] proposed a nonperturbative proof for the ''A''-theorem, which has gained acceptance.<ref>{{Cite journal | last1 = Reich | first1 = E. S. | doi = 10.1038/nature.2011.9352 | title = Proof found for unifying quantum principle | journal = Nature | year = 2011 | pmids2cid = | pmc =211729430 }}</ref><ref name="komargodski">{{Cite journal | last1 = Komargodski | first1 = Z. | last2 = Schwimmer | first2 = A. | doi = 10.1007/JHEP12(2011)099 | title = On renormalization group flows in four dimensions | journal = Journal of High Energy Physics | volume = 2011 | issue = 12 | yearpages = 201199 | pmidyear = | pmc =2011 |arxiv = 1107.3987 |bibcode = 2011JHEP...12..099K | s2cid = 119231010 }}</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 journal | last1 = Curtright | first1 = T. | last2 = Jin | first2 = X. | last3 = Zachos | first3 = C. | title = Renormalization Group Flows, Cycles, and c-Theorem Folklore | doi = 10.1103/PhysRevLett.108.131601 | journal = Physical Review Letters | volume = 108 | issue = 13 | year = 2012 | pmid = | pmc = 22540692|arxiv = 1111.2649 |bibcode = 2012PhRvL.108m1601C | page=131601| s2cid = 119144040 }}</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).

==See also==

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[[Category:Conformal field theory]]

[[Category:Renormalization group]]

[[Category:Quantum field theory]]

[[Category:Theoretical physics]]

[[Category:Mathematical physics]]

[[Category:Theorems in mathematicalquantum physicsmechanics]]