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.

*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>{{cite journal | last1 = Zamolodchikov | first1 = A. B. | authorlinkauthor-link = Alexander Zamolodchikov | year = 1986 | title = "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory | url = http://www.jetpletters.ac.ru/ps/1413/article_21504.pdf | format = PDF | journal = JETP Lett | volume = 43 | issuepages = 730–732 | pagesbibcode = 730–7321986JETPL..43..730Z }}</ref> which lends the name ''C'' to the theorem.

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

Until[[John recently,Cardy]] itin had1988 notconsidered beenthe possiblepossibility to prove an analoggeneralise ''C''-theorem into higher-dimensional quantum field theory. ItHe is knownconjectured that atin fixedfour pointsspacetime ofdimensions, the RGquantity flow,behaving monotonically ifunder suchrenormalization functiongroup existsflows, itand willthus noplaying morethe berole equalanalogous to the central charge {{mvar|c}} in two dimensions, butis rathera certain anomaly coefficient which came to abe differentdenoted quantityas {{mvar|a}}.<ref>{{cite journal | last1 = NakayamaCardy | first1 = YJohn | year = 20151988 | title = ScaleIs invariancethere vsa conformalc-theorem invariancein |four url =dimensions? | journal = Physics ReportsLetters B | volume = 569215 | issue = 4| pages = 1–93749–752 | doi=10.1016/j0370-2693(88)90054-8|bibcode =1988PhLB.physrep.2014215.12.003749C }}</ref> For this reason, the analog of the ''C''-theorem in four dimensions is called the '''''A''-theorem'''.

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

In 2011perturbation theory, Zoharthat Komargodskiis andfor Adamrenormalization Schwimmerflows ofwhich thedo [[Weizmannnot Institutedeviate ofmuch Science]]from proposedfree a proof fortheories, the ''A''-theorem, whichin hasfour gaineddimensions acceptance.<ref>{{Citewas journalproved |by last1[[Hugh =Osborn]] Reichusing |the first1local =renormalization E.group Sequation. | doi = 10.1038/nature.2011.9352 | title = Proof found for unifying quantum principle | journal = Nature | year = 2011 | pmid = | pmc = }}</ref><ref name="komargodski">{{Citecite journal | last1 = KomargodskiOsborn | first1 = Z.Hugh | last2year = Schwimmer | first2 = A. | doi = 10.1007/JHEP12(2011)0991989 | title = OnDerivation renormalizationof groupa flowsFour-Dimensional inc four dimensionsTheorem | journal = JournalPhysics ofLetters High Energy PhysicsB | volume = 2011222 | issue = 12 1| yearpages = 201197 | pmid doi= | pmc = |arxiv = 110710.3987 1016/0370-2693(89)90729-6|bibcode = 2011JHEP1989PhLB..222.12..099K97O }}</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>{{Citecite journal | last1 = CurtrightIan | first1 = T. Jack| last2 = JinOsborn | first2 = X.Hugh | last3year = Zachos1990 | first3title = C.Analogs |for titlethe =c RenormalizationTheorem Groupfor Flows,Four-Dimensional Cycles,Renormalizable and c-TheoremField FolkloreTheories | doiurl = 10.1103https:/PhysRevLett/cds.108cern.131601 ch/record/205908| journal = PhysicalNuclear ReviewPhysics LettersB | volume = 108343 | issue = 13 3| yearpages = 2012647–688 | pmid doi= 22540692| pmc = |arxiv = 111110.2649 1016/0550-3213(90)90584-Z|bibcode = 2012PhRvL1990NuPhB.343..108m1601C647J | page=131601}}</ref>) RG flows of theories in 4 dimensions andHowever, the questionproblem of whether scale invariance implies conformal invariance, isfinding a fieldproof ofvalid activebeyond researchperturbation andtheory notremained allopen questionsfor aremany settled (circa 2013)years.

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 | s2cid = 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 | pages = 99 | year = 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 = 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.

==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]]