C-theorem: Difference between revisions

Revision as of 07:26, 22 March 2016

Revision as of 14:54, 15 January 2017

Line 8:

==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 [[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.]] (| authorlink = 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.ac.ru/ps/1413/article_21504.pdf | format = PDF | journal = JETP Lett'' '''| volume = 43''', pp| issue = | pages = 730–732. }}</ref> which lends the name ''C'' to the theorem.

==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>{{cite journal | last1 = Nakayama, | first1 = Y. (| year = 2015). "| title = Scale invariance vs conformal invariance", ''| url = | journal = Physics Reports'' '''| volume = 569''' | issue = | pages = 1–93 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 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 | pmid = | pmc = }}</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 | year = 2011 | pmid = | pmc = |arxiv = 1107.3987 |bibcode = 2011JHEP...12..099K }}</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 = |arxiv = 1111.2649 |bibcode = 2012PhRvL.108m1601C }}</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).