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m Unnecessary category as its a mainly RG topic. |
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</ref> using the local renormalization group equation. However, the problem of finding a proof valid beyond perturbation theory remained open for many 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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