Content deleted Content added
Cuzkatzimhut (talk | contribs) |
m Remove blank line(s) between list items per WP:LISTGAP to fix an accessibility issue for users of screen readers. Do WP:GENFIXES and cleanup if needed. Discuss this at Wikipedia talk:WikiProject Accessibility#LISTGAP |
||
Line 2:
In [[theoretical physics]], specifically [[quantum field theory]], '''''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 than theories at low energies and that information is lost as we flow from the former to the latter.
==Two-dimensional case==
Line 12 ⟶ 11:
==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.
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).
| |||