C-theorem

In theoretical physics, specifically quantum field theory, C-theorem states that there exists a positive real function, $C(g_{i}^{{}},\mu )$, depending on the coupling constants of the quantum field theory considered, $g_{i}^{{}}$, and on the energy scale, $\mu _{{}}^{{}}$, which has the following properties:

• $C(g_{i}^{{}},\mu )$ 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 $g_{i}^{*}$, the function $C(g_{i}^{*},\mu )=C_{*}$ 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

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 theories, Zamolodchikov's C-function is equal to the central charge of the corresponding conformal field theory,^[1] 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 c, but rather to a different quantity a.^[2] 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.^[3][4] (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.^[5]) 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

• Conformal field theory

References

1. Zamolodchikov, A. B. (1986). "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory, JETP Lett 43, pp 730–732.
2. Nakayama, Y. (2015). "Scale invariance vs conformal invariance", Physics Reports 569 1-93.
3. Reich, E. S. (2011). "Proof found for unifying quantum principle". Nature. doi:10.1038/nature.2011.9352.
4. Komargodski, Z.; Schwimmer, A. (2011). "On renormalization group flows in four dimensions". Journal of High Energy Physics. 2011 (12). arXiv:1107.3987. Bibcode:2011JHEP...12..099K. doi:10.1007/JHEP12(2011)099.
5. Curtright, T.; Jin, X.; Zachos, C. (2012). "Renormalization Group Flows, Cycles, and c-Theorem Folklore". Physical Review Letters. 108 (13). arXiv:1111.2649. Bibcode:2012PhRvL.108m1601C. doi:10.1103/PhysRevLett.108.131601.