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In theoretical physics, specifically quantum field theory, Zamolodchikov's 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:

  • 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.

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] and roughly counts the degrees of freedom of the system.

Until recently, it had not been possible to prove an analog C-theorem in higher-dimensional quantum field theory. However, in 2011, Zohar Komargodski and Adam Schwimmer of the Weizmann Institute of Science proposed a proof for the physically more important four-dimensional case, which has gained acceptance.[2][3] (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.[4]) 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[edit]


  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. ^ Reich, E. S. (2011). "Proof found for unifying quantum principle". Nature. doi:10.1038/nature.2011.9352. 
  3. ^ 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. 
  4. ^ 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.