Ole Barndorff-Nielsen

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:

• $C(g_{i}^{},\mu )$ 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.

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, 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. (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.) 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

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