C-theorem

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

• $C(g_{i}^{},\mu )$ decreases monotonically under 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.

If such a so-called C-function exists for a given quantum field theory, it tells us that the RG flow of the theory is irreversible, since the central charge of a CFT characterizes the number of its degrees of freedom (Cardy formula).

Another way to look at this theorem is to say that, since the RG flow is irreversible due to loss of information under renormalization transformations, then the central charge characterizes the number of degrees of freedom of a CFT.

Alexander Zamolodchikov proved that two-dimensional conformal field theory always has a C-function. Moreover, at fixed points, Zamolodchikov's C-function is equal to the central charge of the corresponding conformal field theory.

It has not yet been possible to prove a C-theorem in higher-dimensional quantum field theory.

Zohar Komargodski and Adam Schwimmer of the Weizmann Institute of Science in Rehovot, Israel, put forward a proof in July 2011 which has gradually gained acceptance since its publication.

See also

• Conformal field theory

References

• A.B Zamolodchikov, "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory, JETP Lett.43:730-732,1986.