C-theorem
Encyclopedia
In theoretical physics
, specifically conformal field theory
(CFT), the Zamolodchikov C-theorem states that there exists a positive real function,
, depending on the coupling constant
s of the CFT,
, and on the energy scale, 
, which has the following properties:
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
.
Until recently, it had not been possible to prove a C-theorem in higher-dimensional quantum field theory. This changed in July 2011 when Zohar Komargodski and Adam Schwimmer of the Weizmann Institute of Science in Rehovot, Israel proposed a proof for the physically important four-dimensional case. This proof has gradually gained acceptance since its publication.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, specifically conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
(CFT), the Zamolodchikov C-theorem states that there exists a positive real function,
, depending on the coupling constantCoupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
s of the CFT,
, and on the energy scale, , which has the following properties:
decreases monotonically under renormalization groupRenormalization groupIn theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...
(RG) flow.
- At fixed points of the RG flow, which are specified by a set of fixed-point couplings
, the function 
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
Alexander Zamolodchikov
Alexander Borissowitsch Zamolodchikov is a Russian physicist, known for his contributions to condensed matter physics and string theory.Born near Dubna,...
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
Central charge
In theoretical physics, a central charge is an operator Z that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group -- the subgroup of elements that commute with all other elements of the original group—or to the center of a Lie algebra...
of the corresponding conformal field theory
Conformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
.
Until recently, it had not been possible to prove a C-theorem in higher-dimensional quantum field theory. This changed in July 2011 when Zohar Komargodski and Adam Schwimmer of the Weizmann Institute of Science in Rehovot, Israel proposed a proof for the physically important four-dimensional case. This proof has gradually gained acceptance since its publication.