In physics, the Cardy formula gives the entropy of a

two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal ...

(CFT). In recent years, this formula has been especially useful in the calculation of the entropy of

BTZ black hole The BTZ black hole, named after Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli, is a black hole solution for (2+1)-dimensional topological gravity with a negative cosmological constant. History In 1992 Bañados, Teitelboim and Zanel ...

s and in checking the

AdS/CFT correspondence In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter ...

and the

holographic principle The holographic principle is an axiom in string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a ...

. In 1986 J. L. Cardy derived the formula: :

S=2\pi\sqrt,

Here

c

is the

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 elemen ...

L_0 = ER

is the product of the total energy and radius of the system, and the shift of

c/24

is related to the

Casimir effect In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pred ...

. These data emerge from the

Virasoro algebra In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string t ...

of this CFT. The proof of the above formula relies on

modular invariance In theoretical physics, modular invariance is the invariance under the group such as SL(2,Z) of large diffeomorphisms of the torus. The name comes from the classical name modular group of this group, as in modular form theory. In string theory, m ...

of a Euclidean CFT on the torus. The Cardy formula is usually understood as counting the number of states of energy

\Delta = L_0 + \bar_0

of a CFT quantized on a circle. To be precise, the microcanonical entropy (that is to say, the logarithm of the number of states in a shell of width

\delta \lesssim 1

) is given by :

S_\delta(\Delta) = 2\pi \sqrt + O(\ln \Delta)

in the limit

\Delta \to \infty

. This formula can be turned into a rigorous bound. In 2000, E. Verlinde extended this to certain strongly-coupled (n+1)-dimensional CFTs. The resulting Cardy–Verlinde formula was obtained by studying a radiation-dominated universe with the

Friedmann–Lemaître–Robertson–Walker metric The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe ...

ds^2=-dt^2+R^2(t)\Omega^2_n

where R is the radius of a n-dimensional sphere at time t. The radiation is represented by a (n+1)-dimensional CFT. The entropy of that CFT is then given by the formula :

S=\frac\sqrt,

where E_c is the Casimir effect, and E the total energy. The above reduced formula gives the maximal entropy :

S\le S_=\frac,

when E_c=E, which is the

Bekenstein bound In physics, the Bekenstein bound (named after Jacob Bekenstein) is an upper limit on the thermodynamic entropy ''S'', or Shannon entropy ''H'', that can be contained within a given finite region of space which has a finite amount of energy—or co ...

. The Cardy–Verlinde formula was later shown by Kutasov and Larsen to be invalid for weakly interacting CFTs. In fact, since the entropy of higher dimensional (meaning n>1) CFTs is dependent on exactly marginal couplings, it is believed that a Cardy formula for the entropy is not achievable when n>1.

References

* {{DEFAULTSORT:Cardy-formula Quantum gravity Conformal field theory

See also

References