The Holographic Principle

The number of possible quantum states in a region of flat space is bounded by the exponential of the area of the region in Planck units. That fact together with the Ultraviolet/Infrared connection and Black Hole Complementarity has led physics to an entirely new paradigm about the nature of space, time and locality. One of the elements of this paradigm is the Holographic Principle and its embodiment in AdS space.

Let us consider a region of flat space Γ. We have seen that the maximum entropy of all physical systems that can fit in Γ is proportional to the area of the boundary ∂Γ, measured in Planck units. Typically, as in the case of a lattice of spins, the maximum entropy is a measure of the number of simple degrees of freedom^∗ that it takes to completely describe the region. This is almost always proportional to the volume of Γ. The exception is gravitational systems. The entropy bound tells us that the maximum number of non-redundant degrees of freedom is proportional to the area. For a large macroscopic region this is an enormous reduction in the required degrees of freedom. In fact if the linear dimensions of the system is of order L then the number of degrees of freedom per unit volume scales like 1/L in Planck units. By making L large enough we can make the degrees of freedom arbitrarily sparse in space. Nevertheless we must be able to describe microscopic processes taking place anywhere in the region. One way to think of this is to imagine the degrees of freedom of Γ as living on ∂Γ with an area density of no more than ∼ 1 degree of freedom per Planck area. The analogy with a hologram is obvious; three-dimensional space described by a two-dimensional hologram at its boundary! That this is possible is called the Holographic Principle.

What we would ideally like to do is to have a solution of Einstein's equations that describes a ball of space with a spherical boundary and then to count the number of degrees of freedom. Even better would be to construct a description of the region in terms of a boundary theory with a limited density of degrees of freedom. Ordinarily , it does not make sense to consider a ball-like region with a boundary in the general theory of relativity. But there is one special situation which is naturally ball-like. It occurs when there is a negative cosmological constant; Anti de Sitter space. Thus AdS is a natural framework in which to study the Holographic Principle.

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∗ By a simple degree of freedom we mean something like a spin or the presence or absence of a fermion. A simple degree of freedom represents a single bit of information.