String Theory Demystified

CHAPTER 15 The Holographic Principle 257
Hence we fi nd that 2 3
V/ a
= exp S,
or taking the logarithm of both sides:
3
V/ a V
S = ln( 2 ) = ln 2 3
a
We’ve found what we intuitively expect—the entropy (and by extension the
amount of information) in the region is proportional to the volume. After all we
started off assuming we had a lattice of spins that fi lled the volume—so what else
could we get?
For black holes we found something very different. In that case, the entropy is
directly proportional to the area of the even horizon. So in some sense, gravity must
be different from other interactions. It turns out that the case of a black hole provides
the maximum entropy that a gravitational system can have.
A Qualitative Description of
AdS/CFT Correspondence
The framework of the holographic principle which comes out of string/M-theory is
known as AdS/CFT (anti-de Sitter/conformal fi eld theory) correspondence. We can
quantitatively describe the space-time using AdS space in fi ve dimensions. The
fi ve-dimensional AdS model has a boundary with four dimensions that looks like
fl at space with three spatial directions and one time dimension.
The AdS/CFT correspondence involves a duality, something we’re already
familiar with from our studies of superstring theories. This duality is between two
types of theories:
• Five-dimensional gravity
• Super Yang-Mills theory defi ned on the boundary
By “super” Yang-Mills theory we mean theory of particle interactions with
supersymmetry. The holographic principle comes out of the correspondence between
these two theories because Yang-Mills theory, which is happening on the boundary, is
equivalent to the gravitational physics happening in the fi ve-dimensional AdS geometry.
So the Yang-Mills theory can be colloquially thought of as a hologram on the boundary
of the real fi ve-dimensional space where the fi ve-dimensional gravitational physics is
taking place.