Holographic Principle
The holographic principle states that all information in a volume of space can be encoded on its boundaryâlike a hologram encoding 3D information in 2D. This profound idea, motivated by black hole thermodynamics, suggests that spacetime itself may be emergent from more fundamental physics.
The Bekenstein Bound
The maximum entropy in a region is bounded by its surface area, not its volume:
$$S \leq \frac{A}{4\ell_P^2} = \frac{k_B c^3 A}{4G\hbar}$$
where $A$ is the area of the boundary and $\ell_P$ is the Planck length. This bound is saturated by black holes, which have maximal entropy for a given size.
Why Area, Not Volume?
In normal thermodynamics, entropy is extensiveâproportional to volume. But black hole entropy scales with area. This suggests that the fundamental degrees of freedom live on the boundary, not in the bulk.
The Holographic Principle
't Hooft (1993) and Susskind (1995) formulated the holographic principle:
Statement:
The physics in a region of space can be completely described by degrees of freedom residing on the boundary of that region. The number of degrees of freedom scales as the boundary area in Planck units.
$$N_{DOF} \sim \frac{A}{4\ell_P^2}$$
This means a 3D universe with gravity is equivalent to a 2D theory without gravity on its boundaryâa radical departure from our intuitive understanding of spacetime.
Implications
1. Spacetime is Emergent
If the fundamental theory lives on the boundary, then the extra spatial dimensionâand possibly spacetime itselfâemerges from entanglement and quantum information on the boundary.
2. Quantum Gravity is Finite
The UV divergences plaguing quantum field theory in curved spacetime are artifacts. The true theory on the boundary has a finite number of degrees of freedom.
3. Black Hole Information Preserved
In the boundary theory, black hole formation and evaporation are unitary processes. Information is never lostâit's encoded on the boundary all along.
4. ER = EPR
Maldacena and Susskind (2013) proposed that quantum entanglement ($EPR$) and wormholes ($ER$bridges) are the same phenomenon. Entangled particles are connected by non-traversable wormholes.
AdS/CFT Correspondence
The holographic principle is realized concretely in the AdS/CFT correspondence (Maldacena 1997):
The Duality:
Type IIB string theory on AdSâ Ă Sâ” is exactly equivalent to N=4 Super Yang-Mills theory in 4D.
- Bulk (AdSâ ): 5D gravity + matter
- Boundary (âAdSâ ): 4D conformal field theory (no gravity)
This is a precise mathematical realization of holography: quantum gravity in the bulk is dual to a non-gravitational gauge theory on the boundary.
Dictionary
- Black hole in AdS $\leftrightarrow$ Thermal state in CFT
- Hawking-Page transition $\leftrightarrow$ Confinement/deconfinement
- Bulk distance $\leftrightarrow$ Entanglement entropy
- AdS radius $\leftrightarrow$ Gauge coupling $g_{YM}^2 N$
Holography Beyond AdS/CFT
Does holography apply to our universe (which is not AdS)?
- de Sitter space: Our universe has positive cosmological constant (accelerating expansion). dS/CFT correspondence is less understood but actively researched.
- Flat space holography: Recent work on celestial holography attempts to formulate holography for asymptotically flat spacetimes.
- Cosmology: Does the holographic principle constrain early universe cosmology? Connection to the cosmic entropy bound.
Open question: Is our universe fundamentally holographic? If so, what is the boundary theory? This remains one of the deepest unsolved problems in theoretical physics.