1. The Holographic Principle
The holographic principle states that all information in a volume of space can be encoded on its boundary. A theory of gravity in (d+1) dimensions is equivalent to a non-gravitational theory in d dimensions!
2. AdS/CFT Correspondence
The AdS/CFT correspondence (Maldacena, 1997) is the most concrete realization of holography. It relates:
Corresponds to energy scale E ~ 32 GeV
The Dictionary
| Bulk (Gravity) | Boundary (CFT) |
|---|---|
| Radial coordinate r | Energy scale E |
| Bulk field phi(r, x) | Operator O(x) |
| Field mass m | Operator dimension Delta |
| Graviton | Stress tensor T_uv |
| Black hole | Thermal state (temperature) |
3. Holographic Renormalization Group
The radial direction in AdS corresponds to the RG scale in the CFT. Moving into the bulk = flowing to lower energies (IR).
UV to IR Flow
- UV (boundary): High energy, conformal fixed point
- RG flow: Coupling constants evolve with scale
- IR (deep bulk): Low energy, possibly new fixed point
4. The IR Fixed Point
In the Z2 framework, the deep IR physics determines the cosmological constant and matter content. The IR fixed point is crucial!
Holographic RG Flow
- - Boundary: UV CFT (high energy, short distances)
- - Moving inward: Integrating out high-energy modes
- - Center: IR fixed point (low energy physics)
- - Z2 connection: The IR fixed point determines low-energy observables!
- UV physics: String theory on T^3/Z_2
- IR physics: 4D effective theory with Lambda = 13/19
5. Applications
Quark-Gluon Plasma
AdS/CFT predicts viscosity eta/s = 1/(4 pi), confirmed by RHIC experiments!
Condensed Matter
Holographic superconductors and strange metals modeled via AdS/CFT.
Quantum Information
Entanglement entropy = area of minimal surface (Ryu-Takayanagi).
Black Hole Information
Holography resolves the information paradox: info preserved on boundary.
6. Connection to Z2 Framework
Why Holography Matters for Z2
- - UV/IR connection: High-energy string physics determines low-energy cosmos
- - Non-perturbative: Strong coupling accessible via gravity dual
- - Counting DOF: Holographic c-function counts degrees of freedom
- - Emergent spacetime: 4D spacetime may emerge from boundary theory
7. The Full Picture
Z2 Framework: UV to IR
Exercises
- Explain in your own words why the boundary of AdS corresponds to UV physics.
- The AdS_5 metric is ds^2 = (L^2/z^2)(dz^2 + dx^2). Where is the boundary? The interior?
- What is the CFT dual of a black hole in AdS?
- Why does the c-theorem say c_UV >= c_IR? What happens to degrees of freedom?
- How does holography help understand strong coupling in QFT?