String Theory
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All Roads Lead to String Theory (Polchinski)
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Prior to the First Superstring Revolution
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Early History
| S-Matrix Theory Regge Trajectory | |

Bosonic String Theory
| Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation | |

Supersymmetric Revolution
| Supersymmetry RNS Formalism GS Formalism BPS | |

Superstring Revolutions
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First Superstring Revolution
| GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory | |

Second Superstring Revolution
| T-Duality D-Brane S-Duality Horava-Witten String Theory M-Theory Holographic PrincipleN=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism | |

After the Revolutions
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Phenomenology
| String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology | |

The **Holographic Principle** is a principle that states that all the information within a reigon is completely encoded onto it's boundary.

## Motivating examplesEdit this section

### Edit this section

The Stokes-Navier Theorem states that the work done along a path is exactly equivalent to the flux through the surface it bounds.

$ \oint\vec f\cdot\mbox{d}\vec r=\iint\left(\nabla\times\vec f\right)\cdot\hat n\mbox{d}S $

In other words, the flux through the surface is equal to the work along the boundary. This means that something on the surface has an alternate description on its boundary.

### Stokes' TheoremEdit this section

The Stokes-Navier Theorem is merely a special case of the more general, all-intuitive theorem, known as [[Stokes' Theorem]. It can be stated as:

$ \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. $

Again, on the left - hand - side, the integral is taken along $ \partial\Omega $, but, whereas, on the right - hand - side, it is taken along $ \Omega $, which means that it relates a property on a surface or manifold to an alternate description of that on it's boundary.

### Gauss's TheoremEdit this section

The Gauss's Theorem statesj thaty:

$ \int\!\!\!\!\int\!\!\!\!\int_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)dV= $$ \scriptstyle S $$ (\mathbf{F}\cdot\mathbf{n})\,dS . $

This again means that some information about the region has an alternate description on the boundary.

### A Schwarzschild Black HoleEdit this section

Consider an in - falling observer towards a Black Hole. For simplicity, let's say that it is a Schwarzschild Black Hole. Then, since the time dilation is given by:

$ \frac{\mbox{d}t}{\mbox{d}\tau}= \frac{-c_0^2\mbox{d}t^2}{\mbox{d}s^2} $

For a Schwarzschild Metric, this becomes $ 0 $ at the event horizon, so that this in - falling observer would appear to an external observer as stopping at the event horizon.

Therefore, all activities that happen to this in - falling observer would appear to the external observer as appearing at the event horizon; where - as the in - falling observer itself would perceive it as happening within the event horizon of the black hole.

Again, something in a reigon, has an alternate description on the boundary of this reigon.

### Black Hole EntropyEdit this section

In the framework of Semi-Classical Gravity, Hawking Radiation is the radiation emitted from black holes due to Quantum Mechanicsal processes. The entropy associated with Hawking Radiation is given by (in a convinient system of natural units):

$ S=\frac{A}4 $

This thus means that the entropy of the (three-dimensional) black hole can be expressed in terms of an alternative description, on it's boundary; the area.

## Statement of the theoremEdit this section

From these observations, we can thus state that:

The information in a region is can be completely be described by the information on its boundary.

This statement is known as the **Holographic Principle**.

## Applications in PhysicsEdit this section

The **Holographic Principle** has many implications in Physics, such as AdS/CFT, and it's extensions, such as AdS/QCD, AdS/CMT, and so on.

## A word of caution about the motivating examplesEdit this section

Most of the motivating examples mentioned, except for the one on black holes is not the same sort of the **Holographic Principle** as that which is used in Physics, but merely just "Holographic Statements", as they relate the information about a region to information on it's boundary.