# Topics: Quantum Mechanics - 1.1 Holographic Principle

### 1.1 Holographic Principle

The Holographic Principle and M-theory

To them, I said,

the truth would be literally nothing

but the shadows of the images.

-Plato, The Republic (Book VII)

Holography Through the Ages

Plato, the great Greek philosopher, wrote a series of `Dialogues' which summarized many of the things which he had learned from his teacher, who was the philosopher Socrates. One of the most famous of these Dialogues is the `Allegory of the Cave'. In this allegory, people are chained in a cave so that they can only see the shadows which are cast on the walls of the cave by a fire. To these people, the shadows represent the totality of their existence - it is impossible for them to imagine a reality which consists of anything other than the fuzzy shadows on the wall.

However, some prisoners may escape from the cave; they may go out into the light of the sun and behold true reality. When they try to go back into the cave and tell the other captives the truth, they are mocked as madmen.

Of course, to Plato this story was just meant to symbolize mankind's struggle to reach enlightenment and understanding through reasoning and open-mindedness. We are all initially prisoners and the tangible world is our cave. Just as some prisoners may escape out into the sun, so may some people amass knowledge and ascend into the light of true reality.

What is equally interesting is the literal interpretation of Plato's tale: The idea that reality could be represented completely as `shadows' on the walls.

The Holographic Principle and Modern Physics

In 1993 the famous Dutch theoretical physicist G. 't Hooft put forward a bold proposal which is reminiscent of Plato's Allegory of the Cave. This proposal, which is known as the Holographic Principle, consists of two basic assertions:

Assertion 1
: The first assertion of the Holographic Principle is that all of the information contained in some region of space can be represented as a `Hologram' - a theory which `lives' on the boundary of that region. For example, if the region of space in question is the DAMTP Tearoom, then the holographic principle asserts that all of the physics which takes place in the DAMTP Tearoom can be represented by a theory which is defined on the walls of the Tearoom.

Assertion 2: The second assertion of the Holographic Principle is that the theory on the boundary of the region of space in question should contain at most one degree of freedom per Planck area.

A Planck area is the area enclosed by a little square which has side length equal to the Planck length, a basic unit of length which is usually denoted Lp. The Planck length is a fundamental unit of length, because it is the parameter with the dimensions of length which can be constructed out of the basic constants G (Newton's constant for the strength of gravitational interactions), \${\hbar}\$ (Planck's constant from quantum mechanics), and c (the speed of light). A quick calculation reveals that Lp is very small indeed:

Lp = 1.6 x 10-33 centimeters

To many people, the Holographic Principle seems strange and counterintuitive: How could all of the physics which takes place in a given room be equivalent to some physics defined on the walls of the room? Could all of the information contained in your body actually be represented by your `shadow'?

In fact, the way in which the Holographic Principle appears in M-theory is much more subtle. In M-theory we are the shadows on the wall. The `room' is some larger, five-dimensional spacetime and our four-dimensional world is just the boundary of this larger space. If we try to move away from the wall, we are moving into an extra dimension of space - a fifth dimension. In fact, people have recently been trying to think of ways in which we might actually experimentally `probe' this fifth dimension.

At the heart of many of these exciting ideas is a version of the Holographic Principle known as the adS/CFT correspondence.

Are YOU a Hologram? M-theory and the adS/CFT correspondence

The adS/CFT correspondence is a type of duality, which states that two apparently distinct physical theories are actually equivalent. On one side of this duality is the physics of gravity in a spacetime known as anti-de Sitter space (adS). Five-dimensional anti-de Sitter space has a boundary which is four-dimensional, and in a certain limit looks like flat spacetime with one time and three space directions. The adS/CFT correspondence states that the physics of gravity in five-dimensional anti-de Sitter space, is equivalent to a certain supersymmetric Yang-Mills theory which is defined on the boundary of adS. This Yang-Mills theory is thus a `hologram' of the physics which is happening in five dimensions. The Yang-Mills theory has gauge group SU(N), where N is very large, and it is said to be `supersymmetric' because it has a symmetry which allows you to exchange bosons and fermions. The hope is that this theory will eventually teach us something about QCD (quantum chromodynamics), which is a gauge theory with gauge group SU(3). QCD describes interactions between quarks. However, QCD has much less symmetry than the theory defined on the boundary of adS; for example, QCD has no supersymmetry. Furthermore, we still don't know how to incorporate a crucial property of QCD, known as asymptotic freedom.

Here in DAMTP, we have been working to see if the adS/CFT correspondence can be generalized. Working with collaborators in such far-flung places as the United States, Canada, and Durham, we have managed to show that the duality is still true even when you replace adS with more complicated five-dimensional spacetimes. In particular, we have calculated what happens when you put electric charge in adS, or rotation in adS, or even what happens when you put a certain exotic charge known as `NUT-charge' into adS.

Source: http://www.damtp.cam.ac.uk/user/gr/public/holo/

April 7, 2003 | Scientific American

The Holographic Principle

The world, in a sense, may be a hologram. The idea comes from black hole physics. In the 1970s researchers knew that when an object becomes part of a black hole, two things happen. One, all the detailed information about that object is lost. And two, the surface area of the black hole's event horizon (the point of no return for infalling matter and energy) grows. The first fact seemed to violate the second law of thermodynamics, because one of the lost details was the object's entropy, or the information describing its microscopic parts. But the second fact offered a way out: if entropy must always grow, and a black hole's surface area must too, perhaps for the black hole they are one and the same, and information is somehow stored on the horizon.

Fast forward to 1993. Two particle physicists working separately conclude that the universe itself must store information in a similar way. Quantum mechanics starts with the assumption that information is stored in every volume of space. But any patch of space can become a black hole, nature's densest file cabinet, which stores information in bits of area. Perhaps, then, all that's needed to describe a patch of space, black hole or no, is that area's worth of information. The idea is called the holographic principle, after the way that a hologram encodes 3D information on a 2D surface.

Recently, Raphael Bousso, while at Stanford University, helped formulate a more precise and more broadly applicable statement of the principle that involves light rays. "The world doesn't appear to us like a hologram, but in terms of the information needed to describe it, it is one," Bousso says. "The amazing thing is that the holographic principle works for all areas in all space times. We have this amazing pattern there, which is far more general than the black hole picture we started from. And we have no idea why this works. What this is telling us is, there is a description of the world we should be looking for which will be more economical than the one that we have right now, and will presumably have to do with quantum gravity." --JR Minkel

A principle for quantum gravity

by Raphael Bousso / Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, U.S.A.

Progress in fundamental physics has often been driven by the recognition of a new principle, a key insight to guide the search for a successful theory. Examples include the principles of relativity, the equivalence principle, and the gauge principle. Such principles lay down general properties that must be incorporated into the laws of physics.

A principle can be sparked by contradictions between existing theories. By judiciously declaring which theory contains the elements of a unified framework, a principle may force other theories to be adapted or superceded. The special theory of relativity, for example, reconciles electrodynamics with Galilean kinematics at the expense of the latter.

A principle can also arise from some newly recognized pattern, an apparent law of physics that stands by itself, both uncontradicted and unexplained by existing theories. A principle may declare this pattern to be at the core of a new theory altogether. In Newtonian gravity, for example, the proportionality of gravitational and inertial mass in all bodies seems a curious coincidence that is far from inevitable. The equivalence principle demands that this pattern must be made manifest in a new theory. This led Einstein to the general theory of relativity, in which the equality of gravitational and inertial mass is built in from the start. Because all bodies follow geodesics in a curved spacetime, things simply couldn’t be otherwise. The holographic principle belongs in the latter class. The unexplained “pattern”, in this case, is the existence of a precise, general, and surprisingly strong limit on the information content of spacetime regions. This pattern has come to be recognized in stages; its present, most general form is called the covariant entropy bound. The holographic principle asserts that this bound is not a coincidence, but that its origin must be found in a new theory.

The covariant entropy bound relates aspects of spacetime geometry to the number of quantum states of matter. This suggests that any theory that incorporates the holographic principle must unify matter, gravity, and quantum mechanics. It will be a quantum theory of gravity, a framework that transcends general relativity and quantum field theory. This expectation is supported by the close ties between the covariant entropy bound and the semi-classical properties of black holes. It has been confirmed—albeit in a limited context—by recent results in string theory. The holographic principle conflicts with received wisdom; in this sense, it also belongs in the former class. Conventional theories are local; quantum field theory, for example, contains degrees of freedom at every point in space. Even with a short distance cutoff, the information content of a spatial region would appear to grow with the volume. The holographic principle, on the other hand, implies that the number of fundamental degrees of freedom is related to the area of surfaces in spacetime. Typically, this number is drastically smaller than the field theory estimate.

Thus, the holographic principle calls into question not only the fundamental status of field theory but the very notion of locality. It gives preference, as we shall see, to the preservation of quantum-mechanical unitarity. In physics, information can be encoded in a variety of ways: by the quantum states, say, of a conformal field theory, or by a lattice of spins. Unfortunately, for all its precise predictions about the number of fundamental degrees of freedom in spacetime, the holographic principle betrays little about their character. The amount of information is strictly determined, but not its form. It is interesting to contemplate the notion that pure, abstract information may underlie all of physics. But for now, this austerity frustrates the design of concrete models incorporating the holographic principle. Indeed, a broader caveat is called for. The covariant entropy bound is a compelling pattern, but it may still prove incorrect or merely accidental, signifying no deeper origin. If the bound does stem from a fundamental theory, that relation could be indirect or peripheral, in which case the holographic principle would be unlikely to guide us to the core ideas of the theory. All that aside, the holographic principle is likely only one of several independent conceptual advances needed for progress in quantum gravity.

At present, however, quantum gravity poses an immense problem tackled with little guidance. Quantum gravity has imprinted few traces on physics below the Planck energy. Among them, the information content of spacetime may well be the most profound. The direction offered by the holographic principle is impacting existing frameworks and provoking new approaches. In particular, it may prove beneficial to the further development of string theory, widely (and, in our view, justly) considered the most compelling of present approaches. This article will outline the case for the holographic principle whilst providing a starting point for further study of the literature. The material is not, for the most part, technical. The main mathematical aspect, the construction of light-sheets, is rather straightforward. In order to achieve a self-contained presentation, some basic material on general relativity has been included in an appendix.

In demonstrating the scope and power of the holographic correspondence between areas and information, our ultimate task is to convey its character as a law of physics that captures one of the most intriguing aspects of quantum gravity. If the reader is led to contemplate the origin of this particular pattern nature has laid out, our review will have succeeded.

Source: Bousso, Raphael - The Holographic Principle, p.2

Holographic Space-time