Holographic principle

The holographic principle, began as a speculation about quantum gravity and it is amongst the most unintuitive abstractions in all of physics. It is the culmination of about 2 decades of debate on the fate of information that falls into a black hole. As weird as it is, it has become a part of mainstream theoretical physics. The holographic principle is a violent restructuring of the laws of physics. However, there is not much mathematics necessary to prove it. 

We can begin to understand the holographic principle with an analogy. We begin with a spherical region of space, which contains some physical information in the interior, defined by a mathematical boundary. Of course, the most massive thing that can fit into this region is a black hole, which has it's horizon defined by the boundary to this space. That is the limit on the mass that can be contained in that sphere. Any more mass and it would overflow the boundary of the sphere. However, the question is: is there a limit on the bits of physical information within this sphere? It would be useful for this analogy, to now consider another spherical shell of physical material. This shell has it's own mass and will surround this entire construction. However, this new boundary spherical shell can be squeezed until it fits the interior region perfectly. In other words, the boundary of this new spherical shell can be tuned to fit perfectly into the boundary of our original spherical region. Let us now return to the physical information inside the original sphere. It has some amount of entropy or hidden information. We don't know what this initial entropy was. However, we can know it's final entropy. It will be the same as the entropy of a black hole. It is its area measured in Planck units. Remember the 2nd law of thermodynamics: entropy always increases. That being said, the entropy of the black hole must be greater than the original physical information. 

The consequence of this is as follows: the maximum number of bits of information that can fit into a spatial region is the same as the area of the boundary, measured in Planck units. Thus, for everything that takes place inside a region of space, there is also an equivalent boundary description. The 2-dimensional boundary surface, acts as a hologram, encoding equivalent information to the 3-dimensional interior of that very space. However, this is no ordinary hologram. This hologram is quantum mechanical and can change as time goes on. Indeed, an understanding of this kind of holographic phenomenon will prove to be an intellectual endeavor for physicists. 

Bekenstein bound - 1970s

The holographic principle begins with the idea of the Bekenstein bound. It is a limmitation on what we can know that comes out of black hole thermodynamics. It is a bound to the information that is available to us about what is contained on the other side of a horizon. The amount of information that can be contained in any volume or region of space is finite and proportional to the boundary of that region, measured in Planck units. The number of bits of information that observer can detect cannot be larger than 1/4 the area of the surface (measured in Planck units). This is a limit on the maximum amount of information that can be stored in a finite region of space (which has a finite amount of energy). It is an upper limit on entropy or information. Entropy is considered as information here and is measured in bits. The more bits, than the more degrees of freedom. The Bekenstein bound is the maximum amount of information necessary to perfectly describe a system down to the quantum level. That information must be finite, if the energy in that region or space is also to be finite. Some form of this bound must exist in order for the laws of thermodynamics and general relativity to remain consistent. This bound is an upper limit to the density of information. The implication of this bound is that the world is discrete on the Planck scale (quite the insight for quantum gravity theorists). This is because, if nature were continuous in the Planck scale, some volume could contain an infinite amount of information. Bekenstein discussed this bound in the 1970s, shortly after his discovery that black holes have a finite entropy.

Gerard ‘t Hooft - 1993

One who refused to accept the fact that quantum mechanics was violated in the black hole information paradox and that black holes could truly evaporate away physical information was Gerard ‘t Hooft. ‘T Hooft is a Dutch theoretical physicist, who won the Nobel Prize in Physics in 1991 for his work on the Electroweak interaction. Gerard ‘t Hooft felt that the problems that arose from Stephen Hawking’s work only arose, because the theory of Hawking was a semiclassical theory. In other words, ‘t Hooft thought that, in a full theory of quantum gravity, these problems with information loss would not be present. This will lead him to propose, in 1993, that the quantum fields (the degrees of freedom) around and near a black hole, can be understood, by a theory, formulated in one lower dimension, encoding equivalent physical information about the higher dimensional theory. A degree of freedom is a variable that will evolve in time according to some dynamic law. An example of a degree of freedom is the position of a particle or the value of an electric field. 

The name of Gerard 't Hooft's paper was "Dimensional Reduction in Quantum Gravity". Surprisingly, this paper, initially went unnoticed. This was a speculative paper on quantum gravity by Gerard 't Hooft. This is going to lead to a new idea in physics, called the holographic principle: the maximum entropy in a region of space is its area in Planck units.

Holographic principle

One physicist, is going to take a serious look at the work of Gerard ‘t Hooft. This was Leonard Susskind, a professor of theoretical physics at Stanford University. In 1995, Leonard Susskind, is going to give a precise interpretation of Gerard ‘t Hooft’s speculative work, within the context of string theory. The title of Leonard Susskind's paper was "The World as a Hologram". This will prove to be a major contribution to quantum gravity. This proposal by Susskind, building on work by ‘t Hooft, became known as the “holographic principle”. In string theory, the holographic principle, is the idea that the description of a volume of space and its properties, can be thought of as theoretically equivalent to a theory that lives on the boundary to that space, in one lower dimension. The theories are dual to one another, despite living in a different number of dimensions. Leonard Susskind is going to be the first to give a precise string theory interpretation of the holographic principle. What Susskind proposed, along with his collaborators: Tom Banks, Willy Fischler and Stephen Shenker, was what was known as the BFSS matrix model of string theory. This was a new holographic description and prototype of M-theory. However, the story of holography in quantum gravity will not end here. The most reliable realization of the holographic principle is the AdS/CFT correspondence, which was proposed by Juan Maldacena in 1997 and relates compactifications of string theory to a quantum field theory, in a lower dimension.