Exploring the Holographic Principle: Questions and Answers

[Lostinthought]

Hi,I don't understand the holographic principle,how can the entire physics in a space be done on its boundary,arent we supressing a dimension and there by losing the information contained in it?,and is it neccesary that the boundary be closed?

 

[Naty1]

The holographic principle is important because it DOESN'T make common sense for information to be area rather than volume related...who kew that would happen?? It's NOT an intuitive conclusion anymore than the fixed speed of light is "logical". Physics requires adopting some new ways of thinking. An aspect of these boundaries is that they are relative" different observers will see different boundaries...so just where IS that information? It's RELATIVE.

The entropy of a volume of space IS proportional to the enclosed volume...This is the Beckenstein-Hawking bound.

Maldecena's work involves a string theory defined on one space, and a quantum field theory without gravity defined on the boundary of this space (one less dimension)...also enclosed.

 

[Polyrhythmic]

The entropy of a volume of space IS proportional to the enclosed volume...This is the Beckenstein-Hawking bound.

The Bekenstein-Hawking entropy is proportional to the area of the enclosing surface, not to the volume. That's why it is a manifestation of the holographic principle.

 

[cosmik debris]

The holographic principle is not magic is it? It is just a projection of a higher dimensional space onto a lower one which is done all the time. People use Green's functions and they are similar aren't they?

 

[tom.stoer]

The holographic principle exists in different versions. Up to now it's not as well-established as the principles of relativity b/c there is neither a strict proof nor a direct and practically observable predictions.

But there are several indications that it might indeed be a principle of nature:

1) Bekenstein & Hawking where able to derive a temperature and thermodynamical relations for black holes where surface area and entropy are related. Here a microscopic derivation of entropy in terms of fundamental degrees of freedom is not available. The Beckenstein bound says that the entropy of a certain volume of space scales with its surface, not with the volume.
2) In string theory there is a microscopic derivation of the entropy = state counting in terms of fundamental degrees for certain extremal (unfortunately not realistic) black holes.
3) In string theory the is the AdS/CFT proposal which relates a bulk theory including gravity in AdS specatime and a surface conformal field theory w/o gravity. That means that the volume degrees of freedom are "dual" to surface degrees of freedom living entrely on the boundary of this AdS space (Maldecena).
4) In loop quantum gravity there is a strict derivation of the microscopic state count for a realistic black hole (defined via so-called isolated horizons). The result is that the the state count and therefore the entropy scale with the area, not with the entropy.

 

[tom.stoer]

... and is it neccesary that the boundary be closed?

It depends to which formulation of the holographic principle you are referring: If you want to talk about "information in a certain volume" the defintion of a volume reqauires a closed surface. This surface need not be a physical entity but could in principle be a mathematical concept, only. For a generalized gauge/gravity duality where you say that a theory including gravity (like string theory in AdS spacetime) is dual = mathematically equivalent to a lower-dimensional field theory living on the surface I don't think that the surface must be closed. But for this rather general stement we do not have proof, it's still a conjecture.

Look at quantum mechanics in an L² Hilbert space defined over the reals. This is certainly a continuum theory. But we know that it can be reformulated using harmonic oscillator wave functions which form a complete but discrete set. In addition we can get rid of wave functions at all and use creation and annihilation operators, energy eigenstates etc. That means that a continuum theory is dual to a discrete theory w/o losing any information.

 

[Demystifier]

My understanding of AdS/CFT is rather superficial, but in my understanding, it suggests the following intuitive "explanation" of the holographic principle:

How can a theory living in d-1 dimensions be equivalent to another theory living in d dimensions? The answer is that they both live in d mathematical dimensions, but one of the mathematical dimensions in the first theory has a different physical interpretation. Namely, the number of mathematical dimensions is simply the number of free continuous variables on which the physical quantities depend. The extra free variable in the first theory is the SCALE PARAMETER, appearing in QFT due to a necessity to introduce regularization and renormalization of the theory (which otherwise does not make physical sense). On the technical level, the dependence on this extra parameter is usually described in terms of the renormalization group equations.

 

[Naty1]

The Bekenstein-Hawking entropy is proportional to the area of the enclosing surface, not to the volume. That's why it is a manifestation of the holographic principle.

Of course, as the rest of my post indicates...poor editing on my part!

Another interesting aspect of this holographic concept: All the information in our universe may be displayed on any boundary enclosing our universe...so sayeth Leonard Susskind...and he also points out the holographic principle lends credence to discrete spacetime and finite information storage within any boundary.

 

[med17k]

What if the space has no boundry like a sphere of a torus? If our universe has the topology of the torus , How can holographic principle be applied in this case?

 

[czes]

Look at quantum mechanics in an L² Hilbert space defined over the reals. This is certainly a continuum theory. But we know that it can be reformulated using harmonic oscillator wave functions which form a complete but discrete set. In addition we can get rid of wave functions at all and use creation and annihilation operators, energy eigenstates etc. That means that a continuum theory is dual to a discrete theory w/o losing any information.

By adopting the most conservative approach where the correction terms, that describe the possibility for space-time fluctuations cumulating across long distances and partially compensate for the effects of the phase variations, are taken into account. We exclude the random walk model and most of the holographic models of the space-time foam.
http://arxiv.org/abs/1108.6005

Tamburini et al. have wrote too much , I think. There aren't the analogous fluctuation of the material spacetime as an aether but also there is non-locality in quantum mechanics and there isn't a distance nor space in mathematical holographic Universe. There is a mathematical digital discreteness rather.
I agree with Tom, that a continuum theory is dual to a discrete theory w/o losing any information.
How to speak about a space fluctuation if there isn't a space at all ?