The relation between entropy and probability at quantum levelu

[dhillonv10]

Hi all,

I have been reading some new stuff recently on the holographic principal and I have a question, I seem to understand that entropy and probability must have some relation to each other, I am not sure what exactly but it seems like they just do, can anyone please explain what it is? Now to continue that chain of thought, is the following scenario possible:

The outer edges of the universe exist in 2 dimensions (AdS/CFT) and in the inside we feel 3 dimensions. Now because of this we have holographic noise which Carl Hogan wants to detect at the holometer. We know that our universe had to start with low entropy and there are several theories that pertain to that, but could it be that because of the holographic principal, entropy also emerged? I understand that my question sounds completely absurd but then again there are a lot of absurd things around :) Also if entropy emerged, could it be that probability also emerged from entropy in quantum systems? Thanks for your time.

 

[qsa]

http://en.wikipedia.org/wiki/Entropy_(information_theory)

 

[dhillonv10]

thanks for the link qsa, could you also possibly comment on the emergence of entropy? Thanks.

 

[mitchell porter]

Quantum randomness is holographic noise caused by an entropic force. If that's the idea, then wow, not bad for a random crackpot idea. :-) I could lecture you about how entropy and probability are actually related, or about problems with the ideas of Erik Verlinde and Craig Hogan, but I just wanted to get the main idea into view first.

 

[dhillonv10]

haha thanks mitchell, yes i couldn't say in proper words but you hit the bull's eye.

I think that probability must have emerged from holographic noise and that if you take the universe as a system, at its boundaries (which always keep moving away) you will find no probability because that region has complete information. Inside that system however, because there is holographic noise, quantum systems have randomness. I have more on this idea so would you like to hear it? or perhaps explain how entropy and probability are actually related and problems with the ideas of Erik Verlinde and Craig Hogan?

Thanks,
Vikram

 

[dhillonv10]

actually mitchell, i think the way i presented this idea and the way you understood it are a little different, can you please expand what you wrote in #4 a little more? it'll help me clarify somethings before we proceed. Thanks.

 

[mitchell porter]

You shouldn't imagine that what follows came to me all in an instant. What I did was to condense what you were saying into a slogan, and then tried to make that slogan meaningful.

Let's start with the standard AdS/CFT example of holography, the duality between d=4 N=4 Yang-Mills field theory and Type IIB string theory on the d=10 space "AdS5 x S5". Working from a different direction, Nima Arkani-Hamed and collaborators have come up with a "Grassmannian" representation of that same field theory which doesn't involve space or time.

If you examine Erik Verlinde's plan, you'll see that his fundamental entropic force is supposed to be pre-holographic. "The starting point is a microscopic theory that knows about time, energy and number of states. That is all, nothing more." The conventional holographic principle, in which a field theory in a space is equivalent to a gravitational theory on a space with at least one more dimension, somehow comes later.

So the first idea is that you could try to understand the progression from the Grassmannian to the field theory to the string theory, according to Verlinde's program.

Now for quantum randomness. The conventional attitude in physics is that it's fundamental, doesn't have a deeper explanation, and can't be understood as an extra "noise" term added to deterministic classical dynamics. Edward Nelson's stochastic mechanics tried to reproduce quantum mechanics in that way, but demonstrated that the noise has to be nonlocally correlated. You get this with the quantum potential of Bohmian mechanics, but Bohmian mechanics is artificial because it's only defined with respect to a particular reference frame. That is, if you pick a coordinate system, then you can construct a nonlocal force which will give you the same predictions as the standard quantum wavefunctions. But your formula will only be valid in that coordinate system, whereas fundamental physics is supposed to be independent of coordinate system.

The holographic principle, as realized in AdS/CFT, is an equivalence between two quantum theories: a field theory on the boundary of a space, and a string theory in the bulk of that space. The boundary quantum states already include information about what happens in the extra dimensions, away from the boundary. The simplest way to think about this is to think of the shadows in Plato's cave. If you have an object and hold it close to the cave wall, the shadow will be small, but as you move it further away, the shadow gets larger. In the same way, the further into the bulk an object is from the boundary, the larger its image in the boundary field theory. Since the boundary field theory is "conformal", which includes scale invariance, it allows structures of all sizes, so it can represent objects at an arbitrary distance from the boundary.

You wouldn't normally try to explain quantum mechanics using the holographic principle. The quantum framework is presupposed both by the field theory and the string theory. But what if you tried expressing both theories using Bohm's equations? Yes, it's artificial, but it's also mathematically well-defined, and it's something that no-one has done. What would the relationship be, between the quantum potential on the boundary, and the quantum potential in the bulk? It is an orthodox fact about the holographic principle that locality on the boundary and locality in the bulk are not directly related. Could a purely local interaction in a classical boundary field theory turn into a nonlocal interaction in its holographic image? In that case, could you understand quantum randomness in the bulk theory as arising from the holographic transformation of a local interaction on the boundary? It's an amazing idea and I think it's a new one.

I'm sure this isn't what you were thinking when you posted :-) but it was inspired by the unusual connections you were making.

 

[dhillonv10]

no that was not what i was thinking but the last few lines are exactly what i tried to say, I need a bit more time to look up the stuff you've mentioned before I can reply back with how this idea is evolving :)

- V

 

[atyy]

Without holography, there's conventional speculation about quantum mechanics and thermalization eg. http://arxiv.org/abs/1007.3957 , which also gives a nice overview of the literature in its introduction.

 

[dhillonv10]

thanks for the link atyy.

mitchell: if its okay with you, could we possibly move to a private conversation? perhaps PM? I wouldn't PM anyone without their permission, that's simply rude, if not than that's cool too, i'll just post here.

 

[mitchell porter]

It can't be rude to PM someone; they can always ignore you! Anyway, PM me if you wish.

I'll just add that I found some papers relating AdS/CFT to "stochastic quantization" (http://arxiv.org/abs/0912.2105" , also its ref #1). That's the closest thing to a holographic explanation of quantum theory that I've seen.

 

[dhillonv10]

Also I found another paper describing a lot of what I'll need http://arxiv.org/abs/1103.3427

 

[dhillonv10]

mitchell, I send you a message in PM if you get time please reply back to that, also with the description of local interactions turning into non local ones, i think there's another way to approach the idea through the use of the paper i posted a link to, that paper describes the whole process in terms of PEPS and here's the abstract:

In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated to their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using Projected Entangled Pair States (PEPS). This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models, like a deformed AKLT [1], an Ising-type [2], and Kitaev's toric code [3], both in finite ladders and infinite square lattices. In the latter case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield non-local Hamiltonians. As our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary.

Seems like someone else has already done a lot of the work required, now from what I understand of this, the paper proposes a method to relate interactions (at least entaglement) on the bulk to boundary. Although we need to generalize the following:

- This method is proposed in lattice theory, does it really matter what framework is being used, our concern is holography.

- The holographic framework being proposed, see how quantum randomness arises from that, going back to your original idea

- Is entanglement enough as a type of interaction? Explore how to generalize to all types of interactions, for that the Hamiltonian would have to be modified.

- Explore other implications.

 

[atyy]

I think the holographic principle mentioned in the OP was AdS/CFT.

However, there are other examples of holography, such as the above mentioned paper by Cirac et al. Some discussions of how the various types of holography may be related are found in Gukov et al's http://arxiv.org/abs/hep-th/0403225 and Swingle's http://arxiv.org/abs/0905.1317 .

 

[dhillonv10]

thank you very much for the link atty, after the comments by mitchell the idea that i put up in the OP has evolved, I am trying to focus on what was brought up:

Could a purely local interaction in a classical boundary field theory turn into a nonlocal interaction in its holographic image? In that case, could you understand quantum randomness in the bulk theory as arising from the holographic transformation of a local interaction on the boundary?

 

[mitchell porter]

1) Despite the authors' terminology, the paper on PEPS is not talking about the holographic principle. The holographic principle is about an equivalence between a higher-dimensional theory and a lower-dimensional theory, but their mapping is not an equivalence. You cannot reconstruct the whole of the bulk theory from their boundary theory (see the remark on page 3 about their mapping not being http://en.wikipedia.org/wiki/Injective_function" ). At best, they are describing some more general phenomenon which might correspond to holography in those cases where the mapping is invertible.

2) The holographic principle relates a quantum theory in the bulk space to another quantum theory on the boundary of that space. That is, in holography as conventionally conceived, you will have quantum uncertainty on both sides of the relationship. The only difference is that in the boundary theory, the quantum uncertainty is defined on top of a fixed space, but the bulk theory contains quantum gravity, so the metric and topology fluctuate as well. It's this extension of quantum uncertainty to space and time themselves which Craig Hogan hopes to detect, as "holographic noise".

So be aware that trying to derive quantum mechanics itself from the holographic relationship is a highly unusual idea, and possibly a wrong idea. Just to say it again: in holographic dualities as they are actually studied, there is already quantum randomness on the boundary, which extends to include space-time itself in the bulk theory on the other side of the equivalence. The innovative concept under discussion in this thread, as I understand it, is that we could start without quantum randomness on the boundary, and still end up with the appearance of quantum randomness in the bulk, because of the "local-to-nonlocal" aspect of the holographic mapping.

3) The origins of the holographic principle lie in the study of black holes. Without gravity, the number of possible states in a field theory will increase with energy as a function of spatial volume. But if one of your fields is gravity, then black holes will form at high densities, and the entropy (and therefore the number of possible states) of a black hole is a function of area (the area of the event horizon), not of volume. So a field theory containing gravity has to behave like an ordinary field theory in a space of one less dimension. That is the qualitative statement of the holographic principle, as proposed by http://arxiv.org/abs/gr-qc/9310026" .

It is a very interesting fact that 't Hooft is trying to explain quantum mechanics itself, and not just quantum gravity, using cellular automaton models in one less dimension. So I was wrong to say that the idea of a holographic explanation of quantum mechanics is entirely new; Gerard 't Hooft, a Nobel Prize winner and one of the many parents of the standard model of particle physics, is an exponent of this idea! But it should be understood that this is 't Hooft in the "later Einstein" phase of his career. Einstein spent the last decades of his life working on unified field theories and away from the mainstream of theoretical and experimental physics. With respect to the holographic principle, the 1997 discovery of AdS/CFT by http://arxiv.org/abs/hep-th/9711200" was a new stage in the development of the idea, because for the first time there was a quantitative example. Maldacena wasn't just proposing that a quantum gravity theory is equivalent to some unknown field theory; he was saying, string theory on a particular space is equivalent to a particular, already known field theory.

The years since then have involved the intensive study of these two theories, and the identification of many other dual pairs. But 't Hooft's recent work does not involve any of this. He is proceeding alone in a different direction.

So what about the people who are working on the concrete examples of holographic duality unearthed by Maldacena and others? They aren't trying to explain quantum mechanics. They take it as a given, and instead they use the duality between two quantum theories to learn about both. However, in some of the early papers by Arkani-Hamed et al (who I mentioned earlier in the thread), you will occasionally see the idea that maybe their new framework will even explain quantum mechanics. The usual headline for their approach is that they have abandoned manifest locality, and so perhaps they have found a level of description beyond space-time. But another feature of their Grassmannian formalism is that unitarity is not an input either. Unitarity is the feature which, in quantum mechanics, ensures that the probabilities add to one. So to have discovered a formalism in which unitarity doesn't have to be introduced may mean they have found something more fundamental than quantum mechanics. (Let me mention that their formalism involves twistors, and it was precisely Penrose's ambition to explain quantum mechanics as well as to go beyond space-time.)

In our discussion here we're saying, what if quantum mechanics only applies in the bulk, but the boundary is classical? However, in Arkani-Hamed's recent talks, he interprets their work as the discovery of a third framework, neither string theory (bulk) nor field theory (boundary), but something else outside space-time entirely (twistor space - perhaps it is as simple as that - twistors are the answer). And this is the framework where neither space-time locality nor quantum unitarity is "manifest", i.e. visible - you have to switch to the other perspectives to see them.

So if we take a hint from the stories of Einstein, 't Hooft, and Arkani-Hamed, and say that the best guide to the truth is to focus on the concrete quantitative example of holography which we are lucky enough to have (AdS/CFT), rather than just guessing - then that would imply that the genesis of quantum mechanics is to be found, not in the boundary-to-bulk transformation, but in the transformation from the "third theory" into either of the space-time descriptions, boundary or bulk.

 

[atyy]

Isn't there a Euclidean-Euclidean version of AdS/CFT? In which case the boundary should have a Bohmian interpretation, shouldn't it?

But of course this wouldn't be a derivation of QM, since it the Bohmian interpretation is QM.

 

[atyy]

However, in Arkani-Hamed's recent talks, he interprets their work as the discovery of a third framework, neither string theory (bulk) nor field theory (boundary), but something else outside space-time entirely (twistor space - perhaps it is as simple as that - twistors are the answer). And this is the framework where neither space-time locality nor quantum unitarity is "manifest", i.e. visible - you have to switch to the other perspectives to see them.

Couldn't one say also say that unitarity isn't manifest in the Lagrangian description, compared to the Hamiltonian one?

 

[dhillonv10]

So then could it be that the way Arkani-Hamed described it, the boundary-bulk holographic transformaiton that you (mitchell) postulated before are part of a third framework? That is also very interesting, I understand that the paper I posted before perhaps doesn't have what i am looking for, actually introducing a third choice give one more freedom, now using the Grassmannian formalism, one can describe interactions that occur in both the bulk and boundary, this third choice gives you a framework can independently describe both of them or at least that is the general idea. Thanks for your help atty and mitchell.

 

[dhillonv10]

i just found this paper: http://arxiv.org/abs/physics/0611104 that talks about some very similar concepts that we have explored here:

Holographic Principle and Quantum Physics
Zoltan Batiz, Bhag C. Chauhan
(Submitted on 10 Nov 2006)

The concept of holography has lured philosophers of science for decades, and is becoming more and more popular on several fronts of science, e. g. in the physics of black holes. In this paper we try to understand things as if the visible universe were a reading of a low-dimensional hologram generated in hyperspace. We performed the whole process of creating and reading the hologram of a point in virtual space by using computer simulations. We claim that the fuzzieness in quantum mechanics, in statistical physics and thermodynamics is due to the fact that we do not see the real image of the object, but a holographic projection of it. We found that the projection of a point particle is a de Broglie-type wave. This indicates that holography could be the origin of the wave nature of a particle. We have also noted that one cannot stabilize the noise (or fuzzieness) in terms of the integration grid-points of the hologram, it means that one needs to give the grid-points a physical significance. So we further claim that the space is quantized, which supports the basic assumption of quantum gravity. Our study in the paper, although it is more qualitative, yet gives a smoking gun hint of a holographic basis of physical reality.

What do you guys think??

 

[mitchell porter]

Couldn't one say also say that unitarity isn't manifest in the Lagrangian description, compared to the Hamiltonian one?

Maybe it's a little less manifest. But in the Lagrangian formalism for QFT, you usually want to prove that the S-matrix, not the time evolution operator, is unitary.

You can see a statement about emergence of unitarity in the Grassmannian formalism http://arxiv.org/abs/0907.5418" , in the final section. That's from two years ago, there has been a lot of progress, and I don't know how the authors think about it now. I have heard it stated, in recent lectures, that these amplitudes are the volumes of polytopes (in twistor space, not AdS, I think), and that different integrations correspond to different triangulations in which different properties (such as locality) are manifest. So it would be of interest to know whether there are triangulations in which unitarity is non-manifest.

One might also want to compare these discoveries with Penrose's original hopes for twistor theory. The original spin networks were like Feynman diagrams but they didn't involve complex numbers, a property which facilitated an interpretation in terms of traditional probability theory, but they only gave you back angular momentum. I think Penrose hoped that twistor diagrams would be a calculus in which quantum amplitudes in space-time would emerge from a combinatorics governed by classical probability.

Isn't there a Euclidean-Euclidean version of AdS/CFT? In which case the boundary should have a Bohmian interpretation, shouldn't it?

But of course this wouldn't be a derivation of QM, since it the Bohmian interpretation is QM.

I've seen Euclidean continuations of one or the other side of AdS/CFT (not sure about both), and this may even pertain to the construction of dS/CFT from AdS/CFT. But you can make a Bohmian version of Lorentzian quantum field theory anyway, it's just a matter of gauge fixing: pick a preferred reference frame, use the Schrödinger picture, and then construct the usual Bohmian trajectories from the resulting wavefunctional. You can do the same thing for quantum gravity too - e.g. by reifying the "lapse" and "shift" functions of the ADM formalism. I don't know how you apply Bohmian constructions to Euclidean QFT, however.

 

[mitchell porter]

A simple fact about how the boundary-to-bulk mapping works (or indeed the mapping in the other direction) is that it relates points which are spacelike separated. You can see an example for AdS3 in figure 1 http://arxiv.org/abs/hep-th/0606141" , look under "figures", and look at figure 1, then it's the set of points in the interior of the cylinder lying between the two tilted lens shapes. The same principle should also apply in de Sitter space, where the holographic surface is either the cosmological horizon or the past/future conformal boundary, depending on which version of dS/CFT you use.

The paper in that first link tries to narrow the set of points on the boundary which contribute to what happens at a point in the bulk. They manage to reduce that set to a boundary region whose size depends on the radial distance from the boundary to the bulk point. This is more like the "shadow on the wall of Plato's cave". If you think of the light source as located at the center of the AdS space, so that a shadow is being cast on the boundary, then the size of the shadow of an object increases, the closer the object moves towards the light. But, a very interesting twist: to achieve this in AdS/CFT, that first paper had to use complexified boundary variables. Reminiscent of twistors! And of course twistor space is used in the neither-bulk-nor-boundary "third theory" of Arkani-Hamed et al.

Returning to the idea that boundary locality might be transformed into bulk nonlocality by a holographic mapping: what needs to be constructed is the inverse of the mappings mentioned above, which start with a point in the bulk and relate it to points on the boundary. For the approach using real variables, one can already see what the result will look like - the second link. But the approach using complex variables might be more challenging - to talk about the region of the bulk affected by a point in the complexified boundary, one may need to complexify the bulk as well.

edit: See figure 1 in http://arxiv.org/abs/0710.4334" for the de Sitter case. And they're saying that the de Sitter case, for real variables ... which ought to be the version of holography relevant for the real world ... is related to the anti de Sitter case, for complex variables! - the "complexified boundary" mentioned above. I'll write more when I understand it.

edit #2: Also see page 7 http://arxiv.org/abs/hep-th/0312282" for a boundary-to-bulk map for de Sitter space.

 

[dhillonv10]

thanks for the information mitchell, I will read those papers thoroughly, I also found another very interesting paper: http://arxiv.org/abs/gr-qc/0609128 which describes the following: (pg 3)

Quantum state changes caused by local interactions between mass quanta have non-local consequences throughout the universe. Any local change in the quantum state of the mass distribution within the universe is instantly reflected in changes in the eigensolutions of the Helmholtz wave equation within the universe, as well as in the lattice gas representation of the universe on an observer’s de Sitter horizon. This instantly changes both the probability distribution of the bits of information on the horizon and the corresponding probability distribution of mass quanta throughout the universe. In this way, the holographic principle indicates a mechanism for non-locality in quantum processes throughout the universe.

Here we see a similar process being defined in new terms. So perhaps this approach can be extended to include bulk to boundary transformations through the use of complex variables as mentioned.

 

[dhillonv10]

Perhaps another idea that came to me, and this is a little unrelated to our discussion of holographic mapping, but it is a consequence of it, let's assume what we are talking about is indeed correct. Now we know that entanglement can occur over very large areas, what i mean by that is, two particles can (theoretically) be hundreds of miles away and as long as they are isolated a spin measurement on one of them causes the other to take the opposite measure. Now from our mapping we can deduce that the idea of the particles being spatially separated isn't necessarily true, they appear to be far apart in the holographic image and the non-local interaction that occurs here is actually a local interaction occurring on the boundary.

Also what about the time considerations? Could time be affected by such a transformation? The equations of physics are time invariant and we "feel" time through increase in entropy, a concept that Eddington calls the arrow of time. Another speculation, it may be the case that the instantaneous signalling of time is not instantaneous after all, the time is takes for an interaction to occur in the bulk seems instantaneous (this time we are going the other way) but through the holographic mapping, at the boundary it is actually stretched out or more precisely said, the time is dilated as we would experience in Einstein's GR.

 

[qsa]

thanks for the information mitchell, I will read those papers thoroughly, I also found another very interesting paper: http://arxiv.org/abs/gr-qc/0609128 which describes the following: (pg 3)



Here we see a similar process being defined in new terms. So perhaps this approach can be extended to include bulk to boundary transformations through the use of complex variables as mentioned.

good find, this is very similar to my idea


https://www.physicsforums.com/showthread.php?t=399053&highlight=susskind post #4,

also,

post #50 https://www.physicsforums.com/showthread.php?p=2615567&highlight=susskind#post2615567


I do that in the most direct way which amounts to an extended Buffon's needle(radon) which is the non-complex Penrose transform.

 

[qsa]

dhillonv10, if you want I can PM you some related info with very interesting results.

 

[atyy]

Also what about the time considerations? Could time be affected by such a transformation? The equations of physics are time invariant and we "feel" time through increase in entropy, a concept that Eddington calls the arrow of time. Another speculation, it may be the case that the instantaneous signalling of time is not instantaneous after all, the time is takes for an interaction to occur in the bulk seems instantaneous (this time we are going the other way) but through the holographic mapping, at the boundary it is actually stretched out or more precisely said, the time is dilated as we would experience in Einstein's GR.

There is some discussion in this article by Hubeny and Rangamani about non-locality when in non-equilibrium situations http://arxiv.org/abs/1006.3675 (bottom of p6).

 

[dhillonv10]

qsa: please do, I would very much appreciate that. Also thanks for the links =)

atyy: that was a good find, they are discussing some ideas very similar to what we were speculating. Although they seem to be approaching this through the use of guage/gravity duality. Nonetheless thanks for the link.

 

[atyy]

atyy: that was a good find, they are discussing some ideas very similar to what we were speculating. Although they seem to be approaching this through the use of guage/gravity duality. Nonetheless thanks for the link.

I thought you were talking about holography in AdS/CFT?

 

[dhillonv10]

Yes indeed, the original intent was to see the ADS/CFT type holography however i recently found an impressive paper: http://arxiv.org/abs/gr-qc/0609128 that finishes half the equation so to speak, i have to work out if the inverse of what is mentioned in that paper holds and I would be able to generalize the statement. I would probably working on this approach for a couple of days however if it meets a dead-end, i would have to take the older twistor-theory approach and the paper you mentioned would compliment that line of thought.