the relation between entropy and probability at quantum levelu


by dhillonv10
Tags: entropy, levelu, probability, quantum, relation
dhillonv10
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#1
May21-11, 08:08 PM
P: 83
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.
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qsa
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#2
May21-11, 09:42 PM
P: 362
http://en.wikipedia.org/wiki/Entropy...rmation_theory)
dhillonv10
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#3
May21-11, 09:50 PM
P: 83
thanks for the link qsa, could you also possibly comment on the emergence of entropy? Thanks.

mitchell porter
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#4
May22-11, 06:06 AM
P: 748

the relation between entropy and probability at quantum levelu


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
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#5
May22-11, 02:25 PM
P: 83
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
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#6
May22-11, 04:16 PM
P: 83
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
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#7
May23-11, 12:46 AM
P: 748
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
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#8
May25-11, 12:09 AM
P: 83
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
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#9
May25-11, 12:35 AM
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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
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#10
May25-11, 09:44 PM
P: 83
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
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#11
May26-11, 05:50 AM
P: 748
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" (this paper, also its ref #1). That's the closest thing to a holographic explanation of quantum theory that I've seen.
dhillonv10
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#12
May26-11, 02:45 PM
P: 83
Also I found another paper describing a lot of what I'll need http://arxiv.org/abs/1103.3427
dhillonv10
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#13
May27-11, 07:57 PM
P: 83
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
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#14
May28-11, 12:04 AM
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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
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#15
May28-11, 12:53 AM
P: 83
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
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#16
May28-11, 10:02 PM
P: 748
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 "injective"). 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 Gerard 't Hooft and Leonard Susskind.

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 Juan Maldacena 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
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#17
May28-11, 10:09 PM
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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
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#18
May28-11, 10:26 PM
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Quote Quote by mitchell porter View Post
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?


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