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This thought gave rise to some new questions in my mind.

What are the consequences for:

  1. How would it affect duality i.e. particle, wave property of photons?
  2. How does this statement affect the information theoretical aspect (entropy) of the universe? Update: Given a volume V of space, is the entropy (maximum information that can be store) in this volume changed when this statement is applied?
  3. How is a black hole affect by this statement? Update How is entropy changed inside the black hole?
  4. Could one consequence be that the universe is hologram, since the construction isn’t continues?
  5. Would the smallest quantified space be planck's constant? Is there an equivalent constant for time?

I hope to get some of your feedbacks regarding this statement.

Qmechanic
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Amir Rezaei
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    You'll have to rework your question a bit I think, because as it stands now, it is way to vast. There are many different approaches to discretizing space-time. Loop quantum gravity is one of them, but there are I think other approaches based on causality. There are also other contexts in physics were we discretize space-time to an appropriate level for the problem at hand, meaning that the discrete units are much bigger than the Planck scale. A lot of solutions to problems in mathematical physics for instance follow Mark Kac advice:"Be wise, discretize!". – Raskolnikov Nov 15 '10 at 09:23
  • Since all of your questions are related to the fundamental nature of space-time, I can already say something about point number 5. The motivation to discretize space-time is usually precisely the realization that there probably is something like a Planck scale. The Planck scale for length is the same as that for time (possibly up to a factor $c$) since both are considered equivalent since the advent of relativity. (Unless you try to build a theory that breaks with this tradition. I'm thinking about a recent talk by Sean Caroll about time being more fundamental than space.) – Raskolnikov Nov 15 '10 at 09:29
  • Well my question is more general. I don’t want to go in detail about which theory is behind the discrete space-time. It may be not possible to answer the question since it’s highly coupled to what you are referring. However my questions are more theoretical and don’t refer to a specific “problem at hand”. – Amir Rezaei Nov 15 '10 at 09:36
  • Could you please send me any available link about Sean Caroll talk? – Amir Rezaei Nov 15 '10 at 09:39
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    @Amir: what do you mean you don't want to go into theory? If you have no theory, you have no physics; only science fiction at best, but crackpottery more likely. I am not sure you are at the right place if you want to discuss that. – Marek Nov 15 '10 at 09:53
  • @Marek I answered Raskolnikov about what he said "There are many different approaches to discretizing space-time.". What I meant was that I don’t want to be specific about the theory behind discrete space-time, such as Loop quantum. – Amir Rezaei Nov 15 '10 at 10:07
  • Here's what I was talking about concerning Sean Caroll. There's only slides for now, but a video of the talk could show up one of these days. – Raskolnikov Nov 15 '10 at 10:21
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    @Amir: I read that and I am still not sure what do you want. If you asked "What is light exactly?" then we could give you variously precise answers based on some accepted physical theory. But we can't answer hypothetical questions regarding some hypothetical construction, unless you tell us in what framework you want to work. This is because an answer to each of your questions is highly specific and might differ among theories. So you should make the question more precise to target some concrete theory. Otherwise, I currently see no way of answering that would belong to this forum. – Marek Nov 15 '10 at 10:25
  • @Marek Ok, Then I change the question and base my question on M-theory. I will then get answer about what the consequences are for the universe defined with M-theory – Amir Rezaei Nov 15 '10 at 11:00
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    @Amir: now the question is answerable, except it's too trivial to answer: M-theory / string theory is built on continuous space-time. I doubt that's what you wanted to hear. I suggest you try to think about what exactly you want to know. E.g. question such "What are some approaches to discrete space-time used in modern physics and what are the consequences for [X]?" where X is e.g. one of your five questions here. Or you can be more specific and pick some discrete approach (if you know some), such as LQG. – Marek Nov 15 '10 at 11:14

4 Answers4

5

Let's try and make things more precise, step-by-step.

  1. There's no such thing as "particle-wave duality": the name-of-the-game is "Quantum Field Theory". This paradoxical notion of a possible "duality" only happens when you don't use the appropriate framework to describe your Physics. Therefore, it makes no sense to speculate on what would happen if spacetime were quantized/discrete: in this scenario, the question would be: "Would a quantized/discrete spacetime affect Quantum Field Theory?" And the answer to this question is "No." The reason being that different physical theories have different domains of validity, given by the characteristic energy of the phenomena they describe.
  2. What is the "information theoretical aspect of the universe"?! This is not even appropriately defined, let alone "well defined".
  3. The black hole is the stereotypical object in a quantum gravity theory. So, when you quantize spacetime, you should look at black holes to see what happens. We already know that black holes have Entropy. So, the very first question should be: What does your particular quantization scheme yields for black hole Entropy? The current state-of-the-art, as far as i know, is that all different schemes of quantization of spacetime yield a reasonable answer to this question.
  4. This question, again, is not even appropriately defined, let alone "well defined". Holography has a very precise and well defined meaning in Physics, which is not related to the hologram in a credit card, for example. So, holography does play a role in quantum gravity, the more famous statement being that of AdS/CFT. But, as it stands, your question does not have meaning.
  5. This has already been stablished a long time ago: if you quantize spacetime, the smallest unit of spacetime is given in terms of Natural units.
Daniel
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  • @David, Thank you for your answer. Regarding question 5 I’m asking for an equivalent constant to planck's constant for time. So the question isn’t what is the unit, the question is what is the constant. If such constant exit then it has value and dimension. That is what I asked. I’ll come back to you regarding other points. – Amir Rezaei Nov 15 '10 at 14:31
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    @Amir: i perfectly understood your question (5). If you check the link i sent, you'll see that things such as "Planck Volume" and "Planck Time" exist and are well defined (and have a numerical value, as per your wish). So, in this sense, a notion of "spacetime atom" does exist, even if it's vague. To make it precise, it will of course depend on the particular theory you have at hand. – Daniel Nov 15 '10 at 15:04
  • there is no particle-wave duality, it makes no sense to ask if spacetime is quantized or discrete, discrete spacetime would have no effect on QFT, dude, what are you are talking about? These are some serious misconceptions that you are propagating to the laymen. The informational theoretical aspect of the universe is not well-defined - seriously, have you not read any scientific literature from the past decade? Unless I'm completely misunderstanding you, this answer is riddled with gross mis-characterizations of various issues and their status in modern physics -1. –  Mar 29 '11 at 20:54
  • @Deepak: I know perfectly well what i'm talking about and assure you there's absolutely no misconceptions anywhere in my answer. Having said that though, you don't need to take my 'assurance' as any form of guarantee, but if you're going to make such a strong criticism, it'd be good form for you to point out the differences, the misconceptions you claim to exist. – Daniel Mar 29 '11 at 23:50
  • @Deepak, cont'd: with that out of the way, let me ask you this: Does QFT resolve the so-called wave-particle duality or not? What is the definition of the information theoretical aspect of the unverse? What are the domains of validity of QFT and how would this fare when compared to a possible theory of quantum gravity? Thinking about these questions will definitely lead you into what my reasoning was to answer these questions. – Daniel Mar 29 '11 at 23:54
  • No, QFT does not resolve the wave-particle duality. It cannot do so, because wave-particle duality is a central aspect of Nature. Also, if you do feel that QFT "resolves" wave-particle duality then it is incumbent upon you to explain such a bold statement. The information theoretical aspect of quantum gravity is encoded in the holographic principle, the question of black hole entropy and the covariant entropy bound - see my answer to this question. The nature of any QFT depends on the background manifold on –  Mar 30 '11 at 00:25
  • which it is constructed. For smooth manifolds QFT is the usual beast we are familiar with. For discrete manifolds - such as the graphs which describe the structure of spacetime in LQG, for instance - the notions of operators and observables in QFT have to be altered accordingly to take into account the discontinuous and discrete behavior of physical quantities. In LQG such considerations lead to the notion of a "generalized (gauge) connection" which has support only on 1D manifolds - sort of like the Wilson loops of ordinary QFT and QCD. –  Mar 30 '11 at 00:30
  • Also see papers by Achim Kempf (link1)[http://arxiv.org/abs/gr-qc/9907084] (link2)[http://arxiv.org/abs/hep-th/0404103] (and many other papers) who has done extensive work in trying to understand the the meaning of "discrete" and "continuous" in the context of quantum gravity. The second reference also shows that the existence of a minimal length leads to a cutoff on the maximum information density in a given region. Work along these lines has by now been done independently by several groups and individuals. Point being, the discreteness of geometry and the informational aspect of nature all –  Mar 30 '11 at 00:34
  • tie up into a neat package. As to precisely what quantum gravity reveals about the question of wave-particle duality is something that is not yet very clear (AFAIK), but it in inevitable that it must shed new light on this (century) old paradigm. I hope that clarifies where I am coming from. I am also happy to discuss further any issues you feel are relevant. Admittedly these are complex topics which have no straightforward resolution hence their categorization as "frontier" research. –  Mar 30 '11 at 00:36
  • @Deepak: I really don't intend into dragging this discussion, so i hope this short version of my answer will satisfy you: you can read about wave-particle duality in Penrose's tome, The Road to Reality, chapter 21, in particular in section 21.5. As for QFT, it creates and destroys wave-functions, thus clearing the confusion on the spot. Now, if you want to call the measurement problem by "wave-particle duality", that's your personal choice, but not the community's. – Daniel Mar 31 '11 at 18:18
  • As for your other points, suffices to say that QFTs over curved backgrounds are a hairball. And i would love to see you do QFT over a de Sitter background (as opposed to its more famous cousing, anti-de Sitter): can you define the Cauchy problem for a simple (bosonic, spin 0 — free scalar; or maybe even $\phi^4$) QFT over de Sitter space? If you can do this, would you be able to do very same for Gauge Theories (Yang-Mills)? Maybe our definitions of "frontier research" are slightly different... – Daniel Mar 31 '11 at 18:22
  • if you want to call the measurement problem by wave-particle duality ... what? That's like comparing apples and oranges. What did I say that makes you think I'm equating the two? About QFT on deSitter take as look at Candelas and Raine's 36 year old paper. To quote from the abstract: The massive scalar and Dirac fields quantized on a de Sitter background geometry prove to be exactly soluble models in general-relativistic field theory. Is that what you're asking for? About your other points, they are as silly as these two. –  Mar 31 '11 at 20:38
  • @Deepak: "(…) they are as silly as these two." I haven't used adjectives to characterize your comments so far and i'd have appreciated you keeping this discussion at the Physics level. However, i understand your inability to do so, granted that you don't even appreciate what a 'Cauchy problem' is. So, if you want to keep on trolling, go right ahead — i have better things to do. – Daniel Apr 01 '11 at 09:00
  • @Daniel you are throwing up random terms and concepts to cover the fact that you made gross errors in your answer. Readers are free to decide, after reading my responses and yours, as to where the truth lies. I'll go troll elsewhere now ;) –  Apr 01 '11 at 09:04
  • @Deepak: don't mistake your lack of knowledge for "random facts": the fact that you can't put these arguments together and understand my points says absolutely nothing about their validity (it only speaks about you). It's not my fault you don't know certain aspects of QFT: i even provided a reference that expands on some of my points (on top of giving you more references to clarify further doubts). I have been very honest and forthcoming, but can't fit in ~350 characters what i understand to be the answers you're looking for. – Daniel Apr 01 '11 at 09:12
  • @Daniel my recent experience with my answer to this question on pseudoscalars in E&M has led me to realize what it feels like to be on the receiving end. In hindsight I was too aggressive in critiquing your answer and am guilty of not doing unto you as I would like to be done unto me. In any case, hopefully, we will have another opportunity to clarify these issues in a less confrontational manner. Cheers mate and apologies for my rudeness. –  Apr 02 '11 at 08:14
  • @Deepak: don't sweat it. Living and learning… – Daniel Apr 03 '11 at 21:31
1

There is a recent paper about Noether theoreom on discrete systems which i found pretty interesting, i thought to share;

http://arxiv.org/abs/1103.4785

lurscher
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OK, I found a review article that might be useful to you:

http://cdsweb.cern.ch/record/704227

I quote the abstract here:

We review some modern theories about the structure of space and time, in particular those related to discrete space and time. Following an epistemological method we start from theories which discuss discrete space and time as a mathematical tool to solve physical models. Antother theories look for physical content of the discrete structure of space and time, based in relational theories of space and time which are derived from the relations of some fundamental entities. Finally we present some philosophical positions who try to find the ontological foundation of the relational theories os space and time.

Hope this is the kind of thing you were looking for.

EDIT: Woops, this was misleading, the abstract is in english, but the paper is in spanish. The references in the article are still useful though.

Raskolnikov
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0

Richard Feynman in this famous sixties public Cornell lecture claimed it's easy to prove "physics space cannot be discrete automata", otherwise it will soon violates existing physical observations. But he didn't mention the proof or explanation later on in this lecture series.( https://www.youtube.com/watch?v=-2NnquxdWFk&list=PLS3_1JNX8dEh5YcO-Y05stU0u_T9nqIlF&index=7 )

I thought about this and felt under classical analysis framework, only continuous space and time make velocity (not the other feature - position) possible. If space or time is really discrete ontologically, then like Zeno's logic in his famous Zeno's Paradox, an arrow can never move! The essence of Zeno's paradox's resolution lies in time and space are continuous, thus you can have possible velocity notion via its position's change along with "measuring" its corresponding time interval. If time is discrete automata, then you can only have position along with "counting" its time instants, it's hard to imagine a way here to derive a velocity-like concept.

Is my above reasoning on the right track?

cinch
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