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Emergent statistical manifolds

  1. Jun 24, 2008 #1


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    Does anyone know of any research programs out there that are considering physical structure formation in terms of emergent, self-organizing statistical manifolds, and does so without starting from some reasonable first principles without any structure or preconception of manifold, or ad hoc evolution rules, chose to give the desired result?

    So that not only the geometry is induced, but that the manifold itself - including dimensionality is also induced.

    Could the things we know, spacetime, laws of physics, and the particle phenomenology be explained in terms of interacting/communicating self-organizing "statistical" manifolds, and what is the logic of that construction?

    There are mathematicians doing alot of stuff on this, but they rarely have the perspective of a physicist, although it might be a "physicists toolbox".

    Could such a path be the road to the new logic we may or may not need?

    Ideas or pointers anyone?

  2. jcsd
  3. Jun 24, 2008 #2
    The only thing even vaguely similar to what you suggest -- and of which I'm aware -- is contained in some of Ariel Caticha's preprints on the ArXiv, where he uses maxEnt methods to investigate classical mechanics and, loosely, general relativity from a statistical, information-theoretic viewpoint.
  4. Jun 24, 2008 #3


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    Fra, by "statistical" manifolds, do you mean only the manifolds of usual statistical mechanics?

    Or do you have something more general in mind?

    In a very general sense, would Renate Loll's result of an emergent deSitter space---a simple 4D manifold---be an example of the kind of thing you are asking about?

    Or are you talking about something more like the N-body problem----the position/momentum states of N particles? Roughly what dimensionality do you have in mind?

    Interesting question.
  5. Jun 24, 2008 #4


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    Thanks for your response Shoehorn! I know of Ariels work. His spirit is right in my taste! but while I've read many of his good papers related to that, it is is not entirely satisfactory. I am thinking of something alone that line, but slightly more different. I am however interested to follow his future work. The problem isn't easy, so all kinds of differential progress is interesting.

    Also Ariel's first ideas (as far as I know) is a derivation of classical GR. I see no reason to distinguish classical from quantum. I think they should follow from one construction, so the normal quantization scheme should not be needed. I am looking for one unified formalism where the superposition principle emerges automatically due to it's presumed higher "fitness" to an observer.

    Also I was hoping to find this applied to "discrete manifolds" rather than continous ones.

  6. Jun 24, 2008 #5


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    With statistical manifold I just mean something like a space of probability distributions. On which one can induce various choices of measures and geometry. For example one can talk about the "distance" between two distributions. The kullback leibler divergence for example.The discrete generalisation would be like a space of microstructures.

    What I envision is definitely a generalisation of stat mech. Where the spaces themselves are dynamical.

    For example, one can define the distance of a possible future, relative to the present. And this would rate the probability of such future - given the limitations.

    With dynamical dimensionality I envision something like a procedure to exploit asymmetries in the degrees of freedom, as per a particular procedure. Then the final dimesionality depends on when no more distinguishable asymmetries exists. But I can't described it briefly and exactly yet. This is ideas, relating to this. I was hoping someone else had some partial progress.

    Not that I can claim to have a better answer atm but the CDT construction has too much baggage is not fundamental enough, it doesn't answer my questions. For example it assumes the einstein hilbert action. In my vision this *measure* should follow as a result of a deeper self-organization. Also it doesn't even attempt to reason towards the path integral formalism. In the general thing I imagine, this should be more clearly understood. I think the logic implicit in the "summation of paths" and it's weights can and should find a deeper understanding.

  7. Jun 24, 2008 #6


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    Meaning that a "point" on the manifold corresponds to a probability distribution.

    But the point is that the physical basis "probability" is one of the problem, so I envision it replaced by a construction of a measure which is like a probability, but discrete one. Also I consider the manifolds to be relative (observer dependent).

    In a way the state of the manifolds are like the state of the observer, or the observers knowledge. It would replace the wavefunction. But I expect a connection where standard QM can be understood as an emergent approximation.

  8. Jun 24, 2008 #7


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    I guess something about what they do is attractive, but too many things are still put in manually. One thing that doesn't make sense to me is that - in the context of the suggested information views, measures are somehow treated on uniform basis, and I do not think it's conceptually consistent to reject all prior measures on spacetime while putting in manually the action. IMO the action is a measure on a similar basis as is entropy and many other things. And I think that analysing this exact relation is a key to progress.

    Even the action - that they put in - is as I see it also dynamical. And I don't see it as a simplest possible starting point either? This view of things, suggest that the "space of actions" is not random, so it should be possible to unify states and processes.

    I see action as a estimated measure of distance between the initial and final states. The estimate is due to the idea that the true measure isn't known and the manifold itself is in motion too. I want to straighten out the logic of this in detail. Once that's accomplished I figure the exact interpretation and meaning of stuff like feynmanns path integral and classical statistics should be come more clear, and the question of howto "interpret" the "formal" path integral (ie howto "choose" the measures of integration and it's integration spaces) should be unambigous?

    That such basic things are yet unresolved is weird.

  9. Jun 26, 2008 #8
    I don't know much about statistical mechanics, but I do know that the Kullback-Leibler divergence doesn't work as a metric. At least not if you want your space of probability distributions to be a Riemannian manifold.

    Are you aware of Sun-ichi Amari and his collaborators' work on "information geometry"? They use the Fisher Information Metric to make the space of probability distributions into a smooth Riemannian manifold where they can then do differential geometry. I think the primary reference on this was a book by Amari with a title something like "Differential-geometrical methods in statistics". There may have been another more recent book also...

    Is this sort of work even related to what you're looking for?
  10. Jun 26, 2008 #9


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    I was not aware of Sun-ich Amari but I did a quick goole and yes what he does is definitely related to what I'm talking about. Wether he has the solutions I need to look. I will try to probe his work, thanks! I several papers of him I'll skim!!

    What I am looking for are new angles. To see if somehow has the answers to what I'm working on. I know the basic fischer stuff. But the fischer metric - seen as a measure of geometry is not unique. I have seen Ariel argue that this measure is unique, and while he has a certain consistency, his rules of reasoning behind it are not satisfactory. So I am looking for some more clever constructs that considers "generic measures", that are evolving. So it's unique if you accept certain premises. That's the weak point.

    The kullback-liebler divergence is not the magic measure, it's one measure only. But from a certain point a very basic one.

    I am trying to resolve this, without starting by another non-trivial premise of my choice. Instead I am trying to find an inital, minimal logic, from wich the rest could be constructed with feedback from the environment. Hopefully as the complexity increases, measures will emerge, and this measures should have properties that match observed properties of nature.

    My quest is not to find geometric interpretations of things at all cost, I have other guides, but since alot of current physics are formulated in geometric forms, making a connection to current models requires me to find a way from my starting point. I have no preconceptions of what I am looking for. For example, I see no fundamental reasons why I must arrive at exact riemann metrics, therefore using riemann constraints as a rule of reasoning may be misleading.

    Rather what I have in mind is that the choices of measures have a logic to them, and the rules for constructing measures may find a physical representation. I associate information processing and exchange between an observer and an unknown environment, with a physical system interacting with it's environment.

    Of course guides are that at least in the "effective theory" sense, the phenomenology of the standard model of particle physics, as well as GR must be recovered.

    But since one might suspect that neither GR nor QM are perfect, the logic that hold exactly in QM and GR, may not necessarily by fundamental to a new theory. However any new logic still needs to explain the logic of GR and QM as emergent. And I think it can be done.

    So first is the logic of how we can have emergent manifolds. Next, one can equip that manifold with many measures. But these measures emerge for a reason.

    The other issues is that I do not a priori consider continous manifolds. I consider discrete manifolds. But macroscopic systems will be effectively continous fapp. But I think some of the logic is lost if one a priori assumes that there is a physical sense in the continuum at all scales.

    After all, the notion of real numbers are constructed as a kind of limiting case.

  11. Jun 26, 2008 #10


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    I would expect that one possible question arising to someone reading this is that this is madness or, do I think that one can predict everything by pure thought? No, of course not.

    However, there are clever guessing and random guessing.

    To put it very generally, the predictions of the theory are in certain sense, questions. IE. what questions should we ask? And if the strategy is good, a system acting as per this logic should be very fit and should have better odds at development.

    So I guess my basic assumption is that "the world", meaning life, cosmos and the laws of physics are assembled/evolve as per "constructive logic".

    So in general I ask a question and receive an answer, my response is another question - but which one? What is my next question? Ie. what measure on the space of questions, has emerged so far? and how this this measure evolve and in turn get measured?

  12. Jun 27, 2008 #11


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    > It will suggest that the measures here more or less picked by hand,
    > can be understood as a result of a process. And furthermore, this
    > process is still ongoing and perhaps additional phenomenology can
    > be extracted (not only spaceitme) if this logic is worked out.

    For example, his guessing that the prior information feedback to produce expectations follows the Einsteins equations for the fisher metric, is something that should be possible to explain as an emergent action.

    IE. Rather than exploit GR to find an information theoretic reformulation only. I think a better understanding should be able to explain, WHY einsteins equations looks the way it does, and also answer to wether any possible corrections look like, and how it is unified with QM.

    These are some of this missing parts.

    Last edited: Jun 27, 2008
  13. Jun 27, 2008 #12


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    This missing thing, is also the same thing that's missing in the CDT approach.

    So my dissatisfaction are at a similar level.

    And I think resolving that may also resolve the problem of incorporating matter into the formalisms in a unified way, rather than trying to patch it in with the result of breaking the line of reasoning.

  14. Jun 27, 2008 #13


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    Strange, I wrote a post between 9 and 10, that got lost. I dont know what happend, maybe I closed the window before I submitted it. How silly. Mmm

    Anyway a short summar: it was a pointer to

    -- http://arxiv.org/abs/physics/9808009

    Which I mentioned just as another example of the spirit.

    And the comments in posts 11 and 12 refered to things I wrote... quoted from that paper...

    "1. The appearance of time is a consequence of uncertainty.
    2. Space is infinite dimensional and only on the average appears as four dimensional."

    "We guess: Information is the source of the curvature of hypothesis spaces. That
    is, prior information is the source of the form of the model
    From: The dynamics of how mass-energy curves spacetime are controlled by the
    field equation"

    "guess: The field equation controls the dynamics of how prior information produces models
    From: The field equation for empty space is the Euler-Lagrange equation that characterizes the extremum of the Hilbert action, with respect to the choice of geometry."

    The spirit here is in line with my associations on the logic of EH action, but I want to, not just exploit the logic of it, assume that the EH action applied to fisher metric is the way to go, but rather see the logic of that logic, and thus be able to explain why the EH actions looks like it does, and what the corrections are expected to be.

    Edit: in the lost post I also wrote that what I mean with "constructive logic" is a self-supporting logic, and the idea is that one could consider an "initial state" with completely random logic and random measures, and that should then self-organize because a random logic is not constructive - it's desctructive. I also wrote I associate here to Smolins refelctions on what is the nature of physical law, which I further associate to the logic of logic.

    Then the question is in such a picture, what kind of structure could one imagine forming? Are some structures and logic forms more common? And can the expected emergent logic and measure here, be matched by the actions of GR and standard model? I think it can, that's my hypothesis - and I'm looking for partial progress in this direction.

    Last edited: Jun 27, 2008
  15. Jun 27, 2008 #14


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    So lets say that CDT gets emergent spacetimes from a given logic - that is good, but IMHO consistency of reasoning implies that also the logic must be treated in the same way.

    I want the action, and logic of QM, itself (that CDT takes as premised) to also be shown to be emergent. There is IMHO, possible by the same toke than there is a logic to the emergence of spacetime, a logic to the emergence of the rules(Action forms of GR) that allows emergent spacetmie.

  16. Jun 28, 2008 #15
    I did read the arXiv article mentioned in your link. The idea at a semi-philosophical level is very interesting. Do you know some more recent references about the topic ?
  17. Jun 30, 2008 #16


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    Not really. Ariel Caticha has some papers from 2005 on arxiv, for exmaple http://arxiv.org/abs/gr-qc/0508108. While they are in the similar spirit, I'm not aware of any substantial progress from my point of view that has been published. That's part of why I posted this thread too :)

    I mainly posted the that link to show an example of the type of reasoning I seek. As I see it those paper does reasonably well communicate parts of the spirit of a new program to fundamental physics. But I'm still looking for refinement.

    I think one first partial step, which is also the goal of Ariel Caticha is to show how the action measures of General relativity follows, at some point, from a spontaneous self-organisation. This would (I think) possibly suggest that the action of gravity is itself not a constant form so to speak, it would itself evolve. So Einsteins equations might be more like a state equation notifying a certain type of equilibrium. So one would then assume that those equilibrium conditions do hold at all scales where we have tested it.

    By the same logic one would expect that the standard model of particle physics would appear.

  18. Jun 30, 2008 #17


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    The way I see it, the core of the problem is how the various measures are constructed. To just construct a measure on vauge grounds, and then give it various geometric interpretations is just half the story as I see it.

    We can see Einsteins equation as a description of how one geometry evolves into another one, as per a fixed measure on the space of geometries - the "to be minimized" action defined by the EH action.

    We can see how one distribution evolves into another one, as per some choice of measure on the space of distributions.

    The analogy is somewhat clear, and the evolution doesn't refer to external time, it's more like relational changes in a direction of increasing "probability", but unless one can argue for how measures are constructed it all seems ambigous. Of course the construction process is an interaction process with the environment, so without interaction with the environment there is nothing to drive the construction.

    What I am looking for, is how that perspective can be complemented with the evolution of measures. And it seems that what we have is a hieararchy of measures, so all we have really is an evolving systems of relational measures.

    In a way I see the geodesic equation and einsteins equations as two such related measures. The geodesic is an evolution relative the the geometry, but at the next level the geometry is also evoling as per Einsteins equation, and what about the next level? There seems to be no next level - why? To me that is a relevant question. If there OTOH is a next level, then that means the Einsteins Equation would be interpreted as an equilibrium condition. And then it is still interesting to understand, how come this equilibrium state seems to be favoured?

    But if we think that any given observer can not construct systems of arbitrary complexity, then starting from the minimal complexity perhaps there is a finite logic of construction? and the relation between the measures should hopefully come from the construction.

    Some of these questions are not typically asked in the mentioned papers. So this is one element I think is missing.

  19. Jun 30, 2008 #18


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    In this respect some things in string theory are very remotely attractive, but there's still something that doesn't seem right, and their fundmental starting points and reasoning are alien to me.

    That construction starts from the concept of a "string" in a background space, and the starting point seems ambigous. As I see it, that's somewhat similar to starting with a given measure on a manifold with non-trivial structure.

    I think there should be a way to get around that, and I doubt that the notion of strings fundamental. It breaks the whole relational spirit. It's not jsut the background space that is disturbing, it's the background measure (the string) itself. I would expect that the spaced itself should be emergent is part of the theory, not just as theorist reasoning about what human level consistency requires.

    I can imagine that from pure measure constructive reasoning, string like measures will emerge at some point, the whe whole understanding seems lost/ignore if it's postulated. And if they really aren't fundamental, then what kind of reasoning can we extract from the formalism?

  20. Jul 1, 2008 #19


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    Emergence of quantum logic? (logic of logic)

    One of the ideas I was probing for behind emergent statistical manifolds, and sets of manifols is that if each manifold is equipped with it's own logic, then the logic itself needs can be transported between manifolds, and thus during transformation classical logic may be deformed.

    For example, if one asks about the meaning of (A or B), then if A and B are points of a statistical manifold they correspond to probability distrubutions, or measures. Then the natural notion of "or" is the normal classical logic.

    But clearly, if A and B refer to two different manifolds, the distributions must be transported to the same manifold for a consistent definition of the logic operators.

    Also, the case of conditional probabilities boil down to the logical and and or operators.

    So for example P (x | p) depends on a consistent definition of (x and p). Obviously, what does x and p mean, if they are seem referring to different manifolds. To make sense of that, there must be a relation between the manifolds. And of course there is, but in QM this is postulated between q and p. But if manifolds are emergent there may still be relations to other manifolds.

    In a certain sense one might see the different manifolds as related by various types of phase transitions, and the equilibrium is selected for maximum fitness. This would similarly cause transitions between different logic.

    An interesting parallell to make here with curved spacetime is that the notion of parallell lines requires a notion of parallell transport, since it makes no sense to define a logical notion of parallell, when the points belong to different spaces. Now byt the same token, but slightly generalized the notion of logic could be relative, if seen as a natural measure defined in a manifold. This means that there must be a similar notion of "transport" or "communication" of the logic.

    Anyone know of any papers relating to that?

    What I'm struggling with now is how to implement the most general transitions betwene manifolds. One ide is that the network of communicating manifolds, at least for a steady state approximation conserve information capacity. So one manifold branching off, requires another one to shrink, and it's determined by equilibrium equations.

    But one problem is how to construct the "counting of the set of possible manifolds", which seems to also imply counting the ways to construct a logic of counting. And the sequence would end where no more questions can be distinguished. So the idea is that divergence is prevented by that a discrete manifold, is equipped with finite possible logic. And those questions that can not be seen to give possible distinguishable answers, are not asked until the symmetry is broken by the indeterministic feedback?

    I guess what I'm trying to exploit is that, perhaps if we can understnad the inside "constructive logic", of an observer digging his way into the unknown, we can see that it's behaviour as seen by other observer actually makes sense?

    Perhaps someone working with neural networks or self-organisation in biology has reflected over this? Rather than considering self-organisation based on superimposed external rules, the idea would be that the only rules are some form of "local self-preservation".

  21. Jul 1, 2008 #20


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    Re: Emergence of quantum logic? (logic of logic)

    Is a cobordism the sort of thing you're looking for? Incidentally, when John Baez talks about Topological Quantum Field Theory, he often relates it to defining relationship between cobordisms and Hilbert spaces. He also talks a lot about using Hilbert space as the underlying notion instead of Set -- from some point of view, this would roughly correspond to replacing ordinary higher-order logic with linear algebra.

    Incidentally, formal logic takes a boring approach -- if P and Q are propositions with completely separate domains, you just lift them to the Cartesian product of those domains.
  22. Jul 1, 2008 #21
    String theory is manifestly background-dependent and hence non-relational in a Machian sense. Indeed, it's non-relational by construction.

    In what sense do you mean that a string is a "measure" on the background?
  23. Jul 1, 2008 #22


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    Re: Emergence of quantum logic? (logic of logic)

    That looks potentially interesting, but I need to look close to have a more specific opinion. Thanks for the hint!

  24. Jul 1, 2008 #23


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    Loosely, I am associating a 1D string embedded in n dimensions, as a distinguishable object or "correlation pattern", and more specifically as a possible measure on n-1 dimensions (thought you might object to this associaton). So "the measure" generates the n'th dimension. Or alternatively that there in a restricted sense is a duality between a string in n dimensions and a mesaure on n-1 dimensions. And if measures are emergent in the larger picture, measures can be thought of as "excited", and the excitation can invoke more dimensions. Similarly when some dimensions are not realized they become indistinguishable (not measureable) and collapse. The collapses and inflations I see as relations between different manifolds. So in a certain sense the separate of the evolution of measures on the manifold, and the evolution of hte manifolds itself are possible of a dual nature.

    One way I picture dimensional emergence myself is by hiearchial measures on measures. And at some point the measures themselves, once stable, becomes indexes in a higher dimension.

    Edit: My admittently vauge and unclear association here, reveals that I think that maybe there is a background independent way to arrive at something that looks like strings, but where they aren't fundamental. But since I'm not advocating string theory I do not wish to continue this line of reasoing, I only consider it a simple conceptual association to the "string logic".

  25. Jul 1, 2008 #24


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    Re: Emergence of quantum logic? (logic of logic)

    I checked that link and some of Baez notes and a cobordism seems associate by a transition/relation between two manifolds of one lower dimension, which is a symmetric equivalance relation which is also seen as a larger manifold? Further one associates hilbert spaces to each manifold, including the larger cobordism manifold itself - sort of attempting to treat processes and states uniformly which is good.

    That is somewhat in line with I imagine, and it comes out as an induction step. Each differential progress of a manifold defines a new higher dimensional manifold. I wonder though, what happens with the information content in the manifold? It is a question of mine. In what I imagine, that is a key issue. I like to think that the system or observer who is implementing this reasoning must be able to relate to these structures. Ie. there must at some level be a physical basis for the mathematical ang logical constructions. This is even why I prefer to think of discrete manifolds ultimately built from historical data, but it could be that in the end the contiuum notion could be recovered, but I have hard to use my intuition in terms of continuum stuff when starting from notions of distinguishability.

    Without getting into details I think I can related to that, but one remaining problems are how is the induction treated. I imagine that this induction defines a flow that is related to time. That seems I think also to be somewhat in line with those ideas, but a problem seems to be how to precent the induction processes to go wild and just spin branch off uncontrollably and inflating the information context of thte manifold to the point where it makse no sense? So, one seems to be looking for a rule, by which a manifold naturally induces a set of possible cobordisms (as differential progressions).

    I don't know enough of that work to know wether they have a plausible tentative solution to that?

    Somehow I would imagine that the logic must be constrained by a defining view of the observer. So that the suggested logic is a true intrinsic logic.

    Also, I'm not sure how they start off and introduce hilbert spaces. I get the feeling that the same old QM are more or less iterated on more sophisticated hilbert spaces? If so, I'm not sure how much sense that makes?


    I'm not sure I got that right though, it's the first impression from skimming some Baez pages and the wiki link.

    I guess the idea I have is that if there exits a relational construction scheme for propositions, two propositions asked by the same observer are always related, at least indirectly. Unless they originate from two different observers, but then there is no direct physical basis for defining the basic logical operations on them either.

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