Hi Lawrence,

I read your essay some time ago but only now got around to rating it -- was delayed in moving from ITB/Bandung to Thailand. In any case I very much liked the use of associators/non-associativity in your essay. Two FSU colleagues of mine (Merab Gogberashvili and Vladimir Dzhunshaliev) have worked on trying to include non-associativity into QFT. Merab in particular had some mantra about the connection of different algebras to physical quantities to the effect--> real numbers are connected with mass; complex numbers with charges; non-commutative numbers with spin; non-associative numbers with the quantum wave-function (actually I do not recall exactly what the last connection was but it was something to do with the quantum nature of matter). In any case you might find Merab's work of interest. Sorry it took so long to finally read your essay.

Also I noticed your address (or one of them) is in Hungary. In this regard you might be interested in a conference Elias is arranging in Prague, Sept. 1 --5. It is a fairly large and broad conference but some of the workshops would seem related to your line of work. In any case the conference site is http://www.icmsquare.net/

I hope this is not considered "advertising". Anyway if this post is not accepted we'll know :-).

Best,

Doug

    Doug,

    I have read several papers by Vladimir Dzhunshaliev on octonion field theory, and Merab Gogberashvili is a familiar name as well. Trying to understand how nonassociative mathematics of operators fits into physics is really the hard part. I think that quantum mechanics is purely complex, or C. Of course classical mechanics is R. Gauge theory can be written according to quaternions H. A lot of gauge theory is done though in standard vector form without quaternions. It is interesting though that Maxwell formulated electromagnetism, the first gauge field theory, in quaternions. Field operators in a second quantization act on a Fock space basis to give quantum amplitudes. So we have a relationship that might be heuristically written as π:H --- > C. The question is then whether there is some sort of higher level structure π:O --- > H.

    Spacetime I think offers a clue. A black hole horizon has some quantum uncertainty on a scale near the string or Planck length. There will then be an associative uncertainty with three quantum fields, where one of those fields is identified near the horizon. The standard approach to QFT is to assign a harmonic oscillator at every point in space, impose equal time commutators on that spatial surface with the Wightman criterion for commutation, and work from there. Yet that spatial surface on a small scale will have some noncommutative structure and this will lead to a host of uncertainties in assigning QFT operators. If there are event horizons this should lead to an associative uncertainty.

    The above "maps" between C, H and O, where a similar map π:C --- > R would be the relationship between quantum mechanics and classical mechanics, are really just forms of the Hopf fibration. The relationship between quantum and classical mechanics is of course a difficult subject in its own right. With each of these "ladders" on the Hopf fibration there is some increased uncertainty. Quantum mechanics saved physics from the UV divergence that classical mechanics predicted with the hydrogen atom. Similarly this may protect physics from divergences with black holes, such as the singularity and maybe with the current big problem of firewalls.

    Thanks for the good word. I had a computer crash (virus attack etc) that erased my voting code. I also had it written down on a paper that also went missing. I have not been able to vote on papers for about a week. The FQXi people have so far not serviced my request that it be retransmitted. I have also been a bit slow in reading papers this contest cycle. I see that you have a paper in the list. I seem to remember that last year your paper was riding fairly high, where mine in contrast tanked.

    Cheers LC

    Dear Lawrence,

    The title of your essay intrigued me because I think that Wheeler wanted us to recognise the same thing.

    I read your essay with much interest (I didn't need to understand the math, because the text was clear) and the conclusion that biology and EVEN consciousness will have to play a role is one that I took as essential inmy own contribution.

    After reading the essay it is always informative to read your reactions on the posts, which are very informative. Especially one reaction ( of may 20 02.42 GMT was in full correspondence with my own perception :

    "Any scheme for causality is going to be incomplete, it will not be able to encode ALL physical states. As a result it means that there exists a DEEPER FOUNDATION to the Universe" This incompleteness I am trying to describe in the infinite number of tones of grey between the digital entities zero and one.

    I really hope that can spare some time to read/comment and rate my essay : "THE QUEST FOR THE PRIMAL SEQUENCE" , I am sure you will find some thoughts we share together for a future approach of reality.

    respectfully

    Wilhelmus

    Wilhalmus,

    I suffered recently a big computer virus crash. This machine had my voting code on it. I have not been able to get the code back in spite of my petition to FQXi.

    I think that consciousness has some epiphenomenology with generating the appearance of measurement outcomes. In the MWI context the mind may be what generates the appearance of being on one of the split off worlds. In the Bohr Copenhagen interpretation I think consciousness may play a similar role in the so called collapse. In your idea of created reality (creatality) it might be that this is in some ways a mentally generated illusion.

    I will score essays again once I get my voting code back.

    Cheers LC

    Outstanding Paper. Very Logically based. Extremely competent writing.

    Similar conclusion as Torsten Asselmeyer-Maluga (incomputable).

    I loved the modal logic foundation of this paper, and the expression of Godel's second theorem in Modal logic, which I had not seen before.

    Your proposal to remove associativity as a physical axiom is a profoundly interesting idea. I see how this introduces a different interpretation for quantum nonlocality (similar in some sense to the paper by Ken Wharton?)

    But where you come out clearly ahead in here (my favorite quote from the paper):

    "This argument employs sufficient and necessary conditions in a tensed fashion, forwards and backwards in time, to give a causal chain."

    Brilliant.

    Your figure (on page ? 3) however, seems to assume an irreversible monotonicity in the order of t1, t2 and t3. Is this what you intended to compare to causal set theory? Was this diagram intended to imply a forwards and backwards in time causal chain?

    Nice description of the history of S-Matrix, and a very thought-provoking conclusion regarding black holes.

    There is also a wealth of tid-bits of mathematical hints in the paper: very insightful. A very worthy read.

    Your conclusion that this (causality and undecideability) is a prospect that may play a role in the emergence of biology and even consciousness is very brave. I hope the orthodoxy does not try to dismiss you for this ;-)

    I look forward to seeing more of your work in this area.

      Paul,

      Thanks for the very good word on my paper. I see tht you squeeked in a paper right at the deadline. I have entered several essays and I have generally found that submitting a paper around the middle of the time period for entries is about the best. I will read your paper soon. I had a couple of weeks ago a major virus attack on the machine which held my password. I can't vote for essays until FQXi honors my request to have it retransmitted to me.

      My essay was inspird in large part by reading David Foster Wallace. He was a philosopher who managed to actually say something. He was also a good writer with his novels "The Broom of the System" and "Infinite Jest." I was reading his essay on the refutation of Taylor's argument for fatalism, and that sort of inspired my FQXi essay. I was originally planning to sit out this essay cycle. I also use the past tense with Wallace for he committed suicide in 2008 since he suffered major depression. I got a deeper appreciation for this problem because my brother suffered from this as well, and the past tense indicates his death by suicide a little over a year ago. He was a molecular biologist of some standing in that community.

      I have been in some exchanges with Asselmeyer-Maluga on these issues. There is a theorem that the number of exotic R^4s is uncountable via the Cantor diagonalization and Godel-Turing undecidability. I worked up some calculations on Hopf links and knots with respect to the exotic smoothness of R^4 spaces. He has been on vacation of late, but should have returned this week or so. Generally I tend to leave people at peace during vacations.

      As for your question about the mother-daughter relationship between events, which is in Foster's paper, the modal logic employed has no particular sense of reversibility or irreversibility. One could imagine some register or physical memory which permits one to reverse the direction of this diagram.

      Cheers LC

      Dear Lawrence,

      In particle scenario as undecidability is the negation of probability, the difference in density of the undecidables with the probability density, is proportional to the nonlocal Lagrangian for the actions at distances with the observer, in that the nature of gravity is unexplainable. Thus a string-matter continuum scenario is considered as an alternative in that gravitation emerges as a tensor product of the eigen-rotational string-matter segment.

      With best wishes

      Jayakar

        The undecidablity is a generic result for any causal model. A causal model of any sort can be represented as some modal logical system of necessity and possibility. The exact structure of this model is not given, but only that causality involves necessity and possibility. The next step is to make some possible hypothesis on what is undecidable about explicit causal structures in quantum field theory. This is not a derivation of what is undecidable, which in mathematics is not itself decidable and only found on a case by case basis, but only to offer up possible physical axioms that can be "toggled" to an on or off state.

        LC

        Dear Lawrence,

        I think you hit the nail on the head in your two concluding paragraphs. "The incompleteness of metaphysical models illustrates the general nature of incompleteness that translate to standard physics." ". . . but the existence of physical states implied by this set means that certain physical states can exist for reasons not computed by the "rule book." The heuristic invoked here is that this concerns the nonlocality of quantum gravity and the existence of a new structure. The physical axiom that is proposed to be removed is associativity."

        And, "The incompleteness of modal causal models is argued to justify nonassociativity as a means towards nonlocality of the quantum gravity field. This is a "bottom-up" type of argument, where an incompleteness of a higher level physics requires a more fundamental physics "further down." It is entirely possible this could be used to argue for a "top-down" physics with the emergence of higher level properties. There is a prospect this may play a role in the emergence of biology and even consciousness."

        Yes! I think you're onto something regarding a "top-down" approach towards physics with an eye on the possible emergence of higher level properties. I believe that you have pointed the way towards discovering the emergence of biology and even consciousness from the foundation of physics.

        As a non-specialist (I'm an attorney with a deep interest in the subject) I am encouraged by your conclusions and the direction of your thinking, so although there were sections that were beyond my understanding, I thought it was an outstanding essay. Thank you.

        Best,

        Ralph

          Hi Lawrence,

          I do not think that we have a real understanding of "possibility" in physics. We treat possibility as a metaphysical condition (e.g. in arguments against fatalism), but when we then try to describe what is "possible" we fall back on epistemology - what is possible given these data, etc. "These data" always refer to local observations of circumscribed systems recorded in memories using classical information.

          Our notion of "causality" is similarly local and classical, in the sense that our representations of "cause" and "effect" are written in classical memories, regardless of the notation used. We write down, for example, a "prepared" quantum state that involves some degrees of freedom and not others. Where did this circumscription come from? From local observations.

          If we took non-locality seriously, every "event" would be a quantum state of the entire universe, none of which (for an instantaneous event) is observable from our (or any) perspective. We seem systematically unable to think of events this way, or to envision the universe evolving as a whole. In particular, we do not seem able to envision our own brains/minds/awareness evolving not because of something previous, but as part of what is happening now. This is, however, the situation we are in. We are not observing the universe from the outside, however convenient this may be as an methodological or mathematical assumption.

          Hence I would like to distinguish between "the universe is (quantum) computing its next state" and "the universe is a (quantum) computer." If the universe is a quantum computer, the data structure on which it acts is its own, complete state. But this is not what we intuitively mean by "computing" - we intuitively think of computing as acting on a data structure that is external to the process of computation. A Turing machine, for example, makes intuitive sense as doing computation because the head is distinguished from the tape and is seen as acting upon it. In the case of the universe, we do not have this luxury; it is not a case of some unchanging parts acting on others that are changed.

          Cheers,

          Chris

          Ralph,

          Thanks for the positive word. The incompleteness extends potentially to various levels, including the dichotomy between the quantum and classical world. Complex structures like organisms impose constraints on the micro-causal systems that compose it. How this occurs is not well known. It is the possible that in addition there is a top-down element to this. The physics of molecules or atoms do not predict biological systems, but biological systems as emergent structures impose constraints on how molecules behave.

          The physical universe doubtless has a computational aspect to it. It is less clear that this defines all of physics. The discrete paradigm of reality clearly has some problems or limits. In particular the discrete model of spacetime, such as offered by LQG, implies there are violations of Lorentz symmetry. Recent distant observations of GRBs have found no dispersion predicted by this. Symmetry breaking implies mass or some dispersion due to longitudinal modes. Much of physics can also be expressed according to a path integral which is derived using variational methods. The initial and final state of the system is defined and the intervening states of the system are derived without any causal state by state evolution perspective.

          A path integral has some classical path, which is usually defined by the extremal path for the largest expected outcome. This is one motivation for the einselection paradigm for assigning states as the stable classical configuration for a system. The odd part of this is that quantum mechanics is noncontextual by the Kochen-Specker theorem. However, the path integral implies some sort of classical "shadow" to QM. Classical or macroscopic physics is contextual, and this seems to imply there is a theorem (the KS theorem etc) which in the broader context of physics is undecidable. This would be the case if one considers classical physics as having some reality, even if in a coarse grained perspective.

          I should also be mentioned that chaos theory does not involve halting algorithms. They are recursively enumerable; they do not halt and carry on to an arbitrary level of floating point precision. So certain disciplines of physics already embody theory that involves nonhalting algorithms. Halting algorithms are recursive, and their complements are recursive. Recursively enumerable algorithms have complements which are not decidable. Recursive algorithms are BLOOPs to use Hofstaeder's term and their duals are BLOOPs. Recursively enumerable algorithms are FLOOPs (free loops), but their duals are GLOOPs (Godel loops).

          I wrote an essay for the FQXi contest, which at last check is #4 out of 182, which demonstrates how a causal (state to state with time) perspective of physics will have some level of undecidability. This has some relationship to Hume's naturalist fallacy and to Wallace's refutation of Taylor's fatalism. Such a perspective motivates the suggestion that associativity of operators may be the axiom that can be toggled on or off.

          Trying to understand how nonassociative mathematics of operators fits into physics is really the hard part. I think that quantum mechanics is purely complex, or C. Of course classical mechanics is R. Gauge theory can be written according to quaternions H. A lot of gauge theory is done though in standard vector form without quaternions. It is interesting though that Maxwell formulated electromagnetism, the first gauge field theory, in quaternions. Field operators in a second quantization act on a Fock space basis to give quantum amplitudes. So we have a relationship that might be heuristically written as π:H --- > C. The question is then whether there is some sort of higher level structure π:O --- > H.

          Spacetime I think offers a clue. A black hole horizon has some quantum uncertainty on a scale near the string or Planck length. There will then be an associative uncertainty with three quantum fields, where one of those fields is identified near the horizon. The standard approach to QFT is to assign a harmonic oscillator at every point in space, impose equal time commutators on that spatial surface with the Wightman criterion for commutation, and work from there. Yet that spatial surface on a small scale will have some noncommutative structure and this will lead to a host of uncertainties in assigning QFT operators. If there are event horizons this should lead to an associative uncertainty.

          The above "maps" between C, H and O, where a similar map π:C --- > R would be the relationship between quantum mechanics and classical mechanics, are really just forms of the Hopf fibration. The relationship between quantum and classical mechanics is of course a difficult subject in its own right. With each of these "ladders" on the Hopf fibration there is some increased uncertainty. Quantum mechanics saved physics from the UV divergence that classical mechanics predicted with the hydrogen atom. Similarly this may protect physics from divergences with black holes, such as the singularity and maybe with the current big problem of firewalls.

          There may be a general level of undecidability in physics which tells us that an algorithmic perspective of physics will not be able to define all of physics. The "bit" or qubit perspective of physics is important, but it may not embody all of what might be called physical truth.

          I see that you have an essay. I just got my voting code retransmitted to me. The computer I had it on suffered a big virus meltdown and I lost that for a couple of weeks. I will get to voting this week as time permits.

          Lawrence B. Crowell

          The term possibility is not quantified very well for physics. I read a paper quite some years ago which stated possibility and plausibility were "unknown unknows" in a Bayesian sense. It is a probability that is not computable because there is no Bayesian prior estimate. The first part of my essay is metaphysics meant to motivate the need to "toggle" some physical axiom between the "on and off" condition.

          Nonlocality of field theory is related to topological quantum field theory (TQFT) A TQFT is a quantum field theory up to homotopy. Physically this means that geometric data is removed and real degrees of freedom are determined by topology. This clearly means the TQFT within some homotopy (or topology) is a class of fields up to the diffeomorphism of space (or spactime). This means that local data concerning geometry is removed by taking a quotient with the space of solutions. In doing this we still want to have some concept of locality, which is usually associated with geometric information. Locality is contained in how the manifold is partitioned into pieces, where locality is defined by how those pieces are joined together. In spacetime this can include a space plus time perspective, where spacetime is defined by thin sections with a local time increment. These pieces have then cobordism defined by spatial surfaces.

          A topological quantum field theory may be considered to be any data to every geometric entity from a zero-dimensional point up to an n-dimensional cobordism. The n-functor

          Z:bord^m(n) --- > A_n

          Where A_n is an algebra of dimension n and the bord^m(n) holds for 0

          Hello Lawrence

          Richard Feynman in his Nobel Acceptance Speech (http://www.nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html)

          said: "It always seems odd to me that the fundamental laws of physics, when discovered, can appear in so many different forms that are not apparently identical at first, but with a little mathematical fiddling you can show the relationship. And example of this is the Schrodinger equation and the Heisenberg formulation of quantum mechanics. I don't know why that is - it remains a mystery, but it was something I learned from experience. There is always another way to say the same thing that doesn't look at all like the way you said it before. I don't know what the reason for this is. I think it is somehow a representation of the simplicity of nature."

          I too believe in the simplicity of nature, and I am glad that Richard Feynman, a Nobel-winning famous physicist, also believe in the same thing I do, but I had come to my belief long before I knew about that particular statement.

          The belief that "Nature is simple" is however being expressed differently in my essay "Analogical Engine" linked to http://fqxi.org/community/forum/topic/1865 .

          Specifically though, I said "Planck constant is the Mother of All Dualities" and I put it schematically as: wave-particle ~ quantum-classical ~ gene-protein ~ analogy- reasoning ~ linear-nonlinear ~ connected-notconnected ~ computable-notcomputable ~ mind-body ~ Bit-It ~ variation-selection ~ freedom-determinism ... and so on.

          Taken two at a time, it can be read as "what quantum is to classical" is similar to (~) "what wave is to particle." You can choose any two from among the multitudes that can be found in our discourses.

          I could have put Schrodinger wave ontology-Heisenberg particle ontology duality in the list had it comes to my mind!

          Since "Nature is Analogical", we are free to probe nature in so many different ways. And you have touched some corners of it.

          Regards

          Than Tin

            • [deleted]

            • Post #126

            Than,

            Nature has an analogical quality to it, or what I see as recherché --- as with Bach's "Musical Offering." Certainly one example is how isospin symmetry is applied to the nuclear physics of protons and neutrons in the MeV range and the same symmetry appears in the theory of weak interactions. I base my argument for the need to toggle on or off physical axioms by appealing to a formal incompleteness of any causal model. Godel's theorem is a recursive aspect of mathematics where a predicate acts upon its own Godel number as the object.

            It is taking me a bit of time to get to many of the essays here. Other pressing concerns mount as well. I will try to get to your essay in the near future. I got an email that the contest deadline has been extended a week. That may allow for a bit more time to get to more of the essays.

            Cheers LC

            Dear Lawrence,

            I am almost sure that I rated your essay at an earlier time. For any reason my vote was not recorded or lost when the system was interrupted. So I reproduced the vote with a bonus due to the high level of your replies.

            Good luck,

            Michel

              Michel,

              Thanks. I have been in the process of moving, so my ability to engage FQXi has dropped seriously. I also lost my voting code for a while after the machine I had it on was virus attacked. I got the code back the other day.

              I think there are underlying relationships between F4 as the group for the KS theorem in 4-d and G2 as the automorphism of E8. F4 is a centralizer in E8, which means it is a "constant of the motion" with respect to G2. I have long thought that general relativity and quantum mechanics share some common basis along these lines.

              Cheers LC

              Dear Lawrence,

              One single principle leads the Universe.

              Every thing, every object, every phenomenon

              is under the influence of this principle.

              Nothing can exist if it is not born in the form of opposites.

              I simply invite you to discover this in a few words,

              but the main part is coming soon.

              Thank you, and good luck!

              I rated your essay accordingly to my appreciation.

              Please visit My essay.

                Hi Lawrence,

                Congratulations on a fine essay! But I wonder whether the arguments you advance are against a straw man... pertinent to an abstract and continuous theory of reality, rather than what may actually be a discrete and non-continuous material cosmos. You wrote:

                > Physical systems in some funny sense have a premonition about how to evolve.... There is no information transfer in this process, but in a nonlocal manner a quantum particle "knows" how to evolve by sampling all possible paths

                To apply the Principle of Least Action you need to specify the end state as well as the starting state. It is therefore, more a view "looking back" than "looking forward". In fact, you can take the viewpoint of essayist Don Limuti, and say that particles do not have to have a continuous trajectory, and only have to appear intermittently. With this sort of interpretation, computation remains feasible.

                > Godel's second theorem indicates that any consistent theory is unable to prove its consistency.

                The precondition to Godel's theorem was that the theory be able to formulate *all* of arithmetic, not just finite models of arithmetic. Actual computers have finite word size and memory size, and might be better modeled by Primitive recursive arithmetic, which is provably consistent in Peano arithmetic. So the application of Godel's theorem is valid only for abstract models of computation, not actual discrete computational models of the cosmos.

                > Taylor's argument may be seen as follows...

                As you note, Taylor's argument depends on the Law of the Excluded Middle. Joseph Brenner (who is also an essayist here) has previously written a paper entitled "The philosophical logic of Stéphane Lupasco (1900-1988)" that describes Lupasco's alternate "Logic of the Included Middle" and argues that it better applies to reality. With a different logic I think you could reach a different conclusion about computability.

                > Quantum gravity is then a quaternion theory, or a system of quaternions in the octonions. There is then a hierarchy R -> C -> H -> O, where classical mechanics is real valued, quantum mechanics complex valued, and underneath are quaternion and octonion valued fields and vacuum structures.

                Nice idea. You might appreciate my advocacy of Geometric Algebra for the formulation of a computable cosmos in Software Cosmos, where I take up the simulation paradigm and construct a digital model.

                > The digital model of the universe or "It From Bit" is not decidable. A model of the physical universe encoded by algorithmic means will not compute reality.

                I am most curious whether you think this limitation applies to my picture of a discrete computational model for the cosmos.

                Hugh

                  Hugh,

                  Your comments are interesting and thought provoking. I don't think that nature is strictly continuous or discrete. I think there is some sort of dualism between the two descriptions of reality. I don't think either description is complete.

                  I agree that Godel's theorem involves infinite systems. The use of the Cantor diagonalization implies an infinite set. The use of modal logic and the appeal to Godel's theorem is somewhat qualitative I will admit. I think to do a full formal analysis of this would be an exhausting piece of work. However, the use of "possibility," which at best can be interpreted as probabilities with a weak or no Bayesian priors (an unknown unknown) is potentially itself "infinite," or so large that from a physicist perspective we can consider it infinite. Any algorithmic description of the universe must appeal to primitive recursive functions. The analyst in this view is performing a sort of "cut off," where the "possibility" extends only to some range of estimated probabilities. This may not necessarily cover all of reality or all possible cases. David Hume made a bit of a point about this. Causality can't be reduced to logic. I make a point of a connection between Hume's observation and Godel.

                  The argument concerning the extension of quantum fields from a quaternion basis to octonions is also qualitative. There is no procedure for finding undecidable propositions; the existence of such a putative procedure is itself undecidable. Godel did manage to derive his proof in reference to Diophantine equations, which amounts to deriving a "special case." Again trying to find how the extension from H to O is a matter of Godelian incompleteness would be a huge undertaking. So my approach is to appeal to a physical argument rather than formal mathematics. After all, while I have learned a fair amount of advanced mathematics I am still primarily a physicist .

                  I will try to look at your essay today or in the next few days. From your description it does appear interesting. My time has been terribly constrained the last couple of months. I have to keep in mind that voting ends in about a week.

                  Thanks for the over all positive assessment.

                  Cheers LC

                  Amazigh,

                  One interesting duality principle is with Yangians. This is a duality with fields in a braid description that has connections to twistors.

                  I will try to get to your paper in the near future. My time is pretty limited right now. I have not had much time to engage this contest very much.

                  Cheers LC