Dear Yutaka

Thank you for your question. From the perspective of my essay, stochastic and chaotic dynamics are very different concepts. Let me give an example. In my essay I wrote down the equations of the famous Lorenz model which is chaotic for certain parameter values. For standard chaotic values of the parameters, about 96% of the variance of the model lies in a two-dimensional sub-space of state space. Now one can choose a basis where you retain the dynamical equations in this two-dimensional subspace, but replace the dynamics in the third dimension with a stochastic process. The resulting attractor looks superficially like the Lorenz attractor. However, it differs in one vital regard - all the fractal gaps in the attractor are filled in by the stochastic process.

That is to say, replacing chaotic determinism with stochasticity completely negates my arguments about counterfactual incompleteness (associated with states which lie in the fractal gaps in my cosmological invariant set). Hence my arguments about why the violation of statistical independence is explainable in a suitable nonlinear dynamical framework are nullified if determinism is replaced with stochasticity.

It is for this reason that I am somewhat sceptical of models which attempt to replace real numbers with truncated rationals stochastic noise will work in explaining quantum physics.

In conclusion, there is a vital difference between chaotic and stochastic dynamics, in my opinion.

With regards

Tim

Tim -

An exquisite and erudite exposition on matters far beyond my formal training in math and physics (from some decades ago). I gather that you are positing some level of determinism arising from infinite recursion of fractal attractors. In lay terms, if our frame and timeframe are large enough, we can regain the confidence of determinism from the local instability of chaos, just as statistical mechanics rescues us from the chaos of the independent behaviors of individual particles. Am I following this correctly?

That said, I am dubious that determinism of any sort can be rescued. We can speculate with infinities but we cannot prove anything at all, as the reasoning will always fall short. This verse from the Rubaiyat captures the thought:

XXIX. Into this Universe, and Why not knowing

Nor Whence, like Water willy-nilly flowing;

And out of it, as Wind along the Waste,

I know not Whither, willy-nilly blowing.

Thanks - George Gantz, The Door That Has No Key: https://fqxi.org/community/forum/topic/3494

Dear George

Thank you for your comment.

My goal is to formulate a finite theory of quantum physics where the fractal invariant set model of quantum physics is a smooth limit as a parameter of the finite model goes to infinity.

Finding such smooth limits is highly non-trivial in quantum physics. For example, if you try to discretise the complex Hilbert Space of quantum theory then you violate the Continuity Axiom of Hardy's axioms of quantum theory - and according to his axioms you would revert to classical theory. In this sense quantum theory is the singular and not the smooth limit of a finite discretised theory of Hilbert space as the discretisation goes to zero.

As Michael Berry has discussed, singular limits are quite commonplace in physics and in some sense represent a discontinuous jump when you go from "very large but finite" to truly infinite.

What I am trying to do is find a finite theory of quantum physics which has a smooth and not a singular limit as some parameter goes to infinity. In practice I can achieve this by assuming that the symbolic labels associated with the fractal iterates of the invariant set have periodic structure. This is entirely equivalent to the idea that rational numbers have a periodic representation in terms of their decimal expansions. The larger the periodicity the closer they are to irrationals.

With this I can effectively interpret the invariant set as a finite periodic limit cycle, but with very large periodicity. As discussed (albeit briefly) in the essay, the property of non-computability is then replaced by computational irreducibility. None of the key properties which allow me to reinterpret Bell's theorem are lost in going from strict non-computability to computational irreducibility.

With regards

Tim

Very intriguing essay! The central idea, (which I understood to be) that one might be able to get around Bell's theorem by having aspects of the underlying deterministic theory be uncomputable in a certain precise sense, is very clever. It's much better than a philosophical monstrosity like superdeterminism, too...Still, I admit I did not fully understand all of the technical details. Maybe I will reread it again.

Here's a philosophical question, though. There's how the universe 'really is', and there's the collection of things we can ever know about it; these sets are almost certainly not equivalent. If there is some sort of deterministic theory that underlies quantum mechanics, but it has the property that it 'looks' probabilistic to us because of uncomputability etc, why should we prefer the deterministic theory? I guess it's possible that ideas like this could help with unification, but it seems to me necessary that the proposed unification would suggest some experiment that would distinguish between the different possibilities in order for that unification to be useful.

More generally, how can we ever know the 'true' behavior of quantum mechanics, given all these clever alternatives?

John

Dear John

Certainly a new theory of quantum physics should suggest some hopefully experimentally testable differences from quantum theory.

In the technical paper https://royalsocietypublishing.org/doi/full/10.1098/rspa.2019.0350 on which this essay is based, I present some preliminary ideas on possible differences.

Thanks

Tim

5 days later

Hi Tim,

Thank you for writing a very interesting essay! I certainly fell into the category of 'physicist who finds p-adic numbers exotic'. I have never encountered them but am eager to take a bit of a dive into them.

You certainly raise some very interesting points particularly that undeciability is a property of the underlying state-space of the system and not the physical process occurring in spacetime. Moreover, this lead into a very nice discussion about counterfactuals and free will that I really appreciated.

If I understand correctly, a non-computable theory can violate the Bell inequality. This uncomputable theory is a based on fractal attractors which correspond to the possible eigenstates of the state space being observed i.e determined by the Hamiltonian?

It's an interesting paradigm and am eager to read more. Another question I would ask is how does dissipation alter this paradigm? Does it change the state space where the fractal atractors now change to multiple steady state attractors?

In any case, it was a very thought provoking essay. I hope you have time to take a look at my essay noisy mahcines which considers the limitations of finite resources in undeciable systems.

Thanks,

Michael

    Dear Michael

    Thanks for your kind comments.

    Regarding p-adics, I am reminded of a paper I once read by Herman Bondi who said that if children we were taught special relativity in primary school, as adults we would not find things like length contraction and time dilation the least bit strange or unusual. Similarly, I expect, if we were taught p-adic arithmetic in primary school, we would not find p-adic numbers strange or exotic as adults. Peter Scholze, who won the Fields Medal last year, is quoted as saying that he has got so used to p-adics that now he finds the real numbers really strange and exotic!

    Your letter raises a really interesting and important issue - the role of irreversibility. The fractal attractors I am considering have zero volume and hence zero measure relative to the measure of the Euclidean space in which they are embedded. The classical dynamical systems which generate these attractors asymptotically must therefore be irreversible: start with a finite volume and it shrinks to zero asymptotically.

    What is the origin of this irreversibility? In terms of the attractor geometry, the irreversibility could be localised to some small region of state space, such that when the state of the system goes through this region, state-space volumes shrink a bit. In this way, it is possible for the dynamics to be Hamiltonian almost everywhere. But it cannot be strictly Hamiltonian everywhere. It is tempting to suppose that such irreversibility is associated with space-time singularities, but this is merely a conjecture.

    Regarding your essay, I think I am in agreement with your perspective. Although I am claiming that the universe as a whole has these properties of uncomputability, I don't think it makes sense to think of sub-systems of the universe as approximating the properties of the full system in any way at all. In my essay I refer to the inability of the full system to be fully emulated by a sub-system of the full system as Computational Irreducibility - a phrase that I think Stephen Wolfram coined.

    Of course it is worth noting that in many practical cases, noise can and should be treated as a positive resource. Personally, I think human creativity arises because the brain has been able to harness noise in this way - please see:

    https://www.mdpi.com/1099-4300/22/3/281

    Best wishes

    Tim

    Tim, a most sophisticated essay! I can believe that if anyone could accomplish what you've sought, "to provide some basis for believing that these theories [chaos, quantum, and GR] can be brought closer together through the unifying concept of non-computability", you would be the one to do it!

    You are no fool on that errand, but regarding chaos, the dependence of a chaotic system on initial conditions, combined with multiple vectors of recurrent interaction, just makes for a recurring deterministic system that may, as you point out, eventually break out into simple (deterministic) turbulence. So chaos: deterministic but not always computable. Quantum theory: un-deterministic but computable at least as a probability. And General Relativity: deterministic and computable. I'm not optimistic.

    I didn't understand your reason for thinking "there must also be some deterministic framework underpinning quantum physics."

    Finally, more in my wheelhouse, you quote R. Kane "one is free when there are no constraints preventing one from doing as one wishes" - a poor definition that doesn't distinguish between being determined to wish for something and being merely influenced to wish.

    Overall, congratulations on an impressive essay.

    Dear James

    Thanks for your kind comments.

    Regarding free will. Bell's Theorem involves a mathematical assumption called Free Choice. I have proposed a revised definition called Free Choice on the Invariant Set. This basically means you can't choose to do things which are inconsistent with the laws of physics (the laws of physics in my proposed model derive from the fractal geometry of the invariant set). Put like this, I hope you will agree that this is not an unreasonable definition. We don't say that we are not free because we can't flap our arms and fly like birds!

    However, in this definition one cannot predict ahead of time what choices will violate the laws of physics and which will not - this is linked to the non-computability of the invariant set. So, therefore I ask what a more operational definition of free will might be that evades this difficulty. The one I propose is such an operational definition. It's one I personally use in my day-to-day life.

    If there is one real takeaway message from my essay that I hope will resonate with you is that in physics the assumption of rather unrestricted counterfactual definiteness is something that has not been analysed enough. I think this issue should be discussed more in Philosophy of Physics circles. For example, in my essay I give a reason why Lewis's counterfactual theory of causation might be faulty because of an implicit use of Euclidean distance in state space.

    Best wishes

    Tim

      Tim,

      My intuition suggests that Bell theorem could be formulated also as " UUU - mathematical problem" in physics. Hence, there is some so - called "nonclassical tacit math" behind Bell as well?

      Thank you for essay.

      Michael

      Tim, does randomness (defined as something uncaused or unprovoked) defy the laws of physics? It is used regularly in quantum physics to describe the unpredictable. I maintain that it is the best explanation for nothing happening at all. I suggest "spontaneity" as an explanation for anything from the quantum level to human inspiration, which by definition exceeds the laws of physics, but is more credible than nothingness.

      James

      Personally, I have considerable difficulty with the concept of randomness in fundamental physics (even though it is an incredibly useful concept for many areas of applied physics). If you give me a bit string 010010....01 that you claim has been generated randomly, I will give you a deterministic rule for generating that same bit string. Now some might say that randomness is informationally incompressible determinism. Well I would say that in practice the two may well be indistinguishable. However, at a fundamental level the latter is generated by a deterministic rule and the former, presumably, is not. In my view the sooner we get back to thinking about physics deterministically (even though it may be computationally irreducible determinism) the better!!

      Best wishes

      Tim

      This is a really exciting essay; I'm really intrigued by the connections you suggest between quantum mechanics and chaos theory, and am now keen to learn more about this area.

      I did have one general question about the motivation for this approach. If I understand you correctly, the idea is that by constraining the state of the universe to evolve on some uncomputable fractal subset of state space, we get a natural way to violate statistical independence (without the denial of free will) and thus we can have violations of Bell's inequality without violations of locality.I wonder, though, why you consider it important to avoid violations of locality? As I understand it, the similarity of Schrodinger's equation to the Liouville equations leads you to consider underlying deterministic dynamics, and then since Bell's theorem rules out any underlying deterministic local dynamics, you turn to non-computability as a means of violating statistical independence and thus invalidating Bell's theorem. But an alternative possible route would have been to accept the existence of nonlocality and consider how Schrodinger's equation could arise from an underlying deterministic non-local dynamics - is there a specific reason you chose not to go down this route?

      I also have some questions about the sense in which 'locality' is preserved by your model. First, consider applying the constraint of evolution on a fractal subset to an indeterministic model. Then if the evolution of the universe is constrained to remain on the fractal subset, it would seem that the (non-deterministic) evolution of the universe at any one spacetime point must depend on the (non-deterministic) evolution of the universe at all points spacelike related to that point, as if the points evolved independently and non-deterministically then it would be possible to go off the fractal subset. So constraining the universe to lie on the fractal subset does not, in the absence of determinism, seem to give us a local theory. So now consider a deterministic model as you propose. Here the evolution of the universe is fully determined by the initial state (I am assuming here that by 'deterministic' you are referring to initial-value determinism, as the term is commonly used), and so the constraint you suggest comes down to requiring that the universe has a fine-tuned initial state which ensures that its evolution always remains on the fractal subset. But surely if this sort of fine-tuning is allowed then we can quite easily explain nonlocality without needing to appeal to noncomputability or fractal subspaces - i.e. we can encode the choices of measurements and the measurement results for all Bell experiments which will ever be performed directly into the initial state, and thus produce experimental results which appear non-local even though they are in fact produced from local evolution from this fine-tuned initial state. I think most physicists are not keen to adopt this approach to eliminating nonlocality because it seems unreasonably conspiratorial and fine-tuned - do you think your fractal approach gets round this complaint in some way, and if so, how?

      I was also interested in the approach you take to recovering 'free will.' The distinction you make between defining free will via counterfactuals vs defining free will as the absence of constraint clearly ties into long-standing arguments in philosophy about the nature of free will, and I think there are indeed good arguments in favour of the latter approach even before one comes to the specific theoretical model that you introduce here - indeed I would be fascinated to read a paper discussing the links between your proposal and the body of philosophical literature on this topic!

        Dear Emily

        Thank you for your interesting and important questions. In replying I need to make sure I don't end up writing another paper!

        As far as my motivation is concerned, I did my PhD many years ago in GR (under the cosmologist Dennis Sciama) and in truth the reason why I have got so interested in Bell's Theorem is not because I am interested in Bell's Theorem per se, but rather that I think sorting this out properly is crucial to finding a viable approach to the synthesis of quantum and gravitational physics. These issues of locality vs nonlocality are very subtle: I can violate Bell's Theorem with the postulates "Free Choice on the Invariant Set" and "Factorisation on the Invariant Set". Technically these violate "Free Choice" and "Factorisation" in Bell's Theorem, but I would argue that they do so in a way that is comprehensible from the perspective of space-time causality in GR. To be honest, instead of talking about locality and nonlocality, I would rather ask: Does a nonlinear deterministic fractal state-space approach to quantum physics lead to a better model for synthesising with GR than conventional quantum theory? Well there are certainly some hints that it might!

        Regarding fine-tuning, I sometimes use the analogy with BCC TV Doctor Who's Tardis. From the outside it is a tiny 1950s Police Box. From the inside it is an incredibly spacious space-ship. A Cantor Set has a similar property. From the outside it has measure zero with respect to the measure of the embedding Euclidean Space and therefore appears insignificant. However, from the inside it has the Cardinality of the continuum and therefore appears incredibly massive and spacious!!

        So it all depends how you define this notion of "fine-tuned". From the intrinsic (technically Haar) measure of the invariant set, it is not at all fine tuned. Moreover using the p-adic metric (discussed in my essay), points which do not lie on the invariant set are necessarily distant from points which do, even though from a Euclidean point of view the distance between such points may appear tiny.

        Finally, you are much more knowledgeable about the philosophy literature on free will than me. I would love to discuss with you how my ideas may or may not connect up to this literature. You will see in the essay that I flagged one of the key ideas in Lewis's theory of counterfactual causality as potentially being incorrect if the metric on state space is not Euclidean. I'm sure there is much to discuss here!!

        Best wishes

        Tim

        Dear Tim,

        It was a pleasure reading your essay. and the valuable insights it gives. Would you happen to know if Connes' non-commutative geometry formalism would classify as uncomputable?

        As regards quantum gravity, I beg to submit that I have made important progress recently, and proposed the theory of Spontaneous Quantum Gravity, described for example in my paper

        Nature does not play dice at the Planck scale

        Independent of anything to do with the present contest, I will value your critique of my theory. I would like to reach out to many physicists with a request to examine this theory.

        Many thanks,

        Tejinder

        Dear Tejinder

        Connes comments in his book that non-commutative differential geometry provides a way to represent the measure of fractal sets such as the Julia set. Since these fractionally dimensioned sets are non-computable, then non-commutative geometry and non-computable geometries are related.

        Hence, perhaps there are some interesting connections to be made between my invariant set model and Connes' non-commutative models of quantum physics.

        Having said that, non-commutative geometry is not an area of mathematics of which I have any great knowledge.

        I will take a look at your essay. The title sounds very appealing to me!!

        Best wishes

        Tim

          Thanks Tim! This connection between non-computable geometries and non-commutative geometries is extremely interesting! I did not know about it. My new quantum theory of gravity builds on Connes' non-commutative geometry and Adler's trace dynamics.

          Best,

          Tejinder

          Dear Tim,

          what a wonderful essay and it rings a bell.

          Now since 8 years I studied general fractals (better known as wild embeddings) to get a spacetime representation for quantum states. You also discussed this interesting relation and you are right, there is a direct link between such objects and general relativity as well. The space of leaves of a foliation is also a non-commutative geometry (a la Connes) and this space is related to wild embeddings (or fractals) as well.

          Amazingly, these connections are naturally related to the structure of spacetime (I mean the 4-manifold). If you choose another differential structure then you get automatically these connections. But enough for now.

          I enjoyed very much reading your essay and gave them the highest score.

          Maybe you are also interested to ream my essay

          Because of Corona, I was to late this year....

          Best wishes

          Torsten

          Hi Tim,

          I read your essay a while back and I have not seen any comment from you on mine. Since there is some overlap in our topic emphases; it would be nice to have your opinion.

          All the Best,

          Jonathan

          Write a Reply...