Undecidability, Fractal Geometry and the Unity of Physics by Tim Palmer

Hi Simon

Thanks for your kind comments.

Yes indeed, if the question you ask of a chaotic system somehow probes its asymptotic states (e.g. "will three objects collide eventually") then one can (likely) reformulate the question in terms of the state-space geometry of these asymptotic states - and my claim is that such geometric questions are typically undecidable. However, the sensitivity of simple finite-time forecasts to the initial state is not itself a illustration of non-computability.

Indeed, I would say that my results (e.g. producing a viable model which can violate statistical independence without falling foul of the usual objections to such violation) arises because of this uncomputable property of chaotic dynamics. I should emphasise that in this picture, uncomputability leaves its mark on finite approximations to such dynamics in the form of computational irreducibility (the system can't be emulated by a simpler system). Hence we can still produce viable finite models where (6) is satisfied.

Re Lewis, my belief is that the potential pitfalls of unconstrained counterfactual reasoning have not been given sufficient attention in studying the foundations of physics. In this we are being beguiled by our intuition. You may be interested a recent paper of mine:

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

which tries to explain why we are so beguiled.

Finally, in classical GR with a cosmic censorship hypothesis, the singularities seem to be hidden from sight and are therefore not as ubiquitous as they might be - if only we could prove it! - in Navier-Stokes!

Tim

Dear Seth

Thanks for your input. I fully agree that complex numbers are central to quantum theory.

To understand the emergence of complex numbers in my fractal model, could I refer you to the technical paper recently published in Proc. Roy. Soc.A (open access):

https://royalsocietypublishing.org/doi/10.1098/rspa.2019.0350

on which this essay is based - some aspects of which are summarised in the Appendix to the essay. In particular, I construct a particular fractal geometric model of (what I call) the state-space invariant set based on the concept of fractal helices (see Fig 4 in Section 3 of the paper). At a particular fractal iteration, the trajectory segments of the helix evolve to specific clusters in state space - these clusters representing measurement outcomes/ eigenstates of observables. I then describe this helical structure symbolically (BTW symbolic dynamics is a powerful tool in nonlinear dynamical systems theory for describing dynamics on fractal attractors topologically). In the case of two measurement outcomes, the symbolic descriptions of the helix are then given by finite bit strings. Now in Section 2 of the paper, I show than I can define multiplicative complex roots of unity in terms of permutation/negation operators on these bit strings. A very simple illustration of this is to take the bit string

S={a_1, a_2}

where a_1, a_2 in {1, -1} - as a representation of a pair of trajectories labelled symbolically by which of the two distinct clusters ("1" and "-1") they evolve. Now define the operator i by

i S = {-a_2, a_1}.

Then i^2=-1 if -S={-a_1, -a_2}.

In fact, even more I can define quaternionic multiplication and hence Pauli spin matrices (and hence Dirac gamma matrices) in terms of certain permutation/negation operators on longer finite bit strings. See the paper for more details.

This answers half of your question - about complex multiplication. The second half of your question - relating additive properties of such bit strings to the additive properties of complex numbers - is something I am currently writing up. It turns out that to do this I have to extend the number-theoretic properties of trigonometric functions which play a vital role in the particular discretisation of the Bloch sphere described in the paper cited above - see also below - to number-theoretic properties of hyperbolic functions. Whilst the former provide a natural way to discretise rotations in physical space, the latter provide a natural way to discretise Lorentz transformations in space time! In this way, I have some belief that the properties of the invariant set are more primitive than those of space-time, with the prospect of the latter emerging from the former. With the current lockdown, I should have a draft paper shortly! With this, I will have a complete answer to your question.

However, a crucially important point in all this is that I do not, and will not, recover in this way the full *continuum* field of complex numbers, but only a particular discrete subset (essentially those complex numbers with rational squared amplitudes and rational phase angles). These complex numbers play an important role in my model for describing the symbolic properties of the helix in a probabilistic way. Number theoretic properties of trigonometric functions applied to these discretised complex numbers provide the basis for my description of quantum complementarity (and indeed the Uncertainty Principle - see Section 2e of paper above). However, in my model there is no requirement for these complex numbers to be arithmetically closed. Such arithmetic closure arises at the deeper deterministic level and this can be described by the arithmetically closed p-adic integers, these providing the basis for a deterministic dynamic on the invariant set. (There is a rich theory of deterministic dynamical systems based on the p-adics.)

All this means that in describing my fractal model from a probabilistic perspective, I can and do (in the paper above) use the formalism of complex Hilbert vectors (and associated tensor products). However, these vectors are required, by the discretised nature of the helix of trajectories in state space, to have squared amplitudes which are rational numbers and complex phases which are rational multiples of pi. Importantly, almost all elements of the complex Hilbert Space *continuum* have no (ontic) correspondence with probabilistic descriptions of the invariant set helices.

My own view is that quantum theory's dependence on the *continuum* of complex numbers (i.e. through the continuum complex Hilbert space) is the origin of its deep conceptual problems, e.g. as arises in trying to understand the meaning of the Bell Theorem or the sequential Stern-Gerlach experiment, or the Mach-Zehnder interferometer, or GHZ, or....you name it!!. Indeed I think quantum theory's dependence on the complex continuum is the origin of the difficulties we have reconciling quantum theory and general relativity theory. Of course, in quantum theory, we don't have a deterministic underpinning and so breaking the arithmetic closure of Hilbert Space is a real theoretical problem. However, in a model where there is a deeper deterministic basis, breaking the arithmetic closure of Hilbert space in this way doesn't matter a jot - since it's not a fundamental description of the underlying theory!! Here, in my view, we physicists have been overly beguiled by one aspect of the beauty of mathematics - the complex continuum field C!!

Recall that in mathematics, C arose as a tool for solving polynomial equations. Perhaps we need to retrace our steps and ask whether taking this tool onboard wholesale for describing the equations of fundamental physics could actually now be causing us some big problems (the utility of C notwithstanding)! Perhaps we imported a virus which has rather grown over the centuries and now completely permeates the core of fundamental physics making it impossible to make vigorous leaps forward! The real-number continuum virus doesn't matter in classical physics, because discretised approximations can come arbitrarily close to the continuum limit. However, the complex-number continuum does matter in a much more essential way in quantum theory. Recall in Lucien Hardy's axioms for quantum theory, the complex continuum plays a central and inviolable role - in complete contrast with classical theory. Hence in order to find a discretised theory of quantum physics, which I think should be an important goal for physical theory, quantum theory must be a singular and not a smooth limit as the discretisation goes to zero. My deterministic model has this property.

I am going to pick up on one other point in your correspondence. You say that I try to reconcile quantum theory with classical mechanics. I don't really see my proposal as "classical" in the following sense. The dynamics of classical chaos are differential (or difference) equations and the fractal attractor is an asymptotic set of states on which, classically, one never actually arrives, at least from a generic initial condition in state space. However, from this classical perspective there is no essential/ontological difference between a state which is "almost" on the attractor, and one on the attractor precisely.

By contrast, here I am postulating a primitive role for this fractal geometry (rather than the differential equations). Because of this, as I try to discuss in the essay, the p-adic metric may be a better yardstick of distance in state space than the familiar Euclidean metric. The p-adic metric certainly does distinguish between points which are not on the fractal and those that are, no matter how close such points may be from a Euclidean perspective. In this sense although my model is certainly motivated by classical deterministic chaos, I would not call it classical.

There is much more to be teased out of this model and I feel I am rather at the beginning of a journey with it, rather than the end.

Thanks again for your interest. Not sure how much you will have been enlightened, but I hope you see where I am coming from, at least!

Tim

Dear Flavio

Thank you for your kind remarks. Nicolas Gisin and I have already discussed some of the matters discussed in your essay and whilst I do agree that your and Nicolas's ideas are very thought provoking, I would say that we are not in complete agreement.

Let me start by remarking that I fully agree that it is possible to treat chaotic classical deterministic systems by some finite indeterministic approximation. In fact this is exactly what we do in modelling climate:

https://www.nature.com/articles/s42254-019-0062-2

which is to say that we approximate a set of deterministic chaotic partial differential equations by a finite deterministic numerical approximation and represent the unresolved remainder of the system by constrained stochastic noise. It works well!

However, I do not believe this approach will work for quantum physics, if one believes that the complex Hilbert Space of quantum theory is somehow fundamental, the reason being (Lucien) Hardy's Continuity Axiom. By virtue of this axiom, the continuity of Hilbert Space is fundamental to quantum theory.

Put this way, the continuum appears to play a more vital role in quantum theory than it does in classical theory. This suggests that if we seek some finite theory of quantum physics - which I certainly do seek - then the resulting theory will have to be radically different from quantum theory (even with a stochastic collapse model) and will not be just some approximation to it.

I can in fact state this a little more precisely. I believe that by virtue of Hardy's Continuity Axiom, quantum theory will have to be a singular limit and not a smooth limit of a finite discretised theory of quantum physics, as the discretisation scale goes to zero.

In my essay I attempt to suggest a deterministic (i.e. not indeterministic) alternative to quantum theory in which quantum theory is a singular limit (as a certain fractal gap parameter goes to zero). However, until potential departures from quantum theory can be experimentally tested, and perhaps this day is not so far away, who knows whether this really is the right way forward.

Having said this, there are clearly many points of commonality between our essays and I look forward to discussing these with you sometime!

Best wishes

Tim

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

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

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

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

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