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

Thanks Jeffrey for these kind remarks. I know what you mean about Pitowski's work. However, on (what I call) the invariant set, the relevant measures can be described by elementary finite frequentist probability theory. The mathematics underpinning undecidability arises only when considering counterfactual states which do not lie on the invariant set. My approach, is simply to deny ontic reality to such states by postulating the primacy of the invariant set. Without this, I think one would indeed be drawn to consider non-measurable sets as did Pitowski. However, the concept of non-measurability does not seem to make physical sense to me - as the famous Banach-Tarski paradox clearly indicates.

PS Reference [18} - my paper arXiv:1804.01734 on invariant set theory - has now been accepted to appear in Proceedings of the Royal Society A.

Thanks Fabien. Good questions.

My fractal model has a free parameter N. In the singular limit N=infinity all the fractal gaps close up and the state-space geometry is classical. However, for any finite value of N, no matter how big, the Bell counterfactuals lie in the fractal gaps and the state-space geometry is non-classical. Michael Berry has written about how old theories of physics are often the singular limits of new theories as some parameter of the new theory is set to zero or infinity.

The Cantor Set underpins the Lorenz attractor. Imagine a point X on the Cantor Set and perturb it with a perturbation delta X drawn randomly using the measure of the Euclidean line in which the Cantor Set is embedded. Then the perturbation almost certainly perturbs the point off the Cantor Set.

Such a perturbation can be thought of as corresponding to one of my counterfactuals: although I live in a world where I did X, what would have happened if I had instead done X+delta X? Suppose the delta X is dynamically unconstrained - i.e. something you just make up in your head without consideration of whether it satisfies the laws of physics - then if the world associated with X lies on the invariant set, the world associated with X+delta X almost certainly does not and so the answer to the question "what would have happened?" is undefined.

Tim Palmer re-uploaded the file Palmer_FXQi_Palmer_1.pdf for the essay entitled "Undecidability, Fractal Geometry and the Unity of Physics" on 2020-04-17 07:52:00 UTC.

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