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
