Dear Prof. McHarris,
I enjoyed your essay and thank you for some interesting new ideas and enabling me to recall others. My Ph.D. focused on nonlinear dynamics in the mid-1990s but I veered into neuroscience, machine learning, astrophysics and now theory. So its been a while since I have thought in these terms.
I wish that you had provided a little more insight into how you think nonlinear dynamics is coming into QM. Perhaps you can point me to some of your key papers (or I could of course just go dig them up).
In our symmetry-based derivations of the Feynman rules
(Goyal P., Knuth K.H., Skilling J. 2010. Origin of complex quantum amplitudes and Feynman's rules, Phys. Rev. A 81, 022109. arXiv:0907.0909v3 [quant-ph].)
we find that associativity of combining measurement sequences in parallel results in additivity. So I am left wondering where nonlinearity would come in. With such a basic symmetry forcing linearity, it is hard to imagine room for nonlinearity. Though as Cristi Stoica points out in his comments above, the classical equations can still be nonlinear, and nonlinear equations can often be broken into linear ones in sufficiently high dimensions.
Despite this, your essay reminded me of work by Pedrag Cvitanovic where he used (classical) periodic orbit theory to derive the energy levels of the Helium atom. I have not kept up with this work, so I do not know where it has led. But your essay made me remember this. At the time, I felt it was quite impressive, and should have shaken the quantum foundations more than it has seemed to.
I also remember coding fractal generating programs based on Barnsley's book Fractals Everywhere. It was always interesting to me that you could think of these processes in two ways: one by an deterministic iterative process and second by a random iterative process. Watching fractal ferns appear on my Amiga computer dot by dot reminded me of electrons appearing on a screen after passing through a slit apparatus. I always wondered whether some nonlinear or iterative process could be behind such confounding behavior.
And now you have sparked a thought for which I thank you and the FQXi people for setting up this essay contest. In my essay, I discuss a new way to think about electron behavior in terms of interactions. Perhaps I should take some time to consider an electron "moving" through two slits by influencing atoms along the walls of the barrier. Could it be that these simple patterns of influence lead to rules guided by the boundary conditions to give rise to the diffraction pattern electron by electron. I have considered this before, and worked for a short while to see what I could work out. But now I am once again emboldened by the fact that I now recall the fractal ferns coming into focus. Perhaps with this perspective, I can make some headway.
Thank you again
Kevin