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Duda Jarek

Time crystal (self organizing into periodic process) - is electron its example?

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Time crystals first described by Frank Wilczek in 2012 have got a lot of attention, recent popular review: https://physicsworld.com/a/time-crystals-enter-the-real-world-of-condensed-matter/

If I properly understand, they would like a lowest energy state spontaneously self-organizing into a periodic process - and propose sophisticated e.g. solid state experiments, or ping-pong of Bose-Einstein condensate, which don't really seem to satisfy this defining requirement (?)

But Louis De Broglie has already postulated in 1924 that with electron's mass there comes some ≈10^21 Hz intrinsic oscillation: E = mc^2 = hf = hbar ω, obtained if using E=mc^2 rest mass energy in stationary solution of Schrödinger's equation: ψ=ψ0 exp(iEt / hbar).  
Similar oscillations come out of solution of Dirac equation - called Zitterbewegung ("trembling motion"). Here is one of its experimental confirmation papers - by observing increased absorption when ticks of such clock agree with spatial lattice of silicon crystal target: https://link.springer.com/article/10.1007/s10701-008-9225-1

Electron can be created together with positron from just 2 x 511keV energy of EM field - after which it (the field?) should self-organize into these ≈10^21 Hz intrinsic oscillations.

So can we call electron an example of time crystal?
What other examples of lowest energy state self-organizing into periodic process are there?

ps. Beside self-organization of the lowest energy state into periodic motion (I don't see they got? in contrast to electron), they alternatively want "period doubling": that system oscillating with T period, self-organizes into 2T period process - breaking discrete time symmetry (invariance to shift by T).
So these popular Couders' walkers recreating many quantum phenomena in classical systems (slides with links) also have period-doubling (can they be classified as time crystals?) - here is such plot from this paper, horizontal axis is time, lower periodic process is for liquid surface - externally enforced by some shaker, upper periodic process shows droplet trajectory - self organizing into twice larger period than enforced:


But generally it seems very valuable to find analogies between spatial and temporal phenomena like crystals here.
Great tool for that is Ising model: Boltzmann ensemble among spatial sequences, what mathematically is very similar to Feynman path (temporal) ensemble of QM - using this mathematical similarity, for Ising model we get Born rule, Bell violation, or analogues of quantum computers.
What other phenomena can be translated between spatial and temporal dimensions?

Edited by Duda Jarek

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