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For quite a while in the first half of the 20th century, many physicists tried to concoct some manner of unified theory to explain all known phenomenon, a lot of them using geometric theories (ie teleparallelism, affine gravity, asymmetric metric theories, Kaluza-Klein, etc). While their main goal was the unification of general relativity and electromagnetism, there was also the hope that it would explain quantum phenomenon, by outputting some discrete sets of solutions.

As far as I know, none of them succeeded in that domain. What I want to know is, was there ever any hope of it? Can a classical theory, be it a field theory, of extended objects or geometric, have a discrete spectrum, or does it by its very structure (perhaps via some theorems on differential equations or whatever else) force it to have a continuous spectrum for all variables? Also just in case, did any such theory ever succeed at least in that aspect, even if it failed in other aspects to describe quantum mechanics?

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    $\begingroup$ Can you clarify what you mean by a "quantum-like effect"? For example, some would say that the most essential quantum-like effect is entanglement, which results in the failure of local hidden variable theories, making the answer to your question a straightforward no. Do you count any theory including anything discrete in it as "quantum"? $\endgroup$ – knzhou Feb 13 at 8:23
  • $\begingroup$ Well I'm considering things in the viewpoint of the era, before entanglement and other such phenomenon were a thing (I know that anything quantum probability related cannot be explained by a classical theory). So my point is more about the discreteness of ie energy, angular momentum, etc for a classical system. $\endgroup$ – Slereah Feb 13 at 8:25
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    $\begingroup$ Hmm, well, theories with topological solitons do have classes of solutions which each have a minimum energy. That has a very rich history, going back to Kelvin's attempt to explain the discrete kinds of atoms as "vortex knots in ether". $\endgroup$ – knzhou Feb 13 at 8:28
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    $\begingroup$ Quantum analogs are a thing, eg using soundwaves or bouncing droplets (cf en.wikipedia.org/wiki/Hydrodynamic_quantum_analogs for the latter - there are some pretty cool videos out there of interacting bouncing droplets); such experiments can capture some, but not all features of quantum systems $\endgroup$ – Christoph Feb 13 at 9:26
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    $\begingroup$ If you mean probabilistic effects a paper by Kirkpatrick, Quantal behavior in classical probability (arxiv.org/abs/quant-ph/0106072) claims to do so . With a deck of cards he can recreate incompatibility and value-indeterminacy of variables, the non-existence of dispersion-free states, etc of quantum mechanics. $\endgroup$ – ahmetselcuk Feb 13 at 9:52
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At present they are called deterministic or hidden variable theories, and they try to show that the standard model elementary particles are composite and there is a deterministic theory from which the standard model and quantum mechanics emerge, similar to the emergence of thermodynamics to statistical mechanics.

As far as I know none have been successful over the total experimentally supported points of the standard model.

One of the proposals is by G. 't Hooft (who is still a member on this site, but inactive for some years)

t Hooft has "deviating views on the physical interpretation of quantum theory".He believes that there could be a deterministic explanation underlying quantum mechanics. Using a speculative model he has argued that such a theory could avoid the usual Bell inequality arguments that would disallow such a local hidden variable theory. In 2016 he published a book length exposition of his ideas

An early theory is Bohmian mechanics, again trying to get the mathematical results of quantum mechanics with an underlying deterministic model.

As far as I know it is successful at non relativistic quantum mechanics, that is why it is called an interpretation of quantum mechanics. Again AFAIK people are still working on trying to find a relativistic version that will do the same.

As for your title:

Can classical theories exhibit quantum-like effects?

The notes coming out of musical instruments are quantum like effects, so yes there are classical equations that give specific energies etc. But to describe quantum theories without the quantum mechanical postulates, by assuming an underlying deterministic level, is a more difficult story.

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  • $\begingroup$ This is a good answer; i’d also like to add that while Bohmian Mechanics is deterministic, it’s still quite distinct to classical mechanics, as a result of the manifest non-locality present in the guiding equation where the instantaneous velocity one particle depends on the position of all the others. $\endgroup$ – Thatpotatoisaspy Feb 13 at 10:43
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You may be interested in Stochastic Electrodynamics :

https://en.wikipedia.org/wiki/Stochastic_electrodynamics

https://arxiv.org/abs/1205.0916

https://arxiv.org/abs/1903.00996

I've read several papers on that classical theory, and I must admit that it shaked strongly my belief in Quantum Mechanics! Since then, I no longer see QM the same way as before and feel more perplexed about it.

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