On computing quantum waves exactly from classical action

[akhmeteli]

TL;DR

An interesting article: "the Schrödinger equation can be
solved exactly based only on classical least action"

Lohmiller W, Slotine J-J. 2026 On computing quantum waves exactly from classical action. Proc. R. Soc. A482:20250413

Abstract:
"We show that the Schrödinger equation can be solved exactly based only on classical least action. Fundamental postulates of quantum mechanics can in turn be derived directly from this construction. The results extend to the relativistic Klein-Gordon, Pauli, and Dirac equations, and suggest a smooth transition between physics across scales. Most quantum mechanics problems have classical versions which involve multiple least action solutions. The associated classical multipaths stem either from the initial position or momentum distribution, or from branch points, generated, e.g. by a multiply connected manifold (double slit experiment), by spatial inequality constraints (particle in a box), or by a singularity (Coulomb potential). We show that the exact Schrödinger wave function ψ can be constructed by combining this classical multi-valued action φ with the classical density ρ, computed analytically from φ along each extremal action path. The construction is general and does not involve any semi-classical approximation. Quantum wave collapse at measurement can be derived from the classical density change. Entanglement corresponds to a sum of classical particle actions mapping to a tensor product of spinors. The results also provide a simpler computational alternative to Feynman path integrals, as they use only a minimal subset of classical paths."

 

[Demystifier]

arXiv link: https://arxiv.org/abs/2405.06328

 

[akhmeteli]

Sorry, I did not provide the link. The article is actually in open access.

 

[Peter Morgan]

Also of interest is an article in MIT News and that someone on 𝕏 pointed out that a preprint by Frank J Tipler is somewhat similar (I communicated a link to Tipler's article and a version of what follows to the corresponding author, Slotine, yesterday). I think of this article in ProcRoySocA as a very welcome contribution to now decades-long attempts to normalize quantum physics by elaborating classical physics.
That's self-serving on my part because I have an article in that literature, "An algebraic approach to Koopman classical mechanics", (arXiv or paywalled in AnnPhys2020), but hopefully it's not a huge stretch to frame Lohmiller's and Slotine's contribution so. The particular construction —using least action, Koopman, or some other formalism— seems to me not specially significant except that a least action approach may be more compelling insofar as path integrals are more of the times than the kind of algebraic methods I prefer.
What I think is of the essence is that we can construct a more powerful classical physics that

1. includes contextuality by using the Poisson bracket to extend the algebra of canonical transformations to include transformations between different phase spaces, representing different experimental contexts, and
2. includes a 'quantum noise' (~fluctuations~) that is different from 'thermal noise' because of different Lorentz transformation properties (quantum noise is Lorentz invariant, as it manifestly is in QFT's vacuum state, whereas thermal noise is not invariant under boosts; the Unruh effect shows that transformations to accelerated coordinates, which are not Lorentz transformations, are enough to 'convert' quantum noise to 'quantum noise & thermal noise', so we know that the difference between quantum and thermal noise is both vital and tenuous.)

There is a third difference between quantum physics and classical physics, which I think is worth preserving (instead of changing classical physics to minimize or eliminate it): that quantum physics insists on only positive frequency components (more strictly, the Hamiltonian operator —the Hermitian generator of time-like evolution— is bounded below), whereas classical physics allows both positive and negative frequency components. This has the consequence that classical physics does not have some analytic properties that are very useful and natural in quantum physics but that are less essential from a measurement theory perspective. An analogy can be made with the difference between the 'analytic signal' and the 'real signal' in signal analysis, making it fairly natural to say that "QM is an analytic form of CM+" (with 'CM+' being the term I use for Classical Mechanics extended to include contextuality & quantum noise.) I think it's worth preserving this difference of analyticity because the measurement theories of the analytic quantum physics and of the non-analytic classical physics are equally powerful, so we can use whichever one is most effective when we describe or model a given experiment. As we get used to the difference, I expect comparison between the different types of models will illuminate the physics that underlies them both.
I rehearse all this because all three of these aspects do not seem to me transparent in the formalism adopted by Lohmiller and Slotine and yet from an algebraic measurement theory perspective I think they almost leap out. That's easy to say now, in 2026, after I have developed the ideas by giving ~20 research seminars about my AnnPhys2020 and other published articles since 2020. I'm told the most accessible of my presentations (with the ideas changing only slowly, none of them are even close to perfect) is a Zoom colloquium I gave for North South University in Dhaka, “A Dataset & Signal Analysis Interpretation of Quantum Mechanics”. My preferences are of course inessential, except insofar as people here may find it helpful to consider the contrast between Lohmiller and Slotine's approach and mine.

 

[bhobba]

Some nice papers/videoes here - well worth the time to read/watch.

Thanks
Bill