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

 

[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

Critique of the article: https://arxiv.org/abs/2605.02621
I would need some time to find out whether the article is wrong or the critique...

 

[Demystifier]

I have not studied the papers in detail, but at a first look the critique looks much more plausible.

 

[akhmeteli]

Hossenfelder has blogged about the article, called it mathematically erroneous (actually, she was somewhat blunter ).

 

[bhobba]

Hossenfelder has blogged about the article, called it mathematically erroneous (actually, she was somewhat blunter ).

I like her, but sometimes she can be, how to put it, sensationalist. Although I think she is right, saying things like "researchers don't fully understand QM", which, while correct, IMHO, is misleading for beginners.

Best to let the usual considered responses just proceed without the hoopla, e.g., "To this day, researchers don’t fully understand quantum physics". I have a very simple, intuitive model of ordinary QM (not QFT). It is wrong because ordinary QM is not the non-relativistic limit of QFT.

https://arxiv.org/abs/1712.06605

So what is the actual issue? IMHO is the far less sensationalist claim, from the paper above:

"Section 6 emphasises the fact that the standard KG field is built from two fields which in the NR limit, represent the particle and the antiparticle. This, by itself, is not new and exists in several textbooks, including my own. But it assumes importance in the context of Eq. 194 (see the link for the actual equation), which I claim no one understands, despite it being the key equation in QFT that makes the formalism work. The fact that the path integral for, ostensibly, a single relativistic particle actually describes the propagation of two particles is the key issue here and the discussion in Sec. 6 provides the backdrop for it."

About equation 194 in section 6:

"That is to say, no one has found a simple, physical argument suggesting why the left and right-hand sides of Eq. 194 should be equal without doing fairly elaborate calculations. This means we do not quite understand the conceptual basis of QFT, and the structural implications of combining the principles of quantum theory and special relativity, in spite of its remarkable success as a working tool."

This elaboration is far less sensational than just saying "we do not understand QM." Sure, it is an issue; we can formally justify it, but for some, formal understanding is not the same as understanding. I am happy with a formal understanding, but others disagree. As usual, further research will likely clarify the issue, but only if we understand what it actually is.

I tend to side with Witten, who thinks QFT is just math, like calculus is just math. At rock bottom, we are led to Wigner's famous article about the unreasonable effectiveness of mathematics. Of course, that QFT requires a cutoff to actually work suggests it too, like ordinary QM, has issues we have found tricks like renormalisation to tame, but in going from ordinary QM to the continuum limit of a field (which seems to be the source of the issues) may need modification. Again, that is for future research and the often associated hubbo bubo.

To be fair, we are all human.

Added later:

Just purchased his two textbooks, which delve further into what I think are important issues.

Thanks
Bill