Jussi Lindgren, Jukka Liukkonen (2019), Scientific Reports 9, 19984 (2019) — DOI-10.1038/s41598-019-56357-3
Key takeaways
- The authors claim that the Schrödinger equation (and relativistic wave equations) can be derived rather than postulated, as a log-transformed Hamilton-Jacobi-Bellman equation of a stochastic optimal control problem on Minkowski spacetime, with the imaginary unit arising from the Minkowski metric.
Seeded from the abstract and a skim, before the reading in NOTE-003. What follows is what the work says about itself.
The paper aims to explain the origin of the complex (imaginary) structure of quantum mechanics. It argues that relativistic invariance, spacetime geometry and a demand for linearity are the key ingredients. Starting from a stochastic control problem, the authors derive the Stueckelberg covariant wave equation, obtain a Telegrapher’s equation from it, and from there recover the relativistic and nonrelativistic quantum equations. The stated payoff is insight into quantum mechanics from a coordinate-invariant stochastic optimization problem instead of postulates.
Standing in the record
Filed by the reading-time triage of 2026-09-25: 1,015 seconds of active reading over 2 sessions in the papers-feed tracker. It was filed Deferred, unread. NOTE-003 is the close reading of 2026-09-25, and it placed the work: Rejected — the claim that the i comes from the Minkowski metric depends on the non-standard √g, not √(−g). The complex noise it also needs leaves no real stochastic process or optimization behind. Nelson (1966), Guerra–Morato (1983) and, for the control-theory side, Kappen (2005) cover the real content better.
When seeded, before the reading in NOTE-003, the entry gave a priority for a deeper reading: medium — moderate owner time (t=1015 s) and a short paper the skim covers well; a deeper read is worth it only to audit the linearization assumptions against Nelson-style stochastic mechanics.
What the seeding asked a deeper reading to check (NOTE-003 is that reading):
- Strongest claim: derivation “from first principles”. Check which assumptions do the work: the log transform and the specific noise-mass coupling are chosen to make the HJB equation linear, which is the standard Hopf-Cole/path-integral-control trick. Is anything beyond that, plus a Wick rotation justified by the metric, being derived?
- The anti-PBR remark rests on a single citation (Rizzi 2018); check whether the paper argues it or only asserts it.
- ML link (honest, indirect): the same HJB/log-transform linearization underlies path-integral control and Schrödinger-bridge formulations used in diffusion generative models. Useful as a physics-side instance of that correspondence, not as an ML result.
Access when seeded: Full open-access HTML from nature.com read (abstract, section heads, main derivation opening, Conclusion and Discussion); DOI, authors and date (online 2019-12-27) confirmed via Crossref. No arXiv version found: arXiv search for title terms and for author “Lindgren, Jussi” returned no matching paper, so arxiv is “none” to the best of that search. Keywords are the journal’s subject tags, not author keywords (none listed).