Quantum Mechanics Surprisingly Based on Real Numbers
VLM writes:
For about a century, quantum mechanics, QM, has required the use of complex numbers, but according to a recently published peer reviewed paper in PRL, given a large amount of mathematical fooling around, it's possible to do QM solely with real numbers. That's very unexpected and very cool and the future is going to be very interesting!
https://arxiv.org/abs/2503.17307
https://journals.aps.org/prl/abstract/10.1103/4k13-sdjh
There's been an effort in recent years to formulate quantum mechanics without complex numbers, slowly chipping away at existing QM theory. This paper solves the last remaining problem, and now AFAIK the entire QM system works, or is compatible, or at least not disproven yet, when doing QM calcs using only real number space. The conclusion of the abstract kind of says it all "Thus, we argue that real-valued quantum mechanics cannot be falsified, and therefore the use of complex numbers is a matter of convenience."
I read, or tried to read the paper, back in March last year when the free preprint was uploaded to arxiv and I see it successfully made it through peer review and was published in PRL three weeks ago. AFAIK, nobody has successfully shot it down in the last three weeks, so this interesting project is likely to be successfully completed. Apparently, they have done it!
Effects:
This is one of those accomplishments like when a century ago they proved 1+1=2 solely using set theory. That specific example is not immediately useful, a century later, and note that set theory accomplishment from the 1910s is still pretty useless IRL, but the point of the effort was that they successfully connected two "very large theories" that were not supposed to be connectable. It's not supposed to be possible to QM without complex numbers. A somewhat more famous and wildly profitable theory unification would be when the separate theory of electrical fields and theory of magnetic fields were connected (see Maxwell, etc) I'd say the theory of electromagnetism has had quite an effect on the world since it was discovered... The accomplishment in the paper will probably has an impact in between the two examples above.
Trivially, textbooks will need to be updated such that QM does not have a mandatory inherent requirement for complex numbers, although the formulation is simpler if you use them. Sort of like you don't have to use logarithms to multiply numbers, but it sure is easier to manufacture slide rules if you use logarithms to multiply numbers...
I always found it philosophically distasteful that the most accurate model we have for subatomic particles IRL, requires imaginary sqrt(-1) math. It only took a century to prove it does not. The main paper itself is only about five pages and seems pretty clear, its the dozen page long appendix thats mathematically a bit of a beast to get thru.
There will probably be an unpredictable effect on unification theories. I'd sure be excited to hear a news story in a year or two about some dude who has been sitting on an experimental pet string theory or loop quantum gravity or some other misc theory for years (decades?) that would unify gravity and QM into one theory that "was obviously wrong because it required a QM expressible without complex numbers" well its 2026 and we got one of those now, so really big things might happen really soon. Or maybe not.
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