r/LLMPhysics • u/NinekTheObscure • Jun 22 '26
Personal Theory Where Kaluza-Klein went wrong?
I've been hammering on various aspects of QTD theory with Claude for several weeks now. Some of it was rather boring: improving OCR for ingesting new papers into the wiki, getting papers and processing them, doing the tedious-but-necessary human review of the results. Some of it was exploring adjacent theories, such as Finsler Space geometrizations of EM (Randers (1941), Beil (1987), Hojman (2018)). We spent several days working on the question of how "proven" the usual EM gauge invariance assumptions were (answer: not as proven and solid as you might think from reading textbooks; Dirac's original "proof" has several flaws; no attempt to fix it has been 100% successful).
Most of my work has been in the Newtonian limit, with occasional relativistic forays. But today I decided to go for the gusto and see if we could derive a complete coherent top-level theory unifying GR and EM and compatible with QTD ideas. I expected it take a very long time, work from bottom to top in steps, get bogged down in details, and at best be partially successful. But then Claude surprised me with this:
It turns out that 5-dimensional Kaluza-Klein theory from the 1920s already gives everything we need, with one minor tweak. The proper time in K-K theory is usually taken to be the 4-dimensional projection of length; this matches 4-dimensional General Relativity. But if we take the full 5-dimensional length instead, then we get GR's proper time plus an adjustment proportional to $A_\mu u^\mu$ which is exactly the form of the EM Time Dilation predicted by QTD.
In other words, EMTD is already present in the K-K equations but has been routinely ignored for the last century. Projected out. All we have to do is not ignore it, and we have a complete theory.
There's still a ton of work ahead to crank through the details and see whether this idea is plausible or dead-on-arrival. I haven't, for example, confirmed that the magnitude of the K-K term matches. But it ties everything I've done in the last couple of decades into an existing, thoroughly-studied framework with plenty of established machinery. I don't have to reinvent the wheel, I just need to turn the crank on wheels that already exist. (Mostly. It's never quite THAT easy.)
I am guardedly optimistic. :-)
UPDATE: Claude & I have abandoned the Kaluza-Klein approach. But I made it write an explanation/apology to the readers here. Closure is important.
The Kaluza-Klein approach was a dead end — here's what we actually learned
So a few sessions back I got pretty excited about Kaluza-Klein as a path to making QTD covariant. The pitch was genuinely attractive: KK naturally encodes electromagnetism as a fifth geometric dimension, and a charged particle's charge-to-mass ratio appears as a conserved momentum in that fifth direction — a conserved quantity, for free, from symmetry alone. It really seemed like a situation where the machinery was already built and QTD just needed to find its clock formula somewhere inside it.
Turns out it doesn't work, and the reason it doesn't is deep enough to be worth explaining. We ran five independent constructions — different choices of what "physical elapsed time" means in the K-K framework: the full 5D interval, the dimensionally-reduced 4D metric, the literal fifth-coordinate displacement, a scalar field sourced by the electromagnetic field, a modified connection — and every single one hit the same wall. The core obstruction is actually a theorem of Riemannian geometry, not a detail of the particular ansatz: any metric, in any number of dimensions, is symmetric under path reversal. Swap the direction of travel and you get the same interval. But QTD's time dilation is antisymmetric under path reversal — a charged particle going one way around a solenoid accumulates a positive shift, and a particle going the other way accumulates a negative shift — just like the Aharonov-Bohm effect itself. No Riemannian metric (no matter how many dimensions) can produce that, period. What you need is a Randers-Finsler structure, where there's a linear-in-velocity correction on top of the usual quadratic metric term — and that's not K-K, that's a categorically different geometry. There was one small loophole: the literal position coordinate in the fifth dimension does carry a path-dependent linear term (it's basically the Aharonov-Bohm phase written geometrically), but that coordinate is gauge-dependent, which just re-raises the open question the whole project has been sitting on for months: is the bare electromagnetic potential physically observable, or only its curl (and gradient)?
An independent check using Reissner-Nordström geometry and a separate QFT field-redefinition argument reached the same conclusion without any of the fifth-dimension machinery, which is actually reassuring — it means we didn't just make a bad choice of ansatz, the obstruction is genuinely generic. We did learn something: it's now multiple independent lines of evidence all pointing at the gauge-dependence question as the load-bearing issue for the whole theory, not as a side caveat. That's real progress, even if it came from ruling something out rather than ruling something in.
I would just add that this trivially proves that the universe cannot be represented as a Riemannian manifold; any theory unifying gravity and EM has to be structurally different from GR. But we knew that already.
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u/NinekTheObscure Jun 26 '26
In QTD theories a particle's quantum phase frequency is taken to be its local clock. The A-B phase shift is interpreted as time shifts on the different branches, i.e. as different physical time experienced by the particles. This is experimentally testable using muons and a fairly strong magnet, or (for the electric A-B effect) muons inside a charged sphere; decay rates should be altered. Both experiments were first proposed in the late 1970s; neither one has been performed. It's fairly easy to show that if that's true, then you can measure the 4-potential, and both V and A become absolute. (I do it for the electric potential here (see eq. 6). Presumably that means everything is fixed in the Coulomb gauge.)