[gnramires]

I've recently learned about Kaluza–Klein theories (just curious wiki browsing), do they unify electromagnetism and gravity in the sense that electromagnetism also "works by changing the geometry of spacetime" ? How does this relate to QFT?

[elashri]

> do they unify electromagnetism and gravity in the sense that electromagnetism also "works by changing the geometry of spacetime" ?

Yes, somehow actually. KK theories introduce an extra spatial dimension beyond our usual (3+1) which is postulated to be "compactified". This just means that it's curled up on itself at such a tiny scale that we don't directly perceive it. The way this extra dimension is curled and shaped affects the geometry of the overall 5-D spacetime. How it is connected to EM is that now these geometrical variations in the 5-D spacetime, when viewed from our 4-D perspective, manifest as the EM force and its associated field.

So you can say like in GR which have gravity arises as consequence of geometry, it is in KK that EM is consequence of geometry. However, the geometry and details are different.

> How does this relate to QFT?

Not much in the sense that they can provide useful information to each other. QFT describes forces in terms of interactions mediated by particles (e.g., photons for EM). KK, while primarily geometrical, give hints that perhaps these force-carrying particles can be associated with specific vibrational modes of the extra dimension. Of course KK theory only include EM and gravity. So we know for sure that we need to go beyond KK. This was the actual motivation for people to think about string theory to expand the original KK work.

[Karellen]

> KK theories introduce an extra spatial dimension beyond our usual (3+1) which is postulated to be "compactified". This just means that it's curled up on itself

You've just explained "compactified" in terms of being "curled up", but that doesn't really help (for me, at least). What does it mean for a dimension to be "curled up"?

[elashri]

I'm sorry if I'm confusing. That's the term I use, usually assuming that it is obvious (like referring to C++ templates to a python developer without explaining what is that).

To be honest, it is hard to visualize, as it is counterintuitive of what we think of space. While I know many people would disagree, I really like the garden hose analogy [1]. The idea is to simply imagine a very long garden hose. From a great distance, it looks like a one-dimensional line. Now you get closer, and realize it has a second dimension, which is its circumference curled around that seemingly 1-D line. An ant walking on the hose can move along its length, but also in a circle around it. The extra dimension in KK is like this circumference. It is tiny, curled up so we don't directly notice it, but still potentially there.

[1] https://www.preposterousuniverse.com/blog/2004/06/30/extra-d...

[Izkata]

This kind of makes me think of a spatial version of the Time Cube.

[gnramires]

Interesting. I ask because I have a suspicion that Quantum theories seem more fundamental than General relativity, because treating geometry within a quantum framework seems very hard and "non-natural", or implausible (e.g. when considering superposition of spacetimes!). While if somehow gravity is a quantum effect (within a simpler space-time framework), that seems much more plausible... but Kaluza-Klein captured my interest in the other direction. Although I'm still thinking the quantum framework is appears to be the correct one, even though the assumptions of GR are very strong (so something like the equivalence principle or some other notable principle needs to break).

[anticensor]

The problem with extra dimensions is that such an increase of dimensions preclude pairwise interaction of particles due to lacking a sound relationship between 2-operand dot product, cross product and 2-operand tensor product in those dimensions.