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Wikipedia (and many other sources) say that by extending the number of spacetime dimensions from four to five, Kaluza–Klein theory unifies general relativity and electromagnetism into a single theory. But why do need to do anything as complicated as adding a new dimension? It's conceptually very easy to write down Maxwell's equations in curved spacetime without doing anything nearly as fancy. How is this any less "unified?"

In the quantum context, a compact extra dimension gives a natural justification for the quantization of electric charge, but Kaluza's original theory was entirely classical.

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  • $\begingroup$ Because it allows for EM to arise from a metric theory in five dimensions. $\endgroup$ – Ryan Unger Jul 19 '17 at 8:06
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"Unification" usually means that you have a single theory - not two coupled theories (and is also not a formally defined notion, so you'll find some people calling theories "unified" that others wouldn't). Maxwellian electrodynamics in curved space is two coupled theories in the sense that it is "electromagnetism + gravity/general relativity", or in the sense that it has two dynamical fields - the metric and the electromagnetic potential.

Kaluza-Klein theory in five dimensions is just gravity, the only dynamical field is the metric, which splits into the four-dimensional metric and vector potential, hence it is unified. Unification is not about the theory being necessarily simple or "not fancy", it's just about there being a single dynamical field from which the multiple dynamical fields being unified arise through an arbitrarily complicated process.

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  • $\begingroup$ Eh, I see what you're getting at, but your distinction between one and two fields seems fairly arbitrary. The metric $g_{\mu \nu}$ in $D$ spacetime dimensions could reasonably be thought of as one field, or $D(D+1)/2$ different fields, or $D^2$ different fields respecting $D(D-1)/2$ constraints, depending on how you count. $\endgroup$ – tparker Jul 20 '17 at 8:08
  • $\begingroup$ The metric and EM field-strength tensors are formally "unified" into one Lorentz-covariant tensor, but this object violates the "spirit" of a tensor - you can't transformation the spacetime coordinates in a way that mixes them together, because doing so would violate the cylinder assumption on which the whole theory rests. By only allowing coordinate transformations within the two sectors, you effectively break the one tensor into two coupled tensors. $\endgroup$ – tparker Jul 20 '17 at 8:08
  • $\begingroup$ Note that Kaluza-Klein theory also isn't "unified" in the technical sense of Grand Unified Theory, because the gauge group isn't a compact simple Lie group. $\endgroup$ – tparker Jul 20 '17 at 8:18
  • $\begingroup$ @tparker Well, I see what you're getting at, but I can't help what people call "unification" in this context. I'm fairly certain that what I wrote here is the intended meaning of "Kaluza-Klein theory unifies gravity and electromagnetism", no matter how consistent it is with other usages of "unification" or how useful/non-arbitrary the notion is to begin with. $\endgroup$ – ACuriousMind Jul 20 '17 at 8:20

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