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I've heard about how, in a gauge theory, the gauge covariant derivative of the field around a closed curve is generally not zero, and this is how you can quantify force or field strength. And that this is the same basic idea as curvature, with the gauge field being equivalent to the connection.

So since gravity is already known to be curvature, we can say that all the forces of nature are curvature in their own way. So what's the significance of that? Is there some deeper reason that we should expect that to be the case? And are the current unification programs based on that similarity?

Qmechanic
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Adam Herbst
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  • This seems to be a related question: @ https://physics.stackexchange.com/q/272902/ – Brendan Darrer Jul 23 '22 at 20:21
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    Curvature is the simplest local invariant (as opposed to a global invariant like the number of holes in spacetime) you can cook up from the data that goes into a physical theory. Once we specify a theory, curvature is something we can measure -- it's a "force." – Charles Hudgins Jul 24 '22 at 18:15
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    @CharlesHudgins That's interesting, can you expand on that or provide a reference that explains that idea? – Adam Herbst Jul 24 '22 at 21:45
  • I would turn it around and say that the significance is that gravity can be thought of as a gauge theory like the other forces (see, e.g.:https://physics.stackexchange.com/questions/71476/gravity-as-a-gauge-theory). And the fact that all the fundamental forces (except, arguably, one associated with the Higgs field) share this broad structure does seem deep indeed, although I can't say more than that. – Rococo Jul 24 '22 at 23:54
  • @AdamHerbst Unfortunately I don't have a reference ready because this is just something I picked up from browsing. But here's a sketch: we want to differentiate something with symmetry in a way that respects the symmetry. That gives us a connection. The connection necessarily couples to the object we're trying to differentiate, and so should be physically observable. How do we observe the connection? Well a connection tells us how things should change as we move around space. The curvature tells us what happens as we move in a loop. So we measure the curvature by traversing a loop... – Charles Hudgins Jul 25 '22 at 00:10
  • Lest this seem extremely abstract, this is literally what happens with the magnetic field $\vec{B}$, which is one part of the EM curvature tensor. We can't measure $\vec{A}$ (the magnetic potential) directly, but we can measure it around a loop. Concretely, if $M$ is a surface, we have $\int_{\partial M} \vec{A} \cdot d\vec{x}= \int_M \nabla \times \vec{A} \cdot d\vec{S} = \int_M \vec{B} \cdot d\vec{S}$. The physically observable thing $\vec{B}$ appears (or at least its flux) when we look at what happens to the potential $\vec{A}$ (itself an avatar of the EM connection) around a loop. – Charles Hudgins Jul 25 '22 at 00:15
  • I don't post this as an answer because I can't speak to any of this with authority. However, in my time looking into these things, I've found that you can't really find a good interpretation written down anywhere (though a professor probably could tell you a good story if you have access to one). This is as much sense as I've been able to make of things on my own. – Charles Hudgins Jul 25 '22 at 00:18
  • One last comment, sorry. It shouldn't bother you that the connection only gives us immediate access to the flux of $\vec{B}$ in the EM example and not $\vec{B}$ itself. Curvature is a $2$-form. It wants to be integrated over a surface. – Charles Hudgins Jul 25 '22 at 00:26

2 Answers2

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When we study non-gravitational fundamental interactions, we distinguish internal symmetries associated with only such interactions from the external symmetries of spacetime. For all fundamental interactions, there is a finite-dimensional Lie group characterizing that interaction's symmetries. In the case of gravity, the Lie derivative of Killing vector fields on the spacetime manifold defines the associated Lie algebra's structure constants; for the other interactions, there is a "space" that plays a role analogous to this manifold, but it's not spacetime itself. Instead, it's a space of legal values for a field over spacetime.

For example, electromagnetism's $U(1)$ symmetry (let's put electroweak unification aside for the moment) is the rotational invariance of $|\phi|$ for $\phi\in\Bbb C$ with $|\phi|:=\sqrt{\phi^\ast\phi}$, or equivalently for $\phi\in\Bbb R^2$ with $|\phi|:=\sqrt{\phi\cdot\phi}$. (I'm denoting the set of values $\phi$ can have at each point in spacetime, say $\Bbb R^4$, so as a function $\phi\in X^{\Bbb R^4}$ for $X=\Bbb C$ or, somewhat less helpfully in QFT, $X=\Bbb R^2$.) So if there is a space which is "curved" in this context by electromagnetism, it is not spacetime per se.

As for unification implications, Wikipedia notes

For ordinary Lie algebras, the gauge covariant derivative on the space symmetries... cannot be intertwined with the internal gauge symmetries... this is the content of the Coleman–Mandula theorem. However, a premise of this theorem is violated by the Lie superalgebras (which are not${}^\dagger$ Lie algebras!) thus offering hope that a single unified symmetry can describe both spatial and internal symmetries: this is the foundation of supersymmetry

${}^\dagger$ emphasis in the original.

J.G.
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In the 1920s–1940s, people developed a unified classical theory of gravity and electromagnetism using just this sort of approach. It's called Kaluza-Klein theory. Some aspects of it even generalize to classical non-abelian Yang-Mills theories (R. Montgomery: Canonical formulations of a classical particle in a Yang-Mills field and Wong's equations). I think I've heard there's some subtle problem with its quantum version that prevents it from describing quantum non-abelian Yang-Mills theories well. That might be one reason it's not talked about much these days.

Vectornaut
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