Geometric interpretation of Electromagnetism

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For gravity, we have General Relativity, which is a geometric theory for gravitation.

Is there a similar analog for Electromagnetism?

This post imported from StackExchange Physics at 2014-03-24 03:29 (UCT), posted by SE-user Mahl-Deneb

retagged Feb 1, 2015

Comment to the question (v1): Do you mean something like (i) EM in curved spacetime, (ii) Kaluza-Klein theory, or perhaps (iii) something along the lines of this question?

This post imported from StackExchange Physics at 2014-03-24 03:29 (UCT), posted by SE-user Qmechanic

Not sure if this helps, but: "However, ordinary Riemannian geometry is unable to describe the properties of the electromagnetic field as a purely geometric phenomenon." (Wikipedia)

This post imported from StackExchange Physics at 2014-03-24 03:29 (UCT), posted by SE-user Glen The Udderboat
I believe 'Gravitation' by Wheeler, Thorne and Meisner has a good introduction to some elementary geometric treatments of Electromagnetism that would certainly benefit the OP.

This post imported from StackExchange Physics at 2014-03-24 03:29 (UCT), posted by SE-user dj_mummy

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1. Kaluza-Klein theory. This is similar to General Relativity, but instead of three space dimensions plus time, there are four space dimensions plus time. The fourth dimension is cyclic, and satisfies some symmetry conditions. The electromagnetic potential appears as the components of the metric in the fourth space dimension. It is usually rejected on the grounds that we can't see the fourth space dimension, or that it is made too small to be seen. In fact, the symmetry conditions along this dimension make it indistinguishable, and moving along it is equivalent to a gauge transformation. So, this is the only evidence predicted by the theory, no matter how large we make the cyclic dimension. Which leads us to

2. Gauge theory. As mentioned by DImension10 Abhimanyu PS, electromagnetism can be described by a gauge theory whose gauge group is $U(1)$; the electromagnetic potential becomes a connection, and the electromagnetic field the curvature associated to the connection. It is in fact the symmetry group of the fourth dimension in Kaluza-Klein theory. For mathematicians, a gauge theory is described in terms of principal bundles, which, if the gauge group is $U(1)$, are in fact 4+1 dimensional spaces, satisfying symmetry conditions like in the Kaluza-Klein theory. So, mathematically, they are equivalent, although there are variations of the Kaluza-Klein theory which cannot be described by a standard gauge theory.

3. Rainich-Misner-Wheeler theory. There is a way to obtain electromagnetism from geometry, in the 4d spacetime of General Relativity. Rainich was able to give in 1925 necessary and suficient conditions that spacetime is curved in a way which corresponds to the electromagnetic field. By Einstein's equation, the spacetime curvature is related to the field. So, Rainich decided to see if one can obtain the electromagnetic field from the curvature, using Einstein's equation. He found some necessary and sufficient conditions for the Ricci tensor, which are of algebraic and differential nature. This works for source free electromagnetism. There is an ambiguity, given by the Hodge duality between the electric and the magnetic fields, for the source free Maxwell equations. So, basically, the field is recovered up to a phase factor called complexion. The idea was rediscovered by Misner and Wheeler three decades later, who combined it with the wormholes of Einstein and Rosen. They interpreted the ends of the wormholes as pairs of electrically charged particles-antiparticles. The electromagnetic field, in this view, doesn't need a source, since the field lines go through the wormhole. While this idea may seem bizarre, it allowed to obtain "charge without charge", and to fix the undetermined phase factor. This model of particles had some issues, for instance it couldn't explain the spin, and Misner and Wheeler abandoned it.

This post imported from StackExchange Physics at 2014-03-24 03:29 (UCT), posted by SE-user Cristi Stoica
answered Sep 5, 2013 by (255 points)
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Yes, there is.

QED

The gauge group for electromagnetism is $U(1)$ in general. In Quantum Electrodynamics, electromagnetism is the curvature of the $U(1)$ bundle (similarly to how the strong interaction is the curvature of the $SU(3)$ bundle, gravity is the curvature of 4-dimensional spacetime, etc.).

In the same context, it's worth noting the covariant derivatives is general relativity are such that $\nabla_\mu-\partial_\mu$ is a relativistic analog of the Newtonian gravitational potential. This is also true in QED, and QFT in general, where to suitable natural some constants, $\nabla_\mu-\partial_\mu=ig_sA_\mu$ is the vector potential (multiplied by the coupling constant and $i$ of course) of the interaction, in this case, of electromagnetism.

Kaluza-Klein theory

In Kaluza-Klein theory, the 5-dimensional metric tensor can be written as

$g_{ab}= \left[ \begin{array} ag_{\mu\nu} +\phi^2A_\mu A_\nu&& \phi^2 A_\mu \\ \phi^2 A_\nu &&\phi^2 \end{array} \right]$

The vector potentials are manifest in the metric itself, scaled by the radion.

More interestingly (for the context of this question), the electric charge is the momentum of the particle in the 5th dimension (which is compactified onto a circle), in other words, the electric charge is a measure of how much the particle revolve around on the compactified dimension.

String theory

In string theory, in addition to both the above interpretations, the electric charge is also equivalent to (in a system of suitable natural units, if you like) the winding number of the string around a compactified dimension. It is to be  noted that this is not so in the Type IIB string theory, which has no gauge group. This is what gives electrodynamics it's U(1) symmetry (the symmetry is however unified into a larger gauge group $SO(32)$ or $E_8\times E_8$ in the respective heterotic theories).

What's interesting is that this description is T-dual to the Kaluza-Klein interpretation (which also holds true in string theory, since Kaluza-Klein theory is just an effective action of classical string theory, without supersymmetry and the nuclear forces).

answered Sep 5, 2013 by (1,950 points)
edited Feb 1, 2015
You don't need the "quantum" in "quantum-electrodynamics".

@drake Yes, of course, you're right that the U(1) is the gauge group of electromagnetism in general. But this answer (albeit a badly written one like all my old answers) is in the context of electromagnetism being the curvature of the U (1) bundle, which is exclusive to QED (and the standard model, etc.).

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