# An equivalence principle for gauge theories

In GR there is the equivalence principal, stating that in an arbitrary gravitational field, no local non-gravitational experiment can distinguish a freely falling, non-rotating system from a uniformly moving system in absence of the gravitational field. Or in mathematical terms, we can always choose a reference system in which the Christoffel symbols vanish.

In gauge theory, we can always find a coordinate system in the internal space, i.e. a section in the principal bundle, such that the connection 1-form vanishes. Let's assume electromagnetism, but it should hold also for the strong and weak interaction. Can we state an 'electromagnetic'-equivalence principle, i.e. that there in no local (i.e. at a specific point) electromagnetic experiment which can distinguish a charge 'falling' in an electromagnetic field from a charge outside every electromagnetic field?

My doubts: In GR the fact that we can choose a reference frame in which the Christoffel symbols vanish is equivalent to the statement of the equivalence principle. I have doubts about the equivalence of the vanishing connection 1-form and the 'electromagnetic equivalence principal', since accelerated charges radiate and their is self interaction. However I only require the equivalence principle locally, i.e. at one point (maybe this saves it). But a spherical mass, e.g. a point mass does not radiate gravitational waves when accelerated, so I assume their is also no self interaction. This is a big difference to electromagnetism.

• How do you want to free fall with an electron?
– user303670
Jun 7, 2021 at 19:39
• With falling I mean, that ab electron moves in a electromagnetic field Jun 7, 2021 at 19:40
• But that's no free fall, as in gravity. In the moving frame of the electron, there will still be a force. You can't gauge it away as in GR. Look at classical electric force. If you move in the same way as a charge in an electric field, how can this be done? There is no frame that can move freely along with it, as there is for a mass freely falling in a gravity field.
– user303670
Jun 7, 2021 at 19:46
• There's an by Weinberg based on scattering amplitudes that gravitational scattering amplitudes (more precisely, exchange of a graviton at tree level by two masses in the low energy limit) is consistent only if the equivalence principle holds. The same constraint applied to $U(1)$ gauge theory leads to conservation of charge. If I have time later I'll try to dig it up and write it as a real answer. Jun 7, 2021 at 20:14
• One of the founders of gauge theory refers in these notes to the vanishing of a gauge field at a single point as 'a kind of generalized principle of equivalence' on P.2215. Interpreting the meaning of this, e.g. as a 'falling electron' which doesn't seem to make sense, is another story. Jun 7, 2021 at 20:23