I'm pretty familiar with the math of principle bundles etc, but I have no intuition for physics at all, so I'm curious how some facts fit together.

When a potential is undetermined up to a gauge, we can choose gauges from a symmetry group on a fiber over the point by a section. For example, elements of $U(1)$ in electromagnetic potential get added to the EM potential. This makes sense, except I have no idea of the following two ideas:

  • Why is the gauge symmetry acting on a potential field, but the connection, here the Lie algebra of the gauge group, is being called the potential?

For example, the electromagnetic four potential is the connection on the $U(1)$ fibers, but it is elements of the group that go into the potential. So I don't see how the Lie algebra is a potential in this case. I'm just confused about the physical significance of these things and how they work together.

  • Where is the symmetry? Obviously the lagrangian changes when we choose a gauge because it is in terms of the potential, so how is this symmetric? Maybe because when we minimize the action the gauge goes away no matter what we choose?

Maybe the first is just a language confusion, and the lie algebra isn't a potential. I have also seen it called a gauge field. But this seems to be the way it is described, as a potential field, despite the mismatch between group elements acting on the potential and the lie algebra being the potential.

I would think of it as saying as we move along fibers, how does the gauge change, making the connection a lie algebra as the tangent space of the group. This makes sense, but then, this doesn't fix the above questions for me.

If you can help, thank you! Maybe there is a good paper or book I can use to see what the math to physics dictionary is, and how these things are used in physical theory.

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    $\begingroup$ The gauge group defines an action on all the fields in the physical theory in such a way the action constructed out of the field is invariant (that is what physicists mean by gauge symmetry). Each field individually transforms according to some representation of the gauge group. In particular the gauge potential which is Lie algebra valued naturally transforms according to the adjoint representation (en.wikipedia.org/wiki/Adjoint_representation) $\endgroup$
    – isometry
    Apr 26 '21 at 11:37
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    $\begingroup$ I don't know how important this is to the question, but when physicists refer to a connection on a principal bundle as a "potential" field (aka gauge field), they don't mean potential energy. If we put on our historian hat and look at the etymology of this confusing terminology, we see that it has roots in electrostatics, where the time-component of a prescribed gauge field can be multiplied by the charge of a test-particle to get a potential energy function for that test-particle. The language evolved from there, and now we have two almost-completely-unrelated meanings of "potential." $\endgroup$ Apr 26 '21 at 13:16
  • $\begingroup$ @ChiralAnomaly Maybe very relevant. It is my thought that this is related to the potential, like the electromagnetic four potential (see bolded statement, maybe this is wrong, or maybe it doesn't generalize). the confusion comes from the fact that it is group elements that act on the potential field, not the connection, and that the connection goes into the covariant derivative to transform with the group. These two things make sense together, but not if you call the connection the potential itself, not for me $\endgroup$
    – rage_man
    Apr 26 '21 at 14:49
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    $\begingroup$ @rage_man yes U(1) appears in many different unrelated ways in physics just as it does in maths. In gauge theory it appears as various subgroups on the standard model gauge group SU(3)xSU(2)xU(1). It appears in the basic QM formalism 'because' the Hilbert space of states is a complex projective space. As such once you have set the norm of a state to 1 (to ensure conservation of probability) there is still an arbitrary U(1) phase change you can make to the state vector. $\endgroup$
    – isometry
    Apr 26 '21 at 15:44
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    $\begingroup$ The book you will probably like ultimately is Nakahara - Geometry, Topology and Physics. It is available as pdf. $\endgroup$
    – isometry
    Apr 26 '21 at 15:50

I don't know exactly the etiquette of answering my own questions, but I think I have cleared it up for me and want to share.

I used this paper a lot: https://arxiv.org/abs/2009.02162 it turns out, my problem is, I didn't know what gauge theory was at all!

The physical ingredients seem to be this:

As @ChiralAnomaly and @duality have discussed with me, a gauge field is not actually a potential, but a field of connection forms that keeps track of the choice of gauge. This is a connection between fibers when the section gives us a choice of gauge from gauge group elements in the fibers. Of course, when this gauge doesn't change from fiber to fiber, the connection is flat, and that's why a dynamical connection has a force from the two form $F=dA$. With this connection we can go into the associated matter field and transform its lagrangian using a covariant derivative to keep the symmetry.

For $\rm U(1)$, I wonder if this is just a special case, because the gauge is also the actual phase of the particle, so the connection is also the actual potential field of the electromagnetic field. This is like saying as we choose a gauge we choose a rescaling of the actual waveform, so the connection tells us how the actual wave behaves. And of course, one component of $\rm U(1)$ symmetry corresponds to an electron, with a charge from $\rm U(1)$ representation. Maybe its the same in other theories that I don't know, like Yang Mills theory, but this clears up for me why the potential coincides with the connection, and its relation to gauge symmetry.

I won't accept my own answer in case a better one is offered, as a matter of courtesy I think.

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    $\begingroup$ Answering own question is perfectly fine $\endgroup$
    – isometry
    Apr 26 '21 at 21:37
  • $\begingroup$ Ok, thank you for the feedback! Does my answer seem right to more discerning eyes? $\endgroup$
    – rage_man
    Apr 26 '21 at 21:50

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