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I am not a physicist. I’ve read some informal stuff about gauge theory and gauge symmetries, however it’s pretty abstract to me.

Can you give the simplest non-trivial example of a gauge theory? E.g. there exist Newtonian/classical gauge theories, then that would be helpful.

Note: I read in another question that “gauge theory” is used not to refer to a theory with a gauge symmetry, but one with a “gauge field” (not sure what that is).

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    $\begingroup$ This link might be helpful. $\endgroup$
    – PolaroidDreams
    Commented Nov 19, 2018 at 8:41
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    $\begingroup$ Did you look at classical electrodynamics? $\endgroup$
    – Toffomat
    Commented Nov 19, 2018 at 13:40
  • $\begingroup$ I just found a great post by @tparker here: physics.stackexchange.com/a/267044/206691, followed by several other good answers. They answer the question, "What, in simplest terms, is gauge invariance?" $\endgroup$
    – Chiral Anomaly
    Commented Nov 22, 2018 at 3:58

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One simple example is if the Lagrangian is a total derivative $$ L(q,\dot{q},t) = \frac{dF(q,t)}{dt} .$$ This has a gauge symmetry $$\delta q(t)~=~\varepsilon(t),$$ where $q(t)$ plays the role of a gauge field and $\varepsilon(t)$ is a gauge parameter.

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    $\begingroup$ 3D abelian Chern-Simons theory $S[A] =\int \!A\wedge \mathrm{d}A$ is another simple example. $\endgroup$
    – Qmechanic ♦
    Commented Nov 19, 2018 at 10:20
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    $\begingroup$ ...and any Lagrangian of the form $L(q_1,q_2)=f(\dot q_1-\dot q_2,q_1-q_2)$, with $\delta q_i(t)=\varepsilon(t)$. $\endgroup$
    – AccidentalFourierTransform
    Commented Nov 20, 2018 at 0:22
  • $\begingroup$ $\uparrow$ Right. $\endgroup$
    – Qmechanic ♦
    Commented Nov 20, 2018 at 0:41

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