Simplest example of a gauge theory?

Question

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).

I just found a great post by @tparker here: https://physics.stackexchange.com/a/267044/206691, followed by several other good answers. They answer the question, "What, in simplest terms, is gauge invariance?" — Chiral Anomaly, Nov 22 '18 at 03:58

score 1 · Answer 1 · answered Nov 19 '18 at 09:56

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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.

answered Nov 19 '18 at 09:56

Qmechanic

201,751

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3D abelian Chern-Simons theory $S[A] =\int !A\wedge \mathrm{d}A$ is another simple example. – Qmechanic Nov 19 '18 at 10:20
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...and any Lagrangian of the form $L(q_1,q_2)=f(\dot q_1-\dot q_2,q_1-q_2)$, with $\delta q_i(t)=\varepsilon(t)$. – AccidentalFourierTransform Nov 20 '18 at 00:22
$\uparrow$ Right. – Qmechanic Nov 20 '18 at 00:41

Simplest example of a gauge theory?

1 Answers1