# A What is the difference between gauge potential and gauge connection?

#### binbagsss

and when are they the same thing?
In quite simple terms.

Many thanks

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#### vis_insita

I would say a "gauge connection" is simply a connection on a principal fiber bundle, i.e. a Lie algebra-valued 1-form $\omega$ that maps fundamental vector fields to their generators, and transforms as $R^\star_g\omega = ad(g^{-1})\omega$ under the right-actions of the structure group (the "gauge group" in physicist's terminology). (But I'm not sure I heard the term "gauge connection" before, so maybe something else is meant?) A gauge potential is what you get if you pull $\omega$ back to the base space via a local section. In other words a gauge potential is defined on spacetime with respect to a local "choice of gauge". So, both are closely related, but different concepts.

Unfortunately it is difficult to explain all this without introducing a lot of jargon first. A book I like, that explains all this in quite simple terms is David Bleecker, "Gauge Theory and Variational Principles".

• dextercioby

#### vanhees71

Gold Member
I think it's the same thing. It's just discussed in different language by theoretical and mathematical physicists. For a physicist the gauge potential is introduced to extend a partial derivative to a derivative covariant under gauge transformations, and this leads him to introduce a gauge potential in the derivative,
$$\partial_{\mu} \rightarrow \partial_{\mu} + \mathrm{i} g \mathcal{A}_{\mu},$$
where $\mathcal{A}_{\mu}$ is a Lie-algebra valued vector field.

From the mathematical point of view this introduces an affine connection on the fiber bundle with the $\mathcal{A}_{\mu}$ the connection coefficients (like the Christoffel symbols in affine differentiable manifolds in differential geometry).

#### vis_insita

I believe what mathematicians call a "connection on a principal bundle" is the object $\omega$ I defined above, which lives on the total space of the bundle. Physicists usually only discuss the pull-back $\sigma^\star\omega$ to the base space ($\sigma$ a local section), which in your notation would be $\sigma^\star\omega = \mathcal{A}_\mu \mathrm{d}x^\mu$, and which posesses the characteristic transformation law of a gauge potential

$$\mathcal{A}'_\mu \mapsto g(\mathcal{A}_\mu + \partial_\mu)g^{-1}$$

under "changes of gauge" $\sigma(x) \mapsto \sigma(x)g(x)$.

Those are not exactly the same, but really very closely related.

• vanhees71