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

  • Thread starter binbagsss
  • Start date
1,182
8
and when are they the same thing?
In quite simple terms.


Many thanks
 
9
14
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".
 

vanhees71

Science Advisor
Insights Author
Gold Member
13,450
5,349
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).
 
9
14
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.
 

Want to reply to this thread?

"What is the difference between gauge potential and gauge connection?" You must log in or register to reply here.

Related Threads for: What is the difference between gauge potential and gauge connection?

Replies
6
Views
6K
Replies
2
Views
3K
Replies
20
Views
1K
Replies
11
Views
4K
Replies
1
Views
2K
Replies
4
Views
10K
  • Posted
Replies
3
Views
1K
Top