Talk:Gauge covariant derivative

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Isn't it so that the gauge covariant derivative and the covariant derivative of GR are a partition of the mathematical covariant derivative. In GR the vector bundle is the tangent bundle and the gauge group is GL(4, R), which could be reduced to SO(3, 1). The gauge cov. der. is for all vector bundles which are not the tangent bundle. Right? --MarSch 10:50, 21 October 2005 (UTC)

The term covariant derivative is often used to describe connections on other vector bundles as well. However, this is not the agreed upon usage for Wikipedia because covariant derivative has a very precise meaning in physics. Instead, the intrinsic description of a "gauge" covariant derivative can be found in the article Koszul connection (which is fairly standard nomenclature between mathematicians and physicists). Silly rabbit 15:48, 27 June 2006 (UTC)

[edit] Request

Hi linas,

Is there any way you would be willing to write something up on a gauge connection? Right now that page is just a redirect to gauge theory. A gauge connection (unless I'm mistaken) is like a connection form adapted to a particular gauge. It then behaves in a special way under gauge transformations. I know this can be viewed in the language of principal bundles to a certain extent. But I don't think that's the way physicists (or many mathematicians) view them.

The "functors":

(Gauge connection) + (gauge fields) -> (gauge covariant derivative)

(G-principal connection form) + (representation of G) -> (covariant derivative)

are "naturally equivalent".

Cheers, Silly rabbit 15:40, 27 June 2006 (UTC)

Hi, yes they are, and speaking from experience, one of the stumbling blocks is to match up the vocabulary and notation used in the physics gauge theory texts with the notation prefered by geometers (Cartan connection, connection form). I presumed most practicing physicists working in this area eventually learn both, and I understand why geometers don't like (and don't learn) the physics notation: its "dirty", suitable only if you actually have to do a calculation, not terribly intuitive, ugly if you have to state a general principle.

I'll have to think about how to present the two (or three, counting GR) notations side-by-side. The book "Gravition" by Wheeler, etal is the only text I know that does this; however its a frusrating read, its rather loose. I've been procrastinating on another promised task (for someone with a small Erdos number!, so I better get cracking!); so nothing before the 4 July weekend. linas 19:24, 27 June 2006 (UTC)

Perhaps I'm being glib. This article partially describes the physics notion, incompletely, in a slightly confused way. The physicists "gauge field" is the value of the connection on some (horizontal, of course) section of the fiber bundle. The physicists "local gauge transformation" is a description of what happens when one moves from one section to another. The word "local" is used to emphasize that the section can be choosen arbitrarily at every point (as long as its differentiable in the end). Physicists almost always assume the trivial bundle, so that sections are well-defined. A section is often chosen by "fixing a gauge", that is, the section is specified as those points on the bundle such that the connection is irrotational, or divergence free, or whatever. I still get confused by torsion but perhaps this can be an opportunity to ponder.

Hmm gauge transformation fixed gauge fixing a gauge choice of gauge section horizontal section vertical subspace choice of gauge non-Abelian gauge field gauge field field strength field curvature linas 20:00, 27 June 2006 (UTC)

I don't find that you're being glib. We need to expand connections out into the physics realm. So my tacked-on "natural equivalence" was more for the purposes of getting this kind of thinking on track. (See, it worked.) Anyway, yes "dirty" is good. Do your worst (if you decide to take up the gauntlet at all). Silly rabbit 00:36, 30 June 2006 (UTC)