added pointer to the early discussion of
and a scan of their Table I (here)
]]>added publication data to:
added pointer to
added pointer to
Will give this book a page of its own…
]]>added some doi-s to the references and brought them into chronological order
]]>added pointer to the exposition
(updated on the arXiv this week) here and elsewhere
]]>I have added a section Basic Idea of the Definition of Fiber Bundles prompted by this PO question
]]>Jim, if you allow then I’ll drop out of this discussion. If you go back through our exchange I think it is hard (or else unfair) to argue that I stated anything which is not standard.
]]>Not sure, Jim, where we are miscommunicating. A -gauge field is a -principal connection and every -principal connection has an underlying -principal bundle. That’s all. It’s a standard fact. (You have co-authored various articles involving this fact, if I may say so.)
]]>a G-gauge field on some space X is itself a G-principal bundle over X equipped with G-principal connection
I’m lost already - granted that, I would go on without protest
But yes, that’s what it is. What physicists call -gauge fields are mathematically -principal connections. What they call the “instanton sector” of the gauge field is mathematically the class of the underlying -principal bundle.
how do you write a gauge field in an action?
For a -gauge field for a semisimple Lie group with Killing form , the canonical action functional on it (the “Yang-Mills action functional”) is the integral over Riemannian base space of the curvature 2-form wedged with its Hodge dual
Moreover, for a path in , thought of as the trajectory of a charged particle, then the interaction functional is the holonomy of the connection along this path
]]>Hey Jim,
let’s get on the same page here, there are two levels of bundles here:
a -gauge field on some space is itself a -principal bundle over equipped with -principal connection;
A gauge transformation is an isomorphism of such -principal bundles with -principal connections over .
Now what is it that a gauge field over is itself a section of? This must be some “second order” bundle over , namely something such that a section of this second order bundle is itself an ordinary bundle (with connection).
Such “second order bundles” which are such that its sections are themselves ordinary -principal bundles are called -gerbes . Or -fiber 2-bundles, to have a more conceptual term.
So if we forget the connections for a moment then:
the “field 2-bundle” for -gauge fields on is the 2-bundle , hence the trivial -gerbe on ;
a section of the “field 2-bundle” is equivalently a map and that is equivalently an ordinary -principal bundle on .
Imho, I think it is a disservice to say a bundle IS a field.
It is a basic fact that a gauge field is a bundle equipped with a connection. For discrete gauge theory the connection disappears and then a gauge field is precisely a bundle. This fact was very much amplified back then by Freed-Quinn 93 (if it takes appeal to third party auhority) Since it is a true fact, I can’t see why it would be a disservice in a scientific context to state it.
Bundles are the global structure in whihc physical fields live.
Not gauge fields. Gauge fields are sections of gerbes. I know that this is not widely advertized fact (due to the focus on perturbation theory), but it is easily seen to be true.
]]>Started an entry in “category:motivation” on fiber bundles in physics.
(prompted by this Physics.SE question)
]]>