Re: This Week's Finds in Mathematical Physics (Week 210)
From: Kwok Man Hui (kmhui_at_math.utexas.edu)
Date: 01/30/05
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Date: Sun, 30 Jan 2005 08:42:47 +0000 (UTC)
> John Baez wrote:
>
>
> Anyway, the real idea behind electromagnetism is that sitting over
> each point in spacetime is a U(1) torsor. If a particle is sitting at
> some point in spacetime, its phase is not really a numbers: it's an
> element of the U(1) torsor sitting over that point! To get a *number*,
> we have to carry the particle around a loop! Its phase will change when
> we do this, so we get *two* points in a U(1) torsor, and their difference
> is an element of U(1).
>
> So while it sounds far-out, the key mathematical structure in electromagnetism
> is a bunch of U(1) torsors, one over each point in spacetime. This is called
> a "principal U(1) bundle" or sometimes just a "U(1) bundle" for short.
>
> If we wanted to describe some force other than electromagnetism, we could
> take this whole setup and replace U(1) with some other group. In fact,
> this idea works great: it's the main idea behind gauge theories, which do
> an excellent job of describing all the forces in nature.
>
> To set up a gauge theory, the first thing you need to do is pick a
> group G and pick a "principal G-bundle" over spacetime. Spacetime
> will be some manifold X. A principal G-bundle over X is gadget that
> assigns a G-torsor to each point of X. A G-torsor is a space where if
> you pick two points in it, you get an element of G which describes their
> "difference".
>
> Anyway, in gauge theory the forces of nature are described by "connections"
> on principal G-bundles. Let's say we have a principal G-bundle P which
> assigns to each point x of our manifold a G-torsor P(x). Then a
> Anyway, the concept of relative phase, or difference in phase, is nicely
> captured by the concept of a "torsor". A unit complex number is a point
> on the unit circle in the complex plane. This circle is a group since
> we can multiply unit complex numbers and get unit complex numbers back.
> This group is called U(1). Like a dial, U(1) has standard names for
> all the points on it - and it has one god-given special point, the
> identity element, namely the number 1.
>
In order to develop a gauge field theory, should one have a Larangian
which is invariant under local symmetry transformations and should have
covariant derivative to define connection? Define Lie algebra-valued
forms? The whole physical point of the theory is the bosonic or fermionic
interaction picture, right?
I look up the definition of gauge field theory from the online
encyclopedia: http://en.wikipedia.org/wiki/Gauge_field_theory
Charles Hui
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