Re: gauge theory (alternative descriptions)

On Apr 26, 9:26 am, r_n_t...@xxxxxxxxx wrote:
phi, A, E, B are Lie vector valued;

what does this mean?

That its components are not numeric but reside in a Lie algebra.

Hence, if (Y_a: a=1,...,G) is the basis of the underlying Lie algebra,
where G is its dimension, then phi = phi^a Y_a, A = A^a Y_a, E = E^a
Y_a, B = B^a Y_a; where each B^a, E^a, A^a is, itself, a 3-vector. The
x-component of B, for instance is the Lie vector B^x = B^{ax} Y_a,
where B^{ax} is the x-component of B^a.

The fields Maxwell terms "electric displacement" (D) and "magnetic
strength" (H) are, in contrast, dual Lie vectors. So the Lie index
goes in the downstairs position, D_a, H_a. Likewise for the charge
e_a. All this comes out of the Lorentz law. To contract e with E and B
in e (E + v x B), you need e's index downstairs, while E and B's are
upstairs. To make Gauss' law consistent (div D = e), you need D's
index downstairs, like e's; similarly for the Maxwell equation (curl H
- dD/dt = J).

Then you can see the true significance of the oft-neglected Lorentz
relation D = epsilon E: epsilon is a Lie algebra metric, up to
proportionality. It's lowering the index.

What's the "underlying Lie algebra" here? Let's pick the strong
force gauge with su(3) as the internal symmetry algebra. If "A"
is in su(3), then I can see that AxA would be in the enveloping
algebra of su(3),

You've just answered your own question here.

but curl A involves differentials, so it has to be in something bigger, right?

No. The a-component of curl A is the curl of A^a: (curl A)^a = curl
(A^a).

In the language of differential forms, the x,y,z components of A fit
together with phi into the 1-form:
A_x dx + A_y dy + A_z dz - phi dt.
The differential of this is just
d(A_x dx + A_y dy + A_z dz - phi dt)
= (dA_y/dx - dA_x/dy) dx^dy - (dA_x/dt + d(phi)/dx) dx^dt
+ (dA_z/dy - dA_y/dz) dy^dz - (dA_y/dt + d(phi)/dy) dy^dt
+ (dA_x/dz - dA_z/dx) dz^dx - (dA_z/dt + d(phi)/dz) dz^dt.
In 3-vector form, using the notation
dr = (dx,dy,dz); dS = (dy^dz,dz^dx,dx^dy)
this becomes
d(A.dr - phi dt) = (curl A).dS - (dA/dt + grad phi).dr^dt.

You can work out the quadratic terms, likewise, as per your indication
above. The z-component of AxA, for instance, is
(AxA)^z = A_x A_y - A_y A_x = [A_x, A_y].
The second equality is where we exploit the fact that we're working in
the covering algebra.

I've run accross this site before. A lot of interesting (and well
written) material there. What's the story behind the "federation"?
..just curious.

It was supposed to be the research archive site adjoining the non-
fiction book trilogy "The Federation Series", but has gotten a bit
larger. Part of that is that the 2nd book, "The Fourth Wave" was going
to delve a little into future physics; which is quite understandable,
given what the central theme of the book is

http://federation.g3z.com/FedSeries/FourthWave.htm

Of course, it's difficult to accurately say what physics is going to
be in the future ... um ... without making it happen, as a
consequence! But either way whether that happens or not: of necessity,
a description of future physics removed from the zeitgeist of the
present and past entails a significant point of departure from the
cliche'd points of view prevalent in the 20th and early 21st
centuries; and a point of departure even from the cliche'd accounts
that is commonly given of what future physics is going to be. Hence,
the dissonance you might pick up reading the material is by design.

To answer your question directly: the third book forecasts the rise of
the World Federation, which might be considered also as a culmination
of Einstein's *other* dream. He was well-known as a staunch advocate
of worldwide political unification and probably on his account (in
part) several states in the US actually passed resolutions during his
later life calling on such unification.

Both the archive and series are precursors to a bona fide interactive
blog site dedicated to these issues and to the latest research in
advanced and theoretical math, physics, computer science and other
fields. In turn, the blog is a precursor (still a long ways off) to a
foundation that will lie at the root of what will become known as the
"World Federation Organization". In a way, this organization will be a
continuation of what had been formerly known as the World Federalist
Society (if memory serves me correctly), which Einstein was a member
of and (I believe) even the leader of for a while.

The only affiliation I list in the set of papers I actually publish,
for now (a non-empty set), is just "The Federation Archive" and its
web address under my name. The personal stuff is going to be moved off
to the side under a "personal profile" page, once the site opens up.
There will probably be multiple people accessing and using it in a
similar way with their own profile pages.

The invitation will be tendered to some of the best and the brightest
to sign on and contribute when that time comes. But the research part
of the site is meant to be neither a substitute for nor clone of
either the Wikipedia or ArXiv.

.