Re: gauge theory (alternative descriptions)

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.

Let me linger on this point just a little bit longer.
For each "a", B^{ax} is a function R^4 -> C.
Let's call the ring of such functions "X1".
The lie algebra we're dealing with is over "X1",
let's move to it's univeral enveloping algebra and call that "X2";
phi = phi^a Y_a and B^x = B^{ax} Y_a, so both are *in* X2;
(calling B^x a Lie vector is a little confusing).
B=(B^x,B^y,B^z) is in a 3-dim vector space over "X2"; let's call that
"X3".
each B^a is also in "X3".

[interesting comments on electric displacement,contraction,...
deleted]
we can revsist this later.

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've jumped disciplines here. The properties of differentiation
haven't been introduced yet in this setting. You have
"curl" : a map "X3" -> "X3"
"d/dt" : a map "X3" -> "X3"
"grad" : a map "X2" -> "X3"

obviously these maps are not *in* "X2" so we're dealing with something
bigger!
component wise all these can be built with the 4 maps :
"d/dx" : "X2" -> "X2"
"d/dy" : "X2" -> "X2"
"d/dz" : "X2" -> "X2"
"d/dt" : "X2" -> "X2"

The "Y_a"'s are also maps "X2"->"X2". It is combining these G+4 in an
alternative way that I've been looking for.

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.

the quadratic part is ok.
A_x and A_y are in in "X2" so A_x A_y and A_y A_x are in "X2" too.

[interesting material on the federation deleted]

...dissonance you might pick up reading the material is by design.

aaah...that's the best part!



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