Re: gauge theory (alternative descriptions)

On May 7, 6:00 pm, r_n_t...@xxxxxxxxx wrote:
For each "a", B^{ax} is a function R^4 -> C.
Let's call the ring of such functions "X1".
etc.

You almost sound like you want to graft a theory of types on top of al
this, a' la a typed programming language or typed logic. OK.

As a space-time function, B^{ax}: M -> R, where M is the spacetime
manifold (which can be treated as R^4 when M is Minkowski space
adapted to Cartesian coordinates). X1 = E(M,R) = E(M) where E(M,N)
denotes the space of C^{infinity{ functions from manifold M to N. It's
possible to expand the treatment to allow for singular functions,
embedding E(M) into a larger space, but that's a digression here.

For each component, B^x: M -> L = Lie(G), where G is the gauge group
and L its Lie algebra. So, for G = SU(3), L = su(3). So, your X2 is
the tensor product E(M) x L of the algebras E(M) and L.

The 3-vector B: M -> L x R^3. So, X3 is E(M) x L x R^3.

The thing you're asking for is an amalgamation of the 2 algebras. This
is gotten from treating the 1-form and 2-form, instead. The potential
1-form in Minkowski space in Minkowski coordinates is
a = (A_x dx + A_y dy + A_z dz - phi dt).
As a function it maps from M to L x Lambda^1(M), where Lambda^1(M) is
the space of 1-forms over the manifold M. The 2-form
f = (E_x dx + E_y dy + E_z dz)^dt
+ (B_x dy^dz + B_y dz^dx + B_z dx^dy)
maps from M to L x Lambda^2(M). So they are, respectively, in
a: E(M) x L x Lambda^1(M)
f: E(M) x L x Lambda^2(M).

The amalgamation you seek out is the simple algebraic relation f = da
+ a^2, and its corresponding Bianchi identity df + af - fa = 0.

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"

All of what you're trying to get at here, and later in your reply, are
subsumed by the considerations just raised.

But, this is actually NOT how the field is formalized mathematically.
What's actually done is to extend the manifold M into a fibre bundle
whose fibre is the group G. The fields are sections over the various
spaces associated with the bundle, not merely functions.

This treatment is not equivalent to one expressed merely in terms of
function spaces, but strictly more general (and robust). It can even
survive the transition to discrete or non-commutative geometries
mostly intact.

.

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