Re: gauge theory (alternative descriptions)

On Apr 18, 9:25 am, r_n_t...@xxxxxxxxx wrote:
Anyway u(1) gauge is probably the simplest example. It would be
good to find a reference where a non-abelian gauge is treated
along the lines of the above example...

Here: phi, A, E, B are Lie vector valued; D, H, J, rho are Lie co-
vector valued. The permittivity epsilon is the gauge group metric
(actually: epsilon c is).

In a universal enveloping algebra of the underlying Lie algebra
B = curl A + A x A
E = -grad phi - dA/dt + A phi - phi A
div B + A.B - B.A = 0
curl E + AxE + ExA + dB/dt + phi B - B phi = 0

Extending the universal covering algebra to the dual space of the Lie
algebra
div D + A.D - D.A = rho
curl H + AxH + HxA - dD/dt + phi D - D phi = J
div J + A.J - J.A + d(rho)/dt + rho phi - phi rho = D.E - E.D + B.H -
H.B.

The Lorentz force law
Force density = rho E + J x B = rho_a E^a + J_a x B^a.
The law for power density = J.E = J_a.E^a.

For a test charge, the force F and power P are
dp/dt = F = e (E + v x B) = e_a (E^a + v x B^a)
dT/dt = P = e (v.E) = e_a (v.E^a)
where the charge (e_a) is a Lie co-vector and p, T represent the
momentum and kinetic energy of the charge.

In first-order form:
d(p + eA)/dt = -div (e(phi - v.A))
d(T + e phi)/dt = d/dt (e(phi - v.A))
where the d/dt on the right and the "div" on the right are taken as
partial derivatives only with respect to the field coordinates in phi
and A (i.e., derivatives taken with v and e constant).

The first order and second order form are consistent only if the
charge is endowed with a time variability, itself, given by
de/dt = e(phi - v.A) - (phi - v.A)e.
The precession is the classical underpinning to "flavor-changing"
interactions.

The most general Lorentz-invariant relations in 4-D (making the Lie
algebra index explicit):
D_a = epsilon_{ab} E^b + theta_{ab} B^b
+ k_{abc} E^b x B^c + 1/2 l_{abc} (E^b x E^c - c^2 B^b x B^c)
H_a = epsilon_{ab} c^2 B^b - theta_{ab} E^b
+ 1/2 k_{abc} (E^b x E^c - c^2 B^b x B^c) + c^2 l_{abc} E^b x B^c
where epsilon, theta are symmetric and k and l are completely anti-
symmetric.

For Yang-Mills fields with a simple Lie group, theta = 0 and
epsilon_{ab} c = K_{ab}/g^2 where K is the Killing metric and g the
coupling coefficient. With a semi-simple Lie group, epsilon is the sum
of the respective contributions from each simple subgroup (e.g. for
U(1) x SU(2) x SU(3) the metric is defined by 3 "coupling
coefficients").

The coupling coefficients, the gauge group metric, and the
permittivity are all synonymous with one another.

For the SU(3) sector theta is taken to be a non-zero multiple of the
SU(3) Killing metric in the standard model.

Nobody says anything about k_{abc} or l_{abc}.

If one takes the permeability to be the coefficients corresponding to
the inverse relation (E,B) <- (D,H) then in the presence of theta,
(epsilon mu) as a matrix product will be smaller than 1/c^2. Things
get more complicated if k and l are present.

Stress tensors can be written down in a form that is independent of
what the constitutive law (and Lagrangian) may be ... so it can be
defined for Yang-Mills gauge fields or more general gauge fields. The
total energy for a static point-like source -- on the assumption that
the stress tensor is Lorentz covariant and the energy is well-defined
-- can be written down in closed form ... independently of what the
Lagrangian is. This result is not well-known ... and (in fact) not
"known" at all (yet).

If the Lagrangian is homogeneous to the first degree in the quadratic
Lorentz field invariants and independent of the cubic Lorentz field
invariants, then the energy is the limiting value of 1/2 (q(r) phi(r))
as r -> 0, where q(r) is the flux of the D field. Otherwise, if I
recall correctly, it's the limiting value of (q(r) phi(r)) as r -> 0
minus the Lagrangian, itself. I'll have the check my notes on this.
But the argument is simple (and almost trivial!)

For the gauge field corresponding to gravity the canonical stress
tensor is non-zero, while the symmetrized stress tensor is 0. The
difference between the two ... in all cases for all gauge fields ...
is related to the divergence of the angular momentum tensor.

References to articles I have on-line:

The Gauge Field Equations in Maxwell Form
http://federation.g3z.com/Physics/index.htm#MaxwellYangMills

The Maxwell Equations for Non-Abelian Gauge Fields
http://federation.g3z.com/Physics/index.htm#Hehl2

The Anatomy of the Electroweak Field
Supplementary article included under:
http://federation.g3z.com/Physics/index.htm#Hehl1

The Gauge-Scalar Fields in Maxwell Form
http://federation.g3z.com/Physics/index.htm#GaugeScalar
(This one is going to be expanded)

The Constitutive Law in Gauge Theory
http://federation.g3z.com/Physics/index.htm#Constitutive
(What replaces/generalizes the Lorentz relations?)

.

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