gauge theory (alternative descriptions)

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Guest

Posted: Mon Apr 14, 2008 4:53 am

I am trying to understand gauge theories in an algebraic setting
(I hope to avoid arguments over what setting is "better",...).

It seems that the basic construct is a combination of two algebras :

one algebra is the commutative algebra generated by the differentials
(d1,d2,d3,d4).

the second algebra is the "internal symmetry" algebra. For the strong
force, this is "su(3)", but over what? not C, but the the ring of
functions A(x,y,z,t) -> C?

the "covariant derivative" is formed by combining the generators of
the two in what seems like a reasonable enough process. I'd like to
see if there are alternative approaches of describing this. Again
my preference is for an algebraic treatment.

Thanks and please excuse the vagueness of the post.

Stephen Blake

Posted: Thu Apr 17, 2008 4:05 am

Guest

r_n_t...@yahoo.com wrote:

Quote:

Dear r_n_t,

I would also like to understand gauge theories and so here is my
current
understanding of what is going on.

A gauge theory is one in which the state space is too big; there are
many
states which correspond to each physical situation. If G is a group
with
elements g, then the big Hilbert space carries a representation U(g).
The
group G has an invariant subgroup H called the gauge group. Since H is
an
invariant subgroup, there is a quotient group G/H. The quotient group
G/H
will show up in the representations of G on the big Hilbert space as
reps,
U(g)=gH (eqn 1).
The subspace of the big Hilbert space that transforms under the
quotient
rep (eqn 1) is spanned by states of the regular rep |gH> so that,
U(g1)|gH>=|g1gH> (eqn 2).
These states |gH> are the physical states. However, when we work in
the big Hilbert space we don't know about the subspace which
transforms
as the quotient rep; we use some states |i> which span the whole of
the big
Hilbert space. We write a state as |psi>=\sum c_i|i> and we have an
equation
of motion F(|psi>)=0 which is satisfied when |psi> lies in the
subspace
which transforms as the quotient rep.

One often reads that "a gauge group is a group of transformations that
leaves invariant every operator representing an observable." If h is
an element of the gauge group H, then the quotient rep is U(h)=H and
so,
U(h)U(g)U(h^-1)=hgh^-1H=hgH=gh1H (for some h1 in H)
and so,
U(h)U(g)U(h^-1)=gH=U(g).

Stephen
http://www.stebla.pwp.blueyonder.co.uk

Alexey Popov

Posted: Thu Apr 17, 2008 4:05 am

Guest

r_n_tsai@yahoo.com wrote:

Quote:

Necessary ingredient is external derivarive. Namely, main agrebra A
is graded commutative algebra with nilpotent derivation d of degree +1.
(this is algebra of differential form in total space of bundle).
Lie group action induces action of lie algebra on A.
This is way on which equivariant cohomology are constructed.
May be this can help you.

Guest

Posted: Fri Apr 18, 2008 4:25 am

Thanks for the responses Alexey and Stephen,

The sort of setting I'm after ideally doesn't get too intertwined
with other subjects. I'd rather not deal with groups, bundles,
cohomology, observables,....if that can be avoided.

I'll try to work out "u(1)" gauge algebra (electomagnetism) as
an example; although keep in mind that there are gaps in the
definitions that I'm not sure how to fill. Let's call this
algebra X. I don't know if X should be over a simple field
(R or C) or something more complicated (functions over R,...).
X can be "generated" by 8 elements : X=<d1,d2,d3,d4,a1,a2,a3,a4>
Here "generated" is along the lines of universal enveloping algebra
(polynomials in the 8 generators). You can think of the di's as
partial derivatives, and ai's as related to electromagnetic 4-
potential;
but that's only for motivation...strictly speaking these are just
generators that obey certain multiplication rule. What are these
rules? let the algebra multiplication be g1g2, then define a second
multiplication (commutation) on the algebra g1*g2=g1g2-g2g1. Then
I think all the rules you need are :

di*dj=0; ai*aj=0; di*aj=bij;

the first just says derivatives commute; the second is because this
is an abelian gauge; the last one is a definition of 16 elements of
the algebra; related to partials of 4-potentials. Higher derivatives
can also be treated as definitions of other elements. I think that's
it.

What can you do with this. Let's try to derive Maxwell's equations!
First define "covariant derivative" :

Di=di+ai;

these are just 4 elements in the algebra

Next define the "field strength" :

Fij=Di*Dj;

These are just 16 elements in the algebra. Because g1*g2=-g2*g1,
only 6 of these are significant; these can be identified with
the electric and magnetic field (E1,E2,E2,B1,B2,B3).

Take the identity g1*(g2*g3)+g2*(g3*g1)+g3*(g1*g2)=0 and
substitute Di's in it and with the right identification
of Fij with E and B you get \Del.E=0 and \Del x E = \partial B /
\partial t
Note that these are 4 equations relating three components at a time.

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...

Rock Brentwood

Posted: Fri Apr 25, 2008 10:22 am

Guest

On Apr 18, 9:25 am, r_n_t...@yahoo.com wrote:

Quote:

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?)

Guest

Posted: Sat Apr 26, 2008 4:26 am

I appreciate the effort that went into this detailed
response, but I think this is still the "standard"
treatment for gauge theory which I think I understand
well enough. What I'm looking for, and perhaps I'm
wrong in expecting to find any such thing, is an
"alternative" description that merges the two algebras
(the algebra of the differentials and the internal
symmetry algebta) into a single algebra.

for example :

Quote:

phi, A, E, B are Lie vector valued;

what does this mean?

Quote:

In a universal enveloping algebra of the underlying Lie algebra
B = curl A + A x A

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

Quote:

References to articles I have on-line:

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

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.

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