On Aug 7, 2011, at 1:20 PM, Paul Zielinski wrote:
There no is question that Kibble, building on the work of Utiyama, did this formally in his 1961 paper. There is no point in denying that. But there is a trick here, having something to do I suspect with the construction of the covariant derivative.
Show us. The covariant derivative is the choice of the connection field that you keep confounding with the gauge transformations because you have not gone through the algebra of the local gauging program with sufficient attention to detail.
I'm not sure why you think this Jack. The gauge field of any gauge theory is essentially a connection field.
You are having trouble parsing my sentences again.
I am?
What don't you get in my
"The covariant derivative is the choice of the connection field that you keep confounding with the gauge transformations"
Let's be explicit about this in the simplest U1 EM case.
This better be good. :-)
Using Cartan's elegant short and sweet notation
Local gauging global U1 to U1(x) induces the connection 1-form A when you require minimal coupling which means
d --> D = d + A
for now I suppress coupling strengths to keep the math simples
In classical U1 gauge theory the "connection 1-form A" is only really "induced" in the sense that it is mathematically constructed to compensate for the spurious effects of locally gauging a group of "rigid" phase transformations on the partial derivatives d.
That's what I said.
I'm not sure whether it was ever specified by Weyl, London, or the others whether such "gauge transformations" of the phase were passive or active.
Irrelevant - makes no difference to the physical reality out there. All detectors connected by a gauge transformation in GR must be locally coincident. An active gauge transformation does not take you from Earth to the Stars. That seems to be part of your misconception. You seem to think that an active gauge transformation is a real physical displacement over a perhaps cosmological distance. For that we need a stable wormhole.
So we're back to square one.
Speak for yourself.
Recall that Weyl's original scale factor was regarded as being purely mathematical in character, and yet he hoped to pull out unified EM theory like a rabbit out of a hat! This is all aside from the fact the it didn't work for other reasons.
It's the extension of the Lagrangian density of the input source field that is the physics - not the redundant gauge transformations.
That is, one starts with a Lagrangian density L(psi, psi,u) that is invariant under the global U1
because of the derivatives in the parameters of the group U1(x) ~ e^iQ@(x) we need the A and this gives a new Lagrangian density
Yes because passive phase transformations don't have any physical meaning. But if that's the case, why would locally gauged phase transformations be compensated by a *physical* field?
Why bother to argue at all? Why not just say, "Abracadabra"?
This idea eludes you. The physics is to keep the total dynamical action integral invariant under the larger local group and the only way to do that is to introduce a compensating gauge potential - a Cartan 1-form that in general is not closed.
That is, given the compensating gauge potential A Cartan 1-form, it's exterior derivative 2-form
F = dA =/= 0
note that
A = AIdx^I
I indices of the Poincare group of 1905 SR for internal symmetries in extra dimensions beyond spacetime
F = FIJdx^I/\dx^J
in U1 EM
FIJ = AI,J - AJ,I = - FJI
in Yang-Mills SU2(weak flavor gauge force), SU3(strong color gauge force) it's
FIJ^a = AI,J^a - AJ,I^a + C^abcA^bIAcJ
where a, b, c are the internal (flavor, color) charge indices
[Qb,Qc] = C^abcQa Lie algebra for charges Qa
a non-closed A 1-form is topologically different from a closed 1-form C = df
A single-valued gauge transformation can never take you from a C to an A - for that you need the Kleinert singular transformations that create and destroy topological defects.
In the case of GR
A = AI(LIF)dx^I = Au(LNIF)dx^u
AI(LIF) = (tetrad)I^uAu(LNIF)
etc
LIF & LNIF are locally coincident
i.e. their separations small compared to local radii of spacetime curvature
LIF is the "inertial frame" objects at rest in it are weightless zero g-force as in Baron Muchhausen's cannonball ride
http:// http://movieclips.com/KSYA-the-adventures-of-baron-munchausen-movie-the-cannonball-ride/
LNIF can be rotating about its center of mass that is still on a timelike geodesic
objects clamped to rotating frame feel inward centripetal reaction force akin to "weight"
e.g. artificial gravity on space-station
http://www.youtube.com/watch?v=q3oHmVhviO8
or it can have translational acceleration from a non-gravity force like a rocket engine firing
http://www.bbc.co.uk/news/science-environment-12573491
and the static LNIF is
It only makes sense from my POV if you specify *active* phase transformations and attach concrete physical meaning to relative phases at spacelike separation.
The idea is that the phases at spacelike separated events are independent because of the local gauging of the group.
Here "phases" are the conjugate variables for the Lie algebra generators of the Lie group.
For example in the Bohm-Aharonov effect the phase shift of the electron's quantum wave function connecting two events is the path-dependent integral of AIdx^I - not the best example because that path is timelike for a real electron. Therefore, there is no unique meaning to the relative phase between any two separated points in the general case unless A is exact
i.e. if A = df
f = 0-form
then the path integral of A is independent of path because of generalized Stokes theorem.
That's the only case where you can define your relative phase.
Now I suppose formally we can use a spacelike path - interesting question.
I think this may be the "original sin" of gauge theory.
Not even wrong in my opinion.
L'(phi, phi,u A) whose global action integral is now gauge invariant under the bigger local group U1(x) with subgroup U1.
Fine.
All local physical observables must be gauge invariant.
So mathematically you need the connection field A to ensure this.
That's what I said.
The 1-1 non-singular topology conserving redundant gauge transformations that do not change any physical reality out there, but simply represent raw data collected by different sets of detectors are
A ---> A' = A + df
where f is a nonsingular 0-form
It's one thing to say that the gauge transformations have no physical meaning, and quite another to say that they leave all local observables invariant.
So what?
In the context of gauge gravity, why can't the connection A be interpreted as simply compensating for the artifacts associated with frame acceleration?
That's what I have been trying to put across to you. The 16 tetrad components connect locally coincident non-accelerating inertial frames with accelerating/rotating non-inertial ones.
Wouldn't that be more direct and natural?
Yes, that's what I have been saying.
Using d^2 = 0 in the absence of singularities (see Kleinert's multi-valued extension where in effect d^2 =/= 0 i.e. a real topodynamical change from mass-energy-stress currrent densities)
F = dA ---> F' = dA' = dA + d^2f = dA = F
so F' = F is a gauge invariant.
OK, but gauge theory interprets A as a *physical field*. It is not treated as a mere connection field.
Correct.
If the gauge transformations have no physical significance, why should compensation of the effects of such transformations on the Lagrangian to restore local gauge invariance give rise to a *physical* field, as opposed to a mere mathematical artifice, which is what a connection really is?
The gauge transformations in GR do have physical meaning, but they do not change the dynamics of the gravity field. Each locally coincident LIF and LNIF makes measurements of the gravity field, e.g. the 4th rank curvature tensor and the theory allows them to construct local gauge invariants from them just like in special relativity.
The gauge transformations connect the shadows on the wall of Plato's Cave for the same Eternal Form Invariant Idea of Objective Reality Out There.
The set of all non-singular f's is the gauge orbit of all physically equivalent solutions that look different to different sets of locally coincident detectors, but the theory allows each set to compute the gauge invariant F's - defining objective reality out there.
Start there, have not read the rest of what you wrote because we are not yet off home plate.
Jack, I feel that you are missing the point here. I have no problem with the mathematical procedure of locally gauging U1 -> U1(x) and then introducing the connection field A to compensate, ensuring local gauge invariance. My problem is with the precise mathematical and physical interpretations of A, and how they are justified.
I have made that very clear in the case of GR where all local frames are coincident detectors solving Einstein's hole paradox.
1) Lorentz group SO1,3 is LIF(Alice) <---> LIF(Bob)
2) T4(x) is LNIF(Carol) <---> LNIF(Ted)
3) Tetrads are all four combinations:
LIF(Alice) <---> LNIF(Ted)
LIF(Alice) <---> LNIF(Carol)
LIF(Bob) <---> LNIF(Ted)
LIF(Bob) <---> LNIF(Carol)
However, in contrast, in the case of internal symmetries A's meaning is in the quantum Bohm-Aharonov effect fringe shifts, but unlike GR there is no direct physical meaning to different "frames" connected by internal gauge transformations that I know of for Bob, Carol, Ted and Alice. Maybe Nick Herbert can think of one? ;-)