Physical significance of gauge invariance

LastOneStanding

I am well aware of how Yang-Mills theory works, which I think was pretty clear from my post since I described that exact process.

samalkhaiat

Quote by LastOneStanding

I am well aware of how Yang-Mills theory works, which I think was pretty clear from my post since I described that exact process.

Then you should also aware of the fact that the gauge bosons (and their interactions) arise naturally from the local gauge principle with no extra input. For theoreticians, this is a BIG DEAL and fundamental.

LastOneStanding

Did you even read my earlier comment I was referring to? It certainly doesn't appear that you did.

the_pulp

Hi, I asked something like this in another thread and the idea I got from that was that if you want a Conservation Law you need a Global Symmetry in the Lagrangian. If you want that Conservation Law to be Local you need that Symmetry to be a Gauge Symmetry. Since this idea has not been precisely state in this thread, Id like to know if you agree with this or if you think that Im wrong.

Right now, this is my understanding about the Physical Interpretation of Gauge Invariance.

dextercioby

Quote by LastOneStanding

[...]Maybe it's splitting hairs, but I wouldn't say, "We have gauge theories, therefore interactions," even if it's a formally valid logical statement. I would say, "We have interacting theories, and they are gauge invariant." [...]

Not excluding general relativity*, all interacting field theories are obtained by starting with free gauge theories and free gauge-less theories. So <we have gauge theories, therefore interactions> is the only valid judgement.

*As a field theory, GR can be derived from gauging the global Lorentz symmetry of a flat-spacetime (see post 24) or by consistent self-couplings of a spin 2 field again on flat space-time (see Bill's post 23 and my comment in post 25 to Bill's post).

Bill_K

*As a field theory, GR can be derived from gauging the global Lorentz symmetry of a flat-spacetime

Actually, what is gauged in the case of General Relativity is not the Lorentz group but the translation group. An infinitesimal coordinate transformation is a position-dependent translation

x^μ → x^μ + ξ^μ(x)

under which the gravitational field undergoes the gauge transformation

h_μν → h_μν + ξ_μ,ν + ξ_ν,μ

dextercioby

Hi Bill,

GR comes from gauging the Lorentz group antisymmetric infinitesimal generators as shown by Utyiama in 1956 (Invariant Theoretical Interpretation of Interaction, Phys.Rev, Vol.101, No.5, page 1597).

It's true that some of his arguments were a little <by hand>, as advocated by Kibble on first page of his article in 1961 (Loreritz Invariance and the Gravitational Field, J. Math. Phys., Vol.2, No.2, page 212), but nonetheless, under reasonable assumptions one can reach GR as a field theory by gauging the Lorentz group generators.

Bill_K

dextercioby, I'm not familiar with Utiyama's work - is it really about the same thing? I did find this review article on arxiv, which mentions it, and has this to say:

He [Utiyama] gauged the Lorentz group SO(1, 3), inter alia. Using some ad hoc assumptions, like the postulate of the symmetry of the connection, he was able to recover GR. This procedure is not completely satisfactory, as is also obvious from the fact that the conserved current, linked to the Lorentz group, is the angular momentum current. And this current alone cannot represent the source of gravity. Accordingly, it was soon pointed out by Sciama and Kibble (1961) that it is really the Poincare' group R4 ⊃× SO(1, 3), the semi-direct product of the translation and the Lorentz group, which underlies gravity. They found a slight generalization of GR, the so-called Einstein-Cartan theory (EC), which relates – in an Einsteinian manner – the mass-energy of matter to the curvature and – in a novel way – the material spin to the torsion of spacetime.

So are they talking about Einstein-Cartan theory instead of GR??

dextercioby

Bill, by merely assuming you have the massless free spin 2 field (h is the linearized metric (linearized perturbation of the g or the Pauli-Fierz field), you already have a gauge symmetry, as can be proven from the representation theory of the Poincaré group (like in the case of a massless spin 1 field, as discussed in Weinberg's book and presented here several times by vanhees71).

Utiyama's idea was different, namely assume there was a matter field in a space-time in which the Lorentz symmetry was gauged (hence the need to introduce both global and local coordinated, i.e. tetrad fields) and from here obtaining a fully covariant theory of interaction between matter and the gravitational field. In a sense, Utiyama was/is the grandfather of supergravity theories of the late 70's and beginning 80's.

I think your assertion <Actually, what is gauged in the case of General Relativity is not the Lorentz group but the translation group> is not quite right, because the local/linear/1st order limit of diffeomorphisms is not a space-time translation, but a space-time roto-translation, i.e. a Poincaré transformation.

dextercioby

Quote by Bill_K

[...]So are they talking about Einstein-Cartan theory instead of GR??

Yes, Poincaré gauge theory as is known today is linked to the Einstein-Cartan gravity, that is the necessary modification/re-formulation/enhancement to known General Relativity to accomodate all sorts of matter fields, in particular spinorial matter fields.

LastOneStanding

Quote by the_pulp

Hi, I asked something like this in another thread and the idea I got from that was that if you want a Conservation Law you need a Global Symmetry in the Lagrangian. If you want that Conservation Law to be Local you need that Symmetry to be a Gauge Symmetry. Since this idea has not been precisely state in this thread, Id like to know if you agree with this or if you think that Im wrong.

No, that's not correct. Gauge symmetries do not give conservation laws (subject to the caveats already discussed earlier in this thread). A global symmetry will gives a conservation law that is local, which implies global as well. This is true because local conservation of X means, "The change in X in some volume is equal to the amount of X flowing out of the boundary of that volume." So if you make the volume all of space, there is no where for X to flow out of (with a few mathematical caveats) and so X is also conserved globally. So, local conservation implies global conservation and so both follow from a global symmetry.

andrien

GR can be derived from gauging the global Lorentz symmetry of a flat-spacetime

I cannot see it.Can you be more specific.thanks.

dextercioby

Quote by andrien

I cannot see it.Can you be more specific.thanks.

I can only invite you to read the posting #26 of this thread (the second paragraph describes by understanding of Utiyama's work) and Utiyama's article referenced above.

the_pulp

No, that's not correct. Gauge symmetries do not give conservation laws (subject to the caveats already discussed earlier in this thread). A global symmetry will gives a conservation law that is local, which implies global as well. This is true because local conservation of X means, "The change in X in some volume is equal to the amount of X flowing out of the boundary of that volume." So if you make the volume all of space, there is no where for X to flow out of (with a few mathematical caveats) and so X is also conserved globally. So, local conservation implies global conservation and so both follow from a global symmetry.

Ok, but if Im not wrong, what you are saying is not coherent to what was said in this other thread (the one I was referring to) http://www.physicsforums.com/showthr...ighlight=Gauge

Do you know what is going on?

Thanks

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LastOneStanding	Physical significance of gauge invariance I am well aware of how Yang-Mills theory works, which I think was pretty clear from my post since I described that exact process.