Maxwell's equations from U(1) symmetry

[Thoros]

I understand that one is able to derive the inhomogenuous pair of Maxwell's equations from varying the field strength tensor Lagrangian.

Now implying the U(1) gauge invariance, how is one led to the Maxwell's equations?

 

[dextercioby]

I understand that one is able to derive the inhomogenuous pair of Maxwell's equations from varying the field strength tensor Lagrangian [...]

No, the homogenous pair results that way. To get the inhomogenous ones you need to add in a term for coupling to (charged) matter.

[...] Now implying the U(1) gauge invariance, how is one led to the Maxwell's equations?

From a geometric perspective, U(1) is embedded in the field.

 

[Thoros]

Sorry about that, you are right about the lagrangian.

From a geometric perspective, U(1) is embedded in the field.

Could you give me a few hints on how to do it explicitly or is this already enough to manage with the task of deriving the equations?

Thanks.

 

[jtbell]

The Wikipedia article on gauge theory gives a short outline of the process:

http://en.wikipedia.org/wiki/Gauge_theory#An_example:_Electrodynamics

 

[Thoros]

Alright so i reached the point where you get an interaction term in the lagrangian density leading to the inhomogenuous pair of Maxwell's equations.

But to me the intrudiction of a covariant derivative is a little confusing. It seems perfectly reasonable to require that physics stay the same under U(1) symmetry. But adding the vector potential and simply implying an ad hoc transformation rule seems unjustifiable. Also, why do we have to add the elementary charge and planks constant to it? Is this required from dimensional analysis?

And from the Wikipedia article i understand, that the homogenuous pair still only arises from the same old field tensor Lagrangian.

Can anyone clear it up for me?

Thanks.

 

[Avodyne]

It seems perfectly reasonable to require that physics stay the same under U(1) symmetry. But adding the vector potential and simply implying an ad hoc transformation rule seems unjustifiable.

It is only justified by the fact that we end up with a theory that agrees with experiment. If gauge theories did not agree with experiment, only mathematicians (and not physicists) would care about them.

You cannot derive Maxwell's equations (or any other physical theory) from pure logic. All you can do is find out which theories agree with experiment, and then subsequently notice any cool mathematical features that they might happen to have (such as gauge invariance).

Actually, that's a little too strong. Quantum electrodynamics is an abelian gauge theory, and by 1950 or so it was well established that it agreed with experiment. This success motivated theoretical investigations of nonabelian gauge theories, which ultimately turned out to be relevant for the description of both strong nuclear and weak nuclear interactions (though it took a long time and travel down several wrong roads before the precise connection was understood).