A Visual Understanding of U(1) symmetry, Charge Conservation, and Noether's Theorem

05 Jun 2009
Posted by doug

The symmetry U(1) has long been known to be represented as a unit circle in the complex plane. So when I was playing with a plane wave EM equation, I generated an image that sure enough had a unit circle in the complex plane, According to Noether's theorem, where there is a symmetry, there is a conserved current. For U(1), we call that charge. Electric charge is quantized, coming only in integral clumps.

Take another look at that circle. How many circles are there in the complex plane? Looks like only one. Because the complex plane is dynamic, one needs to integrate in time to see both the symmetry and the conservation of the circle. There either is one complete circle, or there isn't.

I can imagine a quaternion animation with 4 unit circles in the complex plane. That would mean there were 4 electric charges in the animation. Counting unit circles may be identical to counting electric charges. If this idea pans out, that would be cool. The proposal will be tested as more animations are made of EM equations.

Re: A Visual Understanding of U(1) symmetry, Charge ...

Hey Doug,

This may be a too-matos, to-matos argument, but the conserved "thing" coming from this U(1) symmetry is charge. Also, I've never thought of charge as a conserved current. Did you mean Noether's theorem predicts a conserved quantity, the electric charge?



Re: A Visual Understanding of U(1) symmetry, Charge ...

Hello Lowell:

I completely concur that the conserved thing of U(1) is electric charge. So lets look into your second sentence: "I've never thought of charge as a conserved current." An electric charge density is a scalar, call it q. Multiply that by a 4-velocity, q (\gamma, \gamma \beta_x, \gamma \beta_x, \gamma \beta_x) = (\rho, Jx, Jy, Jz). Each of those current densities must be conserved. After all, if you could kill off a bit of Jz, that would be no different from killing off familiar q. In a different reference frame, we will have different values of Jx, Jy, and Jz. Exactly the same thing happens with mass and mass current densities.

Noether's theorem does predict the conserved quantity of electric charge. Here is how I write the EM action using only real valued quaternions:

S_{EM} = scalar(\int{\sqrt{-g} d^4 x (-\frac{1}{c} J \frac{A}{|A|} - \frac{1}{4 c^2}(\nabla \frac{A}{|A|} - (\nabla \frac{A}{|A|})^*)(\frac{A}{|A|} \nabla  - (\frac{A}{|A|} \nabla)^*))} \quad eq. ~1

U(1) symmetry is taught as the unit circle in the complex plane. No one writes \frac{A}{|A|}, so why did I bother? Quaternions are just 3 complex planes sharing the same real number. So the U(1) symmetry should be apparent even in the current coupling term.

That is all the algebra. The radial idea is that we should be able to visually see both the symmetry and the thing that is conserved because it can be counted (and thus not changed). A circle has a continuous symmetry, and the number of circles can be counted. So I get to see both aspects of Noether's theorem.