# Maxwell Lagrangian

1. Nov 14, 2012

### Petraa

Hello,

Where can I find a good explanation (book) of the derivation via Noether's theorem of the three momentum and angular momentum operators of the usual maxwell lagrangian ?

Thank you!

2. Nov 14, 2012

### dextercioby

This is standard QFT (actually QED) material, any thorough book should have it. Check out a nice treatment in Chapter 2 of F. Gross' "Relativistic Quantum Mechanics and Field Theory", Wiley, 1999.

In purely classical context (no operators), advanced electrodynamics books should also have this.

3. Nov 14, 2012

### Petraa

I've been watching the book and yes, the book treats it but don't deduce them. He just announces and perform some calculations with them

4. Nov 14, 2012

### dextercioby

Can you calculate $T^{\mu\nu}$ and $M^{\lambda}_{~~\mu\nu}$ from the Lagrangian and the general Noether formula which for the energy-momentum 4 tensor reads:

$T^{\mu}_{~~\nu}$ = ($\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}A_{\rho})}$ $-\mathcal{L}\delta^{\mu}_{\lambda}$) X $\frac{\partial x'^{\lambda}}{\partial\epsilon^{\nu}}$,

where

$$x'^{\mu} = x^{\mu} + \epsilon^{\mu}$$

Last edited: Nov 14, 2012
5. Nov 14, 2012

### Petraa

I'll try it.

6. Nov 14, 2012

### Petraa

$$T^{\mu\nu}=-F^{\mu\nu}\partial^{\nu}A_{\rho}+\frac{1}{4}F^{2}g^{\mu\nu}$$

And now? How I relate this to the momentum and total angular momentum operators ?

7. Nov 14, 2012

### dextercioby

The momentum should be $T^{0i}$, just like energy is $T^{00}$. For angular momentum, you should derive the general formula using the linearized version of a general Lorentz transformation (i.e. a linearized space-time rotation):

x'μ=xμμ ν xν, where

ϵμν = - ϵνμ

A minor change

Tμν=−FμρνAρ+1/4 F2gμν

8. Nov 20, 2012

### Meir Achuz

I too would be interested in seeing this for EM angular momentum.
Every place, I have looked seems to use the result in some form without actually deriving it.

9. Nov 25, 2012

### LayMuon

Maggiore "Modern introduction in QFT"