# Angular Momentum Conservation (Lorentz transf. + Noether Th.)

1. Oct 19, 2014

### Breo

1. The problem statement, all variables and given/known data

Prove $$j^{\mu} = j_ {EXTERIOR}^{\mu} + j_ {INTERIOR}^{\mu}$$. Writing $$j_ {EXTERIOR}^{\mu}$$ in terms of the energy-momentum tensor. Prove $$j_ {EXTERIOR}^{\mu}$$ is related to the Orbital Momentum and $$j_ {INTERIOR}^{\mu}$$ to the spin.

Sorry, for the lane shifts.

2. Relevant equations

$$x' \longrightarrow x$$
$$x \longrightarrow \Lambda^{-1}x$$

$$\Lambda = \mathbb{1} - i\epsilon I_D \implies D[\Lambda] = \mathbb{1} - i\epsilon I_R$$

This yields:

$$\phi' (x) = D[\Lambda]\phi(\Lambda^{-1}x) = \phi(x) - i\epsilon I_R\phi(x) + i\epsilon(I_Dx)^{\mu}\partial_{\mu}\phi(x)$$

So in Taylor expansion:

$$\frac {\delta \phi'}{\delta \epsilon} = -iI_R\phi + i(I_Dx)^{\mu}\partial_{\mu}\phi$$

$$\frac {\delta x'}{\delta x} = -i\epsilon I_Dx$$

The conserved current formula:

$$j^{\mu}= \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)} \frac {\delta \phi'}{\delta \epsilon} + \mathcal{L}\frac {\delta x'^{\mu}}{\delta \epsilon}$$

3. The attempt at a solution

Ok, I tried a few things. Trying to use all was given in the course notes but I am always crashing with the next:

The $$\frac {\delta x'}{\delta x} = -i\epsilon I_Dx$$ seems to be wrong for me. If I try this:

$$\frac {\delta x'}{\delta \epsilon} = -i I_Dx$$ Because this must come from the $$\Lambda^{-1}x$$ right?

And setting two indices to the I_R and I_D matrices (there are 6 for each one corresponding to the Lorentz transf. in each angle or rapidity), then I reach the expresion for the internal part, which corresponds to the spin:

$$-iI_{\alpha \beta}( \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)}\phi)$$

plus, $$iI_{\alpha \beta}x^{\beta}T_{\beta}^{\alpha}$$ Which must be wrong! because the indices of x and T must be different to set the antisymmetric part and then prove the orbital momentum conservation.

Help?

2. Oct 20, 2014

### stevendaryl

Staff Emeritus
This problem is worked out in this paper: