Noether current in quantum field theory

[CSpring432]

Homework Statement

Finding Noether current for the given action

Relevant Equations

$$J^{\mu}=\frac{\partial L(\phi, \partial (\phi))}{\partial (\partial _{\mu}(\phi)}(\delta_{\alpha}\phi)-F^{\mu}$$

Hi

Have been trying to solve the below question for a while, wondered if anyone could help.

Considering the action

$$S=\int -\frac{1}{2}\sum^2_{n,m=1} (\partial^{\mu}\phi_{nm}\partial_{\mu}\phi_{mn}+m^2 \phi_{nm} \phi_{mn})dx$$
under the transformation

$$\phi'=e^{\alpha}\phi e^{-\alpha}$$

Find the infinitesimal transformation and associated Noether current, where both $\alpha$ and $\phi$ are real 2x2 matrices.

I've managed to find what (I think) is the infinitesimal transformation:

$$e^{\alpha}\phi e^{-\alpha}\approx \phi-\phi \alpha +\alpha\phi+ \mathcal{O}(\alpha^2)$$
$$\therefore \delta_{\alpha}=[\alpha, \phi]$$

I am however, stumped for calculating the Noether constant. I know that I would have to use the formula

$$J^{\mu}=\frac{\partial L(\phi, \partial (\phi))}{\partial (\partial _{\mu}(\phi)}(\delta_{\alpha}\phi)-F^{\mu}$$

The issue, I think, is calculating the covariant derivatives since the phi terms are matrix elements. Any help would be really appreciated.

 

[Hill]

I think that $\phi$ represents four real fields $\phi_{nm}$ and the first term in the Noether current is
$$\sum^2_{n,m=1}\frac{\partial L(\phi, \partial (\phi))}{\partial (\partial _{\mu}(\phi_{nm}))}(\delta_{\alpha}\phi_{nm})$$