# Homework Help: Bilinears in adjoint representation

1. Apr 8, 2016

### CAF123

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

1)Show that the kinetic term for a Dirac spinor is invariant under the symmetry group $U(N) \otimes U(N)$

2) Show that if $T_a$ are the generators of $O(N)$, the bilinears $\phi^T T^a \phi$ transform according to the adjoint representation.

2. Relevant equations

For 1), $\mathcal L_{kin} = i \bar \phi \gamma^{\mu} \partial_{\mu} \phi$

3. The attempt at a solution
In 1), I considered the case of a Weyl spinor first. This has a kinetic term $i \bar \phi_R \gamma^{\mu} \partial_{\mu} \phi_R$ and if $\phi_R \rightarrow U \phi_R$ then $i \bar \phi_R \gamma^{\mu} \partial_{\mu} \phi_R \rightarrow i\phi^{\dagger}_R U^{\dagger} \gamma_0 \gamma^{\mu} \partial_{\mu} U \phi$ Because $U$ and the gamma matrices act on different spaces, can I just shift the $U$ to the $U^{\dagger}$ and then using $UU^{\dagger}=1$ get the result? The $U(N) \otimes U(N)$ for the Dirac spinors comes about from decomposing a Dirac spinor into its left and right handed components each of which transforms under a 'left handed fundamental representation' or 'right handed fundamental representation' so could write the symmetry group as $U_L(N) \otimes U_R(N)$ (I think).

In 2), how would the generators of O(N) transform?

Thanks!

2. Apr 13, 2016