# 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

### Staff: Admin

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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