Bilinears in adjoint representation

[CAF123]

Homework Statement

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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.

Homework Equations

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

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!

 

[mark726]

1) Yes, you are correct in shifting the U to the U^{\dagger} and using the fact that they are unitary, you can show that the kinetic term is invariant under the symmetry group U(N) \otimes U(N). This is because the gamma matrices act on different spaces, as you mentioned, and therefore commute with the unitary transformations on each space.

2) The generators of O(N) transform under the adjoint representation, which means that they transform as matrices under the group. In this case, the bilinears \phi^T T^a \phi will transform according to the adjoint representation because the generators T^a are transforming as matrices under the group. This can be seen by writing out the transformation of the bilinear explicitly and using the transformation properties of the generators.