I should write the mass term of the Lagrangian with global SO(4) symmetry in tensor representation with anti-symmetric tensors and then diagonalise this term with defining a new set of tensors (self-dual and anti self-dual) and writing the old as a linear combination of those. Here is what I have done so far:

We consider anti-symmetric tensor that transforms in the fundamental rep of SU(4) satisfying $\psi^{ij} = -\psi^{ji}\quad$ and one should write all the invariant mass terms which I thought to be:

$$ \psi^{ij}\psi^{*ij} + \epsilon^{ijkl}\psi_{ij}\psi_{kl}+ \epsilon^{ijkl}\psi_{ij}\psi^*_{kl}+ \epsilon^{ijkl}\psi^*_{ij}\psi^*_{kl} $$

Then one defines a subspace (actually two) of those anti-symmetric tensors imposing that they have to satisfy particular condition: self dual tensor $\psi^{ij}=1/2\epsilon^{ijkl}\psi^*_{kl}$ and anti self-dual tensor as $\psi^{ij}=-1/2\epsilon^{ijkl}\psi^*_{kl}$. I showed that those new tensors are invariant under SO(4) transformations.

But I don't know how to diagonalise the mass term with writing the $\psi$ and $\psi^\dagger$ as a linear combination of those two new tensors. And how to see that now it is diagonal?


1 Answer 1


I think you can write

$$\psi=\psi^++\psi^-$$ $$\psi^\dagger=\psi^+-\psi^-$$

and then you will get something like


Which is decomposed


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