# Can i extend the non-linear realization of the chiral group to $U$ with complex pions?

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In chiral perturbation theory we build a Lagrangian invariant under $SU(2)_L\times{}SU(2)_R$ which acts on the matrix $U$ that accommodates the pion degrees of freedom in the following way

$$U\to{}RUL^{\dagger}$$

where $L\in{}SU(2)_L$, $R\in{}SU(2)_R$ and $U=e^{i\sigma_i\phi_i/f}$ where $f$ is a constant with mass dimensions, $\sigma_i$ are the Pauli matrices and $\phi_i$ are real scalar fields.

Now, this is not a representation of $SU(2)_L\times{}SU(2)_R$ acting on some vector field because the $U$ matrix is a $SU(2)$ matrix and adding to $SU(2)$ matrices doesn't in general give another $SU(2)$ matrix. I have been told that this is rather a *non-linear realization*. I have checked the wiki page but it is beyond my confort zone. In any case, the question I have is very somple. I want to consider a theory with a $\mathcal{U}$ defined analogously to $U$ where now I allow the $\phi_i$ fields to be complex.

Is it legitimate to consider a non-linear realization of $SU(2)_L\times{}SU(2)_R$ on $\mathcal{U}$?

asked Nov 26, 2015

Well, complex  $\phi$would correspond to a non-compact coset space, which should get you a negative kinetic term for the imaginary part of the goldstone fields when you expand out the exponentials. That's a ghost of the bad type, so I don't know if it can be saved.

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