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

+ 4 like - 0 dislike
197 views

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 $\mathcal{U}$?

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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar\varnothing$sicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.