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[tex] \phi \rightarrow \phi' = U \phi U^\dagger [/tex]

Now a couple of stupid questions.

What happens if we consider a correlation function of, say, [itex] \phi \phi^\dagger [/itex]? It seems like we won't have that [itex] \langle \phi' \phi^{\dagger '} \rangle = \langle \phi \phi^\dagger \rangle [/itex].

Also, why do we have to impose that the commutator of two transformations give a transformation? what would go wrong if it didn't?

To take a specific example, susy transformations leave a lagrangian invariant (modulo total derivatives) even before introducing auxiliary fields. And yet, we must introduce auxiliary fields to get the algebra to close. I am trying to understand why this is necessary.

Thanks in advance