The textbook derivation of the quantum effective action (see e.g. Weinberg, vol. 2, sec. 16) and its energy interpretation seems to require that the fields take values from a linear space, as it relies on the Legendre transform of the generating functional of connected Green's functions. Is there a generalization or modification of the concept valid also for theories where fields take values from some nontrivial manifold?
I have specifically in mind an effective theory for Goldstone bosons, living on the coset space of a broken symmetry, such as the Chiral Perturbation Theory in QCD. Say that I want to determine, at least in principle, the one-loop, that is next-to-leading order (NLO), effective action. This should somehow, as usual, require evaluating a one-loop diagram in presence of background fields, and adding counterterms from the NLO Lagrangian. But as said above, it's not even obvious to me how to define the one-loop effective action in the first place.
Note that I am not interested in having a 1PI generating functional for Green's functions in a chosen vacuum. I would like to have a functional that maintains the energy interpretation of the usual quantum effective action, that is gives the minimum energy at a fixed value of the fields.
No comments:
Post a Comment