Effective Action from the Coadjoint Orbit Method

Having obtained a Hamiltonian formulation for Fermi liquids, we perform the next step of the coadjoint orbit method by Legendre transforming the Hamiltonian. Doing so necessarily requires a restriction of our space of states; one that naturally reduces the phase space to only those states containing a sharp Fermi surface. The Legendre transformation is enabled by a Wess-Zumino-Witten term, also known as a Berry phase term, which is invariant under all time-independent canonical transformations. We describe how to implement spacetime and gauge symmetries in this formalism, ending with a perturbative expansion and scaling analysis of the formal action.