The Operator Algebra and the Hamiltonian Formalism

In this chapter we implement the first of the two-step procedure that will end with an effective action for Fermi liquids: the Hamiltonian description of the theory. Starting with the algebra of fermion bilinears, we demonstrate how its dual space, the space of states, can be turned into a phase space with a Poisson bracket. After describing a suitable semiclassical limit for the algebra and the space of states, we choose the most general Hamiltonian for the theory that is compatible with this limit.