Scattering

This chapter is devoted to quantum field theory from the algebraic and geometric viewpoints. Fields never appear in our considerations however in some sense they are present in the theory because we assume that our theory is invariant under spatial and temporal translations. We use translations to define quantum particles and quasiparticles and their scattering. In quantum mechanics, the notion of a particle is primary but from our perspective it is secondary: we define a particle as an elementary excitation of the ground state.One can also consider an elementary excitation of any stationary translationinvariant state, then the elementary excitation is a quasiparticle. We start with a discussion of the scattering of solitons-classical analogs of particles. Then we give general definitions of excitations and elementary excitations in algebraic and geometric approaches. We define Møller matrices and scattering matrix;we study these notions using asymptotic commutativity of the algebra of observables and cluster property. We introduce the notion of Green’s function and express the scattering matrix in terms of these functions (LSZ formula). We define the notion of inclusive scattering matrix and show that it can be expressed in terms of generalized Green’s functions.