ABSTRACT
We have now discussed in some detail the quantization of two free field theories that give approximate descriptions of many of the particles found in Nature. Up to this point, however, free-particle states have been eigenstates of the Hamiltonian; we have seen no interactions and no scattering. In order to obtain a closer description of the real world, we must include new, nonlinear terms in the Hamiltonian (or Lagrangian) that will couple different Fourier modes (and the particles that occupy them) to one another. To preserve causality, we insist that the new terms may involve only products of fields at the same spacetime point: [(ft{x)}4 is fine, but (ft{x)(ft{y) is not allowed. Thus the terms describing the interactions will be of the form
For now we restrict ourselves to theories in which Ttint (= — Ant) is a function only of the fields, not of their derivatives.
