The classical limits of P(phi) sub 2 quantum field

How the structures of special relativity (particles) and classical field theory (waves) can be derived as limits, as Planck's constant tends to zero is discussed in the framework of two dimensional Boson field theory in an effort to reveal the mathematical structures behind wave particle duality. The classical limit problem in quantum mechanics is reviewed and a uniform interpretation of classical limit theories as being identifications of states, i.e. elements of the physical Hilbert space, with expectations which satisfy the appropriate classical equation is established. Two classes of states in interacting P(phi) sub 2 theories are identified: one for which the classical equations are those of special relativity and the other for which the classical equation is the nonlinear Klein-Gordon equation. The fundamental theorem of Hepp in quantum mechanics, many body theory, and P(phi) sub 2 models with a space cutoff are considered. Hepp's methods for proving a classical field limit for P(phi) sub 2 models with space cutoff cannot be extended to models without cutoffs and that much more powerful and difficult techniques, in particular the techniques of Euclidean field theory, are required. A cluster expansion is used to prove that a classical field limit exists for any P(phi) sub 2 model with a convex polynomial P. The problem of a classical particle limit for a P(phi) sub 2 theory is introduced.