Quantization of a Nonlinear Field Equation

Abstract

The physical interpretation of the manifold of solutions of a classical nonlinear field equation is so far an unsolved problem of elementary particle physics. For some equations [1] (λϕ⁴ theory), it has been proved that the solutions develop no singularities for sufficiently smooth initial data. However, it is an open question if these solutions form an infinite dimensional Riemannian space, whose tangent space is an quantum mechanical Hilbert space. Once these questions are solved one can hope to understand the quantum theory of a nonlinear field equation much better. (N.B. Each interacting, fully relativistic field theory is nonlinear.)

Lecture given at NATO advanced Study Institute on Nonlinear Equations in Physics and Mathematics, Istanbul, August 1977.