Quantization of wave equations and hermitian structures in partial differential varieties.
Sufficiently close to 0, the solution variety of a nonlinear relativistic wave equation-e.g., of the form squarevarphi + m(2)varphi + gvarphi(p) = 0-admits a canonical Lorentz-invariant hermitian structure, uniquely determined by the consideration that the action of the differential scattering transformation in each tangent space be unitary. Similar results apply to linear time-dependent equations or to equations in a curved asymptotically flat space-time. A close relation of the Riemannian structure to the determination of vacuum expectation values is developed and illustrated by an explicit determination of a perturbative 2-point function for the case of interaction arising from curvature. The theory underlying these developments is in part a generalization of that of M. G. Krein and collaborators concerning stability of differential equations in Hilbert space and in part a precise relation between the unitarization of given symplectic linear actions and their full probabilistic quantization. The unique causal structure in the infinite symplectic group is instrumental in these developments.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139.
S M
SM
Paneitz
I E
IE
Segal