Quantization as analysis in partial differential varieties

Replacing positive-energy considerations by considerations of invariance under theS-operator, and applying Paneitz' extension of the stability theory of the school of M. G. Krein, a long-sought canonical positive symplectic complex structure in the stable phase space of infinite-dimensional classical field-theoretic systems can be mathematically determined. This almost-Kählerization of the phase space then yields a (positive-definite) infinite-dimensional Riemannian structure that serves to specify formally, and convergently in finite-mode approximation, the physical vacuum measure for functional integrals involved in the associated quantized field. The method applies to a general class of nonlinear wave equations including that of Yang-Mills.Invited talk at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–21, 1981.