The category of Fresnel kernels

Abstract

Symplectic relations and their generating functions have found extensive applications in classical mechanics. In the present paper we undertake the study of the correspondence between generating functions of symplectic relations and kernels of integral operators of quantum theories. As a first step we study this correspondence in the case of linear symplectic relations generated by quadratic functions. The theory is sufficiently complicated even in this simple case. Additional complications must be expected in the general nonlinear theory due to the fact that the composition of regular nonlinear symplectic relations is in general singular and that nonlinear symplectic relations in general do not have global generating functions. The present paper is a continuation of the study of linear symplectic relations undertaken in [2].