Quantization of Fields with Infinite-Dimensional Invariance Groups. III. Generalized Schwinger-Feynman Theory

Abstract

The formal methods of Schwinger and Feynman are applied to nonlinear field theories having elementary vertex functions of arbitrarily high order. In the first half of the paper, familiar theorems are rederived by noncanonical methods. Emphasis is given to purely formal aspects of the theory which may be expected to survive generalization to situations in which standard asymptotic conditions are inapplicable. Since the context in which the field nonlinearities are assumed to appear is that of a non-Abelian infinite-dimensional invariance group, detailed attention is given to the question of a group invariant measure for the Feynman functional integral. It is shown that the physically important part of the measure is not determined by the group. The second half of the paper is devoted to the theory of the propagators and correlation functions which characterize the system when invariant variables are introduced. The existence of a c-number action functional Γ which contains a complete description of all quantum processes is proved. The second variational derivatives of this functional constitute the wave operator for the one-particle propagators (including all radiative corrections), and its higher derivatives are the renormalized vertex functions. A description of the renormalization process is easily carried out in terms of Γ. Finally, the implications which its existence has for quantum gravidynamics are discussed. Because it leads to nonlocal covariant equations for a complex metric tensor the way is open to transmutations of topology at the quantum level.