New measures for nonrenormalizable quantum field theory

Abstract

Generic interactions characteristic of so-called nonrenormalizable scalar and spinor quantum field theories are interpreted as discontinuous perturbations in the sense that the theory does not return to the unperturbed theory as the interaction coupling vanishes. To proceed beyond this interpretation specific alternatives to conventional quantization schemes are developed. Solution of a highly idealized (independent-value), nonrenormalizable scalar field theory automatically entails a formally scale-invariant measure (rather than the conventional translation-invariant measure) in a functional integral formulation, and the success of this measure suggests its use more generally. Such a measure can be motivated (by augmented field theory) on heuristic grounds as taking into account the partial hardcore nature of the interaction responsible for its behavior as a discontinuous perturbation. This modification leads generally to what we call scale-covariant quantization, which can be formulated in terms of unconventional functional differential equations, coupled Green's function equations and operator field equations. Use of affine fields establishes equivalence of these various approaches and enables analogous coupled Green's function equations for models with fermions to be most easily obtained. The basic concepts of this program are illustrated with elementary wave-mechanical examples.