Point to string transformation, or framedependent notion of ?locality?: An attempt to finitize field theories including gravitation

Abstract

A novel approach has been made to the divergence problem in local field theories, in which the notion of “locality” is still retained but loses its absolute meaning, just like “simultaneity”. The basic idea is to introduce a pure-imaginary elementary length into 3-dimensional space, while keeping “time” structureless so as to retain the unitarity of theS-matrix. Consequently, light becomes dispersive at sufficiently short wavelengths, and Lorentz transformation becomes a point-to-string transformation. When reformulated to meet the new Lorentz invariance, all the localfield (in the above sense) theories in a flat space become finite,while retaining their conventional form. This has been demonstrated by the derivation of finitized Coulomb potential and correct high-momentum behavior of quantum-electrodynamic coupling constant. For diagrams including gravitons, evaluation of the superficial degrees of divergence shows that only a restricted number of 1-(and 2-) loop diagrams might be divergent, while those of more than 3 loops are definitely convergent, thus indicating possible renormalizability (or something better) of quantum gravity in Einstein's formalism of general relativity. Since 4-dimensional simple supergravity removes 1-and 2-loop divergence, a combination of the theory and the present one might lead to a more interesting result.