Alternative approach to no integrability field theory

Abstract

This is another paper in a series which illustrates that foregoing the rules of traditional calculus opens up a realm of possibilities. We study the “aesthetic field theory”, although the techniques are more general. We introduce a new integration procedure which is consistent with aesthetic principles. There is a single change function associated with each point rather than many change quantities at a point (there is a change quantity associated with each path going through any point in our previous work). The integration procedure is consistent with the rules of traditional calculus when the integrability equations are satisfied. We find that the new procedure, when applied to the data associated with the loop lattice, leads to multiple maxima and minima on a plane. Another lattice solution (point lattice) does not show evidence of being bounded in certain domains of the region studied. In the summation-over-paths method all the paths from the origin point are needed to calculate the field. Knowledge of the field on one hypersurface is not sufficient to obtain the field on the succeeding hypersurface. In the present approach to no integrability, we can calculate the field on one hypersurface from knowledge of the field on the proceeding hypersurface (without requiring knowledge of the past history). Thus, the new approach enables us to obtain this desirable feature of a hyperbolic theory without requiring the field to be arbitrary on a hypersurface.