Beyond strings, multiple times and gauge theories of area-scalings relativistic transformations

Nottale's special scale-relativity principle was proposed earlier by the author as a plausible geometrical origin to string theory and extended objects. Scale relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations in nature is taken to be the Planck scale, which is not the same as the string scale. Starting with ordinary actions for strings and other extended objects, we show that gauge theories of volume-resolutions scale-relativistic symmetries, of the world volume measure associated with the extended "fuzzy" objects, are a natural and viable way to formulate the geometrical principle underlying the theory of all extended objects. Gauge invariance can only be implemented if the extendon actions in D target dimensions are embedded in D + 1 dimensions with an extra temporal variable corresponding to the scaling dimension of the original string coordinates. This is achieved upon viewing the extendon coordinates, from the fuzzy world volume point of view, as noncommuting matrices valued in the Lie algebra of Lorentz-scale relativistic transformations. Preliminary steps are taken to merge motion relativity with scale relativity by introducing the gauge field that gauges the Lorentz-scale symmetries in the same vain that the spin connection gauges ordinary Lorentz transformations and, in this fashion, one may go beyond string theory to construct the sought-after General Theory of Scale-Motion Relativity. Such theory requires the introduction of the scale-graviton (in addition to the ordinary graviton) which is the field that gauges the symmetry which converts motion dynamics into scaling-resolutions dynamics and vice versa (the analog of the gravitino that gauges supersymmetry). To go beyond the quantum string geometry most probably would require a curved fractal spacetime description (curved from both scaling and motion points of views) with a curvilinear fractal coordinate system. Non-Archimedean geometry and p-adic numbers are essential ingredients comprising the geometrical arena of such extensions of quantum string geometry. (C) 1999 Elsevier Science Ltd. All rights reserved.