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

Nottale's special scale-relativity principle was proposed earlier by the au thor as a plausible geometrical origin to string theory and extended object s. Scale relativity is to scales what motion Relativity is to velocities. T he universal, absolute, impassible, invariant scale under dilatations in na ture is taken to be the Planck scale, which is not the same as the string s cale. Starting with ordinary actions for strings and other extended objects , we show that gauge theories of volume-resolutions scale-relativistic symm etries, of the world volume measure associated with the extended "fuzzy" ob jects, 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 scal ing dimension of the original string coordinates. This is achieved upon vie wing the extendon coordinates, from the fuzzy world volume point of view, a s noncommuting matrices valued in the Lie algebra of Lorentz-scale relativi stic transformations. Preliminary steps are taken to merge motion relativit y with scale relativity by introducing the gauge field that gauges the Lore ntz-scale symmetries in the same vain that the spin connection gauges ordin ary Lorentz transformations and, in this fashion, one may go beyond string theory to construct the sought-after General Theory of Scale-Motion Relativ ity. Such theory requires the introduction of the scale-graviton (in additi on to the ordinary graviton) which is the field that gauges the symmetry wh ich converts motion dynamics into scaling-resolutions dynamics and vice ver sa (the analog of the gravitino that gauges supersymmetry). To go beyond th e quantum string geometry most probably would require a curved fractal spac etime description (curved from both scaling and motion points of views) wit h a curvilinear fractal coordinate system. Non-Archimedean geometry and p-a dic numbers are essential ingredients comprising the geometrical arena of s uch extensions of quantum string geometry. (C) 1999 Elsevier Science Ltd. A ll rights reserved.