Dimensional reduction via dimensional shadowing

1. Uni ed eld theories suggest a N dimensional conguration space with N̄ = N−4 > 0 extra dimensions not perceived in nature. The common heuristic reasons for this proposition are: (i) the volume [Hausdor measure] does not scale like μH(δ,R ) = δ μH(1,R ), where δ is some length scale and 1 stands for its unity. Instead, experience tells us that increasing (or decreasing) the spacial size of an object by δ, changes its volume by approximately δ3, corresponding to a spacial [Hausdor , if not denoted otherwise] dimensionDs = 3; (ii) the number of spacial degrees of freedom D is not N but three, corresponding to a threedimensional vector space; (iii) longrange static potentials around a [conserved] point charge, behaving as r2−DP for D > 2, when r is the distance from the charge, suggest a dimensional value D of approximately three. There is good evidence, that all these parameters coincide and Ds ≈ D ≈ D ≈ 3. According to these observations, physical con guration space is modelled as a product space R = Rs×Rt, where Rt stands for the time continuum . The dimension D of its Cartesian product is [1] D ≥ Ds + Dt ≈ 4. Hence, some kind of dimensional reduction has to e ectively decrease the number of operational attainable dimensions. These may be de ned via the Hausdor dimension, or the maximal number of linear independent vectors of a vector space [this assumes the existence of a vector space], via the distance dependence of potentials, or otherwise. However, it is in no way trivial, that all these de nitions coincide. The common notion of dimensional reduction in the Kaluza-Klein approach assumes compacti cation: Con guration space is assumed as R4×SN̄ , where S is a compact N̄ dimensional manifold. These extra dimensions are assumed to be curled up to very small sizes, such that these additional degrees of freedom could be observed only in the high energy regime. 2. In this brief communication a very di erent approach to dimensional reduction is pursued: con guration space X is assumed to be a fractal embedded in a higherdimensional space R with arbitrary integer dimension D(R ) = N ≥ 4. It is then assumed, that due to some [yet unknown] mechanism, the dimension of the con guration space X is approximately equal to four D(X) ≈ 4.