Fuzzy Dimensions and Planck's Uncertainty Principle for $p$-branes

S.Ansoldi¹

Dipartimento di Fisica Teorica dell'Universita`

Strada Costiera 11, 34014-Trieste, Italy

A.Aurilia²

Department of Physics, California State Polytechnic University

Pomona, CA 91768

E.Spallucci³

Dipartimento di Fisica Teorica dell'Universita`

Istituto Nazionale di Fisica Nucleare, Sezione di Trieste,

Strada Costiera 11, 34014-Trieste, Italy

Abstract:

The explicit form for the quantum propagator of a bosonic $p$-brane, previously obtained by the authors in the quenched-minisuperspace approximation, suggests the possibility of new transitions between $p$-branes of different dimensionality. For such ``$p$-transitions'' to take place, it is sufficient that the Lorentzian metric be replaced by a quadratic form on a Clifford manifold. This substitution introduces into the theory a new source of fuzziness in the form of a quantum of resolution below which it is impossible to resolve the distance between two points. To overlay a minimal (Planckian) resolution over spacetime demands, in turn, an extension of Heisenberg's Principle to the Planck scale; on the other hand, the introduction of the Clifford line element extends the usual relativity of motion to the case of relative dimensionalism of all $p$-cells that make up the spacetime manifold near the Planck scale.