Loop Quantum Mechanics and the Fractal Structure of Quantum Spacetime

S.Ansoldi¹
Dipartimento di Fisica Teorica dell'Università,
Strada Costiera 11, 34014-Trieste, Italy

A.Aurilia²
Department of Physics, California State Polytechnic University
Pomona, CA 91768, USA

E.Spallucci³
Dipartimento di Fisica Teorica dell'Università,
Istituto Nazionale di Fisica Nucleare, Sezione di Trieste,
Strada Costiera 11, 34014-Trieste, Italy

Abstract:

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between $M$-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of ``loop quantum mechanics'', we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate ``evaporates'', and what is left behind is a ``vacuum'' characterized by an effective fractal geometry.