Loop Quantum Mechanics and the Fractal Structure of Quantum Spacetime

TL;DR

This paper explores how loop quantum mechanics suggests that classical spacetime emerges from string condensates at large scales, while near the Planck scale, spacetime exhibits a fractal structure due to condensate evaporation.

Contribution

It proposes a connection between loop quantum mechanics, string theory, and fractal geometry of spacetime at quantum scales.

Findings

Large scale string condensates produce effective Riemannian geometry.

Near the Planck scale, spacetime exhibits fractal geometry.

A link between M-Theory and string loop quantum mechanics is suggested.

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.