Incorporating the Scale-Relativity Principle in String Theory and Extended Objects

TL;DR

This paper explores integrating Nottale's scale-relativity principle into string theory and extended objects, proposing invariance under scale transformations and suggesting new geometric frameworks, with initial steps towards a unified approach.

Contribution

It introduces the concept of scale-relativity invariance in string theory, linking it to vanishing beta functions and proposing a novel geometric perspective involving Weyl-Finsler geometries.

Findings

Scale-relativity invariance is compatible with string beta function conditions.

Preliminary geometric structures akin to Weyl-Finsler geometries are proposed.

Initial steps towards merging motion and scale relativity are outlined.

Abstract

First steps in incorporating Nottale's scale-relativity principle to string theory and extended objects are taken. Scale Relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations, in Nature, is taken to be the Planck scale which is not the same as the string scale. Starting with Nambu-Goto actions for strings and other extended objects, we show that the principle of scale-relativity invariance of the world-volume measure associated with the extended objects ( Lorentzian-scalings transformations with respect to the resolutions of the world-volume coordinates) is compatible with the vanishing of the scale-relativity version of the $\beta$ functions : $\beta^G_{\mu\nu}=\beta^X=0$, of the target spacetime metric and coordinates, respectively. Preliminary steps are taken to merge motion relativity with scale relativity and, in this fashion, analogs of Weyl-Finsler geometries make their appearance. The quantum case remains to be studied.