Spacetime with zero point length is two-dimensional at the Planck scale

TL;DR

This paper proposes a non-local metric modification introducing a zero-point-length in spacetime, leading to a scale-dependent effective dimension that reduces to two at the Planck scale, implying spacetime becomes effectively two-dimensional at quantum gravity scales.

Contribution

It introduces a physical ansatz for a non-local metric tensor that incorporates zero-point-length, revealing a dimensional reduction of spacetime to two dimensions near the Planck scale.

Findings

Effective volume scales as $ ext{ell}_0^{D-2} ext{ell}^2$ at small scales.

Effective dimension decreases from D to 2 as scale approaches Planck length.

Spacetime becomes effectively two-dimensional at the Planck scale.

Abstract

It is generally believed that any quantum theory of gravity should have a generic feature --- a quantum of length. We provide a physical ansatz to obtain an effective non-local metric tensor starting from the standard metric tensor such that the spacetime acquires a zero-point-length $\ell _{0}$ of the order of the Planck length $L_{P}$. This prescription leads to several remarkable consequences. In particular, the Euclidean volume $V_D(\ell,\ell_{0})$ in a $D$-dimensional spacetime of a region of size $\ell $ scales as $V_D(\ell, \ell _{0}) \propto \ell_{0}^{D-2} \ell^2$ when $\ell \sim \ell_{0}$, while it reduces to the standard result $V_D(\ell,\ell_{0}) \propto \ell^D$ at large scales ($\ell \gg \ell_{0}$). The appropriately defined effective dimension, $D_{\rm eff} $, decreases continuously from $D_{\rm eff}=D$ (at $\ell \gg \ell_{0}$) to $D_{\rm eff}=2$ (at $\ell \sim \ell_{0}$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.