Spacetime with zero point length is 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, $D_{\mathrm{eff}}$, decreases continuously from $D_{\mathrm{eff}}=D$ (at $\ell \gg {\ell }_0$) to $D_{\mathrm{eff}}=2$ (at $\ell \sim {\ell }_0$). This suggests that the physical spacetime becomes essentially 2-dimensional near Planck scale.

published version

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