Thermality of the zero-point length and gravitational selfduality

Abstract

It has been argued that the existence of a zero-point length is the hallmark of quantum gravity. In this letter we suggest a thermal mechanism whereby this quantum of length arises in flat, Euclidean spacetime ${\mathbb{R}}^d$. For this we consider the infinite sequence of all flat, Euclidean spacetimes ${\mathbb{R}}^{d^{\prime }}$ with $d^{\prime }\ge d$, and postulate a probability distribution for each $d^{\prime }$ to occur. The distribution considered is that of a canonical ensemble at temperature $T$, the energy levels those of a 1-dimensional harmonic oscillator. Since both the harmonic energy levels and the spacetime dimensions are evenly spaced, one can identify the canonical distribution of harmonic-oscillator eigenvalues with that of dimensions $d^{\prime }$. The state describing this statistical ensemble has a mean square deviation in the position operator, that can be interpreted as a quantum of length. Thus placing an oscillator in thermal equilibrium with a bath provides a thermal mechanism whereby a zero-point length is generated. The quantum-gravitational implications of this construction are then discussed. In particular, a model is presented that realises a conjectured duality between a weakly gravitational, strongly quantum system and a weakly quantum, strongly gravitational system.

15 pages