Thermality of the zero-point length and gravitational selfduality

It has been argued that the existence of a zero-point length is the hallmark of quantum gravity. In this paper, we suggest a thermal mechanism whereby this quantum of length arises in flat, Euclidean spacetime [Formula: see text]. For this, we consider the infinite sequence of all flat, Euclidean spacetimes [Formula: see text] with [Formula: see text], and postulate a probability distribution for each [Formula: see text] to occur. The distribution considered is that of a canonical ensemble at temperature [Formula: see text], 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 [Formula: see text]. 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 realizes a conjectured duality between a weakly gravitational, strongly quantum system and a weakly quantum, strongly gravitational system.