# Minimal length in quantum space and integrations of the line element in Noncommutative Geometry

June 01, 2011

We question the emergence of a minimal length in quantum spacetime, comparing
two notions that appeared at various points in the literature: on the one side,
the quantum length as the spectrum of an operator L in the Doplicher
Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical
noncommutative spacetime; on the other side, Connes' spectral distance in
noncommutative geometry. Although on the Euclidean space the two notions merge
into the one of geodesic distance, they yield distinct results in the
noncommutative framework. In particular on the Moyal plane, the quantum length
is bounded above from zero while the spectral distance can take any real
positive value, including infinity. We show how to solve this discrepancy by
doubling the spectral triple. This leads us to introduce a modified quantum
length d'_L, which coincides exactly with the spectral distance d_D on the set
of states of optimal localization. On the set of eigenstates of the quantum
harmonic oscillator - together with their translations - d'_L and d_D coincide
asymptotically, both in the high energy and large translation limits. At small
energy, we interpret the discrepancy between d'_L and d_D as two distinct ways
of integrating the line element on a quantum space. This leads us to propose an
equation for a geodesic on the Moyal plane.

Keywords:

noncommutative geometry, DFR model, Moyal plane