Minimal Length in Quantum Space and Integrations of the Line Element in Noncommutative Geometry

Abstract

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 (θ-Minkowski); 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.

Minimal length in quantum space and integration of the line element in noncommutative geometry

“…in [27]. Notice that our definition of the length operator L = ∑(dqµ ) 2 heavily relies on the choice of the coordinate system: the dq µ 's are relevant only because the distance can be written as a function of the difference of the coordinates, that is on a flat space.…”

“…In this section, we list the results on the quantum length and the spectral distance between generalized coherent states obtained in [27] for the former, in [8] and [28] for the latter.…”

“…We recall that a standard procedure in noncommutative geometry, consisting in doubling the spectral triple, allows to fruitfully confront the spectral distance with the quantum length. Finally we refine the idea of discrete vs. continuous geodesics in the Moyal plane, introduced in [27]. …”

“…In the following, we review several recent results on the spectral distance and the length operator, mainly from [8,27,28,15]. We begin by recalling in section 2 how to retrieve, in the commutative case Q µν = 0, the Euclidean distance from either the length operator L or Connes' formula (1.8).…”

“…Section 6 presents the strategy developed in [27] in order to compare the quantum length with the spectral distance, despite an obvious discrepancy (the latter vanishes between a state and itself, the former does not as soon as l P = 0). This idea is to compare the quantum length with the spectral distance on a …”