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; 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.

29 pages, 2 figures. Minor corrections to match the published version

References in corpus (5)

• Algebras of distributions suitable for phase-space quantum mechanics. I
• Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime
• Noncommutative spectral geometry, algebra doubling and the seeds of quantization
• Discreteness of area in noncommutative space
• Quantum Field Theory on Quantum Spacetime

Cited by in corpus (14)

• Spectral geometry with a cut-off: topological and metric aspects
• $\kappa$-Poincaré-comodules, Braided Tensor Products and Noncommutative Quantum Field Theory
• Connes distance by examples: Homothetic spectral metric spaces
• On Pythagoras' theorem for products of spectral triples
• A fully covariant information-theoretic ultraviolet cutoff for scalar fields in expanding FRW spacetimes
• Spectral triplets, statistical mechanics and emergent geometry in non-commutative quantum mechanics
• Massive fermions interacting via a harmonic oscillator in the presence of a minimal length uncertainty relation
• A dual formula for the spectral distance in noncommutative geometry
• Revisiting Connes' Finite Spectral Distance on Non-commutative Spaces : Moyal Plane and Fuzzy Sphere
• The $x_i$-eigenvalue problem on some new fuzzy spheres
• Szekeres Universes with GUP corrections
• Thermal effective potential in two- and three- dimensional non-commutative spaces
• A Twisted ${\mathcal{C}}^{\star }$-algebra formulation of Quantum Cosmology with application to the Bianchi I model
• Connes spectral distances, quantum discord and coherence of qubits