Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime

Abstract

We derive new space-time uncertainty relations (STUR) at the fundamental Planck length $L_P$ from quantum mechanics and general relativity (GR), both in flat and curved backgrounds. Contrary to claims present in the literature, our approach suggests that no minimal uncertainty appears for lengths, but instead for minimal space and four-volumes. Moreover, we derive a maximal absolute value for the energy density. Finally, some considerations on possible commutators among quantum operators implying our STUR are done.

Final version published in "Class. Quantum Grav."

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• Can we measure structures to a precision better than the Planck length?
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• Minimal length in quantum space and integrations of the line element in Noncommutative Geometry
• Statistical mechanics of gravitons in a box and the black hole entropy
• Bekenstein-Hawking entropy in expanding universes from black hole theorems
• Entropy, energy and temperature-length inequality for Friedmann universes
• Entropy, temperature and internal energy of trapped gravitons and corrections to the Black Hole entropy
• On a physical description and origin of the cosmological constant
• Planckian corrections to the Friedmann flat equations from thermodynamics at the apparent horizon
• Building a linear equation of state for trapped gravitons from finite size effects and the Schwarzschild black hole case
• Comment on "Can we measure structures to a precision better than the Planck length?"[arXiv:1205.3636], by Sabine Hossenfelder
• Length and distance on a quantum space
• Statistical description of massless excitations within a sphere with a linear equation of state and the dark energy case
• The cosmological constant from Planckian fluctuations and the averaging procedure
• A proposal for Heisenberg uncertainty principle and STUR for curved backgrounds: an application to white dwarf, neutron stars and black holes
• The physical origin of the cosmological constant: continuum limit and the analogy with the Casimir effect
• Graceful exit from inflation to radiation era with rapidly decreasing agegraphic potentials
• The physical nature of the cosmological constant and the decoherence scale in a renormalization-group approach
• Physically viable rotating mass solutions surrounding Kerr black hole
• A Twisted ${\mathcal{C}}^{\star }$-algebra formulation of Quantum Cosmology with application to the Bianchi I model
• The 3D perturbed Schrödinger Hamiltonian in a Friedmann flat spacetime testing the primordial universe in a non commutative spacetime
• Corrections to the Black hole entropy from a Bose Einstein condensate: a semi-classical phenomenological approach