Abstract

Gedanken experiments in quantum gravity motivate generalised uncertainty relations (GURs) implying deviations from the standard quantum statistics close to the Planck scale. These deviations have been extensively investigated for the non-spin part of the wave function, but existing models tacitly assume that spin states remain unaffected by the quantisation of the background in which the quantum matter propagates. Here, we explore a new model of nonlocal geometry in which the Planck-scale smearing of classical points generates GURs for angular momentum. These, in turn, imply an analogous generalisation of the spin uncertainty relations. The new relations correspond to a novel representation of SU(2) that acts nontrivially on both subspaces of the composite state describing matter-geometry interactions. For single particles, each spin matrix has four independent eigenvectors, corresponding to two 2-fold degenerate eigenvalues ħ±(ħ+β)/2, where β is a small correction to the effective Planck’s constant. These represent the spin states of a quantum particle immersed in a quantum background geometry and the correction by β emerges as a direct result of the interaction terms. In addition to the canonical qubits states, |0⟩=|↑⟩ and |1⟩=|↓⟩, there exist two new eigenstates in which the spin of the particle becomes entangled with the spin sector of the fluctuating spacetime. We explore ways to empirically distinguish the resulting "geometric" qubits, |0′⟩ and |1′⟩, from their canonical counterparts.

Highlights

  • The Heisenberg uncertainty principle (HUP) contains the essence of quantum theory.It can be motivated heuristically by the Heisenberg microscope thought experiment, giving ∆xi ∆p j &Received: 5 January 2020Accepted: 14 February 2021Published: 1 March 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claihi δ, j (1)or derived rigorously from the quantum formalism

  • If geometry is truly nonlocal at the Planck scale, we must look for signatures of entanglement between material particles and the quantum spatial background in the statistical correlations between subspaces of multiparticle states

  • We have proposed a new model of quantum nonlocal geometry [10,11,12,13] and have investigated its consequences for both external and internal symmetries

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Summary

Introduction

The Heisenberg uncertainty principle (HUP) contains the essence of quantum theory. It can be motivated heuristically by the Heisenberg microscope thought experiment, giving. A brief recap of the treatment of one- and two-particle systems in canonical quantum mechanics is given in Appendix A In order to distinguish between analogous operators in the canonical and smeared space theories, we use lower case letters to denote former and upper case letters for the latter, e.g., Ŝz and Ŝ2 , as opposed to ŝz and ŝ2 Due to their relatively large size, the explicit forms of Ŝz , Ŝ2 and Ŝ± for the smeared two-particle state are given in Appendix B

The Smeared Space Model of Nonlocal Geometry
Spin in Smeared-Space Quantum Mechanics
One-Particle Systems in Smeared Space
Two-Particle Systems in Smeared Space
Bell States in Smeared Space
Conclusions
Future Work
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