Derivation of the deformed Heisenberg algebra from discrete spacetime

¹ Department of Physics, Aligarh Muslim University - Aligarh-202002, India
² Design and Manufacturing Technology Division, Raja Ramanna Centre for Advanced Technology (RRCAT) Indore-452013, Madhya Pradesh, India
³ Irving K. Barber Faculty of Science, University of British Columbia - 3333 University Way, Kelowna, British Columbia, V1V 1V7, Canada
⁴ Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus Hyderabad-500078, India
⁵ Department of Physics and Astrophysics, University of Delhi - Delhi-110007, India
⁶ Irving K. Barber School of Arts and Sciences, University of British Columbia - Okanagan, Kelowna, British Columbia, V1V 1V7, Canada
⁷ Canadian Quantum Research Center - 204-3002, 32 Ave Vernon, British Columbia, V1T 2L7, Canada
⁸ Department of Mathematical Sciences, Durham University - Upper Mountjoy, Stockton Road, Durham, DH1 3LE, UK
⁹ Faculty of Sciences, Hasselt University - Agoralaan Gebouw D, Diepenbeek, 3590 Belgium
¹⁰ Department of Physics, Jamia Millia Islamia - New Delhi-110025, India

Received: 27 September 2024
Accepted: 8 January 2025

Abstract

Although the deformation of the Heisenberg algebra by a minimal length has become a central tool in quantum gravity phenomenology, it has never been rigorously obtained and is often derived using heuristic reasoning. In this study, we move beyond the heuristic derivation of the deformed Heisenberg algebra and explicitly derive it using a model of discrete spacetime, which is motivated by quantum gravity. Initially, we investigate the effects of the leading order Planckian lattice corrections and demonstrate that they precisely match those suggested by the heuristic arguments commonly used in quantum gravity phenomenology. Furthermore, we rigorously obtain deformations from the higher-order Planckian lattice corrections. In contrast to the leading-order corrections, these higher-order corrections are model dependent. We select a specific model that breaks the rotational symmetry, as the importance of such rotational symmetry breaking lies in the relationship between CMB anisotropies and quantum gravitational effects. Based on the mathematical similarity of the Planckian lattice used here with the graphene lattice, we propose that graphene can serve as an analogue system for the study of quantum gravity. Finally, we examine the deformation of the covariant form of the Heisenberg algebra using a four-dimensional Euclidean lattice.