Generalised uncertainty relations from finite-accuracy measurements

Frontiers in Astronomy and Space Sciences

Astronomy and Space Sciences

Ray Ganardi¹  Marek Miller¹  Tomasz Paterek²  Matthew J. Lake³ 

[1] Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Warsaw, Poland;Department of Physics, Xiamen University Malaysia, Sepang, Malaysia;Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, Gdańsk, Poland;National Astronomical Research Institute of Thailand, Chiang Mai, Thailand;Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand;School of Physics, Sun Yat-Sen University, Guangzhou, China;Department of Physics, Babeş-Bolyai University, Cluj-Napoca, Romania;Office of Research Administration, Chiang Mai University, Chiang Mai, Thailand;

关键词: generalised uncertainty relations;    generalised uncertainty principle;    extended uncertainty principle;    finite-accuracy measurements;    POVM;   

DOI  :  10.3389/fspas.2023.1087724

 received in 2022-11-02, accepted in 2023-01-09,  发布年份 2023

来源: Frontiers

【 摘 要 】

In this short note we show how the Generalised Uncertainty Principle (GUP) and the Extended Uncertainty Principle (EUP), two of the most common generalised uncertainty relations proposed in the quantum gravity literature, can be derived within the context of canonical quantum theory, without the need for modified commutation relations. A generalised uncertainty principle-type relation naturally emerges when the standard position operator is replaced by an appropriate Positive Operator Valued Measure (POVM), representing a finite-accuracy measurement that localises the quantum wave packet to within a spatial region σg > 0. This length scale is the standard deviation of the envelope function, g, that defines the positive operator valued measure elements. Similarly, an extended uncertainty principle-type relation emerges when the standard momentum operator is replaced by a positive operator valued measure that localises the wave packet to within a region σ̃g>0 in momentum space. The usual generalised uncertainty principle and extended uncertainty principle are recovered by setting σg≃ℏG/c3, the Planck length, and σ̃g≃ℏΛ/3, where Λ is the cosmological constant. Crucially, the canonical Hamiltonian and commutation relations, and, hence, the canonical Schrödinger and Heisenberg equations, remain unchanged. This demonstrates that generalised uncertainty principle and extended uncertainty principle phenomenology can be obtained without modified commutators, which are known to lead to various pathologies, including violation of the equivalence principle, violation of Lorentz invariance in the relativistic limit, the reference frame-dependence of the “minimum” length, and the so-called soccer ball problem for multi-particle states.

【 授权许可】

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Copyright © 2023 Lake, Miller, Ganardi and Paterek.