- 1National Astronomical Research Institute of Thailand, Chiang Mai, Thailand
- 2Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, Chiang Mai, Thailand
- 3School of Physics, Sun Yat-Sen University, Guangzhou, China
- 4Department of Physics, Babeş-Bolyai University, Cluj-Napoca, Romania
- 5Office of Research Administration, Chiang Mai University, Chiang Mai, Thailand
- 6Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, Warsaw, Poland
- 7Department of Physics, Xiamen University Malaysia, Sepang, Malaysia
- 8Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics, and Informatics, University of Gdańsk, Gdańsk, Poland
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
1 Introduction
In canonical quantum mechanics the Heisenberg uncertainty principle (HUP) implies a fundamental trade-off between the precisions of position and momentum measurements. 1 It can be introduced heuristically, via the famous Heisenberg microscope thought experiment, giving (Heisenberg, 1927; Heisenberg, 1930)
or derived rigorously from the canonical quantum formalism, yielding (Isham, 1995; Rae, 2002)
The inequality in Eq. 1.2 is exact and, unlike the heuristic uncertainties Δxi and Δpj in Eq. 1.1, Δψxi and Δψpj represent well-defined standard deviations of the probability distributions |ψ(x)|2 and
We emphasise the scale-dependence of the canonical quantum Fourier transform, which is often neglected in standard treatments, by introducing the subscript ℏ. Eq. 1.2 is obtained by combining the Schrödinger-Robertson relation for arbitrary Hermitian operators,
with the canonical position-momentum commutator,
In recent years, thought experiments in quantum gravity research have suggested the existence of generalised uncertainty relations (GURs). By reconsidering Heisenberg’s 1927 gedanken experiment, and accounting for the gravitational interaction between the massive particle and the probing photon, we obtain the generalised uncertainty principle (GUP),
where α0 is a numerical constant of order unity (Maggiore, 1993; Adler and Santiago, 1999; Scardigli, 1999). By minimising the right-hand side with respect to Δpj, the GUP implies the existence of a minimum position uncertainty of the order of the Planck length,
Reconsidering Heisenberg’s arguments in the presence of a constant dark energy density ρΛ = Λc2/(8πG) ≃ 10–30 g.cm−3 (Riess et al., 1998; Perlmutter et al., 1999), or, equivalently, an asymptotically de Sitter background with minimum scalar curvature of the order of the cosmological constant, Λ ≃ 10–56 cm−2 (Ade et al., 2014; Betoule et al., 2014), gives the extended uncertainty principle (EUP),
where η0 is of order one (Bolen and Cavaglia, 2005; Park, 2008; Bambi and Urban, 2008). The EUP implies the existence of a minimum momentum uncertainty of the order of the de Sitter momentum,
Combining both effects yields the extended generalised uncertainty principle (EGUP),
which implies the existence of both minimum length and momentum scales in nature (Bolen and Cavaglia, 2005; Park, 2008; Bambi and Urban, 2008). Like their forebearer Eq. 1.1 all three relations Eqs 1.6–1.8 are heuristic in nature and it remains an open problem how to rigorously derive GURs from within a modified quantum formalism.
Perhaps the simplest way to obtain the GUP, EUP or EGUP, given Eq. 1.4, is to modify the canonical position-momentum commutator Eq. 1.5 and it is clear that a modification of the form
gives rise to an EGUP-type uncertainty relation, at least when both
This leads immediately to the reference frame-dependence of the (supposedly invariant) minimum length. In fact, the situation is even worse since even a redefinition of the position-coordinate origin alters the value of the bound on the right-hand side. This gives rise to a coordinate-dependent “minimum” length, which is clearly unphysical, and which strongly suggests that GUR models based on modified commutation relations are not mathematically self-consistent (Lake, 2020; Lake et al., 2023).
In addition, the modified position-momentum commutator Eq. 1.9 implies a modification of the canonical Heisenberg equation, which immediately gives rise to mass-dependent accelerations for quantum particles, violating the equivalence principle (Tawfik and Diab, 2014; Tawfik and Diab, 2015). Such models also violate Lorentz invariance in the relativistic limit and suffer from the so-called soccer ball problem, so that sensible GUP-compatible multi-particle states cannot be defined (Hossenfelder, 2013; Amelino-Camelia, 2017) 2.
The heuristic, model-independent nature of the gedanken experiments that lead to the relations Eqs. 1.6–1.8, together with the pathologies displayed by modified commutator models, motivate us to consider alternative ways to generate GUP, EUP, and EGUP phenomenology, without modifying the canonical Heisenberg algebra. In this paper, we consider one way in which such a scheme can be implemented from within the canonical quantum formalism. The physical basis of the model is the notion of a finite-accuracy measurement and these are represented mathematically by the construction of appropriate POVM. Roughly speaking, since errors add in quadrature for independent random variables, finite-accuracy measurements of position and momentum with detection “sweet spots” of width σg ≃ lPl and
2 GUR from finite-accuracy measurements described by POVM
In this section, we show that GUP, EUP and EGUP-type uncertainty relations can be derived in an effective model, where position and momentum measurements in canonical quantum theory are not perfectly accurate, and are described by POVM, rather than perfect projective measurements.
Let us begin by replacing the usual position-measurement operator,
where g(x′ − x) is any normalised function, ∫|g(x′ − x)|2d3x′ = 1. These elements satisfy the relations
Finite-accuracy position measurements, conducted on an arbitrary state |ψ⟩, then give rise to the first and second order moments
where
where
where we have used the shorthand notation
In like manner, finite-accuracy momentum measurements may be introduced via the operators
where
where the new action scale β ≠ ℏ is given by
and
Finite-accuracy momentum measurements, conducted on an arbitrary state |ψ⟩, then give rise to the first and second order moments
where
where
where we have again used the shorthand
To obtain a GUP-type relation from Eq. 2.4 we simply take the square root, Taylor expand the right-hand side to first order, and substitute for Δψxi from the HUP Eq. 1.2. Likewise, an EUP-type relation is obtained from Eq. 2.10 by taking the square root, Taylor expanding to first order, and substituting for Δψpj. Next, using the substitutions
where
immediately gives
where we have relabelled ΔψEi ≡ΔψXi and
This proves that GUP- and EUP-type relations can be derived rigorously, from within the canonical quantum formalism, but a remaining criticism of the formulae above is that the uncertainties on the right-hand sides of Eqs 2.13, 2.14 are not equivalent to the uncertainties on the left. Indeed, according to the POVM model, Δψpj and Δψxi are not operationally observable quantities. They arise only in the limits σg → 0 and
Therefore, the EGUP can be rigorously derived within the canonical quantum formalism. The GUP and EUP proper then arise as limits of this more fundamental relation.
We stress that, in this model, ΔψEi ≡ΔψXi and
3 Discussion
We have shown that the three most common GURs studied in the quantum gravity literature, the GUP, EUP, and EGUP, can be derived from within the formalism of canonical quantum mechanics. A GUP-type uncertainty relation is obtained when the standard (projective) position operator is replaced by an appropriate POVM, representing finite-accuracy measurements with error bars of width σg > 0 in real space. In like manner, an EUP-type relation is obtained from finite-accuracy measurements with error bars of width
This work suggests that GUP, EUP, and EGUP phenomenology can be understood in a physically intuitive way, as a simple and natural outcome of finite-accuracy measurements. Such measurements are capable of generating all three GURs and the same phenomenology is obtained, at the level of the uncertainty relations, regardless of whether the limits
We propose that this should give pause for thought to the GUP community. If modified commutators are not necessary for GUP phenomenology, and, after nearly 30 years of research, we are no closer to resolving the pathologies that have afflicted these models since they were first proposed in the mid-1990s, then serious attempts should be made to find alternative mathematical structures that give rise to GURs. These should be capable of generating, via rigorous derivation, the uncertainty relations predicted by model-independent gedanken experiments, but without the problems associated with modified commutation relations.
In this paper, we have proposed one such model, within the context of canonical quantum theory. Another, more radical, alternative is to consider additional quantum mechanical degrees of freedom, not present in the canonical theory, which are capable of describing quantum fluctuations of the background geometry. Such a model was proposed in a recent series of works (Lake, 2019; Lake et al., 2019; Lake et al., 2020; Lake, 2021a; Lake, 2021b) and shares many features with the model described here, including the existence of a new action scale that relates the accuracies of generalised position and momentum measurements,
At first glance, this more radical alternative has nothing to do with the POVM approach described here. It requires extra degrees of freedom associated with the quantum state of the background geometry, contrary to the POVM formalism, which remains entirely within the context of canonical quantum theory. It follows from Stinespring’s dilation theorem (Stinespring, 1955; Paulsen, 2003), however, that the two formalisms are equivalent if we assume the particular values, σg ≃ lPl and
The POVM approach describes a quantum measurement of finite accuracy. The minimum resolution of the measurement may be due to technical limitations, or it can reflect the fact that the minimum length and momentum scales are fundamentally related. We postulate that in a universe with both fundamental and technological limitations to measurement accuracy, the complete description of a realistic quantum measurement should be a POVM extension of the model presented in (Lake, 2019; Lake et al., 2019). We expect that this would give rise to two additional contributions to the position and momentum variances, i.e.,
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
This work was supported by the Natural Science Foundation of Guangdong Province, grant no. 008120251030.
Acknowledgments
ML would like to acknowledge the Department of Physics and Materials Science, Faculty of Science, Chiang Mai University, for providing research facilities.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Footnotes
1In classical error analysis the term “precision” is used to refer to the statistical spread of the results whereas the term “accuracy” refers to the discrepancy between the measured value of a quantity and its true value. In keeping with this general usage, we use the term precision to refer to the quantum mechanical uncertainty and accuracy to refer to the width of the error bars associated with each individual measurement.
2In Amelino-Camelia (2017) an ingenious solution to the soccer ball problem was proposed. In this approach, the generalised momentum operators of a given modified commutator model are defined to be the generators of “generalised spatial translations.” The unitary transformation
References
Ade, P. A. R., Aghanim, N., Armitage-Caplan, C., Arnaud, M., Ashdown, M., Atrio-Barandela, F., et al. (2014). Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys. 571, A16. arXiv:1303.5076 [astro-ph.CO]. doi:10.1051/0004-6361/201321591
Adler, R. J., and Santiago, D. I. (1999). On gravity and the uncertainty principle. Mod. Phys. Lett. A 14, 1371–1381. doi:10.1142/s0217732399001462
Amelino-Camelia, G. (2017). Planck-scale soccer-ball problem: A case of mistaken identity. Entropy 19 (8), 400. arXiv:1407.7891 [gr-qc]. doi:10.3390/e19080400
Bambi, C., and Urban, F. R. (2008). Natural extension of the generalized uncertainty principle. Quant. Grav. 25, 095006. [gr-qc]. doi:10.1088/0264-9381/25/9/095006
Betoule, M., Kessler, R., Guy, J., Mosher, J., Hardin, D., Biswas, R., et al. (2014). Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. Astron. Astrophys. 568, A22. [astro-ph.CO]. doi:10.1051/0004-6361/201423413
Bolen, B., and Cavaglia, M. (2005). (Anti-)de Sitter black hole thermodynamics and the generalized uncertainty principle. Gen. Rel. Grav. 37, 1255–1262. doi:10.1007/s10714-005-0108-x
Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik Z. Physik 43, 172–198. doi:10.1007/BF01397280
Hossenfelder, S. (2013). Minimal length scale scenarios for quantum gravity. Living Rev. Rel. 16, 2. arXiv:1203.6191 [gr-qc]. doi:10.12942/lrr-2013-2
Isham, C. J. (1995). Lectures on quantum theory: Mathematical and structural foundations. London: Imperial College Press.
Kempf, A., Mangano, G., and Mann, R. B. (1995). Hilbert space representation of the minimal length uncertainty relation. Phys. Rev. D. 52, 1108–1118. arXiv:hep-th/9412167 [hep-th]. doi:10.1103/PhysRevD.52.1108
Lake, M. J. (2020). “A new approach to generalised uncertainty relations,” in Touring the Planck scale: Antonio Aurilia memorial volume, Fundamental theories of Physics, Springer. Editor P. Nicolini Accepted for publication arXiv:2008.13183v1 [gr-qc]. doi:10.48550/arXiv.2008.13183
Lake, M. J. (2019). A solution to the soccer ball problem for generalized uncertainty relations. Ukr. J. Phys. 64 (11), 1036. [gr-qc]. doi:10.15407/ujpe64.11.1036
Lake, M. J. (2021). How does the Planck scale affect qubits? Quantum Rep. 3 (1), 196–227. arXiv:2103.03093 [quant-ph]. doi:10.3390/quantum3010012
Lake, M. J., Miller, M., Ganardi, R. F., Liu, Z., Liang, S. D., and Paterek, T. (2019). Generalised uncertainty relations from superpositions of geometries. Quant. Grav. 36 (15), 155012. [quant-ph]. doi:10.1088/1361-6382/ab2160
Lake, M. J., Miller, M., and Liang, S. D. (2020). Generalised uncertainty relations for angular momentum and spin in quantum geometry. Universe 6, 56. doi:10.3390/universe6040056
Lake, M. J. (2021). Why space could be quantised on a different scale to matter. SciPost Phys. Proc. 4, 014. arXiv:2005.12724 [gr-qc]. doi:10.21468/SciPostPhysProc.4.014
Lake, M. J. (2023). “Problems with modified commutators,” in Generalized uncertainty relations: Existing paradigms and new approaches, Front. Astron. Space Sci. Editors M. J. Lake, S. D. Liang, and T. Harko
Maggiore, M. (1993). A Generalized uncertainty principle in quantum gravity. Phys. Lett. B 304, 65–69. doi:10.1016/0370-2693(93)91401-8
Nielsen, M. A., and Chuang, I. L. (2000). Quantum computation and quantum information. Cambridge: Cambridge University Press.
Park, M. i. (2008). The generalized uncertainty principle in (A)dS space and the modification of hawking temperature from the minimal length. Phys. Lett. B 659, 698–702. arXiv:0709.2307 [hep-th]. doi:10.1016/j.physletb.2007.11.090
Paulsen, V. (2003). Completely bounded maps and operator algebras. Cambridge, U.K.: Cambridge University Press.
Perlmutter, S., Aldering, G., Goldhaber, G., Knop, R. A., Nugent, P., Castro, P. G., et al. (1999). Measurements of Ω and Λ from 42 high-redshift supernovae. Astrophys. J. 517, 565–586. doi:10.1086/307221
Riess, A. G., Filippenko, A. V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P. M., et al. (1998). Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038. doi:10.1086/300499
Robertson, H. P. (1929). The uncertainty principle. Phys. Rev. 34, 163–164. doi:10.1103/PhysRev.34.163
Scardigli, F. (1999). Generalized uncertainty principle in quantum gravity from micro - black hole Gedanken experiment. Phys. Lett. B 452, 39–44. doi:10.1016/s0370-2693(99)00167-7
Schrödinger, E. (1930). About Heisenberg uncertainty relation. Bulg. J. Phys. 26, 193. [Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 19, 296 (1999)] [quant-ph/9903100].
Stinespring, W. F. (1955). Positive functions on 𝐶*-algebras. Proc. Am. Math. Soc. 6, 211–216. doi:10.1090/s0002-9939-1955-0069403-4
Tawfik, A. N., and Diab, A. M. (2015). A review of the generalized uncertainty principle. Rept. Prog. Phys. 78, 126001. [physics.gen-ph]. doi:10.1088/0034-4885/78/12/126001
Keywords: generalised uncertainty relations, generalised uncertainty principle, extended uncertainty principle, finite-accuracy measurements, POVM
Citation: Lake MJ, Miller M, Ganardi R and Paterek T (2023) Generalised uncertainty relations from finite-accuracy measurements. Front. Astron. Space Sci. 10:1087724. doi: 10.3389/fspas.2023.1087724
Received: 02 November 2022; Accepted: 09 January 2023;
Published: 08 February 2023.
Edited by:
Daniele Oriti, Ludwig-Maximilians-University Munich, GermanyReviewed by:
Olaf Lechtenfeld, Leibniz University Hannover, GermanyGiulia Gubitosi, University of Naples Federico II, Italy
Copyright © 2023 Lake, Miller, Ganardi and Paterek. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Matthew J. Lake, matthewjlake@narit.or.th