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) Δ x i Δ p j ≳ ℏ 2 δ j i , (1.1) or derived rigorously from the canonical quantum formalism, yielding (Isham, 1995; Rae, 2002) Δ ψ x i Δ ψ p j ≥ ℏ 2 δ j i . (1.2) The inequality in Eq. 1.2 is exact and, unlike the heuristic uncertainties Δx ⁱ and Δp _j in Eq. 1.1, Δ _ψ x ⁱ and Δ _ψ p _j represent well-defined standard deviations of the probability distributions |ψ(x)|² and | ψ ̃ ℏ ( p ) | 2 , respectively, where the momentum space representation of the particle wave function is given by the ℏ-scaled Fourier transform of its position space representation: ψ ̃ ℏ p = 1 2 π ℏ 3 ∫ ψ x e − i ℏ p . x d 3 x . (1.3) 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, O ̂ 1 and O ̂ 2 (Robertson, 1929; Schrödinger, 1930), Δ ψ O 1 Δ ψ O 2 ≥ 1 2 | ⟨ ψ | O ̂ 1 , O ̂ 2 | ψ ⟩ | , (1.4) with the canonical position-momentum commutator, x ̂ i , p ̂ j = i ℏ δ j i I ̂ . (1.5)

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), Δ x i ≳ ℏ 2 Δ p j δ j i 1 + α 0 2 G ℏ c 3 Δ p j 2 , (1.6) where α ₀ is a numerical constant of order unity (Maggiore, 1993; Adler and Santiago, 1999; Scardigli, 1999). By minimising the right-hand side with respect to Δp _j , the GUP implies the existence of a minimum position uncertainty of the order of the Planck length, l Pl = ℏ G / c 3 ≃ 1 0 − 33 c m .

Reconsidering Heisenberg’s arguments in the presence of a constant dark energy density ρ _Λ = Λc ²/(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), Δ p j ≳ ℏ 2 Δ x i δ j i 1 + 2 η 0 Λ Δ x i 2 , (1.7) where η ₀ 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, m dS c = ℏ Λ / 3 ≃ 1 0 − 56 g . cm s − 1 . This is physically reasonable since it is the minimum momentum that a canonical quantum particle can possess, when its wave function is localised within the asymptotic de Sitter horizon, which is comparable to the present day radius of the Universe r U ( t 0 ) ≃ l dS = 3 / Λ ≃ 1 0 28 c m .

Combining both effects yields the extended generalised uncertainty principle (EGUP), Δ x i Δ p j ≳ ℏ 2 δ j i 1 + α 0 2 G ℏ c 3 Δ p j 2 + 2 η 0 Λ Δ x i 2 , (1.8) 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 x ̂ i , p ̂ j = i ℏ δ j i I ̂ ↦ X ̂ i , P ̂ j = i ℏ δ j i I ̂ + α 0 2 G ℏ c 3 P ̂ j 2 + 2 η 0 Λ X ̂ i 2 (1.9) gives rise to an EGUP-type uncertainty relation, at least when both ⟨ P ̂ j ⟩ ψ = 0 and ⟨ X ̂ i ⟩ ψ = 0 (Kempf et al., 1995). Here, we use capital letters to denote modified operators, which generate modified commutators, and lower case letters to denote their canonical quantum counterparts. However, the assumption above is problematic since, even if both ⟨ P ̂ j ⟩ ψ = 0 and ⟨ X ̂ i ⟩ ψ = 0 in a given frame of reference, a simple shift of coordinate origin or a Galilean velocity boost of the observer alters the numerical value of the associated Schrödinger-Robertson bound: Δ ψ X i Δ ψ P j ≥ ℏ 2 δ j i 1 + α 0 2 G ℏ c 3 Δ ψ P j 2 + ⟨ P ̂ j ⟩ ψ 2 + 2 η 0 Λ Δ ψ X i 2 + ⟨ X ̂ i ⟩ ψ 2 . (1.10) 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 σ _g ≃ l _Pl and σ ̃ g ≃ m dS c , respectively, give rise to the GUP and EUP, to first order in the relevant Taylor expansion. These individual relations may then be combined to give the EGUP.

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, x ̂ , with POVM elements corresponding to the result x: E ̂ x ≔ ∫ g x ′ − x | x ′ 〉 〈 x ′ | d 3 x ′ , (2.1) where g(x′ − x) is any normalised function, ∫|g(x′ − x)|²d³x′ = 1. These elements satisfy the relations E ̂ x † E ̂ x ≥ 0 and ∫ E ̂ x † E ̂ x d 3 x = I ̂ , as required, so that Eq. 2.1 defines a standard POVM in canonical quantum mechanics (Nielsen and Chuang, 2000). From here on, we refer to g as the “envelope function” of the measure. For spherically symmetric functions the envelope is centred on the value x, and, for the sake of concreteness, we may imagine |g(x′ − x)|² as a three-dimensional Gaussian distribution with mean x and standard deviation σ _g .

Finite-accuracy position measurements, conducted on an arbitrary state |ψ⟩, then give rise to the first and second order moments 〈 E x 〉 ψ = ∫ x 〈 ψ | E ̂ x † E ̂ x | ψ 〉 d 3 x = 〈 x 〉 g + 〈 x 〉 ψ , 〈 E x 2 〉 ψ = ∫ x 2 〈 ψ | E ̂ x † E ̂ x | ψ 〉 d 3 x = 〈 x 2 〉 g + 〈 x 2 〉 ψ , (2.2) where ⟨ x n ⟩ f ≔ ∫ x n | f ( x ) | 2 d 3 x with f(x) = g(x) or ψ(x). Since |g(x′ − x)|² is a normalised function centred on x′ = x, ⟨x⟩ _g = 0, and the corresponding variance is given by Δ ψ E x 2 = Δ ψ x 2 + σ g 2 , (2.3) where σ g ≔ σ g i e i and σ g i denotes the width of |g|² in each coordinate direction x ⁱ . By spherical symmetry, σ g i = σ g for all i, and we may rewrite Eq. 2.3 in terms of the individual components as Δ ψ E i 2 = Δ ψ x i 2 + σ g 2 , (2.4) where we have used the shorthand notation Δ ψ E i ≡ Δ ψ E x i .

In like manner, finite-accuracy momentum measurements may be introduced via the operators E ̂ p ≔ ∫ g ̃ p ′ − p | p ′ 〉 〈 p ′ | d 3 p ′ , (2.5) where ∫ | g ̃ ( p ′ − p ) | 2 d p ′ = 1 , but it is important to note that there is no intrinsic relation between the functions g and g ̃ , which may be chosen independently for a given POVM model. Nevertheless, if both |g|² and | g ̃ | 2 represent Gaussian distributions, which is perhaps the most natural choice for an envelope function, then g and g ̃ are related via a Fourier transform, g ̃ p ′ − p = ∫ g x ′ − x e i β x ′ − x . p ′ − p d 3 x ′ , (2.6) where the new action scale β ≠ ℏ is given by β ≔ 2 σ g σ ̃ g , (2.7) and σ ̃ g is the standard deviation of | g ̃ | 2 . However, it is equally important to note that there is nothing fundamental about the relation Eq. 2.6. Unlike the ℏ-scaled Fourier transform relating the position and momentum space representations of the quantum wave function, Eq. 1.3, the β-scaled transform relates the “envelope functions” of the model.

Finite-accuracy momentum measurements, conducted on an arbitrary state |ψ⟩, then give rise to the first and second order moments 〈 E p 〉 ψ = ∫ p 〈 ψ | E ̂ p † E ̂ p | ψ 〉 d 3 p = 〈 p 〉 g + 〈 p 〉 ψ , 〈 E p 2 〉 ψ = ∫ p 2 〈 ψ | E ̂ p † E ̂ p | ψ 〉 d 3 p = 〈 p 2 〉 g + 〈 p 2 〉 ψ , (2.8) where ⟨ p n ⟩ f ≔ ∫ p n | f ̃ ( p ) | 2 d 3 p with f ̃ ( p ) = g ̃ ( p ) or ψ ̃ ℏ ( p ) . Since | g ̃ ( p ′ − p ) | 2 is normalised and centred at p′ = p, ⟨p⟩ _g = 0, and Δ ψ E p 2 = Δ ψ p 2 + σ ̃ g 2 , (2.9) where σ ̃ g ≔ σ ̃ g j e j and σ ̃ g j denotes the width of | g ̃ | 2 in each momentum space direction p _j . Again employing spherical symmetry, σ ̃ g j = σ ̃ g for all j, Eq. 2.9 may be rewritten in terms of the individual components as Δ ψ E j 2 = Δ ψ p j 2 + σ ̃ g 2 , (2.10) where we have again used the shorthand Δ ψ E j ≡ Δ ψ E p j .

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 Δ _ψ x ⁱ 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 Δ _ψ p _j . Next, using the substitutions σ g ≔ 2 α 0 l Pl , σ ̃ g ≔ 6 η 0 m dS c , (2.11) where l Pl ≔ ℏ G / c 3 , m dS c ≔ ℏ Λ / 3 , (2.12) immediately gives Δ ψ X i ≳ ℏ 2 Δ ψ p j δ j i 1 + α 0 2 G ℏ c 3 Δ ψ p j 2 , (2.13) Δ ψ P j ≳ ℏ 2 Δ ψ x i δ j i 1 + 2 η 0 Λ Δ ψ x i 2 , (2.14) where we have relabelled Δ _ψ E _i ≡Δ _ψ X ⁱ and Δ ψ E j ≡ Δ ψ P j , for convenience. These expressions are formally analogous to the heuristic relations, Eqs. 1.6, 1.7, respectively, but with Δp _j and Δx ⁱ on the right replaced by the well-defined standard deviations Δ _ψ p _j and Δ _ψ x ⁱ .

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, Δ _ψ p _j and Δ _ψ x ⁱ are not operationally observable quantities. They arise only in the limits σ _g → 0 and σ ̃ g → 0 , respectively, in which both Eqs 2.13, 2.14 reduce to the standard HUP Eq. 1.2. This objection can be overcome, however, by first substituting for Δ _ψ x ⁱ from Eq. 1.2 in Eq. 2.4 and then again for Δ _ψ p _i from Eq. 2.10. This gives rise to an uncertainty relation between the observable standard deviations, Δ _ψ E _i ≡Δ _ψ X ⁱ and Δ ψ E j ≡ Δ ψ P j . It is straightforward to show that, taking the square root, Taylor expanding to first order, and neglecting the final term of order σ g σ ̃ g ≃ l Pl . m dS c , this relation reduces to Δ ψ X i Δ ψ P j ≳ ℏ 2 δ j i 1 + α 0 2 G ℏ c 3 Δ ψ P j 2 + 2 η 0 Λ Δ ψ X i 2 . (2.15) 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, Δ _ψ E _i ≡Δ _ψ X ⁱ and Δ ψ E j ≡ Δ ψ P j represent the physically observable precisions, obtained from generalised position and momentum measurements with finite accuracies σ _g > 0 and σ ̃ g > 0 . By contrast, the canonical Hamiltonian is determined by the canonical (projective) position and momentum operators, x ̂ and p ̂ , via H ̂ = p ̂ 2 / ( 2 m ) + V ( x ̂ ) , where the former obey the canonical Heisenberg algebra: [ x ̂ i , p ̂ j ] = i ℏ δ j i I ̂ , [ x ̂ i , x ̂ j ] = 0 , [ p ̂ i , p ̂ j ] = 0 . This leaves the canonical Heisenberg and Schrödinger equations unchanged and neatly evades the pathologies that afflict modified commutator models (Lake, 2020; Hossenfelder, 2013; Tawfik and Diab, 2014; Tawfik and Diab, 2015; Lake et al., 2023).

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 σ _g > 0 in real space. In like manner, an EUP-type relation is obtained from finite-accuracy measurements with error bars of width σ ̃ g > 0 in momentum space. These can be combined to give a relation that is formally analogous to the EGUP and the standard EGUP is recovered by setting σ _g ≃ l _Pl, the Planck length, and σ ̃ g ≃ m dS c , where m dS = ( ℏ / c ) Λ / 3 is the de Sitter mass.

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 ( Δ ψ X i ) min = σ g and ( Δ ψ P j ) min = σ ̃ g are fundamental, or merely effective, as an outcome of an imperfect measurement scheme.

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, β ≔ 2 σ g σ ̃ g ≃ 1 0 − 61 ℏ (*). The fundamental difference between the two models is the existence of new degrees of freedom in the latter. From this, it follows that the new action scale β implies a modified de Broglie relation of the form p′ = ℏ k + β(k′ − k), where, here, p′ denotes the observable momentum. Heuristically, the non-canonical term β(k′ − k) can be interpreted as an additional momentum “kick,” transferred to the canonical wave function by a quantum fluctuation of the background. The interested reader is referred to (Lake, 2020; Lake, 2019; Lake et al., 2019; Lake et al., 2020; Lake, 2021a; Lake, 2021b; Lake et al., 2023) for further details.

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, σ _g ≃ l _Pl and σ ̃ g ≃ m dS c , and hence the relation (*) above. The POVM picture results from tracing out the x′ (p′) degrees of freedom associated with quantum fluctuations of the background and the x′ (p′) degrees of freedom appear as a consequence of dilating the POVM.

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., σ g 2 + σ h 2 and σ ̃ g 2 + σ ̃ h 2 , respectively, where g is the fundamental smearing function that models the quantum indeterminacy of space-time, and h is the envelope function of a realistic detector. In the limit σ _h ≫ σ _g , σ ̃ h ≫ σ ̃ g , which corresponds to all present-day measurements, the latter are expected to dominate the former.