"Quantum-Spacetime Phenomenology"
Giovanni Amelino-Camelia 
1 Introduction and Preliminaries
1.1 The “Quantum-Gravity problem” as seen by a phenomenologist
1.2 Quantum spacetime vs quantum black hole and graviton exchange
1.3 20th century quantum-gravity phenomenology
1.4 Genuine Planck-scale sensitivity and the dawn of quantum-spacetime phenomenology
1.5 A simple example of genuine Planck-scale sensitivity
1.6 Focusing on a neighborhood of the Planck scale
1.7 Characteristics of the experiments
1.8 Paradigm change and test theories of not everything
1.9 Sensitivities rather than limits
1.10 Other limitations on the scope of this review
1.11 Schematic outline of this review
2 Quantum-Gravity Theories, Quantum Spacetime, and Candidate Effects
2.1 Quantum-Gravity Theories and Quantum Spacetime
2.2 Candidate effects
3 Quantum-Spacetime Phenomenology of UV Corrections to Lorentz Symmetry
3.1 Some relevant concepts
3.2 Preliminaries on test theories with modified dispersion relation
3.3 Photon stability
3.4 Pair-production threshold anomalies and gamma-ray observations
3.5 Photopion production threshold anomalies and the cosmic-ray spectrum
3.6 Pion non-decay threshold and cosmic-ray showers
3.7 Vacuum Cerenkov and other anomalous processes
3.8 In-vacuo dispersion for photons
3.9 Quadratic anomalous in-vacuo dispersion for neutrinos
3.10 Implications for neutrino oscillations
3.11 Synchrotron radiation and the Crab Nebula
3.12 Birefringence and observations of polarized radio galaxies
3.13 Testing modified dispersion relations in the lab
3.14 On test theories without energy-dependent modifications of dispersion relations
4 Other Areas of UV Quantum-Spacetime Phenomenology
4.1 Preliminary remarks on fuzziness
4.2 Spacetime foam, distance fuzziness and interferometric noise
4.3 Fuzziness for waves propagating over cosmological distances
4.4 Planck-scale modifications of CPT symmetry and neutral-meson studies
4.5 Decoherence studies with kaons and atoms
4.6 Decoherence and neutrino oscillations
4.7 Planck-scale violations of the Pauli Exclusion Principle
4.8 Phenomenology inspired by causal sets
4.9 Tests of the equivalence principle
5 Infrared Quantum-Spacetime Phenomenology
5.1 IR quantum-spacetime effects and UV/IR mixing
5.2 A simple model with soft UV/IR mixing and precision Lamb-shift measurements
5.3 Soft UV/IR mixing and atom-recoil experiments
5.4 Opportunities for Bose–Einstein condensates
5.5 Soft UV/IR mixing and the end point of tritium beta decay
5.6 Non-Keplerian rotation curves from quantum-gravity effects
5.7 An aside on gravitational quantum wells
6 Quantum-Spacetime Cosmology
6.1 Probing the trans-Planckian problem with modified dispersion relations
6.2 Randomly-fluctuating metrics and the cosmic microwave background
6.3 Loop quantum cosmology
6.4 Cosmology with running spectral dimensions
6.5 Some other quantum-gravity-cosmology proposals
7 Quantum-Spacetime Phenomenology Beyond the Standard Setup
7.1 A totally different setup with large extra dimensions
7.2 The example of hard UV/IR mixing
7.3 The possible challenge of not-so-subleading higher-order terms
8 Closing Remarks

5 Infrared Quantum-Spacetime Phenomenology

Work on Planck-scale quantum-spacetime phenomenology is a rather recent development, with a significant effort taking place over only little more than a decade. But one can already make a distinction between “traditional” and “novel” quantum-spacetime phenomenology approaches. The proposals I have reviewed in the previous two Sections 3 and 4 cover the scope of the “traditional” approach, considering UV effects that could be relevant for observations in astrophysics and/or in controlled-laboratory experiments. I devote this and the next Section 6 to the “novel” idea that Planck-scale quantization of spacetime could have valuable phenomenological implications in some IR regimes and/or that the tests could rely on cosmology.

Considering that these “novel” areas of quantum-spacetime phenomenology are in a preliminarily exploratory phase I will adopt a lower standard in the selection of topics, meaning that I will even mention some proposals that have not fully established a link to a definite scheme of spacetime quantization and/or have not fully established the availability of sensitivities that could be compellingly linked with the introduction of spacetime quantization at the Planck scale. I will rather rely on an (inevitably subjective) assessment of whether the relevant proposals provide valuable first steps in the direction of establishing in the not-so-distant future robust Planck-scale quantum-spacetime phenomenology.

5.1 IR quantum-spacetime effects and UV/IR mixing

In the long [508, 475], and so far inconclusive, search for quantum gravity and quantum spacetime the main strategy was inspired by the discovery paradigm of the 20th century, the “microscope paradigm” with discovery potential measured in terms of the shortness of the distance scales probed. But recent research has raised the possibility that by quantizing spacetime at the Planck scale one might have not only some new phenomena in a far-UV regime, but also some new phenomena in a “dual” IR regime. Actually, as compellingly stressed in Ref. [176], our present understanding of black-hole thermodynamics, and particularly the scaling S ∝ R2 of the entropy of a black hole of radius R, suggests that such effects of “UV/IR mixing” may be inevitable. It is on the basis of apparently robust hypotheses concerning the behavior of quantum gravity in the UV (Planckian) regime that we arrive at this quadratic dependence, which is surprising with respect to what one might expect in particular in quantum field theory, where cubic scaling (S ∝ R3) naturally arises. But this feature originating from the UV sector clearly should have its most profound implications in the large-distance/IR regime since the difference between quadratic dependence on the radius and cubic dependence on the radius becomes more and more significant as the radius is increased.35

Another argument in favor of UV/IR mixing is found considering a popular intuition for quantum spacetime, which relies on the introduction of an uncertainty principle for spacetime itself (in addition to the Heisenberg one, which acts in phase space). The link with UV/IR mixing can be already seen simply by considering a principle of the form δxδy ≥ λ2∗ for spatial coordinates, with λ ∗ plausibly on the order of the Planck length. This type of uncertainty relation would evidently imply that small uncertainty in x should require large uncertainty in y, and this suggests a link between probing short distance scales (small δx) and probing large distance scales (large δy).

For this last point we have more than general arguments: computations in a noncommutative spacetime compatible with this sort of uncertainty relations, the “canonical spacetime”, with noncommutativity of coordinates governed by [xμ, xν] = i𝜃μν, have found explicit manifestations of UV/IR mixing. This is particularly evident when analyzing mass renormalization within the most popular formalization of quantum field theories in such canonical noncommutative spacetimes. At one loop one finds terms in mass renormalization of the form [213Jump To The Next Citation Point, 516Jump To The Next Citation Point, 397Jump To The Next Citation Point] (for a “Φ4 scalar field theory”)

renorm 1 g2Λ2eff 1 g2m2 Λ2eff 4 Δ m2 = ------2---− ------2- log --2--+ 𝒪 (g ), (78 ) 32 π 32 π m
where Λeff is a peculiar cutoff that can be expressed in terms of a standard UV cutoff Λ, the “noncommutativity matrix” 𝜃μν and the momentum qμ of the particle as follows
2 --------1-------- Λeff = Λ −2 + q (𝜃2)ρσq . (79 ) ρ σ
Removing the cutoff Λ (Λ → ∞) one is left with Λ2eff = 1∕[qρ(𝜃2)ρσqσ], so that
2 Δren2orm ∼ ----g----- + g2m2 log [m2q (𝜃2)ρσq ]. (80 ) m qρ(𝜃2)ρσqσ ρ σ
These power-law and logarithmic IR features are the result of the UV implications of noncommutativity, which manifest themselves in a rearrangement of the renormalization procedure [213, 516, 397Jump To The Next Citation Point]. In general, the presence of such sharp features in the IR may be of some concern, since they have not (yet) been observed. And these concerns are more serious in the cases where these features are sharpest. However, it should be noticed that different choices of the matrix 𝜃 μν produce very different types of IR behavior, and it is well established that in the presence (at least in the UV sector) of supersymmetry only the logarithmic IR features survive (the power-law corrections are removed by one of the standard supersymmetry-induced cancellation mechanisms). The least virulent IR scenario is obtained by assuming the presence of UV supersymmetry and choosing a “light-like” noncommutativity matrix [19Jump To The Next Citation Point, 90] (μν μν ρσ 𝜃μν𝜃 = 𝜖μνρσ𝜃 𝜃 = 0), so that the main IR feature is a modification of the on-shell relation of the form
( ) m2 ≃ E2 − p2 + χ𝜃 m2 log E--+-⃗p-⋅ ˆu𝜃 . (81 ) m
The unit vector ˆu 𝜃 describes a preferential direction [19] determined by the matrix 𝜃μν, while the dimensionless parameter χ𝜃 also allows for an expected [397] dependence of the magnitude of the effect on the specific particle under study: since the IR feature is found in the renormalization procedure, and this in turn has an obvious dependence on the interactions of a given field with other fields in the theory, the coefficient of the logarithmic IR correction has different value for different fields.

Note that in the IR regime (small p) one can rewrite (81View Equation) as follows

m2 ≃ E2 − p2 + χ m ⃗p ⋅ ˆu , (82 ) 𝜃
so that the effect ultimately amounts to a correction that is linear in momentum. Clearly, this is a scenario in which the IR implications of UV/IR mixing are particularly soft.

Interestingly, canonical noncommutativity is not the only quantum-spacetime proposal that can motivate the study of UV/IR mixing. This is suggested by the perspective on the semi-classical limit of LQG that provided motivation for the quantum-spacetime model of Refs. [33Jump To The Next Citation Point, 34Jump To The Next Citation Point], that also inspired the models considered in Refs. [154Jump To The Next Citation Point, 155Jump To The Next Citation Point, 69Jump To The Next Citation Point]. In this LQG-inspired scenario one finds [33Jump To The Next Citation Point, 34Jump To The Next Citation Point] modifications of the dispersion relation that are linear in momentum in the IR regime, and this has motivated a phenomenology based36 on the IR dispersion relation [154Jump To The Next Citation Point, 155Jump To The Next Citation Point, 69Jump To The Next Citation Point]

2 2 2 m ≃ E − p + χ ˆpm p , (83 )
where χ ˆp is a phenomenological parameter37 analogous to χ 𝜃.

5.2 A simple model with soft UV/IR mixing and precision Lamb-shift measurements

The long-wavelength behavior of the two scenarios for “soft UV/IR mixing” summarized here in Eq. (82View Equation) and Eq. (83View Equation) evidently differ only because of the fact that invariance under spatial rotations (lost in Eq. (82View Equation)) is preserved by the scenario described in Eq. (83View Equation). Therefore, one could simultaneously consider the two scenarios, by observing that the characterization of Eq. (82View Equation) in terms of χ 𝜃 and ˆu𝜃 is applicable to the scenario of Eq. (83View Equation) by replacing ˆu𝜃 with ˆp ≡ ⃗p∕p and replacing χ 𝜃 with χˆp. However, in light of the limited scope of my review of results on “soft UV/IR mixing”, I shall be satisfied with a simplified description, assuming space-rotation invariance and limiting my focus to the effects of dispersion relations of the form

2 m2 ≃ E2 − p2 + ξ m--p , (84 ) Ep
where I also introduced a change of definition of the dimensionless coefficient, rescaling it in a way that might be relevant for connecting the IR effects with the Planck scale (χ → ξm ∕Ep, which provides no loss of generality if ξ is allowed to be particle dependent).

The phenomenology of models such as this requires a complete change of strategy with respect to the phenomenology of quantum-spacetime UV effects that I discussed in previous Sections 3 and 4 of this review. Whereas the typical search for those UV effects relied on low-precision high-energy data, for the type of IR effects that I am now considering the best options come from high-precision low-energy data. A first example of this was given in Ref. [155Jump To The Next Citation Point], most notably with a (however brief) discussion of how a dispersion relation of type (84View Equation) could be relevant for Lamb-shift measurements. Indeed, assuming Eq. (84View Equation) holds for the electron, then one should have a modification of the energy levels of the hydrogen atom. And in light of the high precision of certain Lamb shift measurements (which Ref. [155] assesses as being better than one part in 105, see also, e.g., Refs. [328, 546]) one can use this observation to place valuable limits on parameters such as ξ (and χ) for the electron.

5.3 Soft UV/IR mixing and atom-recoil experiments

Evidently the ansatz (84View Equation) is such that if particles of different mass had the same value38 of ξ then the effect would be seen more easily for heavier (more massive) types of particles.

I find particularly striking the case of measurements of the recoil of cesium (and rubidium) atoms. For cesium one would assume, following Eq. (84View Equation), that

m2 ≃ E2 − p2 + ξCsm2 -p--, (85 ) MP
where ξ Cs is the ξ parameter for the case of cesium atoms.

The measurement strategy we proposed in Ref. [69Jump To The Next Citation Point] for testing Eq. (85View Equation) with atoms is applicable to measurements of the “recoil frequency” of atoms with experimental setups involving one or more “two-photon Raman transitions” [548Jump To The Next Citation Point]. The strategy of the analysis is best described by setting aside initially the possibility of Planck-scale effects, and looking at the recoil of an atom in a two-photon Raman transition from the perspective adopted in Ref. [548Jump To The Next Citation Point], which provides a convenient starting point for the Planck-scale generalization that is of interest here. One can impart momentum to an atom through a process involving absorption of a photon of frequency ν and (stimulated) emission, in the opposite direction, of a photon of frequency ν′. The frequency ν is computed taking into account a resonance frequency ν ∗ of the atom and the momentum the atom acquires, recoiling upon absorption of the photon: ν ≃ ν + (h ν + p )2∕(2m ) − p2∕(2m ) ∗ ∗, where m is the mass of the atom (e.g., mCs ≃ 124 GeV for cesium), and p is its initial momentum. The emission of the photon of frequency ν ′ must be such as to de-excite the atom and impart to it additional momentum: ν′ + (2hν∗ + p)2∕(2m ) ≃ ν∗ + (h ν∗ + p )2∕(2m ). Through this analysis one establishes that by measuring Δ ν ≡ ν − ν ′, in cases in which ν∗ and p can be accurately determined, one actually measures h ∕m for the atoms:

Δ ν h --------------= --. (86 ) 2ν∗(ν∗ + p∕h ) m
This result has been confirmed experimentally with remarkable accuracy. A powerful way to illustrate this success is provided by comparing the results of atom-recoil measurements of Δ ν ∕[ν∗(ν∗ + p∕h)] and of measurements [277Jump To The Next Citation Point] of 2 α, the square of the fine structure constant. 2 α can be expressed in terms of the mass m of any given particle [548Jump To The Next Citation Point] through the Rydberg constant, R ∞, and the mass of the electron, me, in the following way [548Jump To The Next Citation Point]: α2 = 2R ∞ m-h- mem. Therefore, according to Eq. (86View Equation) one should have
2 -----Δ-ν------= -α---me-mu- , (87 ) 2ν ∗(ν∗ + p∕h ) 2R ∞ mu m
where m u is the atomic mass unit and m is the mass of the atoms used in measuring Δ ν∕[ν (ν + p∕h )] ∗ ∗. The outcomes of atom-recoil measurements, such as the ones with cesium reported in Ref. [548Jump To The Next Citation Point], are consistent with Eq. (87View Equation) to an accuracy of a few parts in 109. The fact that Eq. (86View Equation) has been verified to such a high degree of accuracy proves to be very valuable, since it turns out [69Jump To The Next Citation Point] that modifications of the dispersion relation of type (85View Equation) require a modification of Eq. (86View Equation). Following Ref. [69Jump To The Next Citation Point] one easily finds
2 ν∗(h ν∗ + p) m Δ ν ≃ ------m------+ ξCs M---ν∗, (88 ) P
and in turn in place of Eq. (87View Equation) one has
[ ( )] Δ ν ( m ) m α2 me mu -------------- 1 − ξCs ----- -------- = -------- --- . 2ν∗(ν∗ + p∕h) 2MP hν ∗ + p 2R ∞ mu m
This equation has been arranged so that on the left-hand side it is easy to recognize that the small quantum-spacetime effect in this specific context receives a sizable “amplification” by the large hierarchy of energy scales m ∕(hν∗ + p), which in typical experiments of the type here of interest can be [548Jump To The Next Citation Point] of order 9 ∼ 10.

This turns out to be just enough to provide the desired “Planck-scale sensitivity”: one easily finds that combining the measurements on cesium reported in Ref. [548] and the determination of α2 reported in Ref. [277], one can establish [69] that ξ = − 1.8 ± 2.1 Cs.

It is interesting that, besides tests of IR modifications of the dispersion relation, these atom-recoil studies can also be used to investigate possible IR modifications of the law of conservation of momentum. An example of such an analysis is given in Ref. [89].

5.4 Opportunities for Bose–Einstein condensates

The use of atoms in quantum-spacetime phenomenology immediately confronts us with issues that are presently beyond the reach of available theoretical results. A legitimate expectation is that quantum-spacetime effects for atoms could be weaker than for the particles that compose atoms, as a result of the sort of “average-out effects” that one is often expected in the quantum-spacetime literature. This would have to be modeled by introducing an extra suppression factor (a sort of “compositeness factor”) in addition to the Planck-scale suppression that is standard in quantum-spacetime phenomenology. Analyses not making room for such an additional suppression might overestimate the Planck-scale-sensitivity reach of the relevant experiments. On the other hand we are at present not sure whether such compositeness-suppression factors are truly needed, or at least if they are needed in all contexts and in all quantum-spacetime models. For example, it is not unreasonable to imagine that in appropriate quantum-spacetime models, when we achieve the ability to analyze them in detail, we might find that as long as a particle is to be handled as a quantum state (far from its classical limit) then it might be irrelevant for the magnitude of quantum-spacetime effects whether the particle is composite or “fundamental”.

This issue of compositeness will surely gradually take an important role in quantum-spacetime research, but at present it is at a very preliminary stage of investigation, and I shall therefore set it aside. However, do note that if particles composed of a very large number of constituent particles experience Planck-scale effects unsuppressed by their compositeness, then not only atoms but also (and perhaps more powerfully) Bose–Einstein condensates could prove to be a very valuable opportunity for quantum-spacetime phenomenology.

And it is noteworthy that in the recent quantum-spacetime-phenomenology literature there has already been a surge of interest in the possibilities offered by Bose–Einstein condensates, as seen in Refs. [542, 472, 139Jump To The Next Citation Point, 138Jump To The Next Citation Point]. In particular, Refs. [139, 138] study Bose–Einstein condensates adopting a perspective on soft UV/IR mixing that is closely related to the one discussed for atoms in the previous Section 5.3.

5.5 Soft UV/IR mixing and the end point of tritium beta decay

Perhaps the most tempting opportunity for the phenomenology of UV/IR mixing comes from studies of the low-energy beta decay spectrum of tritium, 3H → 3He + e− + ¯νe, which have produced so far some rather puzzling results [545Jump To The Next Citation Point, 370Jump To The Next Citation Point]. It is well understood (see, e.g., Refs. [121, 174]) that these puzzles could be addressed by introducing deformed rules of kinematics. And it is intriguing that studies conducted near the endpoint of tritium beta decay are the only known way to accurately investigate the properties of neutrinos in a non-relativistic (non-ultrarelativistic) regime, where their momenta could be comparable to their (tiny) masses. So, it would seem to be a very natural opportunity for advocating UV/IR mixing as a possible explanation. However, the evidence available so far is not very encouraging for the hope of attributing the magnitude of the reported anomalies to IR effects induced by the Planck scale. Still, it is noteworthy that specifically the simple model for soft UV/IR mixing that I described in the previous Sections 5.2 and 5.3 has just the right structure for producing the sort of anomalies that are being reported, as was first stressed in Ref. [154Jump To The Next Citation Point].

The main point of Ref. [154Jump To The Next Citation Point] is centered on the properties of the function K (E ) conventionally used to characterize the Kurie plot of tritium beta decay:

[∫ ]1∕2 K (E ) = dp ν p2ν δ (Q − E − Eν) , (89 )
where Q is the difference between initial and final masses of the process, Q ≃ M3H − M3He − me (and, therefore, Q is the sum of the neutrino energy, E ν, and the kinetic energy of the electron, E).

Using standard dispersion relations one finds

[ ∘ --------2----2-]1∕2 K (E ) = (Q − E ) (Q − E ) − m ν) , (90 )
which does not fit well with the available data near the endpoint [545, 370]. It was observed in Ref. [154Jump To The Next Citation Point] that instead using a modified dispersion relation of type (84View Equation), for negative ξ, one obtains better agreement, but this requires that ξm2ν∕Ep have a value of a few eV. In turn this implies a value39 of ξ that is extremely large with respect to the natural quantum-spacetime estimate ξ ∼ 1, and as a result the case for a quantum-spacetime interpretation is rather weak at present. Still, this exciting experimental situation deserves to be further pursued: perhaps we are modeling soft UV/IR mixing correctly but we have developed the wrong intuition about the role the Planck scale should play, or perhaps one should look at alternative ways to model UV/IR mixing.

5.6 Non-Keplerian rotation curves from quantum-gravity effects

In addition to precision measurements on particles of peculiarly low momentum, another very clear opportunity for UV-IR mixing is provided by data on the behavior of gravity on very large distance scales. And in that context speculating about new-physics phenomena is fully justified by the observed non-Keplerian features of the rotation curves of galaxies or clusters [183]. These non-Keplerian features are usually interpreted as motivation for introducing dark matter (or other non-quantum-gravity new physics, such as MOND [416]), but, in light of the recent awareness of the possibility of UV/IR mixing, it is legitimate to speculate that they may be at least in part due to quantum-spacetime effects.

The perspective one might adopt in trying to profit from this opportunity is similar to when one works within standard quantum field theories and derives an “effective potential” (usually obtained through the calculation of loop contributions) that corrects the tree-level classical potential.

Interestingly, the type of modifications of dispersion relations that have been motivated by quantum-spacetime research do automatically suggest that the Newtonian potential should receive some corresponding corrections. In fact, the Newtonian potential is produced by a static point source when the field that mediates the force described by the potential has energy-momentum space (inverse) propagator − 1 2 2 G (E,p) = E − p. In general, if the field that mediates the force has a different propagator, −1 G def(E,p), the Newtonian potential produced at the spatial point ⃗r by a point-like mass M, located at the origin, is replaced by the potential obtained by computing [283]

∫ d3p V(⃗r) = L2pM ----Gdef (0,⃗p) ei⃗p⋅⃗r , (91 ) 2π2
i.e., the potential is the spatial Fourier transform of the propagator evaluated at E = 0.

A more articulated argument for modifications of the Newton potential at large distances from a quantum-spacetime perspective has been put forward as part of the mentioned research program on “asymptotic safety”. This is done in Ref. [469Jump To The Next Citation Point], which indeed adopts as a working assumption the availability of a quantum field theory of gravity whose underlying degrees of freedom are those of the spacetime metric, defined nonperturbatively as a fundamental, “asymptotically-safe” theory. Obtaining definite predictions for the rotation curves of galaxies or clusters within this formalism is presently well beyond our technical capabilities. However, preliminary studies of the renormalization-group behavior provide encouragement for a certain level of analogy between this theory and non-Abelian Yang–Mills theories, and, relying in part on this analogy, Ref. [469] argued that one could obtain non-Keplerian features from renormalization.

5.7 An aside on gravitational quantum wells

Another opportunity for studies of UV/IR mixing is provided by measurements performed on neutron quantum states in the gravity field of the Earth, such as the striking ones reported in Refs. [428Jump To The Next Citation Point, 429Jump To The Next Citation Point]. I have nothing to report on this that would fit the main focus of this review, concerning Planck-scale quantum pictures of spacetime, but it seemed worth mentioning this nonetheless, especially in light of the fact that this class of low-energy studies (candidates for the investigation of UV/IR mixing) have already been analyzed from the perspective of some quantum spacetimes, even though so far all such studies have introduced spacetime quantization at scales that are very far from the Planck scale (much lower energy scales, much greater distance scales).

Since I am already here diverting from the main theme of the review, I shall be satisfied confining the discussion of quantum-spacetime studies of the gravitational quantum well to the particularly interesting points made in Refs. [118Jump To The Next Citation Point, 102Jump To The Next Citation Point, 483Jump To The Next Citation Point, 137Jump To The Next Citation Point]. The studies in Refs. [118Jump To The Next Citation Point, 102Jump To The Next Citation Point, 483Jump To The Next Citation Point] all assumed “canonical noncommutativity” of spacetime coordinates:

[xj,xk ] = i𝜃jk , [xj,x0] = i𝜃j0, (92 )
where I separated the space/space noncommutativity (𝜃 ⁄= 0 jk) from the space/time noncommutativity (𝜃j0 ⁄= 0).

And Refs. [118Jump To The Next Citation Point, 102Jump To The Next Citation Point, 483Jump To The Next Citation Point] agree on the fact that pure space/space noncommutativity (𝜃j0 = 0) has no significant implications for the gravitational quantum well. However, Ref. [483] notices that with space/time noncommutativity (𝜃 ⁄= 0 j0) there are tangible consequences for the gravitational quantum well so that in turn one can use the measurement results of Refs. [428Jump To The Next Citation Point, 429Jump To The Next Citation Point] to put bounds on space/time noncommutativity,40 although only at the level −9 2 𝜃j0 < 10 m (whereas interest from the Planck-scale-quantum-spacetime side would focus in the neighborhood of 𝜃j0 ∼ 10 −70 m2).

Refs. [118, 102] make the choice of combining space/space noncommutativity with a noncommutativity of momentum space:

[pj,pk] = iψjk. (93 )
It then turns out that this noncommutativity of momentum space does tangibly affect the analysis of the gravitational quantum well. So that in turn one can use the measurement results of Refs. [428Jump To The Next Citation Point, 429Jump To The Next Citation Point] to place bounds at the level −6 2 ψjk < 10 eV.

Ref. [137Jump To The Next Citation Point] is an example of analysis of the gravitational quantum well not from the viewpoint of spacetime noncommutativity, but rather from the viewpoint of the scheme of spacetime quantization introduced in Refs. [323, 322], which is centered on a modification of the Heisenberg principle

[x ,p ] = iδ (1 + βp2 ). (94 ) j k jk
The parameter β does turn out [137] to affect the analysis of the gravitational quantum well, and using the measurement results of Refs. [428, 429] one can place bounds at the level β < 10−18m2.

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