"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

4 Other Areas of UV Quantum-Spacetime Phenomenology

Tests of Lorentz symmetry, and particularly of the form of the dispersion relation, probably make up something on the order of half of the whole quantum-spacetime-phenomenology literature. The other half is spread over a few other, evidently less developed, research lines. Nonetheless, for some of these other research lines the literature has reached some nonnegligible maturity, and even those that are at preliminary stages of development could be precious potential opportunities for quantum-spacetime research.

Evidently, the most challenging part of this review work concerns these other components of quantum-spacetime-phenomenology research, since it is harder to summarize and organize intelligibly the results and scopes of research programs that are still in early stages of development. But it is also the part of this review that could be most valuable, since there is already some work [52, 308, 485, 294] attempting to summarize and review, although more concisely than done here in Section 3, the results obtained by the quantum-spacetime phenomenology of Planck-scale modified dispersion relations.

In reporting on the work done in these other quantum-spacetime-phenomenology reseach lines, I shall use as one of the guiding concepts the one of assessing whether a given research program concerns UV quantum-spacetime effects or IR quantum-spacetime effects. The typical situation for a UV quantum-spacetime effect is that it takes the form of correction terms that grow with the energy of the particles, and whose significance is therefore increasingly high as the energy of the particles increases. For any given standard-physics (no-quantum-spacetime) prediction A0, this will take the general form

( ) En A0 −→ A0 1 + η#--n (with context∕theory specific numerical factorη# ) (37 ) E p

This is the type of quantum-spacetime effect that one traditionally one to be inevitably produced by any form of spacetime quantization, and is the focus of this section. The possibility of “IR quantum-spacetime effects”, effects that are due to Planck-scale spacetime quantization but are significant in some deep-IR regime, came to the attention of the community only rather recently, emerging mainly from work on “IR/UV mixing in quantum spacetime”, and I shall focus on it in the next Section 5.

4.1 Preliminary remarks on fuzziness

In this review, as natural for phenomenology, I am primarily looking at quantum-spacetime effects from the perspective of the type of pre-quantum-spacetime laws that they affect (so we have “departures from classical spacetime symmetries”, “violations of the quantum-mechanical coherence”, and so on). And our experimental opportunities are such that the main focus is on how spacetime quantization could affect particle propagation (and, for a restricted sample of phenomenological opportunities, interactions among particles). For this section on “other UV quantum-spacetime effects” a significant role (particularly noticeable in Sections 4.2, 4.3, and 4.5 will be played by the idea that quantum-spacetime effects may introduce an additional irreducible contribution to the fuzziness of the worldlines of particles.

This should be contrasted to the content of Section 3, which focuses mainly on phenomenological proposals involving mechanisms for systematic departures from the currently-adopted laws of propagation of (and interaction among) particles. In most cases such systematic effects amount to departures from the predictions of Lorentz symmetry (such as a systematic dependence on the velocity of a massless particle on its energy, which would produce a systematic difference between the arrival times of high-energy and low-energy photons that are simultaneously emitted).

If it ends up being the case that the correct quantum-spacetime picture does not provide us any such systematic effects, then we will be left with non-systematic effects, i.e., “fuzziness” [57Jump To The Next Citation Point]. In looking for such effects we can be guided by the intuition that spacetime quantization might act as an environment inducing apparently random fluctuations in certain observables. For example, by distance fuzziness one does not describe an effect that would systematically gives rise to larger (or smaller) distance-measurement results, but rather one describes a sort of new uncertainty principle for distance measurements.

This distinction between systematic and nonsystematic effects can easily be characterized for any given observable Xˆ for which the pre-quantum-spacetime theoretical prediction can be described in terms of a “prediction” X and, possibly, a fundamental (ordinarily quantum mechanical) “uncertainty” δX. The effects of spacetime quantization in general could lead [57Jump To The Next Citation Point] to a new prediction X ′ and a new uncertainty δX ′. One would attribute to quantum spacetime the effects

′ (ΔX )QG ≡ X − X (38 )
′ (δX )QG ≡ δX − δX . (39 )
One can speak of purely systematic quantum-spacetime effects when (ΔX ) ⁄= 0 QG and (δX ) = 0 QG, while the opposite case, (ΔX )QG = 0 and (δX )QG > 0, can be qualified as purely non-systematic. It is likely that for many observables both types of quantum-spacetime effect may be present simultaneously, but it is natural that at least the first stages of development of a quantum-spacetime phenomenology on an observable Xˆ be focused on one or the other special case ((δX )QG = 0 or (ΔX )QG = 0). Clearly, the discusions of effects given in Section 3 were all with (δX ) = 0 QG, while for most of the proposals discussed in this section the main focus will be on the effects characterized by (ΔX )QG = 0.

4.2 Spacetime foam, distance fuzziness and interferometric noise

The scenarios for spacetime fuzziness that are most studied from a quantum-spacetime perspective are intuitively linked to the notion of “spacetime foam”, championed by Wheeler and studied extensively in the quantum-gravity literature, more or less directly, for several decades (see, e.g., Refs. [547, 203, 178, 281, 150, 250, 553]). From a modern perspective one is attempting to characterize the physics of matter particles as effectively occurring in an “environment” of short-distance quantum-gravitational degrees of freedom. And one may expect that for propagating particles with wavelength much larger than the Planck length, when it may be appropriate to integrate out these short-distance quantum-gravitational degrees of freedom, the main residual effect of short-distance gravity would indeed be an additional contribution to the fuzziness of worldlines.

While in full-fledged quantum-spacetime theories, such a LQG, such analyses are still beyond our reach, one can find partial encouragement for this intuition in recent progress on the understanding of quantum gravity in 3D (2+1-dimensional) spacetime. Studies such as the ones reported in Refs. [75Jump To The Next Citation Point, 237Jump To The Next Citation Point, 412, 152, 259Jump To The Next Citation Point, 238Jump To The Next Citation Point, 313, 441Jump To The Next Citation Point] establish that for 3D quantum gravity (exploiting the much lower complexity than for the 4D case) we are able to perform the task needed for studies of spacetime foam: we can actually integrate out gravity, reabsorbing its effects into novel properties for a gravity-free propagation of particles. And foaminess is formalized in the fact that this procedure integrating out gravity leaves us with a theory of free particles in a noncommutative spacetime, Refs. [75, 237, 259, 238Jump To The Next Citation Point], specifically a spacetime with “Lie-algebra noncommutativity”

γ [xα, xβ] = iκ αβxγ (40 )
(in particular the choice of κγαβ as the Levi-Civita tensor is the one suggested by the direct derivation given in Ref.  [441]). In other words, upon integrating out the gravitational degrees of freedom, the quantum dynamics of matter fields coupled to 3D gravity is effectively described [238] by matter fields in a noncommutative spacetime, a fuzzy/foamy spacetime.

While the only direct/deductive derivations of such results are for 3D quantum gravity, it is natural to take that as a starting point for the study of real 4D quantum gravity, whereas analogous results are still unavailable. And a sizable literature has been devoted to the search of possible experimental manifestations of “spacetime foam”. Several subsections of this section concern related phenomenological proposals. I start with spacetime-foam test theories, whose structure renders them well suited for interferometric tests.

4.2.1 Spacetime foam as interferometric noise

The first challenge for a phenomenology investigating the possibility of spacetime foam originates in the fact that Wheeler’s spacetime foam intuition, while carrying strong conceptual appeal, cannot on its own be used for phenomenology, since it is not characterized in terms of observable properties. The phenomenology then is based on test theories inspired by the spacetime-foam intuition.

A physical/operative definition of at least one aspect of spacetime foam is given in Refs. [51Jump To The Next Citation Point, 54Jump To The Next Citation Point, 53Jump To The Next Citation Point, 433Jump To The Next Citation Point] and is well suited for a phenomenology based on interferometry27. According to this definition the fuzziness/foaminess of a spacetime is established[51Jump To The Next Citation Point, 54Jump To The Next Citation Point, 53Jump To The Next Citation Point, 433Jump To The Next Citation Point] on the basis of an analysis of strain noise28 in interferometers set up in that spacetime. In achieving their remarkable accuracy, modern interferometers must deal with several classical-physics strain noise sources (e.g., thermal and seismic effects induce fluctuations in the relative positions of the test masses). And importantly strain noise sources associated with effects due to ordinary quantum mechanics are also significant for modern interferometers (the combined minimization of photon shot noise and radiation pressure noise leads to a noise source that originates from ordinary quantum mechanics [486Jump To The Next Citation Point]). One can give an operative definition [51Jump To The Next Citation Point, 53Jump To The Next Citation Point] of fuzzy/foamy spacetime in terms of a corresponding additional source of strain noise. A theory in which the concept of distance is fundamentally fuzzy in this operative sense would be such that the read-out of an interferometer would still be noisy (because of quantum-spacetime effects) even in the idealized limit in which all classical-physics and ordinary-quantum-mechanics noise sources are completely eliminated/subtracted.

4.2.2 A crude estimate for laser-light interferometers

Before even facing the task of developing test theories for spacetime foaminess in interferometry it is best to first check whether there is any chance of using realistic interferometric setups to uncover effects as small as expected if introduced at the Planck scale. A first encouraging indication comes from identifying the presence of a huge amplifier in modern interferometers: a well-known quality of these modern interferometers is their ability to detect gravity waves of amplitude ∼ 10−18 m by carefully monitoring distances of order 4 ∼ 10 m, and this should provide opportunities for an “amplifier” that is of order 22 10.

This also means that our modern interferometers have outstanding control over noise sources, which is ideal for the task at hand, involving scenarios for how quantum-spacetime effects may contribute an additional source of noise in such interferometers. Clearly, the noise we could conceivably see emerging from spacetime quantization should be modeled in terms of some random vibrations. Evidently random vibrations are particularly difficult to characterize. For example there is in general no spendable notion of “amplitude” of random vibrations. The most fruitful way to characterize them, also for the purposes of comparing their “intensity” to other non-random sources of vibration that might affect the same system, is by using the power spectral density. Let me introduce some notation, which will prove useful when I move on to discuss crude models of quantum-spacetime-induced noise. For this I simple-mindedly consider the readout of an interferometer as h(t), given by the position x(t) of a mirror divided by a reference length scale L (h (t) = x (t)∕L), and adjust the reference frame so that on average x (t) vanishes, μx = 0. Given some rules for fluctuations of this readout one can indeed be interested in its power spectral density Σ (ω), in principle computable via [486Jump To The Next Citation Point]

∫ ∞ −2πiωτ Σ (ω) = −∞ dτ μ [h(t)h(t+τ)]e , (41 )
where μ[h(t)h(t+ τ)] depends only on τ and is the value expected on average for h(t)h (t + τ) in the presence of the vibration/fluctuation process of interest in the analysis.

Having characterized the noise source in terms of its power spectral density we can then easily compute some primary characteristics, such as its root mean square deviation σh, which for cases of zero-mean noise, such as the one I am considering, will be given by the expectation of 2 h. This can be expressed in terms of the power spectral density as follows [486]

∫ ∞ σ2 = μh2 = dω Σ(ω ). (42 ) h −∞
In experimental practice, for a frequency-band limited signal (fmax) and a finite observation time (Tobs), this relation will take the shape
∫ fmax σ2h ≃ dω Σ(ω ). (43 ) 1∕Tobs

In modern interferometers such as LIGO [9Jump To The Next Citation Point, 1Jump To The Next Citation Point] and VIRGO [157Jump To The Next Citation Point, 12Jump To The Next Citation Point] the power spectral density of the noise is controlled at a level of −44 −1 Σ (ω) ∼ 10 Hz at observation frequencies ω of about 100 Hz, and in turn this (also considering the length of the arms of these modern interferometers) implies [9, 1, 157, 12] that for a gravity wave with 100 Hz frequency the detection threshold is indeed around 10−18 m.

The challenge here for quantum-spacetime phenomenologists is to characterize the relevant quantum-spacetime effects in terms of a novel contribution [QG ] Σ (ω ) to the power spectral density of the noise. If at some point experimentalists manage to bring the total noise Σ (ω), for some range of observation frequencies ω, below the level predicted by a certain quantum-spacetime test theory, then that test theory will be ruled out.

Is there any hope for a reasonable quantum-spacetime test theory to predict noise at a level comparable to the ones that are within the reach of modern interferometry? Well, this is the type of question that one can only properly address in the context of models, but it may be valuable to first use dimensional analysis, assuming the most optimistic behavior of the quantum-spacetime effects, and check if the relevant order of magnitude is at all providing any encouragement for the painful (if at all doable) analysis of the relevant issues in quantum-spacetime models.

To get what is likely to be the most optimistic (and certainly the simplest, but not necessarily the most realistic) Planck-scale estimate of the effect, let us assume that quantum-spacetime noise is “white noise”, Σ[QG](ω) = Σ 0 (frequency independent), so that it is fully specified by a single dimensionful number setting the level of this white noise. And since Σ carries units of −1 Hz one easily notices [54Jump To The Next Citation Point] a tempting simple naive estimate in terms of the Planck length and the speed-of-light29 scale: Σ0 ∼ Lp∕c, which, since Lp ∕c ∼ 10−44 Hz− 1, encouragingly happens to be just at the mentioned level of sensitivity of LIGO-VIRGO-type interferometers. This provides some initial encouragement for a phenomenology based on interferometric noise, though only within the limitations of a very crude and naive estimate.

4.2.3 A simple-minded mechanism for noise in laser-light interferometers

My next task is going beyond assuming for simplicity that the quantum-spacetime noise be white and beyond adopting a naive dimensional-analysis estimate of what could constitute a Planck-scale level of such a noise. The ultimate objective here would be to analyze an interferometer in the framework of a compelling quantum-spacetime theory, but this is beyond our capabilities at present. However, we can start things off by identifying some semi-heuristic pictures (the basis for a test theory) with effects introduced genuinely at the Planck scale that turn out to produce strain noise at the level accessible with modern interferometers.

Having in mind this objective, let us take as a starting point for a first naive picture of spacetime fuzziness the popular arguments suggesting that the Planck scale should also set some absolute limitation on the measurability of distances. And let us (optimistically) assume that this translates to the fact that any experiment in which a distance L plays a key role (meaning that one is either measuring L itself or the observable quantity under study depends strongly on L) is affected by a mean square deviation σ2 L.

It turns out to be useful [51Jump To The Next Citation Point, 53Jump To The Next Citation Point] to consider this 2 σL as a possible stepping stone toward the strain-noise power-spectrum estimate. And a particularly striking picture arises by assuming that the distances L between the test masses of an interferometer be affected by Planck-length fluctuations of random-walk type occurring at a rate of one per Planck time (∼ 10−44 s), so that [51Jump To The Next Citation Point, 53Jump To The Next Citation Point]

σ2L ≃ LpT ≃ LpL [random walk case], (44 )
where T is the time scale over which the experiment monitors the distance L, assuming the use of ultrarelativistic particles (T ≃ L).

It is noteworthy that σ2L ≃ LpT can be motivated independently (without having in mind the idea of such effective spacetime fluctuations) on the basis of some aspects of the quantum-gravity problem [50]. And the study of certain quantum-spacetime pictures that have been of interest to the quantum-gravity community, such as the κ-Minkowski noncommutative spacetime of Eq. (4View Equation), provide some support for this random-walk picture: from [xj, t] = iLP xj one could guess roughly a law of the form σ2x ∼ δxδt ∼ LP x.

Some arguments inspired by the “holography paradigm” for quantum gravity [433Jump To The Next Citation Point, 430Jump To The Next Citation Point, 170Jump To The Next Citation Point] suggest even weaker effects, characterized by

2 4∕3 2∕3 σL ≃ Lp L [holography -inspired case]. (45 )
Interestingly, this ansatz σ2 ≃ L4∕3L2∕3 L p had been independently proposed in the quantum-gravity literature on the basis of a perspective on the quantum-gravity problem (see Ref. [432, 319, 209]), which originally in no way involved spacetime fuzziness.

Probably the most conservative (and pessimistic) expectation for spacetime fuzziness one can find in the quantum-spacetime literature is the one omitting any opportunity for amplification by the involvement of a long observation time (see, e.g., parts of Refs. [249, 293])

σ2L ≃ L2p [weakest case]. (46 )

The random-walk case is the most typical textbook study case for random noise. Its power spectral density goes like ω− 2, so one should have

[QG;rw] LP Σ L ≃ ω2- , (47 )
which gives
σ2 1 ∫ fmax L L T σ2h ∼ --L ≃ --- dω --P ≃ -P--obs (48 ) L2 L2 1∕Tobs ω2 L2
(so, for L ∼ Tobs one indeed finds σ2 ∼ LP L L).

Analogously, one can associate to the “holographic noise” of Eq. (45View Equation) a power spectral density going as −5∕3 ω, so one should have

[QG;holo] L4∕3 Σ L ≃ --P5∕3-, (49 ) ω
which indeed gives 2∕3 σ2L ≃ L4p∕3Tobs ≃ L4∕p3L2∕3.

And, finally, for the σ2L ≃ L2p case of Eq. (46View Equation), a rough but valuable approximate description of the power spectral density goes like ω− 1, so one should have

[QG;weak] L2P ΣL ≃ ---, (50 ) ω
which indeed gives 2 2 σL ≃ Lp.

It is tempting to obtain from these estimates of the quantum-spacetime-induced distance uncertainty an estimate for the quantum-spacetime-induced strain noise, by simply dividing by the square of the length of the arms of the interferometer, Σ [QG ] = Σ [QG ]∕L2 L. This would be the way to proceed if we were converting distance noise into strain noise, but really here we are obtaining a rough estimate of strain noise from an estimate of distance uncertainty, and I shall therefore proceed in some sense sub judice (see in particular my comments below concerning the large number of photons collectively used for producing the accurate measurements of a modern interferometer). Assuming that indeed Σ[QG] = Σ[QG]∕L2 L, and taking as reference value an observation frequency of ω ∼ 100 Hz, one would get for the three cases I discussed the following estimates of strain noise at 100 Hz, for arm lengths of a few kilometers:

Σ [QG;weak] ∼ 10−78 Hz −1 , Σ [QG;holo] ∼ 10−52 Hz− 1, Σ [QG;rw] ∼ 10−38 Hz −1. (51 )
These estimates are rather naive but it is nonetheless interesting to compare them to the levels of noise control achieved experimentally. As mentioned, around 100 Hz both LIGO and VIRGO achieve noise control at the level of strain noise of Σ ∼ 10−44 Hz −1, so estimates like Σ[QG;weak] and Σ [QG;holo] would be safe, but the estimate Σ [QG;rw] must be excluded: the estimate Σ[QG;rw] would assign more noise of quantum-spacetime origin than the total noise that LIGO and VIRGO managed to control (which would include the hypothetical quantum-spacetime-induced noise). In spite of the crudeness of the derivations I discussed so far, this does give a rather worthy input for those who fancy the random-walk picture, as I shall stress in Section 4.2.4.

Before I get to that issue, let me stress that there is a possible source of confusion for terminology (and content) in the literature. In the quantum-gravity literature there has been some discussion for several years of “holography-inspired noise” in the sense of my Eq. (49View Equation) and of Refs. [433Jump To The Next Citation Point, 430Jump To The Next Citation Point, 170Jump To The Next Citation Point]. More recently, a different mechanism for quantum-spacetime-induced noise, also labeled as “holography inspired”, was proposed in a series of papers by Hogan [287Jump To The Next Citation Point, 286Jump To The Next Citation Point, 288Jump To The Next Citation Point]. There is no relation between the two “holography-inspired” proposals for quantum-spacetime-induced interferometric noise. I do not think it is particularly important at the present time to establish which (if either) of the two proposals is more directly inspired by holography. I must instead stress that the holographic noise of Refs. [433Jump To The Next Citation Point, 430, 170Jump To The Next Citation Point] is a rather mature proposal, centered on Eq. (49View Equation) and meaningful at least as a quantum-spacetime test theory in the sense I just described. Instead it is probably fair to describe the alternative version of holographic noise more recently proposed in Ref. [287, 286Jump To The Next Citation Point, 288Jump To The Next Citation Point] as a young proposal still looking for some maturity: it does not amount to any however-wild variant of the description of interferometric noise I summarized here, and actually it is claimed [288Jump To The Next Citation Point] to be immune not only to the sort of interferometric noise I discussed in this Section but also to all other effects that have been typically associated with spacetime quantization in the literature. It would be a quantum-spacetime picture whose effects “can only be detected in an experiment that coherently compares transverse positions over an extended spacetime volume to extremely high precision, and with high time resolution or bandwidth” [288] . Evidently some work is still needed on the conceptual aspects (as a rigorous theory of spacetime quantization) and on the phenomenological aspects (as a computably predictive and broadly applicable test theory of spacetime quantization) of this proposal. Only time will tell if this present lack of maturity is due to intrinsic unsurmountable limitations of the proposal or is simply a result of the fact that the proposal was made only rather recently (so there was not much time for this maturity to be reached). I should note that at some point, in spite of its lack of maturity, this proposal started to attract some pronounced interest in relation to reports by the GEO600 interferometer [550] of unexplained excess noise [373]: it had been claimed [286] that Hogan’s version of holographic noise could match exactly the anomaly that was being reported by GEO600. However, it appears that experimenters at GEO600 have recently achieved a better understanding of their noise sources, and no unexplained contribution is at this point reported (this is at least implicit in Ref. [462] and is highlighted at External Link The brief season of the “GEO600 anomaly” (at some point known among specialists as the “mystery noise”) is over.

4.2.4 Insight already gained and ways to go beyond it

At the present time the “state of the art” of phenomenologically-spendable descriptions of Planck-scale-induced strain noise does not go much beyond the simple-minded estimates I just described in relation to Eqs. (47View Equation), (49View Equation), and (50View Equation). But some lessons were nonetheless learned, as usually happens even with the most humble phenomenology. And these lessons do point toward some directions worthy of exploration in the future. In this section I highlight some of these lessons and possible future developments.

Among the few steps of simple derivation, which I described in Section 4.2.3, evidently much scrutiny should be particularly directed toward the assumption [QG ] [QG ] 2 Σ = Σ L ∕L: I motivated some candidate forms for Σ[LQG] using essentially the sort of arguments that usually allow us to establish uncertainty principles for single particles, such as the ones taking as a starting point a postulated noncommutativity of single-particle coordinates; however, the strain noise [QG] Σ relevant for our interferometers is not at all a single-particle feature. Let me use the example of random-walk fuzziness for illustrating how the relationship between single-particle quantum-spacetime arguments and interferometric strain noise could be more subtle than assumed in [QG] [QG] 2 Σ = ΣL ∕L. For this, I shall follow Ref. [57Jump To The Next Citation Point] (a similar thesis was also reported in Ref. [170]). I specialize the more general idea of random-walk quantum-spacetime fuzziness in the sense of assuming that each single photon in an interferometer experiences a random-walk path: a random Planck-length fluctuation per Planck-time would affect the path of each photon of the beam. This would imply in particular that as a photon goes from one mirror of the interferometer to the other, over a distance L, it reaches its destination with an uncertainty corresponding to σ2L ∼ LPT ∼ LP L. However, the interferometer (and this is key to its outstanding sensitivity) does not depend on determining the position of each single photon in the beam: on the contrary the key observable is the average position of the photons composing the beam, which may be viewed as the putative “position of the mirror” (when such a beam reaches the mirror). If L is now viewed as the distance between positions of mirrors defined in this way, rather than as the distance of propagation of an individual photon, then evidently the result is an estimate σ2L ∼ LPT ∼ LP L∕N γ, where N γ is the (very large!) number of photons contributing to each such determination of the “position of the mirror”.

While the noise levels produced by a random-walk ansatz assuming [QG ] [QG ] 2 Σ = Σ L ∕L are, as stressed in Section 4.2.3, already ruled out by the achievements of LIGO and VIRGO, this single-particle picture of a random-walk scenario, which evidently leads us to assume

[QG] [QG ] -ΣL--- Σ = N γL2 (52 )
is still safely compatible with the noise results of LIGO and VIRGO, thanks to the large N γ suppression.

This observation is not specific to the random-walk scenario. A similar N γ suppression could naturally be expected for the holographic noise scenario of Eq. (49View Equation). As discussed in the previous Section 4.2.3, that holographic noise scenario would be safe from LIGO/VIRGO bounds even without the Nγ suppression. (In some sense that holographic noise scenario would turn into unpleasantly “too safe from LIGO/VIRGO”, i.e., probably beyond the reach of any foreseeable interferometric experiment, if it were to take into account the plausible N γ suppression).

Concerning the scenario for weak quantum-spacetime-induced fuzziness, the one of Eq. (50View Equation), contemplating the possibility of an N γ suppression is of mere academic interest: those noise levels are so low, even without the possible additional N γ suppression, that we should exclude their testability for the foreseeable future.

But for random-walk noise and for the holographic-noise scenario of Eq. (49View Equation) this issue of a possible N γ suppression needs to be investigated and understood. This is probably not for the LIGO/VIRGO season: LIGO and VIRGO have not found any excess noise so far, and at this point it is unlikely they will ever find it. But a completely new drawing board for phenomenology would materialize with the advent of LISA [282]: LISA will operate at lower observational frequencies ω than LIGO/VIRGO-type interferometers, which is important from the quantum-spacetime perspective since both random-walk noise, as described by Eq. (47View Equation), and the holographic-noise scenario of Eq. (49View Equation) predict effects that increase at lower observational frequencies.30 The outcome of such LISA quantum-spacetime-noise studies may then depend on issues such as the possible N γ suppression.

I should also stress that the analysis of these opportunities for quantum-spacetime phenomenology from experiments operating at low observational frequencies ω, is perhaps the most significant and most robust conceptual achievement of the sort of phenomenology of spacetime foam that I am discussing in this Section. When these pictures were first proposed it was seen by many as a total surprise that one could contemplate Planck-scale effects at frequencies of observation of only 100 Hz. The naive argument goes something like “Planck-scale-induced noise must be studied at Planck frequency”, and the Planck frequency is E ∕¯h ∼ 1043 Hz p. However, in analyzing actual pictures of quantum-spacetime fuzziness, even the simple-minded ones described above, one becomes familiar with well-known facts establishing (and we should expect this lesson to apply even to more sophisticated picture of quantum-spacetime-induced fuzziness) that discrete fluctuation mechanisms tend to produce very significant effects at low observational frequencies ω, with typical behaviors of the type ω−|α|, even when their charateristic time scale is ultrashort.

4.2.5 Distance fuzzyness for atom interferometers

Since the phenomenology of the implications of spacetime foam for interferometry is at an early stage of development, at the present time it may be premature to enter into detailed discussions of what type of interferometry might be best suited for uncovering quantum-spacetime/Planck-scale effects. Accordingly, in Section 4.2 I focus by default on the simplest case of interferometric studies, the one using a laser-light beam. However, in recent times atom interferometry has reached equally astonishing levels of sensitivity and for several interferometric measurements it is presently the best choice. Laser-light interferometry is still preferred for certain well-established techniques of interferometric studies of spacetime observables, as in the case of searches for gravity waves, and the observations I reported above for the phenomenology of strain noise induced by quantum-spacetime effects appear to be closely linked to the issues encountered in the search for gravity waves. However, it seems plausible that soon there will be some atom-interferometry setups that are competitive for gravity-wave searches (see, e.g., Refs. [526, 208]). This in turn might imply that searches of quantum-spacetime-induced strain noise could rely on atom interferometry.

The alternative between light and matter interferometry might prove valuable at later more mature stages of this phenomenology. It is likely that different test theories will give different indications in this respect, so that atom interferometry might provide the tightest constraints on some spacetime-foam test theories, whereas laser-light interferometry might provide the best constraints on other spacetime-foam test theories. A key aspect of the description of Planck-scale effects for atom interferometry to be addressed by the test theories (and hopefully, some day, by some fully-developed quantum-spacetime/quantum-gravity theories) is the role played by the mass of the atoms. With respect to laser-light interferometry, the case of atom interferometry challenges us with at least two more variables to be controlled at the theory level, which are the mass of the atoms and their compositeness. How do these two aspects of atom interferometry interface with the quantum-spacetime features that are of interest here? Do they effectively turn out to introduce suppressions of the relevant effects or on the contrary could they be exploited to see the effects? For none of the quantum-spacetime models that are presently studied have we reached a level of understanding of physical implications robust enough for us to answer confidently these questions. Perhaps we should also worry about (or exploit) another feature that is in principle tunable in atom interferometry, which is the velocity of the particles in the beam.

4.3 Fuzziness for waves propagating over cosmological distances

Interferometric studies of spacetime-foam are another rare example of tests of quantum-spacetime effects that can be conducted in a controlled laboratory setup (also see Section 3.13). Astrophysics may turn into the most powerful arena for this type of study. Indeed, the studies I discussed in the previous Section 4.2, which started toward the end of the 1990s, inspired a few years later some follow-up studies from the astrophysics side. As should be expected, the main opportunities come from observations of waves that have propagated over very large distances, thereby possibly accumulating a significant collective effect of the fuzziness encountered along the way to our detectors.

4.3.1 Time spreading of signals

An implication of distance fuzziness that one should naturally consider for waves propagating over large distances is the possibility of “time spreading” of the signal: if at the source the signal only lasted a certain very short time, but the photons that compose the signal travel a large distance L, affected by uncertainty σ2L, before reaching our detectors the observed spread of times of arrival might carry little trace of the original time spread at the source and be instead a manifestation of the quantum-gravity-induced σL. If the distance L is affected by a quantum-spacetime uncertainty σL then different photons composing the signal will effectively travel distances that are not all exactly given by L but actually differ from L and from each other up to an amount σL.

Again, it is of particular interest to test laws of the type discussed in the previous Section 4.2, but it appears that these effects would be unobservably small even in the case that provides the strongest effects, which is the random-walk ansatz σ2L ≃ LpL ≃ LpT (assuming ultrarelativistic particles, for which L is at least roughly equal to the time duration T of the journey). To see this let me consider once again gamma-ray bursts, which often travel for times on the order of 17 10 s before reaching Earth detectors and are sometimes characterized by time structures (microbursts within the burst) that have durations as short as −3 10 s. Values of 2 σL as small as σ2L ∼ c210− 8 s2 could be noticeable in the analysis of such bursts. However, the estimate σ2L ≃ cLpT only provides σ2L ∼ c210− 27 s2 and is, therefore, much beyond our reach.

I shall comment in Section 4.8 on an alternative formulation of the phenomenology of quantum-spacetime-induced worldline fuzziness, the one in Ref. [490Jump To The Next Citation Point] inspired by the causal-set approach (the approach on which Section 4.8 focuses).

4.3.2 Fuzziness from nonsystematic symmetry-modification effects

As an alternative way to model spacetime fuzziness, there has been some interest [431Jump To The Next Citation Point, 72Jump To The Next Citation Point, 37Jump To The Next Citation Point] in the possibility that there might be effects resembling the ones discussed in Section 3, which are systematic deviations from the predictions of Poincaré symmetry, but are “nonsystematic” in the sense discussed at the beginning of this section. The possibility of fuzziness of particle worldlines governed by E ∕Ep, mentioned in the previous Section 4.3.1, is an example of such nonsystematic violations of Poincaré symmetry.

These speculations are not on firm ground on the theory side, in the sense that there is not much in support of this among available results on actual analysis of formalizations of spacetime quantization. But it is legitimate to expect that this might be due to our limited abilities in mastering these complex formalisms. After all, as suggested in Ref. [431Jump To The Next Citation Point], if spacetime geometry is fuzzy then it may be inevitable for the operative procedures that give sense to the notion of energy and momentum of a particle to also be fuzzy.

This sort of picture could have tangible observational consequences. For example, it can inspire, as suggested in Refs. [74Jump To The Next Citation Point, 57Jump To The Next Citation Point, 72Jump To The Next Citation Point], scenarios such that spacetime fuzziness effectively produces an uncertainty in the velocity of particles of order E ∕E p. This would give rise to a magnitude of these nonsystematic effects comparable to the one discussed in Section 3.8 for the corresponding systematic effects. After a journey of ∼ 1017 s the acquired fuzziness of arrival times could be within the reach [74Jump To The Next Citation Point] of suitably arranged gamma-ray-burst studies. However, there is no significant effort to report here on establishing bounds following this strategy.

There are instead some studies of this phenomenological picture [431Jump To The Next Citation Point, 72Jump To The Next Citation Point, 37Jump To The Next Citation Point], which take as a starting point the possibility, discussed in Sections 3.4 and 3.5, of modifications of the dispersion relation leading to modifications of the threshold requirements for certain particle-production processes, such as the case of two incoming photons producing an outgoing electron-positron pair. Refs. [431, 72Jump To The Next Citation Point, 37Jump To The Next Citation Point] considered the possibility of a non-systematic quantum-spacetime-induced deformation of the dispersion relation, specifically the case in which the classical relation 2 2 2 E = p + m still holds on average, but for a given particle with large momentum ⃗p, energy would be somewhere in the range of

m2 |η |⃗p2 m2 |η| ⃗p2 |⃗p| + ----− ------≤ E ≤ |⃗p| + ----+ ------, (53 ) 2|⃗p| 2 Ep 2|⃗p | 2 Ep
with some (possibly Gaussian) probability distribution. A quantum-spacetime theory with this feature should be characterized by a fundamental value of η, but each given particle would satisfy a dispersion relation of the type
2 2 E ≃ |⃗p| + m---+ &tidle;ηp⃗-, (54 ) 2|⃗p| 2Ep
with − |η| ≤ &tidle;η ≤ |η |.

In analyses such as the one discussed in Section 3.4 (for observations of gamma rays from blazars) one would then consider electron-positron pair production in a head-on photon-photon collision assuming that one of the photons is very hard while the other is very soft. To leading order, for the soft photon only, the energy 𝜖 is significant (for an already small 𝜖 the actual value of &tidle;η will not matter in leading order). So, the soft photon can, in leading order, be treated as satisfying a classical dispersion relation. In a quantum-spacetime theory predicting such non-systematic effects, the hard photon would be characterized both by its energy E and its value of &tidle;η. In order to establish whether a collision between two such photons can produce an electron-positron pair, one should establish whether, for some admissible values of &tidle;η+ and &tidle;η − (the values of &tidle;η pertaining to the outgoing positron and the electron respectively), the conditions for energy-momentum conservation can be satisfied. The process will be allowed if

m2- &tidle;η-E3-- η&tidle;+-+--&tidle;η−-E3- E ≥ 𝜖 − 4 𝜖E + 16 𝜖E . (55 ) p p
Since &tidle;η, &tidle;η+ and &tidle;η− are bound by the range − |η| to |η|, the process is only allowed, independent of the value of &tidle;η, if the condition
m2 3|η |E3 E ≥ ---− -------- (56 ) 𝜖 8 𝜖Ep
is satisfied. This condition defines the actual threshold in the non-systematic-effect scenario. Clearly, in this sense the threshold is inevitably decreased by the non-systematic effect. However, there is only a tiny chance that a given photon would have η&tidle;= |η|, since this is the limiting case of the range allowed by the nonsystematic effect, and unless &tidle;η = |η |, the process will still not be allowed even if
m2- 3|η|-E3- E ≃ 𝜖 − 8 𝜖Ep . (57 )
Moreover, even assuming η&tidle;= |η|, the energy value described by (57View Equation) will only be sufficient to create an electron positron pair with η&tidle;+ = − |η| and &tidle;η− = − |η |, which again are isolated points at the extremes of the relevant probability distributions. Therefore the process becomes possible at the energy level described by (57View Equation) but it remains extremely unlikely, strongly suppressed by the small probability that the values of &tidle;η, &tidle;η+ and &tidle;η− would satisfy the kinematical requirements.

With reasoning of this type, one can easily develop an intuition for the dependence on the energy E, for fixed value of 𝜖 (and treating &tidle;η, &tidle;η+ and η&tidle;− as totally unknown), of the likelihood that the pair-production process can occur: (i) when (56View Equation) is not satisfied the process is not allowed; (ii) as the value of E is increased above the value described by (57View Equation), pair production becomes less and less suppressed by the relevant probability distributions for &tidle;η, &tidle;η+ and &tidle;η−, but some suppression remains up to the value of E that satisfies

m2 3|η|E3 E ≃ --- + --------; (58 ) 𝜖 8 𝜖Ep
(iii) finally for energies E higher than the one described by (58View Equation), the process is kinematically allowed for all values of &tidle;η, &tidle;η+ and &tidle;η−, and, therefore, the likelihood of the process is just the same as in the classical-spacetime theory.

This describes a single photon-photon collision taking into account the nonsystematic effects. One should next consider that for a hard photon traveling toward our Earth detectors from a distant astrophysical source there are many opportunities to collide with soft photons with energy suitable for pair production to occur (the mean free path is much shorter than the distance between the source and the Earth). Thus, one expects [72Jump To The Next Citation Point, 37Jump To The Next Citation Point] that even a small probability of producing an electron-positron pair in a single collision would be sufficient to lead to the disappearance of the hard photon before reaching our detectors. The probability is small in a single collision with a soft background photon, but the fact that there are, during the long journey, many such pair-production opportunities renders it likely that in one of the many collisions the hard photon would indeed disappear into an electron-positron pair. Therefore, for this specific scheme of non-systematic effects it appears that a characteristic prediction is that the detection of such hard photons from distant astrophysical sources should start being significantly suppressed already at the energy level described by (57View Equation), which is below the threshold corresponding to the classical-spacetime kinematics.

It is interesting [57Jump To The Next Citation Point, 72Jump To The Next Citation Point, 74Jump To The Next Citation Point] to contemplate in this case the possibility that systematic and nonsystematic effects may both be present. It is not unnatural to arrange the framework in such a way that the systematic effects tend to give higher values of the threshold energy, but then the nonsystematic effects would allow (with however small probability) configurations below threshold to produce the electron-positron pair. And for very large propagation distances (very many “target soft photons” available) the nonsystematic effect can essentially erase [72] the systematic effect (no noticeable upward shift of the threshold).

I illustrated the implications of nonsystematic effects within a given scenario and specifically for the case of observations of gamma rays from blazars. One can implement the non-systematic effects in some alternative ways and the study of the observational implications can consider other contexts. In this respect I should bring to the attention of my readers the studies of non-systematic effects for ultra-high-energy cosmic rays reported in Refs. [37, 310, 106].

Combinations of systematic and nonsystematic effects can also be relevant [57, 74Jump To The Next Citation Point] for studies of the correlations between times of arrival and energy of simultaneously-emitted particles. For that type of study both the systematic and the nonsystematic effects could leave an observable trace [74] in the data, codified in the mean arrival time and the standard deviation of arrival times found in different energy channels.

4.3.3 Blurring images of distant sources

The two examples of studies in astrophysics of quantum-spacetime-induced distance fuzziness I discuss in Sections 4.3.1 and 4.3.2 have only been moderately popular. I have left as last the most intensely studied opportunity in astrophysics for quantum-spacetime-induced distance fuzziness. These are studies essentially looking for effects blurring the images of distant sources.

It is interesting that these studies were started by Ref. [367Jump To The Next Citation Point], which cleverly combined some aspects of Refs. [51Jump To The Next Citation Point, 54Jump To The Next Citation Point, 53Jump To The Next Citation Point, 433Jump To The Next Citation Point], providing the main concepts for the proposal summarized in Section 4.2, and some aspects of Ref. [66Jump To The Next Citation Point], providing the main concepts for the proposal summarized in Section 3.8. Ref. [367Jump To The Next Citation Point] was interested in the same phenomenology of distance fuzziness introduced and analyzed for controlled interferometers in Refs. [51Jump To The Next Citation Point, 54Jump To The Next Citation Point, 53Jump To The Next Citation Point, 433Jump To The Next Citation Point], but looked for opportunities to perform analogous tests using the whole Universe as laboratory, in the sense first introduced in Ref. [66Jump To The Next Citation Point].

Gradually over the last decade this became a rather active research area, as illustrated by the studies reported in Refs. [367Jump To The Next Citation Point, 434Jump To The Next Citation Point, 189Jump To The Next Citation Point, 464Jump To The Next Citation Point, 363Jump To The Next Citation Point, 171Jump To The Next Citation Point, 167Jump To The Next Citation Point, 402Jump To The Next Citation Point, 514Jump To The Next Citation Point, 403Jump To The Next Citation Point, 404Jump To The Next Citation Point, 520Jump To The Next Citation Point, 456Jump To The Next Citation Point, 405Jump To The Next Citation Point].

The phenomenological idea is powerfully simple: effects of quantum-spacetime-induced spacetime fuzziness had been shown [51Jump To The Next Citation Point, 54Jump To The Next Citation Point, 53Jump To The Next Citation Point, 433Jump To The Next Citation Point] to be potentially relevant for LIGO/VIRGO-like (and LISA-like) intereferometers, exploiting not only the distance-monitoring accuracy of those interferometers, but also the fact that such accuracy on distance monitoring is achieved for rather large terrestrial distances. Essentially the Universe gives much larger distances for us to monitor [66], and although we can monitor them with accuracy inferior to the one of a LIGO/VIRGO-like intereferometer, on balance, the astrophysics route may also be advantageous also for studies of quantum-spacetime-induced spacetime fuzziness [367Jump To The Next Citation Point].

As for Refs. [51, 54, 53, 433], reviewed in Section 4.2, the core intuition here is that the quantum-spacetime contribution to the fuzziness of a particle’s worldline might grow with propagation distance. And collecting the scenarios summarized in Eqs. (47View Equation), (49View Equation), and (50View Equation), one arrives at a one-parameter family of phenomenological ansätze for the characterization of this dependence of fuzziness on distance

σ2L ≃ L2PαL2− 2α , (59 )
with 1∕2 ≤ α ≤ 1.

An assumption shared by most explorations [367Jump To The Next Citation Point, 434Jump To The Next Citation Point, 189, 464, 363, 171Jump To The Next Citation Point, 167, 402, 514Jump To The Next Citation Point, 403, 404, 520Jump To The Next Citation Point, 456, 405] of this phenomenological avenue is that from Eq. (59View Equation), that there would also follow an associated uncertainty in the specification of momenta

α 1+α α 1+α δE ∼ LP E , δp ∼ L P p . (60 )
I must stress that this (however plausible) deduction of the heuristic arguments has not been confirmed in any explicit model of spacetime quantization. And it plays a crucial role in most astrophysics tests of distance fuzziness: from Eq. (60View Equation) it is easy to see [367Jump To The Next Citation Point] that it follows that (assuming a classical-wave description is still admissible when such effects are nonnegligible) there should be a mismatch between uncertainty in the group velocity and in the phase velocity of a classical wave, and this in turn proves to be a very powerful tool for the phenomenology. During a propagation time T = L ∕vg (vg being the group velocity) the phase of a wave advances by Δ ϕ = 2π(vp∕vg)(L ∕λ) (where vp is the phase velocity and λ is the wavelength). There are two schools of intuition concerning how quantitatively spacetime fuzziness should scramble the phase of a wave. According to Ref. [367] and followers the effect should go as
E αL E1+ α δϕ ≃ ----- = -----L, (61 ) Epαλ E αp
whereas according to Ref. [434, 171Jump To The Next Citation Point] and followers, the effect should grow more slowly with the distance of propagation, going like
E-- 1−α δϕ ≃ Eα L . (62 ) p

As first observed in Ref. [514Jump To The Next Citation Point], the alternative formulas (61View Equation) and (62View Equation) should be improved to account for redshift. For the case of Eq. (62View Equation) Ref. [514Jump To The Next Citation Point] proposes the following

( ) E 1 − α ∫ ∞ − α 1 − q0 δϕ ∼ --α------ dz (1 + z)L 1 − √---------- , (63 ) E p H0q0 0 1 + 2zq0
where q0 is the decelaration parameter, q0 = Ω0 ∕2 − Λ ∕(3H20 ) and L is the luminosity distance, L = [zq + (1 − q )(1 − √1-+-2zq-)]∕(H q2) 0 0 0 0 0 (Λ, H 0 and Ω 0 here denote, as usual, respectively the cosmological constant, the Hubble parameter and the matter fraction).

Evidently, this phenomenology still has a few too many quantitative details subject to further scrutiny and a few too many alternative scenarios. This is the result of the fact that work on actual formalizations of spacetime quantization, while encouraging the general intuitive picture, has been unable to provide detailed guidance. And the heuristic arguments based on these preliminary studies have been unable to narrow the range of possibilities. But pursuing this path further appears to be an exciting opportunity for quantum-spacetime phenomenology, and we should, therefore, persevere. In particular, based on the (however alternative) estimates given by Eqs. (61View Equation), (62View Equation), and (63View Equation), several authors (see, e.g., Refs. [171Jump To The Next Citation Point, 514Jump To The Next Citation Point, 520Jump To The Next Citation Point]) have concluded that a phenomenology based on blurring of the images of distant sources can provide Planck-scale sensitivity for a rather broad range of possible phenomenological test theories and for values of α significantly greater than 1∕2, possibly [520Jump To The Next Citation Point] going all the way up to values of α close to 1. In Ref. [514Jump To The Next Citation Point] one even finds a preliminary data analysis suggesting that for observations of quasars there might be a trend towards lower observed Strehl ratios with increasing redshift, which would provide encouragement for the hope of discovering quantum-spacetime-induced image blurring.

The main opportunities appear to be provided by observations of distant quasars [171, 514, 520], whose dimensions are small and are rather abundantly observed at high redshift.

4.4 Planck-scale modifications of CPT symmetry and neutral-meson studies

Investigations of spacetime symmetries and distance fuzziness are evidently relevant for some of the core features of the idea of spacetime quantization. My next task concerns CPT-symmetry tests, and the possibility that indirectly some scenarios for the quantization of spacetime might affect CPT symmetry.

A complication, but also an opportunity, for quantum-spacetime-motivated tests of CPT symmetries comes from the fact that CPT symmetry should be and is tested independently of the quantum-spacetime motivation. From this perspective the situation is somewhat analogous to that discussed earlier concerning quantum-spacetime-motivated tests of Lorentz symmetry. The quantum spacetime literature can provide special motivation for probing CPT symmetry in certain specific ways, but there is already plenty of motivation, even without quantum-spacetime research, for testing CPT symmetry as broadly as possible [389, 439, 234].

Also, in this case, the Standard Model Extension provides a much appreciated and widely adopted formalization, finding a good balance between the desire of searching for violations of CPT symmetry (and/or, as mentioned, violations of Lorentz symmetry) within the confines of quantum field theory but allowing for both effects that have been discussed from the quantum-spacetime perspective and effects for which so far there is no quantum-spacetime motivation. I shall focus here on the hypothesis of quantum-spacetime-induced and Planck-scale-magnitude CPT violation effects, so I shall not review the broad subject of CPT violation within the Standard Model Extension, for which readers can find valuable reviews and perspectives in Refs. [345, 117, 180, 339, 299, 341, 346] (see also parts of Ref. [395]).

Another issue that should always be kept in mind in relation to CPT symmetry is the fact that it can be derived as a theorem for local quantum field theories with Lorentz invariance. In approaches based on local field theory, it is natural to perform combined studies of CPT and Lorentz symmetry.31 However, the notion of spacetime quantization at the Planck scale involves some aspects of nonlocality (at least the notion of points that coincide with accuracy better than the Planck length is typically abandoned) and in most quantum-spacetime studies of the fate of CPT symmetry the expectation is that these aspects of non-locality may be primarily responsible for the conjectured violations of CPT symmetry.

I shall not attempt to summarize here the results on violations of CPT symmetry arising from spacetime quantization not introduced at the Planck scale (but rather at some much lower scale), for which readers can find valuable starting points to the related literature in Refs. [162, 43, 417, 496, 28] and references therein.

Consistent with the scope of this review, I shall focus exclusively on scenarios for violations of CPT symmetry based on nonclassicality (“quantization”) of spacetime introduced at the Planck scale. As a result of some technical challenges, mentioned in Section 2.2.2, this literature can only rely on preliminary theory results, but does suggest convincingly that Planck-scale sensitivity to quantum-spacetime-induced violations of CPT symmetry is within our reach.

4.4.1 Broken-CPT effects from Liouville strings

In the case of the test of CPT symmetry it is easier for me to start by discussing the availability of Planck-scale sensitivity, postponing briefly some comments on test theories based on the idea of spacetime quantization.

There is a sizable literature establishing that CPT symmetry can be tested with Planck-scale sensitivity in the neutral-kaon and the neutral-B systems (see, e.g., Refs. [219Jump To The Next Citation Point, 220Jump To The Next Citation Point, 298Jump To The Next Citation Point, 108Jump To The Next Citation Point]). It turns out that in these neutral-meson systems there are plenty of opportunities for Planck-scale departures from CPT symmetry to be amplified. In particular, the neutral-kaon system hosts the peculiarly small mass difference between long-lived and short-lived kaons −15 |ML − MS |∕ML,S ∼ 7⋅10 and other small numbers naturally show up in the analysis of the system, such as the ratio |Γ L − Γ S|∕ML,S ∼ 1.4 ⋅ 10−14. And for certain types of departures from CPT symmetry the inverse of one of these small numbers amplifies the small CPT-violation effect [219, 220Jump To The Next Citation Point, 298, 108Jump To The Next Citation Point]. In particular, this mechanism turns out to provide sufficient amplification for Planck-scale effects, inducing a difference of order 2 M K0∕Ep between the terms on the diagonal of the 0 K, ¯0 K mass matrix (exact classical CPT symmetry would require the terms on the diagonal to be identical). It should be noticed that MK0 ∕Ep ∼ 10 −19, which is not overwhelmingly smaller than |ML − MS |∕ML,S ∼ 7⋅10−15.

A much studied quantum-spacetime description of violations of CPT symmetry is centered on the mentioned Liouville-strings approach [221Jump To The Next Citation Point, 220Jump To The Next Citation Point], particularly with its description of spacetime foam and its non-classical description of time, involving a non-trivial role for the Liouville field [224]. This model is, in particular, the reference for the analysis of Planck-scale limits on quantum-spacetime-induced CPT violation reported by the CPLEAR collaboration on the basis of studies of neutral kaons [13Jump To The Next Citation Point] (also see the related results reported using neutral-kaon data gathered at the particle-physics laboratory in Frascati [13Jump To The Next Citation Point, 534Jump To The Next Citation Point, 205Jump To The Next Citation Point]). Interestingly, the Liouville-string model hosts both departures from CPT symmetry and decoherence, and I find it most convenient to discuss it in the later part of this section devoted to decoherence studies.

Let me highlight a recent development that is in part inspired by these Liouville-string studies. It was recently observed (primarily in Refs. [112Jump To The Next Citation Point, 113Jump To The Next Citation Point]) that quantum-spacetime scenarios with violations of CPT symmetry might also require some corresponding modifications of the recipe for obtaining multiparticle states from single-particle states for identical particles. This may apply in particular to the neutral-kaon ¯ K0 − K0 system, since standard CPT transformations take K0 into ¯ K0 but violations of CPT symmetry are likely to also induce a modification of the link between K0 and K¯0.

Refs. [112Jump To The Next Citation Point, 113Jump To The Next Citation Point] proposed a phenomenology inspired by this argument and based on the following parametrization of the state |i > initially produced by a ϕ-meson decay:

|i > ∝ ((|KS (p ),KL (− p) > − |KL (p),KS (− p) > ) + ω (|KS (p),KS (− p) > − |KL (p),KL (− p) >)),(64 )
where the complex parameter ω essentially characterizes the level of contamination of the state |i > by the (otherwise unexpected) C-even component |K (p),K (− p) > − |K (p),K (− p) > S S L L.

Stringent constraints on ω can be placed by performing measurements of the chain of processes ϕ → KK → XY, in which first the ϕ meson decays into a pair of neutral kaons and then one of the kaons decays at time t1 into a final state X, while the other kaon decays at time t2 into a final state Y. By following this strategy the KLOE experiment [534Jump To The Next Citation Point, 522] at DAΦNE is setting [204, 205Jump To The Next Citation Point] experimental limits on ω at the level −3 10 (−3 |Re (ω )| < 10, −3 |Im (ω)| < 10).

It is not easy at present to establish robustly what level of sensitivity to ω could really amount to Planck-scale sensitivity, but it is noteworthy that there are semi-quantitative/semi-heuristic estimates based on a certain intuition for spacetime foam suggesting [112, 113, 398Jump To The Next Citation Point] that sensitivities in the neighborhood of −3 ω ∼ 10, −4 ω ∼ 10 could already be significant.

4.4.2 Departures from classical CPT symmetry from spacetime noncommutativity at the Planck scale

Another formalism for spacetime quantization at the Planck scale where violations of CPT symmetry have been discussed to some extent is “κ-Minkowski spacetime noncommutativity” [391, 374, 70Jump To The Next Citation Point]. A first hint that this might be appropriate comes from the fact that the κ-Minkowski formalism is one of those providing support for the possibility of modifications of the dispersion relation of the form m2 ≃ E2 − ⃗p2 + λE ⃗p2∕2, with λ on the order of the Planck length. It may be relevant for the relation between particles and antiparticles (for which CPT symmetry is a crucial player) that for the values of E allowed by the dispersion relation for given |⃗p| one does not recover the ordinary result (with its traditional two solutions of equal magnitude and opposite sign); instead, one finds that the two solutions E+, E − are given by

λ ∘ -------- E± ≃ − -⃗p2 ± m2 + ⃗p2 . (65 ) 2
The fact that the solutions E+ and E− are not exactly opposite may suggest that one should make room for a mismatch δM of the terms on the diagonal of the K0, K¯0 mass matrix, of order
2 |δM | ∼ |E+-| −-|E-− |2M ≃ λ√--⃗p-M-----. (66 ) |E+ | + |E − | M 2 + ⃗p2
The most significant feature of this description of δM is its momentum dependence, and, for given |λ|, |δM | is an increasing function of |⃗p|, quadratic in the non-relativistic limit and linear in the ultra-relativistic limit. Therefore, among experiments achieving comparable δM sensitivity the ones studying more energetic kaons are going to lead to more stringent bounds on λ.

Considering that, as mentioned, neutral-kaon experiments at Φ factories are now sensitive at the level −18 δM ∼ 10 GeV, one infers a sensitivity to this type of candidate quantum-gravity effect that, for kaons of momenta of about 110 MeV (at the ϕ resonance), corresponds to a sensitivity to values of |λ| around 10− 32 m, i.e., not far (just 3 orders of magnitude away) from the Planck scale. Because of the premium on high momenta of this scenario, better limits could be set using experiments with high-momentum kaons Fermilab’s E731 [554, 450]. And studies with neutral B mesons of relatively high momenta could also be valuable from this perspective.

However, we are at a very early stage of understanding of the fate of CPT symmetry in these spacetimes with quantization at the Planck scale. Specifically, for the case of κ-Minkowski spacetime, analyses such as the one in Ref. [70] suggest that CPT symmetry is deformed rather than broken/lost. Indeed, in κ-Minkowski the anomalies one can presently preliminarily see for CPT symmetry are all linked to the peculiarity of P-parity transformations. It appears that in κ-Minkowski P-parity transformations for momenta should not take a momentum ⃗p into − ⃗p, but rather ⃗p → ⊖ ⃗p, where ⊖ ⃗p denotes the “antipode operation”: −λp0 ⊖ ⃗p ≡ − ⃗pe (where λ denotes again the κ-Minkowski noncommutativity length scale).

4.5 Decoherence studies with kaons and atoms

4.5.1 Spacetime foam as decoherence effects and the “α, β, γ test theory”

As stressed earlier in this review the idea of “spacetime foam” appears to appeal to everyone involved in quantum-spacetime research, but this is in part due to the fact that this idea is not really well defined, not by the qualitative intuitive picture proposed by Wheeler. In order to set up a phenomenology for effects induced by this spacetime foam, it is necessary to provide for it physical/experimentally-meaningful characterizations. I already discussed one possible such characterization, given in terms of distance fuzziness and associated strain noise for interferometry. Another attempt to physically characterize spacetime foam can be found in Refs. [220Jump To The Next Citation Point, 221Jump To The Next Citation Point] (other valuable perspectives on this subject can be found in Refs. [108, 251]), focusing on the possibility that the rich dynamical properties of spacetime foam might act as a decoherence-inducing environment.

The main focus of Refs. [220Jump To The Next Citation Point, 221Jump To The Next Citation Point] has been the neutral-kaon system, whose remarkably delicate balance of scales provides opportunities not only for very sensitive tests of CPT symmetry, but also for very sensitive tests of decoherence. Refs. [220, 221] essentially propose a test theory, based on the mentioned Liouville-strings idea, for spacetime-foam-induced decoherence in the neutral-kaon system. This test theory adopts the formalism of density matrices and is centered on the following evolution equation for the neutral-kaon reduced density matrix ρ:

∂tρ = i[ρ,H ] + δH ρ, (67 )
where H is an ordinary-quantum-mechanics Hamiltonian and δHmn (with indices m, n running from 1 to 4: {m, n } ∈ {1,2,3,4}) is the spacetime-foam-induced decoherence matrix, taken to be such that δH1n = δH2n = δHn1 = δHn2 = 0, while δH34 = δH43 = − 2β, δH33 = − 2α, and δH44 = − 2γ. Therefore, the test theory is fully specified upon fixing H and giving some definite values to the parameters α, β, γ.

It should be stressed that this test theory necessarily violates CPT symmetry whenever δH ⁄= 0. Additional CPT violating features may be introduced in the ordinary-quantum-mechanics Hamiltonian H, by allowing for differences in masses and/or differences in widths between particles and antiparticles. Therefore, this test theory is an example of a framework that could be used in a phenomenology looking simultaneously for departures from CPT symmetry of types admissible within ordinary quantum mechanics and for departures from CPT symmetry that require going beyond quantum mechanics (by allowing for decoherence). It is noteworthy that the two types of CPT violation (within and beyond quantum mechanics) can be distinguished experimentally.

Concerning more directly decoherence, various characterizations of the effects of this test theory have been provided, and in particular a valuable description of how significant the decoherence effects are (depending on the values given to α, β, γ) is found looking at how the rate of kaon decay into a pair of pions, R2 π, evolves as a function of time. This time evolution will in general take the form

R2π(t) = CSe −Γ St + CLe −Γ Lt + 2CI e−(Γ L+ Γ S)t∕2cos[(mL − mS )t − ϕ], (68 )
where the indices S,L, I stand respectively for short-lived, long-lived, interference, and the combination √ ------ ζ ≡ 1 − CI ∕ CSCL provides a good phenomenological characterization of the amount of decoherence induced in the system [398Jump To The Next Citation Point].

Using data gathered by the CPLEAR experiment [13], one can set bounds on α,β,γ at the levels −17 α ∼ 10 GeV, − 19 β ∼ 10 GeV, and −21 γ ∼ 10 GeV. A comparable limit on γ has been placed by DAΦNE’s KLOE experiment, and in that case the analysis was based [398, 534, 205] on entangled kaon states.

I should stress that this is clearly a quantum-spacetime picture (at least in as much as it models spacetime foam) and the objective of the associated research program is to introduce quantum/foamy properties of spacetime at the Planck scale, but it is at present still unclear which levels of sensitivity to α, β, γ would correspond to foaminess of spacetime at the Planck scale. We are still unable to perform a derivation starting from foaminess at the Planck scale and deriving corresponding values for α, β, γ. It is nonetheless encouraging that the present experimental limits on these (dimensionful) parameters are in a neighborhood of the Planck-scale-inspired quantification −19 mK ∕Ep ∼ 10 (but it should be noticed that as much “Planck-scale inspiration” should be attributed, for example, to the scale m2 ∕E2 ∼ 10−38 K p).

4.5.2 Other descriptions of foam-induced decoherence for matter interferometry

Another attempt to characterize spacetime foam as a decoherence-inducing medium was developed by Percival and collaborators (see, e.g., Refs. [452Jump To The Next Citation Point, 453Jump To The Next Citation Point, 454Jump To The Next Citation Point]). This approach assumes that ordinary quantum systems should all be treated as open systems due to neglecting the degrees of freedom of the spacetime foam, but, rather than a formalization using density matrices, Refs. [452, 453, 454Jump To The Next Citation Point] adopt a formalism in which an open quantum system is represented by a pure state diffusing in Hilbert space. The dynamics of such states is formulated in terms of “Primary state diffusion”, an alternative to quantum theory with only one free parameter, a time scale τ 0, which one can set to be the Planck time Lp ∕c.

One way to charaterize τ0 is through a formula for the proper time interval for a timelike segment, which is given by [454Jump To The Next Citation Point]

Δs ≃ |Δξ (x )|2 + Δ ξ(x)√ τ-, (69 ) 0
where Δ ξ(x ) are point-dependent fluctuations induced by the foaminess/quantization of spacetime, which are modelled within the proposed theory.

A key characteristic of this picture would be [454Jump To The Next Citation Point] a suppression of the interference pattern for interferometers using beams of massive particles (such that the original beam is first split and then reunited to seek an interference pattern). The suppression increases with the mass of the particles, so it could more easily be tested with atom interferometers (rather than neutron interferometers). Unfortunately, a realistic analysis of an interferometer in the relevant primary-state-diffusion formalism is much beyond the level of answers one is (at least presently) able to extract from the primary-state-diffusion setup. Ref. [454Jump To The Next Citation Point] considered resorting to some simple-minded simplifications, including the assumption that the Hamiltonian be given by the mass together with projectors onto the wave packets in the arms of the interferometer, neglecting the kinetic-energy terms. Within such simplifications one does find that values of τ0 at or even a few orders of magnitude below the Planck time would leave an observably large trace in modern atom interferometers. However, these simplifications amount to a model of the interferometer that is much too crude (as acknowledged by the authors themselves [454]) and this does not allow us to meaningfully explore the possibility of genuine Planck-scale sensitivities being achieved by this strategy. Note that by taking τ0 as the Planck time it is not obvious that the effects are being introduced genuinely at the Planck scale, since the nature of the effects is characterized not only by τ0 but also by other aspects of the framework, such as the description of the fluctuations. Moreover, even if all other aspects of the picture were understood, the crudity of the model used for matter interferometers would still not allow us to investigate the Planck-scale-sensitivity issue.

Recently, Ref. [498] and Ref. [541Jump To The Next Citation Point] presented somewhat different pictures of quantum-gravity-induced decoherence for atom interferometers. Several aspects of the Percival setup are maintained but different interpretations are applied in some aspects of the analysis. For example, Ref. [541Jump To The Next Citation Point] removes some of the assumptions adopted by Percival and collaborators, particularly in relation to the description of the “quantum fluctuations” of the metric, and proposes an estimate of the amount of suppression of the interference pattern,32 that is perhaps more intriguing from a phenomenology perspective, since it would suggest that the effect is just beyond present sensitivities (but within the reach of sensitivities achievable by atom interferometers in the not-so-distant future). For these recent proposals one is still (for reasons analogous to these just discussed for the Percival approach) unable to meaningfully explore the issue of “genuine Planck-scale sensitivity”, but they may represent a step in the direction of a more detailed description of spacetime foam, if intended as fluctuations of the metric.

4.6 Decoherence and neutrino oscillations

The observations briefly discussed in the previous Section 4.5 that are relevant for the study of manifestations of foam-induced decoherence in some laboratory experiments (neutral-meson studies, atom interferometers) can very naturally be applied to neutrino astrophysics as well, as discussed in Ref. [400Jump To The Next Citation Point] and references therein (see also Refs. [109, 23, 422, 241]). Also in the neutrino context it is natural to attempt to develop test theories codifying the intuition that spacetime foam may act as an environment, so that neutrino observations would have to be analyzed considering the relevant neutrino system as an open system. And the evolution of the neutrino density matrix could be described (in the same sense as the description in Eq. (67View Equation) for neutral-meson systems) by an evolution equation of the type

∂tρ = i[ρ,H ] + δH ρ . (70 )
It is argued in Ref. [400Jump To The Next Citation Point] that such a formalization of the effects of spacetime foam should generate a contribution to the mass difference between different netrinos, and could give rise to neutrino oscillations constituting a “gravitational MSW effect”.

As an alternative to the setup of Eq. (70View Equation) one could consider [400Jump To The Next Citation Point, 401Jump To The Next Citation Point] the possibility of random (Gaussian) fluctuations of the background spacetime metric over which the neutrinos propagates. For the random metric one can take [400Jump To The Next Citation Point, 401Jump To The Next Citation Point] a formalization of the type

( 2 2 ) gμν = − (a1 + 1) + a2 − a3(a1 + 1) + a2(a4 + 1) (71 ) − a3(a1 + 1) + a2(a4 + 1) − a23 + (a4 + 1)2
and enforce [400Jump To The Next Citation Point, 401Jump To The Next Citation Point] for the random Gaussian variables ai a parametrization based on parameters σi (one per ai) such that ⟨ai⟩ = 0 and ⟨aiaj⟩ = δijσi. These fluctuations of the metric are found [400Jump To The Next Citation Point, 401Jump To The Next Citation Point] to induce decoherence even when the neutrinos are assumed to evolve according to a standard Hamiltonian setup,
∂tρ = i[ρ,H ]. (72 )
But the decoherence effects generated in this framework with standard Hamiltonian evolution in a nonstandard (randomly-fluctuating) metric, are significantly different from the ones generated with the nonstandard evolution equation (70View Equation) in a standard classical metric. In particular, in both cases one obtains neutrino-transition probabilities with decoherence-induced exponential damping factors in front of the oscillatory terms, but in the framework with evolution equation (70View Equation) the scaling with the oscillation length (time) is naturally linear [400Jump To The Next Citation Point, 401Jump To The Next Citation Point], whereas when adopting standard Hamiltonian evolution in a fluctuating metric it is natural [400Jump To The Next Citation Point, 401] to have quadratic scaling with the oscillation length (time).

The growing evidence for ordinary-physics neutrino oscillations, which one expects to be much more significant than the foam-induced ones, provides a formidable challenge for the phenomenology based on these test theories for foam-induced decoherence in the neutrino sector. Some preliminary ideas on how to overcome this difficulty are described in Ref. [400]. From the strict quantum-spacetime-phenomenology perspective of requiring one to establish that the relevant measurements could be sensitive to effects introduced genuinely at the Planck scale, these neutrino-decoherence test theories must face challenges already discussed for a few other test theories: there is at present no rigorous/constructive derivation of the values of the parameters of these test theories from a description (be it a full quantum-spacetime theory or simply a toy model) of effects introduced genuinely at the Planck scale, so one can only express these parameters in terms of the Planck scale using some dimensional-analysis arguments.

4.7 Planck-scale violations of the Pauli Exclusion Principle

A case for Planck-scale sensitivity was recently made [97Jump To The Next Citation Point, 99Jump To The Next Citation Point] for the hypothesis of possible violations of the Pauli Exclusion Principle. This has still not been metabolized by an appreciably-wide quantum-gravity community, but it certainly deserves to be highlighted briefly in this review, since the chances for gradually gaining a strong impact on quantum-spacetime phenomenology are rather high.

As observed already a few times in this review, the spin-statistics theorem assumes a classical spacetime with ordinary locality. Therefore, it is legitimate to speculate that small departures from the implications of the spin-statistics theorem may arise in a quantum spacetime. Some earlier suggestions that this might be the case can be found, e.g., in Refs. [98, 163, 86], but the setup then was not such that one could see an emerging case for Planck-scale sensitivity.

The recent studies reported in Refs. [97Jump To The Next Citation Point, 99Jump To The Next Citation Point] investigated this issues assuming the specific form of spacetime noncommutativity given by

[xi,xj] = 0, [x0,xi] = iχ𝜖kijnkxj , (73 )
where nk are the components of a fixed spatial unit vector and the deformation length scale χ can be taken to be on the order of the Planck length.

It is rather easy to show that this form of noncommutativity imposes a corresponding modification of the “flip operator”, i.e., the operator that is used for symmetrization (anti-symmetrization) purposes in the commutative-spacetime case. In turn this gives rise to a deformed description of bosons and fermions. And the end result is that certain transitions that would be Pauli-forbidden in a commutative spacetime are actually allowed, although at a small rate (suppressed by the smallness of χ).

Computing these rates on the basis of Eq. (73View Equation) is at present only possible by relying on an uncomfortable number of simplifying assumptions [97, 99], but the outcome is nonetheless intriguing since it suggests that sensitivity to values of χ on the order of the Planck length is within reach. This exploits the high sensitivity toward possible violations of the Pauli Exclusion Principle at ongoing experiments, such as Borexino [107] and VIP [105].

4.8 Phenomenology inspired by causal sets

Most of the quantum-spacetime phenomenology of this past decade has been inspired by results on spacetime noncommutativity and/or LQG. But several other approaches are getting closer to inspiring phenomenological programs. I share the view of many quantum-spacetime phenomenologists who are looking at the approach based on causal dynamical triangulations [45, 371, 46, 47, 372, 49] as a maturing opportunity for inspiring the phenomenology work. And first indications are coming from the “asymptotic safety approach” [544, 466, 212, 469Jump To The Next Citation Point, 468], on which I shall comment in relation to a tangible proposal for phenomenology later in this review. Certainly in recent years we have seen a blossoming phenomenology emerging from the the causal-set program.

I place here an aside on this recent phenomenology inspired by the causal-set program, which also allows me to return, from a different perspective, on the important subject of non-systematic effects, already briefly discussed in Section 4.3.2. Indeed, because of the perspective that guides that research program, most (if not all) new effects predicted within the causal set program will be of non-systematic type.

Causal sets are a discretization of spacetime that allows the symmetries of GR to be preserved in the continuum approximation [131, 470, 284]. And causal sets can be used to construct simple models suitable for exploring possible manifestations of fuzziness of quantum spacetime. Moreover, the causal set proposal has recently been combined with the loop representation to formulate “causal spin foams” [392], thereby establishing a link to an already mature source of inspiration for quantum-spacetime phenomenology.

Clearly, some of the manifestations one must expect from a causal-set setup fall within the class of phenomena already briefly described in Section 4.3.1: at a coarse-grained level of analysis a causal-set background should introduce an intrinsic limitation to the accuracy of lengths and durations. Several recent works were aimed at formalizing and modeling these aspects of fuzziness for propagation [311, 312, 215]. The preliminary indications that are emerging appear to suggest that, if discreteness is indeed introduced at the Planck scale, the effects are very soft (hard to detect). Nonetheless we do already have a few examples of studies aiming for tangible predictions to be compared to actual data: for example, Ref. [490] reports a causal-set-inspired analysis of possible fuzziness of arrival times (the sort of effects already discussed in Section 4.3.1), relevant for studies conducted by gamma-ray telescopes.

An intriguing effect of random fluctuations in photon polarization can also be motivated by the causal-set framework [186]. The presently-available models of this causal-set-induced effect are to be viewed as very crude/preliminary, particularly since the present understanding of the framework is still not at the point of providing a definite model of photons propagating on a causal set background (from which one could derive the polarization-fluctuation feature). Still, this appears a very promising direction, especially since experimental information on CMB polarization is improving quickly and will keep improving in the coming years.

Presently the most tangible phenomenological plans inspired by the causal-set framework revolve around an effect [214Jump To The Next Citation Point, 458] of Lorentz invariant diffusion in the 4-momentum of massive particles. This is an Ornstein–Uhlenbeck process, a diffusion process on the mass-shell that results in a stochastic evolution in spacetime. An intuitive picture for this mechanism was given in Ref. [214Jump To The Next Citation Point], by considering a classical particle, of mass m propagating on a random spacetime lattice. The particle would then be constrained to move from point to point, but the discretization is such that in order to “reach the next point” (remaining on the lattice) the particle must ‘swerve’ slightly, also adapting its velocity v to the swerving (also see Ref. [457] for a comparison of possible variants of the description of particle propagation in causal-set theory). The change in velocity amounts to the particle jumping to a different point on its mass shell. The net result of this swerving is that [214Jump To The Next Citation Point, 316Jump To The Next Citation Point, 396Jump To The Next Citation Point] a collection of particles initially with an energy-momentum distribution ρ (p ) will diffuse in momentum space along their mass shell according to the equation

∂ρ- 2 1- μ ∂τ = 𝒦 ∇ 𝒫ρ − m p ∂ μρ, (74 )
where [214, 316Jump To The Next Citation Point, 396Jump To The Next Citation Point] 𝒦 is the diffusion constant, 2 ∇ 𝒫 is the Laplacian in momentum space on the mass shell of the particle, τ is the proper time, and ∂μ is an ordinary spacetime derivative.

The tightest limit on 𝒦 is 𝒦 < 10 −61 GeV3, and was obtained [316] from limits on the amount of relic-neutrino contribution to hot dark matter. This follows from the observation that energy on the mass shell is bound from below by the mass, so that particles close to rest, when swerving, can essentially only increase their energy.

Interestingly these 𝒦-governed effects can also be relevant [396Jump To The Next Citation Point] for some of the phenomenology already discussed in this review, concerning the threshold requirements for certain particle-physics processes. Essentially the expected implications of swerving for these threshold analyses is similar to that already discussed in Section 4.3.2: in any given opportunity of interaction between a hard photon and a soft photon the swerving can effectively raise or lower (from the perspective of the asymptotically-incoming states) the threshold requirements for pion production. However, it appears that the bound −61 3 𝒦 < 10 GeV, if applicable also to protons,33 brings the magnitude of such effects safely beyond the reach [396] of ongoing cosmic-ray studies.

4.9 Tests of the equivalence principle

4.9.1 Aside on tests of the equivalence principle in the semiclassical-gravity limit

I am focusing in this review on tests motivated by (and on effects modeled within) proposals of spacetime quantization at the Planck scale, but concerning tests of the equivalence principle inspired by quantum-spacetime models there is some merit in making a small digression on tests of the equivalence principle in the semiclassical limit of quantum gravity (where, by construction, no quantum-spacetime effects could be seen). This will allow me to compellingly set-up the issue of testing the equivalence principle from a general quantum-gravity perspective and specifically from the perspective of spacetime quantization at the Planck scale.

As already discussed briefly in Section 1, there is a long tradition of phenomenological studies, concerning the semiclassical-gravity limit, based on a “gravity version” of the Schrödinger equation of the form

[ ( ) ] -1--- ⃗ 2 ∂ ψ-(t,⃗r) − 2M ∇ + MG ϕ(⃗r) ψ (t,⃗r) = i ∂t , (75 ) I
describing the dynamics of matter (with wave function ψ (t,⃗r), inertial mass MI and gravitational mass MG) in an external gravitational potential ϕ (⃗r). Some of the most noteworthy results obtained within this framework are the interferometric studies of the type first set up by Colella, Overhauser and Werner [177], which establish that the Earth’s gravitational field is strong enough to affect the evolution of the wave function ψ in an observably-large manner, and the more recent evidence [428Jump To The Next Citation Point] that ultracold neutrons falling towards a horizontal mirror do form gravitational quantum bound states.

Of relevance here is the fact that some of the issues that have been most extensively considered by researchers involved in these studies concern the equivalence principle. This is signaled by the adoption of separate notation for inertial and gravitational mass in Eq. (75View Equation). In principle the gravitational mass MG governs the accrual of gravity-induced phases, while the inertial mass MI intervenes in determining the ratio between wave vector and velocity vector in the Galilean limit (p ≪ m). And even for MG = MI the mass does not factor out of the free-fall evolution of the quantum state (but for MG = MI one at least recovers [535] a complete identification between the effects of gravitation and the effects of acceleration).

Besides neutrons, these studies can also be performed with atoms [14]. And, interestingly, one can also perform rather similar analyses in studying neutrino oscillations, finding (see, e.g., Refs. [252, 272, 148] and references therein), that gravity may induce neutrino oscillations if different neutrino flavors are coupled differently to the gravitational field, thereby violating the equivalence principle.

4.9.2 On the equivalence principle in quantum spacetime

Evidently, searches of possible violations of the equivalence principle in the semiclassical-gravity limit of quantum gravity have significant intrinsic interest. And some of these tests of the equivalence principle in semiclassical-gravity limit also find explicit motivation in approaches to the study of the full quantum-gravity problem: most notably the string-theory-inspired studies reported in Refs. [521, 195, 196, 194, 193, 192], and references therein, predict violations of the equivalence principle in the semiclassical-gravity limit.

Returning to the main subject of this review, I should stress that the idea of spacetime quantization at the Planck scale provides a particularly crisp motivation for testing the equivalence principle. The simplest way to see this comes from observing the role of absolute and ideally sharp locality in the role that the equivalence principle plays in classical gravity, in contrast to the large class of qualitatively very severe (though tiny) anomalies for locality that the various known scenarios for spacetime quantization (starting with spacetime noncommutativity for example) provide. Unfortunately, our present level of mastery of the relevant formalisms often falls short of allowing us to investigate the fate of the equivalence principle. Therefore, I will briefly describe one illustrative example of promising attempt to model how spacetime foam could affect the equivalence principle. This is the objective of recent studies, reported in Ref. [263Jump To The Next Citation Point] and references therein, in which spacetime foam is modeled in terms of small fluctuations of the metric on a given background metric.34 The analysis of Ref. [263Jump To The Next Citation Point], which also involves an averaging procedure over a finite spacetime scale, ends up motivating the study of a modified Schrödinger equation of the form

[ ( 1 ) kl kl ] − ---- (δ + &tidle;α )∂k∂l − m ϕ(⃗r) ψ (t,⃗r) = i ∂tψ(t,⃗r), (76 ) 2m
where the tensor &tidle;αkl is a characterization of the spacetime foaminess, and it is natural to consider the tensor &tidle;mkl,
kl− 1 1- kl kl (&tidle;m ) ≡ m (δ + &tidle;α ), (77 )
as an anomalous inertial mass tensor that depends on the type of particle and on the fluctuation scenario. The particle-dependent rescaling of the inertial mass provides a candidate key manifestation of foam-induced violations of the equivalence principle to be sought experimentally, in ways that are once again exemplified by the COW experiments.

This very recent proposal illustrates a type of path that could be followed to introduce violations of the equivalence principle originating genuinely from spacetime quantization at the Planck scale: one might find a way to describe spacetime foaminess n terms of effects of genuinely Planckian size, and then elaborate the implications of this spacetime foaminess for the equivalence principle. The formalization adopted in Ref. [263] is still too crude to allow such an explicit link between the Planck-scale picture of spacetime foam and the nature and magnitude of the effects, but provides a significant step in that direction.

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