## 2 Quantum-Gravity Theories, Quantum Spacetime, and Candidate Effects

Before getting to the main task of this review, which concerns phenomenology proposals, it is useful to summarize briefly the motivation for studying certain candidate quantum-spacetime effects. The possible sources of motivation come either from analyses of the structure of the quantum-gravity problem or from what is emerging in the development of some theories that have been proposed as candidate solutions of the quantum-gravity problem. As already stressed, my main focus here is on effects that can be linked to spacetime quantization at (about) the Planck scale, and particularly the ones that were involved in the two-way interface that materialized over this last decade between phenomenologists and theorists working on the LQG approach and spacetime noncommutativity.In the first part of this section, I offer a few comments on some of the approaches being pursued in the study of the quantum-gravity problem, mostly focusing on whether or not they support a quantum-spacetime picture and the role played by the Planck scale. This part focuses primarily on LQG and spacetime noncommutativity, but I also comment briefly on critical string theory and other approaches.

Then in the second part of this section I list some key candidates as phenomena that could characterize the quantum-spacetime realm. This list is only very tentative but it seems to me we cannot do any better than this at the present time. Indeed, compiling a list of candidate quantum-spacetime effects is not straightforward. Analogous situations in other areas of physics are usually such that there are a few new theories that have started to earn our trust by successfully describing some otherwise unexplained data, and then often we let those theories guide us toward new effects that should be looked for. The theories that are under consideration for the solution of the quantum-gravity problem, and for a “quantum” (non-classical) description of spacetime, cannot yet claim any success in the experimental realm. Moreover, even if nonetheless we wanted to use them as guidance for experiments, the complexity of these theories proves to be a formidable obstruction. In most cases, especially concerning testable predictions, the best we can presently do with these theories is analyze their general structure and use this as a source of intuition for the proposal of a few candidate effects. Similarly, when we motivate the search of certain quantum-spacetime features on the basis of our present understanding of the quantum-gravity problem we are in no way assured that they should still find support in future better insight on the nature of this problem, but it is the best we can do at the present time.

### 2.1 Quantum-Gravity Theories and Quantum Spacetime

#### 2.1.1 Critical String Theory

The most studied approach to the quantum-gravity problem is a version of string theory that adopts supersymmetry and works in a “critical” number of spacetime dimensions. If this main-stream perspective turned out to be correct it would be bad news for quantum-spacetime phenomenologists, since the theory is formulated in classical Minkowski background spacetime. It would be bad news for phenomenology in general because (critical, supersymmetric) string theory is a particularly soft modification of current theories, and the new effects that can be accommodated by the theory are untestably small, if all the new features are indeed introduced (as traditionally assumed) at a string scale roughly given by the Planck scale.

String theory is the natural attempt from a particle-physics perspective, but other perspectives on the quantum-gravity problem remain unimpressed, particularly considering that most results of string theory still only apply in a fixed background Minkowski spacetime. And it is interesting to notice how the most careful analyses performed even adopting a string-theory perspective end up finding that the case for applicability to the quantum-gravity problem is still rather weak (see, e.g., Ref. [257]).

This not withstanding there has been in recent years a more vigorous effort of development of a string-inspired phenomenology, with inspiration found in mechanisms that are, however, outside the traditional formulation of string theory. This string-inspired phenomenology does not involve spacetime quantization and often does not refer explicitly to the Planck scale, so I shall not discuss it in detail in this review (although there will be scattered opportunities, at points of this review, where it becomes indirectly relevant). The possibility that received the most attention in recent years is the one of “large” extra dimensions [80, 375, 552, 84, 85, 480]. The existence of extra dimensions can be conceived even outside string theory, but it is noteworthy that in string theory the criticality criterion actually requires extra dimensions. If the extra dimensions, as traditionally assumed, have finite size on the order of the Planck length, then one ends up having associated Planck-scale effects for the low-energy realm, where our experiments and observations take place. This would be a classic exercise for quantum-gravity phenomenology but it appears that the Planck-scale suppression of these extra-dimension effects is so strong that they really could not ever be seen/tested. The recent interest in the “large extra dimensions” scenario originates from the observation that dimensions of size much larger than the Planck length (but still microscopic), while not particularly natural from a string-theory perspective, may well be allowed in string theory [80, 375, 552, 84, 85, 480]. And for some choices of number and sizes of extra dimensions a rich phenomenology is produced.

Most other phenomenological proposals inspired by string theory essentially make use of
the fact that, at least as seen by a traditional particle physicist, string theory makes room
for several new fields. The new effects are indeed of types that are naturally described by
introducing new fields in a classical spacetime background, rather than quantum-spacetime
features, and the magnitude of these effects is not naturally governed by the Planck
scale.^{9}

In spite of these profound differences there are some points of contact between the Planck-scale quantum-spacetime phenomenology, which I am here concerned with, and this string phenomenology. In a quantum spacetime it is necessary to reexamine the issue of spacetime symmetries, and certain specific scenarios for the fate of Lorentz symmetry come into focus. From a different perspective and in a technically different way one also finds reasons to scrutinize Lorentz symmetry in string phenomenology: it is plausible [347] that some string-theory tensor fields (most likely some of the new fields introduced by the theory) could acquire a nonzero vacuum expectation value, in which case evidently one would have a “spontaneous breakdown” of Lorentz symmetry. I shall also comment on the possibility that spacetime quantization might affect the equivalence principle. Again, from a different perspective and in a technically different way, one also finds reasons to scrutinize the equivalence principle in string phenomenology. And again it is typically due to the extra fields introduced in string theory: most notably some scenarios involving the dilaton, a scalar partner to the graviton predicted by string theory, produce violations of the equivalence principle (see, e.g., Ref. [193]).

I should stress here, because of the scope of this review, that the idea of a quantum spacetime is not completely foreign to string theory. It is presently appearing at an undigested and/or indirect level of analysis, but it is plausible that future evolutions of the string-theory program might have a primitive/fundamental role for spacetime quantization. So far the most studied connection with quantum-spacetime ideas comes from a mechanism analogous to the emergence of noncommutativity of position coordinates in the Landau model (see, e.g., Ref. [101]) that is found to be applicable to the description of strings in the presence of a constant Neveu–Schwarz two-form (“”) field [213, 516]. It should be stressed that these cases of “emerging noncommutativity” (effective descriptions applicable only in certain specific regimes) do not amount to genuine nonclassicality of spacetime. Still, these recent string-theory results do create a point of contact between research (and particularly phenomenology) on fundamental spacetime noncommutativity and string theory, with the peculiarity that from the string-theory perspective one would not necessarily focus (and typically there is no focus) on the case of noncommutativity introduced at about the Planck scale, since it is instead given in terms of the free specification of the field .

For the hope of a possible future reformulation of string theory in some way that would accommodate a primitive role for spacetime nonclassicality my impression is that the key opportunities should be seen in results suggesting that there are fundamental limitations for the localization of a spacetime event in string theory [532, 269, 44, 332]. The significance of these results on limitations of localizability in string theory probably has not been appreciated sufficiently. Only a few authors have emphasized the possible significance of these results [551], but I would argue that finding such limitations in a theory originally formulated in a classical spacetime background may well provide the starting point for reformulating the theory completely, perhaps codifying spacetime quantization at a primitive level.

#### 2.1.2 Loop Quantum Gravity

The most studied theory framework providing a quantum description of spacetime is LQG [476, 96, 502, 524, 93]. The intuition of many phenomenologists who have looked at (or actually worked on) LQG is that this theory should predict quite a few testable effects, some of which may well be testable with existing technologies. However, the complexity of the formalism has proven so far to be unmanageable from the point of view of obtaining crisp physical predictions. Among the many challenges I should at least mention the much debated “classical-limit problem”, which obstructs the way toward a definite set of predictions for the quasi-Minkowski (or quasi-deSitter, or quasi-FRW) regime, which is where most of the opportunities for phenomenology can be found.

However, one may attempt to infer from the general structure of the theory motivation for the study of
some candidate LQG effects. And, as I shall stress in several parts of this review, this type
of attitude has generated a healthy interface between phenomenologists and LQG theorists.
Most of the relevant proposals are ignited indeed by the quantum properties of spacetime in
LQG, which appear to be primarily codified in a discretization of the area and volume
observables [477, 95, 476] In particular, several studies (see later in this review) have argued
that the type of discretization of spacetime observables usually attributed to LQG could be
responsible^{10}
for Planck-scale departures from Lorentz symmetry.

In addition to a large effort focused on the fate of Lorentz symmetry, there has also been a rather large effort focused on early-Universe cosmology inspired by LQG. Among the appealing features of this cosmology work I should at least mention “singularity avoidance”. For the LQG approach, there might be no alternative to avoiding the big-bang singularity, since indeed, at least as presently understood, LQG describes spacetime has a fundamentally discrete structure governed by difference (rather than differential) equations. This discreteness is expected to become a dominant characteristic of the framework for processes involving comparably small (Planckian) length scales, and in particular it should inevitably give rise to a totally unconventional picture of the earliest stages of evolution of the Universe. Attempts at developing a setup for a quantitative description of these early-Universe features have been put forward in Refs. [125, 94, 126, 92] and references therein, but one must inevitably resort to rather drastic approximations, since a full LQG analysis is not possible at present.

For other areas of phenomenology discussed in this review the influence of LQG has been less direct, but it appears safe to assume that it will inevitably grow in the coming years. To give a particularly striking example, let me mention the many proposals here discussed that concern spacetime fuzziness. It is evident that LQG gives a fuzzy picture of spacetime (in the sense discussed more precisely in later parts of this review), and it would be of important guidance for the phenomenologists to have definite predictions for these features. Even just a semiheuristic derivation of such effects is beyond the reach of our present understanding of LQG, but it will come.

#### 2.1.3 Approaches based on spacetime noncommutativity

The idea of having a nonclassical fundamental description of spacetime is central to the study of spacetime noncommutativity. The formalization that is most applied in the study of the quantum-gravity/quantum-spacetime problem is mainly based on the formalism of “quantum-groups” and essentially assumes that the quantum properties of spacetime should be at least to some extent analogous to the quantum properties of phase space in ordinary quantum mechanics. Ordinary quantum mechanics introduces some limitations for procedures intending to obtain a combined determination of both position and momentum, and this is formalized in terms of noncommutativity of the position and momentum observables. With spacetime noncommutativity one essentially assumes that spacetime coordinates should not commute [211, 391, 374, 384, 70, 98] among themselves, producing some limitations for the combined determination of more than one coordinate of a spacetime point/event. This has been the formalization of spacetime noncommutativity for which the two-way interface between theory and phenomenology, which is at center stage in this review, has been most significant.

Looking ahead at the future of quantum-spacetime phenomenology, it appears legitimate to hope that another, perhaps even more compelling, candidate concept of noncommutative geometry, the one championed by Connes [185, 184], may provide guidance. At present the most studied applications of this notion of noncommutative geometry are focused on giving a fully geometric description of the standard model of particle physics, with the noncommutativity of geometry used to codify known properties of particle physics in geometric fashion, while keeping spacetime as a classical geometry.

Going back to the quantum-group-based description of spacetime noncommutativity I should stress that, so far, the most significant developments have concerned attempts to describe the Minkowski limit of the quantum-gravity problem, i.e., a noncommutative version of Minkowski spacetime (spacetimes that reproduce classical Minkowski spacetime in the limit in which the noncommutativity parameters are taken to 0). Some related work has also been directed toward quantum versions of de Sitter spacetime, but very little about spacetime dynamics and only at barely an exploratory level. This should change in the future. But at the present time this situation could be described by stating that most work on spacetime noncommutatvity is considering only one half of the quantum-gravity problem, the quantum-spacetime aspects (neglecting the gravity aspects). Because of the double role of the gravitational field, which in some ways is just like another (e.g., electromagnatic) field given in spacetime but it is also the field that describes the structure of spacetime, in quantum-gravity research the idea that this classical field be replaced by a nonclassical one ends up amounting to two concepts: some sort of quantization of gravitational interactions (which might be mediated by a graviton) and some sort of quantization of spacetime structure. At present one might say that only within the LQG approach are we truly exploring both aspects of the problem. String theory, as long as it is formulated in a classical (background) spacetime, focuses in a sense on the quantization of the gravitational interaction, and sets aside (or will address in the future) the possible “quantization” of spacetime [551]. Spacetime noncommutativity is an avenue for exploring the implications of the other side, the quantization of spacetime geometry.

The description of (Minkowski-limit) spacetime in terms of (quantum-group-based) spacetime noncommutativity has proven particularly valuable in providing intuition for the fate of (Minkowski-limit/Poincaré) spacetime symmetries at the Planck scale. Also parity transformations appear to be affected by at least some schemes of spacetime noncommutativity and this in turn provides motivation for testing CPT symmetry.

Unfortunately, spacetime fuzziness, which is the primary intuition that leads most researchers to noncommutativity, frustratingly remains only vaguely characterized in current research on noncommutative spacetimes; certainly not characterized with the sharpness needed for phenomenology.

#### 2.1.4 Other proposals

I shall not attempt to review the overall status of quantum-gravity research. The challenge of reviewing and offering a perspective on quantum-spacetime phenomenology is already overwhelming. And according to the perspective of this phenomenological approach the central challenge of quantum-gravity research is to find the first experimental manifestations of the quantum-gravity realm. The different formalisms proposed for the study of the quantum-gravity problem can be very valuable for this objective, but only in as much as they provide intuition for the type of new effects that might characterize the quantum-gravity realm. In practice, at least for the next few decades, what will be compared to data will be simple test theories inspired by our understanding of the quantum-gravity problem or by the intuition obtained in the study of formal theories of quantum gravity. The possibility of comparing a full quantum-gravity theory directly to experiments appears to be for a still distant future, as a result of the complexity of these theories (which prevents us from deriving testable predictions).

I have invested a few pages on string theory, LQG and spacetime noncommutativity for different reasons. Providing some reasonably detailed comments on string theory was encouraged, in spite of the lack of a fundamental role for spacetime quantization, by its prominent role in the quantum-gravity literature. And, as stressed above, LQG and spacetime noncommutativity are particularly relevant for this review because the scenarios of spacetime quantization these approaches consider/derive have been a particularly influential source of intuition for proposals in quantum-spacetime phenomenology. Moreover, it is within the LQG and spacetime-noncommutativity communities that we have, so far, witnessed the most significant examples of the healthy two-way cross-influence between formal theory and phenomenology.

I shall not offer comparably detailed comments on any other quantum-gravity formalism, but there are a few that I should mention because of the significance of their role in quantum-spacetime phenomenology. First of all let me mention the noncritical “Liouville string theory” approach championed by Ellis, Mavromatos and Nanopoulos [221, 223, 65, 399]. This is a variant of the string-theory approach that (unlike the main-stream critical-string-theory approach) adopts the choice of working in “noncritical” number of spacetime dimensions, and describes time in a novel way. As will be evident in several points of this review, Ellis, Mavromatos, Nanopoulos and collaborators have developed noncritical Liouville string theory from a perspective that admirably keeps phenomenology always at center stage, and this has been a key influence on several quantum-spacetime-phenomenology research lines.

Another approach for which there is by now a rather sizable research program aimed at phenomenological consequences is the one based on “discrete causal sets” [131, 470]. This is an approach of spacetime discretization that exploits the fact that a Lorentzian metric determines both a geometry and a causal structure and also determines the metric up to a conformal factor. One can then take the causal structure as primary, and start with a finite set of points with a causal ordering, recovering the conformal factor by counting points. Several opportunities for phenomenology are then produced by the discretization of spacetime.

Still, on the subject of approaches in which a role is played by spacetime discretization I should also bring to the attention of my readers the recent developments in the study of causal dynamical triangulations [45, 371, 46, 47, 372, 49]. Through causal dynamical triangulations one gives an explicit, nonperturbative and background-independent, realization of the formal gravitational path integral on a given differential manifold. And some of the results obtained within this approach already provide elements of valuable intuition for quantum-spacetime phenomenology, as exemplified by the results providing [48] first evidence for a scale-dependent spectral dimension of spacetime, varying from four at large scales to two at scales on the order of the Planck length. These “running spectral dimensions” could have very significant applications in phenomenology, and early signs that this might indeed be the case can be found in the debate reported in Refs. [424, 505, 425] concerning the implication for primordial gravity waves.

Also particularly important for quantum-spacetime phenomenology is the program of asymptotically-safe quantum gravity. This is an attempt at the nonperturbative construction of a predictive quantum field theory of the metric tensor centered on the availability of a non-Gaussian renormalization-group fixed point [544, 466, 212]. There are a few perspectives from which this asymptotic-safety program is influencing part of the research on quantum-spacetime phenomenology. As an example of phenomenology work that was directly inspired by asymptotic safety, I should mention the expectation that quantum-gravity effects might also be important in a large-distance regime [469], with possible relevance for phenomenology. I shall comment on this later in this review, also in relation to the idea of “UV/IR mixing” as a possibility that appears to be plausible even within other perspectives on quantum gravity and quantum spacetime. And there are significant indications (see, e.g., Ref. [468]) that ultimately the description of spacetime in a quantum gravity with asymptotic safety will be a quantum-spacetime description. Also significant for quantum-spacetime phenomenology is the whole idea of running gravitational couplings, which is central to asymptotic safety. As mentioned we tentatively assume that quantum-spacetime effects originate at the Planck scale, but the Planck scale is computed in terms of (the IR value of) Newton’s constant and might give us a misleading intuition for the characteristic scales of spacetime quantization.

There are also some perspectives on the quantum-gravity problem that at present I do not see as direct opportunities for quantum-spacetime phenomenology, but certainly are playing the role of “intuition builders” for the phenomenologists, affecting the perception of the quantum-gravity problem that guides some of the relevant research. Among these I should mention the rather large literature on the “emergent gravity paradigm” (see, e.g., Refs. [103, 538, 443, 513, 555, 499, 297]). This literature actually contains a variety of possible way through which gravity could be described not as a fundamental aspect of the laws of nature, but rather as an emergent feature. A simple analogy here is with pion-mediated strong interactions, which emerge from the quantum chromodynamics of quarks and gluons at low energies.

And I should mention as another potential “intuition builder” for the phenomenologists a class of studies that in various ways place dissipation in connection with aspects of the quantum-gravity problem (see, e.g., Refs. [518, 296]).

### 2.2 Candidate effects

From the viewpoint of phenomenologists, the theory proposals I briefly considered in Section 2.1.4 (all still lacking any experimental success) can only serve the purpose of inspiring some test theories suitable for comparison to data.

In this Section, I will briefly motivate a partial list of possible classes of effects that could characterize the quantum-gravity/quantum-spacetime realm. And indeed in compiling such a list, one ends up using both intuition based on the general structure of the quantum-gravity problem and intuition based on what has been so far understood of theories that predict or assume spacetime quantization.

Both the analysis of the general structure of the quantum-gravity problem and the analysis of proposed approaches to the solution of the quantum-gravity problem provide a rather broad collection of intuitions for what might be the correct “quantization” of spacetime (see, e.g., Refs. [406, 532, 269, 44, 332, 442, 211, 20, 432, 50, 249, 489]), and in turn this variety of scenarios produces a rather broad collection of hypothesis concerning possible experimental manifestations of spacetime quantization.

#### 2.2.1 Planck-scale departures from classical-spacetime symmetries

From a quantum-spacetime perspective it is natural to expect that some opportunities for phenomenology might come from tests of spacetime symmetries. It is relatively easy to test spacetime symmetries very sensitively, and it is natural to expect that introducing new (“quantum”) features in spacetime structure would affect the symmetries.

Let us consider in particular the Minkowski limit, the one described by the classical Minkowski
spacetime in current theories: there is a duality one-to-one relation between the classical Minkowski
spacetime and the classical (Lie-) algebra of Poincaré symmetry. Poincaré transformations are smooth
arbitrary-magnitude classical transformations and it is, therefore, natural to subject them to
scrutiny^{11}
if the classical Minkowski spacetime is replaced by a quantized/discretized version.

The most active quantum-spacetime-phenomenology research area is indeed the one considering possible Planck-scale departures from Poincaré/Lorentz symmetries. One possibility that has been considered in detail is the one of some symmetry-breaking mechanism affecting Poincaré/Lorentz symmetry. An alternative, which I advocated a few years ago [58, 55], is the one of a “spacetime quantization” that deforms but does not break some spacetime symmetries.

Besides the analysis of the general structure of the quantum-gravity problem, encouragement for these Poincaré/Lorentz-symmetry studies is also found within some of the most popular proposals for spacetime quantization. As mentioned, according to the present understanding of LQG, the fundamental description of spacetime involves some intrinsic discretization [476, 502], and, although very little of robust is presently known about the Minkowski limit of the theory, several indirect arguments suggest that this discretization should induce departures from classical Poincaré symmetry. While most of the LQG literature on the fate of Poincaré symmetries argues for symmetry violation (see, e.g., Refs. [247, 33]), there are some candidate mechanisms (see, e.g., Refs. [75, 237, 503]) that appear to provide opportunities for a deformation of symmetries in LQG.

A growing number of quantum-gravity researchers are also studying noncommutative versions of Minkowski spacetime, which are promising candidates as “quantum-gravity theories of not everything”, i.e., opportunities to get insight on some, but definitely not all, aspects of the quantum-gravity problem. For the most studied examples, canonical noncommutativity,

and -Minkowski noncommutativity, the issues relevant for the fate of Poincaré symmetry are very much in focus, and departures from Poincaré symmetry appear to be inevitable.^{12}

#### 2.2.2 Planck-scale departures from CPT symmetry

Arguments suggesting that CPT violation might arise in the quantum-gravity realm have a long tradition [279, 445, 540, 446, 42, 222, 298, 345, 117] (and also see, e.g., the more recent Refs. [21, 423, 330]). And, in light of the scope of this review, I should stress that specifically the idea of spacetime quantization invites one to place CPT symmetry under scrutiny. Indeed, locality (in addition to unitarity and Lorentz invariance) is a crucial ingredient for ensuring CPT invariance, and a common feature of all the proposals for spacetime quantization is the presence of limitations to locality, at least intended as limitations to the localizability of a spacetime event.

Unfortunately, a proper analysis of CPT symmetry requires a level of understanding of the formalism that is often beyond our present reach in the study of formalizations of the concept of quantum spacetime. In LQG one should have a good control of the Minkowski (classical-) limit, and of the description of charged particles in that limit, and this is still beyond what can presently be done within LQG.

Similar remarks apply to spacetime noncommutativity, although in that case some indirect arguments relevant for CPT symmetry can be meaningfully structured. For example, in Ref. [70] it is observed that certain spacetime noncommutativity scenarios appear to require a deformation of (parity) transformations, which would result in a corresponding deformation of CPT transformations.

In the mentioned quantum-spacetime picture based on noncritical Liouville string theory [221, 224], evidence of violations of CPT symmetry has been reported [220], and later in this review I shall comment on the exciting phenomenology that was inspired by these results.

#### 2.2.3 Decoherence and modifications of the Heisenberg principle

It is well established that the availability of a classical spacetime background has been instrumental to the successful tests of quantum mechanics so far performed. The applicability of quantum mechanics to a broader class of contexts remains an open experimental question. If indeed spacetime is quantized there might be some associated departures from quantum mechanics. And this quantum-spacetime intuition fits well with a rather popular intuition for the broader context of quantum-gravity research, as discussed for example in Refs. [280, 361].

Some of the test theories used to model spacetime quantization have been found to provide motivation for departures from quantum mechanics in the form of “decoherence”, loss of quantum coherence [432, 50, 246]. A description of decoherence has been inspired by the mentioned noncritical Liouville string theory [221, 224], and is essentially the core feature of the formalism advocated by Percival and collaborators [452, 453, 454].

The possibility of modifications of the Heisenberg principle and of the de Broglie relation has also been much studied in accordance with the intuition that some aspects of quantum mechanics might need to be adapted to spacetime quantization. Although the details of the mechanism that produces such modifications vary significantly from one picture of spacetime quantization to another [322, 22, 122], one can develop an intuition of rather general applicability by noticing that the form of the de Broglie relation in ordinary quantum mechanics reflects the properties of the classical geometry of spacetime that is there assumed. More precisely, the de Broglie relation reflects the properties of the differential calculus on the spacetime manifold, since ordinary quantum mechanics describes the momentum observable in terms of a derivative operator (assuming the Heisenberg principle holds), which, acting on wave functions with wavelength , leads to the de Broglie relation . In a nonclassical (“quantum”) spacetime one must adopt new forms of differential calculus [500, 390], and as a result the description of the momentum observable and its relation to the wavelength of a wave must be reformulated [322, 22, 122, 63].

While the possibility of spacetime quantization provides a particularly direct logical line toward modifications of laws of quantum mechanics, one should consider such modifications as natural for the whole quantum-gravity problem (even when studied without assuming spacetime quantization). For example, in string theory, assuming the availability of a classical spacetime background, one finds some evidence of modification of the Heisenberg principle (the “Generalized Uncertainty Principle” discussed, e.g., in Refs. [532, 269, 44, 332, 551]).

#### 2.2.4 Distance fuzziness and spacetime foam

A description that is often used to give some intuition for the effects induced by spacetime quantization is Wheeler’s “spacetime foam”, even though it does not amount to an operative definition. Most authors see it as motivation to look for formalizations of spacetime in which the distance between two events cannot be sharply determined, and the metric is correspondingly fuzzy. As I shall discuss in Section 4, a few attempts to operatively characterize the concept of spacetime foam and to introduce corresponding test theories have been recently developed. And a rather rich phenomenology is maturing from these proposals, often centered both on spacetime fuzziness per se and associated decoherence.

Unfortunately, very little guidance can be obtained from the most studied quantum-spacetime pictures. In LQG this type of experimentally tangible characterization of spacetime foam is not presently available. And remarkably even with spacetime noncommutativity, an idea that was mainly motivated by the spacetime-foam intuition of a nonclassical spacetime, we are presently unable to describe, for example, the fuzziness that would intervene in operating an interferometer with the type of crisp physical characterization needed for phenomenology.

#### 2.2.5 Planck-scale departures from the equivalence principle

The possibility of violations of the equivalence principle has not been extensively studied from a quantum-spacetime perspective, in spite of the fact that spacetime quantization does provide some motivation for placing under scrutiny at least some implications of the equivalence principle. This is at least suggested by the observation that locality is a key ingredient of the present formulation of the equivalence principle: the equivalence principle ensures that (under appropriate conditions) two point particles would go on the same geodesic independent of their mass. But it is well established that this is not applicable to extended bodies, and presumably also not applicable to “delocalized point particles” (point particles whose position is affected by uncontrolled uncertainties). Presumably also the description of particles in a spacetime that is nonclassical (“quantized”), and, therefore, sets absolute limitations on the identification of a spacetime point, would require departures from some aspects of the equivalence principle.

Relatively few studies have been devoted to violations of the equivalence principle from a quantum-spacetime perspective. Examples are the study reported in Ref. [149], which obtained violations of the equivalence principle from quantum-spacetime-induced decoherence, the study based on noncritical Liouville string theory reported in Ref. [227], and the study based on metric fluctuations reported in Ref. [263].

Also the broader quantum-gravity literature (even without spacetime quantization)
provides motivation for scrutinizing the equivalence principle. In particular, a strong
phenomenology centered on violations of the equivalence principle was proposed in the
string-theory-inspired studies reported in Refs. [521, 195, 196, 194, 193, 192] and
references therein, which actually provide a description of violations of the equivalence
principle^{13}
at a level that might soon be within our experimental reach.

Also relevant to this review is the possibility that violations of the equivalence principle might be a by-product of violations of Lorentz symmetry. In particular, this is suggested by the analysis in Ref. [338], where the gravitational couplings of matter are studied in the presence of Lorentz violation.