Quantum Gravity Phenomenology
Giovanni Amelino-Camelia
Dipartimento di Fisica Università di Roma "La Sapienza" and Sez. Roma1 INFN
P.le A. Moro 2, 00185 Roma , Italy
I review the present status of the development of Quantum Gravity Phenomenology. Among the accomplishments of this young research area I stress in particular the significance of studies which established that some appropriate data analyses provide sensitivity to effects introduced genuinely at the Planck scale. The objective of testing/falsifying theories that provide comprehensive solutions to the quantum-gravity problem appears to be still rather far, but we might soon be in a position to investigate some "falsifiable quantum-gravity theories of not everything".
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I Introduction and preliminaries
I.1 The "Quantum-Gravity problem", as seen by a phenomenologist
I.2 Prehistory of quantum-gravity phenomenology
I.3 Genuine Planck-scale sensitivity and the dawn of quantum-gravity phenomenology
I.4 The simplest/clearest example of genuine Planck-scale sensitivity
I.5 Characteristics of the experiments
I.6 The role of test theories
I.7 The role of falsifiable quantum-gravity theories of not everything
I.8 Schematic outline of this review
I.9 Sensitivities rather than limits
II Some candidate quantum-gravity effects
II.1 Planck-scale departures from classical-spacetime symmetries
II.2 Planck-scale departures from CPT symmetry
II.3 Distance fuzziness and spacetime foam
II.4 Decoherence
II.5 Planck-scale departures from the Equivalence Principle
III Poincaré-symmetry tests
III.1 Some relevant concepts
III.1.1 The Minkowski limit
III.1.2 Three perspectives on the fate of Poincaré symmetry at the Planck scale
III.1.3 Aside on Doubly-Special Relativity
III.2 Preliminaries on test theories with modified dispersion relation
III.2.1 With or without standard quantum field theory?
III.2.2 Other key features of test theories with modified dispersion relation
III.2.3 A test theory for pure kinematics
III.2.4 A test theory based on low-energy effective field theory
III.2.5 A DSR test theory
III.2.6 More on "pure-kinematics" and "field-theory-based" phenomenology
III.3 Photon stability
III.3.1 Broken-symmetry analysis
III.3.2 DSR analysis
III.4 Pair-production threshold anomalies and gamma-ray observations
III.5 Photo-pion production threshold anomalies and the cosmic-ray spectrum
III.6 Pion non-decay threshold and cosmic-ray showers
III.7 Vacuum Cerenkov and other anomalous particle interactions
III.8 In-vacuo dispersion for photons
III.9 Quadratic anomalous in-vacuo dispersion for neutrinos
III.10 Implications for the Bahcall-Waxman neutrino bound
III.11 Implications for neutrino oscillations
III.12 Synchrotron radiation and the Crab Nebula
III.13 Birefringence and polarization
III.14 Testing modified dispersion relations in the lab
III.15 On test theories without modified dispersion relations
IV Other areas of (Planck-scale-perturbative) Quantum Gravity Phenomenology
IV.1 Spacetime foam, distance fuzziness and interferometric noise
IV.1.1 Spacetime foam as interferometric noise
IV.1.2 A dimensional-analysis estimate for laser-light interferometers
IV.1.3 A simple-minded mechanism for noise in laser-light interferometers
IV.1.4 Insight already gained and ways to go beyond it
IV.1.5 Distance fuzzyness for atom interferometers
IV.2 Implications of distance fuzziness for waves propagating over large distances
IV.2.1 Time spreading of signals
IV.2.2 Disappearance of interferometric fringes
IV.3 Planck-scale modifications of CPT symmetry and neutral-meson studies
IV.3.1 Momentum-independent broken-CPT effects
IV.3.2 Momentum-dependent broken-CPT effects
IV.3.3 CPT violation and multiparticle states
IV.3.4 Elements of a test theory with deformed CPT symmetry
IV.4 Decoherence studies with kaons and atoms
IV.4.1 Spacetime foam as decoherence effects and the "α,β,γ test theory"
IV.4.2 Other descriptions of foam-induced decoherence for matter interferometry
IV.5 Decoherence and neutrino oscillations
IV.6 Nonsystematic symmetry-modification effects
IV.7 Probing the classical-gravity limit
IV.8 Probing the Equivalence Principle beyond the classical-gravity limit
IV.9 Quantum-Gravity Cosmology
IV.9.1 An alternative to inflation
IV.9.2 Probing the trans-Plankian problem with modified dispersion relations
IV.9.3 String-inspired probes of the trans-Planckian problem
IV.9.4 Randomly-fluctuating metrics and the cosmic microwave background
IV.9.5 Non Keplerian rotation curves from quantum-gravity effects
IV.9.6 A semiclassical Wheeler-DeWitt-based description of the early Universe
IV.9.7 No-singularity cosmology from strings and loops
V Quantum-Gravity phenomenology beyond the standard setup
V.1 More on the standard setup: building a case for Planck-scale perturbations
V.2 A totally different setup with Large Extra Dimensions
V.3 Adapting to the presence of IR/UV mixing
V.4 The possible challenge of not-so-subleading higher-order terms
VI Closing remarks
Our present description of the fundamental laws of Nature is based on two disconnected pieces: "quantum mechanics" and "general relativity". On the quantum-mechanics side our most advanced formalism is relativistic quantum field theory, which turns out to be an appropriate formalization of (special-)relativistic quantum mechanics. This theory essentially neglects gravitational effects. As far as relativistic quantum mechanics is concerned, gravitation essentially does not exist. Indeed the formalism of relativistic quantum field theory does not make any room for a dynamics of spacetime and the experiments which have formed our trust in relativistic quantum mechanics are all experiments in which spacetime-dynamics effects are expected to be negligible at the presently-achievable levels of experimental sensitivity. So the absence of a role for gravitation in relativistic quantum field theory has never caused inadequacy of this formalism in reproducing the results of the relevant class of measurements. The gravitational interactions between "fundamental" particles are much much weaker than the other known interactions, and can therefore be neglected in all experimental contexts in which we study these particles.
On the gravity side our present description is based on general relativity. This theory neglects all quantum properties of particles. General relativity is a classical-mechanics theory. Our trust in general relativity has emerged in experimental studies and observations in which gravitational interactions cannot be neglected, such as the motion of planets around the Sun. Planets are "composed" of a huge number of fundamental particles, and the additive nature of gravitational charge is such that the gravitational charge of a planet (basically its mass/energy) is very large, in spite of the fact that each composing fundamental particles carries only a small gravitational charge. For a planet its gravitational charge is usually dominant with respect to its other charges. Moreover, a planet satisfies the conditions under which quantum theory is in the classical limit: in the description of the orbits of the planets the quantum properties of the composing particles can be safely neglected.
General relativity and relativistic quantum mechanics do have some "shared tools", such as the concepts of spacetime and fundamental particles, but they handle these entities in profoundly different manner. The differences are indeed so profound that it might be natural to expect only one or the other language to be successful, but instead, because of the type of experiments in which they have been tested so far (two sharply separated classes of experiments in which drastic and complementary approximations are allowed), they are both extremely successful.
Of course, while somewhat puzzling from a philosopher's perspective, all this would not on its own amount to a genuine scientific problem. In the experiments we are presently able to perform and at the level of sensitivities we are presently able to achieve there is no problem. But a scientific problem, which may well deserve to be called "quantum-gravity problem", is found if we consider, for example, the structure of the scattering experiments done in particle-physics laboratories. Of course there are no surprises in the analysis of processes with, say, an "in" state with two particles each with an energy of 1012eV, and some appropriate multiparticle out state. Relativistic quantum mechanics makes definite (probabilistic but definite) predictions for the results of this type of measurement procedures, and our experiments fully confirm the validity of these predictions. We are presently unable to redo the same experiment having as "in" state two particle with energy of, say, 1030eV (i.e. energy higher than the Planck scale Ep∼1028eV), but, nonetheless, relativistic quantum mechanics makes a definite prediction for these conceivable (but presently undoable) experiments. Of course, this relativistic-quantum-mechanics prediction completely neglects gravity. But for collisions of particles of 1030eV energy the gravitational interactions predicted by general relativity are very strong and gravity should not be negligible, contrary to the "desires" of relativistic quantum mechanics. On the other hand also the quantum properties predicted for the particles by relativistic quantum mechanics (for example the fuzzyness of their trajectories) cannot be neglected, contrary to the "desires" of the classical mechanics of our present description of gravity. One could naively attempt to apply both theories simultaneously, but it is well established that such attempts do not produce anything meaningful (for example by encountering uncontrollable divergences).
This "transplankian collisions" picture is one (not necessarily the best) way to introduce a quantum-gravity problem. But is the conceivable measurement procedure I just discussed truly sufficient to introduce a scientific problem? One ingredient in missing: the measurement procedure is conceivable but presently we are unable to perform it. Moreover, one could argue (at least with some license to rigor) that we can reliably predict that mankind will never be able to perform the measurement procedure I just discussed. There appears to be no need to elaborate predictions for the outcomes of that measurement procedure. However, it is easy to see that the measurement procedure I just discussed, which does not itself provide us a genuine scientific problem, contains the elements of a genuine scientific problem. One relevant point can be made considering the experimental/observational evidence we are gathering about the "early Universe". This evidence strongly supports the idea that in the early Universe particles with energies comparable to the Planck energy scale Ep were abundant, and that these particles played a key role in those early stages of evolution of the Universe. This does not provide us with opportunities for "good experiments" (controlled repeatable experiments), but it does represent a context in which proposals for the quantum-gravity/Planck-scale realm could be tested. Different scenarios for the physical theory that applies in the quantum-gravity realm could be compared on the basis of their description of the early Universe. The detailed analysis of a given physical theory for the quantum-gravity realm could allow us to establish some characteristic predictions for the early Universe and for some traces about those early stages of evolution of the Universe which should be accordingly expected in our present observations. The theory would be testable on the basis of those predictions for our present observations. These early-Universe considerations therefore provide an opportunity for comparison between the predictions of a quantum-gravity theory and measurement results. And it might not be necessary to resort to cosmology: the fact that (in setting up the quantum-gravity problem) we have established some objective limitations of our present theories implies that some qualitatively new effects will be predicted by the new theory that applies to the quantum-gravity realm. These effects will likely dominate in that realm (in particular, they will affect profoundly the results of measurements done on particles with Planck-scale energy), but, as usual, they will always be present. For processes involving particles with energy E much smaller than Ep the implications of a typical quantum-gravity theory will be rather marginal but not alltogether absent. The magnitude of the associated effects will typically be suppressed by some small overall coefficient, perhaps given by the ratio E/Ep, small but different from zero.
We therefore do have a genuine "quantum-gravity problem", and this problem has been studied for more than 70 years stachelearly . Unfortunately most of this research has been conducted assuming that no guidance could be obtained from experiments. But of course, if there is to be a "science" of the quantum-gravity problem, this problem must be treated just like any other scientific problem, seeking desperately the guidance of experimental facts, and letting those facts take the lead in the development of new concepts. Clearly physicists must hope this works also for the quantum-gravity problem, or else abandon it to the appetites of philosophers.
It is unfortunately true that there is a certain level of risk that experiments might never give us any clear lead toward quantum gravity, especially if our intuition concerning the role of the tiny Planck length (Lp≡1/Ep∼10−35m, the inverse of the huge Planck scale in natural units) in setting the magnitude of the characteristic effects of the new theory turns out to be correct. But even if the new effects were really so small we could still try to uncover experimentally some manifestations of quantum gravity. This is hard, and there is no guarantee of success, but we must try. I shall stress that some degree of optimism could be inspired by considering, for example, the prediction of proton decay within certain modern grandunified theories of particle physics. The decay probability for a proton in those theories is really very small, suppressed by the fourth power of the ratio between the mass of the proton and grandunification scale (a scale which is only some three orders of magnitude smaller than the Planck scale), but meaningful experiments have been devised.
While the possibility of a "quantum gravity phenomenology" polonpap could be considered, on the basis of these arguments, even already in the early days of quantum-gravity research, it is only over the last decade that a significant effort has been directed in this direction. My task here is to review the accomplishments and provide some outlook on the development of this quantum-gravity-phenomenology research.
The concept of a "quantum-gravity-phenomenology literature" only materialized very recently, at some point during this past decade of exponential growth of the field. But there had been previously some (however rare) examples of studies that admit description as early examples of analyses that make contact with experiments/observations and are relevant for the understanding of the interplay between general relativity and quantum mechanics.
Some of the works produced by Chandrasekhar in the 1930s already fit this criterion. In particular, the renowned Chandrasekhar limit chandralimit , which describes the maximum mass of a white-dwarf star, was obtained introducing some quantum-mechanical properties of particles (essentially Pauli's exclusion principle) within a general-relativistic gravitational-collapse-type analysis.
A fully rigorous derivation of the Chandrasekhar limit would of course require "quantum gravity", but not all of it: it would suffice to master one special limit of quantum gravity, the "classical-gravity limit", in which one takes into account the quantum properties of matter fields (particles) in presence of rather strong spacetime curvature (treated however classically). By testing experimentally the Chandrasekhar-limit formula one is therefore to some extent probing (the classical-gravity limit of) quantum gravity.
Also relevant for the classical-gravity limit of quantum gravity are the relatively more recent studies of the implications of the Earth's gravitational field in matter-interferometry experiments. Experiments investigating these effects have been conducted since the mid 1970s and are often called "COW experiments" from the initials of Colella, Overhauser and Werner who performed the first such experiment cow . The main target of these studies is the form of the Schrödinger equation in presence of the Earth gravitational field, which could be naturally conjectured to be of the form
[−(12MI)→∇2+MGϕ(→r)]ψ(t,→r)=i∂ψ(t,→r)∂t (1)
for the description of the dynamics of matter (with wave function ψ(t,→r)) in presence of the Earth's gravitational potential ϕ(→r). [In (1) MI and MG denote the inertial and gravitational mass respectively.]
The COW experiments exploit the fact that the Earth's gravitational potential puts together the contributions of a very large number of particles (all the particles composing the Earth) and, as a result, in spite of its per-particle weakness, the overall gravitational field is large enough to introduce observable effects.
Valuable reading material relevant for these COW experiments can be found in Refs. sakurai ; gasperiniEP ; dharamEP . While the basic message is that a gravity-improved Schrödinger equation of the form (1) is indeed essentially applicable, some interesting discussions have been generated by these COW experiments, particularly as a result of the data reported by one such experiment cowEPviol (data whose reliability is still being debated), which some authors have interpreted a a possible manifestation of a violation of the Equivalence Principle.
In the same category of studies relevant for the classical-gravity limit of quantum gravity I should mention some proposals put forward mainly by Anandan (see, e.g., Ref. anan1 ), already in the mid 1980s, and some very recent remarkable studies which test how the gravitational field affects the structure of quantum states (for example in Ref. natureNeutron it is established that ultracold neutrons falling towards a horizontal mirror form gravitational quantum bound states).
In works published in the 1980s and early 1990s, one finds the first examples of ''prehistorical"111Particularly the studies reported Refs. ehns ; elmn and Refs. perci1 ; perci2 (and actually all the studies I mention in this subsection) should be viewed, as I stress again later in this review, as pioneer studies which were setting the stage for the advent of the (somewhat different but related) techniques which allowed quantum-gravity phenomenology to flourish over the last decade. I label them as "prehistorical" only to emphasize that there is no record of them in contemporary descriptions of the status of quantum-gravity research (they were overlooked by the most influential quantum-gravity reviews of the time). phenomenological studies contemplating candidate quantum-gravity effects that are not simply obtained by combining classical general relativity and ordinary quantum mechanics, i.e. candidate effects for a quantum-gravity realm that is beyond the classical-gravity limit. One example is provided by the studies of Planck-scale-induced CPT-symmetry violation and violations of ordinary quantum mechanics reported in Refs. ehns ; elmn and references therein (also see, for what concerns mainly the CPT-symmetry aspects, Refs. huetpesk ; kostcpt ; floreacpt ), which are particularly relevant for the analysis of data cplear on the neutral-kaon system. A second example are works perci1 ; perci2 considering violations of ordinary quantum mechanics of a type describable in terms of the "primary-state-diffusion" formalism, with results that could be relevant for atom interferometry.
The "prehistorical" works I mentioned in the previous subsection truly were already rather significant: some were essentially establishing that there are good opportunities for the investigation of the classical-gravity limit of quantum gravity (which could provide valuable constraints on quantum-gravity model building) and works such as the ones in Ref. elmn and Ref. perci2 were showing that, at least in some cases, it is possible to gain experimental high-sensitivity access to effects that could naturally be part of the correct quantum gravity. But, in spite of their objective significance, these studies did not manage to have an impact on the overall development of quantum-gravity research. For example, all mainstream quantum-gravity reviews up to the mid 1990s still only mentioned the "experiments issue" in the form of some brief remarks amounting to the folkloristic slogan "to test Planck scale effects we would need to build a particle accelerator all around our galaxy".
This fact that up to the mid 1990s the possibility of a quantum-gravity phenomenology was mostly ignored resulted in large part from a common phenomenon of "human inertia" that affects some scientific communities, but a small part was also played by a meaningful technical observation: the studies available up to that point relied on models with the magnitude of the effect set by a free dimensionless parameter, and at best the sensitivity of the experiment was at a level such that the dimensionless parameter could be described as a ratio between the Planck length and one of the characteristic length scales of the relevant physical context. It is of course true that this kind of dimensional-analysis reasoning does not amount to really establishing that the relevant candidate quantum-gravity effect is being probed with Planck-scale sensitivity, and this resulted in a perception that such studies, while deserving some interest, could not be described genuinely as probes of the quantum-gravity realm. For some theorists a certain level of uneasiness also originated from the fact that the formalisms adopted in studies such as the ones in Ref. elmn and Ref. perci2 involved rather virulent departures from quantum mechanics.
Still it turned out that those pioneer attempts to investigate the quantum-gravity problem experimentally were setting the stage for what then became quantum-gravity phenomenology. The situation started to evolve rather rapidly when in the span of just a few years, between 1997 and 2000, several analyses were produced describing different physical contexts in which effects introduced genuinely at the Planck sale could be tested. It started with some analyses of observations of gamma-ray bursts at sub-MeV energies grbgac ; gampul ; schaefer , then came some analyses of large laser-light interferometers gacgwi ; bignapap ; nggwi , quickly followed by the first discussions of Planck scale effects relevant for the analysis of ultra-high-energy cosmic rays kifune ; ita ; gactp and the first analyses relevant for observations of TeV gamma rays from Blazars ita ; gactp ; aus (also see Ref. billetal ).
In particular, the fact that some of these analyses (as I discuss in detail later) considered Planck-scale effects amounting to departures from classical Lorentz symmetry played a key role in their ability to have an impact on a significant portion of the overall quantum-gravity-research effort. Classical Lorentz symmetry is a manifestation of the smooth (classical) light-cone structure of Minkowski spacetime, and it has long been understood that by introducing new "quantum features" (e.g. discreteness or noncommutativity of the spacetime coordinates) in spacetime structure, as some aspects of the "quantum-gravity problem" might invite us to do, the Lorentz symmetry may be affected. And the idea of having some departure from Lorentz symmetry does not necessarily require violations of ordinary quantum mechanics. Moreover, by offering an opportunity to test quantum-gravity theories at a pure kinematical level, these "Lorentz-symmetry-test proposals" provided a path toward testability that appeared to be accessible even to the most ambitious theories that are being considered as candidates for the solution of the quantum gravity problem. Some of these theories are so complex that one cannot expect (at least not through the work of only a few generations of physicists) to extract all of their physical predictions, but the kinematics of the "Minkowski limit" may well be within our reach. An example of this type is provided by Loop Quantum Gravity crLIVING ; ashtNEW ; leeLQGrev ; thieREV , where one is presently unable to even formulate many desirable physics questions, but at least some (however tentative) progress has been made gampul ; mexweave ; thiemLS ; kodadsr in the exploration of the kinematics of the Minkowski limit.
From a pure-phenomenology perspective the late-1990s transition is particularly significant, as I shall discuss in greater detail later, for what concerns falsifiability. Some of the late-1990s phenomelogy proposals concern effects that one can imagine honestly deriving in a given quantum-gravity theory. Instead the effects described for example in studies such as the ones reported in Ref. elmn and Ref. perci2 were not really derived from proposed solutions of the quantum-gravity problem but rather they were inspired by some paths toward the solution of the quantum-gravity (the relevant formalisms were not really manageable to the point of allowing a rigorous derivation of the nature and size of the effects under study, but some intuition for the nature and size of the effects was developed combining our limited understanding of the formalisms and some heuristics). Such a line of reasoning is certainly valuable, and can inspire some meaningful "new physics" experimental searches, but if the results of the experiments are negative the theoretical ideas that motivated them are not falsified: when the link from theory to experiments is weak (contaminated by heuristic arguments) it is not possible to follow the link in the opposite direction (use negative experimental results to falsify the theory). Through further developments of the work that started in the late 1990s we are now getting close to taking quantum-gravity phenomenology from the mere realm of searches of quantum-gravity effects (which are striking if they are successful but have limited impact if they fail) to the one of "falsification tests" of some theoretical ideas. This is a point that I am planning to convey strongly with some key parts of this review, together with another sign a maturity of quantum-gravity phenomenology: the ability to discriminate between different (but similar) Planck-scale physics scenarios. Of course, in order for a phenomenology to even get started one must find some instances in which the new-physics effects can be distinguished from the effects of old-physics theories, but a more mature phenomenology should also be able to discriminate between similar (but somewhat different) new-physics scenarios.
Together with some (however slow) progress toward establishing the ability to falsify models and discriminate between models, this past decade of quantum-gravity phenomenology has also shown that the handful of examples of "Planck-scale sensitivities" that generated excitement between 1997 and 2000 were not a "one-time lucky streak": the list of examples of experimental/observational contexts in which sensitivity to some effects introduced genuinely at the Planck scale is established (or found to be realistsically within reach) has continued to grow at a steady pace, as the content of this review will indicate, and the number of research groups joining the quantum-gravity-phenomenology effort is also growing rapidly. And it is not uncommon for recent quantum-gravity reviews ashtereview ; crHISTO ; leePW ; carlip ; leebook (also see Ref. penrosebook ), even when the primary focus is on developments on the mathematics side, to discuss in some detail (and acknowledge the significance of) the work done in quantum-gravity phenomenology.
So far my preliminary description of quantum-gravity phenomenology has a rather abstract character. It may be useful to now provide a simple example of analysis that illustrates some of the concepts I have discussed and makes absolutely clear the fact that effects genuinely introduced at the Planck scale could be seen.
For these objectives it serves me well to consider the Planck-scale effect codified by the following energy-momentum (dispersion) relation
m2≃E2−→p2+η→p2(E2E2p) , (2)
where Ep denotes again the Planck scale and η is a phenomenological parameter. This is a good choice because convincing the reader that I am dealing with an effect introduced genuinely at the Planck scale is in this case effortless. It is in fact well known (see, e.g., Ref. thooftDiscret ) that this type of E−2p corrections to the dispersion relation can result from discretization of spacetime on a lattice with E−1p lattice spacing222The idea of a rigid lattice description of spacetime is not really one of the most advanced for quantum-gravity research, but this consideration is irrelevant at this stage: in order to get this phenomenology started we first must establish that the sensitivities we have are sufficient for effects as small as typically obtained from introducing structure at the Planck scale. The smallness of the effect in (2) is clearly representative of the type of magnitude that quantum-gravity effects are expected to have, and the fact that it can also be obtained from a lattice with E−1p spacing confirms this point. It is at a later stage of the development of this phenomenology that we should become concerned with testing "plausible quantum-gravity models" (if that means anything). Still it is noteworthy that, as discussed in some detail later in this review, some modern quantum-gravity-research ideas, such as the one of spacetime noncommutativity, appear to give rise to the same type of effect, and actually in some cases one is led to considering effects similar to (2) but with a weaker (and therefore more testable) Planck-scale suppression, going like E−1p rather than E−2p..
If such a modified dispersion relation is part of a framework where the laws of energy-momentum conservation are unchanged one easily finds kifune ; ita ; aus ; gactp significant implications for the cosmic-ray spectrum. In fact, the "GZK cutoff", a key expected feature of the cosmic-ray spectrum, is essentially given by the threshold energy for cosmic-ray protons to produce pions in collisions with CMBR photons. In the evaluation of the threshold energy for p+γCMBR→p+π the correction term η→p2E2/E2p of (2) can be very significant. Whereas the classical-spacetime prediction for the GZK cutoff is around 51019eV, at those energies the Planck-scale correction to the threshold turns out kifune ; ita ; aus ; gactp to be of the order of ηE4/(ϵE2p), where ϵ is the typical CMBR-photon energy. For positive values of η, even somewhat smaller333Of course the quantum-gravity intuition for η is η∼1. than 1, this amounts to an observably large shift of the threshold energy, which should easily be seen (or excluded) once the relevant portion of the cosmic-ray spectrum becomes well known.
Of course, the same effect is present and is even more significant if instead of a E−2p correction one introduces in the dispersion relation a correction of E−1p type.
The quality of data on the ultra-high-energy portion of the cosmic-ray spectrum has improved very significantly in recent times, especially as a result of observations performed at the Pierre Auger Observatory. In a later part of this review I shall discuss the indications that are emerging form these observations and their relevance for some aspects of quantum-gravity phenomenology.
Let me here use this cosmic-ray context also as an opportunity to discuss explicitly a first example of the type of "amplifier" which is inevitably needed in quantum-gravity phenomenology. It is easy to figure out polonpap ; gactp that the large ordinary-physics number that acts as amplifier of the Planck-scale effect in this case is provided by the ratio between a ultra-high-energy cosmic-ray proton ultra-high energy, which can be of order 1020eV, and the mass (rest energy) of the proton. In this case the amplification is not achieved by gaining sensitivity to the collective result of many minute effects (this is the case of other quantum-gravity-phenomenology proposals), but rather by finding a context in which a single new-physics effect is strongly amplified. The same Planck-scale threshold analysis of the process p+γ→p+π for protons of laboratory-accessible energies of, say, 1TeV produces a completely negligible Planck-scale correction (since the ratio of energy versus mass for a 1TeV proton is "only" of order 103).
The first step for the search of experiments relevant for quantum gravity is of course the estimate of the characteristic scale of this new physics. This is a point on which we have relatively robust guidance from theories and theoretical arguments: the characteristic scale at which non-classical properties of spacetime physics become large (as large as the classical properties they compete with) should be444I do not plan to review here all the arguments that single out the Planck length, but I shall offer some related observations in parts of Section V. There is truly a large number of, apparently independent, arguments that all converge to this scale, and it is certainly fair to characterized the indication of this characteristic scale as the most robust element of guidance for quantum-gravity phenomenology. Still one should take notice of some recent studies, on which I shall comment briefly in parts of Section V, that are finding ways to effectively increase the size of the quantum-gravity characteristic length scale: those arguments are not in any way "natural" (on the contrary they are often rather contrived) but they still do provide a meaningful invitation toward keeping at least a minimum level of prudence with respect to the assumption that the Planck scale should be the characteristic scale of quantum-gravity effects. the Planck length Lp∼10−35m (or equivalently its inverse, the Planck scale Ep∼1028eV).
The next step requires some intuition for the type of effects that quantum-gravity theories might predict. Unfortunately, in spite of more than 70 years of theory work on the quantum-gravity problem, and a certain proliferation of theoretical frameworks being considered, there is only a small number of physical effects that have been considered within quantum-gravity theories. Moreover, most of these effects concern strong-gravity/large-curvature contexts, such as black-hole physics and big-bang physics, which are exciting at the level of conceptual analysis and development of formalism, but of course are not very promising for the actual (experimental) discovery of manifestations of non-classical properties of spacetime and/or gravity.
Actually, it might be preferable to focus on the implications of quantum gravity for the "Minkowski limit" (no curvature, at least at some coarse-grained level of description of spacetime). The effects are likely to be less significant than, say, in black hole physics (in some aspects of black hole physics quantum-gravity effects might even dominate over classical-physics effects), but for the Minkowski limit the quality of the data we can obtain is extremely high, and in some cases can compensate for an expected large suppression of quantum-gravity effects, a suppression which is likely to take the form of some power of the ratio between the Planck length and the wavelength of the particles involved.
The presence of these suppression factors on the one hand reduces sharply our chances of actually discovering quantum-gravity effects, but on the other hand simplifies the problem of figuring out what are the most promising experimental contexts, since these experimental contexts must enjoy very special properties which would not go easily unnoticed. For laboratory experiments even an optimistic estimate of these suppression factors leads to a suppression of order 10−16, which one obtains by assuming (probably already using some optimism) that at least some quantum-gravity effects are only linearly suppressed by the Planck length and taking as particle wavelength the shorter wavelengths we are able to produce (∼10−19m). In astrophysics (which however limits one to "observations" rather than "experiments") particles of shorter wavelength are being studied, but even for the highest energy cosmic rays, with energy of ∼1020eV and therefore wavelengths of ∼10−27m, a suppression of the type Lp/λ would take values of order 10−8. It is mostly as a result of this type of considerations that traditional quantum-gravity reviews considered the possibility of experimental studies with unmitigated pessimism. However, the presence of these large suppression factors surely cannot suffice for drawing any conclusions. Even just looking within the subject of particle physics we know that certain types of small effects can be studied, as illustrated by the example of the remarkable limits obtained on proton instability. The prediction of proton decay within certain grandunified theories of particle physics is really a small effect, suppressed by the fourth power of the ratio between the mass of the proton and grandunification scale, which is only three orders of magnitude smaller than the Planck scale. In spite of this horrifying suppression, of order [mproton/Egut]4∼10−64, with a simple idea we have managed to acquire full sensitivity to the new effect: the proton lifetime predicted by grandunified theories is of order 1039s and quite a few generations of physicists should invest their entire lifetimes staring at a single proton before its decay, but by managing to keep under observation a large number of protons (think for example of a situation in which 1033 protons are monitored) our sensitivity to proton decay is dramatically increased. In that context the number of protons is the (ordinary-physics) dimensionless quantity that works as "amplifier" of the new-physics effect.
Outside of particle physics more success stories of this type are easily found: think for example of the brownian-motion studies conducted already a century ago. Within the 1905 Einstein description one uses Brownian-motion measurements on macroscopic scales as evidence for the existence of atoms. For the Brownian-motion case the needed amplifier is provided by the fact that a very large number of microscopic process intervenes in each single macroscopic process that is being measured.
It is hard but clearly not impossible to find experimental contexts in which there is effectively a large amplification of some small effects of interest. And this is the strategy that is adopted polonpap in the quantum-gravity-phenomenology attempts to gain access to the Planck-scale realm.
The fact that the formalisms being considered as solutions for the quantum-gravity problem (think for example of String Theory and Loop Quantum Gravity) are so complex that very little is understood of their truly physical implications has of course profound implications for the development of quantum-gravity phenomenology (and in turn of course affects the structure of this review). One typically does not compare with experimental data the predictions of some given quantum-gravity theory, but rather compares to data the predictions of an associated "test theory", a model that is inspired by some features we do understand (usually not more than qualitatively or semi-quantitatively) of the original theory but casts them within a simple framework that is well suited for comparison to experiments (but for which there is no actual guarantee of full equivalence to the original theory).
These test theories are needed to bridge the gap between the experimental data and our present understanding of the relevant formalisms. And their careful development is in itself an interesting challenge, since they should capture as much as possible the structure of the original theories and be of the widest possible applicability (test theories with enough structure to be applicable in the analysis of a large class of experimental contexts), but in adding structures to these test theories one should limit as much as possible the risk of assuming properties that could turn out not to be verified once we understand the original formalisms better.
Essentially in working in quantum-gravity phenomenology one must first develop some intuition for some candidate quantum-gravity effects (both from the structure of the formalisms that are being considered in the search of a solution to the quantum-gravity problem and, in a few cases, also directly from the structure of the quantum-gravity problem) and then develop some test theories of these candidate effects to be used as guidance for experimental searches. And by making reference to some carefully tailored test theories we also essentially establish a common language in assessing the progresses made in improving the sensitivity of experiments, a language that must be suitable for access both from the side of experimentalists and from the side of those working at the development of the quantum-gravity theories.
Let me also try to characterize an extra layer of theory work, to be located somewhere between test theories and proposed full solutions of the quantum-gravity problem. Test theories are exclusively a tool to be used in bridging the gap between the real of experiments and the study of proposed full solutions of the quantum-gravity problem, so they still intend to contribute to the traditional strategy of quantum-gravity research, which essentially reflects the hope of solving all the aspects of the quantum-gravity problem in one step, with a single ingenious theory proposal. The difficulties encountered by proposed full solutions to the quantum-gravity problem in making contact with experiments can however be interpreted as an indication that perhaps this traditional strategy might be too ambitious: by seeking formalisms suitable for a full solution of the quantum-gravity problem one may well achieve remarkable conceptual breakthroughs, but then the complexity of those formalisms obstructs the path toward the realm of experiments. It might be wiser to at least direct part of our efforts toward the study of theories focusing on some specific aspects of the quantum-gravity problem, in addition to the traditionally large effort directed toward to the study of full solutions of the quantum-gravity problems. This alternative strategy would essentially look at the quantum-gravity problem as something that must be solved putting together a few "small steps" rather than making a single huge jump.
The possibility to contemplate such "quantum-gravity theories of not everything" is actually rather natural considering the fact that the "quantum-gravity problem" can indeed be described in terms of several "subproblems", each challenging us perhaps as much as some full open problems of other areas of physics. To mention just a few of these "subproblems" let me notice that: (i) it appears likely that the solution of this problem requires a nonclassical description of spacetime geometry, (ii) quantum gravity might have to be profoundly different (from an "information-theory perspective") from previous fundamental-physics theories, as suggested by certain analyses of the evolution of pure states in a black-hole background, (iii) the perturbative expansions that are often needed for the analysis of experimental data might require the development of new techniques, since it appears that the ones that rely on perturbative renormalizability might be unavailable, and (iv) we must find some way to reconcile general-relativistic background independence with the apparent need of quantum mechanics to be formulated in a given background spacetime.
Well-known examples of "quantum-gravity theories of not everything" are theories formulated in noncommutative spacetimes. These can be used to explore the possibility that the correct "fundamental" description of spacetime should not rely on a conventional classical smooth geometry. Clearly quantum-gravity should require us to do much more than merely introduce spacetime noncommutativity. And most of the proposed theories with spacetime noncommutativity actually assume a fixed background noncommutative spacetime, so there is not even room for genuine gravitational interactions. However, while these limitations are certainly significant, the study of theories in noncommutative spacetime could give a very valuable contribution to the overall solution of the quantum-gravity problem. If we actually managed to establish that a certain type of spacetime noncommutativity actually does apply at least in some limit of quantum-gravity then this in turn could provide some valuable guidance555These remarks are also based on the much-discussed "double role" of gravitational fields: in general-relativistic theories, the gravitational field is not just used to describe "gravitational interactions" but also characterizes the structure of spacetime itself. If we acquire robust intuition on some properties of spacetime this can guide us toward some constraints that should be in general imposed on the properties of gravity fields. toward the full solution of the quantum-gravity problem.
From the perspective of quantum-gravity phenomenology these "quantum-gravity theories of not everything" are particularly appealing since they are a bit simpler than full solutions of the quantum-gravity problem, and it appears more realistic to be able to analyze them deeply enough to establish some of their falsifiable predictions. Of course for a quantum-gravity phenomenologist no other property of a theory is more important than falsifiability. As I shall stress later in this review we are very close to establishing that field theories in noncommutative spacetimes are "falsifiable quantum-gravity theories of not everything" and there are other types of "quantum-gravity theories of not everything" for which it appears legitimate to be hopeful with respect to the prospects of falsifiability.
A list of candidate quantum-gravity effects, in the sense I just described, is given in the next Section. The rest of this review attempts to describe the status of searches of these candidate quantum-gravity effects. Section III focuses on effects that amount to Planck-scale departures from Lorentz/Poincaré symmetry, which is the type of effects on which the most energetic phenomenology effort has been so far directed. The content of Section III has some overlap with another Living Review in Relativity mattinLRR that described the status of modern tests of Lorentz symmetry, and therefore was in part also devoted to quantum-gravity-motivated such tests. My perspective will however be rather different, focused on the quantum-gravity-motivated searches and also using the example of Lorentz/Poincaré-symmetry tests to comment on the level of maturity reached by quantum-gravity phenomenology for what concerns the falsification of (test) theories and the discrimination between different but similar theories. Then in Section IV I describe the status of the rest of quantum-gravity phenomenology, i.e. searches of candidate quantum-gravity effects that are not (at least not necessarily) linked with departures from Lorentz/Poincaré symmetry. Both Section III and Section IV focus on the standard setup of quantum-gravity-phenomenology studies, which essentially relies on a perturbative approach, primarily characterized by the smallness of the Planck length, so that the magnitude of the effects is set (in leading order) by some power of the ratio between the Planck length and another relevant length scale In Section V I give some examples of mechanisms that might cause this standard setup to be inapplicable, including the possibility of IR/UV mixing and the possibility of large extra dimensions. Section VI offers some closing remarks.
In providing my description of the present status of quantum-gravity phenomenology I shall here rather conservatively make reference to the sensitivities that are within reach for certain classes of experiments/observations. The analysis of sensitivities was the traditional exercise a decade ago, in the early days of modern quantum-gravity phenomenology, since the key objective then was to establish that quantum-gravity phenomenology has the right to "exist", by showing that indeed sensitivity to effects introduced genuinely at the Planck scale was achievable. In light of the observation I already reported in Subsection I.D (and several other observations reported later in this review) the "case for existence" of quantum-gravity phenomenology is at this point well settled. We are now entering a more mature phase in which for some candidate quantum-gravity effects the development of suitable test theories is at a rather advanced stage of development and a preliminary effort of placing experimental bounds ("limits") on the parameters of these test theories is under way.
At the time of writing this first version of the "Living Review of quantum-gravity phenomenology" the transition "from sensitivities to limits" is however not yet complete. I will describe the transition as completed when the quantum-gravity-phenomenology community has developed the tools to establish experimental limits with the level of robustness found in other areas of phenomenology (such as particle-physics phenomenology), whereas in this transition period too often limits are claimed in the literature on the basis of a single little-understood experimental result (often a single observation in astrophysics) and most of the test theories are not yet developed to the point of rendering really meaningful the effort of placing limits on their parameters. This is of course a key issue, and throughout this review I will find opportunities to discuss in more detail my concerns and offer some remarks that are relevant for the needed transition "from sensitivities to limits". I will also stress some examples in which the experimental situation is actually rather robustly established, and I will provide some examples of analyses in which a reasonably well developed test theory is already available. But the "default mode" of this first version of the review provides descriptions of sensitivities only. I do plan to keep this really a "living" review, and with each update readers should find the emphasis gradually going more and more from sensitivities to experimental limits.
Compiling a list of candidate quantum-gravity effects is actually not straightforward. Analogous situations in other areas of physics are usually such that there are a few new theories which 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 cannot yet claim any success in the experimental realm. Moreover, even if we wanted to used them as guidance for experiments the complexity of these theories proves to be a formidable obstruction. What we can do with these theories is to look at their general structure and use this as a source of intuition for the proposal of a few candidate effects. Other candidate quantum-gravity effects are suggested by the general structure of the quantum-gravity problem itself: It happens to be the case that by looking at the type of presently-unanswered questions for which quantum-gravity is being sought, one is led to considering a few candidate effects.
Many arguments mead ; venekonmen ; padma ; dopl1994 ; ahlu1994 ; ng1994 ; gacmpla ; garay ; casadio based on the general structure of the quantum-gravity problem suggest that the correct fundamental description of spacetime should involve some form of "quantization", such as in discrete or noncommutative spacetime pictures. This in turn encourages us to contemplate the possibility of Planck-scale violations of the smooth classical symmetries of classical spacetimes. 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 symmetrie. Poincaré transformations are smooth arbitrary-magnitude classical transformations and it is rather obvious that they should be put under scrutiny 666Of course, a space with some elements of quantization/discreteness may have classical continuous symmetries, but only if things are arranged in an ad hoc manner areanew . Typically quantization/discretization of spacetime observables does lead to departures from classical spacetime symmetries areanew . So clearly spacetime-symmetry tests should be a core area of quantum-gravity phenomenology, but to be pursued with the awareness that spacetime quantization does not automatically affect spacetime symmetries (it typically does but not automatically). if the classical Minkowski spacetime is replaced by a quantized/discretized version.
The most active quantum-gravity-phenomenology research area is indeed the one considering possible Planck-scale departures from Poincaré/Lorentz symmetry. One possibility that of course 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 gacdsr , is the one of a "spacetime quantization" which deforms but does not break (in a sense that I discuss in the next section) some spacetime symmetries.
Besides the analysis of the general structure of the quantum-gravity problem, encouragement for these Poincaré/Lorentz-symmetry studies is found also within some of the most popular approaches to the quantum-gravity problem. In the standard formulation of String Theory, which is a theory formulated in a classical background spacetime, there is of course no a priori reason to expect departures from classical Poincaré symmetry when the classical Minkowski background is adopted. Still in recent years there has been much discussion of the possibility that a relevant antisymmetric tensor might have a vacuum expectation dougnekr and give rise to departures from Poincaré symmetry.
According to the present understanding of Loop Quantum Gravity the fundamental description of spacetime involves some intrinsic discretization crLIVING ; ashtNEW ; leeLQGrev , and, although very little of robust is presently known about the Minkowski limit of the theory, several indirect arguments suggest that this discretization ends up inducing departures from classical Poincaré symmetry. While most of the Loop-Quantum-Gravity literature on the fate of Poincaré symmetries argues for symmetry violation (see e.g. Refs. gampul ; mexweave ), there are some candidate mechanisms (see e.g. Refs. kodadsr ; jurekkodadsr ) that appear to provide opportunities for a deformation of symmetries in Loop Quantum Gravity.
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, [xμ,xν]=iθμν, and κ-Minkowski noncommutativity, [xm,t]=ixm/κ , [xm,xl]=0, the issues relevant for the fate of Poincaré symmetry are very much in focus, and departures from Poincaré symmetry appear to be inevitable777In the case of canonical noncommutativity evidence of some departures from Poincaré symmetry are found both if θμν is a fixed tensor dougnekr or a fixed observer-independent matrix chaichan ; thetanoether . The possibility to preserve classical Poincaré symmetry is instead still not excluded in what was actually the earliest approach dopl1994 based on [xμ,xν]=iθμν, where for θμν researchers are seeking a formulation with richer algebraic properties..
Perhaps the most intelligible evidence of a Planck-scale effect would be a violation of CPT symmetry. CPT symmetry is in fact protected by a theorem in our current (Minkowski-limit) theories, mainly as a result of locality and Poincaré symmetry. The fact that the structure of the quantum-gravity problem invites us to consider spacetimes with some element of nonlocality and/or departures from Poincaré symmetry clearly opens a window of opportunity for Planck-scale violations of CPT symmetry.
Unfortunately a proper analysis of CPT symmetry requires a level of understanding of the formalism that is often beyond our present reach in quantum-gravity research. In Loop Quantum Gravity one should have a good control of the Minkowski (classical-) limit, and of the description of charged particles in that limit, and this is well beyond what can be presently done within Loop Quantum Gravity.
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. gacmaj it is observed that certain types of spacetime noncommutativity appear to require a deformation P (parity) transformations, which would of course result in a corresponding deformation of CPT transformations.
Within the most popular formulation of the String Theory ("critical", supersymmetric) one sees no a priori reasons to encounter departures from CPT symmetry, at least as long as the theory is formulated in a classical Minkowski background. The mentioned mechanism involving a vacuum expectation value for a tensor field, besides inducing departures from Poincaré symmetry, of course (because of the role that Poincaré symmetry plays in the CPT theorem) would also open the possibility of CPT violations. And there is a variant of String Theory, the so-called Liouville String Model emn ; emnchaos , which is characterized primarily by the fact that, unlike the most followed formulation of String Theory, it does not adopt the "critical number of spacetime dimensions" criterion, where evidence of violations of CPT symmetry has been reported elmn .
The fact that the structure of the quantum-gravity problem suggests that the classical description of spacetime should give way to a nonclassical one at scales of order the Planck scale has been used extensively as a source of inspiration concerning the proper choice of formalism for the solution of the quantum-gravity problem. A description that is often used to give some intuition for the effects induced by spacetime nonclassicality is Wheeler's "spacetime foam", which however does not amount to a definition (at least not a scientific/operative definition). As I shall discuss in parts of Section IV, a few attempts to characterize operatively the concept of spacetime foam and to introduce corresponding test theories have been recently developed. Unfortunately, for spacetime foam (at least when characterized in this way) very little guidance can be obtained from the most studied quantum-gravity theories. Once again String Theory could not provide guidance for these studies as long as it is formulated in a classical Minkowski background (although things might chance if/when a suitable formalization of the "Generalized Uncertainty Principle" venekonmen ; wittenPT is found). In Loop Quantum Gravity this type of physics 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, an interferometer with the type of crisp physical characterization needed for phenomenology.
For approaches to the quantum-gravity problem which assume that Quantum Mechanics should be revised in merging with General Relativity, one of the most popular effects is decoherence. This may be also motivated using heuristic arguments, based mainly on quantum field theory in curved spacetimes, which suggest that black holes radiate thermally, with an associated "information-loss problem".
The most followed approaches to the study of the quantum-gravity problem all essentially assume (at present) that quantum mechanics should not be significantly revised, and are therefore not in a position to motivate decoherence studies. A description of decoherence has been inspired by the mentioned Liouville String Model emn ; emnchaos , and is essentially the core feature of the formalism advocated by Percival and collaborators perci1 ; perci2 .
Various perspectives on the quantum-gravity problem appear to suggest departures from one or another (stronger or weaker) form of the Equivalence Principle. One of the simplest such arguments is based on the observation that locality is a key ingredient of the present formulation of the Equivalence Principle. In fact, the Equivalence Principle ensures that (under appropriate conditions) two point particles would go on the same geodesic independently 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), and presumably also particles in a spacetime that is nonclassical and therefore sets absolute limitations on the identification of a spacetime point.
Violations of the Equivalence Principle are however not (yet) much studied from a quantum-gravity phenomenology perspective and are mostly ignored in main-stream quantum-gravity research. A counter-example are the string-theory-inspired studies reported in Ref. stringsEP ; damour and references therein, which actually provide a description of violations of the Equivalence Principle at a level which might soon be within our experimental reach.
The largest area of quantum-gravity-phenomenology research concerns the fate of Poincaré symmetry at the Planck scale. In particular, a large effort is focused on one of the implications of Poincaré symmetry: the form of energy-momentum (dispersion) relation.
Before discussing some actual phenomenological analyses it is appropriate to do some preparatory work. This will include some comments on the "Minkowski limit of Quantum Gravity", which I have already referred to but should be discussed a bit more carefully in preparation for this section. And I find appropriate to give a brief but self-contained description of "doubly-special relativity", which is a scenario with deformation of Poincaré symmetry, alternative to the standard symmetry-breaking scenarios. This is not going to present arguments in favour (or against) of the doubly-special-relativity proposal, but simply to introduce it sharply enough to be meaningfully considered in the context of the phenomenological analyses.
And still before going to the discussion of phenomenological analyses, in light of the central role played by the dispersion relation, it will be appropriate to do some preparatory work, mainly to motivate some test theories which I will use in characterizing the sensitivities achievable with certain types of data analyses.
In our current conceptual framework Poincaré symmetry emerges in situations that allow the adoption of a Minkowski metric throughout. This situations could be described as the "classical Minkowski limit".
It is not inconceivable that quantum gravity might admits a limit in which one can assume throughout a (expectation value of the) metric of Minkowski type, but some Planck-scale features of the fundamental description of spacetime (such as spacetime discreteness and/or spacetime noncommutativity) are still not completely negligible. This "nontrivial Minkowski limit" would be such that essentially the role of the Planck scale in the description of gravitational interactions can be ignored (so that indeed one can make reference to a fixed Minkowski metric), but the possible role of the Planck scale in spacetime structure/kinematics is still significant. In particular, the description of Planck-scale corrections to particle-physics processes will be a key characteristic of the Minkowski limit of quantum gravity.
It is of course not obvious that the correct quantum gravity should admit such a nontrivial Minkowski limit. With the little we presently know about the quantum-gravity problem we must be open to the possibility that the Minkowski limit would actually be trivial, i.e. that whenever the role of the Planck scale in the description of gravitational interactions can be neglected (and the metric is Minkowskian at least on average) one should also neglect the role of the Planck scale in spacetime structure/kinematics. But the hypothesis of a nontrivial Minkowski limit is worth exploring: it would be "extremely kind" of quantum gravity to admit such a limit, since it might open a wide range of opportunities for accessible experimental verification, as I shall stress in what follows.
When I mention a result on the theory side concerning the fate of Poincaré symmetry at the Planck scale clearly it must be the case that the authors have considered (or attempted to consider) the Minkowski limit of their preferred formalism.
It is probably fair to state that each quantum-gravity research line can be connected with one of three perspectives on the problem: the particle-physics perspective, the General-Relativity perspective and the condensed-matter perspective.
From a particle-physics perspective it is natural to attempt to reproduce as much as possible the successes of the Standard Model of particle physics. One is tempted to see gravity simply as one more gauge interaction. From this particle-physics perspective a natural solution of the quantum-gravity problem should have its core features described in terms of graviton-like exchange in a background classical spacetime. Indeed this structure is found in String Theory, the most developed among the quantum-gravity approaches that originate from a particle-physics perspective.
The particle-physics perspective provides no a priori reasons to renounce to exact Poincaré symmetry, since Minkowski classical spacetime is an admissible background spacetime, and in classical Minkowski there cannot be any a priori obstruction for classical Poincaré symmetry. Still, a break up of Lorentz symmetry, in the sense of spontaneous symmetry breaking, is of course possible, and this possibility has been studied extensively over the last few years, especially in String Theory where this might occur naturally when a certain String-Theory antisymmetric tensor field acquires a vacuum expectation value (see, e.g., Ref. dougnekr and references therein).
Complementary to the particle-physics perspective is the General-Relativity perspective, whose core characteristic is the intuition that one should firmly reject the the possibility to rely on a background spacetime crLIVING ; leeLQGrev . According to General Relativity the evolution of particles and the structure of spacetime are selfconsistently connected: rather than specify a spacetime arena (a spacetime background) beforehand, the dynamical equations determine at once both the spacetime structure and the evolution of particles. Although less publicized, there is also growing awareness of the fact that, in addition to the concept of background independence, the development of General Relativity relied heavily on the careful consideration of the in-principle limitations that measurement procedures can encounter888Think for example of the limitations that the speed-of-light limit imposes on certain setups for clock synchronization and of the contexts in which it is impossible to distinguish between a constant acceleration and the presence of a gravitational field.. In light of the various arguments suggesting that, whenever both quantum mechanics and General Relativity are taken into account, there should be an in-principle Planck-scale limitation to the localization of a spacetime point (an event), the general-relativity perspective invites one to renounce to any direct reference to a classical spacetime dopl1994 ; ahlu1994 ; ng1994 ; gacmpla ; garay . Indeed this requirement that spacetime be described as fundamentally nonclassical ("fundamentally quantum"), so that the in-principle measurability limitations be reflected by a corresponding measurability-limited description of spacetime, is another element of intuition which is guiding quantum-gravity research from the general-relativity perspective. This naturally leads one to consider discretized spacetimes, as in the Loop Quantum Gravity approach, or noncommutative spacetimes.
Results obtained over the last few years indicate that this general-relativity perspective naturally leads, through the emergence of spacetime discreteness and/or noncommutativity, to some departures from classical Poincaré symmetry. Loop quantum gravity and other discretized-spacetime quantum-gravity approaches appear to require a description of the familiar (classical, continuous) Lorentz symmetry as an approximate symmetry, with departures governed by the Planck scale. And in the study of noncommutative spacetimes some Planck-scale departures from Lorentz symmetry appear to be inevitable.
The third possibility is a condensed-matter perspective on the quantum-gravity problem (see, e.g., Refs. volovik ; laugh ), in which spacetime itself is seen as a sort of emerging critical-point entity. As stressed already in the previous section, condensed-matter theorists are used to describe the degrees of freedom that are measured in the laboratory as collective excitations within a theoretical framework whose primary description is given in terms of much different, and often practically inaccessible, fundamental degrees of freedom. Close to a critical point some symmetries arise for the collective-excitations theory, which do not carry the significance of fundamental symmetries, and are in fact lost as soon as the theory is probed somewhat away from the critical point. Notably, some familiar systems are known to exhibit special-relativistic invariance in certain limits, even though, at a more fundamental level, they are described in terms of a nonrelativistic theory. So clearly from the condensed-matter perspective on the quantum-gravity problem it is natural to see the familiar classical continuous Poincaré symmetry only as an approximate symmetry.
If the fate of Poincaré symmetry at the Planck scale is nontrivial the simplest (and perhaps most likely) possibility is the one of broken Poincaré symmetry, in the same sense that other symmetries are broken in physics. But in recent years a growing numbers of authors within the quantum-gravity community have considered the alternative possibility of a deformation (rather than breaking) of Poincaré symmetry, in the sense of the "doubly-special-relativity" (DSR) proposal I put forward a few years ago gacdsr . I have elsewhere gacdsr007 attempted to expose the compellingness of this possibility. For the purposes of this review of course it's compellingness is only marginally relevant, and it is instead important to characterize the idea sharply enough to allow consideration from the quantum-gravity-phenomenology perspective. In several recent quantum-gravity-phenomenology analyses the DSR perspective is considered alongside the standard symmetry-breaking perspective, and the language established in this subsection will allow me, in later parts of this section, to discuss how this is done. DSR will also be mentioned in some of my comments concerning the level of maturity so far reached by quantum-gravity phenomenology, both because the DSR scenario is a Planck-scale proposal that could be falsified by forthcoming experimental studies (so it provides an example of the ability of quantum-gravity phenomenology to falsify theory ideas), and because certain classes of observations are relevant both for the DSR idea and for standard Poincaré symmetry breaking and depending on the outcome they could favour one or the other possibility (so these contexts are examples of the ability of quantum-gravity phenomenology to discriminate between related but different Planck-scale pictures).
The doubly-special-relativity scenario was proposed gacdsr as a sort of alternative perspective on the results on Planck-scale departures from Lorentz symmetry which had been reported in numerous articles grbgac ; gampul ; kifune ; ita ; gactp ; aus ; mexweave between 1997 and 2000. These studies were advocating a Planck-scale modification of the energy-momentum dispersion relation, usually of the form E2=p2+m2+ηLnpp2En+O(Ln+1pEn+3), on the basis of preliminary findings in the analysis of several formalisms in use for Planck-scale physics. The complexity of the formalisms is such that very little else was known about their physical consequences, but the evidence of a modification of the dispersion relation was becoming robust. In all of the relevant papers it was assumed that such modifications of the dispersion relation would amount to a breakdown of Lorentz symmetry, with associated emergence of a preferred class of inertial observers (usually identified with the natural observer of the cosmic microwave background radiation).
It appeared possible gacdsr to draw an analogy between these developments and the developments which led to the emergence of Special Relativity, now more than a century ago. In Galilei Relativity there is no observer-independent scale, and in fact the energy-momentum relation is written as E=p2/(2m). As experimental evidence in favour of Maxwell equations started to grow, the fact that those equations involve a fundamental velocity scale appeared to require the introduction of a preferred class of inertial observers. But in the end we figured out that the situation was not demanding the introduction of a preferred frame, but rather a modification of the laws of transformation between inertial observers. Einstein's Special Relativity introduced the first observer-independent relativistic scale (the velocity scale c), its dispersion relation takes the form E2=c2p2+c4m2 (in which c plays a crucial role for what concerns dimensional analysis), and the presence of c in Maxwell's equations is now understood as a manifestation of the necessity to deform the Galilei transformations.
It is not implausible that we might be presently confronted with an analogous scenario. Research in quantum gravity is increasingly providing reasons of interest in Planck-scale modifications of the dispersion relation, and, while it was customary to assume that this would amount to the introduction of a preferred class of inertial frames (a "quantum-gravity ether"), the proper description of these new structures might require yet again a modification of the laws of transformation between inertial observers. The new transformation laws would have to be characterized by two scales (c and λ) rather than the single one (c) of ordinary Special Relativity.
These observations that motivated the proposal of the DSR scenario do not automatically guide us toward a mathematical formalism for its implementation, but from a physics perspective do provide a rather sharp picture. With Doubly-Special Relativity one looks for a transition in the Relativity postulates, which should be largely analogous to the Galilei-Einstein transition. Just like it turned out to be necessary, in order to describe high-velocity particles, to set aside Galilei Relativity (with its lack of any characteristic invariant scale) and replace it with Special Relativity (characterized by the invariant velocity scale c), it is at least plausible that, in order to describe ultra-high-energy particles, we might have to set aside Special Relativity and replace it with a new relativity theory, a DSR, with two characteristic invariant scales, a new small-length/large-momentum scale in addition to the familiar velocity scale.
A theory will be compatible with the DSR principles if there is complete equivalence of inertial observers (Relativity Principle) and the laws of transformation between inertial observers are characterized by two scales, a high-velocity scale and a high-energy/short-length scale. Since in DSR one is proposing to modify the high-energy sector, it is safe to assume that the present operative characterization of the velocity scale c would be preserved: c is and should remain the speed of massless low-energy particles999Note however the change of perspective imposed by the DSR idea: within Special Relativity c is the speed of all massless particles, but Special Relativity must be perceived as a low-energy theory (as viewed from the DSR perspective) and in taking Special Relativity as starting point for a high-energy deformation one is only bound to preserving c as the speed of massless low-energy particles. Only experimental data could guide us toward the operative description of the second invariant scale λ, although its size is naturally guessed to be somewhere in the neighborhood of the Planck length Lp.
As a result of the "historical context" described in the preceeding subsection most authors have explored the possibility that the second relativistic invariant be introduced through a modifications of the dispersion relation. This is a reasonable choice gacdsr ; kowadsr ; gacroxkowa ; leedsr ; jurekDSRnew ; dsrnature ; rainbowDSR but it would be incorrect gacdsr007 at present to identify (as often done in the literature) the DSR proposal with the proposal of observer-independent modifications of the dispersion relation. For example the dispersion relation might not be modified but there might instead be an observer-independent bound on the accuracy achievable in the measurement of distances.
Considering the present very early stage of development of the DSR idea, it is clearly not possible at present to direct toward it a standard phenomenological programme. For a fully meaningful DSR phenomenology a minimum requisite would be the availability of a theoretical framework whose compatibility with the DSR principles was fully established and characterized in terms of genuinely observable features. Since we are not ready for that gacdsr007 , one might perhaps consider postponing all reasoning about phenomenology to better times (ahead?). However, as shown in parts of what follows, one can nonetheless obtain valuable insight from the development of (however incomplete and however ad hoc) "toy DSR test theories". And indeed here I will use one example of such a "toy test theory", introduced in the next subsection, to provide a crisper physical characterization of the concept of a DSR theory and to illustrate some general features of DSR phenomenology, such as the incompatibility of a decay threshold with the DSR principles (which I shall discuss as an example of opportunity to falsify the DSR idea).
So far the main focus of Poincaré-symmetry tests planned from a quantum-gravity-phenomenology perspective has been on the form of the energy-momentum dispersion relation. Indeed, certain analyses of formalisms that could be relevant for the solution of the quantum-gravity problem provide encouragement for the possibility that the Minkowski limit of quantum gravity might indeed be characterized by modified dispersion relations. However, the complexity of the formalisms that motivate the study of Planck-scale modifications of the dispersion relation is such that one has only partial information on the form of the correction terms and actually one does not even establish robustly the presence of modifications of the dispersion relation. Still, in some cases, most notably within some Loop-Quantum-Gravity studies and some studies of noncommutative spacetimes, the "theoretical evidence" in favour of modifications of the dispersion relations appears to be rather robust.
This is exactly the type of situation that I mentioned earlier in this review as part of a preliminary characterization of the peculiar type of test theories that must at present be used in quantum-gravity phenomenology. It is not possible to compare to data the predictions for the dispersion relation of, say, Loop Quantum Gravity and noncommutative geometry because these theories do not yet make definite predictions for the dispersion relation. What we can compare to data are some test theories inspired by the little we do believe we understand of the structure of the dispersion relation in the relevant theories.
And the development of such test theories requires a delicate balancing act. If we only provide them with the structures we do understand of the original theories they will be has sterile as the original theories. So we must add some structure, make some assumptions, but do so with prudence, limiting as much as possible the risk of assuming properties that could turn out not to be verified once we understand the relevant formalisms better.
As this description should suggest, there has been of course a proliferation of test theories adopted (sometimes unfortunately only implicitly newjourn ) by different authors, each reflecting a different intuition on what could or could not be assumed. Correspondingly, in order to make a serious overall assessment of the experimental limits so far established with quantum-gravity phenomenology of modified dispersion relations, one should consider a huge zoo of parameters. Of course, even the parameters of the same parametrization of modifications of the dispersion relation if analyzed using different assumptions about other aspects of the test theory should really be treated as different/independent sets of parameters.
For this first version of the Living Review on quantum-gravity phenomenology I shall be content with considering 3 illustrative examples of test theories, chosen in such a way to represent possibilities that are qualitatively very different, and representative of the breadth of possibilities that are under consideration. These 3 test theories will provide the language for me to describe the sensitivity to Planck-scale effects that is within the reach of certain experimental analyses.
Before I give a description of these 3 reference test theories I should discuss at least the most significant among the issues that must be considered in setting up any such test theory with modified dispersion relation.
One of the most important issues to be considered in setting up a test theory with modified dispersion relation concerns the choice of whether or not to assume that the test theory should be a standard low-energy effective quantum field theory.
A significant portion of the quantum-gravity community is in general, justifiably, skeptical about the results obtained using low-energy effective field theory in analyses relevant for the quantum-gravity problem. After all the first natural prediction of low-energy effective quantum field theory in the gravitational realm is a value of the energy density which is some 120 orders of magnitude greater than allowed by observations101010And the outlook of low-energy effective field theory in the gravitational realm does not improve much through the observation that exact supersymmetry could protect from the emergence of any energy density. In fact, Nature clearly does not have supersymmetry at least up to the TeV scale, and this would still lead to a natural prediction of the cosmological constant which is some 60 orders of magnitude too high..
However, just like there are numerous researchers who are skeptical about any results obtained using low-energy effective field theory in analyses relevant for the quantum-gravity problem, there are also quite a few researchers interested in the quantum-gravity problem who are completely serene in assuming that all low-energy (sub-Planckian) quantum-gravity effects should be describable in terms of effective field theory.
In order to understand how such different intuitions have come to be adopted it may be useful for me to stress that there is here in the background a delicate "order-of-limits issue". As mentioned the expectation that there should be departures from classical-spacetime symmetries at the Planck scale is not uncommon, and there are also many quantum-gravity researchers that expect quantum mechanics (and in particular relativistic quantum field theory) not to be exactly applicable at the Planck scale. However, the data analyzed in quantum-gravity phenomenology are gathered at energy scales much lower than the Planck scale, and we know that in some limit (a limit that characterizes our most familiar observations) the field-theoretic description and Lorentz invariance will both hold. So we would need to establish whether experiments that are sensitive to Planck-scale departures from Lorentz symmetry could also be sensitive to Planck-scale departures from the field-theoretic description of dynamics. As an example, let me mention the possibility (not unlikely in a context which is questioning the fate of Lorentz symmetry) that quantum gravity would admit a field-theory-type description only in reference frames in which the process of interest is essentially occurring in its center of mass (no "Planck-large boost" mg10qg1 with respect to center-of-mass frame). The field theoretic description could emerge in a sort of "low-boost limit", rather than the expected low-energy limit. The regime of low boosts with respect the center-of-mass frame is often indistinguishable with respect to the low-energy limit. For example, from a Planck-scale perspective, our laboratory experiments (even the ones conducted at, e.g. CERN, DESY, SLAC…) are both low-boost (with respect to the center of mass frame) and low-energy. However, some contexts that are of interest in quantum-gravity phenomenology, such as the collisions between ultra-high-energy cosmic-ray protons and CMBR photons, are situations where all the energies of the particles are still tiny with respect to the Planck energy scale, but the boost with respect to the center-of-mass frame could be considered to be "large" from a Planck-scale perspective (the Lorentz factor γ with respect to the proton rest frame is much greater than the ratio between the Planck scale and the proton mass γ=E/mproton≫Ep/E).
The concerns for the applicability of low-energy effective quantum field theory are strengthened by looking at the literature available on the quantum pictures of spacetime that provide motivation for the study of modified dispersion relations, which as mentioned usually involve either noncommutative geometry or Loop Quantum Gravity. The construction of field theories in noncommutative spacetimes requires the introduction of several new technical tools, which in turn lead to the emergence of several new physical features, even at low energies. I guess that these difficulties arise from the fact that a spacetime characterized by an uncertainty relation of the type δxδy≥θ(x,y) never really behaves has a classical spacetime, not even at very low energies. In fact, according to this type of uncertainty relation, a low-energy process involving soft momentum exchange in the x direction (large δx) should somehow be connected to the exchange of a hard momentum in the y direction (δy≥θ/δx), and this feature cannot be faithfully captured by our ordinary field-theory formalisms. For the so-called "canonical noncommutative spacetimes" one does obtain a plausible-looking field theory dougnekr , but the results actually show that it is not possible to rely on an ordinary effective low-energy quantum-field-theory description because of the presence111111The issues encountered in dealing with the IR/UV mixing may be related to the concerns I mentioned above for the large-boost limit of quantum gravity. In a theory with IR/UV mixing nothing peculiar might be expected for, say, a collision between two photons both of MeV energy, but the boosted version of this collision, where one photon has, say, energy of 100TeV and the other photon has energy of 10−2eV, could be subject to the IR/UV mixing effects, and be essentially untreatable from a low-energy effective-field-theory perspective. of "IR/UV mixing" dougnekr ; susskind ; gianlucaken (a mechanism such that the high-energy sector of the theory does not decouple from the low-energy sector, which in turn affects very severely gianlucaken the outlook of analyses based on an ordinary effective low-energy quantum-field-theory description). For other (non-canonical) noncommutative spacetimes we are still struggling in the search of a satisfactory formulation of a quantum field theory lukieFT ; gacmich , and it is at this point legitimate to worry that such a formulation of dynamics in those spacetimes does not exist.
And the assumption of availability of an ordinary effective low-energy quantum-field-theory description finds also no support in Loop Quantum Gravity. Indeed, so far, in Loop Quantum Gravity all attempts to find a suitable limit of the theory which can be described in terms of a quantum-field-theory in background spacetime have failed. And on the basis of the arguments presented in Ref. gampulDENSI it appears plausible that in several contexts in which one would naively expect a low-energy field theory description Loop Quantum Gravity might instead require a density-matrix description.
In order to be applicable to a significant ensemble of experimental contexts a test theory should of course specify much more than the form of the dispersion relation. In light of the type of data that we expect to have access to (see later) besides the choice of working within or without low-energy effective quantum field theory there are at least 3 other issues that the formulation of such a test theory should clearly address:
(i) is the modification of the dispersion relation "universal"? or should one instead allow for different modification parameters for different particles?
(ii) in presence of the modified dispersion relation between the energy E and the momentum p of a particle should we still assume the validity of the relation v=dE/dp between the speed of a particle and its dispersion relation?
(iii) in presence of the modified dispersion relation should we still assume the validity of the standard law of energy-momentum conservation?
Unfortunately on these three key points the quantum-spacetime pictures which are providing motivation for the study of Planck-scale modifications of the dispersion relation are not providing much guidance yet.
For example, in Loop Quantum Gravity, while we do have some evidence that the dispersion relation should be modified, we do not yet have a clear indication concerning whether the law of energy-momentum conservation should also be modified and we also cannot yet robustly establish whether the relation v=dE/dp should be preserved.
Similarly in the analysis of noncommutative spacetimes we are close to establishing in rather general terms that some modification of the dispersion relation is inevitable, but other aspects of the framework have not yet been clarified. While most of the literature for canonical noncommutative spacetimes assumes dougnekr ; susskind that the law of energy-momentum conservation should not be modified, most of the literature for κ-Minkowski spacetime argues in favour of a modification of the law of energy-momentum conservation. There is also still no consensus on the relation between speed and dispersion relation, and particularly in the κ-Minkowski literature some departures from the v=dE/dp relation are actively considered Kosinski:2002gu ; Mignemi:2003ab ; Daszkiewicz:2003yr ; jurekREV . And at least for canonical noncommutative spacetimes the possibility of a nonuniversal dispersion relation is considered extensively.
Concerning the relation v=dE/dp it may be useful to stress that it can be obtained assuming that a Hamiltonian description is still available, v=dx/dt∼[x,H(p)], and that the Heisenberg uncertainty principle still holds exactly ([x,p]=1→x∼∂/∂p). The possibility of modifications of the Hamiltonian description is of course an aspect of the debate on "Planck-scale dynamics" that was in part discussed in Subsection. IV.B.1. Concerning the Heisenberg uncertainty principle, there has been a significant amount of work on modifications of the type [x,p]=1+F(p) both inspired by some string-theory results venekonmen ; stringsHP and by some noncommutative-geometry results kappaHP .
With so many possible alternative ingredients to mix one can of course produce a large variety of test theories. As mentioned, I intend to focus on 3 illustrative example of test theories for my characterization of achievable experimental sensitivities.
My first example is a test theory of very limited scope, since it is conceived to only describe pure-kinematics effects. This will restrict strongly the class of experimental that can be analyzed in terms of this test theory, but the advantage is that the limits obtained on the parameters of this test theory will have rather wide applicability (they will apply to any quantum-gravity theory with that form of kinematics, independently of the description of dynamics).
The first element of this test theory, introduced from a quantum-gravity-phenomenology perspective in Refs. grbgac ; aemn1 , is a "universal" (same for all particles) dispersion relation of the form
m2≃E2−→p2+η→p2(EnEnp) , (3)
with real η of order 1 and integer n (>0). This formula is compatible with some of the results obtained in the Loop-Quantum-Gravity approach and reflects some results obtained for theories in κ-Minkowski noncommutative spacetime.
Already in the first studies grbgac that proposed a phenomenology based on (3) it was assumed that even at the Planck scale the familiar description of "group velocity", obtained from the dispersion relation according to v=dE/dp, would hold.
And in other early phenomenology works kifune ; ita ; gactp ; aus based on (3) it was assumed that the law of energy-momentum conservation should not be modified at the Planck scale, so that, for example, in a a+b→c+d particle-physics process one would have
Ea+Eb=Ec+Ed , (4)
→pa+→pb=→pc+→pd . (5)
In the following I will refer to this test theory as the "PKV0 test theory", where "PK" reflects its "Pure-Kinematics" nature, "V" reflects its "Lorentz-symmetry Violation" content, and "0" reflects the fact that it combines the dispersion relation (3) with what appears to be the most conservative set of assumptions concerning other key aspects of the physics: universality of the dispersion relation, v=dE/dp, and unmodified law of energy-momentum conservation.
The restriction to pure kinematics has the merit to allow us to establish constraints that are applicable to a relatively large class of quantum-gravity scenarios (different formulations of dynamics would still be subject to the relevant constraints), but it also restricts severely the type of experimental contexts that one can analyze. The desire to be able to analyze a wider class of experimental contexts is therefore providing motivation for the development of test theories more ambitious than the PKV0 test theory, with at least some elements of dynamics. This is of course completely reasonable, as long as one proceeds with awareness of the fact that, in light of the situation on the theory side, for test theories adopting a given description of dynamics there is a risk that we may eventually find out that none of the quantum-gravity approaches which are being pursued is reflected in the test theory.
When planning to devise a test theory that includes the possibility to describe dynamics of course the first natural candidate (not withstanding the concerns reviewed in Subsection III.B.1) is the framework of low-energy effective quantum field theory. In this subsection I want to discuss a test theory which is indeed based on low-energy effective field theory, and has emerged primarily from the analysis reported in Ref. rob (which is rooted in part in the earlier Ref. gampul ). This test theory explores the possibility of a linear-in-Lp modification of the dispersion relation
m2≃E2−→p2+η→p2LpE , (6)
i.e. the case n=1 of Eq. (3). The implications of introducing this dispersion relation within the framework of low-energy quantum field theory have been explored in some detail both for fermions and photons. It became quickly clear that in such a setup universality cannot be assumed, since one must at least accommodate a polarization dependence for photons: in the field-theoretic setup it turns out that when right-circular polarized photons satisfy the dispersion relation E2≃p2+ηγp3 then necessarily left-circular polarized photons satisfy the "opposite sign" dispersion relation E2≃p2−ηγp3. The analysis rob of spin-1/2 particles does not lead to an analogous helicity dependence, but rather invites one to introduce two independent parameters η+ and η− to characterize the modification of the dispersion relation for electrons.
While the formalism is compatible with the possibility to introduce further independent parameters for each additional fermion in the theory (so that, e.g., protons would have different values of η+ and η− with respect to the ones of electrons) I shall here for simplicity assume "universality for spin-1/2 particles", so that for all spin-1/2 particles the dispersion relation takes the form
m2≃E2−→p2+η+→p2(EEp) , (7)
in the positive-helicity case, and
m2≃E2−→p2+η−→p2(EEp) , (8)
in the negative-helicity case, with the same values of η+ and η− for all spin-1/2 particles.
In the following I will refer to this test theory as the "FTV0 test theory", where "FT" reflects its adoption of a "low-energy effective Field Theory" description, "V" reflects its "Lorentz-symmetry Violation" content, and "0" reflects the "minimalistic" assumption of "universality for spin-1/2 particles".
While at more advanced stages of analysis of this framework it will of course be appropriate to remove the restrictions of "universality for spin-1/2 particles", in this initial stage of investigation it also makes sense newjourn to consider occasionally a further restriction on the parameter space (which amounts to considering a further simplified test theory that could be labeled "FTV00") such that η−=−η+≡ηf, i.e. with a single universal parameter ηf for spin-1/2 particles and the parametrization arranged in such a way that the modification of the dispersion relation has the same magnitude for both helicities but opposite sign. This is inspired newjourn by the fact that, as mentioned, for photons this effective-field-theory setup only allows modifications of the dispersion relation that are polarization dependent and have the same magnitude for both polarizations but opposite sign. By endowing spin-1/2 particles with the analogous property one realizes the natural-looking assumption that the Planck-scale effects are such that in a beam composed of randomly selected particles the average speed in the beam is still governed by ordinary special relativity (the Planck-scale effects average out summing over polarization/helicity).
As one can infer from my remarks on the status of development of theories compatible with the doubly-special-relativity idea, we are not ready to do any "real" DSR phenomenology. In order to claim one was doing real DSR phenomenology we should at least have some theoretical frameworks whose compatibility with the DSR principles was fully established and characterized in terms of genuinely observable features. However, at least from some perspectives the development of "toy DSR test theories" can be valuable. The exercise of developing such "toy test theories" can be used to provide a crisper physical characterization of the concept of a DSR theory and allows to clarify some general features of the phenomenology (as in the case of the incompatibility of a decay threshold with the DSR principles discusses later in this section).
Even working within the limited scopes of setting up some "toy DSR test theories" is not really easy, especially since we have very little guidance to argue about what are good and what are bad DSR test theories. For my purposes here, just because of the limited scopes that at present a "DSR test theory" can have, I could take any one of the examples in the literature. The one I adopt is particularly familiar to me since I used it to illustrate the DSR idea in my original papers in Refs. gacdsr . This is a "limited theory" in that it only concerns gacdsr ; sethdsr ; dsrphen the laws of transformation of the energy-momentum observables121212Of course, an analogous "limited theory" (with the same formulas) could be articulated for the frequency/wavelength observables gacdsr007 ; dsrIJMPrev and for laws governing the composition and onshellness of frequencies/wavelengths.. It assumes that the energy-momentum dispersion relation is observer independent (and "universal"; same dispersion relation for all types of fundamental particles) and takes the following form in leading order in the deformation parameter λ
E2≃→p2+m2+λ→p2E . (9)
This dispersion relation is clearly an invariant of space rotations, but it is not an invariant of ordinary boost transformations. Its invariance (to leading order) is ensured adopting standard space-rotation generators
Rj=−iϵjklpk∂∂pl . (10)
and a deformed action for boost generators
Bj≃ipj∂∂E+i(E+λ2→p2+λE2)∂∂pj−iλpj(pk∂∂pk) . (11)
For the limited "phenomenology of kinematics" that some authors have been doing with this limited "toy DSR test theory" the only remaining ingredient to be specified is the one linking incoming energy-momenta to outgoing ones, as intended in a law of conservation of energy-momentum. Let us start considering processes with two incoming particles, a and b, and two outgoing particles, c and d. The special-relativistic kinematic requirements for such processes are Ea+Eb−Ec−Ed=0 and pa+pb−pc−pd=0, but these clearly gacdsr would not be observer-independent laws in light of (11). Working in leading order actually one finds several acceptable131313The conservation laws that must be satisfied by physical processes should gacdsr be covariant under the transformations that relate the kinematic properties of particles as measured by different observers (all observers should agree on whether or not a certain process is allowed). alternative possibilities for the deformation of the law of conservation of energy-momentum. For my example of DSR test theory I will adopt
Ea+Eb+λpapb≃Ec+Ed+λpcpd , (12)
pa+pb+λ(Eapb+Ebpa)≃pc+pd+λ(Ecpd+Edpc) . (13)
Analogous formulas can be obtained gacdsr ; dsrphen for any process with n incoming particles and m outgoing particles. In particular, in the case of a two-body particle decay a→b+c the laws
Ea≃Eb+Ec+λpbpc , (14)
pa≃pb+pc+λ(Ebpc+Ecpb) . (15)
provide an acceptable (observer-independent, covariant according to the observer-independence of (11)) possibility.
I shall handle both the PKV0 and the DSR test theory as pure-kinematics test theories while FTV0 will be my example of test theory based on effective low-energy quantum field theory. Before starting my characterization of experimental sensitivities in terms of the parameters of these test theories I find appropriate to add a few remarks warning about some difficulties that are inevitably encountered in doing phenomenology with these test theories.
For the pure-kinematics test theories some key difficulties originate from the fact that sometimes an effect due to modification of dynamics can take a form that is not easily distinguished from a pure-kinematics effect, and other times one deals with an analysis of effects that appear to be exclusively sensitive to kinematics but then at the stage of converting experimental results into bounds on parameters some level of dependence on dynamics arises. An example of this latter possibility will be provided by my description of particle-decay thresholds in test theories that violate Lorentz symmetry. The derivation of the equations that characterize the threshold requires only the knowledge of the laws of kinematics. And if according to the kinematics of a given test theory a certain particle at a certain energy cannot decay then observation of the decay allows to set robust pure-kinematics limits on the parameters. But if the test theory predicts that that certain particle at that given energy can decay then by not finding such decays we are not in a position to truly establish pure-kinematics limits on the parameters of the test theory. If the decay is kinematically allowed but not seen it is possible that the laws of dynamics prevent it from occurring (small decay amplitude).
Of course, by adopting low-energy quantum field theory this type of limitations are removed, but other issues must be taken into account, particularly in association with the fact that the FTV0 quantum field theory is not renormalizable. Quantum-field-theory-based descriptions of Planck-scale departures from Lorentz symmetry can only be developed with a rather strongly "pragmatic" attitude. In particular, for the FTV0 test theory, with its Planck-scale suppressed effects at tree level, some authors (notably Refs. suda1 ; suda2 ; tuning ) have argued that the loop expansion could effectively generate additional terms of modification of the dispersion relation that are unsuppressed by the cut-off scale of the (nonrenormalizable) field theory. Of course, the parameters of the field theory can be fine-tuned to eliminate the unwanted large effects, but the needed level of fine tuning is usually rather unpleasant. While certainly undesirable, this severe fine-tuning problem'should not discourage us from considering the FTV0 test theory, at least not at this early stage of the development of the relevant phenomenology. Actually some of the most successful theories used in fundamental physics are affected by severe fine tuning. It is not uncommon to eventually discover that the fine tuning is only apparent, that some hidden symmetry is actually "naturally" setting up the hierarchy of parameters. And it appears that some symmetry principles could also stabilize the FTV0 field theory pospeSYMM .
The first example of Planck-scale sensitivity that I discuss is the case of a process which is kinematically forbidden in presence of exact Lorentz symmetry but becomes kinematically allowed in presence of certain departures from Lorentz symmetry. It has been established (see, e.g., Refs. tedOLDgood ; gacpion ; seth ; orfeupion ) that when Lorentz symmetry is broken at the Planck scale there can be significant implications for certain decay processes. At the qualitative level the most significant novelty would be the possibility for massless particles to decay. And certain observations in astrophysics, which allow us to establish that photons of energies up to ∼1014eV are not unstable, can be then used tedOLDgood ; gacpion ; seth ; orfeupion to set limits on schemes for departures from Lorentz symmetry.
To describe this phenomenological programme for my purposes it suffices here to consider the process γ→e+e−. Let us start from the perspective of the PKV0 test theory, and therefore adopt the dispersion relation (3) and unmodified energy-momentum conservation. One easily finds a relation between the energy Eγ of the incoming photon, the opening angle θ between the outgoing electron-positron pair, and the energy E+ of the outgoing positron (of course the energy of the outgoing electron is simply given by Eγ−E+). Setting n=1 in (3) one finds that, for the region of phase space with me≪Eγ≪Ep, this relation takes the form
cos(θ) ≃ E+(Eγ−E+)+m2e−ηEγE+(Eγ−E+)/EpE+(Eγ−E+) , (16)
where me is the electron mass.
The fact that for η=0 Eq. (16) would require cos(θ)>1 reflects the fact that, if Lorentz symmetry is preserved, the process γ→e+e− is kinematically forbidden. For η<0 the process is still forbidden, but for positive η high-energy photons can decay into an electron-positron pair. In fact, for Eγ≫(m2eEp/|η|)1/3 one finds that there is a region of phase space where cos(θ)<1, i.e. there is a physical phase space available for the decay.
The energy scale (m2eEp)1/3∼1013eV is not too high for testing, since, as mentioned, in astrophysics we see photons of energies up to ∼1014eV that are not unstable (they clearly travel safely some large astrophysical distances). The level of sensitivity that is within the reach of these studies therefore goes at least down to values of (positive) η of order 1 and somewhat smaller than 1. This is what one describes as "Planck-scale sensitivity" in the quantum-gravity phenomenology literature: having set the dimensionful deformation parameter to the Planck-scale value the coefficient of the term that can be tested is of order 1 or smaller. Specifically for the case of the photon-stability analysis it is however rather challenging to transform this Planck-scale sensitivity into actual experimental limits.
Within PKV0 kinematics, for n=1 and positive η of order 1, it would have been natural to expect that photons with ∼1014eV energy would not be stable. But the fact that the decay of 1014eV photons is allowed by PKV0 kinematics of course does not guarantee that these photons should rapidly decay. It depends on the relevant probability amplitude, whose evaluation goes beyond the reach of kinematics. Still it is likely that these observations are very significant for theories that are compatible with PKV0 kinematics. For a theory that is compatible with PKV0 kinematics (with positive η) this evidence of stability of photons imposes the identification of a dynamical mechanism that essentially prevents photon decay. If one finds no such mechanism the theory is "ruled out" (or at least its parameters are severely constrained).
Of course a completely analogous calculation can be done within the the FTV0 test theory, and there one can easily arrive at the conclusion that the FTV0 description of dynamics should not suppress significantly the photon-decay process. However, as mentioned, consistency with the effective-field-theory setup requires that the two polarizations of the photon acquire opposite-sign modifications of the dispersion relation. We observe in astrophysics some photons of energies up to ∼1014eV that are stable over large distances, but as far as we know those photons could be all, say, right-circular polarized (or all left-circular polarized). This evidence of stability of photons therefore is only applicable to the portion of the FTV0 parameter space in which both polarization should be unstable (a subset of the region with |η+|>|ηγ| and |η−|>|ηγ|).
For both the test theories that I am using as examples of schemes with violations of Lorentz symmetry the evidence of stability of photons turns out to be of some (however limited) relevance. Instead for my example of test theory that just deforms (rather than break) Lorentz symmetry, the DSR test theory I adopted, this type of phenomenology could not possibly lead to any experimental limits. A threshold-energy requirement for particle decay (such as the Eγ≫(m2eEp/|η|)1/3 mentioned above) cannot be introduced as an observer-independent law, and is therefore incompatible with the DSR principles. Different observers assign different values to the energy of a particle and therefore in presence of a threshold-energy requirement for particle decay a given particle should be allowed to decay according to some observers while being totally stable for others. Evidence of stability of the photon will therefore not be significant from a DSR perspective. In reverse, since any theory compatible with the DSR principle will not have particle-decay thresholds, by looking for particle-decay thresholds one could falsify the DSR idea. And this is a rare example of analysis within the reach of quantum-gravity phenomenology that could really fully falsify a certain conjecture for Planck-scale physics.
It may be useful to see explicitly how in my chosen illustrative example of DSR test theory the possibility of photon decay is prevented. It originates from the fact that, as mentioned, compliance with the DSR principles imposes that whenever one has a modification of the dispersion relation there should also be a corresponding modification of the law of energy-momentum conservation. Indeed one such modified law of energy-momentum conservation was discussed as part of my chosen illustrative example of DSR test theory. For the case of γ→e+e− that modification of energy-momentum conservation amounts to Eγ≃E++E−−λ→p+⋅→p−, →pγ≃→p++→p−−λE+→p−−λE−→p+. And using these in place of ordinary conservation of energy-momentum one ends up with a result for cos(θ) which is still of the form (A+B)/A but now with A=2E+(Eγ−E+)+λEγE+(Eγ−E+) and B=2m2e. Evidently this formula always gives cos(θ)>1, consistently with the fact that γ→e+e− is forbidden in DSR.
Another opportunity to investigate Planck-scale departures from Lorentz symmetry is provided by certain types of energy thresholds for particle-production processes that are relevant in astrophysics. This is a very powerful tool for quantum-gravity phenomenology kifune ; ita ; gactp ; aus , and in fact at the beginning of this review I chose the evaluation of the threshold energy for photopion production, p+γCMBR→p+π, as the basis for a strong claim that indeed it is now rigorously established that some experimental contexts have sensitivity to effects introduced genuinely at the Planck scale.
I shall discuss the photopion production threshold analysis in more detail in the next subsection. Here I consider instead the electron-positron pair production process, γγ→e+e−.
The threshold for γγ→e+e− is relevant for studies of the opacity of our Universe to photons. In particular, according to the conventional (classical-spacetime) description, the infrared diffuse extragalactic background should give rise to strong absorption of "TeV photons" (here understood as photons with energy 1TeV<E<30TeV), but this prediction can be strongly affected by effects that violate Lorentz symmetry.
To show that this is the case, let me start once again from the perspective of the PKV0 test theory, and analyze a collision between a soft photon of energy ϵ and a high-energy photon of energy E which might produce an electron-positron pair. Using the dispersion relation (3) (for n=1) and the (unmodified) law of energy-momentum conservation, one finds that for given soft-photon energy ϵ, the process γγ→e+e− is allowed only if E is greater than a certain threshold energy Eth which depends on ϵ and m2e, as implicitly codified in the formula (valid for ϵ≪me≪Eth≪Ep)
Ethϵ+ηE3th8Ep≃m2e . (17)
The special-relativistic result Eth=m2e/ϵ corresponds of course to the η→0 limit of (17). For |η|∼1 the Planck-scale correction can be safely neglected as long as ϵ≫(m4e/Ep)1/3. But eventually, for sufficiently small values of ϵ (and correspondingly large values of Eth) the Planck-scale correction cannot be ignored.
This provides an opportunity for pure-kinematics test: if a 10TeV photon collides with a photon of 0.03eV and produces an electron-positron pair the case n=1, η∼−1 for the PKV0 test theory is ruled out. A 10TeV photon and a 0.03eV photon can produce an electron-positron pair according to ordinary special-relativistic kinematics (and its associated requirement Eth=m2e/ϵ), but they cannot produce an electron-positron pair according to PKV0 kinematics with n=1 and η∼−1.
For positive η the situation is somewhat different. While negative η increases the energy requirement for electron-positron pair production, positive η decreases the energy requirement for electron-positron pair production. In some cases where one would expect electron-positron pair production to be forbidden the PKV0 test theory with positive η would instead allow it. But once a process is allowed there is no guarantee that it will actually occur, not without some information on the description of dynamics (that allows us to evaluate cross sections). As in the case of photon decay, one must conclude that a pure-kinematics framework can be falsified when it predicts that a process cannot occur (if instead the process is seen) but it cannot be falsified when it predicts that a process is allowed.
Concerning the level of sensitivity that we can expect to achieve also in this case one can robustly claim the Planck-scale sensitivity is within our reach. This, as anticipated above, is best seen considering the "TeV photons" emitted by some blazars, for which (as they travel toward our Earth detectors) the photons of the infrared diffuse extragalactic background are potential targets for electron-positron pair production. In estimating the sensitivity achievable with this type of analyses it is necessary to take into account the fact that, besides the form of the threshold condition, there are at least three other factors that play a role in establishing the level of absorption of TeV photons emitted by a given Blazar: our knowledge of the type of signal emitted by the Blazar (at the source), the distance of the blazar, and most importantly the density of the infrared diffuse extragalactic background.
The availability of observations of the relevant type has increased very significantly over these past few years. For example for the blazar "Markarian 501" (at a redshift of z=0.034) and the Blazar "H1426+428" (at a redshift of z=0.129) robust observations up to the 20-TeV range have been reported ahaMKcut501 ; h1426 , and for the blazar "Markarian 421" blazar (at a redshift of z=0.031) observations of photons of energy up to 45 TeV has been reported tev45 , although a more robust signal is seen once again up to the 20-TeV range krennMKcut ; ahaMKcut .
The key obstruction for translating these observations into an estimate of the effectiveness of pair-production absorption comes from the fact that measurements of the density of the infrared diffuse extragalactic background are very difficult, and as a result our experimental information on this density is still affected by large uncertainties voelk ; berezin .
The observations do show convincingly that some absorption is occurring ahaMKcut501 ; h1426 ; tev45 ; krennMKcut ; ahaMKcut , and is particularly significant from this perspective the analysis of the combined X-ray/TeV-gamma-ray spectrum for the Markarian 421 blazar, as discussed in particular in Ref. steckXRAY . The X-ray part of the spectrum allows to predict the TeV-gamma-ray part of the spectrum in a way that is rather insensitive on our poor knowledge of the source. This in turn allows to establish in a source-independent way that some absorption is occurring.
For the associated quantum-gravity-phenomenology analysis the fact that some absorption is occurring does not allow to infer much: the analysis will become more and more effective as the quantitative characterization of the effectiveness of absorption becomes more and more precise (as measured by the amount of deviation from the level of absorption expected within a classical-spacetime analysis that would still be compatible with the observations). And we are not yet ready to make any definite statement about this absorption levels. This is not only a result of our rather poor knowledge of the infrared diffuse extragalactic background, but it is also due to the status of the observations, which still presents us with some apparent puzzles. For example, it is not yet fully understood why, as observed by some authors ahaMKcut501 ; krennMKcut ; ahaMKcut ; voelk , there is a difference between the absorption-induced cutoff energy found in data concerning Markarian 421, Ecutoffmk421≃3.6TeV, and the corresponding cutoff estimate obtained from Markarian-501 data, Ecutoffmk501≃6.2TeV. And the observation of TeV γ-rays emitted by the blazar H1426+428, which is significantly more distant than Markarian 421 and Markarian 501, does show a level of absorption which is higher than the ones inferred for Markarian 421 and Markarian 501, but (at least assuming a certain description h1426 of the infrared diffuse extragalactic background) the H1426+428 TeV luminosity "seems to exceed the level anticipated from the current models of TeV blazars by far" h1426 .
Clearly the situation requires further clarification, but it seems reasonable to expect that within a few years we should fully establish facts such as "γ-rays with energies up to 20 TeV are absorbed by the infrared diffuse extragalactic background"141414While some observers understandably argue that the the residual grey areas that I discussed impose us to still be extremely prudent, even at the present time one could legitimately describe as robust newjourn the observational evidence indicating that some γ-rays with energies up to 20 TeV are absorbed by the infrared diffuse extragalactic background. And some authors (see. e.g., Ref. steckIRSTRONG ) actually see in the presently-available data an even sharper level of agreement with the classical-spacetime picture, which would translate in having already at the present time achieved Planck-scale sensitivity.. This would in turn imply that at least some photons with energy smaller than ∼200meV can create an electron-positron pair in collisions with a 20TeV γ-ray, and within the PKV0 test theory, with n=1, this would translate into a limit η≥−50 (i.e. either η is positive or η is negative with absolute value smaller than 50). This means that this strategy of analysis will soon take us robustly at sensitivities that are less than a factor of a 100 away from Planck-scale sensitivities, and it is natural to expect that further refinements of these measurements will eventually take us at Planck-scale sensitivity and beyond.
The line of reasoning needed to establish whether this Planck-scale sensitivity could apply to pure-kinematics frameworks is somewhat subtle. One could simplistically state that when we see a process which is forbidden by a certain set of laws of kinematics then those laws are falsified. However, this statement is correct only when we have full knowledge of the process, including a full determination energy-momenta of the incoming particles. In the case of the absorption of multi-TeV gamma rays from blazars it is natural to assume that this absorption be due to interactions with infrared photons, but we are not in a position to exclude that the absorption be due to higher-energy background photons. We should therefore contemplate the possibility that the PKV0 kinematics be implemented within a framework in which the description of dynamics is such to introduce a large-enough modification of cross sections to allow absorption of multi-TeV blazar gamma rays by background photons of energy higher than 200meV. Therefore, a truly conservative experimental bound on a pure-kinematics framework cannot be placed by mere observation of some absorption of multi-TeV blazar gamma rays.
This concerns are of course not applicable to test theories which do provide a description of dynamics, such as the FTV0 test theory, with its effective-field-theory setup. However, for the FTV0 test theory one must take into account the fact that the modification of the dispersion relation carries opposite sign for the two polarizations of the photon and might have an helicity dependence in the case of electrons and positrons. So also in the case of the FTV0 test theory, as long as observations only provide evidence of some absorption of TeV gamma rays (without much to say about the level of agreement with the amount of absorption expected in the classical-spacetime picture), and are therefore consistent with the hypothesis that only one of the polarizations of the photon is being absorbed, only rather weak limits can be established.
The case of photon-stability analyses, discussed in the previous subsection, provides an example in which the DSR principles lead to a definite (unavoidable) falsifiable prediction. For what concerns the evaluation of energy thresholds for particle-production processes the conclusions one can draw from a general DSR perspective are somewhat weaker, but definite results are obtained within the specific context of a given DSR test theory, such as my chosen illustrative example of DSR test theory.
From a general DSR perspective one should first of all notice that when the equivalence of inertial frames is preserved the threshold conditions must be written as a comparison of invariants. For example, it is no accident that in special relativity the threshold condition for γγ→e+e− takes the form Eϵ≥m2e. In fact, m2e is of course a special-relativistic invariant and Eϵ is also an invariant (for the head-on collision of a photon with four momentum Pμ, such that P0=E, and a photon with four momentum pμ, such that p0=ϵ, one finds, also using the special-relativistic dispersion relation, that Pμpμ=2Eϵ). The fact that in constructing a DSR framework, by definition, one must also insist on the equivalence of inertial frames, implies that in any genuine DSR framework the threshold conditions must also be written as a comparison of invariants.
In the broken-symmetry analysis of threshold conditions, which I discussed above, a key role is played by a correction term of the form E3/Ep that ends up being added to the classical-spacetime formula Eϵ≥m2e. In a DSR framework a term of the type E3/Ep will be allowed only if it can be obtained as part of a genuine invariant, and this turns out to require dsrgzk a discouraging level of fine tuning of the form of the dispersion relation. An explicit example of such a fine-tuned DSR test theory (with the same overall structure of the DSR test theory I here adopted in Subsection III.B.5, but with ad hoc formulation of the dispersion relation) can be found in Ref. dsrgzk .
As an indication of the fact that (unless one indeed arranges things in an extremely ad hoc way) typically DSR test theories will not predict quantitatively large modifications of threshold equations for particle-production processes, let me discuss the relevant analysis in the specific context of the DSR test theory I here adopted in Subsection III.B.5. The prescriptions for the modification of the law of energy-momentum conservation given in Subsection III.B.5 for the case of γγ→e+e− take the form
E+ϵ−λ→P⋅→p≃E++E−−λ→p+⋅→p− , →P+→p+λE→p+λϵ→P≃→p++→p−+λE+→p−+λE−→p+ (18)
where I denoted with →P the momentum of the photon of energy E and I denoted with →p the momentum of the photon of energy ϵ.
Using these (18) and the dispersion relation of my chosen DSR test theory one obtains (keeping only terms that are meaningful for ϵ≪me≪Eth≪Ep)
Eth≃m2eϵ , (19)
i.e. one ends up with the same result as in the special-relativistic case.
This indicates that there is no large threshold anomaly in the DSR test theory I am considering. Actually this test theory does predict a small threshold anomaly, but truly much smaller than obtained in some symmetry-breaking scenarios, such as the one of the PKV0 test theory. If, rather than working within the approximations allowed by the hierarchy ϵ≪me≪Eth≪Ep, one derives a DSR threshold formula of more general validity within the DSR test theory, one does find a result which is different from the special-relativistic one, but the difference is indeed quantitatively much smaller than in the case of the PKV0 test theory.
This result for the evaluation of energy thresholds for particle-production processes illustrates a more general aspect of the differences between broken-symmetry scenarios and scenarios in which spacetime symmetries are only "deformed", in the sense prescribed by the DSR concept. In comparing two frameworks with apparently similar new structures, but one of the broken-symmetry type and one of the DSR type, it is typical to find that for the majority of new-physics effects the size of these effects in the broken-symmetry scenario is larger than in the DSR scenario. As one could perhaps expect intuitively, the act of deforming a symmetry structure is (if all other conditions are roughly the same) softer than the act of breaking a symmetry structure.
In the preceding subsection I discussed the implications of possible Planck-scale effects for the process γγ→e+e−, but of course this is not the only process in which Planck-scale effects can be important. In particular, there has been strong interest kifune ; ita ; gactp ; aus ; tedOLDgood ; gacpion ; orfeupion ; nguhecr in the analysis of the "photopion production" process, pγ→pπ. As I already stressed in Subsection I.D, interest in the photopion-production process originates from its role in our description of the high-energy portion of the cosmic-ray spectrum. The "GZK cutoff" feature of that spectrum is essentially obtained as the threshold energy for cosmic-ray protons to produce pions in collisions with CMBR photons. The argument suggesting that Planck-scale modifications of the dispersion relation may affect significantly the estimate of this threshold energy is of course completely analogous to the one discussed in the preceding subsection for γγ→e+e−. However, the derivation is somewhat more tedious: in the case of γγ→e+e− the calculations are simplified by the fact that both outgoing particles have mass me and both incoming particles are massless, whereas for the threshold conditions for the photopion-production process one needs to handle the kinematics for a head-on collision between a soft photon of energy ϵ and a high-energy particle of mass mp and momentum →kp producing two (outgoing) particles with masses mp,mπ and momenta →k′p,→kπ. The threshold can then be conveniently gactp characterized as a relationship describing the minimum value, denoted by kp,th, that the space momentum of the incoming particle of mass mp must have in order for the process to be allowed for given value ϵ of the photon energy:
kp,th≃(mp+mπ)2−m2p4ϵ+ηk2+np,th4ϵEnp(m1+np+m1+nπ(mp+mπ)1+n−1) (20)
(dropping terms suppressed by both the smallness of E−1p and the smallness of ϵ or mp,π).
Notice that whereas in discussing the pair-production threshold relevant for observations of TeV gamma rays I had immediately specialized (3) to the case n=1, here I am contemplating values of n that are even greater than 1. One could of course admit n>1 also for the pair-production threshold analysis, but it would be a mere academic exercise, since it is easy to verify that in that case Planck-scale sensitivity is within reach only for n not greater than 1. Instead (as I briefly stressed already in Subsection I.D) the role of the photopion-production threshold in cosmic-ray analysis is such that even for the case of values of n as high as 2 (i.e. even for the case of effects suppressed quadratically by the Planck scale) Planck-scale sensitivity is not unrealistic. In fact, using for mp and mπ the values of the masses of the proton and the pion and for ϵ a typical CMBR-photon energy one finds that for negative η of order 1 (effects introduced at the Planck scale) the shift of the threshold codified in (20) is gigantic for n=1 and still observably large ita ; gactp for n=2.
For negative η the Planck-scale correction shifts the photopion-production threshold to higher values with respect to the standard classical-spacetime prediction, which estimates the photopion-production threshold scale to be of about 51019eV. Assuming151515Most studies suggest berezin that indeed the highest energy cosmic rays are protons. An alternative is the one of heavy nuclei, but it would not change much the Planck-scale analysis. The heavy nuclei can undergo photodisintegration when interacting with CMBR photons. And it just happens to be true that the photodisintegration threshold is reached when the energy of typical heavy nuclei, say Fe, is ∼5×1019eV, i.e. just about the value of the photopion-production threshold expected for cosmic-ray protons. This renders most of the observations used in the quantum-gravity-phenomenology of the cosmic-ray spectrum rather insensitive to the (however limited) residual uncertainty between protons and heavy nuclei as candidates for the highest-energy cosmic rays. that the highest energy observed cosmic rays are protons, when the spectrum reaches the photopion-production threshold one should first encounter a pileup of cosmic rays with energies just in the neighborhood of the threshold scale, and then above the threshold the spectrum should be severely depleted. The pileup results from the fact that protons with above-threshold energy tend to loose energy through photopion production and slow down until their energy is comparable to the threshold energy. The depletion above the threshold is the counterpart of this pileup (protons emitted at the source with energy above the threshold tend to reach us, if they come to us from far enough, with energy comparable to the threshold energy).
The availability in this cosmic-ray context of Planck-scale sensitivities for values of n all the way up to n=2 was already fully established by the year 2000 ita ; gactp . The debate then quickly focused on establishing what exactly the observations were telling us about the photopion-production threshold. The fact that the AGASA cosmic-ray observatory announced agasa evidence of a behaviour of the spectrum that was of the type expected in the Planck-scale picture generated a lot of interest. However, more recent cosmic-ray observations, most notably the ones reported by the Pierre Auger observatory augerdata , appear to show absolutely no evidence of unexpected behaviour. There is even some evidence augersources suggesting that to the highest-energy observed cosmic rays one can associate some relatively nearby sources, and that all this is occurring at scales that fit very well in the standard picture of the photopion-production threshold, without Planck scale effects.
These results reported by the Pierre Auger Observatory are already somewhat beyond the "preliminary" status, and within a year or two we should have at our disposal very robust cosmic-ray data, which should be easily converted into actual experimental bounds on the parameters of Planck-scale test theories.
Even for pure-kinematics test theories this type of data analysis is rather strongly relevant. For example, the kinematics of the PKV0 test theory forbids (for negative η of order 1 and n≤2) photopion production when the incoming proton energy is in the neighborhood of 51019eV and the incoming photon has typical CMBR energies. Already the observations reported so far by the Pierre Auger Observatory provide strong evidence that instead for 51019eV protons and CMBR photons the photopion-production process is allowed. As the quality of data improves this deduction will of course become more and more robust. For reasons that I already stressed (for other contexts) previously in this review, in order to establish a robust experimental limit on pure-kinematics scenarios using the role of the photopion-production threshold in the cosmic-ray spectrum it would be necessary to also exclude that other background photons (not necessarily CMBR photons) be responsible for the observed cutoff161616We are here dealing again with the limitations that pure-kinematics particle-reaction analyses suffer when the properties of the incoming particles are not fully under control. The pure kinematics of the PKV0 test theory definitely forbids (for negative η of order 1 and n≤2) pion production resulting from collisions between a 51019eV proton and a CMBR photon. But it allows pion production resulting from collisions between a 51019eV proton and more energetic photons, and in order to exclude that possibility one ends up formulating assumptions about dynamics (low density of relevant photons may be compensated by unexpected increase in cross section).. In light of the quality of the data we expect from a few years of running of the Pierre Auger Observatory it appears likely that such a level of understanding of the cosmic-ray spectrum will be achieved in the not-so-distant future.
For the FTV0 test theory, since it goes beyond pure kinematics, one is not subject to similar concerns. However, the fact that it admits the possibility of different effects for the two helicities of the incoming proton, and definitely predicts a correlation of the sign of the effect with the polarization of the incoming ("target") photon, complicates and renders less sharp this type of cosmic-ray analyses. It does raise however some intriguing possibilities: for example, exploiting the possibility of helicity dependence of the Planck scale effect for protons one can rather naturally end up with a scenario that predicts a pileup/cutoff structure somewhat similar to the one of the standard classical-spacetime analysis, but softer as a result of the fact that only roughly half of the protons would be allowed to loose energy by photopion production. Also the possibility to explore this type of scenarios may well be within the reach of a few years of running the Pierre Auger Observatory.
Finally for what concerns my chosen DSR test theory I should just stress that, for reasons that are completely analogous gacdsr ; dsrphen to the ones I discussed earlier for the case of the pair-production threshold and gamma-ray observations, its predictions for the GZK threshold are indistinguishable from the ones of an ordinary classical-spacetime analysis (the original GZK analysis). Within my chosen DSR test theory the value of the photopion-production threshold is actually different from the one obtained in an ordinary classical-spacetime analysis, but the differences are far too small to be observably meaningful in this cosmic-ray context.
Also relevant for the analysis of cosmic-ray observations is another aspect of the possible implications of Planck-scale departures from Lorentz symmetry: the possibility of a suppression of pion decay at ultrahigh energies. While in some cases departures from Lorentz symmetry allow the decay of otherwise stable particles (as in the case of γ→e+e−, discussed above, for appropriate choice of values of parameters), it is indeed also possible for departures from Lorentz symmetry to either introduce a threshold value of the energy of the particle above which a certain decay channel for that particle is totally forbidden colgla , or introduce some sort of suppression of the decay probability which increases with energy and becomes particularly effective above a certain threshold value of the energy of the decaying particle gacpion ; orfeupion . This may be relevant for the description of the air showers produced by cosmic rays, whose structure depends rather sensitively on certain decay probabilities, particularly the one for the decay π→γγ.
The possibility of suppression at ultrahigh energies of the decay π→γγ has been considered from the quantum-gravity-phenomenology perspective primarily adopting PKV0-type frameworks gacpion ; orfeupion . Using the kinematics of the PKV0 test theory one easily arrives gacpion at the following relation between the opening angle ϕ between the directions of the momenta of the outgoing photons, the energy of the pion (Eπ) and the energies (E and E′=Eπ−E) of the outgoing photons:
cos(ϕ) = 2EE′−m2π+3LpEπEE′2EE′+LpEπEE′ . (21)
This relation shows that at high energies the phase space available to the decay is anomalously reduced: for given value of Eπ certain values of E that would normally be accessible to the decay are no longer accessible (they would require cosθ>1). This anomaly starts to be noticeable at pion energies of order (m2π/Lp)1/3∼1015eV, but only very gradually (at first only a small portion of the available phase space is excluded).
This is rather striking since there is a report dedenko of experimental evidence of anomalies for what concerns the structure of the air showers produced by cosmic rays, particularly for what concerns their longitudinal development. And it has been argued in Ref. dedenko that these unexpected features of the longitudinal development of air showers could be explained in terms of a severely reduced decay probability for pions of energies of 1015eV and higher. This is still to be considered a very preliminary observation, not only because of the need to acquire data of better quality on the development of air showers, but also because of the role gacpion that our limited control of nonperturbative QCD has in setting our expectations for what air-shower development should look like without new physics.
Therefore, while certainly this is an intriguing opportunity, it is too early to argue that there is robust evidence of new physics in the observations of the development of air showers. And actually the uncertainties on the experimental and on the theory side are such that at present it is difficult to establish what level of sensitivity to Planck-scale effects is within the reach of this strategy of analysis.
One point which is clear is that this is only relevant for broken-symmetry scenarios. It is in particular easy to verify that in the DSR test theory that I am considering a cancellation of Planck-scale effects of the type of the ones I already discussed in Subsections III.C and III.D also occurs in the analysis of the process π→γγ.
The quantum-gravity-phenomenology analyses I have reviewed so far are in many ways the most significant Planck-scale studies of departures from Lorentz symmetry and among the most significant quantum-gravity-phenomenology analyses overall. By considering the possibility of a photon-decay threshold one ends up establishing that quantum-gravity phenomenology really has already within its reach the ability to completely falsify a Planck-scale picture (since, as explained above, evidence of such a photon-decay threshold would rule out DSR completely). The analyses of the pair-production threshold for gamma rays, of the photopion-production threshold for cosmic rays, and of a possible non-decay threshold for pions, are cases in which the data relevant for the Planck-scale effect under study are actually data that can be perceived as providing some encouragement for new physics. One can indeed legitimately argue aus that the observed level of absorption of TeV gamma rays is lower than expected, and may suggest "new physics" (but, as stressed in Subsection III.C, taking into account several factors, and especially our poor knowledge of the infrared background radiation, it ends up being perhaps even more reasonable to prudently state that there is no evidence of anomalies at present). And for the photopion-production cutoff of the cosmic-ray spectrum there has been for several years a popular interpretation of then available data as possible manifestation of new physics (but, as stressed in Subsection III.D, the recent Auger-observatory data, while still to be treated prudently as preliminary, appear to suggest that no new physics is needed to describe the cosmic-ray spectrum). And for what concerns the possibility of a pion non-decay-threshold, as I just stressed in the preceding subsection, there are some preliminary data that admit interpretation as a manifestation of the relevant effect.
The ability to falsify and to discover are of course the key qualities of a phenomenology. The particle-interaction analyses described in Subsections III.C,III.D,III.E show that quantum-gravity phenomenology does have the ability to discover its target new physics, so much so that some (however tentative) "experimental puzzles" have been considered and are being considered from the quantum-gravity perspective.
While the distinction of involvement in the discussion of some (apparent) "experimental puzzles" has earned most of the spotlight, within quantum-gravity-phenomenology discussions, for the particle-interaction analyses described in Subsections III.C,III.D,III.E, it is of course important to consider the implications of Planck-scale departures from Lorentz symmetry, and particularly Planck-scale modifications of the dispersion relation, for all possible particle interactions. And a very valuable type of particle interactions to be considered are the ones that are forbidden in a standard special-relativistic setup but could be allowed in presence of Planck-scale departures from Lorentz symmetry. These interactions could be called "anomalous particle interactions", and in the analysis of some of them one does find opportunities for Planck-scale sensitivity.
For a comprehensive list (and more detailed discussion) of these anomalous-particle-interaction analyses, which are relevant for the whole subject of the study of possible departures from Lorentz symmetry (within or without quantum gravity), readers can rely on Refs. mattinLRR ; tedAP and references therein.
I will here just briefly mention one more significant example of anomalous particle interaction that is relevant from a quantum-gravity-phenomenology perspective: the "vacuum Cerenkov" process, e−→e−γ, which in certain scenarios with broken Lorentz symmetry is allowed above a threshold value of electron energy, just in the same sense and by the same mechanism of the process γ→e−e+ (which is another example of anomalous particle interaction) I discussed in Subsection III.C.
Since we have no evidence at present of vacuum-Cerenkov processes the relevant analyses are of the type that sets limits on the parameters of some test theories. And, as in all cases in which one sets limits on the basis of the prediction of a new threshold, the analysis requires a test theory that also describes dynamics, because (as I stressed already earlier in this section) the limits set on parameters will depend on how likely the process is to occur when the conditions established by the kinematical threshold are satisfied. Indeed the FTV0 test theory has been used in analyses of the vacuum-Cerenkov process, and there actually, if one arranges for opposite-sign dispersion-relation correction terms for the two helicities of the electron, one can in principle have helicity-changing e−→e−γ at any energy (no threshold), but estimates performed mattinLRR ; tedAP within the FTV0 test theory show that the rate is extremely small at low energies.
Above the threshold for helicity-preserving e−→e−γ the FTV0 rates are substantial, and this in particular would allow an analysis with Planck-scale sensitivity that relies on observations of 50-TeV gamma rays from the Crab nebula. The argument is based on several assumptions (but all apparently robust) and its effectiveness is somewhat limited by the combination of parameters allowed by FTVO setup and by the fact that for these 50-TeV gamma rays we observe from the Crab nebula we can only reasonably guess a part of the properties of the emitting particles. According to the most commonly adopted model the relevant gamma rays are emitted by the Crab nebula as a result of inverse Compton processes, and from this one infers mattinLRR ; tedAP that for electrons of energies up to 50 TeV the vacuum Cerenkov process is still ineffective, which in turn allows one to exclude certain corresponding regions of the FTV0 parameter space.
A wavelength dependence of the speed of photons is obtained from a modified dispersion relation, if one assumes the velocity to be still described by v=dE/dp. In particular, from the dispersion relation of the PKV0 test theory one obtains (at "intermediate energies", m<E≪Ep) a velocity law of the form
v≃1−m22E2+ηn+12EnEnp . (22)
It is rather significant that on the basis of this formula one would find that two simultaneously-emitted photons should reach the detector at different times if they carry different energy. And this time-of-arrival-difference effect can be significant grbgac in the analysis of short-duration gamma-ray bursts that reach us from cosmological distances. For a gamma-ray burst it is not uncommon171717Up to 1997 the distances from the gamma-ray bursters to the Earth were not established experimentally. By a suitable analysis of the gamma-ray-burst "afterglow" it is now possible to establish the distance from the gamma-ray bursters to the Earth for a significant portion of all detected bursts. 1010 light years (∼1017s) is not uncommon. that the time travelled before reaching our Earth detectors be of order T∼1017s. Microbursts within a burst can have very short duration, as short as 10−3s, and this means that the photons that compose such a microburst are all emitted at the same time, up to an uncertainty of 10−3s. Some of the photons in these bursts have energies that extend at least up to the GeV range, and for two photons with energy difference of order ΔE∼1GeV a ΔE/Ep speed difference over a time of travel of 1017s would lead to a difference in times of arrival of order Δt∼TΔEEp∼10−2s which is significant (the time-of-arrival differences would be larger than the time-of-emission differences within a microburst). So clearly, for n=1, through this strategy, sensitivity to |η|∼1 ("Planck-scale sensitivity") is within our reach. And indeed it is well established glast that the sensitivities achievable with the next generation of gamma-ray telescopes, such as GLAST glast , could allow to test very significantly (22) in the case n=1, by possibly pushing the limit on |η| far below 1.
This in-vacuo-dispersion analysis of gamma-ray bursts is extremely popular within the quantum-gravity-phenomenology community. Its most important quality is the very limited number of assumptions on which it relies. It comes very close to being a direct test of a Planck-scale modification of the dispersion relation, and in fact it applies essentially in the same way (and essentially with the same effectiveness) to all the 3 test theories that I am considering, with only some relatively minor differences. For the DSR test theory one might give priority to positive values of η (positive λ/Lp), since with negative η our leading-order test theory does not appear to admit extension to a test theory at all order in the Planck (λ) scale dsrphen . Also, of course, in comparing the DSR test theory to the PKV0 test theory within this context, with the availability of a large collection of relevant events one could look for evidence of a preferred frame (in the PKV0 test theory the magnitude of the effects will depend not only on the source-Earth distance but also on the state of motion of the source and the Earth with respect to the preferred frame).
In comparing the PKV0 and the FTV0 test theories, one could exploit the fact that whereas for the PKV0 (and the DSR) test theory the Planck-scale-induced time-of-arrival difference would affect a multi-photon microburst by setting a difference in the "average arrival time" of the signal in different energy channels, within the FTV0 test theory one would expect a dependence of the time-spread of a microburst that grows with energy. This originates of course from the polarization dependence imposed by the structure of the FTV0 test theory: for low-energy channels the whole effect will anyway be small, but in the highest-energy channels the fact that the two polarizations travel at different speed will manifest itself as spreading in time of the signal, without any net average-time-of-arrival effect for an ideally unpolarized signal. Since there is evidence that at least some gamma-ray bursts are somewhat far from being ideally unpolarized, one could also exploit a powerful correlation: within the FTV0 test theory one expects to find some bursts with sizeable average-time-of-arrival differences between energy channels, but for those bursts the differences in time spreading in the different channels should be less visible, and some bursts with much less average-time-of-arrival differences between energy channels but a clearly noticeable difference in time spreading in the different channels.
And since polarization-sensitive observations of gamma-ray bursts are possible one could look directly for the polarization dependence predicted by the FTV0 test theory: one would expect to find that for all bursts one of the polarizations tends to arrive at earlier times with respect to the other polarization.
Clearly these in-vacuo gamma-ray dispersion studies provide us at present with the cleanest opportunity to look for Planck-scale modifications of the dispersion relation. Unfortunately they are not very powerful: they do provide us comfortably with Planck-scale sensitivity to linear (n=1) modifications of the dispersion relation, but are unable to probe significantly the case of quadratic (n=2) modifications. As I shall stress in the next subsection one can achieve Planck-scale sensitivity to quadratic Planck-scale modifications of the dispersion relation by following the same strategy but considering neutrinos instead of photons.
For what concerns the applicability to a wide range of quantum-gravity scenarios with modified dispersion relations this phenomenology programme is also uniquely powerful. In particular, it is completely insensitive to the whole issue about description of dynamical aspects of quantum gravity. Still one should prudently be aware of the fact that it might be inappropriate to characterize these studies as tests that must apply to all quantum-gravity pictures with modified dispersion relations. Most notably, the assumption of obtaining the velocity law from the dispersion relation through the formula v=dE/dp may or may not be valid in a given quantum-gravity picture. Validity of the formula v=dE/dp essentially requires that the theory is still "Hamiltonian", at least in the sense that the velocity along the x axis is obtained from the commutator with an Hamiltonian (v∼[x,H]), and that the Heisenberg commutator preserves its standard form ([x,p]∼ℏ so that x∼∂/∂p). Especially this second point is rather significant since heuristic arguments of the type also used to motivate modified dispersion relations suggest stringsHP ; kappaHP that the Heisenberg commutator might have to be modified in the quantum-gravity realm.
For what concerns challenges for data analysis, as for other types of studies based on observations in astrophysics I discussed earlier, the fact that the source of the signal is not under our control is an inconvenience also for these gamma-ray-burst in-vacuo-dispersion studies. But even in this respect the situation here is advantageous, since the availability of some clear markers of the in-vacuo-dispersion effect can be exploited to achieve results that are essentially free from uncertainties originating from our lack of knowledge of the sources, at least in cases when the analysis can rely on a large data sample. For example, it is not implausible piranKARP that the engine mechanism causing the burst of gamma rays be such to introduce a correlations at the source between the energy of the emitted photons and time of their emission. On a single observation of gamma-ray-burst event such at-the-source correlations would be essentially indistinguishable from the effect we expect from in-vacuo dispersion, which indeed is a correlation between times of arrival and energies of the photons. However, the structure of in-vacuo dispersion is such that the effect should grow linearly181818Throughout the schematic discussion I offer in this section my comments for simplicity make reference to a Minkowski-flat Universe Of course, the correct analysis must take into account redshift (and this may show up in more than one point piranRedSh ), but this does not change matters in a qualitatively significant manner: even for the most distant bursts the implications of red-shift essentially amount piranRedSh to a correction factor of order 1. with the distance travelled by the signal, and this can of course turn into a powerful background-removal tool when the data sample includes several gamma-ray bursts.
Using these observations even a simple comparison of (average) times of arrival in different energy channels of a microburst (within a longer gamma-ray burst) can be a rather powerful tool of analysis. And there is room for implementing more sophisticated strategies of statistical analysis, providing even more powerful tools for the search (or falsification) of this Planck-scale effect. A noteworthy example of the possibility of more sophisticated statistical analyses is offered in Ref. magic , which essentially relies on a best-fit technique determining as our best guess for the amount of dispersion that affected a certain observed signal the one that would correspond to the most plausible description of the form of the signal as emitted by the source. Actually very little is known of the shape of a gamma-ray-burst time profile at the source, but one can still use the observation that the fine time structure of a signal would be "distorted" by the conjectured energy dependence of the speed of photons. One can therefore attempt to remove the "signal distortion" (here simply intended as anything that modifies some characteristic properties of the signal) by applying to each photon with energy E in the observed signal the appropriate "inverse time shift". If we had a sizeable amount of information on the properties of the signal at the source this type of technique could be formidably powerful. And it is still a valuable tool in spite of the fact that we know very little of the properties of the signal at the source. For example, some insight can be gained by exploiting the fact magic that a pulse of electromagnetic radiation can be diluted (so that its power, the energy per unit time, decreases) as it propagates through a dispersive medium: this can be used to determine the appropriate "inverse time shift" as the one that maximizes power in a given time interval.
Bursts of gamma rays are ideally suited for in-vacuo-dispersion studies, but as mentioned the fact that one expects to observe them only up to relatively low energies renders them only suitable for studies of linear (n=1) Planck-scale modifications of the dispersion relation. Hoping to probe more broadly the form of the dispersion relation with in-vacuo dispersion studies, one can of course consider other astrophysics signals.
One possibility aemn1 is the one of 1987a-type supernovae; however such supernovae are typically seen at distances not greater than some 105 light years. Then a sensitivity comparable to the one achievable in the study of photons from gamma-ray bursters would require observation of 10-TeV particles from a 1987a-type supernova, and it is unlikely that significant statistics (with rich time structure) would be available for such supernova studies in the near future. So using 1987a-type supernovae one might have serious difficulties to achieve Planck-scale sensitivity for linear (n=1) modifications of the dispersion relation, and clearly they cannot provide an opportunity to go beyond linear order.
Another possibility is the observation of blazars, which are typically at shorter distances (in most cases in the hundreds of megaparsecs) and have a poorer time structure than gamma-ray bursts, but have been observed up to energies of tens of TeVs. For blazars Planck-scale sensitivity for linear (n=1) modifications of the dispersion relation is robustly established, and actually one of the presently most competitive limits on in-vacuo dispersion has been established through an analysis billetal of a TeV-gamma-ray short-duration flare from the "Markarian 421" blazar. However, also in this case it appears to be impossible to probe the dispersion relation beyond the linear (n=1) order.
The prospect of observing photons with energies up to 1018eV using air showers rissehomola is very exciting, and should be pursued very forcefully, but it represents an opportunity whose viability still remains to be fully established.
The most advanced plans for in-vacuo-dispersion studies with sensitivity up to quadratic (n=2) Planck-scale modifications of the dispersion relation actually exploit ellisgrqc9911055 ; kingNeutri ; gacneutrinos ; piranNeutriNat ; gacNeutriNat once again the extraordinary properties of gamma-ray bursters, but their neutrino emissions rather than their production of gammas. Indeed, according to current models grbNEUTRINOnew ; waxneutri , gamma-ray bursters should also emit a substantial amount of high-energy neutrinos. Some neutrino observatories should soon observe neutrinos with energies between 1014 and 1019 eV, and one could either (as it appears to be more feasible piranNeutriNat ; gacNeutriNat ) compare the times of arrival of these neutrinos emitted by gamma-ray bursters to the corresponding times of arrival of low-energy photons or compare the times of arrivals of different-energy neutrinos (which however might require larger statistics than it seems natural to expect). According to a rather robust estimate piranNeutriNat it should be possible to use these observations to probe the form of the neutrino dispersion relation up to quadratic (n=2) correction terms, but this will require the combined analysis of several gamma-ray bursts observed both as gamma and as neutrino signals.
The next opportunity for tests of Planck-scale modifications of the dispersion relation that I intend to discuss combines some elements of the points made in the last few subsections, since it concerns neutrino astrophysics and it uses the implications of modified dispersion relations for the threshold conditions of certain particle-physics processes. It is relevant for a bound proposed by Bahcall and Waxman bw on the basis of the observation that the same particles which we observe as high-energy cosmic rays should also lead to neutrino production at the source. Using the observed cosmic ray fluxes Bahcall and Waxman derive a bound (the "Bahcall-Waxman bound") on the flux of high-energy neutrinos that can be revealed in astrophysics observatories. It is easy to verify uncpa2 that Planck-scale deformations of the dispersion relation can modify the Bahcall-Waxman bound, and this in turn could be turned into limits on the parameters of the relevant Planck-scale test theories.
The Bahcall-Waxman neutrinos are produced by cosmic rays before they escape the source, particularly through chains of processes initiated by a processes of the type p+γ→X+π, with a cosmic-ray proton interacting with photons of the source and producing pions (π) and other particles (X). The produced pions then in turn lead to neutrino production through the decays π+→μ++νμ and μ+→e++νe+¯νμ. The implications of modified dispersion relations for the processes p+γ→X+π are of course analogous to the ones I already discussed for the particular case p+γ→p+π in Subsection III.E: for example, with PKV0 kinematics one finds that, for negative η, the threshold for these processes is shifted higher than in the standard classical-spacetime picture. The result is that for a proton of energy, say, ∼1019eV which is escaping from a source (e.g. an AGN), the photons of the source that could be eligible for production of charged pions would be all those with energy ≥ϵmin∼0.01eV, according to the standard classical-spacetime picture, whereas with PKV0 kinematics with η∼−1 one finds uncpa2 that only photons with energy ≥ϵη∼−1min∼109eV can interact with a 1019eV proton in a p+γ→X+π process.
Since in a typical source the abundance of photons with ϵ≥109eV is much smaller (by several orders of magnitude) than the abundance of photons with ϵ≥0.01eV, the Planck-scale effect would induce a very sizeable reduction in the probability that a charged pion be produced before the proton escapes the source, and in turn this leads to a decrease in the expected high-energy neutrinos flux, which in the case of the PKV0 test theory with n=1 and η∼−1 is a reduction by many orders of magnitude uncpa2 below the level set by the standard Bahcall-Waxman bound.
So for negative η the Bahcall-Waxman bound would be strengthened. The reverse is true for the positive-η case: according to the PKV0 kinematics with positive η one finds that photons in the source with energies even below 0.01eV are viable targets for the production of charged pions by protons with energy ∼1019eV, and as a result one expects a higher level of production of high-energy neutrinos than in the standard Bahcall-Waxman description.
For both choices of sign of η this type of analysis very clearly uncpa2 has the potential for Planck-scale sensitivity, at least for the case n=1. And actually the data analysis would be easier in the positive-η case, since the signature to look for would be a violation of the standard Bahcall-Waxman bound. In the negative-η case one would instead expect that the actual observations of high-energy neutrinos fall much short of saturating the standard Bahcall-Waxman bound, but would still be compatible with the standard Bahcall-Waxman bound (results compatible with the stronger of two bounds are of course also compatible with the softer one). In itself this would be a result of difficult interpretation, but it could be used as a consistency check for the indications emerging from other experimental studies. For example if there was evidence of reduced absorption of gamma rays, and one was considering an interpretation in terms of a Planck-scale-induced upward shift of the pair-production threshold, it would be valuable to look for evidence of a flux of high-energy neutrinos at levels much below the saturation of the Bahcall-Waxman bound.
It is well established colgla ; brust ; emnEscape ; gamboneutri ; winsta that flavour-dependent modifications to the energy-momentum dispersion relations for neutrinos may lead to neutrino oscillations even if neutrinos are massless. This point is not directly relevant for the three test theories I have chosen to use as frameworks of reference for this review. Both the PKV0 test theory and my chosen DSR test theory adopt universality of the modification of the dispersion relation. And also the FTV0 test theory describes flavour-independent effects (its effects are "nonuniversal" only for what concerns polarization/helicity). Still I had to at least mention this possibility both because clearly flavour-dependent effects may well attract gradually more interest from quantum-gravity phenomenologists (some valuable analysis have already been produced; see, e.g., Refs. mattinLRR ; tedAP and references therein), and because even for researchers focusing on flavour-independent effects it is of course important to be familiar with constraints that may be set on flavour-dependent scenarios (those constraints, in a certain sense, provide motivation for their adoption of flavour independence).
Most studies of neutrino oscillations induced by violations of Lorentz symmetry were actually not motivated by quantum-gravity research (they were part of the general Lorentz-symmetry-test research area) and assumed that the flavour-dependent violations would take the form of a flavour-dependent speed-of-light scale colgla , which essentially corresponds to the adoption of a dispersion relation of the type (3), but with n=0, and flavour-dependent values of η. A few studies have considered the case191919Also noteworthy is the analysis reported in Ref. carcorNEUTRI , which argues that neutrino oscillations may play a role also for other aspects of quantum-gravity phenomenology, in addition to their use in relation to flavour-dependent Planck-scale modifications of the dispersion relation. n=1 (still with flavour-dependent η), which is instead mainly of interest from a quantum-gravity perspective202020This is in part due to the fact that "naive quantum gravity" is not a renormalizable theory, and as a result the restriction to power-counting renormalizable correction terms (which is standard outside quantum-gravity research) is expected not to be necessarily applicable to quantum-gravity research., and found brust ; emnEscape ; winsta that for n=1 from (3) one naturally ends up with oscillations lengths that depend quadratically on the inverse of the energies of the particles (L∼E−2), whereas in the case n=0 (flavour-dependent speed-of-light scale) such a strong dependence on the inverse of the energies is not possible brust . In principle, this opens an opportunity for the discovery of manifestations of the flavour-dependent n=1 case through studies of neutrino oscillations brust ; winsta ; however, at present there is no evidence of a role for these effects in neutrino oscillations and therefore the relevant data analyses produce bounds brust ; winsta on flavour dependence of the dispersion relation.
In a part of the next section I shall comment again on neutrino oscillations, but for what concerns the possible role of quantum-gravity-induced decoherence (rather than Lorentz-symmetry violations).
Another opportunity to set limits on test theories with Planck-scale modified dispersion relations is provided by the study of the implications of modified dispersion relations for synchrotron radiation. The starting point of these analyses jaconature ; tedsteck ; stefaCrab is the observation that in the conventional (Lorentz-invariant) description of synchrotron radiation one can estimate the characteristic energy Ec of the radiation through a semi-heuristic derivation jackson leading to the formula
Ec≃1R⋅δ⋅[vγ−ve] , (23)
where ve is the speed of the electron, vγ is the speed of the photon, δ is the angle of outgoing radiation, and R is the radius of curvature of the trajectory of the electron.
Assuming that the only Planck-scale modification in this formula should come from the velocity law (described using v=dE/dp in terms of the modified dispersion relation), one finds that in some instances the characteristic energy of synchrotron radiation may be significantly modified by the presence of Planck-scale modifications of the dispersion relation. This originates from the fact that, for example, according to (22), for n=1 and η<0, an electron cannot have a speed that exceeds the value vmaxe≃1−(3/2)(|η|me/Ep)2/3, whereas in special relativity ve can take values arbitrarily close to 1.
As an opportunity to test such a modification of the value of the synchrotron-radiation characteristic energy one can attempt to use data jaconature on photons emitted by the Crab nebula. This must be done with caution since the observational information on synchrotron radiation being emitted by the Crab nebula is rather indirect: some of the photons we observe from the Crab nebula are attributed to sychrotron processes, but only on the basis of a (rather successful) model, and the value of the relevant magnetic fields is also not directly measured. But the level of Planck-scale sensitivity that could be within the reach of this type of analysis is truly impressive: assuming that indeed the observational situation has been properly interpreted, and relying on the mentioned assumption that the only modification to be taken into account is the one of the velocity law, one could jaconature ; stefaCrab set limits on the parameter η of the PKV0 test theory that go several orders of magnitude beyond |η|∼1, for negative η and n=1, and even for quadratic (n=2) Planck-scale modifications the analysis would fall "just short" of reaching Planck-scale sensitivity ("only" a few orders of magnitude away from |η|∼1 sensitivity for n=2).
However, the assumptions of this type of analysis, particularly the assumption that nothing changes but the velocity law, cannot even be investigated within pure-kinematics test theories, such as the PKV0 test theory. Synchrotron radiation is due to the acceleration of the relevant charged particles and therefore implicit in the derivation of the formula (23) is a subtle role for dynamics newjourn . From a quantum-field-theory perspective the process of synchrotron-radiation emission can be described in terms of Compton scattering of the electrons with the virtual photons of the magnetic field, and its analysis is therefore rather sensitive even to details of the description of dynamics in a given theory. Indeed, essentially this synchrotron-radiation phenomenology has focused on the FTV0 test theory and its generalizations, so that the established formalism of quantum field theory can be used to investigate the assumptions. The results stefaCrab confirm the availability of Planck-sensitivity for some combinations of parameters of the FTV0 test theory, but cannot be translated into a strict limit on some specific parameters as a result of the fact that actually (in addition to the other mentioned uncertainties in the interpretation of the data) we presently have no elements to exclude the possibility that only one of the helicities of the electron is emitting the radiation, and it might even be that the radiation is not at all produced by electrons but rather by positrons.
As I stressed already a few times earlier in this review, the FTV0 test theory, as a result of a rigidity of the adopted effective-field-theory framework, necessarily predicts birefringence, by assigning different speeds to different photon polarizations. Birefringence is a pure-kinematics effect, so it can be also included in straightforward generalizations of the PKV0 test theory, if one assigns different dispersion relation to different photon polarizations and then assumes that the speed is obtained from the dispersion relation via the standard v=dE/dp relation.
I have already discussed some ways in which birefringence may affect other tests of dispersion-inducing (energy-dependent) modifications of the dispersion relation, as in the example of searches of time-of-arrival/energy correlations for observations of gamma-ray bursts. The most direct and characteristic implications of birefringence are for polarization measurements and these too can be very powerful to constrain test theories. A key point is that whereas with polarization-dependent but still energy-independent photon speeds one ends up with a simple rotation of the polarization plane, proportional to the distance travelled by the light wave, characteristic of optical birefringence, in cases such as the one hosted by the FTV0 test theory, in which birefringence is accompanied by an energy dependence of the speed of photons, any polarization present in a short-duration burst of photons emitted by the sources is gradually eliminated by the birefringence mattinLRR ; gleiser .
Experimental limits on this effect can therefore be derived using observations of polarized light from distant galaxies mattinLRR ; gleiser ; jackbire . Planck-scale sensitivities can be definitely achieved in this way, and actually one can go (for "n=1" linear modifications of the dispersion relation) significantly beyond Planck-scale sensitivity; in particular, the analysis reported in Ref. gleiser leads to a limit of |ηγ|<2⋅10−4 on the parameter ηγ of the FTV0 test theory. An even more significant limit on the ηγ parameter (of the order of ηγ<10−14) could be inferred from reported observation polarGRB of polarized MeV gamma rays in the prompt emission of the gamma-ray burst GRB021206. However, the report of Ref. polarGRB has been challenged (see e.g. Ref. nopolarGRB ), and should therefore be presently excluded from any analysis of experimental bounds on parameters. But further improvements of limits on "birefringence accompanied by in-vacuo dispersion" are likely to be accessible relatively soon, as better techniques for polarization observations in astrophysics are being developed.
Over this past decade there has been growing awareness of the fact that data analyses with good sensitivity to effects introduced genuinely at the Planck scale are not impossible, as once thought, but are instead rather conventionally doable in an (however limited) ensemble of contexts in astrophysics. However, the fact that there are also a handful of examples of controlled laboratory experiments with good sensitivity to effects introduce genuinely at the Planck scale is a part of the know-how of quantum-gravity-phenomenology practitioners of which physicists from other areas are for the most part unaware. In this subsection and in parts of the next section I will provide some examples of these controlled laboratory experiments.
In this subsection my focus is on laboratory experiments testing Planck-scale modifications of the dispersion relation. One class of such experiments uses laser-light interferometry to look for in-vacuo-dispersion effects. In Ref. gaclaem two examples of interferometric setups were discussed in some detail, with the common feature of making use of a frequency doubler, so that part of the beam would be for part of its journey through the interferometer at double the reference frequency of the laser beam feeding the interferometer. The setups must of course be such that the interference pattern is sensitive to the fact that, as a result of in-vacuo dispersion, there is a nonlinear relation between the phase advancement of a beam at frequency ω and a beam at frequency 2ω.
For my purposes here it suffices to discuss briefly one such interferometric setup. Specifically let me give a brief description of a setup in which the frequency (or energy) is the parameter characterizing the splitting of the photon state, so the splitting is in energy space (rather than the more familiar splitting in configuration space, in which two parts of the beam actually follow geometrically different paths). The frequency doubling could be accomplished using a "second harmonic generator" shgref so that if a wave reaches the frequency doubler with frequency ω then, after passing through the frequency doubler, the outgoing wave in general consists of two components, one at frequency ω and the other at frequency 2ω.
If two such frequency doublers are placed along the path of the beam at the end one has a beam with several components, two of which have frequency 2ω: the transmission of the component which left the first frequency doubler as a 2ω wave, and another component which is the result of frequency doubling of that part of the beam which went through the first frequency doubler without change in the frequency. Therefore, the final 2ω beam represents an interferometer in energy space.
As shown in detail in Ref. gaclaem the intensity of this 2ω beam takes a form of the type
I(2ω)=Ia+Ibcos(α+(k′−2k)L) , (24)
where L is the distance between the two frequency doublers, Ia and Ib are L-independent (they depend on the amplitude of the original wave and the effectiveness of the frequency doublers gaclaem ), the phase α is also L-independent and is obtained combining several contributions to the phase (both from the propagation of the wave and introduced by the frequency doublers gaclaem ), k is the wave number corresponding to the frequency ω through the dispersion relation, and k′ is the wave number corresponding to the frequency 2ω through the dispersion relation (since the dispersion relation is Planck-scale modified one expects departures from the special-relativistic result k′=2k).
Since the intensity only depends on the distance L between the frequency doublers through the Planck-scale correction to the phase, (k′−2k)L, by exploiting a setup that allows to vary L one should rather easily disentangle gaclaem the Planck-scale effect. And one finds gaclaem that the accuracy achievable with modern interferometers, such as LIGO and VIRGO, is sufficient to achieve Planck-scale sensitivity (e.g. sensitivity to |η|∼1 in the PKV0 test theory with n=1). It is of course rather optimistic to assume that the accuracy achieved in standard interferometers would also be achievable with this peculiar setup, particularly since it would require the optics aspects of the setup (such as lenses) to work with that high accuracy simultaneously with two beams of different wavelength. Moreover, it would require some very smart technique to vary the distance between the frequency doublers without interfering with the effectiveness of the optics aspects of the setup. So in practice we would not be presently capable of using such setups to set Planck-scale-sensitive limits on in-vacuo dispersion, but the fact that the residual obstructions are of rather mundane technological nature encourages us to think that in the not-so-distant future tests of Planck-scale in-vacuo dispersion in controlled laboratory experiments will be possible.
Besides in-vacuo dispersion another aspect of the physics of Planck-scale modified dispersion relations that we should soon be able to test in controlled laboratory experiments is the one concerning anomalous thresholds, at least in the case of the γγ→e+e− process which I already considered from an astrophysics perspective in Subsection III.D. It is in fact not so far from our present technical capabilities to set up collisions between 10TeV photons and 0.03eV photons, thereby reproducing essentially the situation of the analysis of blazars that I discussed in Subsection III.D. And notice that with respect to the analysis of observations of blazars such controlled laboratory studies would give much more powerful indications. In particular, for the analysis of observations of blazars that I discussed in Subsection III.D a key limitation on our ability to translate the data into experimental bounds on parameters of a pure-kinematics framework was due to the fact that (even assuming we are indeed seeing absorption of multiTeV photons) the astrophysics context does not allow us to firmly establish whether the absorption is indeed due to the infrared component of the intergalactic background radiation (as expected) or instead is due to a higher-energy component of the background (in which case the absorption would instead be compatible with some corresponding Planck-scale pictures). If collisions between 10TeV photons and 0.03eV photons in the lab do produce pairs, since we would in that case have total control on the properties of the particles in the in state of the process, then we will have firm pure-kinematics bounds on the parameters of certain corresponding Planck scale test theories (such as the PKV0 test theory).
These laboratory studies of Planck-scale-modified dispersion relations could of course be adapted also to the FTV0 test theory, by simply introducing some handles on the polarization of the photons that are placed under observation.
Readers for which this review is the first introduction to the world of "quantum-gravity phenomenology" might be surprised that this long section, announced by an ambitious title on "tests of Poincaré symmetry", really was very much centered on probing the form of the energy-momentum dispersion relation. Other aspects of the implications of Poincaré symmetry did intervene, such as the law of energy-momentum conservation and its deformations (and the form of the interaction vertices and their deformations), and are in part probed through the data analyses that I reviewed, but the feature that clearly is at center stage is the structure of the dispersion relation. The reason for this is rather simple: scientists that recognize themselves as "quantum-gravity phenomenologists" will consider a certain data analysis as part of the field if one can robustly establish for that analysis the availability of Planck-scale sensitivities, in the sense I described above, and this task has been most successfully accomplished (for what concerns Poincaré symmetry) in cases that involve the form of the dispersion relation.
Part of this originates from the fact that in order to claim Planck-scale sensitivity one needs to rely on some sort of model of the effect. For example, for the type of modifications of the dispersion relation that I considered in this section we have at present rather robust evidence of their applicability in certain noncommutative pictures of spacetime, where the noncommutativity is very clearly introduced at the Planck scale. And several independent (although all semi-heuristic) arguments suggest that the same general type of modified dispersion relations should apply to the "Minkowski limit" of Loop Quantum Gravity, a framework where a certain type of discretization of spacetime structure is introduced genuinely at the Planck scale. Unfortunately, the relevant frameworks are so complex that one does not manage to analyze the Poincaré-symmetry sector beyond building a "case" (and not a waterproof case) for modified dispersion relations.
Of course a broader range of Poincaré symmetry tests could be valuable for quantum-gravity research, but without the support of a model (in the sense I just discussed for modified dispersion relations) it is very hard to argue that the relevant effects are being probed with sensitivities that are significant from a Planck-scale perspective. Think for example of a framework, such as the one adopted in Ref. colgla , in which the form of the dispersion relation is not modified, one still has dispersion relations of the type E2=c2#p2+m2#, but with a different value of the velocity scale c# for different particles. This is not necessarily a picture beyond the realm of possibilities one could consider from a quantum-gravity perspective, but it is basically impossible to estimate what accuracy must be achieved in measurements of, say, cproton−celectron, in order to reach Planck-scale sensitivity. Some authors qualify as "Planckian magnitude" of this type of effects the case in which the dimensionless parameter has value of the order of the ratio of the mass of the particles involved in the process versus the Planck scale (as in cproton−celectron∼(mproton±melectron)/Ep) but this arbitrary criterion clearly does not amount to establishing genuine Planck-scale sensitivity.
Still it is true that the general structure of the quantum-gravity problem and the structure of some of the approaches to the solution of the problem suggest that the Minkowski limit of quantum gravity might host a rather wide range of departures from classical Poincaré symmetry. Correspondingly a broad range of Poincaré-symmetry tests can be of interest, and, while not necessarily valuable when obtaining negative/"nothing-new" results (since without a genuine case for Planck-scale sensitivity negative results do not translate into intelligible constraints on quantum-gravity model building), they could end up providing a decisive hint for the search of quantum gravity if they ever do stumble upon robust evidence of some sort of departures from classical Poincaré symmetry.
I shall not review here this broader Poincaré-tests literature, since it is not specific to the quantum-gravity problem (these are tests that could be done and in part were done even before the development of research on Poincaré symmetries from within the quantum-gravity community) and it has already been reviewed very effectively in Ref. mattinLRR . Let me just stress that for these broad searches of departures from Poincaré symmetry one of course needs test theories with many parameters. Formalisms that are well suited for a systematic programme of such searches are already at a rather advanced stage of development sme1 ; sme2 ; blumnew ; laemSME ; gacmex , and in particular the "standard-model-extension" framework sme1 ; sme2 has reached a high level of adoption of preference for theorists and experimentalists as the language in which to characterize the results of systematic multi-parameter Poincaré-test data analyses. The "Standard Model Extension" was originally conceived sme1 as a generalization of the Standard Model of particle-physics interactions restricted to power-counting-renormalizable correction terms, and as such it was of limited interest for the bulk of the quantum-gravity community: since "naive quantum gravity" is not a (perturbatively) renormalizable theory many quantum-gravity researchers would be unimpressed with Poincaré symmetry tests restricted to powercounting-renormalizable correction terms. However, over these last few years blumnew most theorists involved in studies of the ''Standard Model Extension" have started to add correction terms that are not powercounting renormalizable.212121A warning to readers: whereas originally the denomination "Standard Model Extension" was universally used to describe a framework implementing the restriction to powercounting-renormalizable correction terms, recently (see, e.g., Ref. blumnew ) some theorists describe as "Standard Model Extension" the generalization that includes correction terms that are not powercounting renormalizable, while they describe as a "Minimal Standard Model Extension" the case with the original restriction to powercounting-renormalizable correction terms. However, some authors (especially experimentalists) still rely on the original description of the "Standard Model Extension", and this may create some confusion (for example experimentalists reporting results on the "Standard Model Extension" are actually, according to the terminology now used by some theorists, describing experimental limits on the "Minimal Standard Model Extension". A good entry point for the literature on limits on the parameters of the "Standard Model Extension" is provided by Refs. mattinLRR ; blumnew .
From a quantum-gravity-phenomenology perspective it is useful to contemplate the differences between alternative strategies for setting up a "completely general" systematic investigation of possible violations of Lorentz symmetry. In particular, it has been stressed (see, e.g., Refs. laemSME ) that violations of Lorentz symmetry can be introduced directly at the level of the dynamical equations, without assuming (as done in the Standard Model Extension) the availability of a Lagrangian generating the dynamical equations. This is of course more general than the Lagrangian approach: for example, the generalized Maxwell equation discussed in Ref. laemSME predicts effects that go beyond the Standard Model Extension. And charge conservation, which automatically comes out from the Lagrangian approach, can be violated in models generalizing the field equations laemSME . The comparison of the Standard-Model-Extension approach and of the approach based on generalizations introduced directly at the level of the dynamical equations illustrates how different "philosophies" lead to different strategies for setting up a "completely general" systematic investigation of possible departures from Lorentz symmetry. By removing the assumption of the availability of a Lagrangian the second approach is "more general". Still no "general approach" can be absolutely general: in principle one could always consider removing an extra layer of assumptions. Of course, as the topics I have reviewed in this section illustrate, from a quantum-gravity-phenomenology perspective it is not necessarily appropriate to seek the most general parametrizations. On the contrary we would like to single out some particularly promising candidate quantum-gravity effects (as in the case of modified dispersion relations) and focus our efforts accordingly.
Test of Poincaré symmetry, and particularly of the form of the dispersion relation, are the big giant among quantum-gravity-phenomenology research lines. Probably something of the order of the half of the whole quantum-gravity-phenomenology literature concerns Poincaré-symmetry tests, while the other half is spread over several other quantum-gravity-phenomenology research lines. And the perception is even stronger for most physicists working outside quantum-gravity-phenomenology: many colleagues essentially identify the concept of quantum-gravity phenomenology and the concept of Poincaré-symmetry tests. This of course renders this section on "other areas of Quantum Gravity Phenomenology" particularly significant for the overall balance of my review effort.
As mentioned earlier in this review, Wheeler's "spacetime foam" intuition for spacetime structure, while carrying strong conceptual appeal, cannot on its own be of any use for phenomenology, since it is not articulated in terms of observable properties. It can however inspire genuine physical pictures, and I will discuss two types of test theories that could provide a description of observable properties of a "spacetime foam".
In this subsection I consider spacetime-foam test theories whose structure renders them well suited for interferometric tests (I will discuss a second type of spacetime-foam test theories in Subsection IV.E). I proposed gacgwi ; bignapap a physical/operative definition of (at least one aspect of) spacetime foam that indeed makes direct reference to interferometry. According to this definition the fuzziness/foaminess of a spacetime is established on the basis of an analysis of strain noise222222Since modern interferometers were planned to look for classical gravity waves (gravity waves are their sought "signal"), it is reasonable to denominate as "noise" all test-mass-distance fluctuations that are not due to gravity waves. I choose to adopt this terminology which reflects the original objectives of modern interferometers, even though this terminology is somewhat awkward for the type of studies I am proposing, in which interferometers would be used for searches of quantum-gravity-induced distance fluctuations (and therefore in these studies quantum-gravity-induced distance fluctuations would play the role of "signal"). 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 which originates from ordinary quantum mechanics saulson ). The operative definition of fuzzy/foamy spacetime which I proposed characterizes the corresponding quantum-gravity effects as an 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-gravity effects) even in the idealized limit in which all classical-physics and ordinary-quantum-mechanics noise sources are completely eliminated/subtracted.
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 ∼10−18m by careful monitoring of distances of order ∼104m, and this should provide opportunities for an "amplifier" which is of order 1022.
However, the correct way to characterize the sensitivity of an interferometer requires gacgwi ; bignapap ; saulson the analysis of the power spectrum of the strain noise which is left over after all the sophisticated noise-reduction techniques have been applied. In modern interferometers such as LIGO and VIRGO this strain power-noise spectrum, ρ(ν), is of order 10−44Hz−1 at observation frequencies ν of about 100Hz, and in turn this (also considering the length of the arms of these modern interferometers) implies gacgwi that for a gravity wave with 100Hz frequency the detection threshold is indeed around 10−18m.
But a realistic fluctuation mechanism of course does not look anything like an ideal wave. An ideal wave deposits all its energy in the frequency band of observation that includes its own frequency of oscillation. Things work differently for other fluctuation mechanisms, and particularly for discrete fluctuation mechanisms. This is why the sensitivity of an interferometer is best characterized in terms of the strain power-noise spectrum, which can be easily used saulson to deduce the level of sensitivity of the interferometer to fluctuations of any form.
So in this context the key task for the quantum-gravity phenomenologist is the estimate of the contribution to the strain-noise power spectrum ρ(ν) which could come from quantum-gravity effects. If at some point experimentalists will manage to bring the noise level below the quantum-gravity prediction the relevant quantum-gravity test theory will be ruled out.
Is there any hope for a quantum-gravity 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 quantum-gravity models, or at least quantum-gravity test theories, but it may be valuable to first use dimensional analysis, assuming the most optimistic behaviour of the quantum-gravity effects, and check if the relevant order of magnitude is at all providing any encouragement for the painful (if at all doable) analysis of quantum-gravity 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-gravity noise is "white noise", ρ(ν)=ρ0 (frequency independent), so that it is fully specified by a single dimensionful number setting the level of this white noise. And since the strain-noise power spectrum carries units of Hz−1 one easily finds gacgwi an impressively simple estimate in terms of the Planck length and the speed-of-light scale: ρ0∼Lp/c, which, since Lp/c∼10−44Hz−1, encouragingly happens to be just at the mentioned level of sensitivity of LIGO-VIRGO-type interferometers.
Assuming for simplicity that the quantum-gravity noise be white and adopting a naive dimensional-analysis estimate of what could constitute a Planck-scale level of such white noise one finds indeed a very encouraging result. But of course this in itself does not guarantee that our modern interferometers truly are sensitive to noise sources introduced genuinely at the Planck scale. In order to argue for "Planck-scale sensitivity" one would like to analyze an interferometer in the framework of a good quantum-gravity theory (but this is beyond our capabilities at present), or at least provide some semi-heuristic picture (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 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 in 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 σ2L.
It turns out to be useful gacgwi ; bignapap to consider this σ2L as a possible step stone toward the strain-noise power-spectrum estimate. And it is in particular rather striking that 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 (∼10−44s), and therefore gacgwi ; bignapap estimating σ2L≃=LpT (where T is the time scale over which the experiment monitors the distance L), one finds gacgwi ; bignapap a Planck-scale-induced strain noise with power spectrum given by232323Details of this derivation are given in Refs. gacgwi ; bignapap . It is however well known that random-walk fluctuations generate σ∝T and that correspondingly the power spectrum of the fluctuations goes like ν−2. ρ(ν)≃LpL−2ν−2. Substituting values that are characteristic of some modern interferometers, i.e. ν∼100Hz and L∼104m this corresponds to strain noise at the level ∼10−38Hz−1, which is well within the reach of the mentioned sensitivity levels of LIGO-VIRGO-type interferometers242424Since according to this estimate quantum-gravity-induced noise becomes increasingly significant as the characteristic frequency of observation is lowered, an appealing alternative to LIGO-VIRGO-type interferometers is provided by cryogenic resonators laemgwi , a type of rigid optical interferometers, which have good sensitivities down to frequencies of about 10−6Hz..
It is perhaps worth mentioning here that the estimate σ2L≃LpT can be motivated even without relying on the random-walk picture I just described, but rather relying on some corresponding semi-heuristic arguments bignapap ; gacmpla on the type of decoherence effects that spacetime foam might induce. And other strategies for a semi-heuristic description decoherence effects caused by spacetime foaminess, such as the one adopted in Ref. ng1994 and the one adopted in parts of Ref. bignapap , actually lead to the different estimate σ2L≃=(L2pT)1/3, and interestingly this would then lead to attributing to the Planck-scale-induced strain noise behaviour of the type ν−5/3 with amplitude at a level that would place it just beyond the reach of interferometers we are presently operating but within the reach of already planned interferometers.
At the present time the "state of the art" of descriptions of Planck-scale-induced strain noise does not go much beyond the simple-minded estimates I just described. And accordingly the assessment of Planck-scale sensitivity for these interferometric studies should still be subject to further scrutiny. In order to develop some intuition for what we should look for in devising more advanced test theories it is probably useful to look in particular at the random-walk picture which I just discussed. That picture clearly describes a Planck-scale fluctuations mechanism: the distance between the mirrors fluctuates indeed by Planck-length amounts at a rate which is fixed by the Planck time. So the fact that the random-walk picture leads to an effect which is within the sensitivity of modern interferometers shows that we can have sensitivity to fluctuation mechanisms of genuine Planckian nature. However, while it is true that the effects are introduced genuinely with Planck-scale magnitude my random-walk picture attributes these fluctuations coherently on macroscopic scales (the width of the beam, the distance between the mirrors, and the size of the mirrors are key macroscopic scales of the context). A more reliable characterization of a "genuine Planck-scale sensitivity" should be based on a picture that provides a comprehensive description of the effects in terms of the Planck scale. And it is not difficult to devise Planck-scale fluctuation mechanisms introduced microscopically that might lead to a much softer effect. As an example of this possibility I discussed in Ref. iucaapap a description in which the overall quantum-gravity-induced interferometric noise was obtained as the net result of quantum-gravity-induced noise in the paths of each of the Nγ individual photons of the laser beam used in the interferometer. If a random Planck-length fluctuation per Planck-time would affect the path of each photon of the beam one would still end up with an effective σ2L∝T, but with respect to the case of coherent fluctuation of the distance L the magnitude of σ2L would be penalized iucaapap by an overall factor of order 1/√Nγ, and correspondingly the result concerning the sensitivity of modern interferometers would go from being a few orders of magnitude better than Planck-scale sensitivity (which we found in the case of coherent Planck-length fluctuations of L) to being, perhaps interestingly, a few orders of magnitude short of Planck-scale sensitivity.
Clearly the case for genuine Planck-scale sensitivity for these interferometric studies must be strengthened, but, in spite of the limitations I discussed, our present simple-minded estimates already clearly provide strong encouragement and allowed us to expose the inadequacy of certain folkloristic notions concerning the use of interferometry for Planck-scale physics. I should stress in particular that 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 100Hz. The naive argument goes something like "Planck-scale noise at Planck frequency", and the Planck frequency is Ep/ℏ∼1043Hz. However in analyzing actual pictures of spacetime fluctuations (even the simple-minded ones described above) one learns that all discrete fluctuations mechanisms tend to affect most significantly the infrared (behaviours of the type ν−|α|). This is after all why the simple random-walk picture I described above gave rise to such (relatively) large noise at 100Hz. That same random-walk noise would become quickly unobservably small at higher frequencies.
Since the phenomenology of the implications of spacetime foam for interferometry is at such 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 to uncover Planck-scale effects. Accordingly the analysis I articulated in this Subsection IV.A focused 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 of gravity waves, and the observations I reported above for the phenomenology of strain noise induced by Planck-scale effects appear to be closely linked to the issues encountered in the search of gravity waves. It seems plausible that soon there will be some atom-interferometry setups that are competitive for gravity-wave searches, and these should then also be valuable for what concerns searches of quantum-gravity-induced strain noise atom interferometry might represent a viable alternative to laser-light interferometry.
It appears to be likely that different test theories might give different indications in this respect, so that in order to most effectively constrain the parameters of some test theories atom interferometry might be preferable whereas for other test theories laser-light interferometry might provide the best limits. Of course 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-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 one more variable to be controlled at the theory level, which is indeed the mass of the atoms. And there are more features to be optimized in atom interferometry, particularly because the velocity of the particles in the beam is not a priori fixed.
As I stressed earlier in this review, it is important to establish firmly that besides observations in astrophysics and cosmology also some laboratory experiments can achieve Planck-scale sensitivity for certain conjectured quantum-gravity effects. In Subsection III.N I already provided some examples of such laboratory tests (in that case relevant for the possibility of Planck-scale departures from Poincaré symmetry) and the previous Subsection IV.A described laboratory tests of Planck-scale "distance fuzziness", of the type that could be induced by a spacetime foam.
In this subsection I go back to the astrophysics arena. indeed, besides laboratory interferometric tests it is natural to consider the possibility to explore distance fuzziness using observations in astrophysics, especially when the object of these observations are 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.
The first 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 an in-principle ("quantum") uncertainty 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.
The difficulties we presently have in making any sort of robust estimate of σL were already stressed in the previous subsection. But in the same spirit in which it was considered in the previous subsection one can hope to gain some insight by considering the ansatz σ2L≃LpL (or equivalently σ2L≃LpT if one prefers to explicitate the dependence on the distance travelled as a dependence on the time duration of the journey).
A good benchmark to start exploring what levels of sensitivity to this type of effect are achievable in astrophysics is provided once again by gamma-ray bursts, which as mentioned often travel for times of the order of 1017s before reaching our Earth detectors and are sometimes characterized by time structures (microbursts within the burst) that have duration as short as 10−4s. Values of σ2