It is shown that the amplitude vanishes in the limit of large momenta, and thus simple dispersion relations are derived. Finally, it is proved that the partial-wave expansion is convergent in the unphysical region, provided the potentials satisfy the same conditions as above. Khuri Phys. Abstract Authors References. Abstract The Fredholm theory of integral equations is used to give a rigorous proof of the analyticity and boundedness of the ordinary nonrelativistic scattering amplitude for a fixed momentum transfer.

Issue Vol. Authorization Required. Log In. Sign up to receive regular email alerts from Physical Review Journals Archive. The than O E 2 m or O Em2 , where m is the mass of the equation of motion becomes particle. To identify turbations. We shall consider cases of a scalar, fermion the effect of the new term on the dispersion relation, we and vector particle. As in the case of optical activ- which is odd under CP T and charge conjugation.

Now the Modifications of dispersion relations for stable parti- equation of motion takes the form cles such as electrons, light quarks, and photons could be searched for using the astrophysical probes [2—5]. Limits on the operators involving electrons and elec- Hence the modified dispersion relation becomes tron neutrinos are especially easy to derive. At high energies i.

One could wave function. How- electron spin Hamiltonian. Similarly, we do not constrain ever, these would not effect the threshold tests which are operators involving second and third generation leptons. On the other hand, they may Note that our limit is comparable to the existing con- play a role in indirect tests as those considered below. The photon operator 4 will also contribute cutoff on the momentum integral.

The latter cannot be because of the electromagnetic interactions inside the nu- lower than the electroweak scale or SUSY breaking scale. Despite the many unresolved theoretical jugation. Colladay and V. Kostelecky, Phys. D55 tween the nuclear spin and external direction, and thus ; S. Coleman and S. Glashow, Phys. Carroll, G. Field and R. Jackiw, So far we have neglected the fact that the low energy Phys.

D41 Amelino-Camelia et al. Amelino-Camelia and T. Piran, Phys. D64 values for the same couplings generated at MPl. With ; G. Amelino-Camelia, Phys. B a simple one-loop analysis of the renormalization group ; S. Sarkar, Mod. A17 Jacobson, S. Liberati and D. Mattingly, Phys. Konopka and S. Major, New J. Gleiser and C. Kozameh, Phys. D64 In summary, we have shown that effective field the- Amelino-Camelia, Nature ; I. Mo- bounds on Planck scale interactions from terrestrial ex- cioiu, M.

For the most studied examples, canonical noncommutativity,. Arguments suggesting that CPT violation might arise in the quantum-gravity realm have a long tradition [ , , , , 42 , , , , ] and also see, e. And, in light of the scope of this review, I should stress that specifically the idea of spacetime quantization invites one to place CPT symmetry under scrutiny. Indeed, locality in addition to unitarity and Lorentz invariance is a crucial ingredient for ensuring CPT invariance, and a common feature of all the proposals for spacetime quantization is the presence of limitations to locality, at least intended as limitations to the localizability of a spacetime event.

Unfortunately, a proper analysis of CPT symmetry requires a level of understanding of the formalism that is often beyond our present reach in the study of formalizations of the concept of quantum spacetime. In LQG one should have a good control of the Minkowski classical- limit, and of the description of charged particles in that limit, and this is still beyond what can presently be done within LQG. Similar remarks apply to spacetime noncommutativity, although in that case some indirect arguments relevant for CPT symmetry can be meaningfully structured.

For example, in Ref. In the mentioned quantum-spacetime picture based on noncritical Liouville string theory [ , ], evidence of violations of CPT symmetry has been reported [ ], and later in this review I shall comment on the exciting phenomenology that was inspired by these results. It is well established that the availability of a classical spacetime background has been instrumental to the successful tests of quantum mechanics so far performed.

The applicability of quantum mechanics to a broader class of contexts remains an open experimental question. If indeed space-time is quantized there might be some associated departures from quantum mechanics. And this quantum-spacetime intuition fits well with a rather popular intuition for the broader context of quantum-gravity research, as discussed for example in Refs. A description of decoherence has been inspired by the mentioned noncritical Liouville string theory [ , ], and is essentially the core feature of the formalism advocated by Percival and collaborators [ , , ].

The possibility of modifications of the Heisenberg principle and of the de Broglie relation has also been much studied in accordance with the intuition that some aspects of quantum mechanics might need to be adapted to spacetime quantization. Although the details of the mechanism that produces such modifications vary significantly from one picture of spacetime quantization to another [ , 22 , ], one can develop an intuition of rather general applicability by noticing that the form of the de Broglie relation in ordinary quantum mechanics reflects the properties of the classical geometry of spacetime that is there assumed.

While the possibility of spacetime quantization provides a particularly direct logical line toward modifications of laws of quantum mechanics, one should consider such modifications as natural for the whole quantum-gravity problem even when studied without assuming spacetime quantization. Most authors see it as motivation to look for formalizations of spacetime in which the distance between two events cannot be sharply determined, and the metric is correspondingly fuzzy.

As I shall discuss in Section 4 , a few attempts to operatively characterize the concept of spacetime foam and to introduce corresponding test theories have been recently developed. And a rather rich phenomenology is maturing from these proposals, often centered both on spacetime fuzziness per se and associated decoherence. Unfortunately, very little guidance can be obtained from the most studied quantum-spacetime pictures. In LQG this type of experimentally tangible characterization of spacetime foam is not presently available.

And remarkably even with spacetime noncommutativity, an idea that was mainly motivated by the spacetime-foam intuition of a nonclassical spacetime, we are presently unable to describe, for example, the fuzziness that would intervene in operating an interferometer with the type of crisp physical characterization needed for phenomenology. The possibility of violations of the equivalence principle has not been extensively studied from a quantum-spacetime perspective, in spite of the fact that spacetime quantization does provide some motivation for placing under scrutiny at least some implications of the equivalence principle.

This is at least suggested by the observation that locality is a key ingredient of the present formulation of the equivalence principle: the equivalence principle ensures that under appropriate conditions two point particles would go on the same geodesic independent of their mass. Relatively few studies have been devoted to violations of the equivalence principle from a quantum-spacetime perspective. Examples are the study reported in Ref. Also the broader quantum-gravity literature even without spacetime quantization provides motivation for scrutinizing the equivalence principle.

In particular, a strong phenomenology centered on violations of the equivalence principle was proposed in the string-theory-inspired studies reported in Refs. Also relevant to this review is the possibility that violations of the equivalence principle might be a by-product of violations of Lorentz symmetry. In particular, this is suggested by the analysis in Ref. This was due both to the relative robustness of associated theory results in quantum-spacetime research and to the availability of very valuable opportunities of related data analyses.

Before discussing some actual phenomenologic al analyses, I find it appropriate to start this section with some preparatory work. This intuition inspires the work on quantum-Minkowski spacetimes, and the analysis of the symmetries of these quantum spacetimes. It is 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 could actually be trivial, i.

But the hypothesis of a nontrivial Minkowski limit is worth exploring: it is a plausible hypothesis and it would be extremely valuable for us if quantum gravity did admit such a limit, since it might open a wide range of opportunities for accessible experimental verification, as I shall stress in what follows. It is 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 GR 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. Still, a breakdown of Lorentz symmetry, in the sense of spontaneous symmetry breaking, is possible, and this possibility has been studied extensively over the last few years, especially in string theory see, e. Complementary to the particle-physics perspective is the GR perspective, whose core characteristic is the intuition that one should firmly reject the possibility of relying on a background spacetime [ , ].

According to GR the evolution of particles and the structure of spacetime are self-consistently 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 GR relied heavily on the careful consideration of the in-principle limitations that measurement procedures can encounter.

This naturally leads one to consider discretized spacetimes, as in the LQG approach or noncommutative spacetimes. The third possibility is a condensed-matter perspective on the quantum-gravity problem see, e. Condensed-matter theories 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-excitation theory, which do not carry the significance of fundamental symmetries, and are, in fact, lost as soon as the theory is probed 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.

Further encouragement for the idea of an emerging spacetime though not necessarily invoking the condensed-matter perspective comes from the realization [ , , ] that the Einstein equations can be viewed as an equation of state, so in some sense thermodynamics implies GR and the associated microscopic theory might not look much like gravity.

As mentioned, an example of a suitable mechanism is provided by the possibility that a tensor field might have a vacuum expectation value [ ]. I have elsewhere [ 63 ] attempted to expose the compellingness of this possibility. Still, because of the purposes of this review, I must take into account that the development of phenomenologically-viable DSR models is still in its infancy. In particular, several authors see, e. Interested readers have available a rather sizable DSR literature see, e. My main task in this Section is to illustrate the differences in relation to this compatibility issue between the broken-symmetry hypothesis and the DSR-deformed-symmetry hypothesis.

The DSR scenario was proposed [ 58 ] as a sort of alternative perspective on the results on Planck-scale departures from Lorentz symmetry that had been reported in numerous articles [ 66 , , , 38 , 73 , , 33 ] between and These studies were advocating a Planck-scale modification of the energy-momentum dispersion relation, usually of the form , 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.

However, it then turned out to be possible [ 58 ] to avoid this preferred-frame expectation, following a line of analysis in many ways analogous to the one familiar from the developments that led to the emergence of special relativity SR , now more than a century ago.

### 1. Introduction

But in the end we discovered that the situation was not demanding the introduction of a preferred frame, but rather a modification of the laws of transformation between inertial observers. It is plausible that we might be presently confronted with an analogous scenario. While the DSR idea came to be 0proposed in the context of studies of modifications of the dispersion relation, one could have other uses for the second relativistic scale, as stressed in parts of the DSR literature [ 58 , 55 , , , , , , , , , , 26 , , , , , , ].

Instead of promoting to the status of relativistic invariant a modified dispersion relation, one can have DSR scenarios with undeformed dispersion relations but, for example, with an observer-independent bound on the accuracy achievable in the measurement of distances [ 63 ]. However, as announced, within the confines of this quantum-spacetime-phenomenology review I shall only make use of one DSR argument, that applies to cases in which indeed the dispersion relation is modified.

This concerns the fact that in the presence of observer-independent modifications of the dispersion relation DSR- relativistic invariance imposes the presence of associated modifications of the law of energy-momentum conservation. More general discussions of this issue are offered in Refs. This dispersion relation is clearly an invariant of classical space rotations, and of deformed boost transformations generated by [ 58 , 63 ].

The issue concerning energy-momentum conservation arises because both the dispersion relation and the law of energy-momentum conservation must be DSR- relativistic.

And the boosts 6 , which enforce relativistically the modification of the dispersion relation, are incompatible with the standard form of energy-momentum conservation. An example of a modification of energy-momentum conservation that is compatible with 6 is [ 58 ]. And analogous formulas can be given for any process with n incoming particles and m outgoing particles. This observation provides a general motivation for contemplating modifications of the law of energy-momentum conservation in frameworks with modified dispersion relations.

And I shall often test the potential impact on the phenomenology of introducing such modifications of the conservation of energy-momentum by using as examples DSR-inspired laws of the type 7 , 8 , 9 , I shall do this without necessarily advocating a DSR interpretation: knowing whether or not the outcome of tests of modifications of the dispersion relation depends on the possibility of also having a modification of the momentum-conservation laws is of intrinsic interest, with or without the DSR intuition.

But I must stress that when the relativistic symmetries are broken rather than deformed in the DSR sense there is no a priori reason to modify the law of energy-momentum conservation, even when the dispersion relation is modified. Indeed most authors adopting modified dispersion relations within a broken-symmetry scenario keep the law of energy-momentum conservation undeformed.

On the other hand the DSR research program has still not reached the maturity for providing a fully satisfactory interpretation of the nonlinearities in the conservation laws.

## OSA | Solution of dispersion relations of multilayer optical fibers: a comprehensive study

For some time the main challenge came in addition to the mentioned interpretational challenges connected with spacetime locality from arguments suggesting that one might well replace a given nonlinear setup for a DSR model with one obtained by redefining nonlinearly the coordinatization of momentum space see, e. When contemplating such changes of coordinatization of momentum space many interpretational challenges appeared to arise.

In my opinion, also in this direction the recent DSR literature has made significant progress, by casting the nonlinearities for momentum-space properties in terms of geometric entities, such as the metric and the affine connection on momentum space see, e. This novel geometric interpretation is offering several opportunities for addressing the interpretational challenges, but the process is still far from complete.

Indeed, certain analyses of formalisms 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. 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-spacetime phenomenology.

What we can compare to data are some simple models inspired by the little we believe we understand of the relevant issues within the theories that provide motivation for this phenomenology. And the development of such models requires a delicate balancing act. If we only provide them with the structures we do understand of the original theories they will be as 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 a proliferation of models adopted 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-spacetime phenomenology of modified dispersion relations, one should consider a huge zoo of parameters. I shall be satisfied with considering some illustrative examples of models, chosen in such a way as to represent possibilities that are qualitatively very different, and representative of the breadth of possibilities that are under consideration.

Before describing actual test theories, I should at least discuss the most significant among the issues that must be considered in setting up any such test theory with modified dispersion relation. This 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 and quantum-spacetime community is rather skeptical of the results obtained using low-energy effective field theory in analyses relevant to the Planck-scale regime. One of the key reasons for this skepticism is the description given by effective field theory of the cosmological constant.

The cosmological constant is the most significant experimental fact of evident gravitational relevance that could be within the reach of effective field theory. And current approaches to deriving the cosmological constant within effective field theory produce results, which are some orders of magnitude greater than allowed by observations.

One would like to confine the new effects to unexplored high-energy regimes, by adjusting bare parameters accordingly, but, as I shall stress again later, quantum corrections produce [ , , , ] effects that are nonetheless significant at accessible low energies, unless one allows for rather severe fine-tuning. On the other hand, we do not have enough clues concerning setups alternative to quantum-field theory that could be used. For example, as I discuss in detail later, some attempts are centered on density-matrix formalisms that go beyond quantum mechanics, but those are however legitimate mere speculations at the present time.

Nonetheless several of the phenomenologists involved, myself included, feel that in such a situation phenomenology cannot be stopped by the theory impasse, even at the risk of later discovering that the whole or a sizable part of the phenomenological effort was not on sound conceptual bases. But I stress that even when contemplating the possibility of physics outside the domain of effective quantum field theory, one inevitably must at least come to terms with the success of effective field theory in reproducing a vast class of experimental data.

The regime of low boosts with respect to the center-of-mass frame is often indistinguishable from the low-energy limit. For example, from a Planck-scale perspective, our laboratory experiments even the ones conducted at, e. Another interesting scenario concerning the nature of the limit through which quantum-spacetime physics should reproduce ordinary physics is suggested by results on field theories in noncommutative spacetimes.

One can observe that a spacetime characterized by an uncertainty relation of the type. For other non-canonical noncommutative spacetimes we are still struggling in the search for a satisfactory formulation of a quantum field theory [ , 64 ], 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 has also been challenged by some perspectives on the LQG approach.

For example, the arguments presented in Ref. In order to be applicable to a significant ensemble of experimental contexts, a test theory should 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, e.

Unfortunately on these three key points, the quantum-spacetime pictures that are providing motivation for the study of Planck-scale modifications of the dispersion relation are not giving us much guidance yet. Similarly, in the analysis of noncommutative spacetimes we are close to establishing rather robustly the presence of modifications of the dispersion relation, but other aspects of the relevant theories have not yet been clarified.

And at least for canonical noncommutative spacetimes the possibility of a nonuniversal dispersion relation is considered extensively [ , ]. And concerning the Heisenberg uncertainty principle I have already mentioned some arguments that invite us to contemplate modifications.

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With so many possible alternative ingredients to mix one can of produce a large variety of test theories. As mentioned, I intend to focus on some illustrative examples 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 strongly restrict the class of experiments 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-spacetime theory with that form of kinematics, independent of the description of dynamics.

The first element of this test theory, introduced from a quantum-spacetime-phenomenology perspective in Refs. This rudimentary framework is a good starting point for exploring the relevant phenomenology. But one should also consider some of the possible variants. For example, the undeformed conservation of energy-momentum is relativistically incompatible with the deformation of the dispersion relation so, in particular, the PKV0 test theory requires a preferred frame.

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Modifications of the law of energy-momentum conservation would be required in a DSR picture, and may be considered even in other scenarios. Evidently, the universality of the effect can and should be challenged. And there are indeed as I shall stress again later in this review several proposals of test theories with different magnitudes of the effects for different particles [ , ]. The restriction to pure kinematics has the merit to allow us to establish constraints that are applicable to a relatively large class of quantum-spacetime scenarios different formulations of dynamics would still be subject to the relevant constraints , but it also severely restricts the type of experimental contexts that can be considered, since it is only in rare instances and only to some extent that one can qualify an analysis as purely kinematical.

Therefore, 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 rather 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 that are being pursued are reflected in the test theory.

When planning to devise a test theory that includes the possibility to describe dynamics, the first natural candidate not withstanding the concerns reviewed in Section 3. In this section I want to discuss a test theory that is indeed based on low-energy effective field theory, and has emerged primarily 17 from the analysis reported by Myers and Pospelov in Ref. Motivated mainly by the perspective of LQG advocated in Ref. Perhaps the most notable outcome of the exercise of introducing such a dispersion relation within an effective low-energy field-theory setup is the observation [ ] that for the case of electromagnetic radiation, assuming essentially only that the effects are characterized mainly by an external four-vector, one arrives at a single possible correction term for the Lagrangian density:.

The formalism is compatible with the possibility of introducing further independent parameters for each additional fermion in the theory so that, e. In some investigations one might prefer to look at particularly meaningful portions of this large parameter space. Before starting my characterization of experimental sensitivities in terms of the parameters of some test theories I find it appropriate to add a few remarks warning about some difficulties that are inevitably encountered. For the pure-kinematics test theories, some key difficulties originate from the fact that sometimes an effect due to the 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 one to set robust pure-kinematics limits on the parameters. But if the test theory predicts that a certain particle at a certain 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.

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If the decay is kinematically allowed but not seen, it is possible that the laws of dynamics prevent it from occurring small decay amplitude. By adopting a low-energy quantum field theory this type of limitations is removed, but other issues must be taken into account, particularly in association with the fact that the FTV0 quantum field theory is not renormalizable. In particular, for the FTV0 test theory, with its Planck-scale suppressed effects at tree level, some authors notably Refs. The parameters of the field theory can be fine-tuned to eliminate 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. In particular, it is already established that supersymmetry can tame the fine-tuning issue [ , ].

If one extends supersymmetric quantum electrodynamics by adding interactions with external vector and tensor backgrounds that violate Lorentz symmetry at the Planck scale, then exact supersymmetry requires that such interactions correspond to operators of dimension five or higher, so that no fine-tuning is needed in order to suppress the unwanted operators of dimension lower than five. Supersymmetry can only be an approximate symmetry of the physical world, and the effects of the scale of soft-supersymmetry-breaking masses controls the renormalization-group evolution of dimension five Lorentz-violating operators and their mixing with dimension three Lorentz-violating operators [ , ].

It has also been established [ ] that if Lorentz violation occurs in the gravitational sector, then the violations of Lorentz symmetry induced on the matter sector do not require severe fine-tuning. The study of Planck-scale departures from Lorentz symmetry may find some encouragement in perspectives based on renormalization theory, at least in as much as it has been shown [ 79 , 78 , , ] that some field theories modified by Lorentz-violating terms are actually rather well behaved in the UV.

The first example of Planck-scale sensitivity that I discuss is the case of a process that is kinematically forbidden in the presence of exact Lorentz symmetry, but becomes kinematically allowed in the presence of certain departures from Lorentz symmetry. It has been established see, e. At the qualitative level, the most significant novelty would be the possibility for massless particles to decay. Let us start from the perspective of the PKV0 test theory, and therefore adopt the dispersion relation 13 and unmodified energy-momentum conservation.

However, specifically for the case of the photon-stability analysis it is rather challenging to transform this Planck-scale sensitivity into actual experimental limits. But the fact that the decay of 10 14 eV photons is allowed by PKV0 kinematics of 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. In principle, models based on pure kinematics are immune from certain bounds on parameters that are also derived also using descriptions of the interactions, and it is conceivable that in the correct theory the actual bound would be somewhat shifted from the value derived within effective quantum field theory.

But in order to contemplate large differences in the bounds one would need to advocate very large and ad hoc modifications of the strength of interactions, large enough to compensate for the often dramatic implications of the modifications of kinematics. The challenge then is to find satisfactory criteria for confining speculations about variations of the strengths of interaction only within a certain plausible range.

To my knowledge this has not yet been attempted, but it deserves high priority. A completely analogous calculation can be done within the FTV0 test theory, and there one can easily arrive at the conclusion [ ] that the FTV0 description of dynamics should not significantly suppress 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.

So far I have discussed photon stability assuming that only the dispersion relation is modified. If the modification of the dispersion relation is instead combined with a modification of the law of energy-momentum conservation the results can change very significantly. If the modification of the dispersion relation and the modification of the law of energy-momentum conservation are not matched exactly to get this result, then one can have the possibility of photon decay, but in some cases it can be further suppressed in addition to the Planck-scale suppression by the partial compensation between the two modifications.

The fact that the matching between modification of the dispersion relation and modification of the law of energy-momentum conservation that produces a stable photon is obtained using a DSR-inspired setup is not surprising [ 63 ]. The relativistic properties of the framework are clearly at stake in this derivation. A threshold-energy requirement for particle decay such as the mentioned above cannot be introduced as an observer-independent law, and is therefore incompatible with any relativistic even DSR-relativistic formulation of the laws of physics.

In fact, different observers assign different values to the energy of a particle and, therefore, in the 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.

Another opportunity to investigate quantum-spacetime-inspired Planck-scale departures from Lorentz symmetry is provided by certain types of energy thresholds for particle-production processes that are relevant in astrophysics.