Several recent studies have concerned the faith of classical symmetries in
quantum space-time. In particular, it appears likely that quantum (discretized,
noncommutative,...) versions of Minkowski space-time would not enjoy the
classical Lorentz symmetries. I compare two interesting cases: the case in
which the classical symmetries are "broken", i.e. at the quantum level some
classical symmetries are lost, and the case in which the classical symmetries
are "deformed", i.e. the quantum space-time has as many symmetries as its
classical counterpart but the nature of these symmetries is affected by the
space-time quantization procedure. While some general features, such as the
emergence of deformed dispersion relations, characterize both the
symmetry-breaking case and the symmetry-deformation case, the two scenarios are
also characterized by sharp differences, even concerning the nature of the new
effects predicted. I illustrate this point within an illustrative calculation
concerning the role of space-time symmetries in the evaluation of
particle-decay amplitudes. The results of the analysis here reported also show
that the indications obtained by certain dimensional arguments, such as the
ones recently considered in hep-ph/0106309 may fail to uncover some key
features of quantum space-time symmetries.Comment: LaTex, 7 pages, talk given at the 2nd Meeting on CPT and Lorentz
  Symmetry (CPT 01), Bloomington, Indiana, 15-18 Aug 200

Amelino-Camelia, Giovanni

English

arXiv

Several recent studies have concerned the faith of classical symmetries in quantum space-time. In particular, it appears likely that quantum (discretized, noncommutative,...) versions of Minkowski space-time would not enjoy the classical Lorentz symmetries. I compare two interesting cases: the case in which the classical symmetries are "broken", i.e. at the quantum level some classical symmetries are lost, and the case in which the classical symmetries are "deformed", i.e. the quantum space-time has as many symmetries as its classical counterpart but the nature of these symmetries is affected by the space-time quantization procedure. While some general features, such as the emergence of deformed dispersion relations, characterize both the symmetry-breaking case and the symmetry-deformation case, the two scenarios are also characterized by sharp differences, even concerning the nature of the new effects predicted. I illustrate this point within an illustrative calculation concerning the role of space-time symmetries in the evaluation of particle-decay amplitudes. The results of the analysis here reported also show that the indications obtained by certain dimensional arguments, such as the ones recently considered in hep-ph/0106309 may fail to uncover some key features of quantum space-time symmetries

Amelino-Camelia, G

CERN Document Server

QUANTUM SPACE-TIME: DEFORMED SYMMETRIESVERSUS BROKEN SYMMETRIES1Giovanni AMELINO-CAMELIAaDipart. Fisica, Univ. Roma “La Sapienza”, P.le Moro 2, 00185 Roma, ItalyABSTRACTSeveral recent studies have concerned the faith of classical symmetries in quantum space-time. In particular, it appears likely that quantum (discretized, noncommutative,...) ver-sions of Minkowski space-time would not enjoy the classical Lorentz symmetries. I comparetwo interesting cases: the case in which the classical symmetries are \broken", i.e. at thequantum level some classical symmetries are lost, and the case in which the classical symme-tries are \deformed", i.e. the quantum space-time has as many symmetries as its classicalcounterpart but the nature of these symmetries is aected by the space-time quantizationprocedure. While some general features, such as the emergence of deformed dispersion re-lations, characterize both the symmetry-breaking case and the symmetry-deformation case,the two scenarios are also characterized by sharp dierences, even concerning the nature ofthe new eects predicted. I illustrate this point within an illustrative calculation concerningthe role of space-time symmetries in the evaluation of particle-decay amplitudes. The resultsof the analysis here reported also show that the indications obtained by certain dimensionalarguments, such as the ones recently considered in hep-ph/0106309 may fail to uncover somekey features of quantum space-time symmetries.1Invited talk given at the 2nd Meeting on CPT and Lorentz Symmetry (CPT 01), Bloomington, Indiana,15-18 Aug 2001. To appear in the proceedings.1 IntroductionIn recent years the problem of establishing what happens to the symmetries of classicalspacetime when the spacetime is quantized has taken central stage in quantum-gravity re-search. In particular, the symmetries of classical flat (Minkowski) spacetime are well veriedexperimentally, so it appears that any deviation from these symmetries that might emergefrom quantum-gravity theories would be subject to severe experimental constraints. As aresult symmetry tests are a key component of the programme of \Quantum-Gravity Phe-nomenology" [1, 2, 3].In this lecture I focus on the faith of Lorentz invariance at the quantum-spacetime level.A large research eort has been devoted to this subject. Most of these studies focus onthe possibility that Lorentz symmetry might be \broken" at the quantum level; however, Ihave recently shown that Lorentz invariance might be aected by spacetime quantization ina softer manner: there might be no net loss of symmetries but the structure of the Lorentztransformations might be aected by the quantization procedure [4, 5]. My primary objectivehere will be the one of drawing a clear distinction between the broken-symmetry and thedeformed-symmetry scenarios.2 Quantum-Gravity PhenomenologyQuantum-Gravity Phenomenology [1] is an intentionally vague name for a new approachto research on the possible non-classical (quantum) properties of spacetime. This approachdoes not adopt any particular belief concerning the structure of spacetime at short dis-tances (e.g., \string theory", \loop quantum gravity" and \noncommutative geometry" areseen as equally deserving mathematical-physics programmes). It is rather the proposal thatquantum-gravity research should proceed just in the familiar old-fashioned way: throughsmall incremental steps starting from what we know and combining mathematical-physicsstudies with experimental studies to reach deeper and deeper layers of understanding of theshort-distance structure of spacetime. Somehow research on quantum gravity has wonderedo this traditional strategy: the most popular quantum-gravity approaches, such as stringtheory and loop quantum gravity, could be described as \top-to-bottom approaches" sincethey start o with some key assumption about the structure of spacetime at the Planckscale and then they try (with limited, vanishingly small, success) to work their way backto the realm of doable experiments. With Quantum-Gravity Phenomenology I would liketo refer to all studies that are somehow related with a \bottom-to-top approach" to thequantum-gravity problem.Since the problem at hand is really dicult (arguably the most challenging problem everfaced by the physics community) it appears likely that the two complementary approachesmight combine in a useful way: for the \bottom-to-top approach" it is important to getsome guidance from the (however tentative) indications emerging from the \top-to-bottomapproaches", while for \top-to-bottom approaches" it might be very useful to be alertedby quantum-gravity phenomenologists with respect to the type of new eects that could bemost stringently tested experimentally (it is hard for \top-to-bottom approaches" to obtaina complete description of low-energy physics, but perhaps it would be possible to dig outpredictions on some specic spacetime features that appear to have special motivation inlight of the corresponding experimental sensitivities).Until very recently the idea of a Quantum-Gravity Phenomenology, and in particular ofattempts of identication of experiments with promising sensitivity, was very far from themain interests of quantum-gravity research. One isolated idea had been circulating from themid 1980s: it had been realized [6, 7, 8] that the sensitivity of CPT tests using the neutral-kaon system has improved to the point that even small eects of CPT violation originating at1the Planck scale2 might in principle be revealed. These pioneering works on CPT tests werefor more than a decade the only narrow context in which the implications of quantum grav-ity were being discussed in relation with experiments, but over the last 3 years several newideas for tests of Planck-scale physics have appeared at increasingly fast pace, leading me toargue [1] that the times might be right for a more serious overall eort in Quantum-GravityPhenomenology. At the present time there are several examples of experimentally accesiblecontexts in which conjectured quantum-gravity eects are being considered, including studiesof in-vacuo dispersion using gamma-ray astrophysics [9, 10], studies of laser-interferometriclimits on quantum-gravity induced distance fluctuations [11, 12], studies of the role of thePlanck length in the determination of the energy-momentum-conservation threshold condi-tions for certain particle-physics processes [13, 14, 15, 16, 17], and studies of the role of thePlanck length in the determination of particle-decay amplitudes [18]. These experimentsmight represent the cornerstones of quantum-gravity phenomenology since they are as closeas one can get to direct tests of space-time properties, such as space-time symmetries. Otherexperimental proposals that should be seen as part of the quantum-gravity-phenomenologyprogramme rely on the mediation of some dynamical theory in quantum space-time; com-ments on these other proposals can be found in Refs. [1, 19, 20, 21, 22, 23].The primary challenge of quantum-gravity phenomenology is the one of establishingthe properties of space-time at Planckian distance scales. However, there is also recentdiscussion of the possibility that quantum-spacetime eects might be stronger than usuallyexpected, i.e. with a characteristic energy scale that is much smaller (perhaps just in the TeVrange!) than the Planck energy. Examples of mechanisms leading to this possibility are foundin string-theory models with large extra dimensions [24] and in certain noncommutative-geometry models [25]. The study of the phenomenology of these models of course is in thespirit of quantum-gravity phenomenology, although it is of course less challenging than thequantum-gravity-phenomenology eorts that pertain eects genuinely at the Planck scale.3 The faith of Lorentz symmetry in quantum space-timeIf the Planck length, Lp, only has the role we presently attribute to it, which is basically therole of a coupling constant (an appropriately rescaled version of the coupling G), no problemarises for FitzGerald-Lorentz contraction, but if we try to promote Lp to the status of anintrinsic characteristic of space-time structure (or a characteristic of the kinematic rules thatgovern particle propagation in space-time) it is natural to nd conflicts with FitzGerald-Lorentz contraction.For example, it is very hard (perhaps even impossible) to construct discretized versionsor non-commutative versions of Minkowski space-time which enjoy ordinary Lorentz sym-metry. Pedagogical illustrative examples of this observation have been discussed, e.g., inRef. [26] for the case of discretization and in Refs. [27, 28] for the case of non-commutativity.Discretization length scales and/or non-commutativity length scales naturally end up ac-quiring dierent values for dierent inertial observers, just as one would expect in light ofthe mechanism of FitzGerald-Lorentz contraction.There are also dynamical mechanisms (of the spontaneous symmetry-breaking type) thatcan lead to deviations from ordinary Lorentz invariance, it appears for example that thismight be possible in string eld theory [29].2The possibility of Planck-scale-induced violations of the CPT symmetry has been extensively consideredin the literature. One simple point in support of this possibility comes from the fact that the CPT theorem,which holds in our present conventional theories, relies on exact locality, whereas in quantum gravity itappears plausible to assume lack of locality at Planckian scales.2Departures from ordinary Lorentz invariance are therefore rather plausible at the quantum-gravity level. Here I want to emphasize that there are at least two possibilities: (i) Lorentzinvariance is broken and (ii) Lorentz invariance is deformed.3.1 Deformed Lorentz invarianceIn order to be specic about the dierences between deformed and broken Lorentz invariancelet me focus on the dispersion relation E(p) which will naturally be modied in eithercase. Let me also assume, for the moment, that the deformation be Planck-length induced:E2 = m2 + p2 + f(p, m; Lp). If the function f is nonvanishing and nontrivial and the energy-momentum transformation rules are ordinary (the ordinary Lorentz transformations) thenclearly f cannot have the exact same structure for all inertial observers. In this case onewould speak of an instance in which Lorentz invariance is broken. If instead f does have theexact same structure for all inertial observers, then necessarily the transformations betweenthese observers must be deformed. In this case one would speak of an instance in which theLorentz transformations are deformed, but Lorentz invariance is preserved (in the deformedsense).While much work has been devoted to the case in which Lorentz invariance is actuallybroken, the possibility that Lorentz invariance might be deformed was introduced only veryrecently by this author [4, 30, 31, 5, 32]. An example in which all details of the deformedLorentz symmetry have been worked out is the one in which one enforces as an observer-indepedent statement the dispersion relationL−2p(eLpE + e−LpE − 2)− ~p2e−LpE = m2 (1)In leading (low-energy) order this takes the formE2 − ~p2 + LpE~p2 = m2 . (2)The Lorentz transformations and the energy-momentum conservation rules are accordinglymodied [5].3.2 Broken Lorentz invarianceThe case of broken Lorentz invariance requires fewer comments since it is more familiar tothe community. In preparation for the analysis reported in the next Section it is usefulto emphasize that the same dispersion relation (2), which was shown in Refs. [4, 5] to beimplementable as an observer-independent dispersion relation in a deformed-symmetry sce-nario, can also be considered [9] as a characteristic dispersion relation of a broken-symmetryscenario. In this broken symmetry scenario the dispersion relation (2) would still be validbut only for one \preferred" class of inertial observers (e.g. the natural CMBR frame) and itwould be valid approximately in all frames not highly boosted with respect to the preferredframe. In highly-boosted frames one might nd the same form of the dispersion relation butwith dierent value of the deformation scale (dierent from Lp). All this follows from the factthat in the broken-symmetry scenario the laws of transformation between inertial observersare unmodied. Accordingly also energy-momentum conservation rules are unmodied.Another scenario in which one nds broken Lorentz invariance is the one of canonicalnoncommutative spacetime, in which the dispersion relation is modied (with dierent de-formation term [33, 34]), but, again, the energy-momentum Lorentz transformation rules arenot modied.34 Illustrative example: photon-pair pion decayIn order to render very explicit the dierences between the broken-symmetry and the deformed-symmetry case in this Section I consider photon-pair pion decay adopting in one case de-formed energy-momentum conservation [5], as required by the deformed Lorentz transfor-mations of the deformed-symmetry case, and in another case ordinary energy-momentumconservation, as required by the fact that the Lorentz transformation rules are unmodiedin the broken-symmetry case, but for both cases I impose the same dispersion relation (2).In the broken-symmetry case, combining (2) with ordinary energy-momentum conser-vation rules, one can establish a relation between the energy E of the incoming pion, theopening angle θ between the outgoing photons and the energy Eγ of one of the photons (theenergy E 0γ of the second photon is of course not independent; it is given by the dierencebetween the energy of the pion and the energy of the rst photon):cos(θ) =2EγE0γ −m2 + 3LpEEγE 0γ2EγE 0γ + LpEEγE 0γ, (3)where indeed E 0γ  E−Eγ . This relation shows that at high energies (starting at energies oforder (m2/Lp)1=3) the phase space available to the decay is anomalously reduced: for givenvalue of E certain values of Eγ that would normally be accessible to the decay are no longeraccessible (they would require cosθ > 1).In the deformed-symmetry case one enforces the deformed conservation rules [5]E = Eγ + E0γ , ~p = ~pγ + ~pγ′ + LpEγ~pγ′ , (4)which, when combined again with (2), give raise to the dierent relationcos(θ) =2EγE0γ −m2 + 3LpE2γE 0γ + LpEγE 02γ2EγE 0γ + 3LpE2γE 0γ + LpEγE 02γ. (5)Here it is easy to check that for all physically acceptable values of Eγ (given the value ofE) one is never led to consider the paradoxical condition cosθ > 1: there is no severeimplication of the deformed-symmetry case for the amount of phase space available for thedecays (certainly not at energies around (m2/Lp)1=3, possibly at Planckian energies).5 Closing remarksAs shown by the illustrative example of calculation presented in the preceding Section, thedierences between the case in which Lorentz invariance is broken and the case in whichLorentz invariance is deformed can be very signicant also quantitatively, concerning thenature and the magnitude of the eects predicted, besides being quite clearly signicant atthe conceptual level.The calculation in the preceding Section also shows that simple dimensional estimatesof the eects induced by deviations from Lorentz invariance are futile. Both in the broken-symmetry case and in the deformed-symmetry case the Planck-scale deformation introducesthe same correction terms, respectively in Eqs. (3) and (5), but in the broken-symmetry caseI found profound implications for pion decay, whereas in the deformed-symmetry case thecorrection terms arranged themselves in a less \armful" manner. This observation appearsto be particularly signicant for the argument recently presented by Brustein, Eichler and4Foa in Ref. [35]: in that paper, by applying dimensional analysis to some aspects of neu-trino physics it was suggested that Planck-scale-induced deviations from ordinary Lorentzinvariance are unlikely. The fact that the analysis reported in Ref. [35] relies on the type ofdimensional arguments which I have here shown to be inclusive, forces us to assume that theconclusions drawn in Ref. [35] are equally unreliable. Certainly the observations reported inRef. [35] provide strong motivations for future dedicated and rigorously quantitative studiesof the relevant aspects of neutrino physics within specic examples of Planck-scale-induceddeviations from ordinary Lorentz invariance.AcknowledgmentsI am grateful for the encouraging feed-back received from several participants to CPT01,and particularly for conversations with Roman Jackiw and Alan Kostelecky.References[1] G. Amelino-Camelia, \Are we at the dawn of quantum-gravity phenomenology?",Lect. Notes Phys. 541, 1 (2000).[2] C. Rovelli, Notes for a brief history of Quantum Gravity, gr-qc/0006061, in \Proceedingsof the 9th Marcel Grossmann Meeting on Recent Developments in Theoretical andExperimental General Relativity, Gravitation and Relativistic Field Theories, Rome,Italy, 2-9 Jul 2000".[3] S. Carlip, Rept. Prog. Phys. 64, 885 (2001).[4] G. Amelino-Camelia, gr-qc/0012051, Int. J. Mod. Phys. D (2001) in press; hep-th/0012238, Phys. Lett. B 510, 255 (2001).[5] G. Amelino-Camelia, gr-qc/0106004.[6] J. Ellis, J.S. Hagelin, D.V. Nanopoulos and M. Srednicki, Nucl. Phys. B 241, 381 (1984).[7] P. Huet and M.E. Peskin, Nucl. Phys. B 434, 3 (1995).[8] V.A. Kostelecky and R. Potting, Phys.Rev. D 51, 3923 (1995); O. Bertolami, D. Colla-day, V.A. Kostelecky and R. Potting, Phys.Lett. B 395, 178 (1997).[9] G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos and S. Sarkar, astro-ph/9712103, Nature 393, 763 (1998).[10] S.D. Biller et al., Phys. Rev. Lett. 83, 2108 (1999).[11] G. Amelino-Camelia, gr-qc/9808029, Nature 398, 216 (1999).[12] G. Amelino-Camelia, gr-qc/0104005.[13] T. Kifune, Astrophys. J. Lett. 518, L21 (1999).[14] W. Kluzniak, Astropart. Phys. 11, 117 (1999).[15] R.J. Protheroe and H. Meyer, Phys. Lett. B 493, 1 (2000).5[16] G. Amelino-Camelia and T. Piran, Phys. Lett. B 497, 265 (2001); Phys. Rev. D64,036005 (2001).[17] Ted Jacobson, Stefano Liberati and David Mattingly, hep-ph/0112207.[18] G. Amelino-Camelia, gr-qc/0107086.[19] R. Brustein, gr-qc/9810063; G. Veneziano, hep-th/9902097; M. Gasperini, hep-th/9907067.[20] I.C. Perival and W.T. Strunz, Proc. R. Soc. A 453, 431 (1997).[21] L.J. Garay, Phys. Rev. Lett. 80, 2508 (1998).[22] D.V. Ahluwalia, gr-qc/9903074, Nature 398, 199 (1999).[23] C. La¨mmerzahl and C.J. Borde, Lect. Notes Phys. 562, 463 (2001).[24] N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998).[25] M. R. Douglas and N. A. Nekrasov, hep-th/0106048.[26] G. ‘t Hooft, Class. Quant. Grav. 13, 1023 (1996).[27] S. Majid and H. Ruegg, Phys. Lett. B 334, 348 (1994).[28] J. Lukierski, H. Ruegg and W.J. Zakrzewski Ann. Phys. 243, 90 (1995).[29] V.A. Kostelecky and R. Potting, Phys.Lett. B 381, 89 (1996).[30] J. Kowalski-Glikman, hep-th/0102098.[31] G. Amelino-Camelia and M. Arzano, hep-th/0105120.[32] N.R. Bruno, G. Amelino-Camelia and J. Kowalski-Glikman, hep-th/0107039.[33] A. Matusis, L. Susskind and N. Toumbas, JHEP 0012, 002 (2000).[34] G. Amelino-Camelia, L. Doplicher, S. Nam and Y.-S. Seo, hep-th/0109191.[35] R. Brustein, D. Eichler and S. Foa, hep-ph/0106309.6

Quantum Space-Time: Deformed Symmetries Versus Broken Symmetries

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