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Proper Definiaon of Quantum Gravity Theory!

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Proper Definiaon of Quantum Gravity Theory! Is String Phenomenology an Oxymoron? Fernando Quevedo ICTP/Cambridge Northeastern November 2017 FQ, arXiv:1612.01569 (CUP to appear) Greatest `Recent’ Discoveries • FluctuaIons of Cosmic Microwave Background (1992) (Nobel Prize 2006) • Dark Energy (1998) (Nobel Prize 2011) • Higgs Discovery (2012) (Nobel Prize 2013) • Gravitaonal waves (2016) (Nobel Prize 2017) Fundamental Theories: Special RelaIvity and Quantum Mechanics Contents 1. Introduction and Conclusions 1 Contents 2. Review of CSRs in d dimensions 3 1. Introduction3. CSRs and Conclusions in String Theory 1 5 2. Review of CSRs in d dimensions 3 3. CSRs in String Theory 5 1. Introduction and Conclusions String theory, just like standard relativistic quantum field theories, has very few model independent 1. Introductionconsequences and Conclusions at low energies. In quantum field theory we can name the existence of anti-particles, the CPT theorem, the running of couplings in terms of the renormalisation group and the identity String theory, justof all like particles standard of relativistic the same quantum type. String field theories theory,, for has vacua very few with model non independent-compact dimensions, ‘predicts’ consequences atgravity low energies. and at In least quantum one neutral field theory scalar, we can the name dilaton,the antisymmetric existence of anti-particles,tensors of different ranks and the CPT theorem, the running of couplings in terms of the renormalisation group and the identity usually also charged matter, and supersymmetry (see for instance [1]). of all particles of the same type. String theory, for vacua with non-compact dimensions, ‘predicts’ By the same nature of a theory (string theory or otherwise) with a naturally large energy scale gravity and at least one neutral scalar, the dilaton, antisymmetric tensors of different ranks and usually also chargedto address matter, the and issue supersymmetry of quantum (see gravity, for instanc it ise [1]).very difficult to identify model independent low- By the sameenergy nature implications of a theory (string subject theory to experimental or otherwise) verification with a naturally which large can energy put to scale test the theory and not to address thejust issue particular of quantum models gravity, or scenarios. it is very difficult to identify model independent low- energy implicationsThe subject purpose to experimental of this note verification is to make which a simple can put but to general test the remark. theory and We not point out a low-energy just particularconsequence models or scenarios. of all string constructions, that is the absence of massless continuous spin representa- The purposetions of this (CSR) note of is the to Poincar´egroup make a simple but [2]. general This fact remark. has no We straightfor point outward a low-energy explanation within standard consequence ofparticle all string physics constructions, field theoretical that is the analysis absence ofand massless is consistent continuous with spin all representa- experiments so far since parti- tions (CSR) ofcles the Poincar´egroup fitting into those [2]. This representations fact has no straightfor have notward been explanation found in withinnature; standard see however [3] or a recent particle physics field theoretical analysis and is consistent with all experiments so far since parti- discussion of their phenomenology. cles fitting into those representations have not been found in nature; see however [3] or a recent One of the most elegant theoretical developments in particle physics, pioneered by Wigner, is discussion of their phenomenology. the description of one-particle states in terms of unitary representations of the four- dimensional One of the most elegant theoretical developments in particle physics, pioneered by Wigner, is the descriptionPoincar´egroup of one-particle states [2] (see in also terms [4]). of unitary representations of the four- dimensional Poincar´egroup [2]One-particle (see also [4]). states are classified according to the quantum numbers of the invariant Casimir µ µ µ µ µνρσ One-particleoperators states areC1 classified= P Pµ accordingand C2 = toW theW quantumµ with P numbeand rsW of= theϵ invariantPν Mρσ Casimirthe momentum and Pauli- µ µ µ µ µνρσ operators C1 =LjubanskyP Pµ and C vectors2 = W respectivelyWµ with P andand MWρσ=theϵ LorentzPν Mρσ generators.the momentumC1 and andC Pauli-2 label the representation Ljubansky vectorsin terms respectively of their and eigenvaluesMρσ the Lorentz that essentially generators. correspondC1 and C2 label to mass the representationm and spin J in a representation in terms of their eigenvalues that essentially correspond to mass m and spin J in a representation with fixed momentum pµ. with fixed momentum pµ. The representations differ according to whether C1 is positive, zero or negative. For massive The representations differ according to whether C1 is positive, zero or negative. For massive particles (C1 > 0) the remaining space-time quantum numbers come from the fact that the sta- particles (C1 > 0) the remaining space-timePoincaré Group: quantum numbers come from(SR and QM) the fact that the sta- 1 bilising or Little Group in four-dimensions is SO(3), the subgroupFor of the massless Poincar´egroup particles (C1 =0) leavingthe momentum can be written p =(E,0, 0,E)andthe bilising or Little Group in four-dimensions1 is SO(3), the subgroup of the Poincar´egroup Wigner 1939 leaving For massless particlesinvariant (C1 =0) a statethe in momentum its rest mass can be frame written describedp =(E,0 by, 0,E a)andthe four-momentcorrespondingum p =( Littlem, 0 Group, 0, 0). is not The only cor- the naively guessed SO(2) but actually the whole Euclidean invariant a state in its rest mass frame described by a four-momentum p =(m, 0, 0, 0). The cor- corresponding Little Group is not onlyMassive parIcles: the naively guessed(Lile group SO(3)) SO(2) but actually the whole Euclidean group in two dimensions E2 or ISO(2). This complicates matters since this group has infinite responding states are the different spin states of the multiplet. This defines a particle in terms of groupresponding in two dimensions states areE2 theor ISO different(2). This spin complicates states of the matters multiplet. since this This group defines has a infinite particledimensional in terms unitary of representations, known as continuous spin representations (CSR), that would 2 2 2 2 dimensionalquantum unitary numbers representations,quantumm, J; numberspµ,s knownwith asm,s = continuous J; pJ,µ,sJ with+1 spin, respresentations=,J andJ, pJ =+1 (CSR),m, . that,J wouldand pcorrespond= m . to a continuous spin-like label on the elementary particles, something that has not been | ⟩ | − −⟩ ··· − − ··· correspond to a continuous spin-likeTachyons? label on the elementary particles, something that has not been observed in nature. observed in nature. A standard way to proceed is to simply restrict to the finite dimensional representations that 1 For massless particles (C1 =0) the momentum can be written p =(E,0, 0,E)andthe A standard way to proceed isMassless parIcles: to simply restrict to the(Lile group E2) finite dimensional representations that correspond to those of SO(2). This defines helicity λ, as the good quantum number which is corresponding Little Group is not only the naively guessed SO(2) but actually the whole Euclidean quantised in half integers. Since the two Casimirs vanish in the reference frame defined by p = correspond to those of SO(2). This defines∞-dimensional representaIons (CSR): not observed ?? helicity λ,–1– as the good quantum number which is group in two dimensions E2 or ISO(2). This complicates matters–1– since this group has infinite 2 quantised in half integers. Since the two Casimirs vanish in the reference frame defined by p = (E,0, 0,E) all observed massless particles are then labelled only by pµ and λ: pµ, λ with λ = dimensional unitary representations, known as continuous spin representations (CSR), that would | ⟩ 2 (E,0, 0,E) all observed massless particles areRestricted Li^le group: O(2) in E2: then labelled only by pµ and λ: pµ, λ with λ = 0, 1/2, 1, . But there is no satisfactory explanation why to restrict only to representations of correspond to a continuous spin-like label on the elementary particles,| something⟩ that has± not been± ··· 0, 1/2, 1, . But there is no satisfactory explanation why to restrict only to representations of SO(2) instead of the full E2. Contrary to the massive case for which matter fields fit into generic ± ± ··· observed in nature. Theories for spins 0,1/2,1: Quantum Field Theories (QFT) representations of SO(3), there are massless representations (infinite dimensional) that are allowed SO(2) instead of the fullAE standard2. Contrary way to to proceed the massive is to simply case for restrict which to matter the finite fields dimension fit intoal gener representationsic that representations of SOcorrespond(3), there to are those massless of SO representations(2). This defines (infinite helicity dimensional)λ, as the good that quantum are allowed numberby the which basic is principles of special relativity and quantum mechanics but do not seem to be realised by the basic principlesquantised of special in half relativity Massless spins 3/2,2: (super) gravity: integers. and Since quantum the two mechanics Casimirs bu vanisht do notin the seemEffecIve Field Theories (EFT) refere tonce be realisedframe definedin nature. by p A= theoretical understanding of this fact is needed. 2 in nature. A theoretical(E,0, understanding0,E) all observed of this massless fact particlesis needed. are then labelled only by pµ and λ: pµ, λ withOverλ the= years the continuous spin representations have been discussed in several different | ⟩ 0, 1/2, 1, . But there is no satisfactory explanation why to restrict only to representationscontexts (see of for instance [6] and references therein) and attempts have been made to describe Over the years the± continuous± ··· spin representations have been discussed in several different contexts (see for instanceSO(2) instead [6] and of references the full E2 therein).
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