Nonlinear Hamiltonian analysis of new quadratic torsion theories Part I. Cases with curvature-free constraints
PPrepared for submission to Phys. Rev. D.
Nonlinear Hamiltonian analysis of new quadratic torsion theoriesI. Cases with curvature-free constraints
W.E.V. Barker,
1, 2, ∗ A.N. Lasenby,
1, 2, † M.P. Hobson, ‡ and W.J. Handley
1, 2, § Astrophysics Group, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK
It was recently found that, when linearised in the absence of matter, 58 cases of the generalgravitational theory with quadratic curvature and torsion are (i) free from ghosts and tachyons and(ii) power-counting renormalisable. We inspect the nonlinear Hamiltonian structure of the eightcases whose primary constraints do not depend on the curvature tensor. We confirm the particlespectra and unitarity of all these theories in the linear regime. We uncover qualitative dynamicalchanges in the nonlinear regimes of all eight cases, suggesting at least a broken gauge symmetry,and possibly the activation of negative kinetic energy spin-parity sectors and acausal behaviour.Two of the cases propagate a pair of massless modes at the linear level, and were interesting ascandidate theories of gravity. However, we identify these modes with vector excitations, rather thanthe tensor polarisations of the graviton. Moreover, we show that these theories do not support aviable cosmological background.
PACS numbers: 04.50.Kd, 04.60.-m, 04.20.Fy, 98.80.-k
I. INTRODUCTION
In light of both theoretical minimalism, and experimen-tal and observational verification, the preferred effectivetheory of gravity is that of Einstein and Hilbert L T = − m p2 R + L M . (1)The gravitational portion L G ≡ L T − L M of the total La-grangian L T is powered by the scalar part of the Riemanncurvature tensor R ≡ R µνµν , which is the de facto gravi-tational field strength and contains second derivatives ofthe metric gravitational potential R ∼ ∂ g + ( ∂g ) . Thematter Lagrangian L M is taken to be minimally coupled.Two approaches to generalising (1) have proven espe-cially popular1. The artificially imposed symmetry of the Levi–Civita connection could be relaxed.2. Higher-order geometric invariants could be addedto the Lagrangian.The first approach leads to a non-vanishing torsion T ijk ,and corresponding non-Riemann curvature R ijkl . TheRoman indices refer to a local Lorentz basis, mediatedby tetrads (vierbein), or equivalent translational gaugefields b iµ . The now independent spin connection maylikewise be cast as a rotational gauge field A ijµ . Interms of these new potentials, the gravitational fieldstrengths T ∼ ∂b + bA and R ∼ ∂A + A are closer tothe Yang–Mills form familiar from the strong and elec-troweak sectors of the standard model: they are linear ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] in first derivatives and the structure constants of thePoincar´e group. By demanding positive parity and free-dom from Ostrogradsky ghosts in combination with thesecond approach, one arrives at the general quadratic L G ∼ m p2 R + R + m p2 T theory L T = − α m p2 R + m p2 T ijk (cid:0) β T ijk + β T jik (cid:1) + β m p2 T i T i + α R + R ij (cid:0) α R ij + α R ji (cid:1) + R ijkl (cid:0) α R ijkl + α R ikjl + α R klij (cid:1) + L M , (2)where R ij ≡ R lilk , R ≡ R ll and T i ≡ T lil . Thesequadratic terms are added to the scalar curvature in-variant for a total of ten dimensionless couplings in thetheory. Note that no scalar invariant can be formed fromthe torsion.The theory (2) has been deeply studied over fourdecades. When linearised on a Minkowski back-ground, the theory is capable of propagating six mas-sive torsion modes ( rotons or tordions ) of spin-parity J P = 0 ± , ± , ± , in addition to the 2 + mode of the mass-less graviton [1]. Following early studies by Neville [2, 3],Sezgin and van Nieuwenhuizen found 12 cases of the the-ory whose propagator poles have positive residues andreal masses, i.e. unitary theories [4, 5]. An exhaustivesurvey by Lin, Hobson, and Lasenby [6, 7] recently foundthat there are 450 unitary cases in total. Of these, 58 arealso power-counting renormalisable (PCR), such that thegraviton and roton propagators tend to p − and p − re-spectively in the ultraviolet limit . Note that Ostrogradsky’s theorem forbids all terms quadratic inthe second-order Riemann tensor except for the Gauss–Bonnetterm; this does not apply to the first-order Riemann–Cartan ortorsion tensors. With the exception of Case *1
9, Case *3
10, Case *4
11 and a r X i v : . [ g r- q c ] J a n Perhaps surprisingly, the restriction from unitary toPCR-unitary cases of (2) universally switches off theEinstein–Hilbert term α = 0 . (3)In the context of (2), this term in isolation constitutesthe Einstein–Cartan theory of gravity. Einstein–Cartantheory is dynamically equivalent to GR when the spintensor of the matter sector vanishes; the linearised the-ory contains only the 2 + graviton. Usually, the quadratic R + m p2 T terms are viewed as corrections to thescalar m p2 R , which may be motivated by analogy toEinstein’s theory at one loop, L G ∼ m p2 R + R . A pure L G ∼ R + m p2 T theory is hard to reconcile with thispicture, and necessitates a great deal of work since onecannot appeal to the viable Einstein–Cartan limit at lowenergies. In some sense, a purely quadratic gravity is ac-tually quite natural. Einstein gravity amounts to a R , gauge theory of diffeomorphisms, whose gauge potentialis g µν . However, the fields b iµ and A ijµ additionallygauge rotations and, by extension, the whole Poincar´egroup R , (cid:111) SO + (1 , quadraic, parity-preserving Poincar´e gaugetheory (PGT q+ ) of gravity, as pioneered by Kibble [8],Utiyama [9], Sciama [10] and others . In this context, aLagrangian quadratic in Yang–Mills field strengths wouldmake an appealing addition to the standard model –should it prove viable in the nonlinear regime.The purpose of this series is to test the nonlinear viabil-ity of the 58 novel theories by probing their Hamiltonianstructure. As a higher-spin gauge theory, the PGT q+ (2)is always singular : this degeneracy of the kinetic Hessiangreatly complicates the Lagrangian analysis, incentivis-ing the Hamiltonian approach. By implementing the al-gorithm of Dirac and Bergmann, we are guaranteed toobtain all propagating degrees of freedom (D.o.F), alongwith all constraints [14]. In the linearised theory, this isespecially easy, and allows us to verify the particle spec-tra and unitarity of the cases obtained in [6, 7]. In thenonlinear case, the algorithm allows us to flag potentiallyfatal pathologies which develop under significant depar-tures from Minkowski spacetime – if this spacetime istaken to be a vacuum, then the nonlinear regime is equiv-alent to that of strong fields. In particular, we rely onthe simple ‘health indicator’ of modified gravity set outby Chen, Nester and Yo: the number and type of con-straints should not change in passing from the linear tononlinear regimes [15, 16]. The motivation for this crite-rion is twofold. Generically, a decrease in the number ofconstraints involves the activation of potentially ghostly Case *2
13 as labelled in [7], J P sectors propagate which vio-late these rules. However, these ‘bad’ modes are understood todecouple at high energies without producing divergent loops [7]. For an excellent series on the dynamical structure of PGT q+ from the Lagrangian perspective, see [1, 11–13]. fields [16]. Moreover, it may be that the nonlinear con-straint structure is itself field dependent: this is thoughtto be associated with the propagation of acausal degreesof freedom [15]. Neither of these qualities is necessarilyfatal unless shown to incur a physical ghostly or acausalD.o.F, but for the purposes of this particular study wewill take the avoidance of them as being desirable.In this paper we will test Case 3, Case 17, Case 20,Case 24, Case *5
25, Case *6
26, Case 28 and Case 32, us-ing the numbering of [7], with the numbering of casespreviously discovered in [6] indicated by (*). These eightcases are most conducive to the Hamiltonian analysis.Specifically, these are the only cases whose primary con-straints are not functions of the curvature. To our knowl-edge, this practical restriction does no more than to easethe evaluation of commutators. We therefore tentativelyview the eight cases to be an representative sample of the58 novel theories .All eight cases fail the prescribed strong-field tests. Insome sense, they do so more dramatically than those‘minimal’ cases of PGT q+ which were previously tested,due to the vanishing of mass parameters [16]. Based onthese results, we find no evidence that the simultaneousimposition of the weak-field PCR and unitarity criteriaremedy the questionable health of PGT q+ in the strong-field regime, as observed in [15–17]. If these findings turnout to be general, it would seem more efficient to performfuture surveys of PGT q+ in the strong-field regime fromthe outset .We are also able to rule the cases out on cosmologicalgrounds, using the scalar-tensor analogue theory whichreplicates the background cosmology of the general ten-parameter PGT q+ [18]. Out of the eight cases, onlyCase 3 and Case 17 propagate massless modes consistentwith long-range gravitational forces, yet their nonlinearcosmological equations are non-dynamical. However, wedo show that these cases are the degenerate limit of anotherwise viable and interesting class of torsion theoriesobtained by imposing two very simple constraints on thecouplings of (2), whose background cosmology perfectlyreplicates that of Einstein’s torsion-free gravity (1), con-formally coupled to a scalar inflaton ξL T = − m p2 R + 112 ξ R + X ξξ − m ξ ξ + L M . (4)Here, the inflaton has kinetic term X ξξ ≡ ∇ µ ξ ∇ µ ξ andmass m ξ . The cosmology resulting from (4) is not scale-invariant due to the mass term, which is fortunate forminimal coupling to cosmological matter. However, itis an interesting surprise that the non-minimal couplingshould be exactly scale-invariant. The failure of Case 3 We mention that none of the eight cases are PCR in the conven-tional sense of [4], i.e. all of them feature a J P propagator whosemomentum power is non-PCR in the IR, and which decouples inthe UV. TABLE I. Nonstandard abbreviations.PGT q+ quadratic, parity-preserving Poincar´egauge theoryPCR power-counting renormalisableD.o.F degrees of freedomPPM primary Poisson matrix(P/S/T)iC (primary/secondary/tertiary) if-constraint(F/S)C (first/second)-classi(P/S/T)(F/S)C (P/S/T)iC which is (F/S)C on the finalshells(P/S)FC sure (primary/secondary) constraint, al-ways FC and Case 17 certainly is not a necessary consequenceof the linearised unitarity and power-counting. Indeed,one of the 58 cases has an excellent cosmological back-ground [18, 19], though an analysis of its Hamiltonianstructure is deferred to the companion paper, since itsprimary constraints depend on curvature.Despite our concerns about the strong-field regime, weare able to confirm the weak-field unitarity of all eightcases. We also obtain linearised dynamics which areconsistent with the particle spectra found in [6, 7]. Wealso offer tighter bounds on the massless particle spectra,identifying the massless modes of Case 3 and Case 17 as vector excitations, rather than the expected tensor po-larisations of the graviton.The remainder of this paper is set out as follows. InSection II we develop the Hamiltonian formulation ofthe ten-parameter theory (2). In Sections III and IVwe apply the Dirac–Bergmann algorithm to each of thelinearised cases, and compare with the constraint struc-ture of the nonlinear theories. In Section V we use effi-cient methods to show that even the cases with masslessmodes cannot support any Friedmann-like cosmologicalequation. Conclusions follow in Section VI. Followingthe conventions of [20] we will use Roman and Greek in-dices from the middle of the alphabet i , j . . . µ , ν . . . torefer to general Lorentz and coordinate indices runningfrom 0 to three, while a , b . . . and α , β . . . strictly runfrom one to three. We use the ‘West Coast’ signature(+ , − , − , − ). Our potentially nonstandard acronyms aredetailed in Table I. II. CONSTRAINED HAMILTONIANA. Gauge theory formulation
Recall that in Einstein’s theory, the covariant derivativeacting on a vector V µ is simply ∇ ν V µ ≡ ∂ ν V µ + Γ µλν V λ , (5)where Γ µνλ ≡ g µσ ( ∂ ν g λσ + ∂ λ g νσ − ∂ σ g νλ ) is the Levi–Civita connection with respect to some metric, itselfdefined by tangent vectors of the coordinate functions g µν ≡ e µ · e ν , on a curved manifold M . The Riemanncurvature tensor is then given by R ναβµ ≡ ∂ [ β Γ να ] µ + Γ λ [ α | µ Γ ν | β ] λ ) . (6)This geometric interpretation of gravity is not strictlynecessary. In generalising to theories with both curva-ture and torsion, we adopt the setup more familiar fromthe standard model, where the underlying manifold isalways flat Minkowski space ˇ M . The metric for gener-ally curvillinear coordinates is then γ µν ≡ e µ · e ν , andwe note that this metric is strictly flat in the sense of(6). Besides the coordinate basis, we define a Lorentzbasis whose inner product always returns the Minkowskimetric components η ij ≡ ˆ e i · ˆ e j . Under this condition,the Lorentz basis is completely free to rotate under theproper, orthochronous Lorentz rotations at each point inˇ M , and is not presumed to follow from any particular co-ordinates (i.e. it is non-holonomic). A vector V i referredto this basis has a covariant derivative D j V i ≡ h µj ( ∂ µ V i + A ikµ V k ) , (7)in which the (inverse) translational gauge field h µi and ro-tational gauge field A ijµ ≡ A [ ij ] µ are introduced to main-tain invariance under general coordinate transformationson ˇ M (i.e. passively interpreted diffeomorphisms) androtations of the Lorentz basis. In this way, the Poincar´egroup is gauged. The inverse translational gauge fieldsatisfies b iµ h νi ≡ δ νµ and b iµ h µj ≡ δ ij . The metric compo-nents in the curved space of Einstein’s theory (such as theflat cosmological metric we will consider in Section V) canbe recovered using g µν ≡ η ij b iµ b jν and g µν ≡ η ij h µi h νj .The gauge invariant measures on M and ˇ M are respec-tively √− g , where g ≡ det g µν , and b ≡ h − ≡ det b iµ .Two field strength tensors are motivated by commutingthe derivative (7) R ijkl ≡ h µk h νl (cid:0) ∂ [ µ A ijν ] + A im [ µ A mjβ ] (cid:1) , (8a) T ikl ≡ h µk h νl (cid:0) ∂ [ µ b iν ] + A im [ µ b mν ] (cid:1) . (8b)These are the Riemann–Cartan curvature and torsiontensors, but are not understood to imbue ˇ M with anygeometry. While the ‘particle physics’ picture is consis-tent with our earlier treatments on this topic [18, 19, 21],we concede that it is far more usual to treat torsion andRiemann–Cartan curvature as complimentary geometricqualities. In the geometric picture, the Riemann space-time M is generalised to a Riemann–Cartan spacetime,rather than specialised to a Minkowski spacetime. Theinterpretations are dynamically indistinguishable, andtranslations between the two are provided in [20, 21]. B. Primary constraints and
In order to transition to the constrained Hamiltonianpicture [14, 20, 22], we first define the canonical momenta
TABLE II. From the 58 unitary, power-counting renormalisable cases of (2), we consider the eight cases whose primaryconstraints do not depend on the Riemann–Cartan curvature. The numbering follows [7], with original numbering of casesfirst identified in [6] indicated by (*). The definitive constraints on the couplings are given, along with extra conditions whichfix the unitarity, separated by a caret ( ∧ ). The methods of [6, 7] provide partial information about the particle spectrum.Propagators may be of A ijµ (blue), or the antisymmetric (green) or symmetric (red) parts of b iµ . Propagator poles are massless(empty circles) or massive (filled circles). Within each J P sector, A ijµ and b iµ modes may be coupled (multiple colours) ortransmuted by gauge transformations (multiple circles). Ultimately, the propagator spectrum is only indicative of the particlespectrum: the actual number of propagating degrees of freedom are shown in the final column, but their field character or J P is indeterminate in the massless case, because poles from multiple J P sectors coincide at the origin of momentum-space. − + − + − + D.o.FCase 3 ˆ α = ˆ α = ˆ α = ˆ α = ˆ β = ˆ β + 2 ˆ β = 0 ˆ α < ∧ ˆ α < ∧ ˆ β < α = ˆ α = ˆ α = ˆ α = ˆ α = ˆ β = ˆ β + 2 ˆ β = 0 ˆ α < α = ˆ α = ˆ α = ˆ α = ˆ α = 0 0 < ˆ β ∧ ˆ α < α = ˆ α = ˆ α = ˆ α = ˆ β = 0 0 < ˆ β ∧ ˆ α < *5
25 ˆ α = ˆ α = ˆ α = ˆ α = ˆ α = ˆ β = 0 0 < ˆ β ∧ ˆ α < *6
26 ˆ α = ˆ α = ˆ α = ˆ α = ˆ α = ˆ β = ˆ β = 0 0 < ˆ β ∧ ˆ α < α = ˆ α = ˆ α = ˆ α = ˆ β = ˆ β = 0 0 < ˆ β ∧ ˆ α < α = ˆ α = ˆ α = ˆ α = ˆ α = ˆ β = 0 0 < ˆ β ∧ ˆ α < as follows π µi ≡ ∂bL G ∂ ( ∂ b iµ ) , π µij ≡ ∂bL G ∂ ( ∂ A ijµ ) . (9)Following [6, 7], we will consider only the gravitationalpart of the Lagrangian, i.e. without any matter L M .Since the field strengths (8a) and (8b) from which (2)is constructed make no reference to the velocities of b k and A ij , the definitions (9) incur 10 primary constraints ϕ k ≡ π k ≈ , ϕ ij ≡ π ij ≈ , (10)so that the conjugate fields b k and A ij are non-physical.Notice that the weak equality is denoted by ( ≈ ). Theconstraints (10) are a consequence of Poincar´e gaugesymmetry; their presence is independent of the couplings,and they are first class (FC). We refer to them as ‘sure’primary, first class (sPFC) constraints. In order to sys-tematically isolate the ‘sure’ non-physical fields, we intro-duce the 3+1 (ADM) splitting of the spacetime, in whicha spacelike foliation is characterised by timelike unit nor-mal n k . Any vector which refers to the local Lorentz ba-sis may be split into components V i = V ⊥ n i + V i whichare respectively perpendicular and parallel to the folia-tion: parallel indices are always denoted with an overbar.In what follows, it is very useful to note the identities b kα h αl = δ kl and b kα h βk = δ βα . The lapse function and shift vector are defined with reference to the non-physicalpart of the translational gauge field using this normal N ≡ n k b k , N α ≡ h αk b k . (11)The remaining momenta can be expressed in the ‘paral-lel’ form ˆ π ki ≡ π αi b kα and ˆ π kij ≡ π αij b kα . In order to reveal the Hamiltonian structure of the theory as natu-rally as possible, the Lagrangian in (2) is best written inthe irreducible form L T = − α m p2 R + (cid:88) I =1 ˆ α I R ijkl P I kl pqij nm R nmpq + m p2 3 (cid:88) I =1 ˆ β I T ijk P I jk nmi l T lnm + L M . (12)where the nine operators P I ...... project out all the ir-reducible representations of SO(1 , b k and A ij (which arenon-physical) and the velocities of all fields (which arenon-canonical) T ikl = T ikl + 2 n [ k T i ⊥ l ] , (13a) R ijkl = R ijkl + 2 n [ k R ij ⊥ l ] , (13b)where such variables are confined to the second termin each case. We are concerned with theories of thequadratic L G ∼ R + m p2 T form (i.e. α = 0), un-der the source-free condition L M = 0. Substituting (12)into (9) and using Eqs. (13a) and (13b), we find that theparallel momenta can be neatly expressed as functions ofthe field strengthsˆ π ki J = ∂L T ∂ T i ⊥ k = 4 m p2 3 (cid:88) I =1 ˆ β I P I ⊥ k mli n T nml , (14a)ˆ π kij J = ∂L T ∂ R ij ⊥ k = 8 (cid:88) I =1 ˆ α I P I ⊥ k pqij mn R mnpq , (14b)where the measure J = b/N on the foliation is strictlyphysical, since b k is divided out by N .Writing the parallel momenta in this form facilitates theidentification of further primary constraints. Beginningwith (14a), we find that the 12 translational parallel mo-menta decompose into four irreducible representations ofO(3). Using the spin-parity notation of [16, 17] we writethese as ˆ π kl = ˆ π kl + n k ˆ π ⊥ l , (15a)ˆ π kl = 13 η kl ˆ π + ∧ ˆ π kl + ∼ ˆ π kl . (15b)In this expansion we identify the 0 + scalar ˆ π , theantisymmetric 1 + vector ∧ ˆ π kl , the 1 − vector ˆ π ⊥ k andsymmetric-traceless 2 + tensor ∼ ˆ π kl . Applying this decom-position to (14a) as a whole, we obtain four functionswhich, with the aid of (13a), are simultaneously definedboth in terms of canonical and non-canonical variables ϕ ≡ J − ˆ π = ˆ β m p2 η kl T kl ⊥ , (16a) ∧ ϕ kl ≡ J − ∧ ˆ π kl −
43 ( ˆ β − ˆ β ) m p2 T ⊥ kl = 13 ( ˆ β + 2 ˆ β ) m p2 T [ kl ] ⊥ , (16b) ϕ ⊥ k ≡ J − ˆ π ⊥ k −
43 ( ˆ β − ˆ β ) m p2 (cid:42) T k = 13 (2 ˆ β + ˆ β ) m p2 T ⊥ k ⊥ , (16c) ∼ ϕ kl ≡ J − ∼ ˆ π kl = ˆ β m p2 T (cid:104) kl (cid:105)⊥ , (16d)where the vector and symmetric-traceless torsion are (cid:42) T k ≡ T iki , T (cid:104) kl (cid:105)⊥ ≡ T ( kl ) ⊥ − η kl η ij T ij ⊥ . (17)In each case, if the combination of coupling constantsappearing in the non-canonical RHS definition vanishes,the canonically defined function on the LHS becomes aprimary ‘if’ constraint (PiC). An analogous constructionis available for (14b), with the 18 remaining momentadecomposing as followsˆ π klm = ˆ π klm + 2 n [ k ˆ π ⊥ l ] m , (18a)ˆ π ⊥ kl = 13 η kl ˆ π ⊥ + ∧ ˆ π ⊥ kl + ∼ ˆ π ⊥ kl , (18b)ˆ π klm = 16 (cid:15) klm ⊥ ˆ π P + (cid:42) ˆ π [ k η l ] m + 43 ˆ π T klm . (18c) These are the 0 + scalar ˆ π ⊥ , antisymmetric 1 + vector ∧ ˆ π ⊥ kl , symmetric 2 + tensor ∼ ˆ π ⊥ kl , and then the 0 − pseu-doscalar ˆ π P , the 1 − vector (cid:42) ˆ π k and 2 − tensor ˆ π T klm . Wewill use ( · P ) to refer to the pseudoscalar part of generaltensors, and ( · T klm ) to refer to the tensor part (withantisymmetry implicit in the first pair of indices, even if · klm ≡ · k [ lm ] ). The six PiC functions from (14b) arethen ϕ ⊥ ≡ J − ˆ π ⊥ + 2 (ˆ α − ˆ α ) R = (ˆ α + ˆ α ) R ⊥⊥ , (19a) ϕ P ≡ J − ˆ π P + 4 (ˆ α − ˆ α ) R P ⊥◦ = (ˆ α + ˆ α ) R P ◦⊥ , (19b) ∧ ϕ ⊥ kl ≡ J − ∧ ˆ π ⊥ kl + 4 (ˆ α − ˆ α ) R [ kl ] = (ˆ α + ˆ α ) R ⊥ [ kl ] ⊥ , (19c) (cid:42) ϕ k ≡ J − (cid:42) ˆ π k + 4 (ˆ α − ˆ α ) R ⊥ k = (ˆ α + ˆ α ) R k ⊥ , (19d) ∼ ϕ ⊥ kl ≡ J − ∼ ˆ π ⊥ kl + 4 (ˆ α − ˆ α ) R (cid:104) kl (cid:105) = (ˆ α + ˆ α ) R ⊥(cid:104) kl (cid:105)⊥ , (19e) ϕ T klm ≡ J − ˆ π T klm − α − ˆ α ) R T ⊥ klm = (ˆ α + ˆ α ) R T klm ⊥ , (19f)where we make further notational definitions R P ⊥◦ ≡ (cid:15) ijk ⊥ R ijk ⊥ , R P ◦⊥ ≡ (cid:15) ijk ⊥ R ⊥ ijk , R kl ≡ R ikli , R ≡ R ii . (20)By this analysis, the possible occurence of primary con-straints is systematically exhausted. C. Secondary constraints and the Hamiltonian
In order to be consistent, a primary constraint shouldnot have any velocity within the final mass shell, so itscommutator with the total
Hamiltonian should weaklyvanish ˙ ϕ ( x ) ≡ (cid:90) d x (cid:110) ϕ ( x ) , H T ( x ) (cid:111) ≈ . (21)The total Hamiltonian is related to the canonical Hamil-tonian, the Legendre-transformed Lagrangian, by theconstraints and their multiplier fields H T ≡ H C + u k ϕ k + 12 u ij ϕ ij + ( u · ϕ ) , (22)where the last term schematically represents any PiCswhich may arise. The canonical Hamiltonian may gen-erally be collected into the insightful Dirac form [23, 24] H C ≡ N H ⊥ + N α H α − A ij H ij + ∂ α D α , (23)i.e. as a linear function of the non-physical fields up to asurface term. The remaining functions which appear in(23) are defined as follows H ⊥ ≡ ˆ π ki T i ⊥ k + 12 ˆ π kij R ij ⊥ k − JL − n k D α π αk , (24a) H α ≡ π βi T iαβ + 12 π βij R ijαβ − b kα D β π βk , (24b) H ij ≡ π α [ i b j ] α − D α π αij , (24c) D α ≡ b i π αi + 12 A ij π αij . (24d)It is clear from (21) and the Dirac form (23) that theconsistency of (10) invokes 10 ‘sure’ secondary first class (sSFC) constriants H ⊥ ≈ , H α ≈ , H ij ≈ . (25)It is important to note that while the sSFCs in (25) arealways enforced, it does not always follow that 2 × H α and H ij are kinematic generators which do not impinge on thedynamics. Thus, in the evaluation of (21), it is sufficientto work purely with the super-Hamiltonian H ⊥ , which is,at length, expanded out using Eqs. (12), (13a) and (13b)to give H ⊥ = m p2 J (cid:88) I =1 ˆ β I (cid:104) T i ⊥ k P I ⊥ k ⊥ li j T j ⊥ l − T imk P I mk nli j T jnl (cid:105) + J (cid:88) I =1 ˆ α I (cid:104) R ip ⊥ k P I ⊥ k ⊥ lip jq R jq ⊥ l − R ipmk P I mk nlip jq R jqnl (cid:105) − n k D α π αk = J (cid:34) ϕ β m p2 + 6 ∧ ϕ kl ∧ ϕ kl ( ˆ β + 2 ˆ β ) m p2 + 6 ϕ ⊥ k ϕ ⊥ k (2 ˆ β + ˆ β ) m p2 + 2 ∼ ϕ kl ∼ ϕ kl ˆ β m p2 + 2 ϕ ⊥ α + ˆ α ) + ϕ P 2 α + ˆ α ) + 2 ∧ ϕ ⊥ kl ∧ ϕ ⊥ kl ˆ α + ˆ α + (cid:42) ϕ k(cid:42) ϕ k ˆ α + ˆ α + 2 ∼ ϕ ⊥ kl ∼ ϕ ⊥ kl ˆ α + ˆ α + 16 ϕ T klm ϕ T klm α + ˆ α ) (cid:35) + J (cid:20)
13 (2 ˆ β + ˆ β ) m p2 T ⊥ kl T ⊥ kl + 13 ( ˆ β + 2 ˆ β ) m p2 (cid:42) T k(cid:42) T k −
16 ˆ β m p2 T P 2 + 169 ˆ β m p2 T T klm T T klm + 16 (ˆ α + ˆ α ) R −
16 (ˆ α + ˆ α ) R P ⊥◦ + 2(ˆ α + ˆ α ) R [ kl ] R [ kl ] + (ˆ α + ˆ α ) R ⊥ k R ⊥ k + 2(ˆ α + ˆ α ) R (cid:104) kl (cid:105) R (cid:104) kl (cid:105) + 169 (ˆ α + ˆ α ) R T ⊥ klm R T ⊥ klm (cid:21) − n k D α π αk . (26)To arrive at the second equality in (26), the non-canonical‘perpendicular’ field strengths appearing in the firstequality are canonicalised at length by the dual PiC def-initions in Eqs. (16a) to (16d) and Eqs. (19a) to (19f),resulting in terms quadratic in the PiC functions, andin the canonical ‘parallel’ field strengths. The result-ing expression is quite lengthy, but can be simplified forany given theory by safely eliminting those PiC functionswhich become constraints. The signs of the remainingquadratic PiC terms are then instrumental in the identi-fication of unconstrained ghosts, since the PiC functionsare schematically of the form ϕ ∼ π + R or ϕ ∼ π + T .The consistency of the PiCs is less straightforward.Generally, the PiCs may be FC or SC within their ownmass shell. In the case that a PiC is FC, (21) provides asecondary if-constraint (SiC). Possibly, the PiC and SiCdo not commute; in that case both become SC within the new mass shell and the consistency of the SiC allows amultiplier to be determined˙ χ ( x ) ≡ (cid:90) d x (cid:16) N (cid:110) χ ( x ) , H ⊥ ( x ) (cid:111) + u · (cid:110) χ ( x ) , ϕ ( x ) (cid:111)(cid:17) ≈ . (27)Otherwise, a tertiary if-constraint (TiC) may be found,and the process continues until the contraint chain fromthe PiC is absorbed by another chain, or by the sSFCs.In the case that a PiC is already SC within the PiC massshell, its chain terminates immediately and two multipli-ers are determined. We note that occasionally, a con-straint may be encountered at some deep level whichretroactively terminates the chain at a shallower point.Only once the algoritm has terminated is it safe to cate-gorise the if-constraints as FC or SC. TABLE III. Spin-parity sectors and associated PiCs, alongwith their kinetic and mass parameters. For completeness,we include the m p2 R term, mediated by α . J P PiC Kinetic parameter Mass parameter0 + ϕ ˆ β α (2 α + ˆ β ) ϕ ⊥ ˆ α + ˆ α − ϕ P ˆ α + ˆ α α + 2 ˆ β + ∧ ϕ kl ˆ β + 2 ˆ β ( α + 2 ˆ β )( α − ˆ β ) ∧ ϕ ⊥ kl ˆ α + ˆ α − ϕ ⊥ k β + ˆ β (2 α + ˆ β )( α − ˆ β ) (cid:42) ϕ k ˆ α + ˆ α + ∼ ϕ kl ˆ β α ( α − ˆ β ) ∼ ϕ ⊥ kl ˆ α + ˆ α − ϕ T klm ˆ α + ˆ α α − ˆ β In the linearised theory [12], the analysis is greatly sim-plified by an understanding of the mass spectrum [1].Only the O (1) parts of the PiC commutators contributeto the evaluation of the multipliers. Such commutatorsare possible only between pairs of PiCs which belong tothe same O(3) irrep, and which are known as conjugatepairs [25]. Conjugate PiCs will fail to commute in thelinear theory only when their common mass parameteris non-vanishing. In this case, if only one PiC in a pairis present, it will still fail to commute with the SiC thatmaintains its consistency. Particularly, the rotational ϕ P and ϕ T klm have no O(3) counterparts in the translationalsector, and are conjugate with their secondaries χ P and χ T klm a priori. In the case of vanishing mass parameters,the PiCs are FC, and a new gauge symmetry is invoked.The PiCs belonging to various O(3) irreps, along withtheir kinetic parameters and linearised mass parametersare listed in Table III.Note that up to this point, our discussion has been fullygeneral, and lays the theoretical foundations and conven-tions for the forthcoming series. The evaluation of Pois-son brackets is made tedious by the dependence of variousquantities on the translational gauge field, as illustratedby the following useful identities ∂n l ∂b kµ ≡ − n k h µl , ∂h νl ∂b kµ ≡ − h νk h µl ,∂b∂b kν ≡ bh νk , ∂J∂b kν ≡ Jh νk , ∂N∂b kν ≡ N n k h ν ⊥ . (28)As a crude measure to simplify the calculations, we ar-tificially restrict our analysis in this paper to theorieswhose PiCs among Eqs. (19a) to (19f) do not dependon R ijkl . It must be emphasised that this does not (toour knowledge) translate into any useful restriction onthe physics. Of the 58 novel theories in [6, 7], eight sat-isfy our criterion: Case 3, Case 17, Case 20, Case 24,Case *5
25, Case *6
26, Case 28 and Case 32. For mostof these cases, we are fortunate that the remaining PiCs among Eqs. (16a) to (16d) also do not depend on T ikl .Case 3 and Case 17 are exceptions to this rule. We detailin Table II our prior understanding of these theories, asencoded by the saturated graviton and roton propaga-tors, linearised on Minkowski spacetime in the absenceof matter. Aside from having torsion-dependent PiCs,Case 3 and Case 17 are particularly interesting as can-didate theories of gravity, as they contain two masslessD.o.F with power in the 2 + part of the propagator – wewill return to this point in Section IV.From our discussion in Section II C, we see that the con-straint structure of the theory depends partially on thecommutators between the PiCs, which form the primaryPoisson matrix (PPM). In Sections III and IV we willuse the structure of the nonlinear PPM as a proxy forthe health of each theory. III. MASSIVE-ONLY RESULTSA. Case *6 Conveniently, the PiCs of the massive theories dependon neither T ijk nor R ijkl , so we will have schematically ϕ ∼ π for both translational and rotational sectors. Bysubstituting the definition of Case *6
26 from Table II into(26) and (22), the total Hamiltonian is seen to take theform H T = b (cid:32) ∧ ϕ kl ∧ ϕ kl ˆ β − ϕ P 2 ˆ α (cid:33) + fields , (29)where we include only the part quadratic in the momenta.The remaining eight PiC functions that do not appearin (29) are primarily constrained, and give rise to thefollowing nonvanishing commutators within the PiC shell (cid:110) ϕ ⊥ i , ϕ ⊥ l (cid:111) ≈ J ∧ ˆ π il δ , (30a) (cid:110) ϕ ⊥ i , ∧ ϕ ⊥ lm (cid:111) ≈ − J (cid:15) ilm ⊥ ˆ π P δ , (30b) (cid:110) ∼ ϕ ij , ∼ ϕ lm (cid:111) ≈ J (cid:104) η i ( l | ∧ ˆ π j | m ) + η j ( l | ∧ ˆ π i | m ) (cid:105) δ , (30c) (cid:110) ∼ ϕ ij , ϕ T lmn (cid:111) ≈ J (cid:104) η ( i | n (cid:15) | j ) lm ⊥ − η ( i | l (cid:15) | j ) mn ⊥ + η ( i | m (cid:15) | j ) ln ⊥ (cid:105) ˆ π P δ , (30d)where δ represents the equal-time Dirac function. Thenonlinear PPM of Case *6
26 is then written: ϕ ϕ ⊥ k ∼ ϕ kl ϕ ⊥ ∧ ϕ ⊥ kl (cid:42) ϕ k ∼ ϕ ⊥ kl ϕ T klm ϕ · · · · · · · · ϕ ⊥ k · ˆ π · · ˆ π ! · · · ∼ ϕ kl · · ˆ π · · · · ˆ π ! ϕ ⊥ · · · · · · · · ∧ ϕ ⊥ kl · ˆ π ! · · · · · · (cid:42) ϕ k · · · · · · · · ∼ ϕ ⊥ kl · · · · · · · · ϕ T klm · · ˆ π ! · · · · ·
51 3 5 1 3 3 5 5 (31)The elements of the matrix schematically represent thenonlinear Poisson brackets between the PiCs. The PiCsare labelled, along with their multiplicities, at the edgesof the PPM. They are arranged so as to divide the ma-trix into translational and rotational blocks, separated by( ). All brackets are restricted to the PiC shell. Com-muting PiCs are dentoted by ( · ). Non-commuting PiCsdenoted as (ˆ π ) are strictly linear combinations of ˆ π kij and ˆ π ki as detailed in Eqs. (30a) to (30d). Generally,these expressions can be quite lengthy, so henceforth weconfine them to Appendix C. Commutators dependingon momenta which (as we shall shortly show) propagatein the final linear theory are denoted by (ˆ π !). These aresignificant as they are presumed to persist even when thefull nonlinear Dirac–Bergmann algorithm is terminated,except possibly on any strongly coupled spacetimes whichmight be found away from Minkowski spacetime. Con-stant terms only arise in brackets between conjugate PiCs( ), and then only if both PiCs have non-vanishing massparameters. Since all the PiC mass parameters vanish inCase *6
26, no constant terms can arise. The linearisedtheory is sensitive only to these constant terms, but wesee from (31) that the conjugate PiCs also commute inthe nonlinear Case *6 χ (cid:91) ⊥ k ≈ − η (cid:91)ml D (cid:91)m ∧ ˆ π (cid:91)kl , (32a) ∧ χ (cid:91) ⊥ kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) , (32b)where quantities linearised on the Minkowski back-ground are denoted with ( (cid:91) ). Also within this shell,we find H (cid:91) ⊥ and H (cid:91) ⊥ k already vanish weakly, while thelinear super-momentum and vector part of the rotational Note also that we use the linearised gauge covariant derivative D (cid:91)i , even to replace the nonlinear coordinate derivative h (cid:91) µi ∂ µ . super-momentum give further sSFCs H (cid:91)α ≈ − h (cid:91) jα η (cid:91)kl D (cid:91)k ∧ ˆ π (cid:91)jl , (33a) H (cid:91)kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) . (33b)The SiCs clearly vanish in the sSFC sub-shell, terminat-ing the algorithm immediately. We find that H (cid:91)α is al-ready implied by H (cid:91)kl , which constitutes a total of threesSFCs. The PiCs are all FC, and the total number ofiPFCs can be read off from (31). Recalling also the 10sPFCs, and counting all FCs twice, we find that there isonly one propagating D.o.F, as expected from Table II1 = 12 (cid:0) − × − × − × (1 + 3 + 5 + 1 + 3 + 3 + 5 + 5)[iPFC] (cid:1) . (34)So, what is this D.o.F? We know that there are 26 unde-termined multipliers, to match each of the iPFCs. Gener-ically, this makes it very difficult to make sense of theequations of motion. However, we can make an educatedguess by noticing that the functions ∧ ϕ kl and ϕ P are notPiCs and in the end, it turns out to be the 0 − tordionwhich is propagating. An application of (21) allows us tofind the velocity of the pseudoscalar part of the torsion˙ T P (cid:91) ≈ − α ˆ π P (cid:91) −
34 ˆ β m p2 (cid:15) (cid:91)jkl ⊥ D (cid:91)j ∧ ˆ π (cid:91)kl . (35)Conveniently, we see that this quantity makes no refer-ence to undetermined multipliers in the final shell. More-over, the same can be said of the acceleration¨ T P (cid:91) ≈ − η (cid:91)jk D (cid:91)j D (cid:91)k T P (cid:91) − β ˆ α m p2 T P (cid:91) , (36)which clearly describes a particle of mass m ≡ (cid:115) | ˆ β || ˆ α | m p , (37)if ˆ β / ˆ α >
0. The unitarity conditions in Table II cannow be decoded. The condition ˆ α < − ghost by inspection of (29), whereas ˆ β < − from becoming tachyonic.In the nonlinear theory, the PPM is no longer empty asshown in (31). We anticipate that the emergent commu-tators will ultimately result in a fundamentally differentparticle spectrum. Particularly, we see from Eqs. (30c)and (30d) that ϕ ⊥ k , ∼ ϕ kl , ∧ ϕ ⊥ kl and ϕ T klm are all demotedfrom iPFCs to iPSCs so long as 0 − is activated. Possibly,0 − becomes strongly coupled on some other privilegedsurface within the final shell, but since the converse isunlikely to be true we conclude that an iPFC generally becomes an iPSC in the nonlinear theory. According toDirac’s conjecture, the FCs are associated with gaugesymmetries. More correctly, every PFC can be used toconstruct a nontrivial gauge generator using the Castel-lani algorithm [29]. We therefore expect that a generatoris generally broken.To see one way in which this might affect the outcome,imagine that none of the sSFCs are degenerate in the fullnonlinear theory, but that they still encode the iSFCs(which therefore need not appear in the final count). Thenonlinear theory would then be expected to propagatetwo D.o.F2 (e.g.) = 12 (cid:0) − × − × − × (1 + 1 + 3 + 5)[iPFC] − (3 + 5 + 3 + 5)[iPSC] (cid:1) , (38)suggesting that somehow one D.o.F from the 1 + sector(i.e. the only J P other than 0 − which is not primarilyconstrained), is generally activated, but becomes stronglycoupled on Minkowski spacetime. It is not clear whatthis would look like, and we emphasise that the spe-cific scenario in (38) is unlikely to be the one which isrealised. The full picture can only be revealed by per-forming the nonlinear Dirac–Bergmann analysis, begin-ning from (31). Following treatments of simpler cases ofPGT q+ in [16], we will not go this far. However, we thinkit likely that any activation of the 1 + sector will damagethe unitarity of the theory, since we see from (29) that ∧ ˆ π kl ∧ ˆ π kl has a negative contribution to the energy, by thesame condition ˆ β < − mode. For further discussion of the ‘positive energytest’, we direct the reader to Appendix B.Finally, (31) may also indicate that the nonlinear the-ory violates causality. We refer to the test based on thetachyonic shock in the nonlinear Proca theory [15], andwhich was also implemented in [16], whereby the PPMrank is required not to depend on the values of the fieldsand their momenta. The motivation for this requirementis as follows. It is easy to see from (21) that the multipli-ers u A and u B of a pair of PiCs ϕ A and ϕ B can be deter-mined in the case that { ϕ A , ϕ B } (cid:54)≈ u A will be nonvanishing if {H C , ϕ B } (cid:54)≈ { ϕ A , ϕ B } →
0. The multiplier u A had betternot have any physical interpretation in that case, since itwould diverge . Unfortunately in the case of PGT q+ , themultipliers can be written in terms of the non-canonicalvelocities through the dual definitions of the PiC func-tions in Eqs. (16a) to (16d) and Eqs. (19a) to (19f). The The problem is somewhat analogous to one of strong coupling. Ifthe prefactor to the kinetic term of a field vanishes (i.e. its massbecomes infinite) on some Σ, the Heisenberg principle suggeststhat quantum fluctuations will diverge on the approach to Σ. We note a caveat here, that this interpretation is strictly truefor theories with nonvanishing mass parameters; more carefulinvestigation of the multiplier interpretation may be warrantedfor the cases at hand. interpretation is then that a tachyonic excitation devel-ops on the approach to Σ. In the case at hand, the non-linear PPM in (31) is populated by momenta, and thelinearised PPM is empty. Thus, Minkowski spacetimeis just such a surface Σ. More generally, when the lin-earised PPM is populated by constant mass parameters,the requirement becomes that the nonlinear PPM pseu-dodeterminant should be positive-definite within the finalshell.
B. Case 28
Since Case 28 has fewer PiCs than Case *6
26, the kineticpart of the Hamiltonian is more extensive H T = b (cid:32) (cid:0) (cid:42) ϕ k(cid:42) ϕ k + 2 ∧ ϕ ⊥ kl ∧ ϕ ⊥ kl (cid:1) ˆ α + 18 ∧ ϕ kl ∧ ϕ kl ˆ β − ϕ P 2 ˆ α (cid:33) + fields , (39)while the PPM has fewer dimensions ϕ ϕ ⊥ k ∼ ϕ kl ϕ ⊥ ∼ ϕ ⊥ kl ϕ T klm ϕ · · · · · · ϕ ⊥ k · ˆ π · ˆ π ˆ π ˆ π ∼ ϕ kl · · ˆ π · ˆ π ˆ π ! ϕ ⊥ · ˆ π · · · · ∼ ϕ ⊥ kl · ˆ π ˆ π · · · ϕ T klm · ˆ π ˆ π ! · · ·
51 3 5 1 5 5 (40)Within the PiC shell, we find that ϕ (cid:91) and ∼ ϕ (cid:91)kl alreadyweakly vanish, leaving the following SiCs χ (cid:91) ⊥ k ≈ − η (cid:91)ml D (cid:91)m ∧ ˆ π (cid:91)kl , (41a) χ (cid:91) ⊥ ≈ − η (cid:91)ml D (cid:91)m(cid:42) ˆ π (cid:91)l , (41b) ∼ χ (cid:91) ⊥ kl ≈ D (cid:91) (cid:104) k(cid:42) ˆ π (cid:91)l (cid:105) , (41c) χ T (cid:91)klm ≈ D (cid:91)m ∧ ˆ π (cid:91) ⊥ kl + 12 D (cid:91) [ l ∧ ˆ π (cid:91) ⊥ k ] m + 34 η (cid:91)m [ k | η (cid:91)ij D (cid:91)i ∧ ˆ π (cid:91) ⊥| l ] j , (41d)which do not give rise to any TiCs. Also within the PiCshell, the following sSFCs appear H (cid:91)α ≈ − h (cid:91) jα η (cid:91)kl D (cid:91)k ∧ ˆ π (cid:91)jl , (42a) H (cid:91)kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) + D (cid:91) [ k(cid:42) ˆ π (cid:91)l ] , (42b) H (cid:91) ⊥ k ≈ η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91) ⊥ kl . (42c)In this case it is easiest to restrict to sub-shells using theSiCs and sSFCs simultaneously. We first note that H (cid:91) ⊥ k restricts ∧ ˆ π (cid:91) ⊥ kl to be solenoidal, dual to the gradient of0a scalar, and thus eliminates two D.o.F. The remainingD.o.F is eliminated by χ T (cid:91)klm . Similarly, χ (cid:91) ⊥ restricts (cid:42) ˆ π (cid:91)k to a solenoidal axial vector, removing one D.o.F. Afurther D.o.F is removed by substituting H (cid:91)kl into H (cid:91)α ,and a final D.o.F is removed by ∼ χ (cid:91) ⊥ kl . Separately, H (cid:91)kl removes three D.o.Fs. All the PiCs and SiCs are FC, andone D.o.F remains, as expected from Table II1 = 12 (cid:0) − × − × (1 + 3 + 2)[sSFC] − × (1 + 3 + 5 + 1 + 5 + 5)[iPFC] − × (1 + 1 + 1)[iSFC] (cid:1) . (43)As with Case *6
26, the no-ghost condition ˆ α < − mode in (39). However, we note that thelinearly-propagating ˆ π P again emerges at the nonlinearlevel in (40), so that a linear gauge symmetry is bro-ken and (43) is not valid sufficiently far from Minkowskispacetime. Whether or not an increase in the propagat-ing D.o.F results in a ghost is not so clear in Case 28 as itwas in Case *6
26. From (39), we see that an activation of ∧ ˆ π kl would endanger nonlinear unitarity by the linear no-tachyon condition ˆ β <
0. However, if either of the vectortordions (cid:42) ˆ π k or ∧ ˆ π ⊥ kl were to propagate, positive-definitecontributions to H T could be ensured by respectively fix-ing ˆ α < α >
0, since ˆ α does not serve to shore upthe linearised unitarity. The key point here, as discussedin Appendix B, is that with our ‘West Coast’ signatureevery contraction on parallel indices introduces a factorof −
1. Therefore, if both vector tordions propate in thenonlinear theory, it would seem that negative kinetic en-ergy contributions to H T are unavoidable. Whatever thestatus of ghosts, we observe that the nonlinear PPM hasfield-dependent rank. C. Case *5 The structure of Case *5
25 has many similarities withthat of Case 28. The Hamiltonian takes the form H T = b (cid:32) (cid:0) ϕ + 9 ϕ ⊥ k ϕ ⊥ k (cid:1) ˆ β + 18 ∧ ϕ kl ∧ ϕ kl ˆ β − ϕ P 2 ˆ α (cid:33) + fields , (44)while the nonlinear PPM is more sparesly populated: ∼ ϕ kl ϕ ⊥ ∧ ϕ ⊥ kl (cid:42) ϕ k ∼ ϕ ⊥ kl ϕ T klm ∼ ϕ kl ˆ π · · · · ˆ π ! ϕ ⊥ · · · · · · ∧ ϕ ⊥ kl · · · · · · (cid:42) ϕ k · · · · · · ∼ ϕ ⊥ kl · · · · · · ϕ T klm ˆ π ! · · · · ·
55 1 3 3 5 5 (45) Within the PiC shell, we have ∼ χ (cid:91)kl ≈ −D (cid:91) (cid:104) k ˆ π (cid:91) ⊥ l (cid:105) (46a) χ (cid:91) ⊥ ≈ ˆ π (cid:91) , (46b) ∧ χ (cid:91) ⊥ kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) , (46c) (cid:42) χ (cid:91)k ≈ π (cid:91) ⊥ k , (46d)and this time, all 10 sSFCs persist in the PiC shell H (cid:91) ⊥ ≈ − η (cid:91)kl D (cid:91)k ˆ π (cid:91) ⊥ l (47a) H (cid:91)α ≈ − h (cid:91) kα D (cid:91)k ˆ π (cid:91) − h (cid:91) kα η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91)kl , (47b) H (cid:91)kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) , (47c) H (cid:91) ⊥ k ≈ ˆ π (cid:91) ⊥ k . (47d)We find that H (cid:91) ⊥ k and H (cid:91)kl each remove three D.o.F,while χ (cid:91) ⊥ removes one D.o.F; the remaining sSFCs andSiCs are then implied, and the PiCs and SiCs are FC.Once again, one D.o.F remains as expected from Table II1 = 12 (cid:0) − × − × (3 + 3)[sSFC] − × (5 + 1 + 3 + 3 + 5 + 5)[iPFC] − × (cid:1) . (48)The discussion now proceeds in much the same way aswith Case 28, since PiC commutators linear in the propa-gating ˆ π P emerge away from Minkowski spacetime. Thistime, it is the tetrad momenta ˆ π and ˆ π ⊥ k which intro-duce extra uncertainty regarding ghosts. If only one ofthese momenta becomes activated, ˆ β may be used toensure it has a positive contribution to H T . Again, thenonlinear PPM rank is field-dependent. D. Case 24
Case 24 has only 16 PiCs, the fewest out of all the caseswe consider. The kinetic part of the Hamiltonian is pro-portionally more complicated H T = b (cid:32) (cid:0) (cid:42) ϕ k(cid:42) ϕ k + 2 ∧ ϕ ⊥ kl ∧ ϕ ⊥ kl (cid:1) ˆ α + 18 ∧ ϕ kl ∧ ϕ kl ˆ β + 4 (cid:0) ϕ + 9 ϕ ⊥ k ϕ ⊥ k (cid:1) ˆ β − ϕ P 2 ˆ α (cid:33) + fields , (49)while the PPM is extremely small: ∼ ϕ kl ϕ ⊥ ∼ ϕ ⊥ kl ϕ T klm ∼ ϕ kl ˆ π · ˆ π ˆ π ! ϕ ⊥ · · · · ∼ ϕ ⊥ kl ˆ π · · · ϕ T klm ˆ π ! · · ·
55 1 5 5 (50)1Within the PiC shell, we have the following SiCs ∼ χ (cid:91)kl ≈ −D (cid:91) (cid:104) k ˆ π (cid:91) ⊥ l (cid:105) (51a) χ (cid:91) ⊥ ≈ ˆ π (cid:91) − η (cid:91)kl D (cid:91)k(cid:42) ˆ π (cid:91)l , (51b) ∼ χ (cid:91) ⊥ kl ≈ D (cid:91) (cid:104) k(cid:42) ˆ π (cid:91)l (cid:105) , (51c) χ T (cid:91)klm ≈ D (cid:91)m ∧ ˆ π (cid:91) ⊥ kl + 12 D (cid:91) [ l ∧ ˆ π (cid:91) ⊥ k ] m + 34 η (cid:91)m [ k | η (cid:91)ij D (cid:91)i ∧ ˆ π (cid:91) ⊥| l ] j , (51d)and the following sPFCs H (cid:91) ⊥ ≈ − η (cid:91)kl D (cid:91)k ˆ π (cid:91) ⊥ l (52a) H (cid:91)α ≈ − h (cid:91) kα D (cid:91)k ˆ π (cid:91) − h (cid:91) kα η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91)kl , (52b) H (cid:91)kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) + D (cid:91) [ k(cid:42) ˆ π (cid:91)l ] , (52c) H (cid:91) ⊥ k ≈ ˆ π (cid:91) ⊥ k + η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91) ⊥ kl . (52d)It is clear from the PiC shell that Case 24 has muchin common with Case 28, and again we will implementthe SiCs and sSFCs simultaneously. Firstly, we find that χ T (cid:91)klm constitutes an overdetermined system in ∧ ˆ π (cid:91) ⊥ kl ,which vanishes and takes with it three D.o.F. Conse-quently, from H (cid:91) ⊥ k , we see that ˆ π (cid:91) ⊥ k must vanish alongwith another three D.o.F, such that H (cid:91) ⊥ and ∼ χ (cid:91)kl vanishautomatically. Similarly, ∼ χ (cid:91) ⊥ kl is an overdetermined sys-tem in ∧ ˆ π (cid:91) ⊥ kl , which vanishes with another three D.o.F; χ (cid:91) ⊥ then causes ˆ π (cid:91) to vanish with one D.o.F. As before,one D.o.F propagates in accordance with Table II1 = 12 (cid:0) − × − × (3 + 3)[sSFC] − × (5 + 1 + 5 + 5)[iPFC] − × (1 + 3 + 3)[iSFC] (cid:1) . (53)It is clear from (49) that any inference of the nonlinearunitarity will just combine the discussions of Case 28 andCase *5
25, while the PPM rank is again field dependent.
E. Case 32
For the first time, we encounter non-vanishing massparameters between the PiCs, specifically in ∧ ϕ (cid:91) ⊥ kl and ϕ T (cid:91)klm . We anticipate the nonvanishing commutatorseven at the linear level (cid:110) ∧ ϕ (cid:91) ⊥ kl , ∧ χ (cid:91) ⊥ ij (cid:111) ≈ O (1) – not-ing that the natural conjugate ∧ ϕ (cid:91)kl is not a PiC – and (cid:110) ϕ T (cid:91)klm , χ T (cid:91)ijn (cid:111) ≈ O (1). These PiCs and SiCs will beSC, allowing for the determination of their multipliers. The kinetic part of the Hamiltonian is H T = b (cid:32) (cid:0) ϕ ⊥ k ϕ ⊥ k + 2 ∼ ϕ kl ∼ ϕ kl (cid:1) ˆ β + 36 ∧ ϕ kl ∧ ϕ kl ˆ β + 2 ˆ β − ϕ P 2 ˆ α (cid:33) + fields . (54)In the PPM, we label the PiCs associated with nonvan-ishing mass parameters by ( ↓ ), producing: ↓ ↓ ϕ ϕ ⊥ ∧ ϕ ⊥ kl (cid:42) ϕ k ∼ ϕ ⊥ kl ϕ T klm ϕ · · · · · · ϕ ⊥ · · · · · · → ∧ ϕ ⊥ kl · · · · · · (cid:42) ϕ k · · · · · · ∼ ϕ ⊥ kl · · · · · · → ϕ T klm · · · · · ·
51 1 3 3 5 5 (55)Thus the PPM of this theory is remarkable, since it re-mains empty even in the nonlinear regime. Within thePiC shell, we find the following SiCs χ (cid:91) ≈ − η (cid:91)kl D (cid:91)k ˆ π (cid:91) ⊥ l , (56a) ∧ χ (cid:91) ⊥ kl ≈ − ˆ β + 2 ˆ β ˆ β − ˆ β ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) + 9 ˆ β ˆ β ( ˆ β − ˆ β )( ˆ β + 2 ˆ β ) ∧ ϕ (cid:91)kl , (56b) (cid:42) χ (cid:91)k ≈ − ˆ π (cid:91) ⊥ k , (56c) ∼ χ (cid:91) ⊥ kl ≈ ∼ ˆ π (cid:91)kl , (56d) χ T (cid:91)klm ≈ β m p2 T T (cid:91)klm . (56e)Note the appearance of field strengths, specifically thetorsion in ∧ ϕ (cid:91)kl and T T (cid:91)klm . Whilst these somewhat com-plicate the analysis, they naturally appear with the massparameters. We also mark the first apparent instance ofa TiC accompanying ∼ χ (cid:91) ⊥ kl . Using the notation ζ ≡ ˙ χ ,this may be written as ∼ ζ (cid:91) ⊥ kl ≈ η (cid:91)ij D (cid:91)i χ T (cid:91) (cid:104) k | j | l (cid:105) , (57)which then vanishes in the SiC shell. The PiC shell con-tains the following sSFCs: H (cid:91) ⊥ ≈ − η (cid:91)kl D (cid:91)k ˆ π (cid:91) ⊥ l (58a) H (cid:91)α ≈ − h (cid:91) kα η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91)kl − h (cid:91) kα η (cid:91)jl D (cid:91)j ∼ ˆ π (cid:91)kl , (58b) H (cid:91)kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) , (58c) H (cid:91) ⊥ k ≈ ˆ π (cid:91) ⊥ k . (58d)2Since two of the PiC chains are known to be self-terminating, the algorithm concludes quite quickly. Aswith Case *5 H (cid:91) ⊥ k and H (cid:91)kl each eliminate threeD.o.F. Another five D.o.F are then removed by ∼ χ (cid:91) ⊥ kl ,with the remaining SiCs and sSFCs automatically sat-isfied. The one remaining D.o.F is again expected fromTable II1 = 12 (cid:0) − × − × (3 + 3)[sSFC] − × (1 + 1 + 3 + 5)[iPFC] − (3 + 5)[iPSC] − × − (3 + 5)[iSSC] (cid:1) . (59)On the whole, the outlook for Case 32 appears morepromising than for the previous cases, because the PPMretains its empty structure (and rank) when passing tothe nonlinear regime. This is just the first hurdle, as thefull nonlinear algorithm would still be required to deter-mine whether further fields become activated. The impli-cations of field activaton are slightly relaxed, comparedto Case *5
25 or Case 28. The linear tachyon conditionˆ β < ∧ ˆ π kl contributesnegative kinetic energy if ˆ β + 2 ˆ β >
0. This can be re-alised even if ∼ ˆ π kl is simultaneously activated. Howeverfor positive kinetic energy it seems ˆ π ⊥ k must be acti-vated on its own or not at all, since ˆ β < F. Case 20
The analysis of Case 20 is quite similar to Case 32.Mass parameters again accompany the PiCs, and we ex-pect ∧ ϕ (cid:91) ⊥ kl , (cid:42) ϕ (cid:91)k and ϕ T (cid:91)klm to not commute with theirrespective SiCs on the final shell. The kinetic part of theHamiltonian is H T = b (cid:32) ϕ ˆ β + 12 ∼ ϕ kl ∼ ϕ kl ˆ β + 36 ∧ ϕ kl ∧ ϕ kl ˆ β + 2 ˆ β + 36 ϕ ⊥ k ϕ ⊥ k β + ˆ β − ϕ P 2 ˆ α (cid:33) + fields , (60)and once again the PPM is empty both before and afterlinearisation: ↓ ↓ ↓ ϕ ⊥ ∧ ϕ ⊥ kl (cid:42) ϕ k ∼ ϕ ⊥ kl ϕ T klm ϕ ⊥ · · · · · → ∧ ϕ ⊥ kl · · · · · → (cid:42) ϕ k · · · · · ∼ ϕ ⊥ kl · · · · · → ϕ T klm · · · · ·
51 3 3 5 5 (61) Within the PiC shell, we first the following SiCs χ (cid:91) ⊥ ≈ ˆ π (cid:91) , (62a) ∧ χ (cid:91) ⊥ kl ≈ − ˆ β + 2 ˆ β ˆ β − ˆ β ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) + 9 ˆ β ˆ β ( ˆ β − ˆ β )( ˆ β + 2 ˆ β ) ∧ ϕ (cid:91)kl , (62b) (cid:42) χ (cid:91)k ≈ − ˆ β + 2 ˆ β ˆ β − ˆ β ˆ π (cid:91) ⊥ k + 9 ˆ β ˆ β ( ˆ β − ˆ β )(2 ˆ β + ˆ β ) ϕ (cid:91) ⊥ k , (62c) ∼ χ (cid:91) ⊥ kl ≈ ∼ ˆ π (cid:91)kl , (62d) χ T (cid:91)klm ≈ β m p2 T T (cid:91)klm . (62e)This time, two TiCs appear, but uppon rearranging bothmay eventually be written in terms of the iSSCs, and aretherefore satisfied automatically ζ (cid:91) ⊥ ≈ η (cid:91)ij D (cid:91)i(cid:42) χ (cid:91)j , (63a) ∼ ζ (cid:91) ⊥ kl ≈ η (cid:91)ij D (cid:91)i χ T (cid:91) (cid:104) k | j | l (cid:105) − D (cid:91) (cid:104) k(cid:42) χ (cid:91)l (cid:105) . (63b)The sSFC content in the PiC shell is largely the same asthat of Case 32, with the only difference marked in thelinear super-momentum H (cid:91)α ≈ − h (cid:91) kα D (cid:91)k ˆ π (cid:91) − h (cid:91) kα η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91)kl − h (cid:91) kα η (cid:91)jl D (cid:91)j ∼ ˆ π (cid:91)kl . (64a)Aided by the additional conjugate pair of constraints,the algorithm terminates even faster than with Case 32:we see that one and five D.o.F are removed by each of χ (cid:91) ⊥ and ∼ ϕ (cid:91) ⊥ kl . As before, the one propagating D.o.F isconfirmed from Table II1 = 12 (cid:0) − × − × (3 + 3)[sSFC] − × (1 + 5)[iPFC] − (3 + 3 + 5)[iPSC] − × (1 + 5)[iSFC] − (3 + 3 + 5)[iSSC] (cid:1) . (65)If positive kinetic energy is a requirement, it seems thatthe momenta ˆ π and ˆ π ⊥ k in combination with one or moreof ∼ ˆ π kl or ∧ ˆ π kl , should not all be activated at the same time. IV. MASSLESS RESULTSA. Case 17
Two theories in Table II – Case 17 and Case 3 – admita pair of massless modes according to the linearised anal-ysis. Beginning with Case 17, we find the Hamiltonian3to have the structure H T = b (cid:32) (cid:0) (cid:42) ϕ k(cid:42) ϕ k + 2 ∧ ϕ ⊥ kl ∧ ϕ ⊥ kl (cid:1) ˆ α − ϕ ⊥ k ϕ ⊥ k + 2 ∼ ϕ kl ∼ ϕ kl ˆ β (cid:33) + fields , (66)As mentioned in Section II C, the evaluation of the PPMis complicated by the appearance of torsion in PiC ∧ ϕ kl belonging to the translational sector. In general, commu-tators between field strengths generate derivatives of theDirac function. In many cases, these derivatives eitherhappen to cancel, or they may be discarded up to a sur-face term within the PiC shell. In any case, we find thatthe full nonlinear PPM can be written purely in terms ofthe parallel momenta as before: ↓ ↓ ↓ ϕ ∧ ϕ kl ϕ ⊥ ϕ P ∼ ϕ ⊥ kl ϕ T klm ϕ · ˆ π · · · · → ∧ ϕ kl ˆ π · ˆ π ˆ π ˆ π ˆ π ϕ ⊥ · ˆ π · · · · → ϕ P · ˆ π · · · · ∼ ϕ ⊥ kl · ˆ π · · · · → ϕ T klm · ˆ π · · · ·
51 3 1 1 5 5 (67)Due to the appearance of mass parameters, we will expect ∧ ϕ (cid:91)kl , ϕ P (cid:91) and ϕ T (cid:91)klm not to commute with their SiCs inthe final shell. Within the PiC shell, we find the followingSiCs χ (cid:91) ≈ − η (cid:91)kl D (cid:91)k ˆ π (cid:91) ⊥ l , (68a) ∧ χ (cid:91)kl ≈ − β ˆ α m p2 ∧ ˆ π (cid:91) ⊥ kl − D (cid:91) [ k ˆ π (cid:91) ⊥ l ] − β m p2 D (cid:91) [ k(cid:42) T (cid:91)l ] , (68b) χ (cid:91) ⊥ ≈ − η (cid:91)kl D (cid:91)k(cid:42) ˆ π (cid:91)l , (68c) χ P (cid:91) ≈ (cid:15) (cid:91)jkl ⊥ D (cid:91)j ∧ ˆ π (cid:91) ⊥ kl − β m p2 T P (cid:91) , (68d) ∼ χ (cid:91) ⊥ kl ≈ ∼ ˆ π (cid:91)kl + 12 D (cid:91) (cid:104) k(cid:42) ˆ π (cid:91)l (cid:105) , (68e) χ T (cid:91)klm ≈ D (cid:91)m ∧ ˆ π (cid:91) ⊥ kl + 12 D (cid:91) [ l ∧ ˆ π (cid:91) ⊥ k ] m + 34 η (cid:91)m [ k | η (cid:91)ij D (cid:91)i ∧ ˆ π (cid:91) ⊥| l ] j − β m p2 T T (cid:91)klm . (68f)Among these, we note that a TiC accompanies ∼ χ (cid:91) ⊥ kl , butmay be expressed in terms of χ T (cid:91)klm by precisely (57). Within the PiC shell, the sSFCs are H (cid:91) ⊥ ≈ − η (cid:91)kl D (cid:91)k ˆ π (cid:91) ⊥ l , (69a) H (cid:91)α ≈ − h (cid:91) kα η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91)kl − h (cid:91) kα η (cid:91)jl D (cid:91)j ∼ ˆ π (cid:91)kl , (69b) H (cid:91)kl ≈ ∧ ˆ π (cid:91)kl + D (cid:91) [ k(cid:42) ˆ π (cid:91)l ] , (69c) H (cid:91) ⊥ k ≈ ˆ π (cid:91) ⊥ k + η (cid:91)jl D (cid:91)j ∧ ˆ π (cid:91) ⊥ kl . (69d)The conjugate pairs together eliminate six, two and 10D.o.F before terminating. As with Case 28, (cid:42) ˆ π (cid:91)k becomessolenoidal due to χ (cid:91) ⊥ and loses one D.o.F, while threeD.o.F are lost by each of H (cid:91)kl and H (cid:91) ⊥ k . In total, twopropagating D.o.F remain as expected from Table II2 = 12 (cid:0) − × − × (3 + 3)[sSFC] − × (1 + 1 + 5)[iPFC] − (3 + 1 + 5)[iPSC] − × (1 + 5)[iSFC] − (3 + 1 + 5)[iSSC] (cid:1) . (70)According to Table II, these two D.o.F should be mass-less, and the power in the 2 + sector of the propagatorinvites speculation as to whether they constitute a gravi-ton. In Section III A we were able to show that the oneD.o.F of Case *6
26 belonged unambiguously to the 0 − sector, but our method cannot be so straightforwardlyapplied to Case 17. This is ultimately related to the factthat the dependence of the PiC ∧ ϕ kl on the parallel torsion T ikl in (16b), survives even when the defining constraintsof Case 17 are imposed on the couplings. In the lineartheory, this results in a conjugate SiC ∧ χ (cid:91)kl which dependson the gradient of the torsion D (cid:91) [ k(cid:42) T (cid:91)l ] . This is problem-atic when it comes to determining the linear multiplier ∧ u (cid:91)kl through the consistency condition (27). The resultis a PDE in the multiplier − β D (cid:91) [ k | η (cid:91)ij D (cid:91)i ∧ u (cid:91) | l ] j + 16 ˆ β ˆ α ∧ u (cid:91)kl ≈ X [ kl ] , (71)where the inhomogeneous part on the RHS results froma second-order Euler–Lagrange variation in the Poissonbracket, and likely involves gradients of the equal-timeDirac function. The remaining six determinable multipli-ers are not affected by this problem; we find with relativeease u P (cid:91) ≈ , (72a) u T (cid:91)klm ≈ η (cid:91) [ k | m η (cid:91)ij D (cid:91)i ∼ ˆ π (cid:91) | l ] j − D (cid:91) [ k ∼ ˆ π (cid:91)l ] m + 316 η (cid:91) [ k | m η (cid:91)ij (cid:0) D (cid:91) | l ] D (cid:91)i(cid:42) ˆ π (cid:91)j − D (cid:91)i D (cid:91)j (cid:42) ˆ π (cid:91) | l ] (cid:1) + 38 D (cid:91)m D (cid:91) [ k(cid:42) ˆ π (cid:91)l ] + 8 ˆ β R T (cid:91) ⊥ klm . (72b)The ambiguity of ∧ u (cid:91)kl is problematic, as this multiplierlingers in the equations of motion. Even worse, theeight indeterminate multipliers u (cid:91) , u P (cid:91) and ∼ u (cid:91)kl associ-ated with the iPFCs also feature prominently. In order4to discover the J P character of the propagating modesusing Hamiltonian methods, one would have to fix thegauge.We can draw some tentative conclusions just from thekinetic part of the Hamiltonian however. We see fromTable II that the linear theory is unitary if only ˆ α < α in (66) would suggest thatˆ α < − mode from becoming aghost. By the same arguments, the 1 + mode should bestrongly coupled within the final shell of the linearisedtheory, since it would otherwise enter with negative ki-netic energy. It is reassuring to see from Table II that themassless propagator does indeed have power in the 1 − ,but not the 1 + sectors. However, it also has power in the2 + sector, possibly inviting speculation that the theorymay contain a spin-two graviton akin to that of Einstein.While ∼ ˆ π kl does feature in (66), it seems unlikely that thismode would independently propagate, since the unitarityof the theory does not depend on ˆ β . We reiterate thatthese conclusions may ultimately depend on the gaugechoice.Finally, without a definite understanding of the propa-gating J P , we are unable to say concretely whether fieldswill be activated or the PPM rank be field-dependent inthe nonlinear theory. It is quite likely that these phe-nomena will occur, since we find in Appendix C that thecommutator of ∧ ϕ kl and ϕ T klm depends on (cid:42) ˆ π k . This hasprecedent, since the 2 − commutator has spoiled all thetheories in Section III, but due to the lingering gaugeambiguity we denote it with (ˆ π ) rather than (ˆ π !) in (67). B. Case 3
It should come as no surprise that Case 3 is a relaxationof Case 17, which admits an extra D.o.F. The kineticpart of the Hamiltonian is given by (66), in addition tothe pseudoscalar term encountered in all the cases of Sec-tion III. This is the usual massive 0 − mode, and comeswith the no-ghost condition ˆ α <
0. The extra conditionˆ β will prevent this mode from being tachyonic. Thenonlinear PPM is: ↓ ↓ ϕ ∧ ϕ kl ϕ ⊥ ∼ ϕ ⊥ kl ϕ T klm ϕ · ˆ π · · · → ∧ ϕ kl ˆ π · ˆ π ˆ π ˆ π ! ϕ ⊥ · ˆ π · · · ∼ ϕ ⊥ kl · ˆ π · · · → ϕ T klm · ˆ π ! · · ·
51 3 1 5 5 (73)It is clear that ϕ P (cid:91) is no longer primarily constrained.The only change to the remaining SiCs of Case 17 is H (cid:91)kl ≈ ∧ ˆ π (cid:91)kl − (cid:15) (cid:91)klm ⊥ η (cid:91)mn D (cid:91)n ˆ π P (cid:91) + D (cid:91) [ k(cid:42) ˆ π (cid:91)l ] , (74) but H (cid:91)α is still satisfied. Overall, only the conjugate ϕ P (cid:91) and χ P (cid:91) pair are removed, leaving three propagatingD.o.F as expected from Table II3 = 12 (cid:0) − × − × (3 + 3)[sSFC] − × (1 + 1 + 5)[iPFC] − (3 + 5)[iPSC] − × (1 + 5)[iSFC] − (3 + 5)[iSSC] (cid:1) . (75)We draw the same conclusions from Case 3 as fromCase 17 regarding the vector nature of the gravitationalparticle. This time however, we note the presence of ˆ π P in the nonlinear PPM, indicating that whatever the mass-less J P , at least one gauge symmetry does not survive inthe nonlinear regime. V. PHENOMENOLOGY
The results of Section IV cast serious doubts on thehealth of even the massless theories considered here, onquite general grounds. We can in fact rule these theo-ries out more conclusively on the basis of their cosmol-ogy. In general, this would be quite an arduous task,requiring a dedicated examination of all four equationsof motion. However, we recently developed a mappingbetween the general quadratic torsion theory (2) and atorsion-free biscalar-tensor theory, which immediately re-veals the cosmological background [18]. We begin withthe spatially flat FRW line elementd s = d t − a d x , (76)where a is the scale factor, normalised to the contem-porary epoch, from which we define the Hubble number H ≡ ˙ a/a . We now align the unit timelike normal n k tobe perpendicular to the spatially flat slicing. Cosmologi-cal isotropy at the background level restricts only the 0 + and 0 − torsion modes to propagate. From these modesrespectively we define a pair of scalar fields φ ≡ T k ⊥ k − H, ψ ≡ (cid:15) ⊥ jki T ijk . (77)These fields transform homogeneously and with the cor-rect weight φ (cid:55)→ Ω − φ , ψ (cid:55)→ Ω − ψ under changes ofphysical scale b iµ (cid:55)→ Ω b iµ . In the usual second-order for-mulation of gravity on the curved spacetime M , it canbe shown that the theory L G (cid:39) (cid:104) ˆ β m p2 + 14 (ˆ α + ˆ α ) φ −
14 (ˆ α + ˆ α ) ψ (cid:105) R + 3(ˆ α + ˆ α ) X φφ − α + ˆ α ) X ψψ + (cid:112) | J µ J µ | + 34 ( α + 2 ˆ β ) m p2 φ −
34 ( α + 8 ˆ β ) m p2 ψ + 38 (ˆ α + ˆ α ) φ + 38 (ˆ α + ˆ α ) ψ − (cid:0) (ˆ α + ˆ α ) + 2(ˆ α + ˆ α ) (cid:1) φ ψ (78a) J µ ≡ (cid:2) (ˆ α − ˆ α ) − (ˆ α − ˆ α ) (cid:3) ψ ∇ µ ( φ/ψ ) − ( α + 2 ˆ β ) m p2 ∇ µ φ, (78b)5perfectly replicates the dynamics of the FRW back-ground. We note that the Ricci scalar R is derivedfrom the Riemann curvature as defined in (6), and whilethe biscalar-tensor theory is strictly torsion-free, the be-haviour of the 0 + and 0 − modes is accurately replicatedby the scalars. The theory (78) is known as the metricalanalogue (MA) of (2), and we have translated it here intothe dimensionless couplings (A3) of (12).We will restrict our attention to the massless theories.Following a reparameterisation to the weightless scalar ζ = √ φ/ψ , we find the MA of Case 3 becomes L G (cid:39) − α X ψψ −
14 ˆ α ψ R + 3 ˆ β m p2 ψ +ˆ α ψ (cid:113)(cid:12)(cid:12) X ζζ (cid:12)(cid:12) −
34 ˆ α ζ ψ . (79)In this frame, we see that the MA can be partitionedinto two. The first three terms in (79) describe a mas-sive but conformally coupled scalar ψ . The fourth andfifth terms describe a quadratic cuscuton ζ , which is con-formally coupled by multiplication with the appropriatepowers of ψ .The quadratic cuscuton is itself remarkable for replicat-ing the cosmological background of the Einstein–Hilbertterm [30] c m p3 (cid:113)(cid:12)(cid:12) X ζζ (cid:12)(cid:12) − c m p4 ζ (cid:39) c c m p2 R, (80)This unlikely-looking relation may be verified by substi-tuting the ζ -equation into the g µν -equation on the LHS of(80), and comparing with the Friedmann equations thatfollow from the g µν -equation on the RHS. We find thatthe bizarre characteristics of the cuscuton can be takenfurther: when we replace the Planck mass with a dynam-ical scalar to obtain the conformally coupled quadraticcuscuton, we replicate the cosmological background ofthe same scalar, conformally coupled to gravity c ψ (cid:113)(cid:12)(cid:12) X ζζ (cid:12)(cid:12) − c ψ ζ (cid:39) c c (cid:16) X ψψ + 112 ψ R (cid:17) . (81)This result is very satisfying, but has fatal implicationsfor the massless theories under consideration. By ap-plying (81), we see that the fourth and fifth terms in(79) dynamically cancel with the first and second terms:the whole kinetic structure of the analogue theory van-ishes! The same problem arises in Case 17, since the ex-tra condition ˆ α = 0 prevents the cancelling terms fromappearing even at the level of (79). In both cases, thegravitational Lagrangian responsible for the cosmologicalbackground is a pure 0 − mass, and so the theories are notviable .Notwithstanding the complete failure of the cases athand, the result (81) suggests an interesting class of the-ories, of which Case 3 is a degenerate special case. Fromthe general quadratic torsion theory (12) we impose α + 2 ˆ β = ˆ α + ˆ α = 0 , (82) noting from (19a) that the second constraint in (82) re-sults in the single 0 + PiC ϕ ⊥ ≈
0. The cosmologicalanalogue then becomes L G (cid:39) − α m p2 R − α + ˆ α ) (cid:16) X ψψ + 112 ψ R (cid:17) + (cid:2) (ˆ α − ˆ α ) − (ˆ α − ˆ α ) (cid:3) ψ (cid:113)(cid:12)(cid:12) X ζζ (cid:12)(cid:12) −
34 (ˆ α + ˆ α ) ζ ψ −
34 ( α + 8 ˆ β ) m p2 ψ . (83)The interpretation of the first equality of (82) is nowclear: it forces the Einstein–Hilbert term to appearequally both in the torsion theory and the cosmolog-ical background analogue . We can then set α = 1to view these theories as additive modifications to theEinstein–Cartan or Einstein–Hilbert theories, respec-tively. In order to apply (81), we will strictly requirethat ˆ α + ˆ α (cid:54) = 0, i.e. that the 0 − mode is not primar-ily constrained according to (19b). Under an appropriaterescaling of ψ to ξ , the cosmological background becomes L G (cid:39) − m p2 R + 112 ξ R + X ξξ − m ξ ξ , (84)i.e. Einstein’s gravity conformally coupled to a scalar ξ ,whose mass is m ξ ≡ − (1 + 8 ˆ β )(ˆ α + ˆ α )8(ˆ α + ˆ α )(ˆ α + ˆ α ) m p2 . (85)The theory (84) is of course widely studied in the contextof inflation [31, 32]. In Einstein’s theory, a non-minimalscalar coupling will tend to run, with the conformal valueof 1 /
12 being a fixed point in the IR. This value is alsoused to preserve causality in a curved background, sinceit prevents a massive scalar from propagating along thelight cone. We have shown that the cosmological back-ground of the conformal scalar emerges as a consequenceof the minimal constraints (82) on the quadratic torsiontheory, where the scalar is interpreted as the 0 − part ofthe torsion, and the 0 + part is primarily constrained.We see also from (85) that the effect of the conformallycoupled 0 − can be removed from the expansion historyaltogether. By setting ˆ α + ˆ α = 0 or ˆ α + ˆ α = 0, themass m ξ becomes infinite and one is left with the cosmo-logical background of the pure Einstein gravity in (1). Byinspecting Eqs. (19a) to (19f), we see that these choicescan be imposed without primarily constraining the tor-sion modes in the general theory, including the 0 − mode.This raises the interesting question of whether torsiontheories allow the cosmological background to be alteredindependently of the the perturbations. Note that Case 3has just such a divergent mass, though the Einstein–Hilbert term never appears in the background analoguebecause of the universal constraint (3) that appears tobe required for power-counting renormalisability. Note that this is not generally guaranteed for general choices ofthe coupling constants, as discussed in [18]. VI. CONCLUSIONS
It was recently shown that among all parity-preservingtheories of the form L G ∼ m p2 R + R + m p2 T , thereare 58 cases which are unitary and power-counting renor-malisable when linearised in the absence of source cur-rents [6, 7]. The linearisation was done around a pre-sumed Minkowski vacuum, since even a cosmological con-stant is excluded as a source. In this work we have con-sidered Case 3, Case 17, Case 20, Case 24, Case *5 *6
26, Case 28 and Case 32 – as detailed in Table II.We have inspected their Hamiltonian structure under thesame conditions, but in both the linear and nonlinear regimes. Our principal findings may be summarised asfollows;1. All eight cases (and indeed all the cases proposedin [6, 7]) feature vanishing mass parameters. Thisgreatly complicates the Hamiltonian analysis, com-pared to the ‘minimal’ cases previously treated inthe literature.2. The number of linear, propagating degrees of free-dom are confirmed from [6, 7] for all eight cases.3. With the exception of Case 17, all eight cases lin-early propagate a massive pseudoscalar mode, andthe unitarity conditions from [6, 7] correspond tothe no-ghost and no-tachyon conditions on thismode.4. The two massless modes propagated by Case 3 andCase 17 are identified with vector , rather than thehoped-for tensor modes.5. With the possible exception of Case 20 and Case 32,all eight cases feature primary constraints whichtransition from first to second class when movingto the nonlinear regime. This signals at least abroken gauge symmetry, and possibly acausal be-haviour and/or activation of any of the primarilyunconstrained spin-parity sectors.6. These primarily unconstrained spin-parity sectorsinclude ghosts in all eight cases, according to thesame conditions that ensure linearised unitarity.7. Case 3 and Case 17 are not viable theories of gravitydespite their massless modes, because they do notsupport a dynamical FRW background.These findings come with various caveats. Principally,while we implement the linearised Dirac–Bergmann al-gorithm to completion in all cases, we do not prosecutethe nonlinear algorithm beyond the second set of linksin the constraint chains. This level of analysis at leastmatches the earlier treatment of less complicated theo-ries, in which all couplings are set to zero except thoseabsolutely necessary to propagate whichever mode is un-der investigation [16]. Consequently, we cannot say forcertain if the strongly coupled sectors and the ghost sec-tors coincide.Separately, our definition of ghost sectors as set out inAppendix B is based on the relevant quadratic momenta appearing as negative contributions to the Hamiltonian.We do not go so far as to quantise the theory and confirmthat there are corresponding physical states which violatethe unitarity of the S-matrix. Additional steps would pre-sumably be required to draw completely safe conclusions,such as adding terms to fix the Poincar´e gauge (and anyother case-specific symmetries), and good ghosts to can-cel the anomalies [14]. Meanwhile at the classical level,we mention that negative kinetic energy does not alwaysimply instability.We have also interpreted acausal behaviour, which islinked to the phenomenon of constraint bifurcation orfield-dependent constraint structure [15], as a pathology.This need not always follow, as has been demonstratedfor some special theories in recent decades [33]. For ex-ample, the characteristic surface of a degree of freedomis allowed to lie outside the light cone if it can be shownthat the field does not carry information [34].Even bearing these caveats in mind, the outlook for theremaining new torsion theories is not substantially im-proved by our results. Of the 58 novel theories in [6, 7],only 19 propagate the two massless degrees of freedom.Four of these additionally propagate a massive 0 − mode,while three instead propagate a massive 2 − mode. Ofthe remaining theories, 23 propagate only a massive 0 − mode. The selection in Table II thus appears reasonablyrepresentative of the linearised particle spectra. Sincefundamental changes to the constraint structure are ob-served throughout most of the sample, we do not findnew cause for optimism in the current study. Possi-bly, the admission of primary constraints dependent onthe Riemann–Cartan curvature will miraculously rem-edy the various problems. Certainly, such constraintswill complicate the analysis. We have already seen inSection IV that field-dependent primary constraints caninvoke derivatives of the equal-time Dirac function. Ul-timately, our findings are consistent with the predictionsof Yo and Nester, who anticipate that generalising thequadratic torsion theory (2) beyond very minimal testcases (most of which also fail) serves only to protract thecalculations [16, 17]. Even so, it might seem prudent toattempt to quantify the chances of future success: weprovide a heuristic discussion along these lines in Ap-pendix D.The tentative vector nature of the massless modes inCase 3 and Case 17 is potentially problematic. We recallthat Poincar´e invariance prohibits a matter amplitudeinvolving soft gravitons of spin J >
2, while J = 0 gravi-tons are ruled out by matter coupling [35]. Odd J aresupposed to give rise to repulsive long-range forces, lead-ing to the expectation of a tensor graviton [20]. Plausibly,the J P character will be gauge dependent, but it is diffi-cult to see how this might change the sign of the Green’sfunction. We will not speculate as to whether this trou-bling feature is generic to the remaining massless cases.Finally, we observed that the theories with masslessmodes could be written off instantly using the scalar-7tensor analogue theory which replicates the backgroundcosmology. As a by-product, our analysis suggested aninteresting new class of quadratic torsion theories whichmimic the background of the conformal inflaton, thoughnot motivated by unitarity or renormalisability. It mustbe emphasised that the catastrophic failure of Case 3and Case 17 is not common to the remaining theoriesin [6, 7]. We mention in particular Case 2, which propa-gates two massless modes and the massive pseudoscalar,and Case 16, a special case in which the pseudoscalaris non dynamical. These theories form a complemen-tary pair to Case 3 and Case 17 in many respects, butthey have an excellent cosmological background. Notonly does the cuscuton force the evolution towards a flatFriedmann solution, but the option exists to tune theearly expansion history through an effective dark radi-ation component [19]. Moreover in Case 2 the mass ofthe propagating pseudoscalar acts as a dark energy term(albeit hierarchical, i.e. not resolving the cosmologicalconstant problem) [18]. Other exact solutions to Case 2and Case 16 include the Schwarzschild vacuum and planegravitational waves. These cases call for a more dedicatedHamiltonian analysis, and will be among the remainingtheories to be addressed in the companion paper. ACKNOWLEDGMENTS
We are grateful to Emine S¸eyma Kutluk and Wei ChenLin for very profitable conversations, and also to Yun-Cherng Lin for valuable computational advice. We wouldalso like to thank Ignacy Sawicki for detailed and help-ful correspondence. This manuscript was improved bythe kind suggestions of Amel Durakovi´c. WEVB is sup-ported by the Science and Technology Facilities Council –STFC under Grant ST/R504671/1, and WJH by a RoyalSociety University Research Fellowship.
Appendix A: Irreducible decomposition of the fields
It is necessary to construct a complete set of idempotentand orthogonal projection operators for the irreducibleparts of the field strengths. For general tensors, this canbe done with the appropriate O(3) Young tableaux, fol-lowing the methods of [36]. The three projections of thetorsion are P mnqijk T mnq = 23 T ijk + 23 T [ j | i | k ] + 23 η i [ j T k ] , (A1a) P mnqijk T mnq = − η i [ j T k ] , (A1b) P mnqijk T mnq = 16 (cid:15) ijkl (cid:15) lmnq T mnq . (A1c) The six projections of the Riemann–Cartan curvature are P mnqpijkl R mnqp = 13 R ijkl + 13 R klij + 23 R [ i (cid:107) [ k |(cid:107) j ] | l ] − η [ i | [ k (cid:107) R | j ] (cid:107) l ] − η [ i | [ k (cid:107) R | l ] (cid:107) j ] + 13 η i [ k | η j | l ] R , (A2a) P mnqpijkl R mnqp = 12 R ijkl − R klij − η [ i | [ k (cid:107) R | j ] (cid:107) l ] + η [ i | [ k (cid:107) R | l ] (cid:107) j ] , (A2b) P mnqpijkl R mnqp = − (cid:15) ijkl (cid:15) mnop R mnop , (A2c) P mnqpijkl R mnqp = η [ i | [ k (cid:107) R | j ] (cid:107) l ] + η [ i | [ k (cid:107) R | l ] (cid:107) j ] − η i [ k | η j | l ] R , (A2d) P mnqpijkl R mnqp = η [ i | [ k (cid:107) R | j ] (cid:107) l ] − η [ i | [ k (cid:107) R | l ] (cid:107) j ] , (A2e) P mnqpijkl R mnqp = 16 η i [ k | η j | l ] R . (A2f)Replicating the numbering used in [16, 17], our originalna¨ıve couplings in (2) are expressible in terms of theirmore meaningful irreducible counterparts according to α ≡ ˆ α + 12 ˆ α + ˆ α , α ≡ ˆ α − ˆ α ,α ≡ ˆ α − ˆ α + ˆ α ,α ≡
12 ˆ α + 12 ˆ α + ˆ α + 12 ˆ α + ˆ α ,α ≡
12 ˆ α −
12 ˆ α + ˆ α − ˆ α ,α ≡ α + 32 ˆ α + 32 ˆ α + ˆ α + 12 ˆ α + ˆ α ,β ≡ ˆ β + 12 ˆ β , β ≡ ˆ β + 12 ˆ β + 32 ˆ β ,β ≡ ˆ β − ˆ β , (A3)where ˆ α I and ˆ β I multiply the I th irreducible quadraticinvariants of curvature and torsion in (12). Appendix B: Ghosts, ranks and signatures
In this appendix, we attempt to elaborate on the mo-tivation of the ‘positive kinetic energy test’, which wastacitly employed in the previous Hamiltonian treatmentof Poincar´e gauge theories [16].Consider the free, vector U(1) theory on ˇ M , withoutany coupling to gravity (and with Cartesian coordinates γ µν ≡ η µν ), fixed to the Feynman gauge L = − F µν F µν −
12 ( ∂ µ A µ ) , (B1)where we have F µν ≡ ∂ [ µ A ν ] . Up to a surface term,(B1) is of course equivalent to L = − ∂ µ A ν ∂ µ A ν , (B2)8which safely propagates four massless polarisations, with-out developing any classical instability (cid:3) A µ = 0 . (B3)Notwithstanding this reasonable behaviour, we see thatthe Hamiltonian of (B2) is unbounded from below H = − π µ π µ + 12 ∂ α A µ ∂ α A µ , (B4)where the momentum is π µ ≡ − ∂ A µ , since the inde-pendent timelike polarisation will have a strictly nega-tive contribution. This is naturally revealed in the 3 + 1picture, which we construct by defining a constant unittimelike normal n µ n µ ≡
1, and (extending our previ-ous overbar notation to holonomic indices) decomposingquantities into the 0 + and 1 − irreps A µ ≡ A ⊥ n µ + A µ , π µ ≡ π ⊥ n µ + π µ . (B5)The Hamiltonian then separates into H = − π ⊥ + 12 ∂ α A ⊥ ∂ α A ⊥ − π µ π µ + 12 ∂ α A µ ∂ α A µ , (B6)where the first and last pairs of terms are respectivelynegative and positive-definite on the null shell defined by(B3). The physical consequence is a loss of unitary: thetimelike states have negative norm. In the U(1) theory,this is usually fixed by imposing a Gupta–Bleuler con-dition on the physical states, which is acceptable sincethe gauge-fixing term in (B1) was added by hand any-way. However, in the theories of gravity under considera-tion, the validity of a Gupta–Bleuler condition is not cer-tain. We note that in the kinetic Hamiltonia of Eqs. (29),(39), (44), (49), (54), (60) and (66), we encounter mixedquadratic forms in the momenta, just as we do with thefirst and third terms of (B6). If such terms are negative-definite and propagating, we tentatively identify themwith a loss of unitarity. We note that without full knowl-edge of both the nonlinear shell and the remaining fieldparts of the Hamiltonian (c.f. second and fourth termsin (B6)), this is quite dangerous. Moreover, as is evidentfrom (B3), such negative-energy sectors do not necessar-ily correspond to classical ghosts.We also mention that the sign of quadratic momentain the 3 + 1 formulation is robust against the choice ofsignature (as indeed it should be). Recall that through-out this article we have used the ‘West Coast’ signature(+ , − , − , − ). The sign of each such term may then beinferred by the tensor rank of the momentum irrep, sinceevery contraction on parallel indices introduces a fac-tor of −
1. Had we chosen the ‘East Coast’ signature( − , + , + , +), these factors would not arise. Instead, wewould have n µ n µ ≡ −
1, whose powers would conspire inthe O(3) decomposition of momenta to have the same ef-fect up to an overall sign in the kinetic Hamiltonian. Thisfinal sign is changed by hand in the kinetic part of theLagrangian, as is customary when changing signature.
Appendix C: Nonlinear Poisson brackets
Case 28
In Eqs. (30a) to (30d) we provide the nonlin-ear commutators of Case *6
26. In this appendix we listthe emergent commutators of the other seven theoriesunder consideration. The commutators of Case 28 read (cid:110) ϕ ⊥ i , ϕ ⊥ l (cid:111) ≈ RHS of (30a) , (C1a) (cid:110) ϕ ⊥ i , ϕ ⊥ (cid:111) ≈ − J (cid:42) ˆ π i δ , (C1b) (cid:110) ϕ ⊥ i , ∼ ϕ ⊥ lm (cid:111) ≈ J η i (cid:104) l(cid:42) ˆ π m (cid:105) δ , (C1c) (cid:110) ϕ ⊥ i , ϕ T lmn (cid:111) ≈ J (cid:20) η in ∧ ˆ π ⊥ lm − η i [ l | ∧ ˆ π ⊥| m ] n − η [ l | n ∧ ˆ π ⊥ i | m ] (cid:21) δ , (C1d) (cid:110) ∼ ϕ ij , ∼ ϕ lm (cid:111) ≈ RHS of (30c) , (C1e) (cid:110) ∼ ϕ ij , ∼ ϕ ⊥ lm (cid:111) ≈ RHS of (30c) , (C1f) (cid:110) ∼ ϕ ij , ϕ T lmn (cid:111) ≈ J (cid:20) (cid:15) (cid:104) i | [ l (cid:107) n ⊥ η | j (cid:105)(cid:107) m ] ˆ π P + 112 (cid:15) (cid:104) i | lm ⊥ η | j (cid:105) n ˆ π P + 38 η (cid:104) i | [ l η m ] n(cid:42) ˆ π | j (cid:105) − η (cid:104) i | n η | j (cid:105) [ l(cid:42) ˆ π m ] (cid:21) δ . (C1g)In the RHS of (C1g), we see that the linearly propagatingˆ π P appears, signalling a definite change in the constraintstructure when passing from linear to nonlinear regimes. Case *5 The nonlinear commutators of Case *5 (cid:110) ∼ ϕ ij , ∼ ϕ lm (cid:111) ≈ RHS of (30c) , (C2a) (cid:110) ∼ ϕ ij , ϕ T lmn (cid:111) ≈ RHS of (30d) . (C2b)Again we see that at least (C2b) is expected to persiston the final shell. Case 24
Similarly for Case 24 we find (cid:110) ∼ ϕ ij , ∼ ϕ lm (cid:111) ≈ RHS of (30c) , (C3a) (cid:110) ∼ ϕ ij , ∼ ϕ ⊥ lm (cid:111) ≈ J η ( i (cid:107) ( l | ∧ ˆ π (cid:107) j ) | m ) δ , (C3b) (cid:110) ∼ ϕ ij , ϕ T lmn (cid:111) ≈ RHS of (C1g) . (C3c)Again we see that at least (C3c) is expected to persist onthe final shell.9 Case 3
The nonlinear commutators of Case 3 are allnew (cid:110) ϕ, ∧ ϕ lm (cid:111) ≈ J ∧ ˆ π lm δ , (C4a) (cid:110) ∧ ϕ ij , ϕ ⊥ (cid:111) ≈ − J ∧ ˆ π ⊥ ij δ , (C4b) (cid:110) ∧ ϕ ij , ∼ ϕ ⊥ lm (cid:111) ≈ J η [ i (cid:107)(cid:104) l | ∧ ˆ π ⊥(cid:107) j ] | m (cid:105) δ , (C4c) (cid:110) ∧ ϕ ij , ϕ T lmn (cid:111) ≈ J (cid:20) (cid:15) [ i | [ l (cid:107) n ⊥ η | j ] (cid:107) m ] ˆ π P + 112 (cid:15) [ i | lm ⊥ η | j ] n ˆ π P − (cid:15) ij [ l ⊥ η m ] n ˆ π P − η [ i | [ l η m ] n(cid:42) ˆ π | j ] − η [ i | n η | j ][ l(cid:42) ˆ π m ] + 14 η i [ l η m ] j (cid:42) ˆ π n (cid:21) δ . (C4d)Since (C4d) also depends on ˆ π P , we believe that it willalso persist on the final shell. Note that (C4d) is also lin-ear in (cid:42) ˆ π k , which we suspect will contribute the masslessmodes in the linear theory. Case 17
The nonlinear commutators of Case 17 are ofcourse mostly the same as Case 3 (cid:110) ϕ, ∧ ϕ lm (cid:111) ≈ RHS of (C4a) , (C5a) (cid:110) ∧ ϕ ij , ϕ ⊥ (cid:111) ≈ RHS of (C4b) , (C5b) (cid:110) ∧ ϕ ij , ϕ P (cid:111) ≈ − J η kl (cid:15) ijk ⊥ (cid:42) ˆ π l δ , (C5c) (cid:110) ∧ ϕ ij , ∼ ϕ ⊥ lm (cid:111) ≈ RHS of (C4c) , (C5d) (cid:110) ∧ ϕ ij , ϕ T lmn (cid:111) ≈ − J (cid:20) η [ i | [ l η m ] n(cid:42) ˆ π | j ] + 14 η [ i | n η | j ][ l(cid:42) ˆ π m ] − η i [ l η m ] j (cid:42) ˆ π n (cid:21) δ . (C5e) Note that (C5e) is linear in (cid:42) ˆ π k , the momentum of thevector graviton. Appendix D: Heuristic outlook
In this appendix we attempt to quantify the chancesof future success, in light of our present results. Let k viable theories be found in a sample of n = 8, drawnfrom a population of N = 58 theories. We may modelthe probability of there being a grand total of K viabletheories in the parent population as P ( K | k, n, N ) ≡ n + 1 N + 1 P hyp ( k | K, n, N ) , (D1)where the probability P hyp ( k | K, n, N ) of drawing k given K follows the standard hypergeometric distribution P hyp ( k | K, n, N ) ≡ (cid:0) Kk (cid:1)(cid:0) N − Kn − k (cid:1)(cid:0) Nn (cid:1) . (D2)Note that we have assumed a uniform prior on K , P ( K | N ) ≡ ( N + 1) − , which may or may not be justi-fied. The pessimistic interpretation of our study wouldbe k = 0, but in that case the probability that K = 0is found to be only 0 .
15 according to (D1). Rather, wewould expect K = 5 ± .