Metric Lagrangians with two propagating degrees of freedom
aa r X i v : . [ g r- q c ] O c t Metric Lagrangians with two propagating degrees of freedom
Kirill Krasnov
School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK (Dated: October 21, 2009)There exists a large class of generally covariant metric Lagrangians that contain only local termsand describe two propagating degrees of freedom. Trivial examples can be be obtained by apply-ing a local field redefinition to the Lagrangian of general relativity, but we show that the class oftwo propagating degrees of freedom Lagrangians is much larger. Thus, we exhibit a large familyof non-local field redefinitions that map the Einstein-Hilbert Lagrangian into ones containing onlylocal terms. These redefinitions have origin in the topological shift symmetry of BF theory, towhich GR is related in Pleba´nski formulation, and can be computed order by order as expansionsin powers of the Riemann curvature. At its lowest non-trivial order such a field redefinition pro-duces the (
Riemann ) invariant that arises as the two-loop quantum gravity counterterm. Possibleimplications for quantum gravity are discussed. PACS numbers: 04.60.-m
Loop divergences in quantum gravity require higherderivative counterterms to be added to the Lagrangian[1], [2]. Such higher derivative terms typically intro-duce new propagating degrees of freedom (DOF) thatgenerally lead to instabilities, see [3] for an emphasis ofthis point. The only known way to avoid these insta-bilities is to have a well-behaved underlying theory de-scribing the new DOF, for example string theory. Thehigher-derivative metric Lagrangian is then an effectiveone obtained by integrating out some underlying non-gravitational DOF. In this letter we show that there ex-ists a potentially attractive alternative: higher deriva-tive counterterms can be added to the gravitational La-grangian without adding new degrees of freedom.Field redefinitions play an important role in our con-struction, so we start by briefly recalling some relevantfacts. Quantum gravity, with its negative mass dimensioncoupling constant, is non-renormalizable in the sense thatan infinite number of counterterms is required to absorball arising divergences. However, while in a typical renor-malizable theory transformations that absorb infinitiesare limited to field and coupling constant multiplicativerenormalizations, the field redefinition freedom availablein a theory with a dimensionful coupling is considerablylarger. Thus, in the case of (pure, i.e with no mattercouplings) quantum gravity one can perform field redefi-nitions of the schematic type h → h + P n G n ∂ n h + . . . ,where h is the graviton field, G is Newton’s constant, anddots denote terms of higher order in the metric pertur-bation. The power of G here is as relevant for the caseof 4 spacetime dimensions, but similar field redefinitions,with an appropriate modification of the power of G areavailable in other dimensions as well. Such field redefini-tions, being local, are known not to change the S-matrixof the theory, see e. g. [4], section 2, as well as [5], section10 for a discussion of this point. The availability of thesefield redefinitions implies that many of the arising coun-terterms are unphysical in the sense that they can be dis-posed off without any effect on the S-matrix. An extreme example of this situation arises when, in spite of diver-gences being present, they can all be removed by localfield redefinitions without affecting the S-matrix. In thiscase one says that the theory is (on-shell) finite. An ex-ample of a finite but power-counting non-renormalizabletheory is given by pure quantum gravity in 3 spacetimedimensions.For later purposes we note that classically a localmetric field redefinition maps the Einstein-Hilbert La-grangian into a complicated metric Lagrangian contain-ing an infinite number of local terms. The new La-grangian, however, still describes just two propagatingDOF. This can be seen by following the Ostrogradskimethod of introducing new variables for higher timederivatives. One then observes that the arising La-grangian is highly degenerate and generates many con-straints that remove all but the DOF of the original sys-tem. It is simplest to see this mechanism at work byconsidering a higher derivative field redefinition appliedto a finite-dimensional dynamical system.A celebrated result of [1] is that one-loop divergencesof pure quantum gravity in 4 spacetime dimensions canbe removed by a local field redefinition and so the theoryis one-loop finite. It was for some time hoped that thefiniteness may persist to all loops, but an explicit two-loop computation [2] showed that the term ( Riemann ) that is not removable by a local field redefinition isneeded to absorb the divergences arising.On the other hand, non-local redefinitions, i.e. involv-ing negative powers of (cid:3) = ∂ µ ∂ µ , generically do changethe S-matrix. Still, an appropriate ghost action can beintroduced to offset their effect, see [5]. However, suchfield redefinitions typically map a local action to a non-local one, and are thus uninteresting for the purpose ofeliminating local counterterms. Indeed, the simplest ex-ample is given by the free field Lagrangian (1 / ∂ µ φ ) ,which, after a redefinition φ → φ + ( O / (cid:3) ) φ , where O is some local operator, goes into a non-local Lagrangiancontaining 1 / (cid:3) .We now show that in the case of gravity (in 4 space-time dimensions) the class of field transformations thatmap a local Lagrangian into again a local one is muchlarger than that consisting of local field redefinitions. Inother words, there exists an (infinite-parameter) familyof non-local field redefinitions that map the Einstein-Hilbert Lagrangian into a generally covariant metric La-grangian containing only local terms. The redefinitioncan be computed order by order in perturbation theoryas follows. At lowest order, it is the local transformation h µν → h µν + αR µν + βη µν R (1)that produces R µν R µν and R invariants. Here η µν is theMinkowski metric, and R µν , R are the Ricci tensor andscalar for the perturbation h µν respectively. At the nextorder our field redefinition produces the ( Riemann ) in-variant as well as other on-shell vanishing ones and isgiven by h µν → h µν + γ (cid:3) ∂ α ∂ β R γδµα R νβγδ , (2)plus a set of local terms. The reason why (2) produces γ Z R ρσµν R αβρσ R µναβ (3)is that this quantity can be written as: Z E µν γ (cid:3) ∂ α ∂ β (cid:18) R γδµα R νβγδ − η µν R ργδα R βργδ (cid:19) , (4)where E µν = R µν − η µν R . This is checked using theeasily verifiable identity ∂ [ α ∂ [ β R ν ] µ ] = 14 (cid:3) R βναµ (5)that holds to first order in the perturbation field. Notethat the reason why the last term in brackets in (4) wasnot included in (2) is that it is proportional to η αβ , andthus gives rise to a local term.The structure of the field redefinition at higher ordersis similar to (2) in that the non-local operator ∂ α ∂ β / (cid:3) is applied to a rank 4 tensor constructed from an appro-priate power of the Riemann curvature tensor (and itscovariant derivatives), plus a set of local terms. Impor-tantly, at higher orders there are also terms containinghigher negative powers of (cid:3) . These are needed to elimi-nate terms arising as powers of lower order non-localities.The above prescription can be carried out order by or-der, but this becomes technically difficult at higher or-ders. Below we present an alternative description of thesame field redefinition that guarantees that it can be ex-tended to any order and gives an algorithmic procedurefor computing it.At every order the non-local field redefinition sketchedintroduces a set of parameters that, after it is applied to the Einstein-Hilbert Lagrangian, translate into pa-rameters of the arising local metric Lagrangian. Whentruncated to any given order, the Lagrangian one ob-tains contains many new DOF stemming from its higher-derivative nature. However, the complete Lagrangianwith its infinite number of local terms describes just twopropagating DOF. To see this we must introduce a dif-ferent and at first unrelated description of this class ofLagrangians.An alternative description of the two propagating DOFmetric Lagrangians is provided by an infinite-parameterfamily of deformations of general relativity first describedin [6], building upon works [7, 8, 9, 10]. One starts withan observation [7] that (complexified) Einstein’s generalrelativity can be rewritten as a generally-covariant theoryof an SO(3 , C ) connection. This suggests generalizations,leading to an infinite-parameter family [6] of theories de-scribing two propagating degrees of freedom (DOF) andcontaining GR. These two propagating DOF gravity the-ories can be rewritten in metric terms and can be shownto be obtainable from GR precisely by the above non-local field redefinitions.These deformations of GR can be described compactlyas follows. Let A i , i = 1 , , , C ) connec-tion and F i = dA i + (1 / ǫ ijk A j ∧ A k be its curva-ture two-form. The action of the theory is just themost general generally-covariant action that can be con-structed for A i . Thus, consider the 4-form F i ∧ F j .Choosing an arbitrary volume 4-form ( vol ) we can write F i ∧ F j = ( vol )Ω ij , with Ω ij being defined only mod-ulo rescalings ( vol ) → α ( vol ) , Ω ij → (1 /α )Ω ij . Intro-duce a scalar-valued function f ( X ) of 3 × X ij that is SO(3 , C )-invariant f ( OXO T ) = f ( X ) , O ∈ SO(3 , C ), holomorphic, and homogeneous ofdegree one f ( αX ) = αf ( X ). This function can be ap-plied to the 4-form F i ∧ F j with the result being a 4-form f ( F i ∧ F j ) = ( vol ) f (Ω), independent of which volume 4-form is used. Thus, we can write a generally-covariantand gauge-invariant action as follows: S [ A ] = Z f ( F i ∧ F j ) . (6)It can then be shown that for a generic f ( · ) this givesa theory that describes 2 (complex) propagating DOF.This can be seen by noting that the phase space of thistheory is parametrized by pairs (spatial projection of theconnection, conjugate momentum). The theory is diffeo-morphism and gauge-invariant which means that thereare 4 + 3 first-class constraints acting on the phase space.With the configuration space being 3 × f ( · ) being the δ -function projecting onto:TrΩ = 12 (TrΩ) . (7)Note that the clause about f ( · ) being generic is impor-tant, for the Lagrangian Tr F ∧ F , which is also in theclass (6), is a total divergence and corresponds to a the-ory without propagating DOF. The description of thetheory given here is new, but can be shown to be equiv-alent to one given in [6].As was realized in [11, 12], the theory (6) can be putinto a form that makes the spacetime metric it describesmore explicit. In the retrospect, this is done via a stan-dard ”duality” trick of introducing a set of new fields thatare later taken to be fundamental, with the fields of theoriginal formulation to be integrated out. The new fieldsin our case are two-form fields B i that are valued in theLie-algebra of SO(3 , C ). The new action is given by: S [ B, A ] = Z B i ∧ F i − V ( B i ∧ B j ) . (8)Here V ( · ) is again a holomorphic, SO(3 , C )-invariant, ho-mogeneous function of order one so that it can be appliedto the 4-form B i ∧ B j . Integrating the two-form field B i out by solving its (algebraic) field equations one getsback (6) with f ( · ) being an appropriate Legendre trans-form of V ( · ). One can now take the two-form field B i tobe fundamental, and eliminate A i completely by solvingits field equations that are algebraic. This converts (8)into a second-order theory for the two-form field B i .The spacetime metric described by the theory becomesalmost manifest in the two-form field formulation (8).Thus, it can be shown that the theory is about the space-time (conformal) metric with respect to which the set oftwo-forms F i (or, equivalently, B i ) is self-dual. It is nothard to show that there is a unique such (conformal)metric, see e.g. [13]. Explicitly, this metric is given by: √− gg µν ∼ ǫ ijk B iµα B jνβ B kρσ ˜ ǫ αβρσ . (9)Introducing the conformal metric (9), the action (8) canbe explicitly rewritten in a second-order form as that ofthe metric plus a set of auxiliary non-propagating fields.This is done by introducing a set of special self-dual ”met-ric” two-forms Σ i that satisfy:Σ i ∧ Σ j ∼ δ ij . (10)These forms are easily constructed by introducing atetrad θ I , I = (0 , i ) for the metric, and taking the self-dual part of the two-form θ I ∧ θ J given by:Σ i = i θ ∧ θ i − ǫ ijk θ j ∧ θ k . (11)It can be shown that the knowledge of two-forms that areself-dual and satisfy (10) is equivalent to the knowledgeof the metric. A general self-dual two-form B i can thenbe written as: B i = b ij Σ j , (12) where b ij are arbitrary scalars. The theory (8) with theconnection A i eliminated via its field equations then be-comes a second-order theory of the metric described byΣ i and the non-propagating scalars b ij . A simple phasespace analysis shows that the theory contains only twopropagating DOF. The scalars can then be integratedout to produce a purely metric theory. This leads to aLagrangian given by an infinite expansion in terms of lo-cal curvature invariants, which describes two propagatingDOF by construction. The above discussion was phrasedin terms of complex spacetime metrics, but appropriatereality conditions can be imposed, and the story repeatsitself for real Lorentzian signature metrics.Thus, we have seen that among all generally-covariantlocal (i.e. containing only local terms) metric La-grangians there is an infinite-parameter subset that de-scribes, as GR, only two propagating DOF. To see thenon-local field redefinitions that relate these Lagrangiansto GR we note that in the formulation (8) the first BFterm possesses a large symmetry B i → B i + Dη i , where η i is a Lie-algebra valued one-form, and D is the co-variant derivative with respect to the connection A i . Asubgroup of this symmetry group is formed by space-time diffeomorphisms. The second, potential term of theaction is only invariant under this diffeomorphism sub-group, and this is the reason why the above ”topologicalshift” transformation is not a symmetry of the whole ac-tion. This is also the reason why (8), unlike BF-theory,has propagating DOF.The topological shift transformation described can beused to map (8) to the Einstein-Hilbert action plus asimple potential term for a set of scalars that are decou-pled from the metric. At the linearized level this wasnoted in [14]. To see this for the full theory, we use theobservation of [15] that the Einstein-Hilbert Lagrangiancan be written in BF form in terms of the two-formsΣ i constructed from the metric. We then note that thetopological shift symmetry can be used to transform anytwo-form field B i into a ”metric” one Σ i = B i + Dη i satisfying (10). A detailed demonstration of this factis beyond the scope of this letter, but it is not hard tosee that the number of parameters available in the one-form field η i , modulo diffeomorphisms and modulo the”gauge” η i → η i + Dφ i , where φ i is a Lie-algebra valuedzero-form, matches precisely the number of ”metricity”equations (10) to be satisfied. It can also be shown, atleast perturbatively around the Minkowski background,that the two-form Σ i arising this way is unique. Thisdiscussion shows that the first BF-term of the action (8)can be written as the Einstein-Hilbert one for the met-ric obtained from B i by the topological shift symmetry.The second term in (8) then becomes a potential for thenon-propagating scalars contained in B i . By their fieldequations these scalars are set to a value corresponding toa minimum of the potential, and decouple, which leavesone with the Einstein-Hilbert action (with a cosmologicalconstant whose value is given by the minimum of V ( · ))for the metric described by Σ i . This shows that thereexists a field redefinition that maps (8) into the Einstein-Hilbert action. The field redefinition in question involvessolving a differential equation for the shift one-form pa-rameter η i , and is thus non-local. It can be computedorder by order perturbatively expanding the metric(s)around the Minkowski background. Details will appearelsewhere. The end result is given by the transformationthat was described in the beginning of this letter, withparameters of the transformation related to those of thepotential V ( · ).To summarize, we have seen that the set of generally-covariant local metric Lagrangians describing two prop-agating DOF is larger than the one obtainable from GRby local field redefinitions, and admits a very compactdescription (6). It is obtainable from GR by special non-local field redefinitions that stem from the topologicalshift symmetry of the BF-part of the action (8).Let us conclude by discussing what the existence ofan (infinitely) large class of two propagating DOF met-ric Lagrangians may imply for the problem of quan-tum gravity. The fact that the ( Riemann ) countertermneeded at two loops is contained in our two propagat-ing DOF Lagrangians suggests that it may be possible todevice a renormalization scheme for gravity so that thecounterterm-corrected Lagrangian remains within twoDOF class at every order of perturbative expansion. Forthis to be possible the class of theories (6) must be closedunder renormalization, which appears plausible, since theLagrangian in (6) is just the most general one compat-ible with gauge and diffeomorphism invariance. Such arenormalization scheme, if possible, would give a quan-tum theory of gravity with two propagating DOF, whichwould be in striking contrast with other quantum grav-ity scenarios (e.g. string theory) that typically introducenew DOF.If this scenario was possible, one would face a ques-tion about implications of the non-local topological shiftsymmetry described. While generically non-local fieldredefinitions do change the S-matrix, our redefinitionsare certainly of a very special nature since a local ac-tion is mapped again into a local one. Therefore, the general conclusion has to be carefully re-examined. Pre-liminary considerations suggest that our non-local trans-formations might not affect the S-matrix. If this was so,then all quantum divergences were disposable without af-fecting the S-matrix, and the quantum theory would befinite. It is of considerable interest to see if this visioncan be realized.The story described is that for pure, i.e. not coupledto any matter sources, gravity. Indeed, coupling to usualtype matter essentially removes the field redefinition free-dom. However, similarly to how the one-loop finitenessresult [1] extends to special matter couplings provided bysupergravity theories, our story may also be applicable togravity coupled to at least certain types of matter. 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