Apparent ghosts and spurious degrees of freedom in non-local theories
AApparent ghosts and spurious degreesof freedom in non-local theories
Stefano Foffa, Michele Maggiore and Ermis Mitsou
D´epartement de Physique Th´eorique and Center for Astroparticle Physics,Universit´e de Gen`eve, 24 quai Ansermet, CH–1211 Gen`eve 4, Switzerland
Abstract
Recent work has shown that non-local modifications of the Einstein equations canhave interesting cosmological consequences and can provide a dynamical origin fordark energy, consistent with existing data. At first sight these theories are plaguedby ghosts. We show that these apparent ghost-like instabilities do not describe ac-tual propagating degrees of freedom, and there is no issue of ghost-induced quantumvacuum decay. a r X i v : . [ h e p - t h ] N ov Introduction
Non-local classical equations and non-local field theories have been sporadically studiedsince the early days of field theory [1]. In general, they present a number of subtle issuesconcerning ghosts and unitarity, and often it is not obvious even how many degrees offreedom they describe, see e.g. [2–6]. In recent years there has been a renewed interest fornon-local models in connection with cosmology. Non-local effective equations of motioncan play an important role in explaining the “old” cosmological constant problem, througha degravitation mechanism that promotes Newton’s constant to a non-local operator [7, 8](see also [9–12]), and non-local cosmological models have interesting observational con-sequences and can play a role in explaining the origin of dark energy [13–26]. Non-localgravity models have also been studied as UV modifications of GR, see e.g. [27–31].In two recent papers [32, 33] we have proposed a non-local approach that allows us tointroduce a mass term in the Einstein equations, in such a way that the invariance underdiffeomorphisms is not spoiled, and we do not need to introduce an external referencemetric (contrary to what happens in the conventional local approach to massive gravity[34–40]). In particular, in [33] has been proposed a classical model based on the non-localequation G µν − d − d m (cid:0) g µν (cid:50) − R (cid:1) T = 8 πG T µν . (1.1)Here d is the number of spatial dimensions (the factor ( d − / d is a convenient normaliza-tion of the parameter m ), (cid:50) = g µν ∇ µ ∇ ν is the d’Alembertian operator with respect tothe metric g µν and, quite crucially, (cid:50) − is its inverse computed with the retarded Green’sfunction. The superscript T denotes the extraction of the transverse part of the tensor,which exploits the fact that, in a generic Riemannian manifold, any symmetric tensor S µν can be decomposed as S µν = S T µν + ( ∇ µ S ν + ∇ ν S µ ), with ∇ µ S T µν = 0 [41, 42]. Theextraction of the transverse part of a tensor is itself a non-local operation, which involvesfurther (cid:50) − operators. For instance in flat space, where ∇ µ → ∂ µ , it is easy to show that S T µν = S µν − (cid:50) ( ∂ µ ∂ ρ S ρν + ∂ ν ∂ ρ S ρµ ) + 1 (cid:50) ∂ µ ∂ ν ∂ ρ ∂ σ S ρσ . (1.2)Again, in eq. (1.1) all (cid:50) − factors coming from the extraction of the transverse part are de-fined with the retarded Green’s function, so that eq. (1.1) satisfies causality. Furthermore,since the left-hand side of eq. (1.1) is transverse by construction, the energy-momentumtensor is automatically conserved, ∇ µ T µν = 0. Both causality and energy-momentumconservation were lost in the original degravitation proposal [7], and in this sense eq. (1.1)can be seen as a refinement of the original idea. However, the explicit appearance ofretarded Green’s function in the equations of motion has important consequences for theconceptual meaning of an equation such as (1.1), as we will discuss below.As shown in [33, 43], eq. (1.1) has very interesting cosmological properties, and inparticular it generates a dynamical dark energy. Since during radiation dominance (RD)the Ricci scalar R vanishes, the term (cid:50) − R starts to grow only during matter dominance(MD), thereby providing in a natural way a delayed onset of the accelerated expansion(similarly to what happens in the model proposed in [13]). Furthermore, this model ishighly predictive since it only introduces a single parameter m , that replaces the cosmo-logical constant in ΛCDM. In contrast, models based on quintessence, f ( R )-gravity, or1he non-local model of [13] in which a term Rf ( (cid:50) − R ) is added to the Einstein action,all introduce at least one arbitrary function, which is typically tuned so to get the desiredcosmological behavior. In our case, we can fix the value of m so to reproduce the observedvalue Ω DE (cid:39) .
68. This gives m (cid:39) . H , and leaves us with no free parameter. We thenget a pure prediction for the EOS parameter of dark energy. Quite remarkably, writing w DE ( a ) = w + (1 − a ) w a , in [33] we found w (cid:39) − .
04 and w a (cid:39) − .
02, consistent withthe Planck data, and on the phantom side.These cosmological features make eq. (1.1) a potentially very attractive dark energymodel. The presence of the (cid:50) − operator raises however a number of potential problemsof theoretical consistency, and the purpose of this paper is to investigate them in somedetail. The crucial problem can already be seen linearizing eq. (1.1) over flat space.Writing g µν = η µν + κh µν , where κ = (32 πG ) / and η µν = ( − , + , . . . , +), the equation ofmotion of this theory takes the form E µν,ρσ h ρσ − d − d m P µν ret P ρσ ret h ρσ = − πGT µν , (1.3)where E µν,ρσ is the Lichnerowicz operator, while P µν ret = η µν − ∂ µ ∂ ν (cid:50) ret , (1.4)and 1 / (cid:50) ret is the retarded inverse of the flat-space d’Alembertian. Apparently, the corre-sponding quadratic Lagrangian is L = 12 h µν E µν,ρσ h ρσ − d − d m h µν P µν P ρσ h ρσ . (1.5)Adding the usual gauge fixing term of linearized massless gravity, L gf = − ( ∂ ν ¯ h µν )( ∂ ρ ¯ h ρµ ),and inverting the resulting quadratic form we get the propagator˜ D µνρσ ( k ) = − i k (cid:18) η µρ η νσ + η µσ η νρ − d − η µν η ρσ (cid:19) − d ( d − im k ( − k + m ) η µν η ρσ , (1.6)plus terms proportional to k µ k ν , k ρ k σ and k µ k ν k ρ k σ , that give zero when contracted witha conserved energy-momentum tensor. The first term is the usual propagator of a masslessgraviton, for d generic. The term proportional to m gives an extra contribution to thesaturated propagator ˜ T µν ( − k ) ˜ D µνρσ ( k ) ˜ T ρσ ( k ), equal to1 d ( d −
1) ˜ T ( − k ) (cid:20) − ik − i ( − k + m ) (cid:21) ˜ T ( k ) . (1.7)This term apparently describes the exchange of a healthy massless scalar plus a ghostlikemassive scalar. In general, a ghost has two quite distinct effects: at the classical level, it cangive rise to runaway solutions. In our cosmological context, rather than a problem this canactually be a virtue, because a phase of accelerated expansion is in a sense an instabilityof the classical evolution. Indeed, ghosts have been suggested as models of phantom darkenergy [44, 45]. The real trouble is that, at the quantum level, a ghost corresponds toa particle with negative energy and induces a decay of the vacuum, through processes2n which the vacuum decays into ghosts plus normal particles, and renders the theoryinconsistent.The main purpose of this paper is to discuss and clarify some subtle conceptual issuesrelated to this apparent ghost-like degree of freedom and to show that, in fact, in thistheory there is no propagating ghost-like degree of freedom. The paper is organized asfollows. In sect. 2 we show that the status of a non-local equation such as eq. (1.1) is thatof an effective classical equation, derived from some classical or quantum averaging in amore fundamental theory. In sect. 3 we show that similar apparent ghosts even appearin massless GR when one decomposes the metric perturbation into a transverse-tracelesspart h TT µν and a trace part η µν s . This is due to the fact that the relation between { h TT µν , s } and the original metric perturbations h µν is non-local. We show that (contrary to somestatements in the literature) the apparent ghost field s is not neutralized by the helicity-0component of h TT µν . Rather, what saves the vacuum stability of GR, in these variables, isthat s (as well as the helicity-0, ± h TT µν ), is a non-propagating field andcannot be put on the external lines, nor in loops. Beside having an intrinsic conceptualinterest, this analysis will also show that the same considerations extend straightforwardlyto the non-local modification of GR that we are studying. Finally, in sect. 4 we will workout the explicit relation between the fake ghost that is suggested by eq. (1.7), and thespurious degrees of freedom that are know to emerge when a non-local theory is writtenin local form by introducing auxiliary fields. Sect. 5 contains our conclusions. A crucial point of eq. (1.1), or of its linearization (1.3), is that they contain explicitly aretarded propagator. This retarded prescription is forced by causality, which we do notwant to give up. We are used, of course, to the appearance of retarded propagators in thesolutions of classical equations. Here however the retarded propagator already appears inthe equation itself, and not only in its solution. Is it possible to obtain such an equationfrom a variational principle? The answer, quite crucially, is no. As already observed byvarious authors [13, 18, 32], a retarded inverse d’Alembertian cannot be obtained from thevariation of a non-local action. Consider for illustration a non-local term in an action ofthe form (cid:82) dxφ (cid:50) − φ , where φ is some scalar field, and (cid:50) − is defined with respect to someGreen’s function G ( x ; x (cid:48) ). Taking the variation with respect to φ ( x ) we get δδφ ( x ) (cid:90) dx (cid:48) φ ( x (cid:48) )( (cid:50) − φ )( x (cid:48) ) = δδφ ( x ) (cid:90) dx (cid:48) dx (cid:48)(cid:48) φ ( x (cid:48) ) G ( x (cid:48) ; x (cid:48)(cid:48) ) φ ( x (cid:48)(cid:48) )= (cid:90) dx (cid:48) [ G ( x ; x (cid:48) ) + G ( x (cid:48) ; x )] φ ( x (cid:48) ) . (2.1)We see that the variational of the action automatically symmetrizes the Green’s function.It is therefore impossible to obtain in this way a retarded Green’s function in the equationsof motion, since G ret ( x ; x (cid:48) ) is not symmetric under x ↔ x (cid:48) ; rather G ret ( x (cid:48) ; x ) = G adv ( x ; x (cid:48) ).The same happens if we take the variation of the Lagrangian (1.5). Writing explicitly theconvolution with the Green’s function as we did in eq. (2.1) we find that it is not possibleto get the term P µν ret P ρσ ret in eq. (1.3). If in the action the term (cid:50) − that appears in P µν isdefined with a symmetric Green’s function, so that G ( x ; x (cid:48) ) = G ( x (cid:48) ; x ), we find the same3reen’s function in the equation of motion. If, in contrast, we use h µν P µν ret P ρσ ret h ρσ in theaction, in the equation of motions we get ( P µν ret P ρσ ret + P µν adv P ρσ adv ) h ρσ .Of course, one can take the point of view that the classical theory is defined by itsequations of motion, while the action is simply a convenient “device” that, through a setof well defined rules, allows us to compactly summarize the equations of motion. We canthen take the formal variation of the action and at the end replace by hand all factors (cid:50) − by (cid:50) − in the equation of motion. This is indeed the procedure used in [13, 46],in the context of non-local gravity theories with a Lagrangian of the form Rf ( (cid:50) − R ).As long as we see the Lagrangian as a “device” that, through a well defined procedure,gives a classical equation of motion, this prescription is certainly legitimate. However,any connection between these classical causal equations of motion and the quantum fieldtheory described by such a Lagrangian is now lost. In particular, the terms in eq. (1.6)or in eq. (1.7) that apparently describe the exchange of a healthy massless scalar plus aghostlike massive scalar are just the propagators that, to reproduce eq. (1.3), after thevariation must be set equal to retarded propagators. Taking them as Feynman propagatorsin a QFT gives a quantum theory that has nothing to do with our initial classical equation(1.3) and that has dynamical degrees of freedom that, with respect to our original problem,are spurious.Thus, eq. (1.1) is not the classical equation of motion of a non-local quantum field the-ory. To understand its conceptual meaning, we observe that non-local equations involvingthe retarded propagator appear in various situation in physics, but are never fundamental.They rather typically emerge after performing some form of averaging, either purely clas-sical or at the quantum level. In particular, non-local field equations govern the effectivedynamics of the vacuum expectation values of quantum fields, which include the quan-tum corrections to the effective action. The standard path integral approach provides thedynamics for the in-out matrix element of a quantum field, e.g. (cid:104) out | ˆ φ | in (cid:105) or, in a semi-classical approach to gravity, (cid:104) out | ˆ g µν | in (cid:105) . The classical equations for these quantitiesare however determined by the Feynman propagator, so they are not causal, since theycontain both the retarded and the advanced Green’s function. This is not surprising, sincethe in-out matrix element are not directly measurable quantities, but only provide inter-mediate steps in the QFT computations. Furthermore, even if ˆ φ is a hermitean operator,its in-out matrix element are complex. In particular, this makes it impossible to interpret (cid:104) out | ˆ g µν | in (cid:105) as an effective metric. In contrast, the in-in matrix elements are real, andsatisfy non-local but causal equations [47, 48], involving only retarded propagators (thatcan be computed using the Schwinger-Keldysh formalism).Similar non-local but causal equations can also emerge from a purely classical averagingprocedure, when one separates the dynamics of a system into a long-wavelength and ashort-wavelength part. One can then obtain an effective non-local but causal equation forthe long-wavelength modes by integrating out the short-wavelength modes, see e.g. [49]for a recent example in the context of cosmological perturbation theory. Another purelyclassical example comes from the standard post-Newtonian/post-Minkowskian formalismsfor GW production [50, 51]. In linearized theory the gravitational wave (GW) amplitude h µν is determined by (cid:50) ¯ h µν = − πGT µν , where ¯ h µν = h µν − (1 / hη µν . In such a radiationproblem this equation is solved with the retarded Green’s function, ¯ h µν = − πG (cid:50) − T µν .When the non-linearities of GR are included, the GWs generated at some perturbativeorder become themselves sources for the GW generation at the next order. In the far-wave4one, this iteration gives rise to effective equations for ¯ h µν involving (cid:50) − .In summary, non-local equations involving (cid:50) − are not the classical equation of motiona non-local QFT (a point already made e.g. in [13, 18, 24, 25, 52]). Even if we can find aclassical Lagrangian whose variation reproduces them (once supplemented with the (cid:50) − → (cid:50) − prescription after having performed the variation), the quantum field theory describedby this Lagrangian has a priori nothing to do with the problem at hand. Issues of quantumconsistencies (such as the possibility of a vacuum decay amplitude induced by ghosts) canonly be addressed in the fundamental theory that, upon classical or quantum smoothing,produces these non-local (but causal) classical equations.So, there is no sense, and no domain of validity, in which the Lagrangian (1.5) canbe used to define a QFT associated to our theory. To investigate whether the classicalequation (1.1) derives from a QFT with a stable quantum vacuum we should identify thefundamental theory and the smoothing procedure that give rise to it, and only in thisframework we can pose the question. It is instructive to see more generally how spurious degrees of freedom, and in particu-lar spurious ghosts, can appear when one uses non-local variables. A simple and quiterevealing example is provided by GR itself. We have already discussed this example inapp. B of [53], but it is useful to re-examine and expand it in this context. We considerGR linearized over flat space. The quadratic Einstein-Hilbert action is S (2)EH = 12 (cid:90) d d +1 x h µν E µν,ρσ h ρσ (3.1)We decompose the metric as h µν = h TT µν + 12 ( ∂ µ (cid:15) ν + ∂ ν (cid:15) µ ) + 1 d η µν s , (3.2)where h TT µν is transverse and traceless, ∂ µ h TT µν = 0, η µν h TT µν = 0. The vector (cid:15) µ could befurther decomposed as (cid:15) µ = (cid:15) T µ + ∂ µ α , where ∂ µ (cid:15) T µ = 0. Under a linearized diffeomorphism h µν → h µν − ( ∂ µ ξ ν + ∂ ν ξ µ ) we have (cid:15) µ → (cid:15) µ − ξ µ while the tensor h TT µν and the scalar s aregauge invariant. Plugging eq. (3.2) into eq. (3.1) we find that (cid:15) µ cancels (as it is obviousfrom the fact that eq. (3.1) is invariant under linearized diffeomorphisms and (cid:15) µ is a puregauge mode), and S (2)EH = 12 (cid:90) d d +1 x (cid:20) h TT µν (cid:50) ( h µν ) TT − d − d s (cid:50) s (cid:21) . (3.3)Performing the same decomposition in the energy-momentum tensor, the interaction termcan be written as S int = κ (cid:90) d d +1 x h µν T µν = κ (cid:90) d d +1 x (cid:20) h TT µν ( T µν ) TT + 1 d sT (cid:21) , (3.4)5o the equations of motion derived from S (2)EH + S int are (cid:50) h TT µν = − κ T TT µν , (3.5) (cid:50) s = κ d − T . (3.6)This result can be surprising, because it seems to suggest that in ordinary massless GRwe have many more propagating degrees of freedom than expected: the components of thetransverse-traceless tensor h TT µν (i.e. 5 components in d = 3) plus the scalar s . Note thatthese degrees of freedom are gauge invariant, so they cannot be gauged away. Furthermore,from eq. (3.3) the scalar s seems a ghost! Of course these conclusions are wrong, and forany d linearized GR is a ghost-free theory; in particular, in d = 3 it only has two radiativedegrees of freedom, corresponding to the ± d ,( d + 1)( d − / SO ( d − s , as well as all the extra components of h TT µν , satisfy a Klein-Gordon rather than aPoisson equation?The answer, as discussed in [53], is related to the fact that h TT µν and s are non-localfunctions of the original metric perturbation h µν . In particular, inverting eq. (3.2) onefinds that s = (cid:18) η µν − (cid:50) ∂ µ ∂ ν (cid:19) h µν = P µν h µν . (3.7)The fact that s , as a function of h µν , is non-local in time means that the initial dataassigned on h µν on a given time slice are not sufficient to evolve s , so a naive countingof degrees of freedom goes wrong. A simple but instructive example of what exactly goeswrong is provided by a scalar field φ that satisfies a Poisson equation ∇ φ = ρ . If wedefine a new field ˜ φ from ˜ φ = (cid:50) − φ , the original Poisson equation can be rewritten as (cid:50) ˜ φ = ∇ − ρ ≡ ˜ ρ , (3.8)so now ˜ φ looks like a propagating degree of freedom. However, for ρ = 0 our originalequation ∇ φ = ρ only has the solution φ = 0. If we want to rewrite it in terms of ˜ φ without introducing spurious degrees of freedom we must therefore supplement eq. (3.8)with the condition that, when ρ = 0, ˜ φ = 0. In other words, the homogeneous plane wavesolutions of eq. (3.8), ˜ φ hom ( x ) = (cid:90) d k (2 π ) (cid:104) a k e ikx + a ∗ k e − ikx (cid:105) (3.9)is fixed uniquely by the original equation, and a k , a ∗ k cannot be considered as free param-eters that, upon quantization, give rise to the creation and annihilation operators of thequantum theory.Exactly the same situation takes place in GR, for the field s and for the the extracomponents in h TT µν . For instance, writing s in terms of the variables entering the (3 + 1)decomposition (and specializing to d = 3) one finds s = 6Φ − (cid:50) − ∇ (Φ + Ψ) , (3.10)6igure 1: The tree-level Feynman graph describing the exchange of a graviton betweentwo matter lines. In this graph, the contribution due to the exchange of the scalar s iscanceled by the exchange of the helicity-0 component of h TT µν .where Φ and Ψ are the scalar Bardeen’s variable defined in flat space (see [53]). Since Φand Ψ are non-radiative and satisfy Poisson equations, s is non-radiative too, and it isjust the (cid:50) − factor in eq. (3.10) that (much as the (cid:50) − in the definition of ˜ φ ), potentiallyintroduces a fake propagating degree of freedom. In order to eliminate such a spuriousdegree of freedom, we must supplement eq. (3.6) with the condition that s = 0 when T = 0, i.e. we must discard again the homogeneous solution of eq. (3.6) (and similarly foreq. (3.5)). This implies that, at the quantum level, there are no creation and annihilationoperators associated to s . Therefore s cannot appear on the external legs of a Feynmandiagram, and there is no Feynman propagator associated to it, so it cannot circulate inthe loops.One might also observe that the contribution to the propagator of s is canceled byan equal and opposite contribution due to the helicity-0 component of h TT µν , see app. A1of [38]. However this only shows that, in the classical matter-matter interaction describedby tree level diagrams such as that in Fig. 1, these contributions cancel and we remain, asexpected, with the contribution from the exchange of the helicity ±
2. This cancellationhas nothing to do with the vacuum stability of GR. Consider in fact the graphs shownin Fig. 2. If s were a dynamical ghost field that can be put on the external line, thesegraphs would describe a vacuum decay process. Such a process is kinematically allowedbecause the ghost s carries a negative energy that compensates the positive energies of theother final particles. There is no corresponding vacuum decay graph in which we replace s by the helicity-0 component of h TT µν , since the latter is not a ghost, and the process isno longer kinematically allowed. In any case, these processes have different final states, sothe positive probability for, e.g., the decay vac → ssφφ shown in Fig. 2 (where φ is anynormal matter field, or a graviton) cannot be canceled by anything.It is interesting, and somewhat subtle, to understand the same point in terms ofthe imaginary part of vacuum-to-vacuum diagrams. The vacuum-to-vacuum diagramscorresponding to the processes of Fig. 2 are shown in Fig. 3. Here, whenever we have adashed line corresponding to s , we indeed have a corresponding graph where this line isreplaced by the propagation of the helicity-0 component of h TT µν , and one might believe thatthese graphs cancel. In fact this is not true, due to a subtlety in the i(cid:15) prescriptions of thepropagators. For a normal particle the usual scalar propagator is − i/ ( k + m − i(cid:15) ) (with7 g s(a) ss(b) φφ s φ Figure 2: Examples of vacuum decay graphs that would be induced by s if we were allowedto put it on the external legs. The wavy lines denotes gravitons, the solid line a genericmatter field φ (or a graviton itself) and the dashed line the would-be ghost field s .our ( − , + , + , +) signature). For a propagating ghost the correct prescription is instead i/ ( k − m + i(cid:15) ). As discussed in [54], this + i(cid:15) choice propagates negative energies forwardin time but preserves the unitarity of the theory and the optical theorem. With a − i(cid:15) choice, in contrast, ghosts carry positive energy but negative norm, and the probabilisticinterpretation of QFT is lost. This latter choice is therefore unacceptable. In our case m = 0 and the sum of the contributions to each internal line of the “healthy” helicity-0component of h TT µν and the ghost s is − ik − i(cid:15) + ik + i(cid:15) (3.11)We see that, because of the different i(cid:15) prescriptions, these two terms do not cancel.Indeed, the ghost contribution to the diagrams of Fig. 3 generates an imaginary part thatcorresponds to the modulus square of the corresponding diagrams in Fig. 2, as requiredby unitarity. In contrast, the contribution to the diagram of Fig. 3 from the helicity-0component of h TT µν has no imaginary part, again in agreement with unitarity, since theprocesses corresponding to Fig. 2, with s replaced by the helicity-0 component of h TT µν , arenot kinematically allowed.To sum up, what saves the vacuum stability in GR is not a cancelation between thecontributions of the ghost s and that of the helicity-0 component of h TT µν . If one treatsthem as propagating degrees of freedom there is no such cancelation, and one reachesthe (wrong) conclusion that in GR the vacuum is unstable. Rather, vacuum stability ispreserved by the fact that the field s , as well as the extra components of h TT µν , are non-radiative. There are no destruction nor creation operators associated to them, and we arenot allowed to put these fields on the external lines or in loops.In other words, the theory defined by eq. (3.3) is not equivalent to that defined bythe quadratic Einstein-Hilbert action (3.1), because the non-local transformation between h µν and { h TT µν , s } introduces spurious propagating modes. We can still describe GR using8 a) (b) Figure 3: The vacuum-to-vacuum diagrams corresponding to the processes shown in Fig. 2.the formulation in terms of { h TT µν , s } , but in this case we must impose on eq. (3.6) theboundary condition that s = 0 when T = 0 (and similarly for the extra components of h TT µν in eq. (3.5)), in order to eliminate these spurious modes. It is now straightforward to make contact between linearized GR and its non-local massivedeformation given by eq. (1.3). Integrating by parts the operator P µν and using eqs. (3.3)and (3.7), the Lagrangian (1.5) can be written as L = 12 h µν E µν,ρσ h ρσ − d − d m ( P µν h µν ) = 12 (cid:20) h TT µν (cid:50) ( h µν ) TT − d − d s ( (cid:50) + m ) s (cid:21) . (3.12)Thus, the non-local term in eq. (1.5) is simply a mass term for the field s . However, inthe original equation of motion (1.3) that this action is supposed to reproduce, the non-local term was defined with the retarded Green’s function. Thus, in order not to introducespurious propagating degrees of freedom, we must simply continue to impose the conditionthat s is a non-radiative field, just as we did in GR. In other words, now the equation ofmotion (3.6) is replaced by ( (cid:50) + m ) s = κ d − T , (3.13)and, just as in eq. (3.9), we must refrain from interpreting the coefficients of the the planewaves e ± ikx with k = m as free parameters that, upon quantization, give rise to creationand annihilation operators. Thus, again, there are no creation and annihilation operatorsassociated to s , which therefore cannot appear on the external lines of graphs such asthose in Fig. 2, not in the internal lines of graphs such as those in Fig. 3, and there isno vacuum decay. Observe that, since the pole of s is now massive while that of thehelicity-0 mode of h TT µν remain massless, there is no cancelation among them in the treegraphs that describe the classical matter-matter interaction, which is therefore modified9t cosmological distances, compared to GR. This is just as we want, since our aim is tomodify classical gravity in the IR. In contrast, the lack of cancelation between s and thehelicity-0 mode has nothing to do with unitarity and vacuum decay. As discussed above,this cancellation does not take place even in the m = 0 case. Graphs such as those inFig. 2 could not be canceled by anything, and the reason why the vacuum decay amplitudein GR is zero is that these graphs simply do not exist, because we cannot put s on theexternal lines. It is instructive to compare the situation with the usual local theory of linearizedmassive gravity. With a generic local mass term, the quadratic Lagrangian reads L FP = 12 h µν E µν,ρσ h ρσ − m b h µν h µν + b h ) . (3.14)Using again the decomposition (3.2), now the action depends also on (cid:15) µ , since the in-variance under linearized diffeomorphisms is broken. Writing (cid:15) µ = (cid:15) T µ + ∂ µ α , the scalarsector now depends both on s and α , with h scalar µν = ∂ µ ∂ ν α + (1 /d ) η µν s . In particular, themass term in eq. (3.14) produces a term proportional to ( b + b )( (cid:50) α ) . For b , b generic,this higher-derivative term gives rise to a ghost, and the Fierz-Pauli tuning b + b = 0is designed so to get rid of it. Indeed, this longitudinal mode of the metric of the form ∂ µ ∂ ν α is nothing but the mode that is isolated using the St¨uckelberg formalism, and thedRGT theory [34, 35] is constructed just to ensure that its equations of motion remainsof second order, even at the non-linear level. The situation is quite different from that ineq. (3.12), where no higher-derivative term is generated, but we just added a mass termto an already non-radiative field. An alternative way of studying the degrees of freedom of a non-local theory is to transformit into a local theory by introducing auxiliary fields, see e.g. [55, 56]. As it has been rec-ognized in various recent papers [18, 24, 57, 58], such “localization” procedure introduceshowever spurious solutions, and in particular spurious ghosts. This is in fact an equiva-lent way of understanding that the apparent ghosts of these theories do not necessarilycorrespond to propagating degrees of freedom. An example that has been much studiedis the non-local model originally proposed by Deser and Woodard [13], which is based onthe action S = 116 πG (cid:90) d x √− gR (cid:2) f ( (cid:50) − R ) (cid:3) , (4.1)for some function f . This can be formally rewritten in local form introducing two fields ξ ( x ) and φ ( x ) and writing S = 116 πG (cid:90) d x √− g { R [1 + f ( φ )] + ξ ( (cid:50) φ − R ) } . (4.2) Observe that, because of the lack of cancelation due to the signs of the i(cid:15) factors in eq. (3.11),the argument proposed in sect. 6.1 of [32] (where the contribution to the vacuum-to-vacuum amplitudecoming from the massive ghost and the massless scalars in eq. (1.7) partially canceled, modulo corrections O ( m /E )), is incorrect. However, we now see that the result is even stronger. The vacuum decayamplitude in the theory with finite m is not suppressed by factors m / Λ (where Λ is the UV cutoff) butis in fact identically zero, because s cannot appear on the external lines of a Feynman diagram, neither for m = 0 nor for m (cid:54) = 0. ξ is a Lagrange multiplier that enforces the equation (cid:50) φ = R , so that formally φ = (cid:50) − R . The kinetic term ξ (cid:50) φ = − ∂ µ ξ∂ µ φ can be diagonalized writing ξ = ϕ + ϕ , φ = ϕ − ϕ , and then S = 116 πG (cid:90) d x √− g { R [1 + f ( ϕ − ϕ ) − ϕ − ϕ ] − ∂ µ ϕ ∂ µ ϕ + ∂ µ ϕ ∂ µ ϕ } , (4.3)and we see that one of the two auxiliary fields ( ϕ , given our signature) is a ghost. However,this apparent ghost is a spurious degree of freedom, as it is immediately understoodobserving that the above formal manipulation hold even when the function f ( x ) is equalto a constant f [57] (or, in fact, even when f = 0). In this case the original action(4.1) is obviously the same as GR with a rescaled Newton constant, and certainly hasno ghost (and, in fact, it has no ghost also for a broad class of functions f ( (cid:50) − R ) [24]).Once again, the point is that eq. (4.3) is equivalent to eq. (4.1) only if we discard thehomogeneous solution of (cid:50) φ = R , and therefore there are no annihilation and creationoperators associated to φ (nor to ξ ). A similar example has been given, for a non-localmodel based on the term R µν (cid:50) − G µν , in [18], where it was also clearly recognized thatthe auxiliary ghost field that results from the localization procedure never exists as apropagating degree of freedom, and does not appear in the external lines of the Feynmangraphs.Exactly the same happens in our model. To define the model we must specify what (cid:50) − actually means. In general, an equation such as (cid:50) U = − R is solved by U = − (cid:50) − R = U hom ( x ) − (cid:90) d d +1 x (cid:48) (cid:112) − g ( x (cid:48) ) G ( x ; x (cid:48) ) R ( x (cid:48) ) , (4.4)where U hom ( x ) is any solution of (cid:50) U hom = 0 and G ( x ; x (cid:48) ) is any a Green’s function of the (cid:50) operator. To define our model we must specify what definition of (cid:50) − we use, i.e. wemust specify the Green’s function and the solution of the homogeneous equation. In ourcase we use the retarded Green’s function, but still we must complete the definition of (cid:50) − by specifying U hom ( x ). A possible choice is U hom ( x ) = 0. Then, in eq. (1.1),( (cid:50) − R )( x ) ≡ (cid:90) d d +1 x (cid:48) (cid:112) − g ( x (cid:48) ) G ret ( x ; x (cid:48) ) R ( x (cid:48) ) . (4.5)A similar choice must be made in the non-local operators which enter in the extraction ofthe transverse part in eq. (1.1). Thus, at the linearized level, with this definition of (cid:50) − ,in eq. (1.3) we have P ρσ ret h ρσ ≡ h ( x ) − (cid:90) d d +1 x (cid:48) G ret ( x ; x (cid:48) )( ∂ ρ ∂ σ h ρσ )( x (cid:48) ) , (4.6)and similarly P µν ret P ρσ ret h ρσ ≡ η µν (cid:20) h ( x ) − (cid:90) d d +1 x (cid:48) G ret ( x ; x (cid:48) )( ∂ ρ ∂ σ h ρσ )( x (cid:48) ) (cid:21) (4.7) − ∂ µ ∂ ν (cid:90) dx (cid:48) G ret ( x ; x (cid:48) ) (cid:20) h ( x (cid:48) ) − (cid:90) d d +1 x (cid:48) G ret ( x (cid:48) ; x (cid:48)(cid:48) )( ∂ ρ ∂ σ h ρσ )( x (cid:48)(cid:48) ) (cid:21) . U = − (cid:50) − R and S µν = − U g µν . Formally, eq. (1.1) can be written as G µν − d − d m S T µν = 8 πG T µν , (4.8)where S µν = S T µν + ( ∇ µ S ν + ∇ ν S µ ). To make contact with eq. (1.3) we linearize overflat space and we use eq. (1.2). Then eq. (4.8) can be rewritten as the coupled system E µν,ρσ h ρσ − d − d m P µν ret U = − πGT µν , (4.9) (cid:50) U = − R . (4.10)Such a local form of the equations can be convenient, particularly for numerical studies,because it transforms the original integro-differential equations into a set of coupled dif-ferential equations. However, exactly as in the example discussed above, it introducesspurious solutions. The choice of homogeneous solution, that in the original non-localformulation amounts to a definition of the theory, is now translated into a choice of initialconditions on the field U ( x ). There is one, and only one choice, that gives back our originalmodels. For instance, if the original non-local theory is defined through eq. (4.5), we mustchoose the initial conditions on U in eq. (4.10) such that the solution of the associatedhomogeneous equation vanishes. In any case, whatever the choices made in the definitionof (cid:50) − , the corresponding homogeneous solution of eq. (4.10) is fixed, and does not rep-resent a free field that we can take as an extra degree of freedom of the theory. In flatspace this homogeneous solution is a superposition of plane waves of the form (3.9), andthe coefficients a k , a ∗ k are fixed by the definition of (cid:50) − (e.g. at the value a k = a ∗ k = 0 ifwe use the definition (4.5) ), and at the quantum level it makes no sense to promote themto annihilation and creation operators. There is no quantum degree of freedom associatedto them.Comparing eq. (4.9) with eq. (1.3) we see that, at the linearized level, that U = P ρσ ret h ρσ .Therefore at the linearized level U is the same as the variable s given in eq. (3.7). The factthat the homogeneous solutions for U does not represent a free degree of freedom meansthat the same holds for s . We therefore reach the same conclusion of the previous section:the homogeneous solutions for s do not describe propagating degrees of freedom, and atthe quantum level there are no creation and annihilation operators associated to it. Non-local modifications of GR have potentially very interesting cosmological consequences.At the conceptual level, they raise however some issues of principle which must be un-derstood before using them confidently to compare with cosmological observations. Inparticular, these equations feature the retarded inverse of the d’Alembertian. The re-tarded prescription ensures causality, but at the same time the fact (cid:50) − appears not onlyin the solution of such equations, but already in the equations themselves, tells us thatsuch equations cannot be fundamental. Rather, they are effective classical equations.Such non-local effective equations can emerge in a purely classical context. Typicalexamples are obtained when integrating out the short-wavelength modes to obtain an12ffective theory for long-wavelength modes. Another example is given by the formalismfor gravitational-wave production, beyond leading order. In both cases one basically re-injects a retarded solution, obtained to lowest order, into the equation governing the next-order corrections. Another way to obtain non-local equations is by performing a quantumaveraging, in particular when working with the in-in expectation values of the quantumfields, and deriving these equations from an effective action that takes into account theradiative corrections. In particular, in semiclassical quantum gravity we can write sucheffective non-local (but causal) equations for an effective metric (cid:104) in | ˆ g µν | in (cid:105) .We have seen (in agreement with various recent works, e.g. [18, 24, 57, 58]) that, if oneis not careful, it is quite easy to introduce spurious degrees of freedom in these models,which furthermore are ghost-like. Basically, this originates from the fact that the kernelof the (cid:50) − operator is non-trivial: the equation (cid:50) − (0) = f does not imply that f = 0but only that f satisfies (cid:50) f = 0. The non-local equations that we are considering onlyinvolve the retarded solutions of equations of the form (cid:50) f = j , for some source j , i.e. f ( x ) = (cid:90) dx (cid:48) G ret ( x ; x (cid:48) ) j ( x (cid:48) ) . (5.1)However, any action principle that (with some more or less formal manipulation, as dis-cussed in sect. 2) reproduces the equation (cid:50) f = j will automatically carry along the mostgeneral solution of this equation, of the form f ( x ) = f hom + (cid:90) dx (cid:48) G ( x ; x (cid:48) ) j ( x (cid:48) ) , (5.2)where (cid:50) f hom = 0 and G ( x ; x (cid:48) ) is a generic Green’s function. In order to recover the solu-tions that actually pertain to our initial non-local theory we must impose the appropriateboundary conditions, that amount to choosing G ( x ; x (cid:48) ) = G ret ( x ; x (cid:48) ) and fixing once andfor all the homogeneous solution. In particular, one should be careful not to use the corre-sponding Lagrangian at the quantum level, and one should not include the correspondingfields in the external lines or in the loops. The corresponding particles, some of whichare unavoidably ghost-like, do not correspond to propagating degrees of freedom in ouroriginal problem, and the quantization of these spurious solutions does not make sense.We have seen in particular how the above considerations apply to the model defined byeq. (1.1). We have found that the apparent ghost signaled by the second term in eq. (1.7)is actually a non-radiative degree of freedom, and we have also seen that in the m → Acknowledgments.
We thank Claudia de Rham, Stanley Deser, Yves Dirian, LaviniaHeisenberg and Maud Jaccard for useful discussions. Our work is supported by the FondsNational Suisse. 13 eferences [1] A. Pais and G. Uhlenbeck, “On Field theories with nonlocalized action,”
Phys.Rev. (1950) 145–165.[2] D. Eliezer and R. Woodard, “The Problem of Nonlocality in String Theory,” Nucl.Phys.
B325 (1989) 389.[3] J. Z. Simon, “Higher derivative lagrangians, nonlocality, problems and solutions,”
Phys.Rev.
D41 (1990) 3720.[4] R. P. Woodard, “Avoiding dark energy with 1/R modifications of gravity,”
Lect.Notes Phys. (2007) 403–433, astro-ph/0601672 .[5] N. Barnaby and N. Kamran, “Dynamics with infinitely many derivatives: TheInitial value problem,”
JHEP (2008) 008, .[6] I. B. Ilhan and A. Kovner, “Some Comments on Ghosts and Unitarity: ThePais-Uhlenbeck Oscillator Revisited,”
Phys.Rev.
D88 (2013) 044045, .[7] N. Arkani-Hamed, S. Dimopoulos, G. Dvali, and G. Gabadadze, “Nonlocalmodification of gravity and the cosmological constant problem,” hep-th/0209227 .[8] G. Dvali, S. Hofmann, and J. Khoury, “Degravitation of the cosmological constantand graviton width,”
Phys.Rev.
D76 (2007) 084006, hep-th/0703027 .[9] G. Dvali and G. Gabadadze, “Gravity on a brane in infinite volume extra space,”
Phys.Rev.
D63 (2001) 065007, hep-th/0008054 .[10] G. Dvali, G. Gabadadze, and M. Shifman, “Diluting cosmological constant ininfinite volume extra dimensions,”
Phys.Rev.
D67 (2003) 044020, hep-th/0202174 .[11] A. Barvinsky, “Nonlocal action for long distance modifications of gravity theory,”
Phys.Lett.
B572 (2003) 109–116, hep-th/0304229 .[12] G. Dvali, “Predictive Power of Strong Coupling in Theories with Large DistanceModified Gravity,”
New J.Phys. (2006) 326, hep-th/0610013 .[13] S. Deser and R. Woodard, “Nonlocal Cosmology,” Phys.Rev.Lett. (2007) 111301, .[14] T. Koivisto, “Dynamics of Nonlocal Cosmology,” Phys.Rev.
D77 (2008) 123513, .[15] T. Koivisto, “Newtonian limit of nonlocal cosmology,”
Phys.Rev.
D78 (2008)123505, .[16] S. Capozziello, E. Elizalde, S. Nojiri, and S. D. Odintsov, “Accelerating cosmologiesfrom non-local higher-derivative gravity,”
Phys.Lett.
B671 (2009) 193–198, . 1417] A. Barvinsky, “Dark energy and dark matter from nonlocal ghost-free gravitytheory,”
Phys.Lett.
B710 (2012) 12–16, .[18] A. O. Barvinsky, “Serendipitous discoveries in nonlocal gravity theory,”
Phys.Rev.
D85 (2012) 104018, .[19] E. Elizalde, E. Pozdeeva, and S. Y. Vernov, “De Sitter Universe in Non-localGravity,”
Phys.Rev.
D85 (2012) 044002, .[20] Y. Zhang and M. Sasaki, “Screening of cosmological constant in non-localcosmology,”
Int.J.Mod.Phys.
D21 (2012) 1250006, .[21] E. Elizalde, E. Pozdeeva, and S. Y. Vernov, “Reconstruction Procedure in NonlocalModels,”
Class.Quant.Grav. (2013) 035002, .[22] S. Park and S. Dodelson, “Structure formation in a nonlocally modified gravitymodel,” Phys.Rev.
D87 (2013) 024003, .[23] K. Bamba, S. Nojiri, S. D. Odintsov, and M. Sasaki, “Screening of cosmologicalconstant for De Sitter Universe in non-local gravity, phantom-divide crossing andfinite-time future singularities,”
Gen.Rel.Grav. (2012) 1321–1356, .[24] S. Deser and R. Woodard, “Observational Viability and Stability of NonlocalCosmology,” JCAP (2013) in press, .[25] P. G. Ferreira and A. L. Maroto, “A few cosmological implications of tensornonlocalities,” .[26] S. Dodelson and S. Park, “Nonlocal Gravity and Structure in the Universe,” .[27] H. Hamber and R. M. Williams, “Nonlocal effective gravitational field equationsand the running of Newton’s G,”
Phys.Rev.
D72 (2005) 044026, hep-th/0507017 .[28] J. Khoury, “Fading gravity and self-inflation,”
Phys.Rev.
D76 (2007) 123513, hep-th/0612052 .[29] T. Biswas, T. Koivisto, and A. Mazumdar, “Towards a resolution of thecosmological singularity in non-local higher derivative theories of gravity,”
JCAP (2010) 008, .[30] L. Modesto, “Super-renormalizable Quantum Gravity,”
Phys.Rev.
D86 (2012)044005, .[31] F. Briscese, A. Marciano, L. Modesto, and E. N. Saridakis, “Inflation in(Super-)renormalizable Gravity,”
Phys.Rev.
D87 (2013) 083507, .[32] M. Jaccard, M. Maggiore, and E. Mitsou, “A non-local theory of massive gravity,”
Phys.Rev.
D88 (2013) 044033, .[33] M. Maggiore, “Phantom dark energy from non-local massive gravity,” .1534] C. de Rham and G. Gabadadze, “Generalization of the Fierz-Pauli Action,”
Phys.Rev.
D82 (2010) 044020, .[35] C. de Rham, G. Gabadadze, and A. J. Tolley, “Resummation of Massive Gravity,”
Phys.Rev.Lett. (2011) 231101, .[36] C. de Rham, G. Gabadadze, and A. J. Tolley, “Ghost free Massive Gravity in theSt¨uckelberg language,”
Phys.Lett.
B711 (2012) 190–195, .[37] S. Hassan and R. A. Rosen, “Resolving the Ghost Problem in non-Linear MassiveGravity,”
Phys.Rev.Lett. (2012) 041101, .[38] S. Hassan, R. A. Rosen, and A. Schmidt-May, “Ghost-free Massive Gravity with aGeneral Reference Metric,”
JHEP (2012) 026, .[39] S. Hassan and R. A. Rosen, “Confirmation of the Secondary Constraint andAbsence of Ghost in Massive Gravity and Bimetric Gravity,”
JHEP (2012)123, .[40] K. Hinterbichler, “Theoretical Aspects of Massive Gravity,”
Rev.Mod.Phys. (2012) 671–710, .[41] S. Deser, “Covariant Decomposition and the Gravitational Cauchy Problem,” Ann.Inst.Henri Poincare (1967) 149.[42] J. J. York, “Covariant decompositions of symmetric tensors in the theory ofgravitation,” Ann.Inst.Henri Poincare (1974) 319.[43] S. Foffa, M. Maggiore, and E. Mitsou, “Cosmological dynamics and dark energyfrom non-local infrared modifications of gravity,” .[44] R. Caldwell, “A Phantom menace?,” Phys.Lett.
B545 (2002) 23–29, astro-ph/9908168 .[45] S. M. Carroll, M. Hoffman, and M. Trodden, “Can the dark energy equation-of-stateparameter w be less than -1?,” Phys.Rev.
D68 (2003) 023509, astro-ph/0301273 .[46] M. Soussa and R. P. Woodard, “A Nonlocal metric formulation of MOND,”
Class.Quant.Grav. (2003) 2737–2752, astro-ph/0302030 .[47] R. Jordan, “Effective Field Equations for Expectation Values,” Phys.Rev.
D33 (1986) 444–454.[48] E. Calzetta and B. Hu, “Closed Time Path Functional Formalism in CurvedSpace-Time: Application to Cosmological Back Reaction Problems,”
Phys.Rev.
D35 (1987) 495.[49] S. M. Carroll, S. Leichenauer, and J. Pollack, “A Consistent Effective Theory ofLong-Wavelength Cosmological Perturbations,” .[50] L. Blanchet, “Gravitational radiation from post-Newtonian sources and inspirallingcompact binaries,”
Living Rev.Rel. (2006) 4.1651] M. Maggiore, Gravitational Waves. Vol. 1. Theory and Experiments . OxfordUniversity Press, 574 p, 2007.[52] N. Tsamis and R. Woodard, “Nonperturbative models for the quantum gravitationalback reaction on inflation,”
Annals Phys. (1998) 145–192, hep-ph/9712331 .[53] M. Jaccard, M. Maggiore, and E. Mitsou, “Bardeen variables and hidden gaugesymmetries in linearized massive gravity,”
Phys.Rev.
D87 (2013) 044017, .[54] J. M. Cline, S. Jeon, and G. D. Moore, “The Phantom menaced: Constraints onlow-energy effective ghosts,”
Phys.Rev.
D70 (2004) 043543, hep-ph/0311312 .[55] S. Nojiri and S. D. Odintsov, “Modified non-local-F(R) gravity as the key for theinflation and dark energy,”
Phys.Lett.
B659 (2008) 821–826, .[56] S. Jhingan, S. Nojiri, S. Odintsov, M. Sami, I. Thongkool, and S. Zerbini, “Phantomand non-phantom dark energy: The Cosmological relevance of non-locally correctedgravity,”
Phys.Lett.
B663 (2008) 424–428, .[57] N. Koshelev, “Comments on scalar-tensor representation of nonlocally correctedgravity,”
Grav.Cosmol. (2009) 220–223, .[58] T. S. Koivisto, “Cosmology of modified (but second order) gravity,” AIP Conf.Proc. (2010) 79–96,0910.4097