Exorcising ghosts in quantum gravity

We show that Ostrogradsky ghosts in higher-derivative quantum gravity are fictitious as they result from an unjustified truncation performed in a complete theory containing infinitely many curvature invariants. The apparent ghosts can then be projected out by redefining the boundary conditions of the theory in terms of an integration contour that does not enclose the ghost poles. One thus obtains a quantum field theory of gravity that is both renormalizable and ghost-free.

Despite the major advances in the quantization of gravity obtained in the past few decades, a deep understanding of quantum gravity in the UV remains a matter of debate. General relativity is known to be non-renormalizable, generating higher curvature invariants in the action, which are required for renormalization [1]. However, by introducing higher-derivative terms, ghosts inevitably appear in the spectrum unless the theory is treated under the effective field theory formalism where the higher derivatives are seen as perturbations [2,3]. The purpose of this paper is to show that ghosts in higher-derivative gravity are only apparent when one truncates the infinite series of curvature invariants. We then show how these ghosts can be removed by means of a suitable boundary condition.
The known issue with higher powers of the curvature invariants is due to Ostrogradsky theorem [4][5][6]. It states that any dynamical system described by differential equations containing time derivatives higher than second order necessarily possesses unbounded energy solutions, dubbed ghosts. The existence of a ghost is not itself an issue, but it becomes a problem when the ghost field interacts with other sectors, which allows for the endless process of transmitting energy from healthy fields to the ghost. At the quantum level, negative energy states are sometimes traded by states with negative norm which is again problematic as it violates the optical theorem [7]. The only known way of evading Ostrogradsky theorem is with degenerate theories, such as f (R), but as we will see, functions of the Ricci scalar are not sufficient for renormalization [8]. We will show yet another way of evading Ostrogradsky theorem in quantum gravity.
When general relativity is quantized at one-loop order, one finds that the divergences are proportional to second order curvature invariants, i.e. terms containing four derivatives [1]. Thus one must start off with the action where M p is the Planck mass and a i are bare parameters, to be able to renormalize general relativity at one-loop. Similarly, the renormalization of general relativity at two-loop order requires terms such as R R [9]. We conclude that to renormalize general relativity at all loop orders it is required the inclusion of infinitely many powers of the curvature invariants to the Lagrangian, leading to the naive conclusion that quantum general relativity is not falsifiable. This is in fact a reflection of the non-renormalizability of general relativity.
This problem is circumvented within the realm of effective field theories [3] (see [10] for a review). In the effective field theory description of quantum gravity, terms in the action are organized in powers of E/M p , where E is the typical energy of the problem. Dimensional analysis shows that higher-order curvatures correspond to higher powers of the E/M p , thus at energies way below the Planck scale the higher powers of the curvature are utterly small and can be treated as tiny perturbations. Thus at any given precision, the infinite series can be truncated, producing only a finite number of free parameters. In this scenario, there is no new degree of freedom, ghost or otherwise, besides the standard graviton and the interaction is that of general relativity. The higher-order terms capture the underlying physics perturbatively and only contribute to the vertices of Feynman diagrams, not to the propagators. As a result, one obtains a theory that can be renormalized, albeit being nonrenormalizable, at every loop order without introducing ghosts to the spectrum, but that only makes sense at energies below the Planck scale.
On the other hand, we could let the fourth derivative terms in (1) take arbitrary values which would make them compete with the Einstein-Hilbert term at the Planck scale [8]. This theory, which is no longer quantum general relativity, came to be known as higher-derivative gravity. In this case, the action with second order curvature invariants is renormalizable to all loop orders and one need not include terms with even higher derivatives, although these would not change the renormalizability of the theory. This theory could be interpreted as a fundamental theory for quantum gravity if it was not for the presence of a ghost in its spectrum [8]. Unitarity can thus be traded by renormalizability. This interpretation has the advantage of having interesting new solutions, such as Starobinsky inflation [11], but its drawback is the instability and the violation of unitarity caused by the ghost. Some solutions to the non-unitarity problem have been proposed [12][13][14][15], but a consistent way of dealing with the classical ghost instabilities remains unknown. We shall argue that this ghost is however absent when one takes into account all possible curvature invariants.
We have seen that higher-derivative gravity must contain fourth derivative terms in the action for renormalizability. Nonetheless, nothing forbids us from including sixth derivative terms or higher, such as curvature terms of third order. In fact, there are infinitely many terms that are allowed by the diffeomorphism symmetry and there is a priori no reason to exclude them from the action. Furthermore, effective field theory requires the inclusion of nth-order curvature invariants to renormalize calculations with n − 1 loops. Since effective field theory is a model-independent formulation of quantum gravity in the IR, any respectful quantum theory of gravity should give rise to operators of arbitrary order in the low-energy regime, unless there is some additional symmetry that forbids the appearance of these terms. Therefore, even though renormalization only requires second order curvature invariants, higher-order operators are necessary for a consistent matching with the effective theory.
The subset of terms (1) is thus obtained by truncating the following infinite series of curvature invariants: where O 2n denotes all curvature invariants containing 2n derivatives and a 2n are their coefficients. In effective field theory, this truncation is well-defined and can be performed at any order without introducing ghosts or any other particle. This is possible because the set of operators O 2n becomes increasingly smaller as we crank up n. But in higher-derivative gravity, a truncation is poorly motivated, although it is sometimes assumed that a theory of the type (1) could arise in one of the many vacua of string theory. We now show that the Ostrogradsky ghost is actually a result of this poor truncation.
A crucial step for proving Ostrogradsky theorem on instability consists in writing an Nth-order differential equation as a system of N coupled first-order differential equationṡ where x n = x (n−1) are phase space coordinates and f n can be expressed in terms of derivatives of the Hamiltonian H(t, x 1 , x 2 , . . . , x N ). In this situation, the Hamiltonian H turns out to depend linearly on N/2 − 1 of its arguments, signaling that H is not positive-definite for a large portion of the phase space. However, as noted in the mathematical literature [16][17][18], an infinite differential equation cannot be written as a system of N = ∞ first-order differential equations by simply applying the same reasoning of the case where N is finite. In particular, the Hamiltonian system for N = ∞ does not correspond to the same problem described by an infinite differential equation, thus one should not expect Ostrogradsky instabilities for differential equations of infinite order. Since the theory (2) contains infinitely many higher-order operators in its action, its equation of motion is described by a differential equation of infinite order. We thus conclude that Ostrogradsky ghosts are necessarily absent in higher-derivative gravity when one includes every possible curvature invariant to the action. Evading Ostrogradsky theorem, however, does not guarantee that other types of ghosts will be absent. Nonetheless, it shows that the Ostrogradsky ghost present in quadratic gravity (1) is an artificial particle that originates from the truncation. We should point out that there are theories whose ambition is to eliminate every ghost, Ostrogradskian or not, from the spectrum [19,20].
As a concrete example, consider quantum gravity in two dimensions as described by the

Polyakov action [21]
The non-local operator 1/ can be seen as an infinite series of diffeomorphism invariants. In fact, one can bring (5) to the form in the region of convergence of the series. The action (5) contains only a healthy scalar degree of freedom in its spectrum. This can be seen by performing a conformal transformation g µν = e 2σ η µν , which leads to [22] Therefore, the infinite action (6) does not contain any ghost resulting from the Ostrogradsky construction. Another example is given by Barvinsky's non-local theory [23]: whereP is a matrix defined in terms of the curvatures, G µν is the Einstein tensor and the non-local piece can again be viewed as an infinite series 1 +P = ∞ n=0 (−1) n n P n+1 . The degrees of freedom of (8) are the same ones of general relativity, thus there is no Ostrogradsky ghost in the spectrum, which corroborates our result. Therefore, the apparent Ostrogradsky ghost that appears in (1) is just a byproduct of the unjustified truncation of S ∞ . Note that truncating S ∞ to a finite order N yields many non-physical particles that are not present in the full theory because the truncation inevitably changes the pole structure of the propagator. Along with these particles, many ghosts are expected to appear as a result of Ostrogradsky theorem. We emphasize that all these particles, regardless of being ghosts, are fictitious as they only appear as an artifact of the truncation. Thus whenever we want to truncate S ∞ , we can very well project out these unphysical poles by suitably choosing an integration contour to define the truncated theory. This is done by first linearizing the equations of motion of the truncated theory around Minkowski, which ultimately leads to the generalized wave equation for the transverse and traceless perturbation h µν 2 : 2 We focus on the homogeneous equation for the sake of the argument, but the method can be easily generalized to the case where matter is present. See [18] for a detailed discussion of the initial value problem of infinite and finite higher-derivative differential equations.
where F ( ) is a generalized wave operator. We can write the perturbation in Fourier space as where C is a contour that is chosen according to the desired boundary conditions. The action of the pseudo-differential operator F ( ) is defined in Fourier space, where Eq. (9) becomes Note that the boundary conditions, as well as the contour C, are part of the definition of the theory. The most standard choice is a contour that encloses all zeros of (9) (or equivalently, all poles of the propagator) and yet satisfies Feynman boundary conditions. But that is not the only choice. We can, for example, choose C without enclosing the ghost poles, keeping them from appearing in the truncated theory [18,24]. The zeros of F (−q 2 ) can be isolated by means of the Weierstrass factorization theorem where g(q 2 ) is a holomorphic function, {λ n } is a sequence of integers and q n are the zeros of F (−q 2 ). From Cauchy theorem, we obtain the solution to (9) restricted to a contour C that does not enclose any ghost h µν = n s=+,− a n,s µν e −iq s n x , where a n,s µν are polarization tensors and (q s n ) 2 are the poles of F (−q 2 ) −1 located inside the contour C, i.e. plane waves corresponding to ghost fields are absent. Given that Ostrogradsky ghosts are spurious as they only appear in the spectrum as an artifact of the truncation, it is natural to make this choice. Projecting out a ghost particle prohibits its corresponding field of having plane wave solutions, thus not leading either to instabilities in the classical theory or to unitarity issues at the quantum level. In particular, the ghost field cannot be written in terms of creation and annihilation operators, thus it never appears in asymptotic states and does not constitute a physical particle. Its only effect is to intermediate a repulsive Yukawa interaction that originates from the particular solution to the inhomogeneous Klein-Gordon equation, i.e. from the Green's function, which for a point particle of mass M reads [25] where The truncated theory is then free of pathologies and yet able to capture the essence of quantum gravity.
A few comments are in order. Firstly, it is important to note that projecting out the ghost particle does not recover renormalization issues. In fact, the functional form of the graviton propagator continues to behave as q −4 , thus the superficial degree of divergence D of Feynman diagrams remains the same [8,26] where d is the number of metric derivatives in the counterterms and n 2r is the number of graviton vertices with 2r derivatives, indicating that the theory is renormalizable. The ghost pole is however absent because we have changed the boundary conditions, namely the contour C, and the momentum q never hits the ghost pole. Secondly, the ghost no longer appears in external legs of Feynman diagrams (asymptotic states), therefore it no longer causes problems with unitarity and the optical theorem is satisfied.
We must also stress that evading the Ostrogradsky instability does not guarantee that other types of ghosts would not originate from some yet unknown mechanism, even when all curvature invariants are present. In fact, if the full theory S ∞ contains more than one pole, one of them is necessarily a ghost should F be analytic 3 [28]. This is easy to see by calculating the residue of the poles in the full theory. Suppose there are M poles (labeled by the index a below) in the propagator of the full theory. Then their residues are given by [28] where and Γ(z) is everywhere non-zero. The contour C a encloses only the pole at z = m 2 a . Thus, as long as F is analytic, η a must change sign for M > 1. In our bottom-up approach, F is analytic by construction as it is defined as an infinite series, leading to the conclusion that the scalar degree of freedom is likely to be a byproduct of the truncation as well, otherwise it would be itself a ghost should it exist in the full theory S ∞ . While a single massive particle whose decay could produce massive scalars and massless gravitons can very well exist in S ∞ , it is clearly impossible to figure out the spectrum of the theory by having at our disposal only an infinite series. If we want to keep the scalar field in the spectrum while insisting that a fundamental theory of quantum gravity can be obtained by the bottom-up construction 3 An example of analytic F includes the polynomial function F (z) = N n=0 c n z n . On the other hand, an instance of a non-analytic generalized wave operator can be given by F (z) = (1 − α log(z))z, which appears when one-loop corrections to general relativity are considered [27]. of the infinite series (2), then there is no option other than looking for a non-analytic F . The most obvious way to make the series (2) non-analytic is by including negative powers of the curvature in the action, which forces the appearance of poles in F . It would then be possible to have multiple poles in the propagator with no ghosts in the spectrum, leading to a theory that is both renormalizable and ghost-free, while keeping interesting solutions such as the scalar degree of freedom responsible for Starobinsky inflation.
Alternatively, should we not be interested in Starobinsky inflation, we can stick to the supposedly simpler series (2) with no negative power of the curvature and project out all undesired modes with the exception of the graviton by means of an integration contour following the same procedure as before. We must note that in the theory (1), it is very natural to pick up a contour that does not enclose the Ostrogradsky ghost because, as we have seen, this ghost is a byproduct of the truncation. On the other hand, the removal of a non-Ostrogradsky ghost in a theory with infinitely many curvature invariants by means of an integration contour might seem less natural, but it is still a legitimate procedure as we always get to choose the boundary conditions.
In this short paper, we argued that Ostrogradsky ghosts that haunt higher-derivative gravity are actually just artificial byproducts of the truncation of the full theory (2), which contains infinitely many curvature invariants. We showed how these fictitious ghosts can be projected out in the truncated theory with the help of an integration contour defined in the Fourier space and which is part of the definition of the theory itself. This allows one to use higher-derivative gravity to study quantum gravity without facing stability or unitarity issues. We also discussed the possibility of building a ghost-free theory containing infinite curvature invariants in its action, without having to get rid of the scalaron responsible for Starobinsky inflation. A more detailed analysis of this last possibility is however needed.