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Is negative kinetic energy meta-stable?

We explore the possibility that theories with negative kinetic energy (ghosts) can be meta-stable up to cosmologically long times. In classical mechanics, ghosts undergo spontaneous lockdown rather than run-away if weakly-coupled and non-resonant. Physical examples of this phenomenon are shown. In quantum mechanics this leads to meta-stability similar to vacuum decay. In classical field theory, lockdown is broken by resonances and ghosts behave statistically, drifting towards infinite entropy as no thermal equilibrium exists. We analytically and numerically compute the run-away rate finding that it is cosmologically slow in 4-derivative gravity, where ghosts have gravitational interactions only. In quantum field theory the ghost run-away rate is naively infinite in perturbation theory, analogously to what found in early attempts to compute vacuum tunnelling; we do not know the true rate.


Introduction
A tentative quantum theory of gravity and matter is obtained writing the most generic action with renormalizable terms, taking into account that the graviton g µν has mass dimension 0. Such action is [1] where R µν is the Ricci tensor, R is the curvature, and L matter contains scalars, fermions and vectors. The first two terms, suppressed by the dimension-less gravitational couplings f 0 and f 2 (in the notation of [2]), are graviton kinetic terms with 4 derivatives. However, a classical degree of freedom with 4 derivatives can be rewritten as 2 degrees of freedom with 2 derivatives, and one of the two (dubbed ghost) has negative kinetic energy [3]. Gravity is no exception. The 4-derivative graviton splits into the massless graviton and a ghostgraviton with mass M 2 = f 2MPl / √ 2. The full action in split form can be found in [4], and the negative kinetic energy can be seen through the following simple argument. Omitting Lorentz indices, the propagator of the 4-derivative graviton is where the minus sign indicates negative kinetic energy. 1 It makes the theory renormalizable, cancelling the graviton propagator at large energy p M 2 . We explore the possibility that the degrees of freedom with negative kinetic-energy are physical, unlike what happens in gauge theories, where similar states are unphysical, introduced as mathematical tools to deal with gauge redundancies.
A classical degree of freedom with positive kinetic energy interacting with negative kinetic energy has run-away solutions, where total energy is conserved while individual energies diverge. Thereby negative kinetic energy is dubbed 'ghost', meaning an unphysical object to be excluded from sensible theories. However, theories with negative and even unboundedfrom-below potential energy can give sensible meta-stable physics around a false vacuum. Can unbounded-from-below kinetic energy similarly give rise to meta-stability?
To explore this issue, we will consider theories featuring some positive-energy degree of freedom q 1 (t) interacting with a ghost q 2 (t) as described by Lagrangians such as as well as the analogous relativistic theory of fields ϕ 1,2 ( x, t) (scalars, for simplicity) with Lagrangian density In both cases the ghost is obtained for ± = −1.
We preliminarily need to address the concerns of those authors who, at this point, dismiss the study with the motivation that an unbounded-from-below Hamiltonian is inconsistent, for example because it allows for classical solutions that hit singularities. These authors also view as inconsistent positive kinetic energies but with unbounded-from-below potentials.
What we want to study is how long the physical system can stay around a "false vacuum", before falling to other regions. In the case of potential meta-stability, the WKB approximation in quantum mechanics shows that the meta-stability time is determined only by the potential barrier, irrespectively of the fate beyond the barrier. The potential beyond the barrier might be unbounded-from-below (giving rise to singular solutions) or have a true minimum: this does not affect the meta-stability time. The fate beyond the barrier depends on possibly unknown high-energy theory. In effective Quantum Field Theories (QFT) one considers extra non-renormalizable terms that stabilise an unbounded-from-below potential. As such operators have negligible impact at low field values, the meta-stability time is computable in terms of low-energy physics.
Returning back from the analogy to the argument of the present study, we want to explore if a theory with negative kinetic energy might similarly be meta-stable up to cosmologically large times. Let us consider, for example, the model in eq. (3). Its Hamiltonian is unboundedfrom-below, but can be modified for example into that is bounded from below and negligibly differs from the original theory at energies E E 0 The energy of the 2nd degree of freedom has a Mexican-hat form that avoids singularities replacing them with a generalization of 'ghost condensation' [14] such that q 2 reaches a constant but finite velocity. The critical energy E 0 plays a role analogous to coefficients of non-renormalizable operators: in the limit where it is much higher than the energies available around the false vacuum, it plays no role until the escape event happens. In the following, we can thereby study the meta-stability issue in the simpler model of eq. (3) where energy is unbounded-from-below.
In order to see if a ghost is really excluded we start studying the problem in the simplest limit, classical mechanics.
It has been noticed that, in classical mechanics, some theories containing an interacting ghost have stable classical solutions with appropriate initial conditions dubbed "islands of stability" [15][16][17][18][19][20][21][22][23][24]. This happens even when interactions are generic enough that no constant of motion forbids interacting ghosts to evolve towards catastrophic run-away instabilities. Rather, ghosts undergo spontaneous lockdown, with energies that vary but remain in a non-trivial restricted range. Studies based on numerical computations of classical time evolution cannot reach cosmological meta-stability times, so an analytic understanding is needed. Extending earlier works [16] we will show that the needed mathematics had been already developed to understand a related problem: why the solar system is meta-stable, despite that no constant of motion forbids planets to escape? Oversimplifying, it has been shown that classical systems that can be approximated as oscillators plus small interactions tend to undergo ordered epicyclelike motions, while large interactions lead to chaos. We will see that this implies that ghosts with large interactions run away, but ghosts with generic small interactions are stable. Weakly coupled theories contain hidden quasi-constants of motion. Since this might appear exotic, in Appendix A we recall that known physical systems exhibit this behaviour: asteroids around the Lagrangian point L 4 and electrons in magnetic fields plus repulsive potentials are described by a ghost degree of freedom, and yet they are meta-stable.
Since classical mechanics does not exclude ghosts, in section 3 we study quantum mechanics, finding that meta-stability persists: a ghost (negative kinetic energy, K-instability) is not qualitatively less meta-stable than a negative potential energy (V -instability).
However, resonances (such as ω 1 = ω 2 in eq. (3)) can lead to ghost run-away even at small coupling, depending on the specific form of the interaction. Studying in section 4 classical field theory we encounter an infinite number of resonances, by expanding a field in Fourier modes. While local field theories can give resonances of benign type, the infinite number of resonances removes the hidden constants of motion. We then perform a statistical analysis showing that systems containing ghosts do not have a thermal state: heat keeps flowing from ghost fields to positive-energy fields, because this increases entropy. We compute the rate of this instability through Boltzmann equations, finding a rate not exponentially suppressed by small couplings. Nevertheless, in the special case of 4-derivative gravity, the graviton ghost has Planck-suppressed interactions which are small enough that the ghost run-away rate is not problematic in cosmology. We validate this analytic understanding through classical lattice simulations.
In section 5 we finally consider relativistic quantum field theory, which is the relevant but most difficult theory. By performing the zero-temperature limit of Boltzmann equations we find a divergent tree level ghost run-away rate. Such divergence arises because the initial vacuum state is Lorentz-invariant, giving rise to an integral over the non-compact Lorentz group that describes a boost of the final state. The same Lorentz integral arose in earlier computations of V -instability tunnelling, but Coleman later argued that that vacuum decay can be computed in terms of a Lorentz-invariant instanton, the 'bounce', and its rate is exponentially suppressed at small coupling. We don't know if something similar holds for K-instability.
Conclusions are presented in section 6.
2 Ghost meta-stability in classical mechanics?
We consider a degree of freedom q(t) in 0+1 dimensions with 4-derivative kinetic term where the first term is quadratic in q and V I contains interactions. We add zero as a perfect square containing an auxiliary degree of freedomq with no kinetic term: Expanding the square cancels both the second-order and the fourth-order kinetic terms leaving The kinetic and mass terms are diagonalised performing the field redefinition obtaining, after an integration by parts We can thereby focus on the toy model of eq. (3) that captures the relevant physics. This classical theory only has one free physical parameter, ω 1 /ω 2 , plus the initial conditions for its time evolution. Indeed, without loss of generality we can rescale q 1 and q 2 to set m 1 = m 2 = 1. By rescaling t we can set ω 1 = 1. Furthermore, classical physics is invariant under a multiplicative rescaling of L, so that we could set λ = 1. To improve readability we keep ω 1 , ω 2 and λ as apparent parameters, but it should be clear that our following analysis is general. The classical equations of motion arë The only constant of motion is the total energy E = E 1 − E 2 + V I , which is conserved, where while E 1 and E 2 are not conserved, e.g.Ė 1 = −λq 2 2 d(q 2 1 )/dt. No conservation law prevents rapid ghost run-away to E 1 , E 2 → ∞. Numerical evolution shows that solutions starting from |E 1 − E 2 | > ∼ V I quickly undergo run-away. On the other hand, for solutions starting from small enough initial energies E 1 , E 2 V I , E 1 (t) and E 2 (t) evolve remaining confined to a small range, for a time longer than what can be numerically computed. 2 Analytic work is needed to understand this surprising phenomenon.

Action-angle variables
A technique used to study perturbed quasi-periodic motions in celestial mechanics is useful. Considering one pair (q, p) of Hamiltonian variables, it is useful to pass to canonical actionangle variables (Θ, J) such that the Hamiltonian only depends on J and motion is immediately solved.
In the simplest case of an harmonic oscillator, this gives where m > 0 (m < 0) for a normal particle (a ghost). The canonical transformation is and its inverse is One can verify that [Θ, J] = (∂Θ/∂q)(∂J/∂p) − (∂Q/∂p)(∂J/∂q) = 1 or more formally write the generator of the canonical transformation In action-angle variables H = ωJ so that motion of a harmonic oscillator is trivially solved by Θ = Θ 0 + ωt, J = E/ω. For a generic anharmonic oscillator, the transformation to action-angle variables such that H depends only on J i cannot be written analytically. Going to action-angle variables for the two free harmonic oscillators, our toy ghost model of eq. (3) becomes and E i = ω i J i ≥ 0. The − signals a ghost. The change of variables makes numerics stable up to longer time scales. Starting from t = 0, fig. 1 shows the time t end at which the ghost run-away happens as function of λ for some fixed initial conditions and given ω 2 /ω 1 . We see a chaotic behaviour at larger λ that sharply starts above some critical value. Fig. 2a shows that, for small λ, J 1 and J 2 remain confined in a well-defined region up to long times, while Θ 1 and Θ 2 evolve almost linearly in time. Analytic work is needed to know if smaller λ leads to meta-stability or to absolute stability. The region in the (J 1 , J 2 ) plane extends with increasing λ until suddenly chaos and ghost run-away take over.
This behaviour is characteristic of near-integrable system. Integrable systems (such as n independent oscillators) are those for which any trajectory evolves along tori in phase space,  rather than filling higher-dimensional sub-spaces up to the whole phase space. Adding small interactions, a near-ordered behaviour persists because the system can be computed perturbatively. In the case of ghosts, this implies their meta-stability. For large coupling the perturbative expansion fails and the system becomes chaotic. If the system contains ghosts, this leads to run-aways.
For small we can analytically solve the equations of motion as power series in . At 0th order in = 2λ/ω 1 ω 2 the equations of motion are solved by We see that J i (t) = J i0 are constant, for both i = {1, 2}. Their equations of motion at 1st order are solved by having defined ω 1−2 = ω 1 − ω 2 etc. The dimension-less expansion parameter is ∼ J/ω, that describes the energy in the interaction term divided by the energy in the free quadratic part of the Hamiltonian. This 1st order approximation fails after some oscillations; nevertheless for small it approximates well the range of (J 1 , J 2 ) covered by the full numerical solution. The 1st order perturbation diverges if ω 1 = ±ω 2 . More in general, higher orders diverge if ω 1 and ω 2 are 'commensurable', namely if the resonance condition N 1 ω 1 + N 2 ω 2 = 0 is satisfied for some integers N 1,2 . 3 A more important problem is that the perturbative series in (or λ) is not convergent but at most asymptotic. Thereby its existence does not imply absolute stability and a more complicated analysis is needed, yielding stability over exponentially long times.

Perturbative Birkhoff series
We consider the toy model described by the Hamiltonian of eq. (17). Rather than finding solutions perturbatively in we follow a more general, equivalent, approach. We seek to 'diagonalise' the classical Hamiltonian. Namely, we search for a canonical transformation J i → J i and Θ i → Θ i such that the Hamiltonian does not depend on Θ i : We perform a generic canonical transformation with generator So, defining f = sin 2 Θ 1 sin 2 Θ 2 one gets If we could solve this equation, all J i would be exact constants of motion and the system would be integrable. However, we can only expand and perturbatively solve eq. (23) in powers of , Since the system is not integrable, the Birkhoff series is only asymptotic and J i are the approximated constants of motion observed in numerics. Because of the periodicity in Θ ≡ (Θ 1 , Θ 2 ), we expand each term in Fourier series, e.g.
The only non-zero coefficient of the Fourier series of

First order in the coupling
Expanding eq. (23) at first order gives The first term involves derivatives of a periodic function with period 2π, by the very definition of the angle variables Θ i . Therefore, its average over a period is zero. Averaging over Θ i we get We next compute the canonical transformation W (1) through the Fourier expansion. We get W for N 1 , N 2 = 0. Summing over the non-vanishing N this means and thereby which gives the extra approximate integral of motion (in addition to energy, an exact constant). At this order the only resonance is ω 1 = ±ω 2 . The perturbative expansion fails close to the resonance. The numerical solution shows that J 1 is an approximate pseudo-integral of motion for small , unless ω 1 ≈ ω 2 .

Generic order in the coupling
Eq. (23) expanded at order n > 1 (n = 1 is special) is At each order n only a finite set of coefficients of W (n) N 1 N 2 are non-zero, since f only has few non-zero Fourier coefficients. The constant term (p 1 = p 2 = 0) allows to find explicitly the Hamiltonian, whereas the other terms give the canonical transformation. We may fix the freedom of performing Θ-only transformations by choosing W which are explicit equations for the Hamiltonian and the canonical transformation at order n in terms of the lower orders. H (n) is a polynomial of degree n + 1 in J 1,2 with coefficients that depend on ω i . 4

Stability estimates
For typical interacting systems, frequencies vary depending on initial conditions and can thereby hit resonances, invalidating the Birkhoff series that guarantees stability. Kolmogorov proved that instability only happens for a sub-set of values of initial conditions that are as rare as rational numbers within real numbers: most initial conditions lead to stable motion. For systems with 2 degrees of freedom and conserved energy this is enough to guarantee exact stability, because there is only one quasi-constant of motion, say J 1 . Any initial condition is 'surrounded' by nearby values so that stability holds. On the other hand, with more than 2 degrees of freedom there are 2 or more quasi-constants J i , so that they can undergo Arnold diffusion: their values slowly drift through the rare instabilities, not being surrounded by stable values. This drift is not visible in perturbation theory because it takes place for 'rational' values of ω i such that perturbation theory fails. Nekhoroshev estimated that the drift is nonperturbatively slow, giving rise to an exponentially large instability time [25].
In concrete systems the meta-stability time can be computed as follows. The perturbative Birkhoff series allows to remove interactions up to an arbitrarily large power k so that the remaining small interaction can destroy stability on long enough time-scales, of order −k . As the Birkhoff series is only asymptotic, stability estimates are obtained by computing up to some high optimal order in the asymptotic expansion. For example, [26] computed the meta-stability time of asteroids around the Lagrangian point L4, that contain a ghost degree of freedom.
In our model, we can compute the time τ n (J in max → J max ) for which we are guaranteed that any evolution starting from We maximise over J max , when possible, having in mind Lyapunov stability, so that Computing at different orders n in the expansion give different J i and different times τ n ; because of the asymptotic character of the Birkhoff series stability is guaranteed up to the largest τ n . Non-conservation of J i happens because interactions δH remain at higher order: where H (≤n) = n k=0 k H (k) includes terms up to order n. The leading-order contribution to the residual is Such term can be computed from its Fourier coefficients The residual time evolution of J i is given by its Hamiltonian equation of motioṅ where, at leading order in the residual, we can approximate ∂/∂Θ i ∂/∂Θ i and thereby avoid re-expressing Θ in terms of Θ in δH. A lower bound on the stability time is obtained by substitutingJ i with its maximal value. Neglecting higher orders in : having used the triangular inequality. Higher orders in weaken the bound in eq. (38) by a factor of 2 [26].

Stability at lowest order
To start, we outline the procedure at lowest order, such that the approximately conserved quantities are simply J i = J i and the remainder in the Birkhoff series simply is the whole interaction To compute the stability time, we use the inequality The region can be abandoned only after a time Its maximal value, achieved for J max = 2J in max , is the Lyapunov stability time:

Stability at generic order
The above discussion is easily generalized at order n. The residual time evolution is bounded by where we included the factor of 2 due to higher orders, maximised over the free index i = 1, 2, and used the fact that the remainder is a homogeneous polynomial in J i of order n + 2. The function β n (ω 1 , ω 2 ) can be computed numerically and diverges close to resonances: The Lyapunov stability time is In view of the asymptotic character of the Birkhoff series, for each value of ρ 0 , there is an optimal order n that gives the strongest bound.
As an example, in fig. 1a we show the stability bound computed for ω 2 /ω 1 = √ 2. The numbers on the curve indicate the optimal order. Some order dominates for a larger range when it contains enhanced denominators. Our example contains enhanced denominators at 7th order (1/(7ω 1 − 5ω 2 )) and 17th order. Something similar happens in fig. 1b, where we consider ω 2 /ω 1 = π 2 . In both cases we keep fixed J 1,2 = 1 at any given order, which approximatively means J 1,2 = 1 for values of λ small enough that the series converges.
For small enough coupling we proved ghost meta-stability up to cosmological times that cannot be probed by numerical studies. We consider a specific model that contains no special features: a similar analysis can be performed for any other model.

Resonances i.e. on-shell processes
The previous perturbative approximation becomes less accurate close to resonances. The most dangerous resonance corresponds to ω 1 = ω 2 , as 1/(ω 1 − ω 2 ) enhancements occur at leading order in the coupling. As a result the Birkhoff series already fails for E int /E free > ∼ (ω 1 − ω 2 )/ω 1,2 , instead of holding, as usual, when the energy in the interaction terms is smaller than the free energy. Numerical solutions in our model with resonant ω 1 ≈ ω 2 (and very small λ such that interactions negligibly modify frequencies) show that a linear combination of J 1,2 fails to be quasi-constant of motion, but remains bounded so that run-aways remain avoided.
We extend analytic techniques to study resonances as they will be important in our subsequent study of classical and quantum field theories. 5 As described in advanced books about analytic mechanics [27], resonant processes can be analytically studied by modifying the Birkhoff normal form into a "resonant normal form" that avoids the enhanced terms by selectively downgrading the goal of cancelling all dependence on the angle variables. One needs to keep those that give resonant combinations, obtaining a more complex but still manageable partially-diagonalised Hamiltonian. Some combinations of J remain quasi-conserved, whereas others evolve as governed by the resonant form.

Example: ghost that remains stable close to resonance
To clarify with a worked example, we reconsider our model of eq. (17) in the resonant case ω 2 → ω 1 . We perform a canonical transformation analogous to eq. (28) (at leading order) 5 By expanding fields into Fourier modes one gets an infinite number of interactions, that always contain giving rise to decays and other on-shell process, using the standard terminology of quantum field theory (when E = ω the resonance condition becomes conservation of energy and momentum). Figure 3: Phase portrait of the auxiliary system close to the resonance. The thick line is the separatrix between the different kinds of motion. The shaded gray region in phase space cannot be accessed with J 1,2 ≥ 0.
but omitting the singular Fourier modes with N 1 = N 2 ≡N , that multiply Θ 1 + Θ 2 . A straightforward but tedious change of variables gives The same result can be re-obtained by expanding eq. (26) in Fourier modes and taking into account that off-diagonal elements of W (1) NN vanish, leaving the Hamiltonian Fourier coefficients H (1) gives again eq. (46), after taking into account that Θ i Θ i . The series expansion is no longer singular at the resonance, so that its first order is accurate at small coupling. We can use it to study the dynamics close to the resonance finding that, since 1 + 1 2 cos 2(Θ 1 + Θ 2 ) > 0, motion remains bounded. This can be better seen by performing the canonical transformation such that, writing ω ≡ (ω 1 + ω 2 )/2, ∆ω ≡ ω 1 − ω 2 , the Hamiltonian becomes H and E are constants of motion, 6 while J is no longer conserved and forms, together with Q, a system with 1 degree of freedom, simple enough that can be analytically studied. The key point is that its Hamiltonian is bounded so that J , despite not constant, is bounded and the action variables J 1,2 are bounded too. The possible motions are shown in fig. 3. Typical trajectories move away from the resonance and then go back to it. J max /J min is generically of order one, with the maximal variation √ 3 obtained for ∆ω = E = 0. For ∆ω sufficiently large, some of the trajectories in phase space oscillate. All trajectories are bounded.
In conclusion, the ghost system with quartic interaction q 2 1 q 2 2 is stable when perturbed around the non-interacting equilibrium point. Away from resonances stability follows from the Birkhoff expansion and the KAM theorem [28,27]; the latter states that away from resonances most trajectories in phase-space are still confined to be toroidal, even in the presence of small interactions. Close to the ω 1 ω 2 resonance, stability follows because the extra system is not a ghost, so its motion is bounded; higher-order resonances are not dangerous because their resonant normal forms remain dominated by leading-order non-resonant terms.

Example: ghost that undergoes run-away close to resonance
The 'safe' situation found in the previous model is not generic. In other models a ghost can become unstable close to resonances. This happens when the auxiliary dynamics that approximates the system close to a resonance is ghost-like and the resonant surface in phase space extending to J → ∞ (at fixed energy/approximate integrals of motion) is attractive. This happens, for example, replacing the quartic interaction q 2 1 q 2 2 with a cubic interaction q 2 1 q 2 . The Hamiltonian in action-angle variables is and the dangerous resonance is ω 2 ≈ 2ω 1 that (loosely speaking) allows for a q 1 → q 1 + q 2 'decay'. The resonant Birkhoff form at first order is The sign of sin(2Θ 1 + Θ 2 ) now qualitatively impacts the system. This can be seen performing withω = (2ω 1 + ω 2 )/4, ∆ω = 2ω 1 − ω 2 . The auxiliary system is now a ghost: the resonant (∆ω = 0) trajectories at fixed E extends to J → ∞, e.g. the trajectory with Q = 0. Moreover, these trajectories are attractive. At the resonance all trajectories are unbounded. Moving away from the resonance some stable KAM tori appear "on one side" for J small enough, but nothing protects stability on the other side (large J ).
Notice that the condition of ghost safety is independent from the condition of bounded-frombelow potential. For example, consider a model with quartic interactions H ⊃ λ (q 2 1 q 2 2 +κq 3 1 q 2 )/2. Close to the resonance ω 2 3ω 1 we find that the ghost is safe for |κ| < 2/ √ 3, despite the potential is unstable for any κ = 0 (for instance along the line q 2 = 1, q 1 → −∞). Conversely, the potential with with quartic interactions H ⊃ λ (q 4 1 + κq 3 1 q 2 ) is stable for any finite value of κ, but the ghost causes run-away for |κ| > 3 √ 3. The above considerations generalize to systems with more degrees of freedom. For instance, let us consider a system of 3 degrees of freedom with interaction q 1 q 2 q 3 , where q 2 is a ghost. The Hamiltonian in action-angle variables is The first-order resonant form close to the dangerous resonance The extra-system Hamiltonian is unbounded and as a consequence the system, on resonance, undergoes ghost run-away.
The discussion of various examples allows to identify a useful general property: only the part of the Hamiltonian at most quadratic in J is typically relevant for stability, since close enough to the origin cubic and quartic interactions dominate over higher orders. In the presence of both cubics and quartics, quartic interactions generically stabilise the otherwise un-safe behaviour of cubic-only interactions. This can be seen by noticing that resonant normal forms of quartic interactions contain stabilising terms ∼ J 2 (as in eq. (46)), that dominate with respect to the dangerous dynamical terms ∼ J 3/2 f (Θ) for sufficiently large J.
In conclusion, ghost stability in classical mechanics is generic at small coupling away from resonances. In most models, resonances do not lead to ghost run-away but only to partial energy flow.

Ghost meta-stability in quantum mechanics?
Moving from classical to quantum mechanics, we again consider the prototype model of eq. (3), described by the Hamiltonian which leads to the Schroedinger equation for the wave-function ψ(q 1 , q 2 ) We remind the following features of the Schroedinger equation in the absence of ghosts and relevant for computing vacuum tunnelling through a potential barrier: 1) the sign of E − V tells in which regions ψ oscillates or gets exponentially suppressed; 2) the vanishing of E − V determines the 'release point' q * on the other side of the potential barrier after which classical motion is unstable; 3) the tunnelling rate is exponentially suppressed by the WKB bounce action W = min where the integral is along the path in multi-dimensional field space that minimises W .
These features are now lost because the ghost appears with an opposite sign in eq. (57). So the classically meta-stable ghost q 2 might become unstable if the wave-function ψ(q 1 , q 2 ) of any state extends along the classically-allowed region q 1 ≈ q 2 reaching the large values where classical motion leads to run-away.

Model computation
In the presence of a ghost an infinite numbers of states have E = 0, or any other value. The same happens, without ghosts, in the presence of a potential like V = ω 2 q 2 /2 + λq 4 /2 with negative λ: despite that V is unbounded-from-below, the lowest-energy bound state is special. We focus on the analogous of this state for the ghost system. In the free theory such groundlike bound state has minimal positive energy and maximal negative energy. Thanks to this property, it might be selected by cosmological evolution. We now show that the ground-like state is meta-stable.
We start by numerically computing the ghost model described by the Hamiltonian of eq. (56). If the coupling λ vanishes it reduces to two decoupled harmonic oscillators, with the usual eigenstates |n 1 , n 2 . The ground-like state is |0, 0 with wave-function ψ 00 (q 1 , q 2 ) = ψ 0 (q 1 )ψ 0 (q 2 ) with ψ 0 (q i ) ∝ e −q 2 i ω i /2 . For λ = 0 the ground-like state is the one that tends to |0, 0 as λ → 0, and that thereby, at small λ, has maximal projection along |0, 0 . Its wave function ψ(q 1 , q 2 ) has no nodes around q 1 ∼ q 2 ∼ 0 and can be computed either numerically solving the Schroedinger eq. (57) or by writing the Hamiltonian H of eq. (56) as a matrix in the |n 1 , n 2 basis and diagonalising it. Matrix elements of the interaction term λq 2 1 q 2 2 /2 are computed using q 1,2 and an unbounded-from-below potential with λ < 0. Inside the barrier at q 1 ∼ q 2 ∼ 0 the wave-function is the usual Gaussian; outside it has an oscillatory pattern with exponentially suppressed amplitude. In our approximation the wave-function is real, but one can compute a more accurate bound-state with complex wave-function such that the exponentially suppressed probability current is out-flowing only. Its flux equals the vacuum decay rate, and the energy eigenvalue acquires a correspondingly exponentially suppressed imaginary part (see e.g. [29]). The ghost case qualitatively differs from the negative-potential case only in the resonant situation ω 1 = ω 2 : the ghost ground-like state does not reduce to |0, 0 as λ → 0.

The WKB approximation
Ghost meta-stability can be understood more in general taking into account that tunnelling can be approximated a la WKB. Writing the wave function as ψ = e iS/ , the Schroedinger equation reduces to the classical Hamilton-Jacobi (HJ) equation plus extra terms 1 2 i ∂ 2 S/∂q 2 i neglected at leading order in the semi-classical expansion, which is enough to approximate vacuum decay at weak coupling.
In Hamiltonian mechanics, eq. (59) is obtained by demanding that S generates a classical canonical transformation such that the transformed Hamiltonian vanishes. Its solution is the classical action S(q, t) = q,t 0,0 L(q cl ) dt computed along the classical particle trajectory going from q = 0 at time t = 0 to q at time t. Thereby the HJ wave equation provides a bridge between waves and particles: S respects the good hidden properties of a classical ghost discussed in section 2. To make better contact with the formalism of section 2 we consider a Hamiltonian H that does not depend on time. Then eq. (59) can also be solved by separating variables as S(q, t) = W (q) − Et where E = H is the constant energy and W generates a canonical transformation to action-angle variables (Θ i , J i ) such that H only depends on J i . The 'reduced action' W satisfies the wave equation The classical change of variables to action-angle coordinates essentially is a 'diagonalization' of the classical Hamiltonian. Eq. (59) (eq. (60)) approximates the time-dependent (time independent) Schroedinger equation eq. (57), with the first (second) form being more useful for computing the propagator (energy eigenstates). The hidden constants of motion that in the classical theory forbid motion into the dangerous region q 1 ≈ q 2 still play a role in the semi-classical approximation. No new dramatically fast ghost instabilities appear in the quantum theory as, going away from the origin q 1 ∼ q 2 ∼ 0, the wave function gets exponentially suppressed by the semi-classical WKB factor W . Having a quantum Hamiltonian in action-angle variables, H = ω(J)J, its eigenstates are the |J states with eigenvalues E = H(J) and wave function Θ|J = e iJΘ/ , so that its periodicity demands J = n with n an integer.
To obtain tunnelling rates we need to compute how the wave-function extends into the classically forbidden region: as well-known it is useful to perform an analytic continuation to Euclidean time, t E = it and solve the Euclidean HJ equation with L E = 1 2 (d q/dt E ) 2 − V E and inverted potential V E = −V . A well-known computational simplification allows to approximate potential tunnelling in the absence of ghosts: the vacuum decay rate is approximated by e −B , where the bounce action B = min W E is computed along the classical Euclidean trajectory in field space that connects the false vacuum to the other side of the potential barrier with minimal W E . For example for the ground state with E → 0 + . This simplification holds in the presence of multiple degrees of freedom, and thereby allows to compute vacuum decay in Quantum Field Theory [30]. A similar result holds in the presence of ghosts only, with the only difference that boundary conditions (normalizable wave-function) now demand picking the opposite-sign solution to the HJ equation. The sign of W is not fixed because H contains p 2 = (∂W/∂q) 2 . For the ground state E → 0 − the bounce action is similar to eq. (61) but with t E → −∞. Equivalently, an opposite-sign Wick rotation is needed to make the Euclidean ghost action positive.
In the presence of positive-energy particles that interact with ghosts, the desired solution to the HJ equation can be found numerically or perturbatively up to q 2 < ∼ ω/λ, (62) but we don't know how to compute vacuum decay bypassing a full solution to the HJ equation [31]. Physically, the new complication arises because we are interested in the ground-like state, which is neither the lowest nor the highest energy state, so that selecting it gets more complicated.

Ghost meta-stability in classical field theory?
A field ϕ( x, t) can be decomposed as an infinite number of Fourier modes q n (t). An infinite numbers of degrees of freedom allows for new phenomena. Some of them make any interacting classical field theory problematic, others are a problem for theories containing ghosts. As ghosts are at most a co-morbidity of the theory, one needs to address and disentangle the new intertwined issues.
1. In order to compute numerically one has to 'regularise' the theory by introducing a cut-off on the number of degrees of freedom, usually realised by a minimal length a, such as a lattice discretisation of space-time. Typical discretised field equations do no conserve energy and can lead to fake run-away behaviours when evolving configurations with excited modes near the cut-off (the ones where energy conservation is badly violated). We will define special discretised classical equations that exactly conserve total energy, but hidden pseudo-constants of motion can be violated by the regularisation.
2. At some moment and in some region of space, some modes can acquire a higher energy density and overcome the energy barrier between stability and instability. In thermal field theories with local minima in the potential this is the well-known thermal tunnelling, characterised by a space-time tunnelling probability density. 7 The same mechanism contributes to ghost instabilities.
3. General initial field configurations tend to thermalise. However, a thermal state is impossible in classical field theory, as each one of the infinite modes should have the same energy ∼ T . In electro-magnetism, this is the well known black-body problem. An interacting field theory gives rise to a cascade of energy towards higher-frequency modes, 7 Some authors claim that they can approximate quantum vacuum decay rate by classically evolving a field starting from quantum-like initial conditions [32,33] and waiting for a large enough energy fluctuation that goes over the potential barrier. However this can only be a rough approximation, because an interacting classical field theory tends to evolve towards a thermal state where energy is equipartitioned among all modes. and the temperature evolves towards T → 0. On a lattice, this cascade stops when the problematic modes at the cut-off thermalise.
4. The above issue is solved by quantum mechanics. For a thermal state, classical field theory only holds for modes with E < ∼ T , and is replaced by quantum field theory for modes with E > ∼ T that get suppressed energy density: 5. Finally, the main new point. Field theory contains an infinite number of modes q n (t) with frequencies ω n , so resonances are always possible. These resonances are the usual on-shell processes such as decays and scatterings. In the presence of ghosts, resonances can lead to partial or total loss of hidden constants of motion as discussed in section 2.4.
In section 4.1 we decompose fields ϕ(x, t) into modes q n (t), and in section 4.2 we perform a stability analysis of the resonances: hidden constant of motion persist up to O(1), but the number of resonance is so large that dangerous energy transfer between normal fields and ghosts can take place. As a consequence, assuming no protection, in section 4.4 we use statistical methods to compute the energy transfer between normal fields and ghost fields. Finally, in section 4.5 we compare analytic results to numerical classical lattice simulations (using the convenient discretised field equations described in appendix C).

Classical equations of motion in momentum space
We consider a scalar field ϕ(x, t) in 1+1 dimensions. In a box 0 ≤ x ≤ L with periodic boundary condition the scalar field is expanded in normal modes q n as We consider a real scalar field, so q −n = q * n . The Lagrangian density L ϕ = (∂ µ ϕ) 2 /2−m 2 ϕ 2 /2+ L I gives the Lagrangian The dx integral is simply given by L times the expansion of L , keeping only those terms such that their e ikx factors multiply to 1. The classical equations of motion arë q n + ω 2 n q n = ∂L I ∂q n .
Classical evolution can be restricted to real q n , which means zero momentum for each mode. The averaged free classical Hamiltonian is so that the classical thermal state with equipartition of relativistic energy corresponds to q n = √ T /ω n , which is the (UV divergent) classical limit of the Bose-Einstein distribution, q n q −n = (1/2 + f n )/ω n with f = 1/(e E/T − 1) → T /E 1 at E T . The extra 1/2 is the purely quantum fluctuation. H is UV divergent both in classical physics at finite temperature T , and in quantum physics.

Analytic study of one ghost resonance in field theory
As a prototypical field theory containing a normal field ϕ 1 interacting with a ghost field ϕ 2 we consider the Lagrangian of eq. (4) where the ghost is obtained setting ± = −1. For simplicity we here compute in 1+1 dimensions, as this is enough to encounter the new key phenomena. The two fields ϕ 1,2 have positive and negative kinetic energy, respectively. We expand each of them in normal modes q n 1 and q n 2 as outlined in the previous section. The interactions among momentum modes q n i are complicated because locality is not manifest. Let us focus on four generic modes: n 1 and n 1 for ϕ 1 and n 2 and n 2 for ϕ 2 . We assume that k n 1 +k n 1 +k n 2 +k n 2 = 0. Then, their interaction term is dx ϕ 2 1 ϕ 2 2 = 4 L (q n 1 q n 1 q n 2 q n 2 + q −n 1 q −n 1 q −n 2 q −n 2 + q n 1 q −n 1 q n 2 q −n 2 + + q n 1 q −n 1 q n 2 q −n 2 + q n 1 q −n 1 q n 2 q −n 2 + q n 1 q −n 1 q n 2 q −n 2 + · · · ).
The frequencies are generically off-resonance but for some choice of momenta they satisfy resonant conditions such as N 1 ω n 1 + N 2 ω n 1 − N 3 ω n 2 − N 4 ω n 2 even for N i = ±1, giving rise to on-shell processes. We isolate a sub-system of four such degrees of freedom q n i . For simplicity we can assume that their initial conditions are real, so that they remain real and we can treat q n = q −n as a single degree of freedom. Moving to action-angle variables and simplifying the notation, we write their pulsations as ω 1,2,3,4 and their actions as J 1,2 (positive energy) and J 3,4 (negative energy). The Hamiltonian of the sub-system is where = 8λ/L. Off-resonance the system is stable, and we now study the possibly dangerous resonant case, assuming ω 1 + ω 2 − ω 3 − ω 4 ≡ ∆ω 0. 8 Close to resonance, the normal resonant form at leading order is We isolate the auxiliary system by the canonical change of variables generated by i.e. 4Q = Θ 1 + Θ 2 + Θ 3 + Θ 4 and J 1 = J /4, J i = J /4 + E i . The resonant form becomes so that E 1,2,3 are constant of motion i.e. all J i vary by a common amount J /4. The important result is that cos 4Q cannot dominate over the sum of other terms, so that this resonance does not lead to ghost run-away, but only to a partial violation up to O(1) factors of the hidden conservation law. This means that the local interaction ϕ 2 1 ϕ 2 2 of field theory gives, when expanded in normal modes, a specific set of interactions among them such that each on-shell resonance allows an order one energy transfer among the modes, but no ghost run-away.

Analytic study of multiple ghost resonances in field theory
We next need to study what is the collective effect of the infinite number of such resonances present in the continuum limit: the number of modes N = L/a diverges when the lattice cut-off a becomes infinitesimally small, or the box size L infinitely large. The Hamiltonian in action-angle variables is an infinite sum of terms like those discussed in the previous section 8 We assume for now that no other combinations are vanishing, so that resonances do not "overlap". In appendix B we show that the case of all frequencies close to each other leads to similar conclusions as the ones discussed here. + n 1 ,n 1 ,n 2 ,n 2 δ 0,n 1 +n 1 +n 2 +n 2 J n 1 J n 1 J n 2 J n 2 sin Θ n 1 sin Θ n 1 sin Θ n 2 sin Θ n 2 , (73) with = 2λ/L(ω n 1 ω n 1 ω n 2 ω n 2 ) 1/2 . The rough argument goes as follows. At small coupling the theory contains 2N quasi-integral of motion: one for each degree of freedom. In the continuum limit the number of resonances scales as N 2 (out of the 4 momenta, 2 combinations are fixed by momentum conservation and resonance condition, i.e. energy conservation). Each resonance produces the partial loss of a quasi-integral of motion E. Asymptotically, all quasi-integrals of motion are lost and the available phase-space is filled up, allowing for ghost run-away.
The argument above can be made more precise. A combination is resonant if the detuning ∆ω ≡ ω n 1 + ω n 1 − ω n 2 − ω n 2 is smaller than the expansion parameter J, where J is the typical value of the actions, e.g. J = T /ω for a thermal state. For finite L the resonance is not exactly satisfied and the expansion parameter is finite. Both quantities go to zero in the continuum limit, so a careful analysis is needed. Let us consider modes up to an UV cut-off k k max . A resonance that would be perfect in the continuum acquires, in view of the discreteness δk = 2π/L, a typical detuning ∆ω ≈ (8π/L)(k max /ω max ). Here ω max is the frequency corresponding to k max having ignored, for simplicity, that it differs for fields ϕ 1 and ϕ 2 if m 1 = m 2 . The fraction of such interactions that are resonant for finite L is f = J/∆ω. This stays finite in the continuum limit, as both and ∆ω scale as 1/L. So the ghost is not protected when f N 2 N i.e. N = L/a 1/f ∼ ω 3 /λT . Then the action J n of one typical microscopic mode can change by order one on a time-scale Γ ∼ λT /ω 2 , linear in λ at leading order. As discussed in the next section, the macroscopic properties of the system evolve on a slower time-scale 1/τ = Γ/N ∼ λ 2 T 2 /ω 5 . As we will see, this is the scale of the instability time. If, instead, there were no microscopic protection for the single modes, the instability time would have been much faster, linear in λ.

The ghost run-away rate
Based on the previous discussion we assume that the extra quasi-conserved energies get violated in field theory by resonances. Then the system evolves statistically, towards the direction that increases total entropy S = S 1 + S 2 , where 1 is the positive-energy sector and 2 is the ghost. We define the ghost temperature T 2 as the average ghost energy E 2 ≤ 0 per degree of freedom, The volume in phase space is easily found in action-angle variables: The factor of (2π) N is the contribution of the angle variables, whereas the remaining factor is the volume of the simplex ω n J n ≤ |E 2 |. Therefore the ghost entropy is up to a T 2 -independent constant. The total entropy S = S 1 + S 2 of the system at fixed total energy E 1 + E 2 is maximal when which can only occur for T 1 → ∞ and T 2 → −∞. Heat flows from the ghost to the positiveenergy system and the thermodynamic evolution eventually causes the run-away on a time-scale τ , that we now compute. We consider a theory in d spatial dimensions with the Lagrangian of eq. (4). To set the formalism, we first assume that both fields ϕ 1,2 have positive kinetic energy. Then, starting from temperatures T 1,2 ≥ 0 they thermalise towards the equilibrium state with a common temperature T = (T 1 + T 2 )/2 via the λϕ 2 1 ϕ 2 2 /2 interaction. The thermalization process can be computed using Boltzmann equations. We consider their well known quantum expression and perform its classical limit, to later compare with numerical classical evolution on a lattice. In order to keep factors explicit it is convenient to express quadri-momenta P µ in terms of wave vectors, P µ = (E, p) = K µ = (ω, k). The Lagrangian L contains no factors, so the mass parameters m 1,2 have dimension 1/time. The contribution of 12 ↔ 1 2 scatterings to the Boltzmann equation for the energy density ρ 1 (assumed to be spatially homogeneous) of ϕ 1 at leading order in the interaction λ iṡ where A = 2 λ is the amplitude; d k = d d k/2ω(2π) 3 is the usual relativistic phase space; one can symmetrise E 1 → (E 1 − E 1 )/2. Finally F depends on particle number densities It vanishes when Bose-Einstein distributions f (E) = 1/(e E/T − 1) realise thermal equilibrium. Total energy is conserved, soρ 2 = −ρ 1 . The quantum Boltzmann eq. (77) has two classical limits: particle and wave. The particle limit corresponds to small occupation numbers f 1 such that 1 + f 1 and f e −E/T . We are here interested in the wave classical limit, that corresponds to large occupation numbers f T /E 1. The classical wave term arises at leading order f 3 [34,35] where In this limit factors cancel leaving the classical Boltzmann equatioṅ where the latter term is 3 F . One can similarly compute the contribution toρ 1 from 11 ↔ 22 scatterings. Furthermore, a gϕ 2 1 ϕ 2 /2 interaction among positive-energy fields ϕ 1,2 gives rise to 2 ↔ 11 decays for m 2 > 2m 1 such thaṫ with A = g 1/2 and in the classical limit.
We can now repeat the computation assuming that ϕ 2 is a ghost. Boltzmann equations again involve a sum over on-shell processes, and the resonance condition among ω's now has an extra − sign when a ghost is involved, see e.g. eq. (28). This is equivalent to telling that ghosts appear with negative energy in the quantum Boltzmann equations. One can re-express the unusual (negative-energy) kinematical integrals in terms of usual (positive-energy) ones by rewriting each ghost wave vector as K µ = −K µ , so that a negative-energy particle in the initial (final) state becomes a positive-energy particle in the final (initial) state. In the limit where each field is thermal, the Bose-Einstein distribution satisfies the identity f (E/T ) = −(1+f (−E/T )), so statistical factors too match those of the positive-energy process, up to an overall − sign when an odd number of ghosts is flipped. Let us consider some examples: • A ϕ 2 1 ϕ 2 2 ghost interaction allows the kinematically open on-shell processes 12 ↔ 1 2 and 11 22 ↔ ∅, that become 12 ↔ 12 and 11 ↔22 . In the classical limit one then haṡ ρ 1 ∝ +T 1 T 2 (T 2 − T 1 ) both in the ghost and the non-ghost cases.
• A ϕ 1 ϕ 2 2 ghost interaction allows the kinematically open on-shell process 122 ↔ ∅, that becomes a 1 ↔22 decay. In the classical limit one then hasρ 1 ∝ +T 2 (T 2 − T 1 ) both in the ghost and the non-ghost cases.
The factors F vanish in the thermal limit with a common temperature, f (E) = 1/(e E/T − 1). However ghosts have E 2 < 0, so that a physical f (E 2 ) ≥ 0 is obtained for T 2 ≤ 0: ghosts must have a negative temperature. 9 We now see the key difference that arises in the presence 9 We verified that the non-equilibrium Kadanoff-Baym formalism (see e.g. [36]) gives the same Boltzmann equations. In particular, for a ghost, the form of its two thermal Wightman propagators is exchanged with respect to positive-energy fields, so that initial-state ghosts are equivalent to final-state normal particles. In this formalism f ≥ 0 because it is the expectation value of a positive number operator.
Previous literature studied possible thermal equilibrium thermodynamics for Lee-Wick resonances with negative classical energy [37][38][39] finding contradictory results. We now see that there is no thermal equilibrium. of a ghost: there is no thermal equilibrium at common T such that the factor F vanishes thanks to detailed balance, because the two systems have opposite-sign energies and thereby temperatures. In all cases listed above this means that the non-ghost system heats up,ρ 1 > 0. This sign of the heat flow agrees with our earlier considerations about increase of entropyṠ ≥ 0: both |T 1 | and |T 2 | increase, as higher temperature allows for more states. Boltzmann equations add that the energy flow rate is proportional to the coupling squared. The purely quantum effect will be studied in section 5. We here study the classical effect, that can be isolated as long as the low-frequency modes excited classically ω < ∼ ω max are separated from the high-frequency modes at which the divergent quantum effect starts giving a larger contribution toρ 1 . In such a case, the quantum contribution is smaller than the classical contribution assuming a cut-off Λ UV > ∼ ω max .
We next compare these analytic results with numerical classical simulations in toy models, and finally provide estimates for situations of physical interests.

Results
First we simulate the classical thermalization among two positive-energy fields, finding that the simulated rate agrees with the rate obtained from Boltzmann equations such as eq. (77).
We next consider a positive-energy field ϕ 1 interacting with a ghost field ϕ 2 . We numerically simulate their time evolution for m 1,2 = 1 and λ = 0.01 in 1 + 1 dimensions on a lattice with spacing a = 0.1 and size L = 200. We start from a thermal-like distribution with T 1,2 = 1 Figure 6: Energy spectrum of the normal field (left) and of the ghost field (right) at some fixed times.
cut at the maximal momentum k max = 200 · 2π/L. This means that each excited mode has an initial amplitude as extracted from the thermal distribution and a random phase. Fig. 5a shows that the system undergoes ghost run-away. Fig. 6 shows the energy spectra of ϕ 1 (left) and ϕ 2 (right) at some selected times. We see that modes at higher k get progressively excited: energy cascades towards the UV giving rise to the usual black-body instability of interacting field theories (see e.g. [40,41]). In order to disentangle this phenomenon (that lowers T 1 ) from ghost run-away (that increases T 1 ) we choose a small enough k max such that modes around the cut-off are still negligibly excited when ghost run-away happens. Each random initial condition with fixed temperatures produces final run-away times that differ by order one. In order to better compare with the analytic approach, that predicts the average energy flowρ 1 between the two fields, we run for a short time many different simulations with the same initial temperatures and average over them. Having assumed one spatial dimension and m 1 = m 2 we can analytically perform the integrals in the Boltzmann eq. (80),ρ where we added, in the second term, the contribution of 11 ↔ 22 scatterings. This process contains a logarithmic IR divergence at vanishing relative velocity between the particles, which is typical of field theory in 1 spatial dimension. 10 Despite this aside issue, fig. 5b shows that the analytic rate agrees with the numerical rate. We can next compute ρ 1 in terms of T 1 and obtain a differential equationṪ 1 = γT 1 T 2 (T 2 − T 1 ) = −Ṫ 2 that can be solved obtaining the average time evolution (one example is plotted in fig. 5a).
We next vary the lattice spacing, box size and number of digits used in the numerics, finding consistent results. We also run for different values of the physical parameters; additional IR divergences arise when a field is massless. Running for special initial conditions, such as starting from a single excited mode, f (E) ∝ δ(E − E 0 ), blocks or delays the ghost run-away until when enough modes can get excited, so that many resonances can happen.
Based on the above experience, we can now consider the more complicated theory of possible physical interest: 4-derivative gravity. First, the resonances caused by cubic interactions present in 4-derivative gravity, while potentially un-safe, are stabilised by quartic and higher interactions, as argued at the end of section 2.4. Then, each resonance causes an O(1) energy flow variation and the system as a whole evolves statistically, as described above. The massive ghost present in 4-derivative gravity only has Planck-suppressed non-renormalizable interactions. Thereby its run-away rate Γ ≡ρ/ρ ∼ T 3 /M 2 Pl is smaller than the Hubble cooling rate H ∼ T 2 /M Pl . As usual, gravitational short-range interactions give negligible effects in big-bang cosmology.
Furthermore, inflation with Hubble constant H roughly behaves as a thermal bath with temperature T ∼ H, producing a spectrum of primordial inflationary fluctuations for the graviton, its ghost, and the other fields.
In conclusion, a ghost undergoes run-away in classical field theory, but in 4-derivative gravity ghost run-away is negligibly slow on cosmological time-scales. 10 In order to isolate the IR divergence, it is useful to to put the 11 ↔ 22 contribution toρ 1 into the form In lattice simulations the IR divergence gets regulated by the finite box size L, so that the lower limit of s integration changes into (2m + 2π/L) 2 4m 2 + 8πm/L.

Ghost meta-stability in quantum field theory?
We again consider a field theory with two scalars ϕ 1 (positive energy) and ϕ 2 (negative-energy ghost) in d space dimensions. We want to compute the purely quantum rate for the qualitatively new processes where particles are emitted from the Lorentz-symmetric vacuum. For example a gϕ 2 1 ϕ 2 /2 interaction allows for the 3-body process ∅ ↔ 11 2.
The rates of such processes can be obtained from the finite-temperature rates discussed in the previous section in the limit T 1 → 0 + , T 2 → 0 − and thereby f 1,2 → 0, 1+f 1,2 → 1. Following the discussion in section 4.4, it is convenient to rewrite the Boltzmann equation in terms of positive energies2 ↔ 11 by definingK 2 = −K 2 . Since f 2 (−E 2 /T 2 ) → −1 the statistical factor at zero temperature is F → −1, while it would be F = 0 for a usual process involving only positive-energy particles. The resulting quantum rate for the 3-body process ∅ ↔ 11 2 contains a UV-divergent integral over K 0 . Similarly, an interaction λϕ 2 1 ϕ 2 2 /2 allows for the 4-body process ∅ ↔ 11 22 that leads to the energy flow rateρ By introducing K ≡ K 1 + K 1 =K 2 +K 2 and s ≡ K 2 it becomeṡ Again, the integral over K 0 is UV-divergent. This new divergence arises because, unlike in the thermal case, the vacuum initial state ∅ is now Lorentz-invariant so that the final state too must be the same in all frames. This is why the rate contains a dK 0 integral over the non-compact Lorentz group. This is the same divergent 'boost' integral discussed by [42,43] (and more recently by [44]). These early studies of vacuum decay considered a theory containing a scalar with positive kinetic energy (no ghost) and assumed that its potential V contains a local minimum e.g. with V = 0 and a deeper minimum with V < 0. The vacuum decay bubble with mass m = 0 can appear with any initial velocity, giving rise to the divergent Lorentz integral [42,43]. Furthermore, by e.g. increasing its radius one obtains field configurations with generic m 2 < 0, that thereby have negative energy with K 2 = (m 2 , 0). Such ghost configurations can be emitted from the vacuum together with one particle with positive energy K 1 = (m 1 , 0), for m 1 + m 2 = 0. Due to relativistic invariance, this process happens with the same amplitude for arbitrarily boosted K 2 and K 1 , giving rise to a divergent dK 0 integral over boosts [44].
One then wonders if both K-instability (ghosts) and V -instability (vacuum tunnelling) proceed with infinite rate, in contradiction with our usual understanding of vacuum tunnelling as exponentially slow [44].
In the case of V -instability, Coleman [30] and more recently [45] interpreted the Lorentz boost divergence as emission of lots of extra quanta i.e. that the naive perturbative computation is not expanding the path integral around the right saddle point. 11 These authors argue that vacuum tunnelling must instead be computed expanding around a Lorentz-invariant 'bounce' configuration, such that an integral over the Lorentz group is not needed because it would be an over-counting of the same configuration. Accepting this argument, the WKB approximation allows to find the desired configuration as the 'bounce' instanton that minimises an effective Euclidean action. The 'bounce' is the solution to the scalar field equations that only depends on the Euclidean r 2 E = x 2 + y 2 + z 2 + (it) 2 (the Euclidean Lorentz group is compact) and has the desired boundary conditions: false vacuum at r → ∞ and over the barrier at r → 0: The resulting vacuum decay rate is exponentially suppressed by the coupling, e −O(1)/λ .
In the ghost case, we do not have a similarly simple formulation nor a positive Euclidean action. Unless a suitable continuation is found, a brute-force computation is needed to establish if the ghost decay rate is exponentially suppressed (restricting the action to Lorentz-invariant field configurations removes field-theory resonances but leads to r-depended frequencies). We speculate that, if the vacuum decay rate will turn out to be exponentially suppressed, the difficulties that seem to hinder unitarity and/or renormalizability of Minkowskian theories with ghosts (see e.g. [49][50][51][52]) will turn out to be similarly suppressed by similar factors.

Conclusions
Systems containing positive kinetic energy K 1 interacting with negative kinetic energy K 2 can undergo a run-away where the total energy E = K 1 + K 2 + V is constant while |K i | → ∞. Thereby negative kinetic energy is considered as unphysical and dubbed 'ghost'. We explored the possibility that negative kinetic energy can be physically acceptable because meta-stable up to cosmological times, similarly to negative potential energy. In order to exclude this possibility we started from the simplest limit (classical mechanics), but we found that a weakly-interacting ghost behaves almost as well as a free ghost: • In section 2 we found that ghosts are meta-stable in classical mechanics. Recent numerical studies rediscovered that, in some cases, energies of individual degrees of free-dom surprisingly remain confined to a finite region despite that no constant of motion imposes such lock-down. Ghost meta-stability is understood using the same mathematical techniques developed in the past centuries to study if multi-body systems like the solar system are stable up to cosmological times despite that individual planets can acquire enough energy to escape. One 'diagonalises' the classical Hamiltonian by performing a perturbative expansion around the limit where each degree of freedom undergoes periodic motion with pulsation ω i . Technically, this means finding a canonical transformation to action-angle variables such that the Hamiltonian does not depend on angle variables. If interactions are strong, outside the convergence radius of the perturbative series, motion is chaotic, planets escape and ghosts run-away. If interactions are weak the perturbative series is convergent: planets undergo quasi-periodic motion with epicycles, and ghosts are stable. The dimension-less expansion parameter is the energy in the interaction term divided by the energy in the free quadratic part of the Hamiltonian. Ghost lock-down within finite regions of phase space is understood as due to hidden quasi-constants of motion present in almost generic theories at weak coupling. Extending towards infinite time reveals an exponentially suppressed run-away rate, that we controlled in some model.
Actually some physical systems are meta-stable ghosts, such as asteroids around the Lagrangian point 4 (appendix A.1) or electrons in magnetic fields plus a destabilising radial force (appendix A.2).
• However the perturbative series contains terms proportional to 1/(N 1 ω 1 −N 2 ω 2 ) where N i are integers that grow at higher orders. One can thereby encounter resonances where such terms are large or divergent. The most dangerous case arises at leading order N 1,2 = 1 when ω 1 = ω 2 . We studied what happens using resonant normal forms: some interactions lead to ghost run-away, others only to order-one violations of hidden quasi-constant of motion. We argued that the latter situation seems quite generic in the presence of multiple interactions.
In order to exclude a ghost, we then moved to less simple limits: • In section 3 we argued that ghosts are meta-stable in quantum mechanics. We first performed a brute-force computation in our toy model. Wave-functions with no nodes (the ground-like state with lowest positive energy and highest negative energy) get exponentially-suppressed away from the origin even into the dangerous new region that leads to ghost run-away (large |K i | and small K 1 + K 2 ). The ghost run-away time is thereby exponentially suppressed at small coupling analogously to usual tunnelling. In general, tunnelling can be approximated in the semi-classical limit, that inherits the good properties of ghosts in classical mechanics. We could however not generalise the WKB simple formula to the ghost case.
• In section 4 we studied classical field theory. The infinite number of degrees of freedom give rise to new phenomena. One is the black-body problem of interacting classical field theories, that complicates our study. More relevant for us is the presence of an infinite number of modes with different frequencies and thereby an infinite number of resonances, that correspond to the usual on-shell decays and scatterings. Each resonance is potentially deadly in the presence of ghosts. By expanding examples of local interactions in terms of momentum modes we found specific resonances that do not immediately lead to runaways, but only to partial loss of hidden constants of motion. Nevertheless we argued that the infinite number of resonances makes ghosts unprotected in the continuum limit. Based on general entropy arguments we found that there is no thermal state when a system with positive temperature T 1 > 0 interacts with a ghost system with negative T 2 < 0: heat keeps flowing such that both |T 1,2 | increase up to infinity. By writing Boltzmann equations in specific models we computed the rate of such process, finding that it is quadratic in the couplings, rather than non-perturbatively suppressed. We validated this finding by evolving classical field theories on appropriate lattice discretizations. In principle both our analytic understanding and the numerics might have missed hidden properties that keep ghosts stable, but various checks do not find evidence in this sense.
We next considered the case of 4-derivative gravity -a renormalizable theory of gravity containing a spin-2 field with negative kinetic energy and gravitational interactions only -finding that the ghost run-away time is negligible on cosmological time-scales.
In order to exclude such ghost, we finally considered the theory currently considered as fundamental.
• In section 5 we considered Relativistic Quantum Field Theory in the presence of a ghost. Since the initial vacuum state is Lorentz invariant (unlike a thermal state), the naive tree-level vacuum decay rate contains a divergent integral over the non-compact Lorentz group, that describes an arbitrary boost of the same final state. We recalled that this same problem was encountered in early computations of vacuum decay due to potential instability: even in the absence of a ghost, negative potential energy gives rise to field configurations that behave as a ghost. Using WKB Euclidean techniques Coleman argued that the vacuum decay rate is finite and exponentially suppressed. We could not extend such tecniques to the case of ghost instability, so we do not know if it is fast (thereby ruling out theories containing ghosts) or exponentially suppressed at small couplings.
It will be important to fully clarify if negative kinetic energy can be meta-stable up to cosmological time-scales, as the negative-energy quantization of 4-derivative gravity would provide a renormalizable theory of quantum gravity.
The corresponding resonant form is The only quasi-integral of motion (in addition to H) is E ≡ J 1 + J 2 − J 3 − J 4 . The Hamiltonian of the extra-system can be easily obtained from eq. (100) and, recalling that the combination E is approximately constant, is found to be bounded (this can be seen by noticing that the absolute value of the oscillatory term in the square brackets is smaller than 4 J 1 J 2 J 3 J 4 and using twice the inequality 2 √ xy < x + y between arithmetic and geometric means).