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Is negative kinetic energy meta-stable?
Christian Gross
a,b
, Alessandro Strumia
a
Daniele Teresia,b, Matteo Zirillia
a (^) Dipartimento di Fisica dell’Universit`a di Pisa b (^) INFN, Sezione di Pisa, Italy
We explore the possibility that theories with negative kinetic energy
(ghosts) can be meta-stable up to cosmologically long times. In clas-
sical 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, lock-
down is broken by resonances and ghosts behave statistically, drifting
towards infinite entropy as no thermal equilibrium exists. We analyt-
ically and numerically compute the run-away rate finding that it is
cosmologically slow in 4-derivative gravity, where ghosts have gravita-
tional 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.
arXiv:2007.05541v2 [hep-th] 15 Jun 2021
Contents
1 Introduction 2
2 Ghost meta-stability in classical mechanics? 5 2.1 Action-angle variables................................ 7 2.2 Perturbative Birkhoff series............................. 9 2.3 Stability estimates.................................. 11 2.4 Resonances i.e. on-shell processes.......................... 14
3 Ghost meta-stability in quantum mechanics? 17 3.1 Model computation.................................. 18 3.2 The WKB approximation.............................. 19
4 Ghost meta-stability in classical field theory? 21 4.1 Classical equations of motion in momentum space................. 22 4.2 Analytic study of one ghost resonance in field theory............... 23 4.3 Analytic study of multiple ghost resonances in field theory............ 24 4.4 The ghost run-away rate............................... 25 4.5 Results......................................... 28
5 Ghost meta-stability in quantum field theory? 31
6 Conclusions 32
A Physical systems described by ghosts 35 A.1 Asteroids around the Lagrangian point L 4..................... 35 A.2 Charged particle in a magnetic field......................... 36
B Resonant form for overlapping resonances 37
C Classical lattice simulations 37
1 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 gμν has mass dimension 0. Such action is [1]
S =
d^4 x
| det g|
[
R^2
6 f 02
1 3 R
2 − R 2
μν f 22
M¯ (^) Pl^2 R + Lmatter
]
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 unbounded- from-below, but can be modified for example into
H =
p^21 2 m 1
p^22 2 m 2
p^22 2 m 2
λ 2 q^21 q^22 (5)
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 replac- ing 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 vac- uum, 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” [15–24]. 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 epicycle- like 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
L = −
q
∂^2
∂t^2
∂^2
∂t^2
q − VI (q, ¨q) (6)
2.1 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 action- angle 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
H = p^2 2 m
mω^2 2 q^2 = ωJ (13)
where m > 0 (m < 0) for a normal particle (a ghost). The canonical transformation is
q =
2 J
mω
sin Θ, p =
2 mωJ cos Θ (14)
and its inverse is
Θ = arccos p √ p^2 + (mωq)^2
, J =
p^2 + (mωq)^2 2 mω
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
W (q, J) =
p dq =
q
mω(2J − mωq^2 ) + J arccos
mωq^2 2 J
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 Ji cannot be written analytically. Going to action-angle variables for the two free harmonic oscillators, our toy ghost model of eq. (3) becomes
H = ω 1 J 1 − ω 2 J 2 + J 1 J 2 sin^2 Θ 1 sin^2 Θ 2 where = 2 λ ω 1 ω 2
and Ei = ωiJi ≥ 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 tend 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,
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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 perturba- tively. 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
Ji(t) = Ji 0 , Θ 1 (t) = Θ 1 (0) + ω 1 t, Θ 2 (t) = Θ 2 (0) − ω 2 t. (18)
We see that Ji(t) = Ji 0 are constant, for both i = { 1 , 2 }. Their equations of motion at 1st order J 1 ′ = −J 10 J 20 sin(2ω 1 t) sin(ω 2 t)^2 , J 2 ′ = J 10 J 20 sin(2ω 2 t) sin(ω 1 t)^2 (19)
are solved by
J 1 (t) = J 10 + J 10 J 20 ω 1 ω 2 − 1 cos(2tω1+2) + ω1+2 (2ω 1 − 2 cos (2tω 1 ) − ω 1 cos(2tω 1 − 2 )) + 2ω^22 8 ω 1 (ω 1 − ω 2 )(ω 1 + ω 2 )
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
We perform a generic canonical transformation with generator
J i′ Θi + W (J′, Θ) i.e. J = J′^ + ∂Θi W, Θ′^ = Θ + ∂J′^ W. (22)
So, defining f = sin^2 Θ 1 sin^2 Θ 2 one gets
H′(J′) = H(J) = ω 1 (J 1 ′ + ∂Θ 1 W ) − ω 2 (J 2 ′ + ∂Θ 2 W ) + f (J 1 ′ + ∂Θ 1 W )(J 2 ′ + ∂Θ 2 W ). (23)
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 ,
W = W (1)^ + ^2 W (2)^ + · · · , H′^ = H + H(1)^ + ^2 H(2)^ + · · ·. (24)
Since the system is not integrable, the Birkhoff series is only asymptotic and J i′ are the approx- imated 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.
W (n)(Θ 1 , Θ 2 ) = −i
∑^ ∞
N 1 ,N 2 =−∞
ei N~ ·Θ~ W (^) N( ~n ), N~ = (N 1 , N 2 ). (25)
The only non-zero coefficient of the Fourier series of f =
N~ eiNiΘi^ f^ N~ are^ f^00 = 1/4,^ f±^2 ,±^2 = f± 2 ,∓ 2 = 1/16, f± 2 , 0 = f 0 ,± 2 = − 1 /8.
2.2.1 First order in the coupling
Expanding eq. (23) at first order gives
H(1)^ = ω 1
∂W (1)
− ω 2
∂W (1)
- J 1 ′J 2 ′f (Θ 1 , Θ 2 ). (26)
The first term involves derivatives of a periodic function with period 2π, by the very definition of the angle variables Θi. Therefore, its average over a period is zero. Averaging over Θi we get
H(1)^ =
J 1 ′J 2 ′
i.e. H′^ = ω 1 J 1 ′ − ω 2 J 2 ′ +
J 1 ′J 2 ′
+ O(^2 ). (27)
We next compute the canonical transformation W (1)^ through the Fourier expansion. We get W 00 (1) = 0 and
W (^) N(1) 1 N 2 =
J 1 ′J 2 ′fN 1 N 2 N 2 ω 2 − N 1 ω 1
for N 1 , N 2 6 = 0. Summing over the non-vanishing N~ this means
W (1)^ =
J 1 ′J 2 ′
8(ω 22 − ω 12 )
[
(ω 1 cos(2Θ 2 ) − ω 1 + ω^22 ω 1 ) sin(2Θ 1 ) + (ω 2 cos(2Θ 1 ) − ω 2 + ω^21 ω 2 ) sin(2Θ 2 )
]
and thereby
J 1 ′ = J 1 +
J 1 J 2
4 ω 1 (ω^21 − ω^22 )
[
cos 2Θ 1
ω^22 − ω^21 + ω^21 cos 2Θ 2
− ω 1 ω 2 sin 2Θ 1 sin 2Θ 2
]
+ O(^2 ) (30)
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.
2.2.2 Generic order in the coupling
Eq. (23) expanded at order n > 1 (n = 1 is special) is
H(n)(J 1 ′, J 2 ′) = ω 1 ∂W (n) ∂Θ 1
− ω 2 ∂W (n) ∂Θ 2
×
[
J 1 ′
∂W (n−1) ∂Θ 2
+ J 2 ′
∂W (n−1) ∂Θ 1
∑^ n−^2
m=
∂W (m) ∂Θ 1
∂W (n−^1 −m) ∂Θ 2
]
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 00 (n )= 0 finding
H(n)^ =
~q+~r=~ 0
(r 2 J 1 ′ + r 1 J 2 ′)f~qW (^) ~r( n−1)+
n∑− 2
m=
~q+~r+~s=~ 0
r 1 s 2 f~qW (^) ~r( m)W (^) ~s( n−^1 −m), (32)
W (^) N( ~n )=
N 2 ω 2 − N 1 ω 1
~q+~r=~p
(r 2 J 1 ′ + r 1 J 2 ′)f~qW (^) ~r( n−1)+
n∑− 2
m=
~q+~r+~s=~p
r 1 s 2 f~qW (^) ~r( m)W ~ (^) s(n−^1 −m)
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 ωi.^4
2.3 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
(^4) Relations such as W (n) −N 1 ,N 2 (ω^1 , ω^2 ) = (−1) nW (n) N 1 ,N 2 (−ω^1 , ω^2 ) allow to compute only for positive^ N^1 ,^2 ≥^ 0, if ωi are left generic. However this produces cumbersome expressions, and computations are more efficiently performed setting ωi to numerical values, such that each term is a short polynomial in J i′.
substituting J˙ i′ with its maximal value. Neglecting higher orders in : ∣∣ ∣∣^ ∂ ∂Θ′ i
δH
~p
∣∣^ ∂
∂Θ′ i
δH ~p(n)
N 1 ,N 2
∣∣N
i(N 2 ω 2 −^ N 1 ω 1 )W^
(n) N^ ~
having used the triangular inequality. Higher orders in weaken the bound in eq. (38) by a factor of 2 [26].
2.3.1 Stability at lowest order
To start, we outline the procedure at lowest order, such that the approximately conserved quantities are simply J i′ = Ji and the remainder in the Birkhoff series simply is the whole interaction δH(0)^ = 2λ
J 1 J 2
ω 1 ω 2 sin^2 Θ 1 sin^2 Θ 2. (39)
To compute the stability time, we use the inequality
|J i′ (t) − J i′ (0)| ≤ t max J i′ ≤Jmax
∣∣ J˙′
i
∣∣ ≤ t 2 λJmax^2 ω 1 ω 2
The region can be abandoned only after a time
t ≥ τ 0 (Jmaxin → Jmax) = ω 1 ω 2 Jmax − Jmaxin 2 λJmax^2
Its maximal value, achieved for Jmax = 2Jmaxin , is the Lyapunov stability time:
τ 0 (Jmaxin ) = ω 1 ω 2 8 λJmaxin
2.3.2 Stability at generic order
The above discussion is easily generalized at order n. The residual time evolution is bounded by
max i,J′ i ≤Jmax
∣∣^ ∂
∂Θi δH
∣∣ ≤ 2 n+1^ max i,J i′ ≤Jmax
N 1 ,N 2
∣Ni(N 2 ω 2 − N 1 ω 1 )W (^) N( ~n+1)
∣ ≡ n+1Jmaxn+2 βn (43)
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 βn(ω 1 , ω 2 ) can be computed numerically and diverges close to resonances:
τn(Jmaxin → Jmax) = Jmax − Jmaxin n+1Jmaxn+2 βn
The Lyapunov stability time is
τn(Jmaxin ) =
βn
(n + 1)n+ (n + 2)n+
ω 1 ω 2 2 λJmaxin
)n+
. (45)
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 =
- 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 − 5 ω 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.
2.4 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 Eint/Efree >∼ (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 subse- quent 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.
2.4.1 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
resonances ∑ i ωin i = ∑ j ωout j giving rise to decays and other on-shell process, using the standard terminology of quantum field theory (when E = ℏω the resonance condition becomes conservation of energy and momentum).
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. Jmax/Jmin 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 12 q^22 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.
2.4.2 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^21 q^22 with a cubic interaction q 12 q 2. The Hamiltonian in action-angle variables is
H = ω 1 J 1 − ω 2 J 2 + J 1
J 2 sin^2 Θ 1 sin Θ 2 (51)
and the dangerous resonance is ω 2 ≈ 2 ω 1 that (loosely speaking) allows for a q 1 → q 1 + q 2 ‘decay’. The resonant Birkhoff form at first order is
H′^ = ω 1 J 1 ′ − ω 2 J 2 ′ −
J 1 ′
J 2 ′ sin(2Θ′ 1 + Θ′ 2 ). (52)
The sign of sin(2Θ′ 1 + Θ′ 2 ) now qualitatively impacts the system. This can be seen performing the canonical transformation E ≡ J 1 ′ − 2 J 2 ′, J ≡ (J 1 ′ + 2J 2 ′)/4, Q ≡ 2Θ′ 1 + Θ′ 2 such that
H′^ = ˜ωE + ∆ωJ −
E
+ 2J
J −
E
sin Q (53)
(^6) In terms of original variables the ‘resonant’ constant of motion E ≡ J 1 ′ − J 2 ′ is
E = J 1 − J 2 + λJ 2 ω^1 J^2 1 ω 2
[ (^) cos 2(Θ 1 −^ Θ 2 ) ω 1 + ω 2 −^
cos 2Θ 1 ω 1 −^
cos 2Θ 2 ω 2
]
with ˜ω = (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-from- below potential. For example, consider a model with quartic interactions H ⊃ λ (q^21 q 22 +κq 13 q 2 )/2. Close to the resonance ω 2 ' 3 ω 1 we find that the ghost is safe for |κ| < 2 /
3, despite the potential is unstable for any κ 6 = 0 (for instance along the line q 2 = 1, q 1 → −∞). Conversely, the potential with with quartic interactions H ⊃ λ′^ (q 14 + κq 13 q 2 ) is stable for any finite value of κ, but the ghost causes run-away for |κ| > 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
H = ω 1 J 1 − ω 2 J 2 + ω 3 J 3 +
J 1 J 2 J 3 sin Θ 1 sin Θ 2 sin Θ 3. (54)
The first-order resonant form close to the dangerous resonance ω 1 − ω 2 + ω 3 ≡ ∆ω ' 0 is
H ' −ω 2 E 2 + ω 3 E 3 + ∆ω
J
J
J
+ E 2
J
+ E 3
sin 3Q (55)
where Ei ≡ J i′ − J 1 ′, Q = (Θ′ 1 + Θ′ 2 + Θ′ 3 )/3 and J = 3J 1 ′. 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.
3 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
H = p^21 2
p^22 2
- V, V = ω^21 q 12 2 − ω^22 q 22 2
λ 2 q 12 q^22 (56)
Figure 4: Iso-curves of the ground-like state wave-function |ψ(q 1 , q 2 )|^2 for different values of the quartic coupling λ between the positive-energy q 1 and the negative-energy q 2. Contour curves are separated by one order of magnitude.
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.
3.2 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 = eiS/ℏ, the Schroedinger equation reduces to the classical Hamilton-Jacobi (HJ) equation
∂S ∂t
= −H
qi, pi =
∂S
∂qi
plus extra terms 12 iℏ∂^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(qcl)^ 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 (Θi, Ji) such that H only depends on Ji. The ‘reduced action’ W satisfies the wave equation
E = H(qi, pi =
∂W
∂qi
) ⇒ W =
pi dqi. (60)
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 in- dependent) 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〉 = eiJΘ/ℏ, so that its periodicity demands J = nℏ 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, tE = it and solve the Euclidean HJ equation with LE = 12 (d~q/dtE)^2 − VE and inverted potential VE = −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 WE 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 WE. For example
B = min WE = min SE = min (^) tlim E→+∞
∫ (^) ~q∗,tE
0 , 0
LE dtE = min
∫ (^) ~q∗
0
dq
2 VE (61)
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 tE → −∞. Equivalently, an opposite-sign Wick rotation is needed to make the Euclidean ghost action positive.
