Ghosts and Kinetic-Sign Instabilities

A ghost is a propagating fluctuation whose kinetic structure has the wrong sign relative to the healthy matter sector. Depending on the quantization prescription, it appears as negative energy, negative norm, or a pole with the wrong-sign residue. The decisive qualifier is propagating: lapse-like multipliers, gauge components, and variables removed by constraints must be eliminated before a sign is interpreted. The same reduction distinguishes a genuine Ostrogradsky mode from a higher-derivative operator whose extra pole lies beyond the effective cutoff.

Required background. Variational and Symplectic Geometry supplies the constraint reduction and physical symplectic form, while Ordinary Differential Equations supplies the normal-mode analysis used to interpret quadratic fluctuations.

Helpful background. Einstein Equations fix the healthy gravitational baseline, Gravitational Actions and Boundary Terms clarify which action is being varied, and Formulations of General Relativity separates gauge, constraint, and evolution variables before the kinetic test.

The reduced quadratic action

Pages in this chapter use one common local quadratic action. After expanding a declared covariant action around a specified background, choosing a gauge-fixed formulation, solving nondynamical constraints, and selecting a solution branch, write the remaining real perturbations as a column $q$. Up to integrations by parts,

$$S_{\mathrm{red}}^{(2)}=\frac12\int dt\,d^3x\,a^3 \left[ \dot q^{\mathsf T}K\dot q -\frac{1}{a^2}\partial_iq^{\mathsf T}G\partial_iq -q^{\mathsf T}M^2q +2\dot q^{\mathsf T}Bq \right].$$

Here $K$, $G$, and $M^2$ are symmetric, while $B$ may be chosen antisymmetric after integrations by parts. They can depend slowly on the background. The action is not a universal property of a theory name: its matrices depend on background, gauge-fixed formulation, branch, matter coupling, and EFT regime. An invertible change of reduced variables sends $K\mapsto R^{\mathsf T}KR$ and cannot change the numbers of positive and negative eigenvalues.

If $K$ is positive definite, canonical normalization is possible locally. If it has a negative eigenvalue, the corresponding physical combination is a ghost. If an eigenvalue vanishes, the quadratic theory has not shown that the mode is healthy; it may instead signal an additional constraint, an accidental symmetry, or strong coupling. This is why the Hamiltonian count on the preceding page comes before the kinetic-sign test.

For cosmological scalar–tensor perturbations, the same logic appears after lapse and shift have been integrated out: the coefficient of the reduced time-derivative term must be positive. The construction and its stability conditions are displayed in Kobayashi 2019, § 3.2.1.

Hamiltonian energy and pole residue

For one slowly varying mode with $B=0$,

$$L=\frac{Z}{2}\dot q^2-\frac{Zc_s^2}{2a^2}(\nabla q)^2-\frac{Zm^2}{2}q^2.$$

The momentum is $p=Z\dot q$, and the quadratic Hamiltonian density is

$$\mathcal H=\frac{p^2}{2Z}+\frac{Zc_s^2}{2a^2}(\nabla q)^2+\frac{Zm^2}{2}q^2.$$

Thus $Z\lt0$ makes the energy unbounded below when this sector can exchange energy with an ordinary positive-energy sector. In Fourier space, the propagator is proportional to

$$\frac{i}{Z\left(\omega^2-c_s^2k^2/a^2-m^2+i0\right)}.$$

Its residue has sign $1/Z$. The absolute overall sign of an isolated free action can be reversed, but the relative sign between coupled sectors cannot. A healthy graviton coupled to matter therefore supplies the reference sign. Agreement between the reduced Hamiltonian and pole residue is an independent check; disagreement usually means that a constraint, normalization, or source coupling was mishandled.

Notice that multiplying a free oscillator action by $-1$ leaves its classical equation $\ddot q+\omega_0^2q=0$ unchanged. A ghost can therefore have a real linear frequency. The problem appears in its negative relative energy or norm and in interactions with positive-energy modes; looking only for exponentially growing free solutions would miss it.

Ostrogradsky instability after constraint reduction

Consider a mechanical Lagrangian $L(q,\dot q,\ddot q)$ whose Hessian with respect to $\ddot q$ is nonzero. Introduce $Q_1=q$ and $Q_2=\dot q$, with

$$P_2=\frac{\partial L}{\partial\ddot q}, \qquad P_1=\frac{\partial L}{\partial\dot q}-\frac{d}{dt}\frac{\partial L}{\partial\ddot q}.$$

Nondegeneracy lets one solve $\ddot q=A(Q_1,Q_2,P_2)$. The Hamiltonian contains $P_1Q_2$ linearly, so it is unbounded as $P_1$ varies. This is the Ostrogradsky construction reviewed in Woodard 2015, §§ 2.2–2.3 and 5.

The theorem does not say that every equation containing higher derivatives propagates a ghost. Gauge symmetry or degeneracy may produce constraints that remove the dangerous canonical pair. Conversely, writing equations in a manifestly second-order form is sufficient neither for a positive kinetic matrix nor for gradient stability. The correct procedure is to compute the full constraint algebra and then inspect the reduced quadratic action.

Fundamental ghosts versus poles above the cutoff

A truncated EFT can contain a higher-derivative correction without claiming that its exact resummation is fundamental. For example,

$$\mathcal L=-\frac12(\partial\phi)^2 -\frac{1}{2M^2}(\Box\phi)^2$$

formally yields two poles when treated exactly. With $p^2=-\omega^2+\mathbf p^2$, its inverse propagator is proportional, up to an overall healthy-sign normalization, to

$$p^2\left(1+\frac{p^2}{M^2}\right),$$

so the poles at $p^2=0$ and $p^2=-M^2$ have opposite residues. But if this Lagrangian is known only as an expansion below $\Lambda_{\mathrm{EFT}}\ll M$, the heavy pole lies outside the theory’s domain. One expands predictions perturbatively in $E^2/M^2$ and does not supply independent initial data for the spurious high-frequency solution. A pole with wrong residue inside the declared EFT regime is different: it invalidates that background or branch unless an additional constraint removes it.

This distinction is not permission to hide an inconvenient mode by lowering the cutoff after the calculation. The cutoff must follow from the operator expansion and canonical normalization independently of the pole one is testing. Matter couplings also matter: loops can regenerate operators and move the scale at which the truncation ceases to be controlled.

Independent checks and limits

Three checks should agree. First, diagonalize $K$ on the reduced constraint surface and record its inertia. Second, compute the signs of residues of the coupled propagator at simple poles. Third, construct the quadratic Hamiltonian in canonically normalized variables. An invertible local field redefinition preserves the kinetic inertia, so an apparent sign change under such a transformation diagnoses a missing matter or source transformation.

In a controlled general-relativistic limit, the two tensor modes must retain the positive Einstein kinetic coefficient while any extra mode either remains healthy, becomes constrained, or decouples above the cutoff. If its kinetic eigenvalue tends to zero while interactions remain finite, the limit is strongly coupled rather than smoothly healthy. The next pages separate that phenomenon from gradient and mass instabilities.

This page establishes theory-side consistency only. Empirical likelihoods and mutable bounds belong to Tests of Gravity; stable low-energy quantum gravity belongs to Quantum Gravity; generic loop technology receives only the bounded bridge needed for the sign test here.

Common pitfalls

Reading a sign before solving constraints. A negative entry in the unreduced kinetic matrix may belong to a gauge or auxiliary component. Only the inertia of the physical reduced matrix diagnoses ghosts.

Calling every tachyon a ghost. A negative mass squared changes the potential and produces low-momentum growth; a ghost has a wrong-sign kinetic term. Their timescales and consequences differ.

Promoting a truncated EFT to an exact theory. Exact treatment manufactures high-frequency solutions that the derivative expansion did not authorize. State the cutoff before interpreting an extra pole.

Exercises

For two reduced modes let

$$K=\begin{pmatrix}1&\epsilon\\ \epsilon&1\end{pmatrix}.$$

Determine when the kinetic sector is ghost free and interpret the limiting values $|\epsilon|=1$.

Solution

The eigenvalues are $1+\epsilon$ and $1-\epsilon$. Both are positive exactly when $\lvert\epsilon\rvert\lt1$. At $\lvert\epsilon\rvert=1$ one eigenvalue vanishes, so the quadratic action alone cannot certify health: the mode may become constrained or, if interactions remain, strongly coupled. For $\lvert\epsilon\rvert\gt1$ one physical eigenvalue is negative and the reduced theory contains a ghost.

Show why the partial-fraction identity $1/[p^2(1+p^2/M^2)]=1/p^2-1/(p^2+M^2)$ does not by itself prove a physical low-energy ghost when the EFT cutoff obeys $\Lambda_{\mathrm{EFT}}\ll M$.

Solution

The second term has the opposite residue, but its pole occurs at momentum of order $M$, outside the declared EFT domain. Low-energy predictions expand the full expression as $p^{-2}(1-p^2/M^2+\cdots)$ and retain only the controlled order. A physical instability would require the wrong-sign pole or its effects to enter below the independently established cutoff.

References

• Kobayashi, Tsutomu. 2019. “Horndeski Theory and Beyond: A Review.” Reports on Progress in Physics 82, 086901. DOI; arXiv:1901.07183.
• Woodard, Richard P. 2015. “The Theorem of Ostrogradsky.” Scholarpedia 10, 32243. DOI; arXiv:1506.02210.