Living with Ghosts

S.W. Hawking; Thomas Hertog

REVIEW

Living with Ghosts

S.W. Hawking, Thomas Hertog

High Energy Physics - Theory · Submitted 11 Jul 2001 · 12 references

Pith's verdict: The free-theory calculation is exact and the α→0 check is real, but the step to gravity is an extrapolation from a free scalar, and the cross-term sign in Eq. 22 deserves a careful look.

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arXiv IDhep-th/0107088 (version 2)

Pith IDLUB6N6V3, Pith's permanent record of this paper and its review

CitationS.W. Hawking, Thomas Hertog. "Living with Ghosts." arXiv:hep-th/0107088 (2001). https://pith.science/paper/LUB6N6V3

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Editor's note

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Quick read on the Hawking–Hertog ghost paper. The thing to know: the free-theory calculation is exact and the α→0 limit is a genuine check, but the headline claim about gravity is an extrapolation from a free scalar. The paper does the fourth-order oscillator carefully — Gaussian path integral, explicit coefficients in Appendix A, and Eq. 22 reduces to the second-order result as α→0. That is real, checkable work. The prescription (states specified by (φ, φ,τ), trace out φ,τ) is clearly stated, and the composition-law argument for why you cannot use φ,ττ is a nice piece of reasoning. The paper also correctly identifies that the ghost problem is an artifact of treating φ and □φ as independent canonical variables, which is the conceptual core and it holds up. Where it is soft: the step from the free oscillator to 'quantum gravity with fourth order corrections can make sense' (Section VI) is not derived. The paper cites [6,7] for interaction-induced ghost-pole instabilities and handles them only with a final boundary condition that it concedes introduces acausality (Section V). No interacting calculation is performed. The authors do acknowledge the early-universe caveat and the loss of unitarity, so those are not new findings — but the abstract's 'unitarity is restored' is stronger than the free calculation supports. One technical thing worth checking: Eq. 22 has a cross term −2mφ1φ2/sinh mT, whereas the standard second-order |A|² has +2mφ1φ2/sinh mT. The quantity being computed is a vacuum sandwich, not |A|², so the comparison is not straightforward — but if the claim is that Eq. 22 reproduces the second-order sandwich, the sign looks off. Worth a careful check before taking the α→0 limit as confirmation. The acausality from the final boundary condition is acknowledged but not quantified beyond the radiation-reaction analogy. That is fine for a first pass, but it is the kind of thing that needs a real calculation before the gravity claim is credible. Who this is for: people working on higher-derivative gravity, Euclidean quantum gravity, or the foundations of the ghost problem. It is a useful paper — the free-theory result is solid and the prescription is clear — but it is not the last word on the interacting case. Recommendation: worth a serious referee. The free-theory calculation is exact and the α→0 check is real, and the conceptual framing is clear enough to be worth engaging with. The interacting case is the open question, and the paper should be read as setting up that problem rather than solving it.

Referee report

Significance

If the argument holds, it would rehabilitate R+R² gravity and its higher-derivative relatives as consistent low-energy effective theories and would reframe a long-standing objection (ghosts imply no unitary S matrix) as a statement about which questions can be posed, rather than a fatal flaw. The free-theory calculation is exact and is a genuine independent check: I verified Eq. (17) by expanding λ1, λ2 and F′ to first order in α, and both the exponent −m(1+mα)φ0² and the prefactor (m/π)^{1/2}(1+mα/2) come out correctly, as does the α→0 limit of the ground-state wave function (Eq. 13). The composition-law argument of §III for why states must be labelled by (φ, φ,τ) rather than (φ, φ,ττ) is clean and, in my view, correct. The paper is also candid in the body that unitarity is lost and that the removal of runaways via a final boundary condition introduces acausality. The weakness is that the headline claim is an extrapolation from a free oscillator to interacting and gravitational theories, and the abstract states it more strongly than the body supports.

Major objections (5)

[Abstract (s7, s8); §IV–§VI] The claim that 'unitarity is restored at the low energies that now occur in the universe' (abstract:s8) rests entirely on the free-theory limit Eq. (22) (abstract:s7). No interacting calculation is performed anywhere in the paper. The authors themselves cite [6,7] for the result that interactions shift the ghost poles into the complex p-plane and produce exponentially growing modes, and §V addresses this only by imposing a final boundary condition that the authors concede produces acausal behaviour on the scale m2⁻¹. The step from the free oscillator to 'quantum gravity with fourth order corrections can make sense' (§VI) is therefore an extrapolation, not a derived result. A concrete test that would make the claim load-bearing: compute the λφ⁴-corrected two-point function (or the one-loop effective action) within the Euclidean prescription, trace over φ,τ as prescribed, and show that the resulting probability remains positive and reduces to the second-order result as α→0. Absent that, the claim should be explicitly scoped to the free theory.
[§IV B, Eq. (22) versus Eq. (21) and Eq. (A9)] As printed, Eq. (22) contains the cross term −2mφ1φ2/sinh mT. This is opposite in sign to what the paper's own preceding results give: Eq. (21) has +2Hφ1φ2 in the exponent, and the small-α amplitude Eq. (A9) contains −m(αφ2,τ+φ2)(αφ1,τ−φ1)/sinh mT, which at φ,τ=0 yields +mφ1φ2/sinh mT in the amplitude and hence +2mφ1φ2/sinh mT in |A|². The standard second-order oscillator propagator squared likewise gives a positive cross term. The prefactors of Eq. (22) and of the square of Eq. (A9) also do not appear to agree. Since Eq. (22) is the sole explicit verification of abstract:s7, the sign and prefactor must be reconciled with Eqs. (21) and (A9); if the printed form is a transcription error, the corrected equation should be given, and if not, the α→0 limit needs to be redone.
[Abstract (s6 versus s8)] The abstract states both that 'one loses unitarity' (abstract:s6) and that 'unitarity is restored at the low energies that now occur in the universe' (abstract:s8). These are in tension as written, and the body supports only the weaker statement that the departures from unitarity become small as α→0. The abstract should say that, not that unitarity is restored. Relatedly, the assertion that 'one can never produce a negative norm state or get a negative probability' (abstract:s6) is demonstrated here only for the free Gaussian measure; the general argument (positivity of the Euclidean measure for the fourth-order free action) should be stated as the basis, with the free oscillator presented as an explicit check rather than as the proof.
[§VI, paragraph on Lorentz invariance] The paper acknowledges the worry that integrating out φ,τ might break Lorentz invariance and answers it in two sentences: 'φ,τ is conjugate to φ,ττ so tracing over φ,τ is equivalent to not observing ✷φ', and hence 'one only considers Feynman diagrams whose external legs are ψ1−ψ2'. This is asserted, not shown, and it is load-bearing for the whole prescription, since a Lorentz-violating set of probabilities would not be a viable replacement for the S matrix. A demonstration is needed that the traced-out probabilities, or the resulting n-point functions, are Lorentz invariant; the statement about external legs being the ψ1−ψ2 combination should be made precise (which combination of ψ1 and ψ2, and why the orthogonal combination is unobservable).
[§VI, concluding remarks on string theory] The sentence 'Thus string theory effectively has ghosts, though this awkward fact is quietly glided over' is a strong claim about a large body of work, asserted without argument or reference, and it is not needed for the paper's derivation. Either substantiate it with a specific calculation or citation, or remove it; as it stands it invites a dispute that is orthogonal to the paper's actual result.

Minor objections (6)

[Whole manuscript] The supplied text has badly degraded typesetting: fractions are collapsed throughout Eqs. (6), (13)–(22), and (A6)–(A9), so that expressions such as the exponent of Eq. (22) cannot be read unambiguously. A cleanly typeset version is needed before the algebra of §IV and Appendix A can be checked line by line.
[§II, Eq. (10)] The canonical momenta quoted in Eq. (10), Πφ = −φ,τττ + (m1²+m2²−2∇²)φ,τ and Π_{φ,ττ} = −φ,τ, do not appear to follow from the Lagrangian (3) under the stated conventions; the Ostrogradski construction applied to (3) seems to give a different expression. Since Eq. (10) is used only to motivate the wave-functional argument, this does not affect the main result, but it should be corrected or the conventions stated explicitly.
[Appendix A, normalization factor N] The statement that 'the normalization factor N is independent of α to first order' is asserted without derivation. A short justification, or the explicit expansion, would let the reader check the prefactor of Eq. (21) and its α→0 limit.
[§V, causality discussion] The claim that a causality violation could never be detected in quantum gravity because of the logarithmic time delay Δt ∼ −m log b is qualitative and unsupported by a calculation or reference. Either give the estimate explicitly or soften the claim.
[§I, Sobolev-norm motivation] The statement that in two dimensions 'any quantum field theory is renormalizable' is stronger than what the Sobolev-norm argument establishes, since that argument bounds entire functions of φ but does not by itself deliver renormalizability. The sentence should be softened or supported by a citation.
[References] Reference [1] gives the publisher as 'Springer-Verlach'; this should read 'Springer-Verlag'.

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The paper

The paper claims

Higher-derivative gravity may survive its ghosts: the negative-norm states that seemed to kill curvature-squared quantum gravity are an artifact of the canonical method, and the probabilities one actually observes stay positive and relax to

The reading

Einstein's action plus curvature-squared corrections is renormalizable, but it appears to carry ghosts — quantum states of negative norm that would make some probabilities negative and the S-matrix non-unitary. Hawking and Hertog argue that the ghost is an artifact of the canonical treatment, which promotes $φ$ and $□φ$ to independent coordinates even though both are determined by $φ$. Quantizing instead with a Euclidean path integral over $φ$ alone, where the fourth-order free action is positive definite, forces states to be labelled by $φ$ and $φ_{,τ}$; because the Lorentzian distribution for $φ_{,τ}$ is unbounded, one must trace over it, which costs unitarity but never produces a negative norm or a negative probability. Working the fourth-order harmonic oscillator exactly, they show that the traced-out transition probability tends to the second-order answer as the coefficient $α$ of the fourth-order term goes to zero, so any departure from unitarity is negligible at the energies that occur in the universe today.

What carries the argument

The load-bearing calculation is the Euclidean path integral for the higher-derivative harmonic oscillator, $S = ∫dτ[ (α^2/2) φ_{,ττ}^2 + (1/2) φ_{,τ}^2 + (1/2) m^2 φ^2 ]$, whose Gaussian integral is done exactly in Appendix A. Splitting $φ = φ_{cl} + φ'$ gives an exponent built from the classical action with coefficient functions $E, F, G, H, K, L$ and a state-independent prefactor fixed by the Jacobian of the change of variables. Two further objects carry the argument: the ground-state wave function $Ψ_0(φ_0, φ_{0,τ})$, which is a bounded Gaussian in Euclidean signature, and its Euclidean conjugate, whose product gives the normalizable distribution $P(φ_0, φ_{0,τ})$. Integrating that distribution over $φ_{0,τ}$ produces the observable probability $P(φ_0)$, and the same trace applied to the sandwiched transition amplitude produces the observable transition probability whose $α → 0$ limit is the second-order result.

Extended reading notes

Core claim

The paper's claim is that a fourth-order (higher-derivative) scalar field theory, and by extension $R + R^2$ gravity, can be given a consistent set of rules for computing probabilities even though its free propagator is the difference of two second-order propagators, one of which enters with the wrong sign. The authors locate the source of the ghost not in the theory but in the canonical procedure: treating $φ$ and $□φ$ as independent variables splits the field into $ψ_1$ and $ψ_2$ with opposite-sign actions, and the $ψ_2$ path integral then diverges. If instead one defines the theory by a Euclidean path integral over $φ$, the free action is positive definite and the integral converges. The price is that a quantum state must be specified by $φ$ and $φ_{,τ}$ on a spacelike surface; the corresponding probability distribution is normalizable in Euclidean signature but grows without bound after Wick rotation in the $φ_{,τ}$ direction, so $φ_{,τ}$ cannot be observed. Tracing over $φ_{,τ}$ on the final surface yields a density matrix rather than a pure state — unitarity is lost — but every probability so obtained is positive and normalized. For the higher-derivative oscillator the traced-out transition probability is computed in closed form and, as $α → 0$, reduces exactly to the second-order expression $P(φ_2,φ_1) = (m/2π) (1/π) ... $ of Eq. (22), i.e. to the ordinary quantum mechanics of a second-order field.

Load-bearing premise

The argument rests on the assumption that the Euclidean path integral over $φ$ alone, with its positive-definite fourth-order free action and with $φ_{,τ}$ traced out, defines the correct Lorentzian quantum theory — and that the $α → 0$ agreement demonstrated for the free oscillator survives once interactions and gravity are switched on.

Editorial extensions

If this is right

Curvature-squared corrections to gravity need not be discarded on ghost grounds: they can be treated as a low-energy effective theory whose predictions match ordinary (second-order) gravity to excellent accuracy at presently accessible energies.
The loss of unitarity is a loss of information, not a loss of positivity — the final state is described by a density matrix, so no experiment can record a negative probability or a negative-norm state.
In the early universe, where the fourth-order terms matter, the relevant quantities are $n$-point functions of metric perturbations (and hence of microwave-background fluctuations), not scattering amplitudes or an S-matrix, so the question of unitarity does not arise there.
Runaway production of positive- and negative-energy modes is removed by the Euclidean path integral's implicit final boundary condition that fields die off, at the cost of acausal behaviour on a timescale set by the inverse ghost mass.
Because the acausal scale in gravity is tied to a mass, the associated time advance is logarithmic in impact parameter, so no standard arrival time exists against which the violation could be registered.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The argument's real hinge is dimensional: if the ghost mass $m_2$ is pushed to the Planck scale, the acausal and non-unitary corrections are suppressed by powers of $E/m_2$, which would make the effective theory quantitatively safe for all accelerator and cosmological data — a suppression the paper asserts but does not estimate.
Tracing out $φ_{,τ}$ is equivalent to declaring that only the combination $ψ_1 - ψ_2$ appears on external legs; this suggests a sharper statement than the paper gives, namely that the ghost is confined to internal lines and never appears as asymptotic radiation, which is the property an effective-field-theory reader would want proved.
The same Euclidean-then-trace prescription could in principle be applied to other theories with wrong-sign kinetic terms (higher-derivative inflation, Lee-Wick models), and would predict a common signature: decoherence without negative probability, i.e. a density matrix whose off-diagonal terms decay at a rate set by the higher-derivative coefficient.
A natural check the paper does not perform is whether the traced-out probability respects Lorentz invariance beyond the free theory; since $φ_{,τ}$ is tied to a chosen time slicing, an interacting calculation could reveal frame dependence that the free oscillator cannot show.

Original abstract

Perturbation theory for gravity in dimensions greater than two requires higher derivatives in the free action. Higher derivatives seem to lead to ghosts, states with negative norm. We consider a fourth order scalar field theory and show that the problem with ghosts arises because in the canonical treatment, $\phi$ and $\Box \phi $ are regarded as two independent variables. Instead, we base quantum theory on a path integral, evaluated in Euclidean space and then Wick rotated to Lorentzian space. The path integral requires that quantum states be specified by the values of $\phi$ and $\phi_{,\tau}$. To calculate probabilities for observations, one has to trace out over $\phi_{,\tau}$ on the final surface. Hence one loses unitarity, but one can never produce a negative norm state or get a negative probability. It is shown that transition probabilities tend toward those of the second order theory, as the coefficient of the fourth order term in the action tends to zero. Hence unitarity is restored at the low energies that now occur in the universe.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,730 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.90 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.84 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Perturbation theory for gravity in dimensions greater than two requires higher derivatives in the free action. Higher derivatives seem to lead to ghosts, states with negative norm. We consider a fourth order scalar field theory and show that the problem with ghosts arises because in the canonical treatment, $\phi$ and $\Box \phi $ are regarded as two independent variables. Instead, we base quantum theory on a path integral, evaluated in Euclidean space and then Wick rotated to Lorentzian space. The path integral requires that quantum states be specified by the values of $\phi$ and $\phi_{,\tau}$. To calculate probabilities for observations, one has to trace out over $\phi_{,\tau}$ on the final surface. Hence one loses unitarity, but one can never produce a negative norm state or get a negative probability. It is shown that transition probabilities tend toward those of the second order theory, as the coefficient of the fourth order term in the action tends to zero. Hence unitarity is restored at the low energies that now occur in the universe.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Perturbation theory for gravity in dimensions greater than two requires higher derivatives in the free action.	background 1.00	0.07	0.03	asserted 1.00
s1	Higher derivatives seem to lead to ghosts, states with negative norm.	background 1.00	0.05	0.03	hedged possibility 0.95
s2	We consider a fourth order scalar field theory and show that the problem with ghosts arises because in the canonical treatment, $\phi$ and $\Box \phi $ are regarded as two independent variables.	contribution 0.71	0.87	0.12	asserted 1.00
s3	Instead, we base quantum theory on a path integral, evaluated in Euclidean space and then Wick rotated to Lorentzian space.	method 0.96	0.86	0.05	asserted 0.99
s4	The path integral requires that quantum states be specified by the values of $\phi$ and $\phi_{,\tau}$.	method 0.97	0.51	0.05	asserted 1.00
s5	To calculate probabilities for observations, one has to trace out over $\phi_{,\tau}$ on the final surface.	method 0.98	0.51	0.07	asserted 1.00
s6	Hence one loses unitarity, but one can never produce a negative norm state or get a negative probability.	result 0.80	0.82	0.53	asserted 0.99
s7	It is shown that transition probabilities tend toward those of the second order theory, as the coefficient of the fourth order term in the action tends to zero.	result 0.99	0.90	0.18	asserted 1.00
s8	Hence unitarity is restored at the low energies that now occur in the universe.	interpretation 0.86	0.68	0.45	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (12)

[1]

A.L. Besse, Einstein Manifolds , Springer-Verlach, Berlin (1987), p.457

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[2]

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[3]

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[4]

E.S. Fradkin and A.A. Tseytlin, Nucl. Phys. B201, 469 (1982)

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[5]

S.W. Hawking, in Quantum Field Theory and Quantum Statistics:Essays in Honor of the 60th Birthday of E.S. Fradkin , eds. A. Batalin, C.J. Isham and C.A. Vilkovisky, Hilger, Bristol, UK (1987)

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[6]

E. Tomboulis, Phys. Lett. 70B, 361 (1977)

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[7]

D.A. Johnston, Nucl. Phys. B297, 721 (1987)

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[8]

A.A. Starobinsky, Phys. Lett. 91B, 99 (1980)

View on Pith 1980
[9]

S.W. Hawking, T. Hertog, H.S. Reall, Phys. Rev. D63, 083504 (2001)

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[10]

S. Coleman in Theory and Phenomenology in Particle Physics , ed. A. Zichichi, New York (1969)

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[11]

M.S. Marinov, Phys. Rep. 60, 1 (1980)

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[12]

J.H. Van Vleck, Proc. Nat. Ac. Sci. USA 14, 178 (1928)

View on Pith 1928

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How later papers used it

68 later papers cite it

Evidence: substantial. 67 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 100 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 2 papers with this label

Following this paper, defined the path integral by specifying the field and its first time derivative on initial and final time slices.

Cancelling the vacuum energy and Weyl anomaly in the standard model with dimension-zero scalar fields (2021) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

59 arXiv papers cite this in 91 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 62 sentences without a confident label

Builds on it 1 paper

We can see that the probability can be integrated over all values of $|γ_0|$ giving a sensible extension of the method in [ref] to the sixth order case.

Tackling Higher Derivative Ghosts with the Euclidean Path Integral · The de Sitter background · 1.00

Uses its method or tool 1 paper

Following Hawking and Hertog [ref] , we define the path integral by specifying $φ$ and its first time derivative $φ_,τ$ on initial and final time slices.

Cancelling the vacuum energy and Weyl anomaly in the standard model with dimension-zero scalar fields · No ghosts · 1.00

Compares with it 1 paper

Both the Cutkowski et al. contour deformation [ref] and also the Hertog and Hawking [ref] approach are, classically, equivalent to adopting stability as a future boundary condition.

A New Formulation of the Initial Value Problem for Nonlocal Theories · Conclusions · 0.91

Cites it as support 5 papers

Since the mass of the ghost is $m∼ηM_p$, as long as $η$ is not fine tuned to values $≪1$, the ghost is massive, can be integrated out and there is no instability in the theory [ref] .

Standard Model in conformal geometry: local vs gauged scale invariance · Weyl integrable geometry and Einstein gravity · 0.99
Because mass states degenerated there is no conflict in this case with unitarity (see [ref] ).

On superluminal fermions within the second derivative equation · Conclusion · 1.00
It does affect the unitarity of the theory, but one does not create such a state in the initial or final state of the theory [ref] .

The fall and the rise of Weyl gauge theory · [1]. Weyl conformal geometry and its symmetry · 0.98

Cites it as background 17 papers

Despite the ubiquity of the effective field theory idea, alternative procedures have been proposed to deal with higher derivative terms in the action [ref] .

Tackling Higher Derivative Ghosts with the Euclidean Path Integral · Introduction · 0.93
These results are familiar from the many discussions of the (four-derivative) Pais-Uhlenbeck oscillator [ref] in the recent literature [ref] .

Cancelling the vacuum energy and Weyl anomaly in the standard model with dimension-zero scalar fields · No ghosts · 0.98
The existence of runaway solutions in higher order theories has been first pointed out by Dirac [ref] in his classic paper, and in the context of quadratic gravity, in [ref] [ref] .

Divergences of the scalar sector of quadratic gravity · Introduction. · 0.87

Discussion

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