PHYSICAL REVIEW D 104, 045010 (2021)
Ostrogradsky instability can be overcome by quantum physics
John F. Donoghue * Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA
† Gabriel Menezes Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23897-000 Serop´edica, RJ, Brazil
(Received 15 June 2021; accepted 27 July 2021; published 16 August 2021)
In theories with higher time derivatives, the Hamiltonian analysis of Ostrogradsky predicts an instability. However, this Hamiltonian treatment does not correspond the way that these theories are treated in quantum field theory, and the instability may be avoided in at least some cases. We present a very simple model which illustrates these features.
DOI: 10.1103/PhysRevD.104.045010
I. INTRODUCTION Quadratic gravity falls in the category which we are discussing here. However, it is a far more complicated In 1850, Ostrogradsky analyzed Lagrangians which theory. We hope that our simple model here illustrates the contained higher time derivatives and showed that such key features for this class of theories. theories are classically unstable [1,2]. Classical instability need not imply quantum instability. A counterexample is the theory of the Dirac equation, where the classical II. THE MODEL Hamiltonian has unbounded negative energies while the Consider a “normal” theory (i.e., without higher deriv- quantized Dirac field is stable with positive energies. For atives) of a complex scalar field χ coupled to a real scalar ϕ higher derivative theories there remains much debate about with the Lagrangian stability, positive energies, unitarity and causality. We here present a simple model representative of a class of theories † L ¼ Lχ þ Lϕ − gϕχ χ ð1Þ and show that the Ostrogradsy instability is not present, as well as showing that the massive excitation carries positive with energy and that the classical limit is normal. We have elsewhere demonstrated unitarity for this class of theories 1 L ¼ ½∂ ϕ∂μϕ − 2ϕϕ [3], and have also discussed causality [4–6]. We also review ϕ 2 μ m these elements below. L ¼ ∂ χ†∂μχ − 2χ†χ − λðχ†χÞ2 ð Þ The work on quantum field theories with higher deriv- χ μ mχ : 2 atives goes back to Lee and Wick [7–9] in the 1960s and The relative masses here are not particulary important, but our discussion incorporates elements of this past work. 2 2 Much of the present interest arises from quadratic gravity, a we will present results for m
2470-0010=2021=104(4)=045010(9) 045010-1 Published by the American Physical Society JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 104, 045010 (2021) Z R L → L ð Þ 4 ½1∂ ∂μ − χ†χ ϕ hd 3 i d x 2 μa a ga Zϕ½χ ¼ ½da e Z R − 4 ½1∂ η∂μη−1 2η2− ηχ†χ with × ½dη e i d x 2 μ 2M g ¼ ð Þ 1 μ 2 1 Za × Zη: 9 L ¼ ∂μϕ∂ ϕ − m ϕϕ − □ϕ□ϕ : ð4Þ hd 2 M2 The first path integral, over aðxÞ, is just the original normal theory, with ϕðxÞ replaced by aðxÞ. The second path Here M is a large mass, very much larger than m and mχ. integral, over ηðxÞ, is the complex conjugate of a normal We invite the reader to consider M as the Planck mass—far massive theory. The remnant of the original higher deriva- beyond the range of ordinary experiments. tive term is the −i instead of þi in the second path integral. We know from work on effective field theories that if we We defer any interpretation to later. For now, let us just treat this new term as a perturbation, it would have a calculate the path integral over η. This is a Gaussian integral negligible effect at low energies, and would not change the and is perfectly well defined. The result is simply the classical wave equation if m ¼ 0. However if we treat this complex conjugate of the usual Gaussian integral. To see as a fundamental theory, the analysis of Ostrogradsky thisR explicitly, we add an real infinitesimal factor of would say that the theory has an instability which renders −ϵ d4xη2 to the exponent to make the result well behaved the theory nonviable even at low energy. This is the simple for large fields. Then we complete the square using model which we wish to analyze. Z 0 4 † η ðxÞ¼ηðxÞ − d xiD−Fðx − yÞχ ðyÞχðyÞð10Þ III. THE LOW ENERGY/CLASSICAL LIMIT Let us define the quantum theory by using path integrals. with ϕ We first focus on the path integral over the field Z 4 − Z ð − Þ¼ d k i ð Þ R iD−F x y 4 2 2 : 11 i d4x½L −gϕχ†χ ð2πÞ k − M − iϵ Zϕ½χ ¼ ½dϕ e hd : ð5Þ This propagator is the complex conjugate of the usual Feynman propagator, changing the sign in the numerator We can now manipulate this a bit. We introduce an and also the sign of the iϵ term in the denominator. The auxiliary field η which, when you integrate it out, repro- 0 integral over η yields duces the same Lagrangian. This is R 4 4 1 † † d xd y gχ ðxÞχðxÞiD− ðx−yÞgχ ðyÞχðyÞ Zη ¼ Ne 2 F : ð12Þ 1 μ 1 2 2 † Lðϕ; ηÞ¼ ∂μϕ∂ ϕ − η□ϕ þ M η − gϕχ χ: ð6Þ 2 2 At low energy, the interaction becomes local and we obtain
R 2 4 g † 2 This results in i d x 2½χ ðxÞχðxÞ Zη ¼ Ne 2M : ð13Þ Z R i d4x½Lðϕ;ηÞ This is just a shift in the quartic interaction of the χ field, Zϕ½χ ¼ ½dϕ ½dη e : ð7Þ with
2 ϕð Þ¼ λ → λ0 ¼ λ − g ð Þ As a next step we can define a new field by x 2 : 14 aðxÞ − ηðxÞ replacing the field ϕ by this combination. The 2M Lagrangian then completely separates and becomes The minus sign in the new contribution is the remnant of the use of expð−iSÞ in the path integral. However, for a 0 1 μ † large mass M, this will not change the sign of λ . Lða; ηÞ¼ ∂μa∂ a − gaχ χ 2 The low energy limit of this theory, quantized using path 1 1 integrals, is then perfectly normal. The resulting classical − ∂ η∂μη − 2η2 − ηχ†χ ð Þ theory for small or vanishing m is then also unchanged. We 2 μ 2 M g 8 colloquially refer to the classical limit as taking ℏ → 0. However in fact ℏ is a fixed constant, and the classical In summary, we have transformed the original theory regime is that with kinematics such that ℏ effects are exactly to unimportant. We will see that at high energy and short
045010-2 OSTROGRADSKY INSTABILITY CAN BE OVERCOME BY … PHYS. REV. D 104, 045010 (2021) wavelengths ℏ effects are crucial. In this theory the classical 2 − 2 − ϵ Σð Þ ∼ g − q i þ 2 ð Þ q 2 log 2 17 limit involves wavelengths much larger than the Compton 32π mχ wavelength of the χ field, much like in usual QED. The most appropriate interpretation of the η path integral such that is as the time-reversed version of a regular path integral. Time-reversal is an antiunitary operation, involving com- 2 Σ ∼ g ≡ γ ð Þ plex conjugation. The Lagrangian itself is time-reversal Im 32π : 18 invariant but in the path integral expðiSÞ changes to expð−iSÞ. Within the path integral, this change is manifest With this result we can look for the high mass pole. It is most importantly in the iϵ in the propagators. These define found at the arrow of causality [4,5]- telling us what is the past light 2 cone and what is the future. We will see that when we q2 ¼ Me ¼ M¯ 2 þ iγ ð19Þ decompose the propagator into time ordered factors, the usual iϵ tells us that positive energies propagate forward in where the real part of the mass is found to be time. Changing the sign on iϵ leads to propagation of 2 positive energies backwards in time. This is described M¯ 2 ¼ ReMe ∼ M2 − ReΣðM2Þð20Þ explicitly in the following section. to first order in g2. In the neighborhood of this pole we use 2 ¼ ¯ 2 þð 2 − ¯ 2Þ IV. HIGH ENERGY q M q M to find the approximate form In this section, we show how the coupling to the χ fields i iDðqÞ ∼ makes the heavy particle decay, that positive energy is 2 q4 q − 2 þ ReΣ þ iγ needed to excite this resonance and we further demonstrate M “ ” −i the backwards in time behavior of the resonance. ∼ : ð21Þ While one often starts the analysis of a theory in the free- q2 − M¯ 2 − iγ field limit with no interactions, here it is important to include the effect of interactions in order to properly The important thing to notice here is that there are two understand the spectrum of the theory. In this regard, it minus sign differences from a normal resonance. The −i in is more similar to the analysis of the electroweak theory, the numerator and the −iγ in the denominator are both of where the interaction with the Higgs boson is included from opposite signs from usual resonances. These combined sign the start in order to get the spectrum correct. In our case differences will lead to the eventual identification of this as here, the coupling to the χ fields is required to provide the time-reversed version of a usual propagator. information on the decay width which is crucial for We can see that this propagator corresponds to expo- understanding the spectrum. nential decay rather than exponential growth by writing it in Consider the ϕ propagator in the original basis, before time ordered form, any field redefinitions have been performed. Including the ð ⃗Þ¼Θð Þ ð ÞþΘð− Þ ð ÞðÞ vacuum polarization, this has the form D t; x t Dfor x t Dback x 22
i with x0 ¼ t. The poles in the complex q0 plane are shown iDðq2Þ¼ : ð15Þ 2 − 2 þ ϵ − q4 þ Σð Þ in Fig. 1. There is the massless pole at q m i M2 q 2 2 q0 − q⃗ þ iϵ ¼ 0 ð23Þ The one loop vacuum polarization has a divergent piece which goes into the renormalization of m2. As noted above which corresponds to q0 ¼ ðωq − iϵÞ with ωq ¼jq⃗j. for convenience we will choose the renormalized value of There are massive poles at m2 to vanish, in which case the finite part of the vacuum polarization is 2 ⃗2 2 q0 − q − mr − iγ ¼ 0 ð24Þ 2 Z 2 2 1 mχ − xð1 − xÞðq þ iϵÞ Σ ð Þ¼− g ð Þ or f q 2 dx log 2 : 16 32π 0 mχ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 γ q0 ¼ E þ iγ ∼ E þ i ð25Þ 2 2 q q 2 Beyond q ¼ 4mχ there will be an imaginary part of the Eq vacuum polarization, which for our purposes is the most pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 important feature. At high q , where we apply this, we have with Eq ¼ q⃗ þ mr . When t>0, we close the contour in the result the lower half plane. This yields the forward propagator
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Here there is the extra change of sign in the denominator 0 q −iγ, which is crucial in making this propagator the time- reversed version of a usual resonance propagator. Reference [24] also discusses particles propagating back- wards in time. This interpretation is reinforced by calculating the Green function with retarded boundary conditions. The loop integrals for the χ fields going into the vacuum polarization x need to be calculated using the in-in formalism, as in Ref. [25]. The result is the same functional dependence, but x with a different iϵ prescription, such that the logarithm is
2 2 log ð−½ðq0 þ iϵÞ − q⃗ Þ 2 ¼ log ð−q − iϵq0Þ 2 2 ¼ log jq j − iπθðq Þðθðq0Þ − θð−q0ÞÞ: ð29Þ
This shifts the location of the poles to the positions indicated in Fig. 2.Fort>0 we pick up the usual massless poles FIG. 1. The location of poles in the complex q0 plane for the Feynman propagator. ð 0 ⃗Þ¼ ð0Þð 0 ⃗ÞðÞ Dret t> ; x Dret t> ; x 30 Z ⃗⃗ ⃗⃗ 3 −iðωqt−q·xÞ iðEqt−q·xÞ γt However, even with these boundary conditions, the Merlin d q e e −2 D ðt; x⃗Þ¼−i − γ e Eq resonance gives a contribution for t<0, for ð2πÞ3 2ω 2ðE þ i Þ q q 2Eq ð 0 ⃗Þ ð26Þ Dret t< ; x ≡ D< ðt; x⃗Þ ret which shows the decaying exponential for the massive Z 3 − ð −⃗⃗Þ ð −⃗⃗Þ d q e i Eqt q·x − γjtj ei Eqt q·x − γjtj term, with the identification 2E 2E ¼ i γ e q − γ e q : ð2πÞ3 2ðE þ i Þ 2ðE − i Þ q 2Eq q 2Eq γ ¼ m Γ: ð27Þ r ð31Þ The term describing propagation backwards in time is 0 obtained for t< by closing in the upper half plane, with q0 the result Z 3 iðωqt−q⃗·x⃗Þ −iðEqt−q⃗·x⃗Þ γjtj d q e e −2 D ðt; x⃗Þ¼−i − γ e Eq : back ð2πÞ3 2ω 2ðE þ i Þ q q 2Eq ð28Þ x x Again we see exponential decay. The other notable feature is that the direction of energy flow is reversed for the high mass resonance. Whereas the normal massless pole prop- agates positive energy forward in time, the high mass resonance propagates it backwards in time. We have elsewhere proposed calling the high mass resonance in this type of theory a Merlin mode [4], named after the wizard in the Arthurian tales who ages backwards in time. This distinguishes it from the more generic phrasing of “ghost,” which is applied to any field with the minus sign in the numerator of the propagator. For example, Faddeev-Popov ghosts have a negative sign in FIG. 2. The location of poles in the complex q0 plane for the the numerator but carry the usual iϵ in the denominator. retarded Green function.
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This also contains decaying exponentials. If we choose to two time derivatives there are extra degrees of freedom use this as a Green function giving the response to an associated with the Lagrangian, and this requires two external source, it would correspond to the propagation of canonical coordinates and the two associated canonical the effect backwards in time. This is related to the micro- momentum. His choices for the coordinates are causality violation on scales of order of the resonance width. ϕ1 ¼ ϕ The width corresponds to the decay into two on-shell χ ϕ ¼ ϕ_ ð Þ particles. These carry positive energy, so that the resonance 2 33 also corresponds to positive energy. We can also see that this resonance requires positive energy in order to be and for the momenta produced by the same reasoning. It is seen as an s-channel 2 χχ¯ → χχ¯ ∂L d ∂L □ þ M _ resonance in . The amplitude for the process is π1 ¼ − ¼ ϕ ∂ϕ_ dt ∂ϕ̈ M2 − 2 M ¼ 2 ð Þ ∼ g ð Þ ∂L □ g D q 2 ¯ 2 : 32 π2 ¼ ¼ − ϕ: ð34Þ q − M − iγ ∂ϕ̈ M2 jMj2 When squared has the same form as a usual The Hamiltonian is formed by resonance, so this yields the characteristic Breit-Wigner shape. The incoming χ fields carry positive energy and one _ _ Hðϕ1; ϕ2; π1; π2Þ¼π1ϕ1 þ π2ϕ2 − L: ð35Þ needs a large positive energy to produce the resonance. With higher derivative theories, there is no guarantee that In writing this Hamiltonian, we must eliminate ϕ̈in terms all quantization methods will yield the same result. The of the canonical coordinates and momenta. This is accom- equivalence of various approaches to quantization has been plished by using the second line of Eq. (34) to write demonstrated only for normal theories. We have used path integral quantization because it is exceptionally clear in this ̈ 2 2 ϕ ¼ ∇ ϕ − M π2: ð36Þ case. However, there are four canonical quantization schemes which we know of which also yield positive The resulting Hamiltonian is energies for the ghost field [7,26–28]. Each requires some modification to traditional canonical quantization. The 2 2 H ¼ π1ϕ2 þ π2ð∇ ϕ − M π2Þ earliest was due to Lee and Wick in the 1960s where they 2 2 proposed a higher derivative theory for a finite version of − Lðϕ1; ϕ2; ∇ ϕ − M π2Þ: ð37Þ QED [7–9]. Their approach was to treat the Pauli-Villars regulator as a dynamical field. The minus sign between the The initial choices of coordinates are then compatible with normal propagator and the Pauli-Villars field then becomes Hamilton equations the essential complication. They used what they called an “indefinite metric” scheme, which modifies the canonical _ ∂H ϕ1 ¼ commutation relations. The result is a massive field with ∂π1 positive energy. While our path integral analysis does not ∂H ϕ_ ¼ ð Þ rely on the specifics of any canonical quantization scheme, 2 ∂π 38 the fact that such schemes exist is welcome. 2 The high energy structure of this theory is intrinsically and with some effort the Hamilton equation quantum. The decay width is crucial for understanding the nature of this resonance. Again we note that while we often ∂H ℏ → 0 ℏ π_ ¼ − ð Þ refer to the classical world as taking , in nature is a 1 ∂ϕ 39 constant and the width is a quantum effect. While the exact 1 magnitude of the width is not important, one cannot neglect can be shown to be equivalent to the Euler-Lagrange its effect. Taking the ℏ → 0 version of the propagator equations. functions is not physically sensible. The Ostrogradsky instability is seen in the first term of the Hamiltonian of Eq. (37). The canonical momentum π1 V. WHAT WOULD OSTROGRADSKY SAY? appears linearly, and there is no other factor of π1 in the The basic point to be noticed is that the Ostrogradsky remainder of the Hamiltonian. This implies that the construction has no resemblance to quantization via path Hamiltonian is not positive definite, and there is no barrier integrals. to making the Hamiltonian negative. One does not need Ostrogradsky’s analysis of the higher derivative energies of order M in order to trigger the instability in this Lagrangian of Eq. (4) starts by noting that with extra analysis.
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To emphasize that the Ostrogradsky construction is not VI. CAUSALITY AND UNITARITY the one relevant for quantum physics, we present the Despite our analysis of the disconnect between following heuristic version of Hamiltonian quantization, Ostrogradsky and quantum phyics, we know that some- related to the indefinite metric quantization schemes of thing has to go wrong in higher derivative theories. Refs. [7,26,27]. We emphasize in advance that this pre- Axiomatic field theorists tell us that propagators cannot sentation does not do justice to the care taken by those 2 fall faster than 1=k . This follows from the Kallen- authors, but it does capture how quantization is different Lehmann representation [29,30] from the Ostrogradsky method. If one starts with the Z separated form for the Lagrangian given in Eq. (8) we d4k η D ðkÞ¼ eik·xh0jTϕðxÞϕð0Þj0i would define the canonical momentum by F ð2πÞ4 Z ∂L 1 ρðsÞ πη ¼ ¼ −η_ ð40Þ ¼ ds ð45Þ ∂η_ π k2 − s þ iϵ which has the opposite sign from usual. Imposing the where ρðsÞ is a positive definite spectral function. If ρ is equal-time quantization conditions never negative, the high energy limit has the form Z ½ηð Þ π ð 0 Þ ¼ ℏδ3ð − 0ÞðÞ 1 1 x; t ; η x ;t i x x 41 D ðkÞ ∼ dsρðsÞ: ð46Þ F k2 π then actually implies the negative of the usual rule, i.e., If ρ is positive definite, this can never vanish. In higher ½ηðx; tÞ; η_ðx0;tÞ ¼ −iℏδ3ðx − x0Þ: ð42Þ derivative theories the propagator falls asymptotically as 1=k4. Therefore at least one of the axioms which goes into (We note that, much like in the path integral analysis, this is this theorem must be violated. the complex conjugate of the usual relation.) To solve this For our simple theory, the defect is in microcausality. It with the usual field decomposition, one would then apply has been known since the time of Lee-Wick and Coleman the negative of the usual commutator for the creation [31] that these theories violate causality. For a clear modern ’ operators, i.e., exposition, see the work of Grinstein, O Connell and Wise [32]. The violation is evident from the factor of −iγ in the ½aðpÞ;a†ðp0Þ ¼ −δ3ðp − p0Þ: ð43Þ propagator for the Merlin mode, and from the interpretation of this mode as propagating backwards in time. We have – If we do this, then the η Hamiltonian which emerges from written sufficiently on this topic elsewhere [4 6] about this the Lagrangian of Eq. (8), i.e., feature that we do not need to repeat that analysis here. However, unitarity survives intact. We have presented a Z 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi formal proof of this elsewhere [3] (see also Lee and Wick d p 2 2 † Hη ¼ − p þ M a ðpÞaðpÞðÞ [7]). However the rationale is quite simple to understand. ð2πÞ3 44 Veltman [33] has shown that the states which appear in the unitarity sum are only the stable states of the theory. actually has positive energy states defined from The unstable states of the theory are not included in the j 0i¼ †j0i p a . While Refs. [7,26,27] provide more analysis asymptotic spectrum. With the heavy unstable Merlin to be convincing that such constructions are sensible, they mode, the result is the same. The unitarity sum includes do involve changing the commutation relations. As far as only the decay products, which in this case are the χ fields. Ostrogradsky is concerned however, the main thing to note One does not include the Merlin mode in the unitarity sum, is that the choice of coordinates and momenta is different. and hence one is not bothered by the unusual minus signs ’ The distinction is in the use of Hamilton s equations. which appear in the analysis of this state. Ostrogradsky has chosen the coordinates to reproduce While there is no need to repeat the formal proof here, ’ Hamilton s equations. Quantum physics does not require there is a simple and explicit example of how unitarity is these. The quantum choice of coordinates is made to manifest in a way which reflects on our treatment of the produce positive energy states—i.e., quanta. The two spectrum above. The s-channel reaction χχ¯ → χχ¯ excites the choices lead to different Hamiltonians. Merlin resonance. The S-wave partial wave amplitude is We have seen that the Ostrogradsky assignment of ϕi and π is not equivalent to the path integral construction of the 2 i ð Þ¼ g ð ÞðÞ quantum theory, or even to the canonical methods such as T0 s 32π D s 47 the Lee-Wick indefinite metric quantization. Quantum physics does not use the Ostrogradsky Hamiltonian as it where DðsÞ is the propagator. Including the fact that near starting point. the resonance the width is given by
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2 lifetime small enough, this appears to be compatible with γ ¼ g ð Þ 32π ; 48 experiment. Further work is needed to better understand the full quantum field theory of these theories. Perhaps a lattice this then has the near-resonance form simulation would be useful in order to provide a non- perturbative study. While higher derivative theories have −g2 1 unusual features, perhaps some of them can still lead to T0 ¼ ð49Þ 32π 2 ¯ 2 g2 reasonable physical theories. q − M − i 32π which satisfies elastic unitarity ACKNOWLEDGMENTS We would like to thank the following for useful comments 2 ImT0 ¼jT0j : ð50Þ or discussions: Simon Caron Huot, G. Dvali, G. ’t Hooft, P. Mannheim, A. Salvio, R. Percacci, I. Shapiro, and K Stelle, Again it is the correlation between the two sign changes and most particularly Bob Holdom and Richard Woodard. characteristic of the Merlin mode which allows unitarity to The work of J. F. D. has been partially supported by the US be satisfied. The asymptotic states here are the χ fields, as in National Science Foundation under Grant No. NSF-PHY18- the Veltman analysis. 20675. The work of G. M. has been partially supported However, the existence of the Merlin mode may require by Conselho Nacional de Desenvolvimento Científico — further changes to field theory practices at higher orders. e Tecnológico CNPq under Grant No. 310291/2018-6 For example, Lee and Wick first showed [7], and we and Fundação Carlos Chagas Filho de Amparoa ` — confirmed in our analysis, that at the two loop order one Pesquisa do Estado do Rio de Janeiro FAPERJ under needs a modification of the contour integral in order to Grant No. E-26/202.725/2018. reproduce the discontinuity which is calculated using the ’ Cutkosky rules. The work of Grinstein, O Connell and APPENDIX: ON EFFECTIVE FIELD THEORY Wise [32] contains an explicit example of how this Lee- Wick contour works. At higher order there may be further When working at low energy, heavy fields can be modifications, or potential problems [34]. It remains for integrated out and we only need to work with those degrees future investigations to better understand the field theory of of freedom which are active at the low energy scale. The these theories. resulting effective Lagrangian can be expanded in a derivative expansion, and so that the low energy world VII. SUMMARY always contains higher derivative interactions describing interactions from the full theory which are suppressed by We have displayed a simple higher derivative model powers of the heavy masses. In this sense, all of our whose path integral quantization avoids the Ostrogradsky theories are effective field theories with higher derivative instability. The main features are the positive energy of the corrections at low enough energy. A simple example is the states, and the decay of the ghost field which removes it effective Lagrangian for the photon at energies well below from the asymptotic spectrum. These features are consistent the electron mass, having integrated out the electron via the with the expectations from early work by Lee and vacuum polarization diagram. The result is Wick, although the simplicity of the model and the path integral analysis shed further light on the physics of such 1 μν α μν theories. L ¼ − FμνF þ Fμν□F þ : ð Þ eff 4 60π 2 A1 The toy model has features which tell us where to look me for problems in higher derivative theories of a type which include both the toy model and quadratic gravity. For this The second interaction yields a contribution to the class of theories, the problems are not negative energies, Lamb shift. nor the Ostrogradsky instability, nor the classical limit. It is important to emphasize that higher derivatives in the However, there still must be some problem, because such effective Lagrangian do not cause any instability problems. theories do not satisfy all the properties of standard Indeed the QED case is the “experimental” verification of quantum field theories. In this analysis, the problem is this, as QED is perfectly stable at low energies. The microcausality. The heavy ghost field becomes a Merlin Ostrogradsky analysis is not relevant for effective field mode with differing signs in the numerator and the theories. One quantizes the effective field theory using only denominator of the propagator. These two minus signs the lowest order Lagrangian, and treats the higher order indicate that the propagation is the T-reversal of a usual operator as a perturbative interaction. If one were to try to resonance. This leads to “dueling arrows of causality,” extrapolate the effective field theory beyond its range of which appears to be the most unusual feature of such validity, one would find that other physics is needed to theories. If the Merlin particle is heavy enough, and its describe accurately the higher energy theory. In the QED
045010-7 JOHN F. DONOGHUE and GABRIEL MENEZES PHYS. REV. D 104, 045010 (2021) example, the form of the vacuum polarization changes to a where κ is a coupling constant. Now the light ϕ field is logarithmic function at higher energy. protected from acquiring a mass by the shift symmetry With this in mind, we can note that there is a variant on ϕ → ϕ þ c, and there will be a simple classical limit our simple model which is much closer to the modern without having to tune the mass to zero. The analysis of application in quadratic gravity. This involves a derivative the path integral of this model proceeds similarly to the interaction, mimicking how gravity couples to matter presentation above, but now the effective low energy proportional to the energy-momentum tensor, interaction from integrating out the heavy field η is proportional to κ2ð□χ†χÞ2=M2. Treated as an effective 1 1 μ † field theory, this suppressed interaction also does not upset L ¼ ∂μϕ∂ ϕ − □ϕ□ϕ − κð□ϕÞχ χ; ðA2Þ hd 2 M2 the stability of the theory.
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