Healthy degenerate theories with higher derivatives
Hayato Motohashi, Karim Noui, Teruaki Suyama, Masahide Yamaguchi, David Langlois
aa r X i v : . [ h e p - t h ] J u l Healthy degenerate theories with higher derivatives
Hayato Motohashi, ∗ Karim Noui,
2, 3, † Teruaki Suyama, ‡ Masahide Yamaguchi, § and David Langlois ¶ Kavli Institute for Cosmological Physics,The University of Chicago, Chicago, Illinois 60637, U.S.A. Laboratoire de Math´ematiques et Physique Th´eorique,Universit´e Fran¸cois Rabelais, Parc de Grandmont, 37200 Tours, France Laboratoire APC – Astroparticule et Cosmologie,Universit´e Paris Diderot Paris 7, 75013 Paris, France Research Center for the Early Universe (RESCEU),Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan Department of Physics, Tokyo Institute of Technology,2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
In the context of classical mechanics, we study the conditions under which higher-orderderivative theories can evade the so-called Ostrogradsky instability. More precisely, we con-sider general Lagrangians with second order time derivatives, of the form L ( ¨ φ a , ˙ φ a , φ a ; ˙ q i , q i )with a = 1 , · · · , n and i = 1 , · · · , m . For n = 1, assuming that the q i ’s form a nondegeneratesubsystem, we confirm that the degeneracy of the kinetic matrix eliminates the Ostrogradskyinstability. The degeneracy implies, in the Hamiltonian formulation of the theory, the exis-tence of a primary constraint, which generates a secondary constraint, thus eliminating theOstrogradsky ghost. For n >
1, we show that, in addition to the degeneracy of the kineticmatrix, one needs to impose extra conditions to ensure the presence of a sufficient number ofsecondary constraints that can eliminate all the Ostrogradsky ghosts. When these conditionsthat ensure the disappearance of the Ostrogradsky instability are satisfied, we show that theEuler-Lagrange equations, which involve a priori higher order derivatives, can be reduced toa second order system.
I. INTRODUCTION
While present observations are compatible with general relativity with a cosmological constant,it is of interest to explore alternative theories as they could provide a more fundamental descriptionof present data or better account for future observations. Among these alternative theories, specialattention has been devoted to Horndeski theory [1], or generalized Galileon theory [2–6], definedby the most general scalar-tensor Lagrangian that yields second-order Euler-Lagrange equationsof motion. Recently it has been pointed out that one can find healthy extensions of Horndeskitheories [7, 8], whose Euler-Lagrange equations involve higher order derivatives (see also [9] for anearlier example of theory “beyond Horndeski” ).To construct a sensible theory with higher order derivatives, one needs to avoid the presenceof additional degrees of freedom (DOF) causing an instability due to the linear dependence ofthe Hamiltonian on momenta, known as the Ostrogradsky instability [10–12]. This instability isinevitable for nondegenerate Lagrangians, unless one introduces “by hand” additional constraintsin order to reduce the phase space [13]. Otherwise, one must turn to degenerate Lagrangiansto find viable theories. Particularly interesting are Lagrangians whose degeneracy is due to thecoupling between a special variable, by which we mean a variable associated with higher order ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] derivatives in the Lagrangian, and regular variables [14]. The degeneracy implies the existence of aprimary constraint and of an associated secondary constraint, which reduce the effective dimensionof phase space and thus eliminate the extra DOF at the source of the Ostrogradsky instability.This can be seen explicitly with the Hamiltonian analysis which has been performed recently [15]for quadratic *1 degenerate higher order scalar tensor (DHOST) theories introduced in [14] (see also[16–18] for subsequent works using the same approach).Given these results, it is worth investigating how, for more general Lagrangians, the degeneracyof the kinetic matrix is related to the elimination of the Ostrogradsky ghosts. In contrast to naiveexpectations, maximal degeneracy of the kinetic matrix, i.e. of order n when n > L ( ¨ φ, ˙ φ, φ ; ˙ q, q )that depend on two variables: φ ( t ) with at most second order derivative, and q ( t ) with at most firstorder derivative (throughout the paper, q denotes a “regular” variable while φ corresponds to a“special” variable). We explain how the Ostrogradsky ghost is eliminated by the constraints in theHamiltonian formulation, and we also show that the Euler-Lagrange equations can be rewrittenas a second-order system. In Sec. III, we consider a Lagrangian L ( ¨ φ, ˙ φ, φ ; ˙ q i , q i ) with multipleregular variables q i and generalize the analysis of the previous section. After discussing the case ofLagrangians L ( ¨ φ a , ˙ φ a , φ a ), depending only on special variables, in Sec. IV, we consider in Sec. V themost general case of a Lagrangian L ( ¨ φ a , ˙ φ a , φ a ; ˙ q i , q i ), depending on multiple special variables φ a and multiple regular variables. In addition to the degeneracy condition, we impose extra conditionsthat guarantee the absence of the Ostrogradsky instability. We present the Lagrangian form ofthese additional conditions and show that they allow the Euler-Lagrange equations to be rewrittenas a second-order system. Sec. VI is devoted to our conclusions. II. LAGRANGIANS WITH SINGLE REGULAR AND SPECIAL VARIABLES
In this section, we consider a Lagrangian of the form L ( ¨ φ, ˙ φ, φ ; ˙ q, q ) (2.1)which depends on two time-dependent variables φ ( t ) and q ( t ). The Lagrangian contains at mostsecond order derivatives of φ whereas q appears at most with first order derivatives. In general, *1 i.e. whose action depends quadratically on the second derivatives of the scalar field. theories of this type are known to exhibit an Ostrogradsky instability. More precisely, φ and q generically obey fourth order and second order equations of motion, respectively *2 , which require6 initial conditions in total, corresponding to three physical DOF. One of these DOF is a ghostwith an energy unbounded from below.Let us now derive the conditions to escape such an instability even when the Lagrangian fea-tures nontrivial second order derivatives (i.e. which cannot be eliminated by simply integrating byparts). We start with a Hamiltonian analysis and find necessary and sufficient conditions for thetheory to be ghost free. To perform the Hamiltonian analysis, we consider the following equivalentLagrangians L (1) eq ≡ L ( ˙ Q, Q, φ ; ˙ q, q ) + λ ( ˙ φ − Q ) , (2.2)and L (2) eq ≡ L ( Q , Q , φ ; Q , q ) + λ ( ˙ φ − Q ) + λ ( ˙ Q − Q ) + λ ( ˙ q − Q ) , (2.3)where the Q ’s are auxiliary variables and the λ ’s are Lagrange multipliers whose associated equa-tions of motion impose that the Q ’s are fixed in terms of ˙ q , ˙ φ and ¨ φ . The two forms are obviouslyequivalent but they possess different advantages for the Hamiltonian analysis.First, in Sec. II A-II C, we investigate the form (2.2), in which the canonical momenta capturethe structure of the highest derivatives in the Lagrangian. This form is useful to understandthe physical meaning of the degeneracy of a theory in terms of additional primary constraintsbetween the momenta. Next, in Sec. II D, we consider the form (2.3), which is easier to generalizeto a Lagrangian with an arbitrary number of variables or of derivatives, by introducing enoughauxiliary variables and Lagrange multipliers so that the final Lagrangian is of the form (2.3) wherethe velocity terms only appear linearly. Then, in Sec. II E, we write the Euler-Lagrange equationsof motion which are a priori higher order. As expected, we show that, when the theory satisfiesthe degeneracy condition, the equations of motion can be reduced to a second order system. Weprovide several specific examples of healthy degenerate Lagrangians in Sec. II F. A. Hamiltonian analysis
Starting from the Lagrangian (2.2), which depends on four variables, we introduce the fourcanonical momenta P , p , π and ρ , associated with Q , q , φ and λ , respectively. The (only nonvan-ishing) elementary Poisson brackets are thus defined by { Q, P } = { q, p } = { φ, π } = { λ, ρ } = 1 . (2.4)From (2.2), it is easy to get P = ∂L∂ ˙ Q ≡ L ˙ Q , p = ∂L∂ ˙ q ≡ L ˙ q , π = ∂L (1) eq ∂ ˙ φ = λ , ρ = ∂L (1) eq ∂ ˙ λ = 0 . (2.5)We thus have two primary constraintsΦ = π − λ ≈ , Ψ = ρ ≈ , (2.6)whose Poisson bracket is nonvanishing: { Φ , Ψ } = − *2 The equation of motion for φ could involve also third derivative of q . Such a third derivative can be eliminatedmaking use of the equation of motion for q . Thus the dynamics is described by a system which is at most secondorder in q and can be up to fourth order in φ . When one can invert (at least locally in the vicinity of any point in phase space) the first twoexpressions in (2.5) to obtain ˙ Q and ˙ q in terms of the momenta, then there is no further primaryconstraint. Considering infinitesimal variations of the momenta with respect to ˙ Q and ˙ q , we canwrite (cid:18) δPδp (cid:19) = K (cid:18) δ ˙ Qδ ˙ q (cid:19) . (2.7)where the matrix K , which we will call the kinetic matrix, is given by K ≡ L ˙ Q ˙ Q L ˙ q ˙ Q L ˙ q ˙ Q L ˙ q ˙ q ! with the notation L xy ≡ ∂ L∂x∂y . (2.8)It is possible to invert (2.7) if and only if K is nondegenerate, i.e. det K = 0.Taking into account the primary constraints (2.6), the Hamiltonian can be written as H = H + πQ with H = P ˙ Q + p ˙ q − L ( ˙ Q, Q, φ ; ˙ q, q ) , (2.9)where the velocities ˙ Q and ˙ q are expressed in terms of the momenta so that H depends only onthe conjugate variables (2.4). For the nondegenerate case, (2.6) are the only primary constraintsand the total Hamiltonian is given by H T = H ( P, p, π, Q, q, φ ) + µ Φ + ν Ψ , (2.10)where µ and ν are Lagrange multipliers.Requiring stability under time evolution (with respect to the total Hamiltonian) of the primaryconstraints leads to fixing the Lagrange multipliers to µ = −{ Ψ , H } = 0 and ν = { Φ , H } . Asa consequence, there are no secondary constraints. Therefore, we have 2 second class primaryconstraints which eliminate 2 initial conditions, and we end up with (8 − / H is linear in π which shows that,without extra constraints, the theory features an instability. This is an illustration of the well-known Ostrogradsky instability. B. Additional primary constraint
The only hope to eliminate such an instability is the existence of an additional primary con-straint. Thus, we will assume, from now on, that the momenta P and p are not independentvariables but, instead, satisfy a relation R ( P, p ; Q, φ, q ) = 0. In general, such a relation definesimplicitly P in terms of p (or the reverse) and it is not always possible to express uniquely andglobally P as a function of p (or the reverse). However, locally, it is always possible to write either P = F ( p, Q, φ, q ) or p = G ( P, Q, φ, q ). In the first case, the theory admits the additional primaryconstraint Ξ ≡ P − F ( p, Q, φ, q ) ≈ , (2.11)whereas in the second case, the additional primary constraint isΠ ≡ p − G ( P, Q, φ, q ) ≈ . (2.12)As long as the analysis is local in phase space, (2.11) and (2.12) are equivalent, except if F (resp. G ) does not depend explicitly on p (resp. P ) in some open set of the phase space. Thus there existsan independent case with the two unrelated primary constraint˜Ξ ≡ P − f ( Q, φ, q ) ≈ , ˜Π ≡ p − g ( Q, φ, q ) ≈ , (2.13)where f and g depend on the coordinates only and not on the momenta.Let us investigate the Hamiltonian structure with, say, the primary constraint (2.11), in additionto the two primary constraints (2.6). The total Hamiltonian is now given by H T = H ( P, p, π, Q, q, φ ) + µ Φ + ν Ψ + ξ Ξ , (2.14)where ξ is a new Lagrange multiplier. Using the Poisson bracket { Φ , Ξ } = F φ , the invariance undertime evolution of the 3 primary constraints gives˙Φ = { Φ , H } − ν + ξF φ ≈ , ˙Ψ = µ ≈ , (2.15)˙Ξ = { Ξ , H } − µF φ ≈ . The first two equations enable us to fix the two Lagrange multipliers µ and ν . Since µ is requiredto vanish, the third equation implies Θ ≡ { Ξ , H } ≈
0. A direct computation shows that Θ is givenby Θ = − π + { Ξ , H } − F φ Q , (2.16)where H = H − πQ has been defined in (2.9). A more explicit expression of Θ is given inAppendix A. The condition Θ ≈ π in terms of the other phase space variables. An important consequence isthat one can get rid of the linear dependence of the Hamiltonian H on π , which signals thatthe Ostrogradsky instability is not present. Note however that the Hamiltonian could still beunbounded from below, but for other reasons.Let us continue our Hamiltonian analysis by requiring the time invariance of Θ:˙Θ = { Θ , H } + ξ { Θ , Ξ } ≈ , (2.17)where we have set µ ≈
0. For the generic case where ∆ ≡ { Θ , Ξ } 6 = 0, the above equation fixes ξ and we have thus determined all Lagrange multipliers. In this case, the theory admits 4 constraintsdenoted generically χ i ∈ (Φ , Ψ , Ξ , Θ) for i = 1 , · · · ,
4. The Dirac matrix D = − F φ { Φ , Θ } − F φ − ∆ { Θ , Φ } (2.18)whose entries D ij = { χ i , χ j } are the Poisson brackets between the constraints is invertible as | det { χ i , χ j }| = ∆ . Thus the constraints are all second-class constraints and we end up with(8 − / φ, Q ) and ( q, p ), as themomenta P and π have been eliminated by solving explicitly the constraints Ξ ≈ ≈ ≡ { Θ , H } does not vanishautomatically and one needs to impose it as a new constraint Γ ≈
0. One then has to continue theprocedure and check whether ˙Γ ≈ χ i ≈
0. Since theskew-symmetric Dirac matrix D is degenerate and contains the nonzero entry { Φ , Ψ } = −
1, oneinfers that it is of rank 2, which means that one can identify two first-class constraints among thefour constraints. As a consequence, we end up with only (8 − × − / = 0. As for the case (2.13), the analysis is a bit different,as we start with the two primary constraints ˜Ξ and ˜Π. The total Hamiltonian is then of the form H T = H + µ Φ + ν Ψ + ξ ˜Ξ + ζ ˜Π , (2.19)where we have introduced the two Lagrange multipliers ξ and ζ . Time invariance of these con-straints fixes ξ and ζ , provided ˜∆ ≡ { ˜Ξ , ˜Π } = L ˙ qQ − L q ˙ Q = 0. In this generic situation, we closethe canonical analysis with 4 constraints ˜ χ i = (Φ , Ψ , ˜Ξ , ˜Π) and a Dirac matrix which is invertible,since | det { ˜ χ i , ˜ χ j }| = ˜∆ . We conclude that all the constraints are second class and the systemcontains 2 DOF. In the special situation where ˜∆ = 0, we may have more constraints or some ofthe constraints may become first class. In both cases, the theory possesses 1 or zero DOF.We conclude that an additional primary constraint leads to the elimination of the unwantedghost-like DOF. C. Degenerate Lagrangians
In the previous subsection, we have assumed the existence of explicit relations between the mo-menta, of the form (2.11), (2.12) or (2.13), which are valid locally. A more intrinsic characterizationof the corresponding Lagrangians is that their kinetic matrix, defined in (2.8), is degenerate.It is immediate to check that each of the conditions (2.11)–(2.13) implies the degeneracy of thekinetic matrix. Indeed, in the case (2.11) for instance, we have the relations L ˙ Q ˙ Q = L ˙ Q ˙ q F ′ ( L ˙ q ) and L ˙ Q ˙ q = L ˙ q ˙ q F ′ ( L ˙ q ) , (2.20)which implies immediately that the determinant of the kinetic matrix vanishes:det K = L ˙ Q ˙ Q L ˙ q ˙ q − L q ˙ Q = 0 . (2.21)For the case (2.12), the same result holds with the replacements q ↔ Q and F → G . Finally, inthe case (2.13), the full kinetic matrix vanishes K = 0, not only its determinant.Conversely, let us now show that det K = 0 implies the existence of a primary constraint ofthe form (2.11) or (2.12), or a set of two primary constraints (2.13). First, let us consider theLagrangians for which L ˙ q ˙ q = ∂p∂ ˙ q = 0 . (2.22)According to the implicit functions theorem, one can find locally (in the vicinity of any point inphase space) a function ϕ such that ˙ q = ϕ ( p, ˙ Q, Q, q, φ ) . (2.23)Consequently, the momentum P which depends a priori on the two velocities ( ˙ q , ˙ Q ) and on thecoordinates ( Q, q, φ ) can locally be expressed as a function P = F ( p, ˙ Q, Q, q, φ ) replacing ˙ q by ϕ .Furthermore, the degeneracy of K implies ∂F/∂ ˙ Q | p = 0. Indeed, if ∂F/∂ ˙ Q | p = 0, one could invokethe implicit functions theorem again and deduce that ˙ Q can be expressed in terms of the momenta( P, p ) and the coordinates, which is in contradiction with the degeneracy of K . We thus conclude P = F ( p, Q, q, φ ) , (2.24)which corresponds precisely to the primary constraint (2.11). To summarize, det K = 0 togetherwith L ˙ q ˙ q = 0 implies that there exists a function F ( p, Q, q, φ ) such that P = F ( p, Q, q, φ ).If L ˙ Q ˙ Q = 0, a very similar analysis enables us to conclude that there exists now a function G ( P, Q, q, φ ) such that p = G ( P, Q, q, φ ) and we recover the primary constraint (2.12). Note thatwhen ∂F/∂p = 0, then necessarily ∂G/∂P = 0 and the two constraints are locally equivalent. Thesefirst two cases apply to degenerate kinetic matrix K which admits only one vanishing eigenvalue.To complete the proof, one must finally consider the cases for which L ˙ q ˙ q = L ˙ Q ˙ Q = 0. Since thematrix K is degenerate, this implies that K in fact vanishes. It is then straightforward to showthat there exist two functions f ( Q, q, φ ) and g ( Q, q, φ ) such that the constraints (2.13) hold. Amore explicit proof of this property is given in the Appendix B without referring to the abstractimplicit functions theorem. We can derive the relations between the functions F , G and the initialLagrangian L via this explicit proof.In summary, we conclude that the condition det K = 0 is equivalent to the existence of primaryconstraints restricting the momenta. Depending on the dimension (one or two) of the kernel of K , the theory admits one or two primary constraints. It amounts to the case (2.11) for L ˙ q ˙ q = 0,the case (2.12) for L ˙ Q ˙ Q = 0, and the case (2.13) for L ˙ Q ˙ Q = L ˙ q ˙ q = 0. As we said previously, theconstraints (2.11) and (2.12) are equivalent when ∂F/∂p = 0 or ∂G/∂P = 0. In practice, onecan check whether a given Lagrangian (2.2) is (Ostrogradsky) ghost-free by using the degeneracycondition det K = 0. Then one can see whether it has 2 DOF or less by checking ∆ = 0 or ˜∆ = 0. D. Alternative Hamiltonian analysis
For completeness, we now perform the Hamiltonian analysis of the Lagrangian (2.3) L (2) eq ≡ L ( Q , Q , φ ; Q , q ) + λ ( ˙ φ − Q ) + λ ( ˙ Q − Q ) + λ ( ˙ q − Q ) , (2.25)which is equivalent to (2.2). Starting with such a formulation has disadvantages. First the canonicalanalysis involves more constraints and thus it could be a priori more complicated than the analysisof (2.2). Second this formulation is more difficult to generalize and to adapt to field theories,including scalar-tensor theories. However, there are important benefits by considering (2.3) in thecontext of this article. As we will see, the total Hamiltonian is explicitly defined as a functionof the phase space variables, and thus there is no need to resort to a local analysis to write theHamiltonian and the constraints. Another benefit is that we can always reduce any Lagrangianwith arbitrary higher derivatives to the form (2.3) where the velocity terms only appear linearly.Let us start with (2.3). The form of the Lagrangian implies that there are initially 8 pairs ofconjugate variables { Q i , P i } = { λ i , ρ i } = { q, p } = { φ, π } = 1 , (2.26)with i ∈ { , , } . It is immediate to see that we have the following 8 primary constraintsΦ = π − λ ≈ , Φ = P − λ ≈ , Φ = p − λ ≈ ,ρ ≈ , ρ ≈ , ρ ≈ , P ≈ P ≈ . (2.27)Contrary to the previous subsection, the Hamiltonian and the total Hamiltonian are now definedglobally and, after simple calculations, one obtains H = − L ( Q , Q , φ ; Q , q ) + πQ + P Q + pQ , (2.28) H T = H + X i =1 ( µ i Φ i + ν i ρ i ) + ξ P + ξ P , (2.29)where µ i , ν i , ξ and ξ are Lagrange multipliers enforcing the primary constraints.To pursue the canonical analysis, we compute the time evolution of the constraints and imposetheir conservation. The simple property { ρ i , Φ j } = δ ij (2.30)implies immediately that time conservation of the six constraints Φ i ≈ ρ i ≈ µ i and ν i . Thus, these primary constraints do not generate secondary constraints. Thisis not the case for P ≈ P ≈
0. Indeed, computing their time derivatives, we obtain twosecondary constraints: χ ≡ ˙ P = { P , H T } = L Q − P ≈ , (2.31) χ ≡ ˙ P = { P , H T } = L Q − p ≈ . (2.32)These constraints are easily interpreted. They simply mean that the momentum P ≈ λ conjugateto Q = ˙ φ is ∂L/∂ ˙ Q and the momentum p ≈ λ conjugate to q is ∂L/∂ ˙ q , as expected.We continue the analysis by computing the time evolution of these two secondary constraintsand we obtain the two conditions˙ χ = { χ , H } + L Q Q ξ + L Q Q ξ ≈ , (2.33)˙ χ = { χ , H } + L Q Q ξ + L Q Q ξ ≈ . (2.34)To simplify notations we have introduced H = H T − ( ξ P + ξ P ). We note that the kineticmatrix K naturally arises here when one identifies Q to ˙ Q = ¨ φ and Q to ˙ q as it should be. As aconsequence, the two previous conditions can be reformulated as follows K (cid:18) ξ ξ (cid:19) = (cid:18) { H, χ }{ H, χ } (cid:19) . (2.35)The end of the analysis depends on the rank of the matrix K .If K is invertible, the system of equations fixes the Lagrange multipliers ξ and ξ and there isno further constraint. It is easy to check that all the constraints are second class. As a consequence,we end up with 10 second class constraints for 8 initial pairs of conjugate variables. This leads to(16 − / K is one-dimensional, in the direction ( u , u ), one obtains the tertiary constraintΞ ≡ u { H , χ } + u { H , χ } , (2.36)where u and u are functions of ( Q i , q, φ ). This constraint is the analog of (2.11) or (2.12) inthe previous analysis. Requiring time invariance of this constraint generically gives one additionalconstraint, which leads to a total of 2 DOF.Finally, when K vanishes, (2.35) implies two constraints { χ , H } ≈ { χ , H } ≈ , (2.37)which are the analog of (2.13) in the previous analysis. The discussion of this case is similar tothat of (2.13) and we end up in general with 2 DOF (or less).In conclusion, we have checked that the two analyses starting from the Lagrangians (2.2) or(2.3) are equivalent. E. Euler-Lagrange equations
We now proceed to study the equations of motion in presence of either of the primary constraints(2.11)–(2.13). For a general Lagrangian of the form (2.1), the Euler-Lagrange equations read L φ − dL ˙ φ dt + d L ¨ φ dt = 0 , (2.38) L q − dL ˙ q dt = 0 . (2.39)Due to the dependence of L on ¨ φ , the equation of motion for q (2.39) in general involves the thirdderivative of φ . As for the equation of motion for φ (2.38), it involves the fourth derivative of φ and the third derivative of q .When the theory is degenerate, the equations of motion can be reformulated as a second ordersystem, as we now show. In order to make the correspondence with the Hamiltonian analysisclearer, we first replace (2.38) and (2.39) by the equivalent Euler-Lagrange equations derived fromthe alternative Lagrangian (2.2): L Q − dL ˙ Q dt = λ , L q − dL ˙ q dt = 0 , L φ = ˙ λ , ˙ φ = Q . (2.40)Let us concentrate on the first two equations which can easily be rewritten in a more explicit wayas K (cid:18) ¨ Q ¨ q (cid:19) = (cid:18) V − λv (cid:19) with (cid:18) Vv (cid:19) = L Q − L ˙ QQ ˙ Q − L ˙ Qq ˙ q − L ˙ Qφ ˙ φL q − L ˙ qQ ˙ Q − L ˙ qq ˙ q − L ˙ qφ ˙ φ ! . (2.41)As the kinetic matrix K is degenerate, it possesses a null vector. Let us assume that L ˙ q ˙ q = 0which corresponds to the case (2.11). Then, as shown in (B2) and (B3), K admits the null vector(1 , − L ˙ q ˙ Q /L ˙ q ˙ q ), which is a function of ( ˙ Q, Q, φ, ˙ q, q ). As a consequence, (2.41) is equivalent to thefollowing two equations: λ = V − L ˙ q ˙ Q L ˙ q ˙ q v , L ˙ q ˙ q ¨ q + L ˙ q ˙ Q ¨ Q = v . (2.42)Note that the first of these equations does not contain second derivatives and determines thevariable λ . Since λ = π , the first equation of (2.42) can be seen as the Lagrangian version of thesecondary constraint (2.16), which we obtained in the Hamiltonian analysis.The equations of motion for φ and q are provided by the second equation in (2.42) and L φ = ˙ λ ,where λ is replaced by its expression in (2.42). To reduce these two equations to a second ordersystem, we need to use explicitly the constraint (2.11), which can be written as L ˙ Q = F ( p, Q, q, φ )with p = L ˙ q . The derivatives of this constraint yield the following useful relations: L ˙ QQ = F p L ˙ qQ + F Q , L ˙ Qq = F p L ˙ qq + F q , L ˙ Qφ = F p L ˙ qφ + F φ ,L ˙ Q ˙ Q = F p L ˙ q ˙ Q , L ˙ Q ˙ q = F p L ˙ q ˙ q . (2.43)From (2.40) one can then express λ in terms of F as follows λ = L Q − dFdt = L Q − F p L q − F Q ˙ Q − F q ˙ q − F φ ˙ φ . (2.44)0Using the above relations, one finds ∂ ˙ λ∂ ¨ Q = L ˙ QQ − F p L ˙ Qq − F Q = F p ( L ˙ qQ − L ˙ Qq ) , (2.45) ∂ ˙ λ∂ ¨ q = L ˙ qQ − F p L ˙ qq − F q = L ˙ qQ − L ˙ Qq . (2.46)Thus, the equation of motion for φ , i.e. L φ = ˙ λ , takes the form( L ˙ qQ − L ˙ Qq )(¨ q + F p ¨ Q ) = w , (2.47)where w depends only on ( ¨ φ, ˙ φ, φ ; ˙ q, q ). When L ˙ qQ = L ˙ Qq , the equation of motion for φ is w = 0:it is second order in φ and does not depend on ¨ q . When L ˙ qQ = L ˙ Qq , one can combine (2.47) withthe equation of motion for q in (2.42), which can be written as L ˙ q ˙ q (¨ q + F p ¨ Q ) = v , (2.48)to obtain an equation of motion for φ of the form E ( ¨ φ, ˙ φ, φ ; ˙ q, q ) ≡ L ˙ q ˙ q w − ( L ˙ qQ − L ˙ Qq ) v = 0 , (2.49)where E can be computed explicitly, although its expression is not simple in general.As a consequence, the equation for φ is always a second order equation which involves at mostthe first derivative of q . Computing the time derivative of this equation enables us to obtaingenerically (when ∂ E /∂ ¨ φ does not vanish) ... φ in terms of up to second derivatives of q and φ .Substituting this last relation in the second equation of (2.42) with Q = ˙ φ leads to a secondorder equation for q as well. This proves that the equations of motion can be recast as a secondorder system. One can deal with the case (2.12) with an analogous procedure and reach the sameconclusions.The remaining case (2.13) L ¨ φ = f ( ˙ φ, φ, q ) , L ˙ q = g ( ˙ φ, φ, q ) (2.50)is simpler to analyze. Indeed, it is obvious that the fourth-order derivatives of φ and third-orderderivatives of q do not appear in the equation of motion (2.38) for φ . Moreover, the terms with... φ cancel as L ˙ φ ¨ φ = f ˙ φ . Since the equation of motion (2.39) for q in this case involves only upto second order derivatives, one thus concludes that the Euler-Lagrange equations form directly asecond order system.In conclusion, degenerate Lagrangians of the form (2.1) are such that their equations of motioncan be reformulated as a system of second order equations for φ and q . This is consistent withthe Hamiltonian analysis which shows that there is no extra degree of freedom in these theories.Nonetheless, it is worth stressing that the Euler-Lagrange equations derived from the Lagrangianin general do not give directly the “minimal” system of equations because they can involve up tofourth-order derivatives of φ , as we saw. Demanding the Euler-Lagrange equations to be secondorder is clearly not a necessary requirement in order to avoid the Ostrogradsky ghost. F. Examples of degenerate theories
To illustrate our previous considerations, we now give some concrete examples of degeneratetheories of the form (2.1).1
Example 1: Linear primary constraint
One can construct degenerate Lagrangians by assuming that the function F appearing in theprimary constraint (2.11) depends linearly on the momentum p , i.e. F ( p, Q, φ, q ) = a ( Q, φ, q ) p + b ( Q, φ, q ) . (2.51)In this case, it is easy to see that the corresponding Lagrangians are of the form L (1) eq ( ˙ Q, Q ; ˙ φ, φ ; ˙ q, q ) = L ( ˙ q + a ˙ Q ; Q, q, φ ) + b ˙ Q (2.52)where L is arbitrary. This is a special case of the toy model considered in Sec. 4 of [20]. Example 2: Factorized Lagrangians
Another class of examples is given by Lagrangians, whose dependence on ˙ Q and ˙ q is factorized,i.e. of the form L (1) eq = L ( ˙ Q ; Q, q, φ ) L ( ˙ q ; Q, q, φ ) . (2.53)Such a Lagrangian leads to a primary constraint (2.11) with F ( t ) = at α , where α = 1 and a anonvanishing function a ( Q, φ, q ), if the functions L and L satisfy the differential equations ∂L ∂ ˙ Q = b L α and (cid:18) ∂L ∂ ˙ q (cid:19) α = ba L , (2.54)where b = b ( Q, φ, q ).Assuming b/a > L and L as L = [(1 − α ) b ˙ Q + c ] − α and L = " α − α (cid:18) ba (cid:19) α ˙ q + c αα − (2.55)where c and c are functions of ( Q, φ, q ) only. Choosing for instance α = 2, c = − b = − a = 2,and c = 0, we obtain the Lagrangian L = 12 ˙ q φ , (2.56)whose Euler-Lagrange equations can be rearranged into ˙ q = C (1 + ¨ φ ) with C being a constant.Similar Lagrangians have been considered in Sec. 7.1 of [21]. Example 3: Linear second derivative
As an example for the case (2.13), we can consider L = ¨ φ f ( ˙ φ, φ, q ) + ˙ q g ( ˙ φ, φ, q ) . (2.57)We note that the terms involving ... φ in the Euler-Lagrange equations vanish identically. However,when multiple variables of the type φ are considered, the Euler-Lagrange equations in generalcontain nonvanishing ... φ terms [19]. We will return to this point in Sec. IV.2 III. LAGRANGIAN WITH MULTIPLE REGULAR VARIABLES AND SINGLESPECIAL VARIABLE
We wish to extend our previous analysis to multiple variables. For pedagogical reasons, inSec. III we start with theories that possess only one special variable φ along with multiple regularvariables q i . In Sec. IV, we consider Lagrangians with multiple special variables only. In Sec. V,we finally study the full generalization with both multiple special and regular variables. In allcases, we assume that the regular subsystem, when present, is by itself nondegenerate, i.e. theirmomenta are all independent. Although there are a few subtleties with multiple variables, theanalysis in this section is very similar to the simpler case studied in Sec. II. Under the assumptionthat the q i ’s form a nondegenerate subsystem, the degeneracy of the kinetic matrix will be shownto be a necessary and sufficient condition for getting rid of the Ostrogradsky ghost. The analysisof this section generalizes the degeneracy condition for the quadratic toy model considered in [14]to general Lagrangians with a single special variable and multiple regular variables.Let us consider a Lagrangian of the form L ( ¨ φ, ˙ φ, φ ; ˙ q i , q i ) ( i = 1 , · · · , m ) . (3.1)As before, it is convenient, in particular for the Hamiltonian analysis, to use the equivalent La-grangian L (1) eq ( ˙ Q, Q ; ˙ φ, φ ; ˙ q i , q i ; λ ) ≡ L ( ˙ Q, Q, φ ; ˙ q i , q i ) + λ ( ˙ φ − Q ) , (3.2)which depends on ( m + 3) variables.Out of ( m + 3) variables, the Lagrange multiplier λ is clearly nondynamical and the correspond-ing DOF is automatically removed by the primary constraints as we shall see below. In general,the theory thus contains ( m + 2) DOF. In order to eliminate another DOF, associated with the Os-trogradsky ghost, one needs additional constraints, which are provided by degenerate Lagrangians.In this case, one ends up with ( m + 1) healthy DOF, which correspond to one DOF associated withthe special variable φ and m DOF associated with the regular variables q i ’s. In that respect, theproblem is very similar to the simpler case studied in Sec. II. A. Constraints
As usual, we introduce the pairs of conjugate variables { Q, P } = { φ, π } = { λ, ρ } = 1 , and { q i , p j } = δ ij . (3.3)The form of the Lagrangian (3.2) implies the existence of two primary constraintsΦ = π − λ ≈ , Ψ = ρ ≈ . (3.4)If there is no further primary constraint, we can proceed exactly as in Sec. II A. In this way, wefind 2 second class constraints that reduce the ( m + 3) initial DOF to ( m + 2) DOF, one of whichis the Ostrogradsky ghost.To obtain ( m + 1) healthy DOF, we need additional constraints, analogous to (2.11)–(2.13).These constraints must kill the ghost, but not a safe degree of freedom like one of the regularvariables. To be certain that we will not eliminate one of the q i ’s variables, we assume, as alreadyemphasized in the introduction of this section, that the subsystem of regular variables is nondegen-erate, i.e. their momenta are all independent. More precisely, we assume that the relation defining3the momenta p i = ∂L/∂ ˙ q i is invertible and then one can locally express the velocities ˙ q i in termsof the momenta p i (and of the remaining phase space variables). This requirement is equivalent toasking that the sub-kinetic matrix L ij defined by L ij ≡ L ˙ q i ˙ q j ≡ ∂ L∂ ˙ q i ∂ ˙ q j (3.5)is non degenerate. A consequence of this hypothesis is that only the case (2.11) can be generalizedto multiple regular variables. Thus, we look for Lagrangians giving a primary constraint of thetype Ξ ≡ P − F ( p i , q i , Q, φ ) ≈ , (3.6)where F is an arbitrary function. This condition is equivalent to the degeneracy of the full ( m + 1)dimensional kinetic matrix K ≡ L ˙ Q ˙ Q L ˙ Qj L i ˙ Q L ij ! with L i ≡ ∂L∂ ˙ q i and L i ˙ Q ≡ L ˙ q i ˙ Q . (3.7)Since the determinant of K is given bydet K = ( L ˙ Q ˙ Q − L ˙ Qi L ij L j ˙ Q ) det L ij (3.8)where L ij L jk = δ ik , the degeneracy of K , together with det L ij = 0, implies L ˙ Q ˙ Q − L ˙ Qi L ij L j ˙ Q = 0 . (3.9)To prove the equivalence between (3.6) and (3.9), one can use a strategy similar to that of Sec. II.First, it is easy to show that (3.9) follows from (3.6). Indeed, (3.6) implies L ˙ Q = F ( L i , Q, q i , φ )which in turn implies L ˙ Q ˙ Q = L i ˙ Q ∂F∂p i and L i ˙ Q = L ij ∂F∂p j . (3.10)To show the converse, one writes the momentum in the form P = F ( p i , q i , ˙ Q, Q, φ ) , (3.11)where the velocities ˙ q i have been replaced by the momenta p i , which is always possible to do since L ij is invertible. If ∂F/∂ ˙ Q | p i = 0, then one could express locally ˙ Q in terms of the momenta, whichwould mean that the Legendre transform ( ˙ Q, ˙ q i ) ( P, p i ) is invertible, in contradiction with thedegeneracy of K . Therefore, F does not depend on ˙ Q and we obtain a primary constraint of thetype (3.6). An alternative and more concrete proof of this equivalence is provided in Appendix B.The Hamiltonian analysis of the theory closely follows that of Sec. II and we will not reproduceit here. It can be easily checked that ˙Ξ ≈ ≈
0. In general,there is no further constraint and one ends up with 4 second class constraints. As a consequence,the theory admits generically ( m + 1) DOF and there is no Ostrogradsky ghost. In some particularcases, there may exist extra (tertiary) constraints and some of the constraints may be first class,as discussed in Sec. II. In such cases, the theory could possess only m degrees of freedom, stillwithout Ostrogradsky ghost.4 B. Euler-Lagrange equations
To show that the Euler-Lagrange equations in degenerate theories reduce to a second ordersystem, we follow the same strategy as in Sec. II E. We first derive the equations of motion associatedto the equivalent Lagrangian (3.2): L Q − dL ˙ Q dt = λ , L q i − dL ˙ q i dt = 0 , L φ = ˙ λ , ˙ φ = Q . (3.12)To avoid confusion, we have returned in this section to the more explicit notation L ˙ q i instead of L i for ∂L/∂ ˙ q i . The first two equations can be reformulated as K (cid:18) ¨ Q ¨ q i (cid:19) = (cid:18) V − λv i (cid:19) with (cid:18) Vv i (cid:19) = L Q − L ˙ QQ ˙ Q − L ˙ Qq j ˙ q j − L ˙ Qφ ˙ φL q i − L ˙ q i Q ˙ Q − L ˙ q i q j ˙ q j − L ˙ q i φ ˙ φ ! . (3.13)The kinetic matrix is degenerate in only one null direction defined by the vector ( − , u i ) with u i = L ij L ˙ q j ˙ Q = ∂F/∂p i , as shown in (B8) and (B9). Thus projecting the previous system in thisnull direction allows us to fix λ to λ = V − u i v i . (3.14)The equations of motion for Q and q i are given by L ij ¨ q j + L ˙ q i ˙ Q ¨ Q = v i , L φ = ˙ λ . (3.15)They involve a priori the third derivative of φ . To get rid of these higher derivatives we make useof the primary constraint L ˙ Q = F ( p i , q i , Q, φ ) with p i = L ˙ q i . Furthermore, the expression of λ (3.14) simplifies to λ = L Q − F p i L q i − F Q ˙ Q − F q i ˙ q i − F φ ˙ φ , (3.16)which gives ∂ ˙ λ∂ ¨ Q = L ˙ QQ − F p i L ˙ Qq i − F Q = F p i ( L ˙ q i Q − L q i ˙ Q ) , (3.17) ∂ ˙ λ∂ ¨ q i = L ˙ q i Q − F p j L ˙ q i q j − F q i = L ˙ q i Q − L q i ˙ Q + F p j ( L ˙ q j q i − L q j ˙ q i ) , (3.18)where we used relations similar to (2.43). As a consequence, the equation of motion for φ , i.e. L φ = ˙ λ , takes the form( L ˙ q i Q − L q i ˙ Q )(¨ q i + F p i ¨ Q ) + F p j ( L ˙ q j q i − L q j ˙ q i )¨ q i = w , (3.19)where w depends only on ( ¨ φ, ˙ φ, φ ; ˙ q i , q i ). One can combine (3.19) with the equation of motion for q i in (3.15), which can be written as ¨ q i = L ij v j − F p i ¨ Q , (3.20)to obtain an equation of motion for φ of the form E ( ¨ φ, ˙ φ, φ ; ˙ q i , q i ) ≡ h ( L ˙ q i Q − L q i ˙ Q ) + F p k ( L ˙ q k q i − L q k ˙ q i ) i L ij v j − w = 0 , (3.21)where E can be computed explicitly, although its expression is not simple in general. Note thatthe coefficient for ¨ Q = ... φ vanishes identically.We conclude that the equation for φ is always second order, and involves at most first derivativesof the q i . Following the same reasoning as in the previous section, taking a time derivative of E allows us to write down ... φ as a function of terms up to second derivatives. Substituting thisexpression of ... φ into the first equation of (3.15), we obtain second order equations for all the q i ’svariables. Thus the degeneracy condition, with L ij invertible, implies that the equations of motionscan be written as a second order system.5 IV. LAGRANGIAN WITH ONLY SPECIAL VARIABLES
Before going to the general analysis for Lagrangians with arbitrary numbers of regular andspecial variables in Sec. V, we discuss in this section the particular case of Lagrangians thatdepend only on special variables. As pointed out in [19], there is a qualitative difference whenwe consider multiple special variables. For L ( ¨ φ a , ˙ φ a , φ a ) with a = 1 , · · · , n , the Euler-Lagrangeequations are in general fourth order, ∂ L∂ ¨ φ a ∂ ¨ φ b .... φ b = (lower derivatives) . (4.1)If the matrix ∂ L/∂ ¨ φ a ∂ ¨ φ b is nondegenerate, one can multiply the above system by its inversematrix to obtain n fourth order EOMs of the form.... φ a = (lower derivatives) , (4.2)which require 4 n initial conditions. In other words, we have 2 n DOF and half of them are Os-trogradsky ghosts, associated with a linear dependence of the Hamiltonian on their canonicalmomenta.If some of the eigenvalues of the matrix ∂ L/∂ ¨ φ a ∂ ¨ φ b vanish, one can take particular linear com-binations of EOMs to eliminate some fourth order derivatives. Let us now consider the maximallydegenerate case for which ∂ L∂ ¨ φ a ∂ ¨ φ b = 0 . (4.3)In that situation, the Lagrangian takes necessarily the form L = X a ¨ φ a f a ( ˙ φ b , φ b ) + g ( ˙ φ b , φ b ) , (4.4)where f a and g are ( N + 1) arbitrary functions of the fields φ b and their velocities ˙ φ b . The highestderivative terms in the EOMs are then third order: E ab ... φ b = (lower derivatives) with E ab ≡ ∂ L∂ ¨ φ a ∂ ˙ φ b − ∂ L∂ ¨ φ b ∂ ˙ φ a = ∂f a ∂ ˙ φ b − ∂f b ∂ ˙ φ a . (4.5)If det E = 0, the system is essentially third order and cannot be reduced to a lower order system.One thus needs to specify 3 n initial conditions and the system still suffers from the Ostrogradskyinstability [19]. This is a simple illustration of the fact that the degeneracy is not a sufficientcondition for eliminating the Ostrogradsky ghost when several special variables are present.To circumvent this problem, a sufficient condition is to require E ab = 0 . (4.6)With the conditions (4.3) and (4.6), the Euler-Lagrange equations for the Lagrangian L ( ¨ φ a , ˙ φ a , φ a )are second order and only 2 n initial conditions are needed, i.e. only n DOF are present. This canbe seen immediately from the fact that (4.6) implies the existence of a function F ( ˙ φ a , φ a ) such that f a = ∂F/∂ ˙ φ a , hence L = dFdt − X a ˙ φ a ∂F∂φ a ( ˙ φ b , φ b ) + g ( ˙ φ b , φ b ) . (4.7)6As a consequence, the second order derivatives are removed from the Lagrangian.To summarize, the first condition (4.3) is the usual degeneracy condition and eliminates thefourth order derivative terms in EOMs. The second condition (4.6) eliminates the third orderderivative terms in EOMs. As one can see from its anti-symmetric nature, the second condi-tion (4.6) only applies when several special variables are present, which is consistent with Ex-ample 3 in Sec. II F. It is straightforward to extend this discussion to Lagrangians of the form L ( φ a ( N ) , · · · , ˙ φ a , φ a ), with derivatives of arbitrary order N . The equations of motion are then oforder 2 N , unless the matrix ∂ L/∂φ a ( N ) ∂φ b ( N ) is degenerate. If this matrix vanishes, the EOMsbecome of order (2 N −
1) in general and the system still suffers from the Ostrogradsky instability.One needs extra conditions similar to (4.6) to get rid of the Ostrogradsky ghosts [19].
V. LAGRANGIAN WITH MULTIPLE REGULAR AND SPECIAL VARIABLES
In this section, we consider the general case of Lagrangians containing both multiple specialvariables φ a and multiple regular variables q i , L ( ¨ φ a , ˙ φ a , φ a ; ˙ q i , q i ) ( a = 1 , · · · , n ; i = 1 , · · · , m ) . (5.1)As in Secs. II and III, we assume that the regular subsystem is by itself nondegenerate *3 . It is alsoconvenient to use the equivalent Lagrangian L (1) eq ( ˙ Q a , Q a ; ˙ φ a , φ a ; ˙ q i , q i , λ a ) ≡ L ( ˙ Q a , Q a , φ a ; ˙ q i , q i ) + λ a ( ˙ φ a − Q a ) , (5.2)where the Lagrange multipliers λ a can be treated as new variables.In general, the Lagrangian (5.1), or equivalently (5.2), describes (2 n + m ) DOF, each specialvariable being associated to 2 DOF. Our goal will be to identify a subclass of Lagrangians that arefree of Ostrogradsky ghosts, which implies that they should contain at most ( n + m ) DOF. A. Hamiltonian analysis
Canonical variables are defined by the following nontrivial Poisson brackets { Q a , P b } = { φ a , π b } = { λ a , ρ b } = δ ab , { q i , p j } = δ ij . (5.3)The Lagrangian induces two sets of n primary constraintsΦ a = π a − λ a ≈ , Ψ a = ρ a ≈ , (5.4)which can be used to eliminate the extra-variables λ a together with their momenta ρ a . If there areno other primary constraints, one can follow the procedure already discussed in Secs. II and III,and one ends up with (2 n + m ) DOF, among which n are Ostrogradsky ghostsIn order to eliminate the Ostrogradsky ghosts, we now assume, generalizing the constraint (3.6)for a single special variable discussed in Sec. III, that there exist n primary constraints of the formΞ a ≡ P a − F a ( p i , q i , Q b , φ b ) ≈ . (5.5)The total Hamiltonian is then given by H T = H + µ a Φ a + ν a Ψ a + ξ a Ξ a with H = P a ˙ Q a + ρ a ˙ λ a + π a ˙ φ a + p i ˙ q i − L (1) eq , (5.6) *3 It is of course possible to consider systems where the variables q i are also degenerate. The present analysis isstraightforward to extend although it would be more involved in practice. µ a , ν a and ξ a are Lagrange multipliers. Requiring the time invariance of the primaryconstraints Φ a and Ψ a , using { Ψ a , Φ b } = δ ab , fixes the Lagrange multipliers µ a and ν a , in particular µ a = 0. And time invariance of the remaining primary constraints Ξ a leads to the following n conditions ˙Ξ a = { Ξ a , H } + ξ b { Ξ a , Ξ b } ≈ . (5.7)The status of this set of conditions depends on the n -dimensional matrix M whose entries are M ab ≡ { Ξ a , Ξ b } . If M is invertible, all the Lagrange multipliers ξ a are determined and there areno secondary constraints. In this case, the system does not have sufficient number of constraintsto eliminate the ghost DOF.In order to get secondary constraints, M must be degenerate. The simplest scenario to get ridof all the Ostrogradsky ghosts is to require that the whole matrix M vanishes *4 { Ξ a , Ξ b } = − ∂F a ∂Q b + ∂F b ∂Q a + ∂F a ∂q i ∂F b ∂p i − ∂F a ∂p i ∂F b ∂q i = 0 . (5.8)In that case, (5.7) implies the existence of n secondary constraintsΘ a ≡ { Ξ a , H } ≈ . (5.9)These constraints fix all the momenta π a in terms of the p i and of the canonical coordinates. SinceΞ a does not contain π a , the set of secondary constraints Θ a is independent of the set Ξ a . Now,these constraints are sufficient to eliminate all the ghost-like DOF.If in addition the matrix ∆ with entries ∆ ab = { Θ a , Ξ b } is invertible, then the primary andsecondary constraints are all second class and we end up with exactly ( n + m ) DOF as required.If det ∆ = 0, then there may be tertiary constraints or there might be first class constraints in thetheory. Thus, Lagrangians with degenerate ∆ have fewer than ( n + m ) DOF and none of them is anOstrogradsky ghost. In conclusion, the conditions (5.5) and (5.8) are sufficient to define ghost-freehigher derivative Lagrangians with multiple special variables. B. Conditions for the Lagrangian to evade the Ostrogradsky instability
In analogy with the results of the previous sections, one can show that the condition (5.5) isequivalent to the degeneracy of the ( n + m )-dimensional kinetic matrix K = (cid:18) L ab L aj L ib L ij (cid:19) , (5.10)where we use the notations L ij ≡ ∂ L∂ ˙ q i ∂ ˙ q j , L ab ≡ ∂ L∂ ˙ Q a ∂ ˙ Q b , L ia ≡ ∂ L∂ ˙ Q a ∂ ˙ q i . (5.11)More precisely, the degeneracy must be of order n , i.e. dim(Ker K ) = n , which can be expressedby the conditions L ab − L ai L ij L jb = 0 . (5.12) *4 It would also be possible to have a nonvanishing matrix M , thus yielding fewer than n secondary constraints.The elimination of all the ghosts would then require the existence of a sufficient number of further (tertiary, etc)constraints. K ) = n is equivalent to the existence of n eigenvectors ( v bα , v iα ) for α ∈ { , · · · , n } such that (cid:18) L ab L aj L ib L ij (cid:19) (cid:18) v bα v jα (cid:19) = 0 = ⇒ ( L ab − L ai L ij L jb ) v bα = 0 , (5.13)where we have used the property that L ij is invertible. Since the v bα form a family of n independent n -dimensional vectors, we conclude that L ab − L ai L ij L jb = 0. Conversely, if L ab − L ai L ij L jb = 0, onecan easily construct at least n null-vectors of K , with their components satisfying v iα = − L ij L jb v bα .This, together with the invertibility of L ij , implies dim(Ker K ) = n .Let us now show the equivalence between (5.5) and (5.12). It is immediate to see that (5.5)implies (5.12) by writing L ab = L ib ∂F a ∂p i and L ia = L ij ∂F a ∂p j , (5.14)which directly follows from (5.5). The converse is proved in a way similar to previous sections. As L ij is invertible, one can write any momentum P a as a function P a = F a ( ˙ Q b , p i , Q, q, φ ). If thereexists a pair ( a, b ) such that ∂F a /∂ ˙ Q b | p i = 0 then one sees immediately that L ab − L ai L ij L jb = ∂F a /∂ ˙ Q b | p i = 0. Thus the functions F a do not depend on the velocities ˙ Q b . A more explicit proofis provided in Appendix B.Finally, let us examine the consequences of the conditions (5.8) for the Lagrangian. Takingderivatives of (5.5) with respect to Q b and q i with the use of (B19), we obtain ∂F a ∂Q b = ∂ L∂ ˙ Q a ∂Q b − ∂ L∂ ˙ q i ∂Q b L ij L aj ,∂F a ∂q i = ∂ L∂ ˙ Q a ∂q i − ∂ L∂ ˙ q j ∂q i L jk L ak . (5.15)Plugging these expressions together with (B19) into (5.8) yields the conditions0 = ∂ L∂ ˙ Q a ∂ ˙ φ b − ∂ L∂ ˙ Q b ∂ ˙ φ a + ∂ L∂ ˙ φ a ∂ ˙ q i L ij ∂ L∂ ˙ q j ∂ ˙ Q b − ∂ L∂ ˙ Q a ∂ ˙ q i L ij ∂ L∂ ˙ q j ∂ ˙ φ b + ∂ L∂ ˙ Q a ∂ ˙ q i L ij ∂ L∂q j ∂ ˙ Q b − ∂ L∂ ˙ Q a ∂q i L ij ∂ L∂ ˙ q j ∂ ˙ Q b + ∂ L∂ ˙ Q a ∂ ˙ q i L ij (cid:18) ∂ L∂ ˙ q j ∂q k − ∂ L∂q j ∂ ˙ q k (cid:19) L kl ∂ L∂ ˙ q l ∂ ˙ Q b , (5.16)where we have explicitly written some second derivatives of L with respect to velocities to avoidconfusion. The converse is also true. Note that the above conditions reduce to (4.6) in the absenceof regular variables.In conclusion, any Lagrangian of the form (5.1) which satisfies the relations (5.12) and (5.16)is free of Ostrogradsky ghosts. These conditions have a clear interpretation from the Hamiltonianpoint of view: they ensure the existence of primary and secondary second class constraints whichenable one to get rid of the Ostrogradsky ghosts. C. Euler-Lagrange equations
We conclude our study of multi-variable Lagrangians by showing that the equations of motioncan be written as a second order system. We follow the same method as in previous simpler cases9starting from the equivalent formulation (5.2) of Lagrangian. Euler-Lagrange equations can bewritten as K (cid:18) ¨ Q a ¨ q i (cid:19) = (cid:18) V a − λ a v i (cid:19) , L φ a = ˙ λ a , Q a = ˙ φ a , (5.17)where K is the kinetic matrix (5.10) and ( V a , v i ) is given by (cid:18) V a v i (cid:19) = L Q a − L ˙ Q a Q b ˙ Q b − L ˙ Q a q j ˙ q j − L ˙ Q a φ b ˙ φ b L q i − L ˙ q i Q b ˙ Q b − L ˙ q i q j ˙ q j − L ˙ q i φ b ˙ φ b ! , (5.18)which is written down in terms of up to the second order derivatives of q and φ . As previously, n among these equations fix the Lagrange multipliers λ a and they correspond to the secondaryconstraints Φ a in the Hamiltonian analysis. The equations of motion for φ a and q i take the form L φ a = ˙ λ a , L ˙ Q a ˙ q i ¨ Q a + L ij ¨ q j = v i , (5.19)where λ a is replaced by its expression λ a ( ˙ Q b , Q b , φ b ; ˙ q i , q i ) (see Appendix B 3).To go further, we first make use of the primary constraints L ˙ Q a = F a ( p i , q i ; Q b , φ b ) with p i = L ˙ q i .Using these constraints, a straightforward calculation shows that the terms proportional to ¨ Q b and¨ q i in the equation of motion for φ a are given by ∂ ˙ λ a ∂ ¨ Q b = L Q a ˙ Q b − ∂F a ∂p i L q i ˙ Q b − ∂F a ∂Q b , (5.20) ∂ ˙ λ a ∂ ¨ q i = L Q a ˙ q i − ∂F a ∂p j L q j ˙ q i − ∂F a ∂q i . (5.21)The equations of motion (5.19) then read ∂ ˙ λ a ∂ ¨ Q b ¨ Q b + ∂ ˙ λ a ∂ ¨ q i ¨ q i + R a = 0 , (5.22)¨ q i = L ij v j − L ij L ˙ Q b ˙ q j ¨ Q b , (5.23)where R a depends only on ( ¨ φ b , ˙ φ b , φ b ; ˙ q j , ˙ q j ). After substituting (5.23) into (5.22), an immediatecalculation shows that the coefficients of the ¨ Q b in the resulting equations are given by ∂ ˙ λ a ∂ ¨ Q b − L ij L ˙ q j ˙ Q b ∂ ˙ λ a ∂ ¨ q i = − ∂F a ∂Q b + ∂F b ∂Q a + ∂F a ∂q i ∂F b ∂p i − ∂F a ∂p i ∂F b ∂q i . (5.24)We recognize in the r.h.s. the Poisson brackets { Ξ a , Ξ b } between the secondary constraints Ξ a .These coefficients are in general nonvanishing, in contrast to the previous cases where the coefficientfor ¨ Q vanishes identically (see(2.49) or (3.21)). This illustrates the role of the extra conditions (5.8)at the level of the equations of motion. Imposing them ensures that (5.24) vanishes and that the¨ Q b = ... φ b terms can be removed from the equations of motion for the φ a . We thus obtain E a ( ¨ φ b , ˙ φ b , φ b ; ˙ q j , ˙ q j ) ≡ ∂ ˙ λ a ∂ ¨ q i L ij v j + R a = 0 . (5.25)Following the same idea as in the previous cases, and taking a time derivative of E a , we can writedown ... φ a in terms of up to second derivatives. Plugging it into (5.23), we obtain a set of secondorder equations of motion for q i ’s. This concludes our analysis.0 VI. CONCLUSION
In this work, we have investigated in which circumstances a classical mechanics Lagrangiancontaining higher order derivatives can escape the generic Ostrogradsky instability. We have shownthat there is a qualitative difference between Lagrangians that contain only one special variableand those with multiple special variables.In the first case, the degeneracy of the kinetic matrix is a necessary and sufficient condition(under the assumption that the regular variables q i form a nondegenerate subsystem) to evade theOstrogradsky instability. The degeneracy of the kinetic matrix is associated with the existence ofa primary constraint in phase space, whose time invariance implies a secondary constraint. Bothconstraints eliminate the would-be Ostrogradsky ghost. This result holds for any number of regularvariables and the degeneracy is expressed by simple conditions on the second derivatives of theLagrangian (see Eq. (3.9)).By contrast, when n ( >
1) special variables are present, the degeneracy, of order n , of the kineticmatrix (expressed by the conditions (5.12)) is not sufficient to eliminate the n Ostrogradsky ghoststhat are present in general. The reason is that the degeneracy of order n induces n primaryconstraints, but requiring the time invariance of these constraints does not necessarily generate n secondary constraints. Therefore, the degeneracy condition is not sufficient in general to get ridof the Ostrogradsky instability. This can however be achieved by imposing additional conditions,such as the vanishing of all Poisson brackets between the primary constraints, which leads tothe presence of n secondary constraints. In the Lagrangian formulation, these conditions can beexpressed as antisymmetric relations between the second derivatives of the Lagrangian with respectto the second or first order time derivatives of the various variables (see Eq. (5.16)).In all cases, we showed how the higher order Euler-Lagrange equations can be rewritten asa second-order system. We also provided some specific examples of ghost-free Lagrangian (seeSec. II F). Although our results apply to Lagrangians describing point particles, we believe thatthe conditions obtained in this paper could be quite useful to construct ghost-free field theoriesinvolving for example several scalar fields and other fields such as the gravitational metric. Itwould thus be interesting to extend the present analysis to field theories, a task which we leave fora future work. ACKNOWLEDGMENTS
We thank the organizers and participants of the workshop “Exploring Theories of Modified Grav-ity” at Kavli Institute for Cosmological Physics at University of Chicago (October 2015), wherethis work was initiated. This work was supported in part by Japan Society for the Promotion ofScience (JSPS) Grant-in-Aid for Young Scientists (B) No. 15K17632 (T.S.), JSPS Grant-in-Aidfor Scientific Research Nos. 25287054 (M.Y.), 26610062 (M.Y.), MEXT Grant-in-Aid for ScientificResearch on Innovative Areas “New Developments in Astrophysics Through Multi-Messenger Ob-servations of Gravitational Wave Sources” Nos. 15H00777 (T.S.) and “Cosmic Acceleration” No.15H05888 (T.S. & M.Y.).
Appendix A: Expression for ∆ In this Appendix we derive the expression of ∆ ≡ { Θ , Ξ } , where Ξ and Θ are defined in (2.11)and (2.16), respectively. Let us start by expressing the secondary constraint Θ asΘ = − π + L Q − F Q ˙ Q − F φ Q − F q ˙ q − F p L q . (A1)1Even if velocities ˙ Q and ˙ q seem to enter in this expression, it is simple to show that Θ is a functionof the phase space variables only.Now, we provide how to obtain ∆ given in (A12) expressed by derivatives of Lagrangian. First,it is straightforward to write down∆ = { Θ , P − F } = Θ Q + Θ P F Q + Θ π F φ − Θ q F p + Θ p F q . (A2)In order to proceed further, we need to know how Θ changes under the infinitesimal variation ofthe canonical variables. To this end, let us perturb (A1). The result is given by δ Θ = L Q ˙ Q δ ˙ Q + L QQ δQ + L Q ˙ q δ ˙ q + L Qq δq + L Qφ δφ − δπ − ˙ Q ( F Qp δp + F QQ δQ + F Qφ δφ + F Qq δq ) − F Q δ ˙ Q − Q ( F φp δp + F φQ δQ + F φφ δφ + F φq δq ) − F φ δQ − ˙ q ( F qp δp + F qQ δQ + F qφ δφ + F qq δq ) − F q δ ˙ q − L q ( F pp δp + F pQ δQ + F pφ δφ + F pq δq ) − F p (cid:16) L q ˙ Q δ ˙ Q + L qQ δQ + L q ˙ q δ ˙ q + L qq δq + L qφ δφ (cid:17) . (A3)Picking up velocity variation part only, we have δ Θ = (cid:16) L Q ˙ Q − F Q − F p L q ˙ Q (cid:17) δ ˙ Q + ( L Q ˙ q − F q − F p L q ˙ q ) δ ˙ q + · · · . (A4)Using the primary constraint (2.11) written in the Language of the Lagrangian formalism, L ˙ Q = F ( L ˙ q , Q, q, φ ), and definition of the conjugate momenta, (A4) becomes δ Θ = ( L Q ˙ q − L ˙ Qq ) (cid:16) F p δ ˙ Q + δ ˙ q (cid:17) + · · · = L Q ˙ q − L ˙ Qq L ˙ q ˙ q ( δp − L ˙ qQ δQ − L ˙ qq δq − L ˙ qφ δφ ) + · · · , (A5)where we used (B5) and L ˙ q ˙ q = 0. As it should be from the fact that Θ is a function of the canonicalvariables, δ Θ has been finally expressed as a linear combination of the infinitesimal variation ofthe canonical variables. Then, we findΘ Q = L QQ − ˙ QF QQ − QF φQ − F φ − ˙ qF qQ − L q F pQ − F p L qQ − ( L Q ˙ q − L ˙ Qq ) L ˙ qQ L ˙ q ˙ q , Θ P = 0 , Θ π = − , (A6)Θ q = L Qq − ˙ QF Qq − QF φq − ˙ qF qq − L q F pq − F p L qq − ( L Q ˙ q − L ˙ Qq ) L ˙ qq L ˙ q ˙ q , Θ p = − ˙ QF Qp − QF φp − ˙ qF qp − L q F pp + L Q ˙ q − L ˙ Qq L ˙ q ˙ q . It is appropriate to make one remark here. Although Θ is a function of the canonical variables,its specification is not unique in the sense that there is ambiguity of expressing Θ in terms of thecanonical variables due to the constraint Ξ ≈
0. For instance, it is always possible to replace all P ′ s appearing in Θ by other variables by using P = F ( p, Q, q, φ ). By the same token, it is equally2allowed to partially keep P in Θ. This ambiguity amounts to adding αδ Ξ to δ Θ where α is anarbitrary function of the canonical variables. Indeed, we can derive the following relation; δ Θ + αδ Ξ = δ { Ξ , H } + αδ Ξ= δ ( { Ξ , H } + α Ξ) − Ξ δα = δ { Ξ , H + α ′ Ξ } − Ξ δα ≈ δ { Ξ , H + α ′ Ξ } , (A7)where α = { Ξ , α ′ } . This shows that adding αδ Ξ is equivalent to replace P in H by F ( p, Q, q, φ )by some amount controlled by α ′ .What remains is to express the derivatives of F in (A2) and (A6) in terms of derivatives ofLagrangian. For the sake of clarity, let us write the primary constraint as L ˙ Q = F ( L ˙ q , x i ) , x = Q, x = q, x = φ. (A8)Taking the first derivative we obtain ∂F∂x i = ∂L ˙ Q ∂x i − F p ∂L ˙ q ∂x i , (A9)and the second derivative yields ∂ F∂x i ∂x j = ∂ L ˙ Q ∂x i ∂x j − ∂L ˙ q ∂x i ∂L ˙ q ∂x j F pp − (cid:18) ∂L ˙ q ∂x i ∂F p ∂x j + ∂L ˙ q ∂x j ∂F p ∂x i (cid:19) − ∂ L ˙ q ∂x i ∂x j F p . (A10)In a similar way, we obtain ∂F p ∂x i = 1 L ˙ q ˙ q (cid:18) ∂L ˙ Q ˙ q ∂x i − F pp L ˙ q ˙ q ∂L ˙ q ∂x i − F p ∂L ˙ q ˙ q ∂x i (cid:19) ,F pp = 1 L q ˙ q (cid:16) L ˙ Q ˙ q ˙ q − F p L ˙ q ˙ q ˙ q (cid:17) , (A11) F p = L ˙ Q ˙ q L ˙ q ˙ q . Plugging the derived expressions into (A2), we finally obtain the following expression for ∆: L q ˙ q ∆ = − ˙ mε abφ ε cdφ δ αβm ˙ q δ γδd ˙ q L ˙ q ˙ a L ˙ qα L ˙ cγ L δβ ˙ b + ε abφ ε cdφ δ αβd ˙ q L q L ˙ q ˙ a L ˙ cα L ˙ bβ ˙ q − ε abφ L q ˙ q L ˙ q ˙ a L ˙ bφ + 2 δ αβq ˙ q L q ˙ q L ˙ Qα L βQ + L q ˙ q ( L ˙ Q ˙ Q L qq + L ˙ q ˙ q L QQ ) . (A12)Here, Einstein summation convention is used and the Roman/Greek letters denote { q, Q, φ } / { q, Q, φ, ˙ q, ˙ Q, ˙ φ } respectively. The ε abc is the totally anti-symmetric matrix with ε qQφ = 1 andthe generalized Kronecker delta δ αβγδ is defined by δ αβγδ = δ αγ δ βδ − δ αδ δ βγ . Appendix B: Degeneracy of kinetic matrix and primary constraints
This Appendix is devoted to showing more explicitly that the degeneracy of the kinetic matrixleads to the existence of primary constraints (see Secs. II C, III A, and V B). We study the caseswith single regular and special variables, multiple regular variables and single special variable, andmultiple regular and special variables.3
1. Single regular and special variables
We provide an alternative and more concrete proof of det K = 0, where K is defined in (2.8),leads to one of the three additional primary conditions (2.11), (2.12) or (2.13). For that purpose,we start writing the degenerate kinetic matrix as follows K = (cid:18) a bb c (cid:19) with ac = b . (B1)Its degeneracy implies that one or two of its eigenvalues are vanishing.Furthermore, K is a real symmetric matrix and thus is diagonalizable. The explicit diagonal-ization depends on whether c and a are vanishing or not. First, if c ≡ L ˙ q ˙ q = 0, K has one nonzeroeigenvalue and can be diagonalized as K = (cid:18) cr bb c (cid:19) = O T (cid:18) c ( r + 1) (cid:19) O, (B2)where r ≡ b/c and O = 1 √ r + 1 (cid:18) − rr (cid:19) = O T = O − . (B3)Now, we are going to show that the degeneracy leads in this case to a constraint (2.11). Indeed,using this eigenbasis of K in (2.7) leads to (cid:18) − δP + rδprδP + δp (cid:19) = (cid:18) c ( r + 1)( rδ ˙ Q + δ ˙ q ) (cid:19) . (B4)We thus arrive at δP − L ˙ q ˙ Q L ˙ q ˙ q δp = 0 , L ˙ q ˙ Q L ˙ q ˙ q δ ˙ Q + δ ˙ q = 1 L ˙ q ˙ q δp. (B5)The function L ˙ q ˙ Q /L ˙ q ˙ q is a priori a function of the velocities ˙ q and ˙ Q . From the Legendre transform,it can be viewed as a function of p and ˙ Q which in fact can be shown to depend only p (and Q, φ, q ).Indeed, when one computes variations of L ˙ q ˙ Q /L ˙ q ˙ q with respect to δ ˙ q and δ ˙ Q first and with respectto δp and δ ˙ Q using (B5), one obtains δ (cid:18) L ˙ q ˙ Q L ˙ q ˙ q (cid:19) = δ ˙ q ∂∂ ˙ q (cid:18) L ˙ q ˙ Q L ˙ q ˙ q (cid:19) + δ ˙ Q ∂∂ ˙ Q (cid:18) L ˙ q ˙ Q L ˙ q ˙ q (cid:19) = δp L ˙ q ˙ q ∂∂ ˙ q (cid:18) L ˙ q ˙ Q L ˙ q ˙ q (cid:19) + δ ˙ Q ∂∂ ˙ Q (cid:20) L ˙ q ˙ Q L ˙ q ˙ q − L ˙ q ˙ Q L ˙ q ˙ q ∂∂ ˙ q (cid:18) L ˙ q ˙ Q L ˙ q ˙ q (cid:19)(cid:21) = δp L ˙ q ˙ q ∂∂ ˙ q (cid:18) L ˙ q ˙ Q L ˙ q ˙ q (cid:19) + δ ˙ Q L q ˙ q ∂∂ ˙ q (det K ) . (B6)As det K = 0, L ˙ q ˙ Q /L ˙ q ˙ q is a function of p only. Thus, the first equation in (B5) gives the primaryconstraint (2.11) with F ′ ( p ) = L ˙ q ˙ Q /L ˙ q ˙ q . The second equation of (B5) is discussed in (A5).Then, the case a ≡ L ˙ Q ˙ Q = 0 in (B1) is treated in a way similar to the previous case, andleads to a primary constraint of the type (2.12). Finally, when K has two vanishing eigenvalues,necessarily K = 0, which leads immediately to constraints of the type (2.13).4
2. Multiple regular variables and single special variable
We show that imposing the degeneracy condition (3.9) and det L ij = 0 to the kinetic matrix(3.7) leads to the existence of the primary constraint (3.6). Using (3.9) and defining u i ≡ L ij L j ˙ Q ,we can write L i ˙ Q = L ij u j , L ˙ Q ˙ Q = L ij u i u j , (B7)which amounts to (3.10). In fact, the ( n + 1)-dimensional vector ( − , u i ) with u i = ∂F/∂p i is anull vector of K . We make use of this null vector to block-diagonalize K as follows K = L ˙ Q ˙ Q L ˙ Qj L i ˙ Q L ij ! = (cid:18) u T Lu u T LLu L (cid:19) = T − (cid:18) CLC (cid:19) T, (B8)where T = (cid:18) ( u T u + 1) − / C − (cid:19) (cid:18) − u T u (cid:19) , T − = (cid:18) − u T u (cid:19) (cid:18) ( u T u + 1) − / C − (cid:19) , (B9)and an m × m matrix C = ( uu T +1) / is the square root of ( u i u j + δ ij ). As the kinetic matrix relatesthe infinitesimal variations as ( δP, δp i ) T = K ( δ ˙ Q, δ ˙ q j ) T , evaluating it in the block-diagonalizedbasis yields (cid:18) − δP + u T δpuδP + δp (cid:19) = (cid:18) uu T + 1) L ( uδ ˙ Q + δ ˙ q ) (cid:19) . (B10)We thus arrive at δP − L ˙ Qi L ij δp j = 0 , L ij L ˙ Qj δ ˙ Q + δ ˙ q i = L ij δp j , (B11)which is precisely a generalization of (B5) to the case with multiple regular variables. We canconfirm that the infinitesimal variation of L ij L ˙ Qj with respect to δ ˙ Q and δ ˙ q i is given by δ ( L ij L ˙ Qj ) = δ ˙ Q ∂ ( L ij L ˙ Qj ) ∂ ˙ Q + δ ˙ q k ∂ ( L ij L ˙ Qj ) ∂ ˙ q k = δp ℓ L ℓk ∂ ( L ij L ˙ Qj ) ∂ ˙ q k . (B12)Thus the first equation of (B11) gives the primary constraint (3.6) with ∂F/∂L i = L ij L ˙ Qj .
3. Multiple regular and special variables
Similarly to Appendix B 2, imposing the degeneracy condition (5.12) and det L ij = 0, the kineticmatrix (5.10) can be block-diagonalized as K = (cid:18) L ab L aj L ib L ij (cid:19) = (cid:18) A T kA A T kkA k (cid:19) = T − (cid:18) CkC (cid:19) T, (B13)where A ia ≡ L ij L ja , k ij ≡ L ij to avoid confusion, T = (cid:18) B − C − (cid:19) (cid:18) − A T A (cid:19) , T − = (cid:18) − A T A (cid:19) (cid:18) B − C − (cid:19) , (B14)5and an n × n matrix B and an m × m matrix C are the square roots of A T A + 1 and AA T + 1,respectively: B = A T A + 1 , C = AA T + 1 . (B15)Since all the eigenvalues for A T A + 1 and AA T + 1 are positive, B and C are well-defined. Further,they are symmetric and have symmetric inverse matrices as their determinants are nonvanishing.Substituting the block-diagonalization (B13) into the relation ( δP a , δp i ) T = K ( δ ˙ Q b , δ ˙ q j ) T , weobtain (cid:18) − δP + A T δpAδP + δp (cid:19) = (cid:18) AA T + 1) k ( Aδ ˙ Q + δ ˙ q ) (cid:19) . (B16)We thus arrive at δP a − L ai L ij δp j = 0 , L ij L aj δ ˙ Q a + δ ˙ q i = L ij δp j . (B17)which is a generalization of (B5) or (B11). We can confirm that the infinitesimal variation of L ij L aj with respect to δ ˙ Q and δ ˙ q i is given by δ ( L ij L aj ) = δ ˙ Q b ∂ ( L ij L aj ) ∂ ˙ Q b + δ ˙ q k ∂ ( L ij L aj ) ∂ ˙ q k = δp ℓ L ℓk ∂ ( L ij L aj ) ∂ ˙ q k . (B18)Thus the first equation of (B17) gives the primary constraint (5.5) with ∂F a ∂p i = L ij L aj . (B19) [1] G. W. Horndeski, Int. J. Theor. Phys. , 363 (1974).[2] A. Nicolis, R. Rattazzi, and E. Trincherini, Phys. Rev. D79 , 064036 (2009), arXiv:0811.2197 [hep-th].[3] C. Deffayet, G. Esposito-Farese, and A. Vikman, Phys. Rev.
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