Stability in the higher derivative Abelian gauge field theory
PPrepared for submission to JHEP
Stability in the higher derivative Abelian gaugefield theory
J.L. Dai, a Department of Physics, Zhejiang University, Hangzhou, 310027, P. R. China b Center of Mathematical Science, Zhejiang University, Hangzhou, 310027, P. R. China
E-mail: [email protected]
Abstract:
We present the derivation of conserved tensors associated to higher-order sym-metries in the higher derivative Maxwell Abelian gauge field theories. In our model, thewave operator of the higher derived theory is a n -th order polynomial expressed in termsof the usual Maxwell operator. Any symmetry of the primary wave operator gives rise toa collection of independent higher-order symmetries of the field equations which thus leadsto a series of independent conserved quantities of derived system. In particular, by the ex-tension of Noether’s theorem, the spacetime translation invariance of the Maxwell primaryoperator results in the series of conserved second-rank tensors which includes the standardcanonical energy-momentum tensors. Although this canonical energy is unbounded frombelow, by introducing a set of parameters, the other conserved tensors in the series can bebounded which ensure the stability of the higher derivative dynamics. In addition, with theaid of auxiliary fields, we successfully obtain the relations between the roots decompositionof characteristic polynomial of the wave operator and the conserved energy-momentum ten-sors within the context of another equivalent lower-order representation. Under the certainconditions, the 00-component of the linear combination of these conserved quantities isbounded and by this reason, the original derived theory is considered stable. Finally, as aninstructive example, we discuss the third-order derived system and analyze extensively thestabilities in different cases of roots decomposition. Corresponding author. a r X i v : . [ phy s i c s . c l a ss - ph ] A ug ontents The research of higher-order derivative systems dates back to the nineteenth century inOstrogradsky’s pioneering work [1] and is still actively studied nowadays such as in theareas of effective low energy theories, astrophysical and cosmological behaviors and themodified gravities [2–9]. This promising approach was first introduced in field theories toget rid of the infinities associated to point particles [10–12] and later, Pais and Uhlenbeckproposed a class of classical higher derivative harmonic oscillators [13]. In these examples,the Lagrangian containing higher derivative terms was been very attractive due to its niceultraviolet behaviour and will result in a renormalizable quantum field theory [14]. How-ever, such Lagrangians yield higher-order equations of motion, which require more initialconditions than in usual dynamical systems and a standard framework for dealing withthese theories on Hamiltonian level is provided by Ostrogradski canonical approach [15–18]. Unfortunately, the Hamiltonian functions obtained in such a way contain terms linearin momenta and are almost always unbounded from below. Therefore with the presence ofhigher derivatives, the systems turn out to be unstable [19–21] and moreover, the existenceof unbounded kinetic terms inevitably lead to runaway solutions if interactions are turnedon. In order to circumvent these problems in a physical allowed sector, various motivationsand techniques have been put forward to avoid the Ostrogradsky ghosts in different higherderivative models [22–30]. For instance, at least in the Pais-Uhlenbeck’s harmonic oscillatorwhich is served as a toy model to understand several important issues related to Ostrograd-sky instabilities, Raidal and Veermae advised that for the purpose of the energy spectrumof the theory be bounded, the ghost degrees of freedom should be necessarily complex [31].– 1 –n this sense, the resulting complex system can be consistently quantized using the rules ofcanonical quantisation which possesses all good properties of the known quantum physicsincluding the positive definite Hamiltonian. In the usual Lee-Wick theories, from the view-point of polynomials with complex conjugate poles, it is possible to construct a unitary S -matrix of gravitational excitations to remove the negative effects of the ghosts [32, 33].On the other hand, in contrast to the classical systems, quantizing the higher derivativedynamics imposes even more constraints. It is thought that higher-order theories wouldpossess propagators having poles with non-positive residues which lead to the appearance ofghost states. At the quantum level, these ghost states have non-positive norms and due tothis, they will violate the causality and spoil the unitarity evolution of the quantum theorywhich is unacceptable physically. By introducing form-factors with an analytic dependenceon the propagating momenta [34], we are able to avoid the unphysical ghosts and thismethod will preserve all fundamental properties of a quantum field theory. Furthermore, inthe non-Hermitian, PT -symmetric model, i.e., symmetric under combined parity reflectionand time reversal, it is essential to modify the dynamical inner product instead of using thestandard Dirac inner product [35–38]. In this manner, we explicitly obtain the self-adjointHamiltonian and its ghost state is reinterpreted as an ordinary quantum state with positive PT norm which gives rise to the standard probabilistic interpretation.In the [39], a special class of linear higher derivative systems is discussed as an al-ternative approach to the problem of Ostrogradsky instability. To be more precise, theoperators of the dynamic equations in these theories are supposed to be factorable in termsof a pair of different second-order operators satisfying some certain conditions. In this way,with the help of auxiliary fields, it is possible to establish two equivalent systems whichmay be thought of as two different representations of the same theory. Then the Noether’stheorem tells us that if the action functional is preserved under the spacetime translations,the system is equipped with canonical energy-momentum tensors and the 00-component isof particular importance since it has the sense of energy density and will lead to the energyconservation law. Especially, for the models of factorable type, applying the Noether’s the-orem, we are capable of acquiring two family of integrals of motion which may be eitherbounded or unbounded depending on the specific values of parameters. As is explainedin [39], the stability of the higher derivative system can be ensured if the 00-component inthis family is positive definite even if the Noether’s canonical energy is unbounded. So far,the efforts of this approach have been focused mostly on various known factorable modelssuch as Pais-Uhlenbeck’s harmonic oscillators, higher derivative scalar fields and Podolsky’sgeneralized electrodynamics.After that, a more general and systematic method was carried out as a guide to in-vestigate the stabilities in a wide class of higher derivative systems named derived typetheories [40–44]. Generally speaking, these derived theories are based on simpler free pri-mary models whose equations of motion only involve first and second order differentialoperators without higher derivatives. In this setting, the wave operator which determinesthe dynamic equations of the higher derivative systems is a polynomial in terms of theprimary wave operator in the lower-order free theory. Then, every symmetry of primarytheory enables us to construct n -parametric series of symmetries of the derived theory if the– 2 –rder of the characteristic polynomial of the wave operator is n . More importantly, thesesymmetries are connected to n independent conserved quantities from the perspective ofmore general correspondence between symmetries and conservation laws which is estab-lished by the Noether’s theorem as well as the Lagrange anchor [45, 46]. Especially, whenthe primary wave operator commutes with the spacetime translation generators, the derivedtheories have n -parametric series of conserved second-rank tensors ( T k ) νµ , k = 0 , , ..., n − and in particular, the k = 0 term corresponds to the usual canonical energy ( T ) of thehigher derivative systems. Now although the canonical energy is unbounded due to thenature of higher derivatives, the linear combination of these tensors ( T k ) νµ may give riseto bounded conserved charge which will stabilize the classical dynamics of derived modelat free level, which also persists at quantum level. Moreover, as demonstrated at lengthin [40, 41], when these conserved tensors are bounded in the free theory, the inclusionof consistent interactions will not spoil the stability of the coupling systems, at least atperturbative level.The paper is organized as follows. In section 2, we start by describing the Lagrangiansfor the higher derivative Maxwell gauge field theories by means of general wave operators.Subsequently, we give a detailed derivation of series of second-rank conserved tensors fromthe higher-order symmetries and investigate the issue of the stability in this derived system.In section 3, according to the different root decompositions of the characteristic polynomials,we set up the formulae of the conserved tensors associated with real and complex roots withthe aid of auxiliary fields. Then as an application, section 4 is devoted to the full analysisof the stabilities in the third-order derived system. The final section of this paper includessome concluding remarks and discussions. Let us start with the Lagrangian density of usual Maxwell electromagnetic theory which isdescribed by the gauge fields A µ in (1+3)-dimensional spacetime as follows S = − (cid:90) F µν F µν d x (2.1)here the metric is g µν = diag(1 , − , − , − which can be used to rise and lower the multi-indices. The dynamic equations of motion of (2.1) are simply present as ∂ µ F µν = 0 (2.2)if we set W µν = δ µν (cid:3) − ∂ ν ∂ µ (2.3)as the primary wave operator, then (2.3) defines the primary free field equation [41] W µν A ν = 0 (2.4)– 3 –ased on this primary model, the most general Lagrangian density of the higher-orderextensions of Maxwell gauge theory is given by S = (cid:90) d xA µ M µν A ν (2.5)here M is termed as wave operator which is a polynomial in the formal variable WM = a n W n + ...... + a W + a W + a (2.6)and the equation of motion contains terms up to the 2 n -th time derivative n (cid:88) l =0 a l W lµν A ν = 0 (2.7)It is well known that the symmetry of a field theory plays a very significant role inmodern physics and it has been regarded as one of the most powerful tools to analyze thebehaviors of the physical gauge systems. In particular, the simplest possible and useful sym-metry of the free field theory is the translation invariance which means that the translationgenerators ∂ µ commute with the primary wave operator in the form of [ ∂ µ , W ] = 0 (2.8)this assumption implies that the derived theory (2.5) enjoys the following higher-ordersymmetries δ ε A µ = ε ν ∂ ν ( W k A ) µ , k = 0 , , ..., n − (2.9)especially, k = 0 corresponds to the spacetime translations invariance of the action of thederived model. Now the Noether’s theorem tells us that each continuous symmetry of theaction (2.9) determines a conserved quantity [40, 41] ∂ µ (Θ k ) µν = ( ∂ ν ( W k A ) µ )( M A ) µ (2.10)in the above expression, we obtain n independent conserved tensors and the k = 0 termcorresponds to the Noether’s canonical energy-momentum tensors, while the other k ≥ terms are different conserved tensors connected to the higher-order symmetries of the gaugefields A µ .To find out the explicit expressions of (Θ k ) µν , at first, a simple calculation shows that ( W k A ) µ = (cid:3) k − ∂ ρ F ρµ , k ≥ (2.11)for convenience, we define (cid:3) − ∂ ρ F ρµ := A µ (2.12)thus in this way, the dynamical equation of motion (2.7) turns out to be n (cid:88) l =0 a l (cid:3) l − ∂ µ F µν = 0 (2.13)– 4 –ubsequently, if l = k in (2.10), it is easy to see that ( ∂ ν ( W k A ) µ )( W k A ) µ = 12 ∂ ν ( (cid:3) k − ∂ ρ F ρλ (cid:3) k − ∂ τ F τλ ) (2.14)on the other hand, when l ≥ k + 1 , in view of ∂ ν ∂ ρ F ρµ − ∂ µ ∂ ρ F ρν = (cid:3) F νµ (2.15)there is no difficulty in evaluating ( ∂ ν ( W k A ) µ )( W l A ) µ =( ∂ ν (cid:3) k − ∂ ρ F ρµ ) (cid:3) l − ∂ λ F λµ = (cid:3) k F νµ (cid:3) l − ∂ λ F λµ + ( (cid:3) k − ∂ µ ∂ ρ F ρν ) (cid:3) l − ∂ λ F λµ = ∂ λ ( (cid:3) k F νµ (cid:3) l − F λµ ) − ( (cid:3) k ∂ λ F νµ ) (cid:3) l − F λµ + ∂ µ ( (cid:3) k − ∂ ρ F ρν (cid:3) l − ∂ λ F λµ ) (2.16)then making using of ∂ λ F νµ + ∂ ν F µλ + ∂ µ F λν = 0 (2.17)and taking into account of the symmetry among the indices λ, µ , we infer that − ( (cid:3) k ∂ λ F νµ ) (cid:3) l − F λµ = 12 ( (cid:3) k ∂ ν F µλ ) (cid:3) l − F λµ (2.18)at this stage, if l = k + 1 we are thus led to
12 ( (cid:3) k ∂ ν F µλ ) (cid:3) k F λµ = 14 ∂ ν ( (cid:3) k F µλ (cid:3) k F λµ ) (2.19)while l ≥ k + 2 , applying the general procedure of integration by parts together with (2.17),we simply have
12 ( (cid:3) k ∂ ν F µλ ) (cid:3) l − F λµ = 12 ∂ ν ( (cid:3) k F µλ (cid:3) l − F λµ ) − (cid:3) k F µλ (cid:3) l − ∂ ν F λµ = 12 ∂ ν ( (cid:3) k F µλ (cid:3) l − F λµ ) + (cid:3) k F µλ (cid:3) l − ∂ µ F νλ = 12 ∂ ν ( (cid:3) k F µλ (cid:3) l − F λµ ) + ∂ µ ( (cid:3) k F µλ (cid:3) l − F νλ ) − ( ∂ µ (cid:3) k F µλ ) (cid:3) l − F νλ (2.20)as well as − ( (cid:3) k ∂ µ F µλ ) (cid:3) l − F νλ = − ( (cid:3) k ∂ µ F µλ ) (cid:3) l − ( ∂ ν ∂ ρ F ρλ − ∂ λ ∂ ρ F ρν )= − ∂ ν ( (cid:3) k ∂ µ F µλ (cid:3) l − ∂ ρ F ρλ ) + ( (cid:3) k ∂ ν ∂ µ F µλ ) (cid:3) l − ∂ ρ F ρλ + ∂ λ ( (cid:3) k ∂ µ F µλ (cid:3) l − ∂ ρ F ρν ) (2.21)furthermore, using (2.15), after a straightforward computation we get ( (cid:3) k ∂ ν ∂ µ F µλ ) (cid:3) l − ∂ ρ F ρλ =( (cid:3) k +1 F νλ ) (cid:3) l − ∂ ρ F ρλ + ( (cid:3) k ∂ λ ∂ µ F µν ) (cid:3) l − ∂ ρ F ρλ = ∂ ρ ( (cid:3) k +1 F νλ (cid:3) l − F ρλ ) − ( (cid:3) k +1 ∂ ρ F νλ ) (cid:3) l − F ρλ + ∂ λ ( (cid:3) k ∂ µ F µν (cid:3) l − ∂ ρ F ρλ )= ∂ ρ ( (cid:3) k +1 F νλ (cid:3) l − F ρλ ) + 12 (cid:3) k +1 ∂ ν F λρ (cid:3) l − F ρλ + ∂ λ ( (cid:3) k ∂ µ F µν (cid:3) l − ∂ ρ F ρλ ) (2.22)– 5 –omparing (2.20) to (2.22) and employing a recursive algorithm, we acquire the followingequation
12 ( (cid:3) k ∂ ν F µλ ) (cid:3) l − F λµ = l − k − (cid:88) i =0 ( 12 ∂ ν ( (cid:3) k + i F µλ (cid:3) l − − i F λµ ) + 2 ∂ µ ( (cid:3) k + i F µλ (cid:3) l − − i F νλ ) − ∂ ν ( (cid:3) k + i ∂ µ F µλ (cid:3) l − − i ∂ ρ F ρλ )+ 2 ∂ µ ( (cid:3) k + i ∂ λ F λµ (cid:3) l − − i ∂ ρ F ρν )) + 12 (cid:3) l − ∂ ν F λρ (cid:3) k F ρλ (2.23)particularly, in the above derivation of (2.23), we have used the identities l − k − (cid:88) i =0 ∂ ρ ( (cid:3) k +1+ i F νλ (cid:3) l − − i F ρλ ) = l − k − (cid:88) i =0 ∂ ρ ( (cid:3) k + i F ρλ (cid:3) l − − i F νλ ) , l − k − (cid:88) i =0 ∂ µ ( (cid:3) k + i ∂ λ F λν (cid:3) l − − i ∂ ρ F ρµ ) = l − k − (cid:88) i =0 ∂ µ ( (cid:3) k + i ∂ λ F λµ (cid:3) l − − i ∂ ρ F ρν ) (2.24)According to these results, we are able to formulate the higher-order conserved tensorsin (2.10) in the form of (Θ k ) µν = n (cid:88) l = k +1 a l ( t k,l ) µν + 12 δ µν a k ( (cid:3) k − ∂ ρ F ρλ (cid:3) k − ∂ τ F τλ ) + k − (cid:88) l =0 a l ( t k,l ) µν (2.25)here l = k + 1 : ( t k,l ) µν = (cid:3) k F νλ (cid:3) k F µλ − δ µν ( (cid:3) k F ρλ ) (cid:3) k F ρλ + (cid:3) k − ∂ ρ F ρν (cid:3) k ∂ τ F τµ ,l ≥ k + 2 : ( t k,l ) µν = (cid:3) k F νλ (cid:3) l − F µλ − δ µν ( (cid:3) k F ρλ ) (cid:3) l − F ρλ + l − k − (cid:88) i =0 ( (cid:3) k + i F µλ (cid:3) l − − i F νλ − δ µν (cid:3) k + i F ρλ (cid:3) l − − i F ρλ + (cid:3) k + i ∂ λ F λµ (cid:3) l − − i ∂ ρ F ρν − δ µν (cid:3) k + i ∂ τ F τλ (cid:3) l − − i ∂ ρ F ρλ ) + (cid:3) k − ∂ ρ F ρν (cid:3) l − ∂ τ F τµ (2.26)analogously, when l ≤ k − , after integration by parts ( ∂ ν ( W k A ) µ )( W l A ) µ = ∂ ν (( W k A ) µ ( W l A ) µ ) − ( W k A ) µ ∂ ν ( W l A ) µ (2.27)and taking a similar tactic in the case of l > k , it is not difficult to re-express the ( W k A ) µ ∂ ν ( W l A ) µ as total derivative terms which give rise to the exact expressions of– 6 – t k,l ) µν , these are l = k − t k,l ) µν = δ µν (cid:3) k − ∂ ρ F ρλ (cid:3) k − ∂ τ F τλ − (cid:3) k − F νλ (cid:3) k − F µλ + 14 δ µν ( (cid:3) k − F ρλ ) (cid:3) k − F ρλ − (cid:3) k − ∂ ρ F ρν (cid:3) k − ∂ τ F τµ ,l ≤ k − t k,l ) µν = δ µν (cid:3) k − ∂ ρ F ρλ (cid:3) l − ∂ τ F τλ − (cid:3) l F νλ (cid:3) k − F µλ + 14 δ µν ( (cid:3) l F ρλ ) (cid:3) k − F ρλ − k − l − (cid:88) i =0 ( (cid:3) l + i F µλ (cid:3) k − − i F νλ − δ µν (cid:3) l + i F ρλ (cid:3) k − − i F ρλ + (cid:3) l + i ∂ λ F λµ (cid:3) k − − i ∂ ρ F ρν − δ µν (cid:3) l + i ∂ τ F τλ (cid:3) k − − i ∂ ρ F ρλ ) − (cid:3) l − ∂ ρ F ρν (cid:3) k − ∂ τ F τµ (2.28) Once obtained the explicit expressions of (Θ k ) µν , we wish to investigate the problem ofstability in (2.5) by introducing n independent parameters β , β , ......, β n − (2.29)and the total series of second-rank energy-momentum tensors of the derived theory undercurrent study reads as [40, 41] Θ µν ( A, β ) = n − (cid:88) k =0 β k (Θ k ) µν (2.30)this family of conserved tensors includes the canonical energy-momentum (Θ ) µν of thederived model (2.5) when β = 1 , β = ...... = β n − = 0 , though it is always unboundedand the other conserved quantities originate from the higher-order symmetries in the set(2.9). In the light of this, the 00-component of this conserved tensor has the meaningsof the energy density of the higher derivative system and the total energy of the derivedtheory is provided by the integral E = (cid:90) d x Θ (2.31)as far as the issue of stability of the higher derivative model is concerned, our strategy is toguarantee the positive definite of the total energy which can be achieved by the requirement Θ ≥ .In the present situation, choosing µ = ν = 0 in (2.25) and with the aid of metric– 7 – µν = diag(1 , − , − , − , it is possible to cast the 00-component of (Θ k ) µν in the form (Θ k ) = − n (cid:88) l = k +1 a l (cid:3) k F ρλ (cid:3) l − F ρλ + 14 k − (cid:88) l =0 a l (cid:3) l F ρλ (cid:3) k − F ρλ − n (cid:88) l = k +2 l − k − (cid:88) i =0 a l ( 12 (cid:3) k + i F ρλ (cid:3) l − − i F ρλ − (cid:3) k + i ∂ µ F µλ (cid:3) l − − i ∂ ρ F ρλ )+ 12 k − (cid:88) l =0 k − l − (cid:88) i =0 a l ( 12 (cid:3) l + i F ρλ (cid:3) k − − i F ρλ − (cid:3) l + i ∂ µ F µλ (cid:3) k − − i ∂ ρ F ρλ )+ n (cid:88) l = k +1 a l (cid:3) k − ∂ ρ F ρ (cid:3) l − ∂ τ F τ + k − (cid:88) l =0 a l (cid:3) k − ∂ ρ F ρλ (cid:3) l − ∂ τ F τλ − k − (cid:88) l =0 a l (cid:3) l − ∂ ρ F ρ (cid:3) k − ∂ τ F τ + 12 a k ( (cid:3) k − ∂ ρ F ρλ (cid:3) k − ∂ τ F τλ ) (2.32)notice that using the equations of motion (2.13) and setting ν = 0 , we have n (cid:88) l = k +1 a l (cid:3) l − ∂ τ F τ = − a k (cid:3) k − ∂ τ F τ − k − (cid:88) l =0 a l (cid:3) l − ∂ τ F τ (2.33)inserting this relation back into (Θ k ) , one can check that the last four terms in (2.32) couldbe rewritten as follows − k − (cid:88) l =0 a l (cid:3) k − ∂ ρ F ρλ (cid:3) l − ∂ τ F τλ − a k ( (cid:3) k − ∂ ρ F ρλ (cid:3) k − ∂ τ F τλ ) (2.34)which permits us to express the total energy density in a more concise and compact way Θ = n − (cid:88) k =0 β k (Θ k ) = n − (cid:88) i,j =0 ( A ij ( a, β ) (cid:3) i F ρλ (cid:3) j F ρλ + B ij ( a, β ) (cid:3) i ∂ ρ F ρλ (cid:3) j ∂ τ F τλ ) (2.35)here A ij ( a, β ) , B ij ( a, β ) are polynomial functions of the variables a l , β k which can be deter-mined from (2.30),(2.32) and (2.34). Inspecting the above formulas, we observe that the 00-component of the energy density is a quadratic form of the formal variables (cid:3) i F ρλ , (cid:3) i ∂ ρ F ρλ ,therefore Θ is positive if A ij ( a, β ) , B ij ( a, β ) are all positive definite matrices (2.36)in other words, once the coefficients a l and parameters β k satisfy these positive definite con-ditions, the original free Abelian derived theory (2.5) admits bounded conserved quantitieswhich thus is considered stable, though its canonical energy is unbounded from below. It is well known that every polynomial in principle can be formulated in terms of its rootsand in this section, we want to establish the relations between the conserved tensors and– 8 –he structure of roots of characteristic polynomial of the higher derivative Maxwell derivedtheory. In fact, under certain assumptions about the roots, it is possible to obtain thebounded 00-component of the conserved quantities which may not been seen directly fromthe general expression (2.6). In order to do so, we suppose the wave operator of the derivedtheory has the following decomposition structure M = n (cid:88) l =0 a l W l = p (cid:89) i =1 ( W − λ i ) p i q (cid:89) j =1 ( W − ( ω j + ¯ ω j ) W + ω j ¯ ω j ) q j (3.1)without loss of generality, we assume a n = 1 and the numbers λ i , ω j , ¯ ω j label different realroots and complex roots. In addition, the numbers p i , q j are the corresponding multiplicitiesand the indices p, q satisfy the condition p (cid:88) i =1 p i + 2 q (cid:88) j =1 q j = n (3.2) Based on the above decomposition, firstly for every real root λ i and complex conjugateroots ω j , ¯ ω j , it is useful to define the new dynamic fields to absorb the higher derivatives ofthe original fields ξ k = p (cid:89) i =1 ,i (cid:54) = k ( W − λ i ) p i q (cid:89) j =1 ( W − ( ω j + ¯ ω j ) W + ω j ¯ ω j ) q j A,η k = p (cid:89) i =1 ( W − λ i ) p i q (cid:89) j =1 ,j (cid:54) = k ( W − ( ω j + ¯ ω j ) W + ω j ¯ ω j ) q j A (3.3)by this construction, when the original fields A µ are subject to the higher derivative fieldequations (2.7), these new component fields of course fulfill the lower-order derived equations ( W − λ i ) p i ξ i = 0 , ( W − ( ω j + ¯ ω j ) W + ω j ¯ ω j ) q j η j = 0 (3.4)for i = 1 , , ..., p and j = 1 , , ..., q . Moreover, we observe that these dynamic equations alsocome from the following action functional S = (cid:90) d x p (cid:88) i =1 ξ i ( W − λ i ) p i ξ i + q (cid:88) j =1 η j ( W − ( ω j + ¯ ω j ) W + ω j ¯ ω j ) q j η j (3.5)at this point, it is an easy exercise to show that the relations (3.3) allow us to establishone-to-one correspondence between the solutions to the higher derivative Maxwell gaugetheory (2.5) and the lower-order dynamical system (3.5). In other words, these two systemsare equivalent and can be viewed as two different representations of the same theory whichare usually called A - and ξ i η i -representations.On the other hand, noting that all the fields ξ i , η j are independent degrees of freedomwhich means the action functional S possesses the following variational symmetries δ ε ξ i = ε µ ∂ µ ( W − λ i ) k ξ i , k = 0 , , ..., p i − (3.6)– 9 –or ξ i and under this condition, it immediately follows that the spacetime translation in-variance of the primary operator W gives us a series of conserved tensors ∂ µ ( T ki ) µν = ∂ ν (( W − λ i ) k ξ i )( W − λ i ) p i ξ i , i = 1 , , ..., p (3.7)then a direct calculation leads to ∂ ν (( W − λ i ) k ξ i )( W − λ i ) p i ξ i = k (cid:88) j =0 p i (cid:88) l =0 C jk ( − λ i ) k − j ∂ ν W j ξ i C lp i ( − λ i ) p i − l W l ξ i (3.8)making using of (2.25), we are able to formulate the second-rank conserved tensors ( T ki ) µν in the form of ( T ki ) µν ( ξ i ) = p i (cid:88) l =0 ( l − (cid:88) j =0 C jk C lp i ( − λ i ) k + p i − j − l (˜ t j,l ) µν + k (cid:88) j = l +1 C jk C lp i ( − λ i ) k + p i − j − l (˜ t j,l ) µν )+ 12 p i (cid:88) l =0 C lk C lp i ( − λ i ) k + p i − l δ µν ( (cid:3) l − ∂ ρ ˜ F iρλ (cid:3) l − ∂ τ ˜ F τλi ) (3.9)for convenience, here we adopt the notations (˜ t j,l ) µν = ( t j,l ) µν ( ξ i ) , (˜ t j,l ) µν = ( t j,l ) µν ( ξ i ) , ˜ F iρλ = ∂ ρ ξ iλ − ∂ λ ξ iρ (3.10)At this stage, let us pay attentions that upon substitution of (3.3) into (3.9), these ( T ki ) µν are just the linear combinations of (Θ k ) µν in (2.25) and by this reason, it is moreconvenient to utilize this description to deal with the issues of stability in original higher-order derived theory. Furthermore, we remark here that a simple observation shows theaction functional is also equipped with the symmetries δ ε ξ i = ε µ ∂ µ ( W k ξ i ) (3.11)and following the same procedure employed above, one can simplify the conserved tensorsas ( T ki ) µν ( ξ i ) = p i (cid:88) l = k +1 C lp i ( − λ i ) p i − l (˜ t k,l ) µν + k − (cid:88) l =0 C lp i ( − λ i ) p i − l (˜ t k,l ) µν + 12 C kp i ( − λ i ) p i − k δ µν ( (cid:3) k − ∂ ρ ˜ F iρλ (cid:3) k − ∂ τ ˜ F τλi ) (3.12) In a similar way, for the fields η j , we have the following independent higher-order symmetrytransformations δ ε η j = ε µ ∂ µ ( W k η j ) (3.13)which are parameterized by the indices k = 0 , , ..., q j − . In analogy to the previousdiscussions, it is evident to see that the corresponding conserved quantities satisfy ∂ µ ( U kj ) µν = ∂ ν ( W k η j )( W − ( ω j + ¯ ω j ) W + ω j ¯ ω j ) q j η j = q j (cid:88) r,s =0 C r,sq j ∂ ν ( W k η j ) W r ( − ω j − ¯ ω j ) s W s ( ω j ¯ ω j ) q j − r − s η j (3.14)– 10 –ere C r,sq j = q j ! r ! s !( q j − r − s )! (3.15)as a consequence, utilizing (2.25), the explicit expressions of the second-rank conservedtensors associated with complex roots take the form of ( U kj ) µν ( η j ) = (cid:88) r + s>k C r,sq j ( − ω j − ¯ ω j ) s ( ω j ¯ ω j ) q j − r − s ( t k, r + s ) µν ( η j )+ (cid:88) r + s 14 ˜ F iµν ˜ F iµν + 12 λ i ξ iρ ξ iρ (4.9)in this manner, the 00-component of the linear combination of ( T i ) is given by T = (cid:88) i =1 β i ( − 14 ˜ F iµν ˜ F iµν + 12 λ i ξ iρ ξ iρ ) (4.10)now under the assumption β i < , λ i < , i = 1 , , (4.11)the contributions of all the component fields are positive which allow us to confirm thestability of the higher derivative Abelian gauge system defined by the third-order waveoperator. – 12 –ext, let us suppose the third-order equation has the simple real root λ and real root λ of multiplicity of 2 M = ( W − λ )( W − λ ) (4.12)which corresponds to the case of p = 3 , p = 1 , p = 2 , q = 0 (4.13)in (3.1) and analogously, we define the following auxiliary fields ξ = ( W − λ ) A, ξ = ( W − λ ) A (4.14)which yield the relations ( W − λ ) ξ = 0 , ( W − λ ) ξ = 0 (4.15)then by virtue of formula (3.12), the second-rank conserved tensors in current situationhave the form ( T ) µν ( ξ ) =( t , ) µν ( ξ ) − λ δ µν ξ ρ ξ ρ , ( T ) µν ( ξ ) = − λ ( t , ) µν ( ξ ) + ( t , ) µν ( ξ ) + 12 λ δ µν ξ ρ ξ ρ , ( T ) µν ( ξ ) =( t , ) µν ( ξ ) + λ ( t , ) µν ( ξ ) − λ δ µν ∂ ρ ˜ F ρλ ∂ τ ˜ F τλ (4.16)in the above expressions, the notations F ρλ = ∂ ρ ξ λ − ∂ λ ξ ρ , ˜ F ρλ = ∂ ρ ξ λ − ∂ λ ξ ρ (4.17)are adopted and after a straightforward calculation of ( t k,li ) µν in (2.26) and (2.28), we arethus led to ( t , ) µν ( ξ ) = ˜ F νλ (cid:3) ˜ F µλ − δ µν ˜ F ρλ (cid:3) ˜ F ρλ + ˜ F µλ (cid:3) ˜ F νλ − δ µν ˜ F ρλ (cid:3) ˜ F ρλ + ∂ λ ˜ F λµ ∂ ρ ˜ F ρν − δ µν ∂ τ ˜ F τλ ∂ ρ ˜ F ρλ + ξ ν (cid:3) ∂ τ ˜ F τµ , ( t , ) µν ( ξ ) = (cid:3) ˜ F νλ (cid:3) ˜ F µλ − δ µν (cid:3) ˜ F ρλ (cid:3) ˜ F ρλ + ∂ ρ ˜ F ρν (cid:3) ∂ τ ˜ F τµ , ( t , ) µν ( ξ ) = δ µν ∂ ρ ˜ F ρλ ξ λ − ˜ F νλ ˜ F µλ + 14 δ µν ˜ F ρλ ˜ F ρλ − ξ ν ∂ τ ˜ F τµ (4.18)for ξ , by making use of the equations of motion from (4.15) (cid:3) ∂ ρ ˜ F ρ − λ ∂ ρ ˜ F ρ + λ ξ = 0 (4.19)we are capable of writing the 00-components of ( T i ) ( ξ ) in a more compact form ( T ) ( ξ ) = − F µν F µν + 12 λ i ξ ρ ξ ρ , ( T ) ( ξ ) = − 12 ˜ F µν (cid:3) ˜ F µν + 12 ∂ µ ˜ F µρ ∂ ν ˜ F νρ + 12 λ ˜ F µν ˜ F µν − λ ξ ρ ξ ρ , ( T ) ( ξ ) = − (cid:3) ˜ F µν (cid:3) ˜ F µν + λ ∂ µ ˜ F µρ ∂ ν ˜ F νρ − λ ξ µ ∂ ρ ˜ F ρµ + 14 λ ˜ F µν ˜ F µν (4.20)– 13 –ubsequently, by introducing a series of parameters β, β and β , the total energy densityof the system is given by T = β ( T ) + β ( T ) + β ( T ) = β ( − F µν F µν + 12 λ ξ ρ ξ ρ ) − β (cid:3) ˜ F µν (cid:3) ˜ F µν − β ˜ F µν (cid:3) ˜ F µν + ( 12 λ β + 14 λ β ) ˜ F µν ˜ F µν + ( 12 β + β λ ) ∂ µ ˜ F µρ ∂ ν ˜ F νρ − β λ ξ µ ∂ ρ ˜ F ρµ − β λ ξ ρ ξ ρ (4.21)it can be shown that for the field ξ , we simply have β < , λ < (4.22)to guarantee the stability of dynamics of ξ and for the field ξ , to ensure the positivedefinite of the quadratic form, it is better to choose β < , β = − β λ , λ < (4.23)indeed, one can verify this assertion through the evaluation of the discriminant in thequadratic form directly. Similarly, in the case of a pair of complex conjugate roots, thelinear combination of ( U ) and ( U ) will not give us a positive conserved tensor unlessthe imaginary part of complex root is set to zero which turns out to be the case we discussedabove.Finally, when the third-order equation possesses real root λ of multiplicity 3 M = ( W − λ ) (4.24)which corresponds to the case of p = 3 , q = 0 , p = 3 (4.25)in (3.1). Then from (3.12), after a direct computation, it is not difficult to derive the explicitformulae of the conserved tensors ( T ) µν =3 λ ( t , ) µν − λ ( t , ) µν + ( t , ) µν − λ δ µν A ρ A ρ , ( T ) µν = − λ ( t , ) µν + ( t , ) µν − λ ( t , ) µν + 32 λ δ µν ∂ ρ F ρλ ∂ τ F τλ , ( T ) µν =( t , ) µν − λ ( t , ) µν + 3 λ ( t , ) µν − λδ µν (cid:3) ∂ ρ F ρλ (cid:3) ∂ τ F τλ (4.26)taking into account of (2.26) and (2.28), the ( t k,li ) µν we need in present case can be worked– 14 –ut in the form of ( t , ) µν = F νλ (cid:3) F µλ − δ µν F ρλ (cid:3) F ρλ + F µλ (cid:3) F νλ − δ µν F ρλ (cid:3) F ρλ + ∂ λ F λµ (cid:3) ∂ ρ F ρν − δ µν ∂ τ F τλ (cid:3) ∂ ρ F ρλ + (cid:3) F µλ (cid:3) F νλ − δ µν (cid:3) F ρλ (cid:3) F ρλ + (cid:3) ∂ λ F λµ ∂ ρ F ρν − δ µν (cid:3) ∂ τ F τλ ∂ ρ F ρλ + A ν (cid:3) ∂ τ F τµ , ( t , ) µν = (cid:3) F νλ (cid:3) F µλ − δ µν (cid:3) F ρλ (cid:3) F ρλ + (cid:3) F µλ (cid:3) F νλ − δ µν (cid:3) F ρλ (cid:3) F ρλ + (cid:3) ∂ λ F λµ (cid:3) ∂ ρ F ρν − δ µν (cid:3) ∂ τ F τλ (cid:3) ∂ ρ F ρλ + ∂ ρ F ρν (cid:3) ∂ τ F τµ , ( t , ) µν = (cid:3) F νλ (cid:3) F µλ − δ µν (cid:3) F ρλ (cid:3) F ρλ + (cid:3) ∂ ρ F ρν (cid:3) ∂ τ F τµ , ( t , ) µν = δ µν (cid:3) ∂ ρ F ρλ A λ − F νλ (cid:3) F µλ + 14 δ µν F ρλ (cid:3) F ρλ − ( F µλ (cid:3) F νλ − δ µν F ρλ (cid:3) F ρλ + ∂ λ F λµ ∂ ρ F ρν − δ µν ∂ τ F τλ ∂ ρ F ρλ ) − A ν (cid:3) ∂ τ F τµ , ( t , ) µν = δ µν (cid:3) ∂ ρ F ρλ ∂ τ F τλ − (cid:3) F νλ (cid:3) F µλ + 14 δ µν (cid:3) F ρλ (cid:3) F ρλ − ∂ ρ F ρν (cid:3) ∂ τ F τµ (4.27)and when expanding the operator (4.24), the equation of motion for the fields A µ becomes (cid:3) ∂ ρ F ρ − λ (cid:3) ∂ ρ F ρ + 3 λ ∂ ρ F ρ − λ A = 0 (4.28)under these constraints, the expressions of 00-components of conserved tensors are morecomplicate ( T ) = − λ F µν F µν + 32 λF µν (cid:3) F µν − F µν (cid:3) F µν − (cid:3) F µν (cid:3) F µν − λ∂ ρ F ρµ ∂ τ F τµ + ∂ ρ F ρµ (cid:3) ∂ τ F τµ + 12 λ A ρ A ρ , ( T ) = − λ F µν F µν + 34 λ (cid:3) F µν (cid:3) F µν − (cid:3) F µν (cid:3) F µν + 12 (cid:3) ∂ ρ F ρµ (cid:3) ∂ τ F τµ − λ ∂ ρ F ρµ ∂ τ F τµ + λ A ρ ∂ τ F τρ , ( T ) = − (cid:3) F µν (cid:3) F µν + 34 λ (cid:3) F µν (cid:3) F µν − λ F µν (cid:3) F µν + 12 λ ∂ ρ F ρµ ∂ τ F τµ + 32 λ (cid:3) ∂ ρ F ρµ (cid:3) ∂ τ F τµ − λ ∂ ρ F ρµ (cid:3) ∂ τ F τµ + λ A µ (cid:3) ∂ τ F τµ (4.29)then to fix the instability of the higher derivative system, we need parameters β i to enter– 15 –he total energy density which can be illustrated as follows T = (cid:88) i =0 β i ( T i ) = − β (cid:3) F µν (cid:3) F µν + ( 34 λ β + 34 λβ − β ) (cid:3) F µν (cid:3) F µν − ( 34 λ β + 14 λ β ) F µν F µν − β F µν (cid:3) F µν + ( 32 λβ − λ β ) F µν (cid:3) F µν − β (cid:3) F µν (cid:3) F µν + ( 12 β + 32 λβ ) (cid:3) ∂ ρ F ρµ (cid:3) ∂ τ F τµ − ( 32 λβ + 32 λ β − λ β ) ∂ ρ F ρµ ∂ τ F τµ + 12 λ β A ρ A ρ + ( β − λ β ) ∂ ρ F ρµ (cid:3) ∂ τ F τµ + λ β A µ (cid:3) ∂ τ F τµ + λ β A ρ ∂ τ F τρ (4.30)now T is positive and bounded only if − β − β − β − β λ β + λβ − β λβ − λ β − β λβ − λ β − ( λ β + λ β ) and β + λβ ( β − λ β ) λ β ( β − λ β ) − ( λβ + λ β − λ β ) λ β λ β λ β λ β are all positive definite matrices. In this paper, we investigate the stability of higher derivative Maxwell gauge field theo-ries from the viewpoint of the n -parameter series of conserved quantities. These conservedquantities can be derived from the higher-order symmetries by the extension of Noether’stheorem if there exists some linear operators commute with the primary wave operator.In particular, we obtain n independent second-rank conserved tensors which are connectedwith the spacetime translation invariance of the action functional and the linear combina-tion of these conserved tensors contains the standard canonical energy-momentum tensors.As a matter of fact, the existence of these additional conserved quantities can be seen as aconsequence of the so-called Lagrange anchor which may be traced back to the quantizationof not necessarily Lagrangian dynamics [47]. In the context of general Lagrangian systemor not, the Lagrange anchor maps the conserved quantities to symmetries for the field equa-tions [48]. More importantly, it should be emphasized that usually the Lagrange anchor isnot unique in higher derivative systems and once the dynamic equations are equipped withmultiple Lagrange anchors, the same symmetry can be linked to different conserved quanti-ties. Moreover, in non-Lagrangian system, the Lagrange anchor gives us a new insight into– 16 –he description of the higher derivative dynamic systems. Indeed, when the field equationsadmit different Lagrange anchors, the inequivalent ones will give rise to the canonically in-equivalent Poisson brackets, therefore the theory turns out to be multi-Hamiltonian in thefirst-order formulation [49]. Especially in the derived theories, a suitable choice of parame-ters brings the corresponding Hamiltonian bounded from below and this classical stabilitycan be promoted to the quantum level which implies the bounded spectrum of energy inquantum theory. Finally, the Lagrange anchor also allows us to systematically add consis-tent interactions into field equations of motion by proper deformation method [50] and inthis sense, the conserved tensors in coupling system are regarded as the deformations of theconserved quantities in free case. It was demonstrated that if the anchor connects the sym-metry with the bounded quantity, the system remains stable upon inclusion of consistentinteractions. In the class of derived theories, generally speaking, the vertices of stable inter-actions are always non-Lagrangian but they can still admit quantization once appropriateLagrange anchor is applied. All of these would be interesting to exploit in future. Acknowledgments The author would like to thank the G.W.Wan for long time encouragements and is gratefulto S.M.Zhu for useful support. References [1] M. Ostrogradsky, Mem. Acad. St. Petersbourg, VI (1850) 385.[2] E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory ofgravity , Nucl. Phys. B 201, 469 (1982).[3] I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective Action in Quantum Gravity , IOP,Bristol,1992.[4] P. Gosselin and H. Mohrbach, Renormalization of higher derivative scalar theory , EPJ. direct4 (2002), 1-10.[5] A. Anisimov, E. Babichev and A. Vikman, B-inflation , JCAP 0506: 006, (2005).[6] R.P. Woodard, Avoiding Dark Energy with /R Modifications of Gravity , Lect. Notes. Phys,720, 403 (2007).[7] E.T. Tomboulis, Renormalization and unitarity in higher derivative and nonlocal gravitytheories , Mod.Phys.Lett. A 30 (2015), 03n04, 1540005.[8] J.R. Villanueva, F. Tapia, M. Molina and M. Olivares, Null paths on a toroidal topologicalblack hole in conformal Weyl gravity , Eur. Phys. J. C, 78 10 (2018) 853.[9] A.A. Salas, A. Molgado and E. Rojas, Hamilton-Jacobi approach for Regge-Teitelboimcosmology , Classical. Quant. Grav. 37 (14) 2020.[10] B. Podolsky, A generalized electrodynamics. I. Nonquantum , Phys. Rev. (2) 62 (1942),68âĂŞ71.[11] B. Podolsky and P. Schwed, Review of a Generalized Electrodynamics , Rev. Mod. Phys. 20, 40(1948). – 17 – 12] A.E.S. Green, Self-energy and interaction energy in Podolsky’s generalized electrodynamics ,Phys. Rev. (2) 72 (1947), 628âĂŞ631.[13] A. Pais and G.E. Uhlenbeck, On field theories with non-localized action , Phys. Rev. 79(1950), 145-165.[14] W. Thirring, Regularization as a Consequence of Higher Order Equations , Phys. Rev. 77(1950), 570.[15] L.V. Belvedere, R.L.P.G. Amaral and N.A. Lemos, Canonical transformations in a higherderivative field theory , Z.Phys.C 66 (1995) 613.[16] T. Nakamura and S. Hamamoto, Higher Derivatives and Canonical Formalisms ,Prog.Theor.Phys. 95 (1996) 469-484.[17] F.J.de Urries and J. Julve, Ostrogradski Formalism for Higher-Derivative Scalar FieldTheories , J.Phys.A 31:6949-6964,1998.[18] J. Gegelia and S. Scherer, Ostrogradsky’s Hamilton formalism and quantumcorrections ,J.Phys.A 43 (2010) 345406.[19] V.V. Nesterenko, On the instability of classical dynamics in theories with higher derivatives ,Phys.Rev.D 75 (2007) 087703.[20] N.G. Stephen, On the Ostrogradski instability for higher-order derivative theories and apseudo-mechanical energy ,J. Sound. Vib 310(3):729-739ïijŇ2008.[21] H. Motohashi and T. Suyama, Third order equations of motion and the Ostrogradskyinstability , Phys.Rev.D 91 (2015) 8, 085009.[22] H.J. Schmidt, Stability and Hamiltonian formulation of higher derivative theories ,Phys.Rev.D 49 (1994) 6354.[23] A. Mostafazadeh, A Hamiltonian formulation of the PaisâĂŞUhlenbeck oscillator that yieldsa stable and unitary quantum system , Phys.Lett.A,375(2):93-98,2010.[24] M. Niedermaier, A quantum cure for the Ostrogradski instability , Ann.Phys 327(2):329âĂŞ358,2012.[25] T. Chen, M. Fasiello, E.A. Lim and A.J. Tolley, Higher derivative theories with constraints :exorcising OstrogradskiâĂŹs ghost , JCAP 02 (2013) 042.[26] D.S. Kaparulin and S.L. Lyakhovich, Energy and Stability of the Pais-Uhlenbeck Oscillator ,arXiv:1506.07422.[27] I. Masterov, An alternative Hamiltonian formulation for the Pais-Uhlenbeck oscillator ,Nucl.Phys.B 902 (2016) 95-114.[28] A. Salvio and A. Strumia, Quantum mechanics of 4-derivative theories , Eur. Phys. J. C 76(2016) 227.[29] M.A. Camachoy, J.A. Vallejo and Y. Vorobiev, A perturbation theory approach to thestability of the Pais-Uhlenbeck oscillator , arXiv:1703.08929.[30] B. Paul, Removing Ostrogradski ghost from degenerate gravity theories , Phys. Rev. D 96,044035 (2017).[31] M. Raidal and H. Veermae, On the Quantisation of Complex Higher Derivative Theories andAvoiding the Ostrogradsky Ghost , Nucl. Phys. B, 916,607-626,2017. – 18 – 32] T.D. Lee and G.C. Wick, Negative Metric and the Unitarity of the S Matrix , Nucl. Phys. B 9(1969) 209-243.[33] T.D. Lee and G.C. Wick, Finite Theory of Quantum Electrodynamics , Phys. Rev. D 2 (1970)1033-1048.[34] M. Asorey, L. Rachwal and I. Shapiro, Unitary Issues in Some Higher Derivative FieldTheories , Galaxies 6 (2018) 1, 23.[35] A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry 3: Equivalence ofpseudoHermiticity and the presence of antilinear symmetries , J. Math. Phys. 43, 3944 (2002).[36] C.M. Bender, Introduction to PT-Symmetric Quantum Theory , Contemp. Phys. 46, 277(2005).[37] C.M. Bender, Making sense of non-Hermitian Hamiltonians , Rep. Prog. Phys. 70, 947 (2007).[38] C.M. Bender and P.D. Mannheim, No-Ghost Theorem for the Fourth-Order DerivativePais-Uhlenbeck Oscillator , Phys.Rev.Lett, 100(11):110402,2007.[39] D.S. Kaparulin, S.L. Lyakhovich and A.A. Sharapov, Classical and quantum stability ofhigher-derivative dynamics , Eur. Phys. J. C 74(10),2014.[40] D.S. Kaparulin, Conservation Laws and Stability of Field Theories of Derived Type ,Symmetry 2019, 11(5), 642.[41] V.A. Abakumova, D.S. Kaparulin and S.L. Lyakhovich, Stable interactions in higherderivative field theories of derived type , Phys. Rev. D 99, 045020,(2019).[42] V.A. Abakumova, D.S. Kaparulin and S.L. Lyakhovich, Stable Interactions between extendedChern-Simons theory and charged scalar field with higher derivatives: Hamiltonianformalism , Russ. Phys. J. 62 (2019).[43] V.A. Abakumova, D.S. Kaparulin and S.L. Lyakhovich, Conservation laws and stability ofhigher derivative extended Chern-Simons , arXiv:1907.02267.[44] V.A. Abakumova, D.S. Kaparulin and S.L. Lyakhovich, Stable interactions between higherderivative extended Chern-Simons and charged scalar field , arXiv:1907.08075.[45] D.S. Kaparulin, Lagrange Anchor for BargmannâĂŞWigner Equations , arXiv:1210.2134.[46] D.S. Kaparulin, S.L. Lyakhovich and A.A. Sharapov, Lagrange Anchor and CharacteristicSymmetries of Free Massless Fields , SIGMA 8 (2012), 021.[47] P.O. Kazinski, S.L. Lyakhovich and A.A. Sharapov, Lagrange structure and quantization ,JHEP 0507 (2005) 076.[48] D.S. Kaparulin, S.L. Lyakhovich and A.A. Sharapov, Rigid symmetries and conservationlaws in non-Lagrangian field theory , J. Math. Phys. 51 (2010) 082902.[49] V.A. Abakumova, D.S. Kaparulin and S.L. Lyakhovich, Multi-Hamiltonian formulations andstability of higher-derivative extensions of 3d Chern-Simons , Eur. Phys. J. C 78(115),2018.[50] D.S. Kaparulin, S.L. Lyakhovich and A.A. Sharapov, Stable interactions via properdeformations , J. Phys. A: Math. Theor. 49 (2016) 155204., J. Phys. A: Math. Theor. 49 (2016) 155204.