Resonance and stability of higher derivative theories of derived type

We consider the class of higher derivative field equations whose wave operator is a square of another self-adjoint operator of lower order. At the free level, the models of this class are shown to admit a two-parameter series of integrals of motion. The series includes the canonical energy. Every conserved quantity is unbounded in this series. The interactions are included into the equations of motion such that a selected representative in conserved quantity series is preserved at the non-linear level. The interactions are not necessarily Lagrangian, but they admit Hamiltonian form of dynamics. The theory is stable if the integral of motion is bounded from below due to the interaction. The motions are finite in the vicinity of the conserved quantity minimum. The equations of motion for fluctuations have the derived form with no resonance. The general constructions are exemplified by the models of the Pais-Uhlenbeck oscillator with multiple frequency and Podolsky electrodynamics. The example is also considered of stable non-abelian Yang-Mills theory with higher derivatives.


Introduction
In 1950, Pais and Uhlebeck first considered the class of relativistic higher derivative theories whose wave operator is a polynomial in the another formally self-adjoint operator of lower order [1]. Such models can be termed the theories of derived type. The general derived model is specified by the constant coefficient polynomial, termed the characteristic polynomial, and the lower order operator, being the primary operator. As far as the wave equations are determined modulo multiplication by a nonzero constant, only the structure of the characteristic polynomial roots is relevant. Depending on the structure of roots and primary operator, the setting of a derived model can describe many long-known higher derivative field theories, see [2] for review. The example is the Podolsky electrodynamics [3], the conformal gravity theories in various dimensions [4,5] also fall in this class. The extended Chern-Simons [6] usually serves as prototype of a gauge derived model in the space-time dimension three. The Pais-Uhlenbeck (PU) oscillator [1] is the best known example of the derived-type higher derivative mechanical model. The derived theories are often considered in the context of studying of various aspects of higher derivative dynamics, including the problem of stability. There is a vast literature on the subject, see the reviews in [7][8][9][10] and references therein.
The recent studies demonstrate that the higher derivative theories are not necessary unstable, even though the canonical energy is unbounded. 1 The stability has been studied from several viewpoints. The non-Hermitian quantum mechanics is used to construct stable quantum theory of the PU model [11][12][13]. The same problem is solved by means of alternative Hamiltonian formulations in [14,15]. The stability of classical paths is studied by numerical simulations and analytical methods in [16][17][18][19], including the estimates of adiabatic invariants [20]. The structure of symmetries and conservation laws of derived type models is studied in [21,22]. It is shown that the bounded integrals of motion can exist in these theories that stabilizes the higher derivative dynamics.
The conclusions about the stability of higher derivative models are mostly related to the theories with simple roots of characteristic polynomial. The models with multiple characteristic polynomial roots usually demonstrate instability already at free level. The simplest example of this phenomenon is provided by the fourth order PU oscillator with multiple frequency. The oscillations resonate, so the motion is unbounded. The theory with resonance has a bounded from below integral of motion, though it does not lead to stability [23]. The alternative Hamiltonian formulations for PU model with resonance are canonically equivalent, and all the possible Hamiltonians are unbounded [14]. The multiple root in the characteristic polynomial has been previously considered as a indication of instability unless the corresponding degree of freedom is a pure gauge. As the example of this phenomenon we mention the the extended Chern-Simons theory with multiplicity two zero root which is stable due to the gauge invariance [22].
In papers [24][25][26], the problem of inclusion of stable interactions is studied from the viewpoint of consistent deformation of equations of motion and conserved quantities. It is demonstrated that the higher derivative theories admit the class of interactions such that preserve a selected conserved quantity of the free model. If a bounded from below quantity is conserved at the interacting level, the dynamics of the non-linear theory is stable. The interacting equations are non-Lagrangian, but the dynamics is explicitly covariant and it still admits the Hamiltonian formulation. For construction of interaction, it is critical that the free theory is stable. The interaction just preserves the model stability. In the class of derived theories with resonance at free level, the dynamics free evolution is unbounded. This means that the interactions have to stabilize originally unstable theory. The phenomenon of stabilization of dynamics by the interaction is well-known in the theories without higher derivatives. For example, the motions of mechanical system can bounded in the vicinity of the unstable equilibrium position once the higher order corrections are accounted for the energy.
In the present paper, we construct the stable interactions for the class of higher derivative theories of derived type with a single multiplicity two root of the characteristic polynomial. The wave operator of general model is given by the square of another formally self-adjoint operator of lower order. The simplest model is the PU oscillator with a resonance solutions at free level. We exploit the idea of stabilization of dynamics by means of interaction. We demonstrate that the interactions can be introduced such that the unbounded energy of the free model becomes bounded from below on the account of the interactions. The dynamics turns out stable in the vicinity of the energy minimum just by the reasons of energy conservation. The equations of motion for fluctuations have derived type without a resonance. The motions are stable in the range of energies below a certain level. Above this level, the dynamics becomes singular. All the proposed stable interactions are non-Lagrangian, but the equations still admit Hamiltonian formulation. The theory of 1 Gauge theories with higher derivatives can have on-shell bounded energy. small fluctuations in the vicinity of stationary solution is Hamiltonian with bounded Hamiltonian.
The general construction is illustrated by two examples: the PU oscillator with multiple frequency, and Podolsky electrodynamics. For the PU oscillator, we detail the structure of the conserved quantities at free and interacting level.
We notice that the free theory admits a two-parameter series of conserved quantities. The canonical energy is included in the series. All the conserved quantities in the series are unbounded. The particular class of interactions that preserves a selected representative in the series of conserved quantities is proposed. If the interaction makes the integral of motion bounded, the non-linear model turns out stable. The equations of motion for small fluctuations and the upper bound are identified for the energy such that the model admits the stable paths. The Podolsky electrodynamics provides a fieldtheoretical example of a similar phenomenon. In this higher derivative theory with unbounded energy of free model, we find the class of stable interactions with complex scalar field. The non-abelian generalization of the proposed interactions are discussed.
The article is organized as follows. In the next section, we consider the conservation laws of the free PU model with multiple frequency. We construct a two parameter series of integrals of motion which includes the canonical energy.
The general representative in the series can be interpreted as the energy of a certain model without higher derivatives.
Proceeding from this interpretation, we construct the class of stable interactions that correspond to the (not necessary Lagrangian) theory with higher derivatives. In section 3, we observe that the theories with bounded energy are stable in certain sense. We identify all the equilibrium positions of theory. The equilibrium position at the origin is unstable, while the additional equilibrium positions can be stable. The motions are bi-harmonic oscillations in the vicinity of the stable equilibrium positions. The oscillation frequencies are different. The estimate for the upper limit of energy is given such that leads to bounded trajectories. In section 4, we consider the issue of stability from the viewpoint of Hamiltonian formalism. It is shown that the interacting theory admits the Hamiltonian formulation with a bounded from below Hamiltonian. In section 5, we consider the field theoretical example -the Podolsky electrodynamics. In section 6, the further generalizations of the proposed constructions are discussed. In conclusion, we summarize the results.

The PU oscillator with a multiple frequency
Consider the mechanical model with two coordinates x(t), y(t) and the action functional The action includes dimensionless parameter k, while m is a mass, and ω is a frequency. The Lagrange equations read These equations can be equivalently written as Obviously, the parameter k drops out of the equations, while it cannot be excluded from the action by adding any total derivative to the Lagrangian. In this sense, the equations (3) admit a one-parameter series of Lagrangians (1). This simple observation has consequences: given the series of actions, any symmetry of the equations leads to the series of the conserved quantities constructed by Noether theorem. This series is important for inclusion of stable interactions.
The linear equations (3) obviously describe the resonating oscillations: y harmonically oscillates with the frequency ω, while the oscillator equation for x includes the same frequency as in the solution for y. The latter is included as the " force" on the r.h.s. of the equation for x So, the motion in the direction of x is unbounded at free level. A different interpretation is possible for the same system (3): y can be considered as an auxiliary variable because it can be expressed in terms of x andẍ: The symbol ≈ means the equality on the mass shell. Once the auxiliary variable y is excluded by means of (4), the remaining coordinate x obeys a single fourth order equation of the derived type It is the PU oscillator [1] with resonance. Equation (5) in itself comes from the action principle with higher derivatives, As we see, the model (1) with any k is equivalent to (5) at the level of equations of motion. The solutions, symmetries and conserved quantities are in one-to-one correspondence for the models (1) and (6). With this regard, one can consider (1) and (6) as two different representations of the same dynamics. The formula (4) expresses y in terms of the derivatives of x,ẍ, whileẏ admits representation in terms ofẋ, ... x: The substitution of y in terms of x andẍ takes equations (2) to (5), and vice versa.
We shall seek for the stable interactions such that the equivalence still exists between the higher derivative representation and the first order form of this dynamics at the level of equations of motion. We shall not require, however, the higher derivative equations to remain Lagrangian upon inclusion of interactions. The key role for our construction is played by the series of conserved quantities parameterized by k involved (1). All these conservation laws are generated by the same symmetry, as we have already mentioned.
The action functional (1), being essentially k-dependent, is invariant with respect to the time translations. The canonical energy of the model is the conserved quantity, This expression represents the energy in terms of the coordinates x, y. It explicitly depends on the parameter k. Given the connection between y and x (4), the conserved quantity can be represented in terms of coordinate x and its higher derivatives As is seen, the constant k is involved in the integral of motion, while the equation of motion (5) is independent of k. This means that k-dependent and k-independent terms conserve separately, i.e.
Equations (8), (9) determine a two-parameter series of the conserved quantities of the PU theory with resonance. Each representative in the series can be considered as the canonical energy of a certain representative of the action functional series (1). This interpretation is not generally true at the higher derivative formalism (5). The canonical energy of the higher derivative theory (6) corresponds to the first entry in the series (9).
The integrals of motion (10) do not result in the stability of the dynamics. The series (8) involves the dynamical variable x in a linear way. Once x, y andẋ,ẏ admit independent initial data, the corresponding conserved quantity is obviously unbounded. As the classical energy does not have a minimum, one can expect the unbounded spectrum of Hamiltonian at quantum level, and the "Ostrogradski ghosts" emerge. Another reason is that the conservation law (8) does not restrict the motion to the vicinity of the critical point x = y = 0. The general solution of the equations (2) read where A, B and a, b are the arbitrary integration constants. The amplitude of the first oscillation (11) is linearly growing with time. The motion is finite only for the special solutions with A = 0. The substitution of the general solution (11) into (8) gives the expression for the conserved quantity in terms of initial data: For any fixed value of the parameter k, the condition J = const does not restrict the amplitudes of oscillations. The motion is unbounded for any value of J. The latter fact explains the instability of model (5) from the viewpoint of structure of its classical paths. Obviously, the absence of lower bound of the general representative of the energy series (8) is a source of the instability. However, the stabilization of dynamics is possible upon inclusion of interaction if the conserved quantity is bounded from below in a vicinity of the critical point.
At the end of this section, we would like to make a remark on the role of the parameter k in the series of action functionals (1) and integrals of motion (8). Obviously, that the different representatives in these series are connected by the coordinate transformation, where s is the real constant parameter. The transformation with s = −k brings k to zero in a general representative in (1), (8). It terms of the higher derivative description, this transformation does not correspond to any local change of the coordinate x. Instead, we have a non-local transformation which is not invertible in the class of local coordinate changes of the coordinate x. In particular, the variational principle (1) with k = 0 does not correspond to any variational principle with higher derivatives and single dynamical variable x.
This fact has several consequences at the non-linear level and in the Hamiltonian formalism, which are discussed in the next two sections.

Stable interactions
In this section, we consider the inclusion of interaction in the Lagrangian (1) such that the coordinate y can be still excluded on shell in terms of x,ẋ,ẍ, though not necessarily in the same way as it has been done in the free theory (4). This would mean that the interactions in the first order theory (1) correspond to the local interactions in the higher derivative equation (5), though the interaction vertex is not necessarily Lagrangian in the higher derivative setup.
The simplest option is to consider the class of interactions in the model (1) where the coordinate y is expressed from the same combination of the Lagrangian equations as in the free theory (4). In the slightly different wording, this means that at the interacting level, the same linear combination of Lagrangian equations defines y as function of x,ẋ,ẍ: where f (x,ẋ,ẍ) is a function of the coordinate x and its derivatives. This function can be nonlinear, unlike the free model (4). The most general Lagrangian that has the property (15) reads where U (x,ẋ), V (z,ż) are the functions of the variables x, z ≡ x + ky and its first time derivativesẋ,ż. The interaction potentials U , V are assumed at least cubic in the variables. It terms of the potentials, the coordinate y is defined on shell as follows The function V (x,ẋ) does not contribute to this equation. The higher derivative equation of motion for a single coordinate x has the following form: where the function f (x,ẋ,ẍ) denotes the right hand side of equation (17). The obtained equation is the obvious deformation of the free PU theory equation (5). The class of interacting theories (16) is not invariant with respect to the transformation (13). This automatically means that the different representatives of the free action functional series (1) give rise to different classes of interacting theories, which are unconnected to each other by any change of coordinates.
Only one representative of the integral of motion series (8) conserves at the interacting level. It is given by the canonical energy of the action functional (16), This quantity admits representation in two equivalent forms. First, the quadratic form of this expression can be brought to the diagonal form. In so doing, we assume 2 that k(k − 1) = 0 Second, the coordinate y can be expressed in terms of x and its derivatives by means of equation (17). In this way, we arrive to the conservation law of the higher derivative theory (18), As is seen from equations (20) and (21), the unbounded contributions come from the terms with squares of x andẋ. The integral of motion of the interacting theory is bounded if the contributions from the interaction potentials U , V compensate the unbounded terms. The simplest example of interactions that meet the stability condition reads where the constants α, β, γ represent the interaction parameters.
In this article, we do not explore the dynamics of non-linear model (18) with the most general potentials U, V such that lead to stability. For the sake of technical simplicity, we mostly focus on the particular class of interaction potentials (22). As we observe below, this class is wide enough to illustrate the general properties of the dynamics at the non-linear level. Given the specific potentials, equation of motion (18) reads The integral of motion of the non-linear theory (23) is given by the expression (21) with the interaction potential (22). 2 The assumption does not restrict generality. As we demonstrate below, only interactions with k > 1 can be stable.
From the viewpoint of stability, the presence or absence of lower bound of energy is relevant. The first two lines of equation (24)  Depending on the structure of the interaction term in the integral of motion, the non-linear theory can be stable or unstable. If the level surfaces of constant energy are bounded in the phase-space, the motions are finite for all the initial data. It is the case of globally stable theory. The globally stable interactions are admitted by the PU oscillator with nondegenerate frequency spectrum [24,27]. For examples of stable interactions in the field theory we refer articles [2,25,26].
There is no analogous way to include the globally stable interaction for the PU oscillator with a multiple frequency.
Under the less restrictive assumptions, the energy can be an unbounded function which admits a local minimum. Then, the motions are stable in the vicinity of the energy minimum. This is a case of so-called stability island. The models with stability island are stable in the range of energies below certain limit. The highest value of energy with bounded isoenergetic surface determines the upper energy limit for the stable paths. In principle, the existence of a stability island is sufficient for construction of quantum theory with quasi-stable states and well-defined vacuum state. The precedents are known of this type for the higher derivative systems [8,18], though not with a resonance at free level. We see that the concept of an island of stability also suits well for the dynamics of interacting theory (23). The free model (5) is unstable.
To get the stability in the interacting theory, the model should have a (local) minimum of energy due to interaction. The motions are bounded in the vicinity of the stable equilibrium position just by virtue of the energy conservation law.
Depending on the value of the interaction parameters α, β, γ, the theory (23) may have one or three stationary solutions. The case γ(k − 1) ≥ 0 is not interesting because the model (23) has a single unstable equilibrium position at the origin. If γ(k − 1) > 0, we have two nonzero stationary solutions: They can be stable or unstable depending on the values of the interaction parameters. Introduce the special notation for the fluctuation in the vicinity of the equilibrium position, 8 The decomposition of the integral of motion (24) in the vicinity of nonzero stationary solution reads The dots denote cubic and higher terms in u and its derivatives. The first term defines the value of energy at the stationary solution. We introduce the special notation for this value, This value is negative for γ > 0, k > 1. The stability properties of the equilibrium position are determined by signature of the quadratic form in the decomposition (27). The energy has minimum if all the coefficients at squares are positive, Hereinafter, it is assumed that the stability conditions are satisfied. In particular, we suppose below that β, γ > 0. The parameter α is not involved in the stability conditions because theẋ 4 term cannot influence the motions in the vicinity of the equilibrium position. However, this term influences the stability properties of the model at higher energies.
The model (23) On this phase-space surface, the conditions of existence and uniqueness of solution to the Cauchy problem for equation (23) are violated. Our analysis shows that relations (23) and (30) are inconsistent. This means that neither true trajectory can be transverse to this surface, no the classical path can lie on the phase-space submanifold (30). The only alternative is that true trajectories begin or end in the vicinity of the surface (30). In the globally stable theories, the classical path are complete. 3 It is not possible for the system (23) as we see. The complete trajectories can exist if the certain level energy surface defined by (24) has no intersection with the phase-space submanifold (30). The stationary solutions (25) are examples of complete paths. They do not lie on the surface (30) because conditions (25) and (30)  Consider the dynamics of small fluctuations in the vicinity of the nonzero stationary solution (25). The linearization of equation (18) in the vicinity of solution (25) reads where u is the fluctuation. By construction, the equation of motion has the PU form. The frequencies of oscillations are determined by the interaction parameters and the constant k. The stability conditions (29) imply that all the coefficients of equation are positive, so the roots of characteristic equations are complex. The frequencies of oscillations for the model (31) read The conditions (29) imply that the frequencies are not equal, so the system (31) has no resonance at the equilibrium position. A simplest way to see the fact is follows. The equations (18) admit alternative formulation without higher derivatives (16). The resonance has place if both the oscillators one and the same frequency. This option is not possible because the system of two free oscillators with one and the same frequency does not allow reformulation in terms of higher derivative PU theory with a single dynamical variable. The solution to the equation of motion is bi-harmonic oscillation, where A, B, a, b are integration constants. The bi-harmonic oscillation is a finite path, so the dynamics should be considered as stable. The account of interaction does not change conclusion of about the stability of motion. The argument is that the dynamics is localized on zero energy surface, which is compact for the energies slightly above the lower bound.
All this means that the dynamics of non-linear theory is stable in the vicinity of equilibrium position.
Let us now specify the stability island. The classical path is singular if the surfaces (24) and (30) are intersect. For regular path conditions (24) and (30) are inconsistent. The regularity condition is met in the range of energies where J min is the minimal value of energy (28), and J max is the minimal value of energy on the singular surface (30).
Expressing the coordinate x on the surface (30), we represent the energy (24) as the function of three variables, The domain of a function is |ẋ| < 1/3kα for α > 0, and all the phase space for α ≤ 0. Otherwise the condition (30) is inconsistent. The arguments in squares in two first lines of (35) are independent initial data, which account the dependence of the energy onẍ, ...
x. The value of these terms can be set zero irrespectively to the x,ẋ terms of third line.
The actual minimum of the function is given by minimum of the quadratic form inẋ 2 in the third line of the expression (35). Three different cases are summarized in the equation below,  (36)). The size of the stability island decreases for negative α, which contributes a negative correction to the kinetic term, and also for small positive α. The latter decrease of the energy limit may seem contrintuitive. We explain it by almost degeneracy of the energy quadratic form (27) at the equilibrium position (the coefficient atu is small). In this case, every small change of the model parameters can have a negative impact on stability.

Hamiltonian formalism
Let us begin with the free PU theory. At first, consider the canonical Hamiltonian formalism for the theory with the action functional (1). The action involves the parameter k, which is not involved in the equivalent higher derivative theory (6), so we aim at clarifying the consequences of this ambiguity in Hamilitonian formalism.
Introduce the canonical momenta p x , p y obeying the canonical The equivalence between the Hamiltonian equations (39), (40) and higher derivative PU theory (5) is easy to see: The momenta p x , p y and auxiliary variable y can be expressed in terms of the coordinate x and its derivatives. The remaining dynamical variable x obeys the higher derivative PU equation (5).
The Hamiltonians (39) depend on the parameter k. This dependence is due to the canonical transformation, where s is the transformation parameter, being real number. The Ostrogradski canonical formulation [28] The Hamiltonian (40) is given by the general representative of the conserved series (9), The Poisson brackets of the phase-space variables x,ẋ,ẍ, ... Let us now focus on the Hamiltonian formulation for the higher derivative equation with interaction (22). Even though the interaction vertices are non-Lagrangian, the equivalent lower formulation admits the action principle (16). So, one can proceed from the action (16). Introduce the canonical momenta Form these equations the velocityẏ can be expressed in terms of p x , p y andẋ, whileẋ is determined as a solution of The Hamiltonian is singular at the surface (30). So, the Hamiltonian dynamics is not smooth in the vicinity of the singular surface even in the case α = 0.
The dynamical variables are fluctuations in the vicinity of nonzero equilibrium position (25), The generalized momentau,v are expressed as follows: The Hamiltonian of the model reads The Hamiltonian is bounded from below if the conditions (29) are met. This means that the theory of small fluctuations is stable at the classical and quantum level.
The canonical coordinates v, p u , p v are expressed in terms the derivatives of the fluctuation u and its derivatives up to the third order, v = 1 The inverse transformation reads (expression foru see in (51)) In terms of derivatives of u, the Hamiltonian (52) takes the form of integral of motion (27). The Poisson bracket reads The coordinate u and velocityu are inevitably have nonzero Poisson bracket for stable interactions. This automatically means that such Hamiltonian does not follow from the Ostrogradki procedure. This is not surprising because the Ostrogradski Hamiltonian of a non-singular higher derivative theory is not bounded from below, while the function (52) is bounded.
The example of the higher derivative PU theory (23) with the multiple frequency tells us that the stable vertices are possible, though they are not necessarily Lagrangian. One more conclusion is that the higher derivative equations with non-Lagrangian interactions can admit Hamiltonian formulation, though it is inequivalent to any Ostrogradski formalism.
If the quantization of the theory of fluctuations (31) is constructed by the means Hamiltonian formulation (50), (52), the classical stubility will persits at quantum level. Being equivalent to the system of two oscillators, this model has the usual equidistant spectrum of energy, and it admits the well defined vacuum state. The unharmonic terms can be accounted for by perturbation theory. As the wave functions of stationary states of harmonic oscillator exponentially decreasing at infinity, the perturbation theory is well defined in each order. In principle, this is sufficient for perturbative construction of the stable quantum theory of the non-linear model (23).

A higher derivative field-theory with the resonance
The Podolsky electrodynamics without Maxwell term provides a simplest example gauge field theory with resonance.
The action reads 4 In the general Podolsky model, the action also includes the Maxwell term, so one of the photons is massless, while another one is massive. The mass spectrum of the model (56) is degenerate: both the subrepresentations are massless. To our knowledge, this theory has not been studied in the literature yet. Maybe it does not attract the interest because the representation with degenerate mass spectrum is non-unitary. We view (56) as a toy model that exemplifies the stability issue in higher derivative field theory with the resonance at free level, leaving aside the interpretation of the non-unitary representation.
Similar to the PU model (6), the theory (56) admits an equivalent formulation without higher derivatives. The analog of the lower derivative action (1) reads where G µν = ∂ µ B ν − ∂ ν B µ , and m is the constant with mass dimension. We introduce m for the reasons of convenience.
The dynamical variables are the vector fields A µ (x), B µ (x). The Lagrange equations read The first equations defines the vector field B µ in terms of derivatives of A, B µ ≈ −m −2 ∂ ν F µν . Then, the second equation means that A should obey the "double massless" Podolsky equation ∂ ν F νµ = 0. Obviously, the mass m is an accessory parameter that does not contribute to the equations, much like the parameter k in the first order equivalent (1) of the PU action (6).
Much like the mechanical analogue (8) of the previous section, the first-order theory (57) admits a two-parameter series of conserved tensors. So, the higher derivative equivalent (56) should also admit the series of conserved quantities.
The parameter of the series is the real number k. The canonical energy-momentum tensor of the higher derivative theory (56) is included in the series (59) for k = 0. The general representative of the series (59) is associated with canonical energy-momentum tensor of the theory (57). The energy density is given by the 00-component of the conserved tensor.
The energy density is unbounded because the field A µ (x) is involved into (59) in a linear way. Because of this observation, the theory (57) is unstable at free level. To stabilize the dynamics at the interacting level, the terms with quadratic dependence on the vector field A µ (x) are needed. To make it in an explicitly covariant way we include the complex scalar field. From this perspective, we slightly deviate from the pattern of the mechanical model considered in the previous section, where no extra degree of freedom is needed for the stabilization at the interacting level.
The theory (57) of the vector fields A µ (x) and B µ (x) admits the following interactions with the complex scalar field ϕ(x), where α, β, γ are coupling constants, and e is electric change. The interaction is consistent 5  The theory (60) corresponds to the model of single higher derivative vector field A µ (x) and scalar field ϕ(x). The field equations read where j µ (ϕ, A) denotes the scalar field charge From these equations, the vector B µ (x) can be expressed on-shell, B µ ≈ −∂ ν (F νµ −k|ϕ| 2 F µν )+kj µ (ϕ, A). Substituting the result into the remaining equations and accounting for ∂ µ j µ ≈ 0, we obtain These equations are non-Lagrangian if k = 0. The conserved tensor of the model (60) is the canonical energy-momentum tensor, i.e.
The conserved tensor of the model (63) is deduced from the expression above by expressing the auxiliary field B µ (x) in terms of the derivatives of A µ (x) on shell.
Consider the issue of stability of the theory (63). The model has a nonzero stationary solution, where θ is the angle of the vacuum, and α, β are interaction parameters. In the vicinity of this solution, the conserved tensor (64) reads: where φ(x) = ϕ(x) − ϕ 0 is the scalar field fluctuation, and B µ = ∂ ν F νµ . The 00-component of the conserved tensor (66) reads where F i = F 0i , G i = G 0i . The summation over the repeated index i, j = 1, . . . , d − 1 is implied. The quadratic form In this range of the coupling parameters α, β, γ, the interacting theory of small fluctuations in the vicinity of stationary solution (65) is stable. We note that for the stable interactions the parameter k should be strictly positive, so the higher derivative theory (63) is inevitably non-Lagrangian at interacting level.
Let us discuss the dynamics of the stable theory (63), (68). The linearization of equations of motion (63) in the vicinity of stationary solution (65) takes the following form In the sector of vector fields, we have the usual Podolsky theory without a resonance. The spectrum of mass of the vector field theory includes the massless state, and the massive state with the mass m β/α. This theory is stable and unitary at free level, see in [21]. The complex scalar field is decomposed into two real components φ + = e −iθ φ + e iθ φ * and The scalars enjoy the Klein-Gordon and d'Alembert equations, respectively, The mass of the first field is √ 2αm, while the second one is massless. Summarizing all the above, we conclude that the solutions of the equation (69)  Let us comment on the "Higgs mechanism" of stabilizing the higher derivative dynamics from slightly different point of view, being unrelated to the existence of the resonant solutions. It is a common wisdom that the PU oscillator with non-degenerate frequency spectrum is equivalent to the system of free harmonic oscillators. The canonical Ostrogradski's energy of the PU action includes the energies of these oscillators with the opposite signs. This corresponds to the canonical energy of the first order action, being a combination of harmonic oscillators with the alternating signs. Corresponding Lagrangian admits the interaction vertices such that the corresponding energy has a local minimum, so the dynamics is stable in the vicinity of this shifted equilibrium point. The Lagrangian reads where ω x , ω y , ω z are the frequencies, ω x = ω y , and α, β > 0 are the coupling constants. The variables x, y can be thought of as the degrees of freedom of the original PU oscillator, while z can be be viewed as the "Higgs mode". Upon inclusion of the interaction, the equilibrium position shifts to the point x, y = 0, z = 1/α. If β/α > 1, the energy is positive in the vicinity of the equilibrium position. This indicates the stability in the vicinity of the equilibrium. Once the frequency spectrum is non-degenerate, the PU equations admit globally stable interactions [21] unlike the degenerate case, while the Higgs-like mechanism results in the local stability in both cases.

Higher derivative Yang-Mills theory
In the previous section we considered a mechanism for including the stable interaction in the abelian higher derivative gauge theory. The key tool for doing that is the series of equivalent actions (57) without higher derivatives involving the parameter k which does not contribute to the equations at free level. The action (57) admits inclusion of k-dependent stable interaction (60) such that the auxiliary fields B can be still excluded on shell. This leads to the field equations with stable interactions, though the vertices are not Lagrangian in the higher derivative picture. This mechanism admits, to some extent, a non-abelian extension. At the level of the first order action (60), the non-abelian generalisation is obvious L = 1 2 (G a µν F aµν + B a µ B ν + 1 2 kG a µν G aµν ) + 1 2 (D µ ϕ a D µ ϕ a + m 2 (αϕ a ϕ a − 1 2 β|ϕ a ϕ a | 2 ) + 1 4 γϕ a ϕ a F b µν F bµν . (72) Here, all the scalars and vectors take values in Lie algebra of certain semisimple Lie group of dimension r, a = 1, . . . , r.
The tensor F denotes the Yang-Mills strength tensor of the field A. The tensors G is the field strength of the vector field B, G µν = D µ B ν − D ν B µ . The Yang-Mills covariant derivative is D = ∂ + A. The Lagrangian is invariant with respect to the Yang-Mills gauge symmetry transformation. The vector multiplet A transforms as connection. The vector B and scalar ϕ transform as tensors. The canonical energy is bounded for the model (72) admits a local minimum. The decomposition of Lagrangian in the vicinity of the energy minimum has the same structure as in the abelian case (63).
This theory has same stability conditions (68). The only subtlety is that the auxiliary field B cannot be explicitly expressed from the equations of motion unlike the abelian case. This can be done only by perturbation theory with respect to the Yang-Mills coupling constant. Once the auxiliary field B is expressed from the equations of motion, the non-Lagragian non-abelian higher derivative gauge theory, which is stable and unitary.

Conclusion
In this article, we consider the issue of stability in the class of the higher derivative theories of derived type with a resonance. The wave operator of the theory is the square of another lower-order operator. We see that this class of models admits a two-parameter series of conserved quantities. One of the entries of the series is the canonical energy, and another one is a different integral of motion. All the conserved quantities are unbounded. This structure of conserved quantities is consistent with the instability of free model. To stabilize the dynamics of theory at the non-linear level, the class of interactions is considered such that preserves the selected conserved quantity of the free model. The conserved quantity of the non-linear theory is bounded from below in the vicinity of the equilibrium due to the interaction. Therefore, the fluctuations are stable in vicinity of the equilibrium . The stable interactions are non-Lagrangian in the higher derivative equations, but the dynamics admit the Hamiltonian formulation. The Hamiltonian is defined by the conserved quantity of the interacting theory. Being bounded from below, the Hamiltonian is not canonically equivalent to any (deformation of the) Ostrogradski one.
The general scheme is illustrated by the PU oscillator of fourth order with coinciding frequencies and by Podolsky electrodynamics with zero mass. Both models are unstable at free level, but they can be stabilized by an appropriate interactions is also described in the original set of variables. In the Podolsky theory, the Higgs field, being the charged scalar, is introduced to explicitly preserve gauge invariance. The interacting model is a theory of higher derivative vector field non-minimally coupled to the charged scalar. The Higgs field has nonzero value at the minimum of energy. The theory of small fluctuations in the vicinity of the energy minimum has a non-degenerate spectrum of mass. The model with non-degenerate mass spectrum is shown to be stable and unitary.
The proposed procedure of inclusion of stable interactions seems admitting further applications. It is consistent with the non-abelian gauge symmetry, and it can be used in the theories of PU type without resonance. Among the possible applications, we can mention various higher derivative models of interest, including gravity. The common feature is that these theories are unstable at free and interacting level (except the class of f (R)-gravity models). The introduction of