Multi-Hamiltonian formulations and stability of higher-derivative extensions of $3d$ Chern-Simons

Most general third-order $3d$ linear gauge vector field theory is considered. The field equations involve, besides the mass, two dimensionless constant parameters. The theory admits two-parameter series of conserved tensors with the canonical energy-momentum being a particular representative of the series. For a certain range of the model parameters, the series of conserved tensors include bounded quantities. This makes the dynamics classically stable, though the canonical energy is unbounded in all the instances. The free third-order equations are shown to admit constrained multi-Hamiltonian form with the zero-zero components of conserved tensors playing the roles of corresponding Hamiltonians. The series of Hamiltonians includes the canonical Ostrogradski's one, which is unbounded. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. This means, the theory admits inequivalent quantizations at the free level. Covariant interactions are included with spinor fields such that the higher-derivative dynamics remains stable at interacting level if the bounded conserved quantity exists in the free theory. In the first-order formalism, the interacting theory remains Hamiltonian and therefore it admits quantization, though the vertices are not necessarily Lagrangian in the third-order field equations.


Introduction
Classical dynamics and quantization of various higher-derivative models are discussed once and again for decades.
Among most frequently studied specific models we can mention Pais-Uhlenbeck (PU) oscillator [1], Podolsky and Lee-Wick electrodynamics [2], [3], [4], higher-derivative extensions of the Chern-Simons model [5], higher-derivative Yang-Mills models [6], conformal gravity [7], various higher-derivative higher-spin fields theories [8], [9], [10], modified theories of gravity [11], including critical gravity [12]. The higher-derivative models often reveal remarkable various properties comparing to the counterparts without higher derivatives. In particular, the inclusion of the higher-derivatives improves the convergency in the field theory both at classical and quantum level in many models. Also the conformal symmetry often requires inclusion of higher derivatives in the field equations.
The higher-derivative dynamics are also notorious for the classical and quantum instability. The key point, where the problem can be immediately seen is that the canonical energy is unbounded for general higher-derivative Lagrangian sys-The paper [19] provides a list of examples of higher-derivative systems admitting multiple Lagrange anchors, including the PU oscillator. By the above mentioned reasons, every model on this list has to be a multi-Hamiltonian system. It has been earlier noticed that the free PU oscillator admits alternative Hamiltonian formulations [32], [33]. It has been observed that the series of canonically inequivalent Hamiltonians includes the bounded ones, while the canonical Ostrogradski Hamiltonian is unbounded. Later, the multi-Hamiltonian formulations of PU oscillator have been re-derived and re-interpreted from various viewpoints in [23], [34], [35], [36], [37], [38]. All these observations can be summarized in the statement that the PU oscillator of order 2n admits the n-parameter series of alternative Hamiltonians and associated Poisson brackets. Once the equations of motion admit Hamiltonian formulation with bounded Hamilton functions, the dynamics is stable classically and quantum-mechanically. It is also worse to notice that the PU oscillator equation of motion admits the interaction vertices such that do not spoil the classical stability [19], [39]. These vertices are non-Lagrangian, while the interacting higher-derivative equations, being brought to the first-order formalism, still remain Hamiltonian with positive Hamilton function [24], [23].
In this way, the PU oscillator equation admits inclusion of interactions such that leave the dynamics stable beyond the free level and admit Hamiltonian formulation. Notice that the stability of PU oscillator with the Lagrangian interaction vertices is studied once and again for decades. In some cases, the model admits isles of stability, see e.g. [40], [41], [42], [43], and [44] for the most recent results and review, while it is unstable in general, unlike the case of above mentioned non-Lagrangian interactions.
If the equations of motion admit a Lagrange anchor, the dynamics have to admit a constrained Hamiltonian formulation [31]. With multiple Lagrange anchors, the dynamics should be multi-Hamiltonian. In general, the construction of Hamiltonian formulation for a given Lagrange anchor is implicit [31]. A direct relation between the Lagrange anchor and corresponding Hamiltonian formalism has been established for the PU oscillator in [23]. In [19], [39], the interactions are introduced, being compatible with the Lagrange anchor. The stable interactions are found by means of the factorization method [19] and proper deformation method [39]. These two methods are equivalent [25] in principle, though they apply different techniques. Recently, more examples has become known of stable interaction vertices in various higher-derivative models with unbounded canonical energy at free level. The examples include PU theory [19], [23], [37], [38], Podolsky electrodynamics [19], and higher-derivative extensions of the Chern-Simons theory [22]. The stable interaction vertices are explicitly covariant in all the field theoretical examples, though they do not follow from the least action principle. The existence of a Lagrange anchor, however, implies that these models have to admit the Hamiltonian description at interacting level.
To the best of our knowledge, no explicit example has been known yet of the higher-derivative field theory admitting multi-Hamiltonian formulation. In this work, we construct the multi-Hamiltonian formulation for higher-derivative extensions of Chern-Simons theory. The canonical unbounded Hamiltonian is included into the two-parametric series of admissible Hamilton functions. The series can also include bounded Hamiltonians in some cases. The existence of a bounded Hamiltonian depends on the parameters in the third-order equations. We also demonstrate that the covariant interactions exist such that the higher-derivative theory still admits bounded Hamiltonian, and therefore it remains stable at interacting level if the free model was stable.
We consider the class of theories of the vector field A = A µ dx µ in 3d Minkowski space with the free action functional Here, d is the de-Rham differential, * is the Hodge star operator, m is a dimensional constant, α 0 , α 1 , α 2 , α 3 , . . . are the dimensionless constant real parameters. Depending on the values of the parameters, the action (1) can reproduce various 3d field theories, including the Chern-Simons-Proca theory [45], [46], topologically massive gauge theory [47], [48], Maxwell-Chern-Simons-Proca model [49], [50], Lee-Wick electrodynamics [3], [4] and extended Chern-Simons [5]. The classical stability of the model (1) is considered in the works [22], [51]. It has been found that the model admits multiple conserved tensors being connected with the time translation by the Lagrange anchors. The anchors are the polynomials in the Chern-Simons operator * d. The set of conserved quantities can include bounded ones. This depends on the roots of the characteristic equation Here, z is considered as a formal complex-valued variable, and α k are the parameters of the model (1). As is established in [51], the model (1) admits a bounded conserved tensor and, hence, it is stable iff all the non-zero simple roots of equation (2) are real, while zero root may have the maximal multiplicity 2, and no roots occur with a higher multiplicity.
In this paper, we focus at the model (1) with at maximum third-order derivatives, i.e. the action reads with α 1 , α 2 being two independent dimensionless parameters. This model has been proposed in [5] as the third-order extension of Chern-Simons theory. The model is obviously gauge invariant. We construct the multi-Hamiltonian constrained formalism for this model at free level. For similar reasons, the more general case (1) has to be a multi-Hamiltonian system, with a broader class of admissible Hamiltonians depending on the structure of roots in (2). As the construction of multi-Hamiltonian formalism becomes more cumbersome with growth of the order of derivatives, we do not go beyond the third-order models in this paper. 2 Let us explain what do we understand by constrained multi-Hamiltonian formalism. At first, notice the obvious fact that the higher-derivative field equations can be always reduced to the first-order derivatives in time by introducing extra fields absorbing higher the time derivatives. The first-order equations are said multi-Hamiltonian if there exists k-parametric series of Hamiltonians H(β, ϕ, ∇ϕ, ∇ 2 ϕ, ∇ 3 ϕ, . . .) and Poisson brackets {ϕ a ( x), ϕ b ( y)} β , with β 1 , . . . , β k being constant parameters and ∇ denoting derivatives by space x, such that the equations constitute constrained Hamiltonian system with The rhs of equations (4) does not depend on the parameters β, while both the total Hamiltonian H T (β) and the Poisson bracket do. In the other wording, changing values of parameters β, we simultaneously change Hamiltonian H T (β) and Poisson brackets {·, ·} β in such a way that the equations of motion (4) remain intact.
Any higher-derivative Lagrangian field theory always admits at least one Hamiltonian formulation which can be con- 2 The multi-Hamiltonian formulation for the gauge-invariant extension of the Chern-Simons model of the fourth order have been constructed in [52].
structed by the Ostrogradski method in the unconstrained case, and by various generalizations [27], [28], [29], [30] developed for the constrained systems. In this paper, we develop the Hamiltonian formalism of higher-derivative field theory in several respects by the example of the model (3). At first, the third-order extension of the Chern-Simons model (3) is shown to admit a two-parameter series of constrained Hamiltonian formulations. The Hamiltonians from this series can be bounded from below in some cases, depending upon parameters α 1 , α 2 , even though Ostrogradski's Hamiltonian of the model is unbounded in all the instances. The second is that the free higher-derivative equations of this model admit inclusion of covariant interactions which do not break the stability if the theory have bounded conserved quantity at free level. Furthermore, the stable theory admits constrained Hamiltonian formulation at interacting level with a bounded Hamilton function.
Let us also remark that the multi-Hamiltonian formulation helps to resolve the discrepancy between classical stability of higher-derivative dynamics and quantum instability which is connected to the unboundedness of canonical Hamiltonian. As it is noticed in [22], the theory will be quantum-mechanically stable, as it is at the classical level.
Let us make some comments on the interactions which do not break the stability of the higher-derivative theory. An example of stable couplings in the model (1) has been noticed in [22] in the case involving massive Proca term, so it is the theory without gauge symmetry. In the present paper, we consider the gauge model (3) and introduce gauge-invariant interaction with spinors. This class of interactions can be viewed as a generalization to the non-minimal stable couplings of d = 4 Podolsky electrodynamics to the spinor matter proposed in ref. [19].
The article is organized as follows. In Section 2, we describe conserved tensors of the third-order model (3). We also relate the existence of bounded conserved tensors with the structure of the corresponding Poincaré group representation. In doing that, we mostly follow the general prescriptions of [22] and [51]. The section is self-contained, however. In Section 3, the multi-Hamiltonian formulation is constructed with the Hamiltonians defined by the conserved tensors of the Section 2.
In Section 4, we introduce the interactions with spin 1/2 such that do not break the stability of higher-derivative theory if the theory is stable at free level. After that, we demonstrate that the higher-derivative interacting theory still admits Hamiltonian formulation in all the instances, even if the vertices are not Lagrangian.

Conserved tensors
For the action (3), the Lagrange equations read The third-order time derivatives are involved in these equations. That is why, the conserved quantities can involve the secondorder time derivatives.
The equations (6) correspond to reducible representation of the Poincaré group. Specifics of the representation depend on the constants α 1 , α 2 . Different cases are distinguished by the structure of roots in the characteristic equation associated to the field equations (6). Here, z is a formal unknown variable, and α 1 , α 2 are the parameters of the model. There are the following different cases distinguished by the structure of roots for the variable z: two simple real nonzero roots, and one simple zero root; (B) α 1 = 0 , α 2 = 0, one simple real nonzero root, and one zero root of multiplicity two; (C) α 1 = 0 , α 2 2 − 4α 1 = 0, one real nonzero root of multiplicity two, and one simple zero root; (D) α 1 = 0 , α 2 = 0, one zero root of multiplicity three; (E) α 1 = 0 , α 2 2 − 4α 1 < 0, two simple complex conjugate roots, and one simple zero root.
In cases A and B, the representation is unitary and reducible. In case A, the representation is decomposed into two irreducible sub-representations. Each one corresponds to a self-dual massive spin 1, while the masses can be different. In case B, the set of sub-representations includes a massless spin 1 and a massive spin 1 subject to a self-duality condition. Cases C and D correspond to reducible indecomposable non-unitary representations. These two options are distinguished by different multiplicity of the multiple real root in eq. (7). In case E, the representation is irreducible and non-unitary. So, one can see that the field equations (6) can describe either unitary or non-unitary representations of the 3d Poincaré group depending on the relations between the parameters α 1 , α 2 .
The third-order field equations (6) admit two-parameter series of on-shell conserved second-rank tensors where β 1 , β 2 are the real constant parameters, and (T a ) µν , a = 1, 2 read Here we use the notation 3 Tensor T 1 is a canonical energy-momentum for the action (3), while T 2 is another independent conserved quantity. As F and G are gauge invariant quantities, the tensor (9) is gauge invariant with any β. Also notice that F i , G i , i = 1, 2 define independent unconstrained Cauchy data for the field equations (6). Once T 1 is linear in G, it is unbounded anyway. The general entry of the series (9) is bilinear in both G and F . So, T (β) can be bounded, in principle, if β 2 = 0.
The conserved tensors of the series (9) are connected to the invariance of the model with respect to the space-time 3 The Minkowski metric is taken with mostly negative signature.
translations if the parameters meet the condition This connection can be traced by the Lagrange anchor method along the same lines as in the paper [51]. From this perspective, any representative of the series (9) satisfying condition (12) can be viewed as energy-momentum.
The 00-component of the conserved tensor T (β 1 , β 2 ) from the series (9) can be bounded or unbounded from below depending of the parameters α involved in the equations (6) and on specific values of β. Once the representation is unitary (that corresponds to the cases A,B in classification (8) Also notice that the canonical energy (T 1 ) 00 is always unbounded.
The conserved tensors are defined modulo on-shell vanishing terms. So, we have equivalence classes of conserved quantities which coincide on-shell, being off-shell different. The choice of specific representative of the equivalence class is a natural ambiguity in the definition of conserved quantity. We mention this ambiguity because it has a natural counterpart in the Hamiltonian formalism considered in the next section. As far as the linear equations (3) admit bilinear gauge invariant conserved tensors (9), it is natural to consider the series up to quadratic on-shell vanishing terms. The most general gaugeinvariant bilinear and symmetric representative in the equivalence class of T µν (β 1 , β 2 ) (9) reads Two real parameters β 3 , β 4 label different representatives of the same equivalence class of conserved tensors, while β 1 , β 2 determine the equivalence class of conserved tensor as such. Only one of two constants β 3 , β 4 is independent. The other one can be absorbed by the multiplication of the equations of motion by the constant overall factor.
In the next section, we construct a multi-Hamiltonian formulation where 00-components of the conserved tensors T µν (β 1 , β 2 , β 3 , β 4 ) (13) serve as Hamiltonians, and all the values of the parameters β 1 and β 2 , being subject to condition (12), are admissible. We consider all the cases in a uniform way, be the Hamiltonian bounded or not.

Multi-Hamiltonian formulation
The multi-Hamiltonian formalism is constructed for the equations (6) with ε ij = ε 0ij being the 2d Levi-Civita symbol. Substituting these variables into (6), we arrive at the following first-order equations in terms of the fields A µ , F i , G i : In terms of fields A, F, G, the evolutionary equations (15)  In the first-order formalism, the equations are invariant under the gauge transformation where ξ is the gauge transformation parameter, being arbitrary function of x. In what follows, it is natural to consider the field A 0 as the Lagrange multiplier associated to the constraint (16). This interpretation is consistent with the gauge transformation (17) which includes the time derivative of the gauge parameter, as it should be for Lagrange multiplier in the constrained Hamiltonian formalism.
In the first-order formalism, the zero-zero component of the conserved tensor (13) reads We treat this quantity as the series of on-shell Hamiltonians parameterized by constants α, β. Off-shell, the Hamiltonian can be a sum of (18) and constraints. We chose the following ansatz for the total Hamiltonian: where β 1 , β 2 , γ are constant parameters. On account of the constraint (16), the quantities (18) and (19) coincide on shell.
The parameter γ is introduced to control the inclusion of the constraint term into the Hamiltonian 4 . The admissible values of the parameters β and γ subject to conditions Here, the first condition implies that the conserved quantity (18) is connected to the invariance of the model (6) (18), (19) and the constraint (16).
Given the series of Hamiltonians (18), (19) and the r.h.s. of the equations (15), we arrive at the system of linear algebraic equations defining the series of Poisson brackets {·, ·} β,γ : The Poisson bracket, being defined by these equations, involves five independent parameters α 1 , α 2 , β 1 , β 2 , γ. The bracket eventually reads The accessory parameter γ controls the constraint terms in the total Hamiltonian (19). As is seen, the same parameter defines the Poisson bracket between the components A i of gauge potential. This parameter does not contribute to the Poisson brackets between the physical observables, being the functions of the gauge-invariant quantities F i , G i , and the strength ε ij ∂ i A j . That is why, γ can be considered as an accessory parameter. Inclusion of γ-terms into total Hamiltonian and brackets allows us to literally reproduce in Hamiltonian form the first-order dynamical equations (15) for all the quantities, be they gauge-invariant or not.
Let us make one more comment on the meaning of the accessory parameter γ which defines the bracket between A i and does not affect on the brackets of gauge-invariant quantities. Notice that the Poisson brackets in gauge theory have the inherent ambiguities. The general study of these ambiguities can be found in ref. [53]. In context of the bracket (22), one of these ambiguities turns out relevant. It is related to the option of redefining the Poisson bracket by adding the bi-vector, being the wedge product of gauge symmetry generator operator to another vector. This redefinition does not affect the brackets between gauge-invariant observables, while it can alter the brackets of non-gauge-invariant quantities. The bracket (22) involves the ambiguous terms of this type, and it is the ambiguity which is controlled by the accessory parameter γ.
The problem of identification of ambiguous terms in the Poisson bracket is a subtle issue. The Poisson bracket (22) is ultralocal between components of A i with no derivatives involved, while the generator of the gauge symmetry for A i (17) involves a derivative. Thus, the ambiguous terms in the Poisson bracket cannot be absorbed by adding the wedge product of the gauge symmetry generator, being a derivative, to another vector, being a polynomial in the partial derivatives ∂ i . The problem is solved by including the inverse Laplace operator ∆ −1 = (∂ i ∂ i ) −1 into the coefficient at the gauge generator.
The space non-locality of this type is usually considered as admissible for the constrained Hamiltonian formalism in the field theory 5 . To represent the bracket (22) between the components of A i in terms of gauge generators, we use the following identical representation for the 2d Levi-Civita tensor ε ij : Substituting ε ij from this relation into rhs of the Poisson bracket for the potential components, we rewrite the bracket in the Here, all the partial derivatives act on argument x in the delta-function. Once the operator ∂ i is a gauge generator for the field A i , the vector V j (γ) parametrizes the ambiguity in the Poisson bracket. Thus, we treat the parameter γ as inherent ambiguity of Poisson bracket in the gauge theory outlined in ref. [53].
Let us summarize all the aspects related to the ambiguity in parametrization of the multi-Hamiltonian formulation of the equations (6). The Hamiltonian and brackets (19), (22) involve 5 parameters. Two of them, α 1 and α 2 , define the original equations (6). The constants β 1 , β 2 parameterize the series of conserved tensors tensors (9). These tensors admit gauge- In this case, one and the same parameter has to control the ambiguity in the Hamiltonian and Poisson bracket. It is the parameter γ. In the free theory, γ can be set to an arbitrary value. This corresponds to the choice of the representative in the equivalence class in the series of Hamiltonian formulations with the Hamiltonian (18), (19) and Poisson bracket (22). Thus, γ is an accessory parameter in the series of Hamiltonian formulations unless the interaction is introduced. We keep γ in the Hamiltonian formulation throughout this section to have the contact with Section 4, where the couplings are introduced with spinors. As we will see, this parameter becomes essential for inclusion of consistent interactions in the non-linear model.
The Hamiltonians in the series (19) can be bounded or unbounded form below. The Hamiltonian H T (β, γ) is on-shell bounded if the parameters meet the conditions In cases A,B in classification (8) (19) with For every β, γ, the Poisson bracket (22) is a non-degenerate tensor, so it has an inverse, being a symplectic two-form. The latter defines the series of Hamiltonian action functionals where H T (β, γ) denotes the total Hamiltonian (19).
For β 2 = 0, we get the non-canonical Hamiltonian actions that still result to the same original equations (6). Different actions in the series (26) are not connected by a canonical transformation. This is obvious because the Hamiltonian in the series (19) can be bounded from below, while the canonical Hamiltonian (27) is always unbounded.
The Poincaré invariance can be questioned of the non-canonical Hamiltonian actions (26), and hence the covariance of the corresponding quantum theory may seem in question. We do not elaborate on this issue here, while we claim that the quantum theory associated to any model in the series (26) is Poincaré-invariant. The argument is that the original higher-derivative theory admits the series of covariant Lagrange anchors [22]. It is the series of anchors which underlies the multi-Hamiltonian formulation (26). One more reason is provided by the fact that every Hamiltonian in the series (18) is 00-component of the second rank tensor (13). All the entries of the series transform in the same way, including Ostrogradski's Hamiltonian.
As we have seen above, the higher-derivative extensions of the Chern-Simons theory admit multi-Hamiltonian formulations. In some cases, the Hamiltonians are bounded. In this section, we provide an example of coupling to spinors such that the theory still has bounded Hamiltonian and therefore it remains stable at interacting level.
In [19], the stable interaction is included for the higher-derivative Podolsky's electrodynamics in the dimension d = 4. The stable interaction is non-Lagrangian in d = 4, while the Hamiltonian formalism is not considered there. So, the possibility could be questioned of the canonical quantization of the interacting model even without gauge invariance. The three-dimensional model admits more options than its four-dimensional counterpart, because (due to the presence of the Chern-Simons term) it can describe a variety of reducible representations of the 3d Poincaré group. Below we introduce the interaction mostly following the lines of [24] with regard to the d = 3 specifics, and then we construct the Hamiltonian formalism for the interacting theory.
Let us introduce coupling of the vector field A and 2-component spinor field ψ a , a = 1, 2 (ψ a stands for conjugate spinor) by imposing the following non-linear field equations Here, J µ = eψγ µ ψ is the current of the spinor field, γ's are the 3d gamma matrices, and D is the covariant derivative, The spinors ψ, ψ are Grassmann odd fields. The spinor field ψ and its conjugate ψ are considered as independent variables.
The real constants g 1 , g 2 , g 3 are dimensionless parameters of interaction. The parameter e is a coupling constant.
In general, the interaction vertices are non-Lagrangian in the equations (28). The Lagrangian case corresponds to g 1 = 0, g 2 = g 3 = 0 in (29). As we shall see, the Lagrangian model is unstable, while the stability can be retained by admitting non-Lagrangian higher-derivative contributions to the interaction, i.e. by g 2 = 0, g 3 = 0. As we shall demonstrate in this section, with non-Lagrangian stable interactions, the equations (28), (29) still admit constrained Hamiltonian formulation with on-shell bounded Hamiltonian.
The consistency of interaction implies that the gauge transformation (17) is complimented by the standard U (1)transformation for the spinor field The non-linear theory describes propagation of the gauge field A coupled to the spinor ψ in the gauge-invariant way.
The equations (28) admit the second-rank conserved tensor where T µν (β 1 , β 2 ) stands for the conserved tensor (9) of the free theory with the parameters β fixed by the interaction constant in the following way We chose the conserved tensor in the form (31) because its 00-component does not involve time derivatives of the spinor field. Once the time derivatives of the spinor filed are not involved in T 00 (g), the conserved tensor still admits by redefinition on-shell vanishing terms that involve the derivatives of the vector field. The structure of this term is analogous to (13), so we do not write these contributions explicitly.
Upon inclusion of interaction, the deformation is still conserved of a single representative from the series of conserved tensors (9) admitted at free level. The parameters β 1 , β 2 in this conserved tensor are fixed by the interaction constants by the formula (32).
The procedure of construction of the conserved tensor (31) is analogous to that from [19], Sec  (12) and (32). Substituting (32) into (12), we get In what follows, we consider the interactions (28), whose parameters satisfy this condition. By this reason, we consider (31) as the energy-momentum tensor of the non-linear theory (28).
The 00-component of the tensor (31) reads Depending on the vales of the parameters g, this quantity can be bounded or unbounded from below. 6 The necessary and sufficient condition for that follows from (25). It reads In case of minimal interaction g 1 = 1, g 2 = g 3 = 0, the equations of motion (28) are Lagrangian. However, the Lagrangian non-linear theory is unstable because the canonical energy of the model is unbounded. Once the condition (35) is satisfied, the model is stable, while the field equations (28) and (29) are non-Lagrangian.
Let us bring the theory (28) to the form of constrained Hamiltonian dynamics. The first-order formulation for the model (28) is constructed in the same way as in the linear case. The variables A i , F i , G i are introduced by the recipe (14) to absorb 6 With the cubic interaction contribution, the conserved tensor (34) is no longer bounded in the strict sense. By saying 'bounded' we mean that the quadratic contribution in the conserved quantity is bounded. The latter property is interpreted as stability of the theory with respect to small fluctuations of initial data, and it is not considered as obstruction to the stability of the model. For example, the energy-momentum of spinor electrodynamics includes cubic term. the time derivatives of A.For these fields, we get three equations of evolutionary typė Obviously, the first pair of equations in this system have the same form as in (15) Equations (36), (37) are complimented by the equations for the spinors from (28): These equations are of the first order from the outset. In this way, we have the first-order formulation for the model (28) which includes equations (36), (37) and (38).
We chose the following ansatz for the total Hamiltonian: where g 1 , g 2 , g 3 are the parameters. On account of (34), the Hamiltonian describes the same conserved quantity as the 00component of the tensor (31). Substituting (39) into (4), (5), we arrive at the system of linear algebraic equations defining the series of Poisson brackets: These relations should take into account the Grassmann parity of the fields, so it is an even Z 2 -graded Poisson bracket. In particular, the brackets are symmetric of the spinor fields ψ, ψ.
Equations (40) are consistent if the interaction parameters satisfy condition (33). The structure of the Poisson bracket, however, depends on the relations between the interaction parameters g 1 , g 2 , g 3 . Below, we focus on the case g 1 = 0, while the other cases can be treated in a similar way. The Poisson bracket, being defined by equations (40), reads The spinor field ψ and its Dirac conjugate ψ † = ψγ 0 are conjugate w.r.t. to the graded canonical bracket, As is seen from these relations, the Poisson bracket is unique in the non-linear theory (28). No free parameters are involved in the Poisson bracket (41), (42) besides the coupling constants g.
With no arbitrary parameters involved in the Hamiltonian formulation, the non-linear theory (28) is not multi-Hamiltonian anymore, while the free limit admits the two-parameter series of Hamiltonian formulations (19), (22). This means, the interaction preserves one of possible Hamiltonian formulations admitted by the free theory. This fact can be explained in various ways. The most simple explanation is that upon inclusion of interaction, the deformation of the unique entry still conserves of the series of tensors (9). The parameters of the series (31) are fixed by the interaction constants in the non-linear theory. It is the sole conserved tensor which defines the unique Hamiltonian at interacting level, while the corresponding Poisson bracket is fixed by the Hamiltonian.
For every g the Poisson bracket (41) is a non-degenerate tensor, so it has an inverse, being a symplectic two-form. The latter defines the Hamiltonian action functional where H T (g) denotes the total Hamiltonian (39). For the minimal interaction g 1 = 1, g 2 = g 3 = 0, we get the standard Ostrogradski action For non-minimal interactions, we have the Hamiltonian action functional (43). This action is not canonically equivalent to (44). The non-minimal interactions are consistent with the bounded Hamiltonian (39), while the Ostrogradski Hamiltonian, which is associated with the minimal interaction, is unbounded in all the instances.
In this way, we see that the higher-derivative field equations (6) are compatible with inclusion of non-minimal explicitly covariant interactions (28) such that the theory still admits the Hamiltonian formalism with bounded Hamiltonian if the model has a bounded conserved quantity at the free level.

Concluding remarks
Let us summarize and discuss the results. First, we have seen that the third-order extension of the Chern-Simons admits a two-parameter series of conserved tensors. If the equations (6)  Hamiltonians in the series. The Ostrogradski Hamiltonian, being included in the series, is unbounded. Third, we introduce explicitly the Poincaré-covariant and gauge-invariant stable interactions in higher-derivative dynamics. If the free theory has a bounded conserved quantity, it is still conserved at interacting level. After that, we demonstrate that the covariant and stable higher-derivative interacting theory admits the canonical formulation with the bounded Hamiltonian.