Higher derivative extensions of $3d$ Chern-Simons models: conservation laws and stability

We consider the class of higher derivative $3d$ vector field models with the field equation operator being a polynomial of the Chern-Simons operator. For $n$-th order theory of this type, we provide a general receipt for constructing $n$-parameter family of conserved second rank tensors. The family includes the canonical energy-momentum tensor, which is unbounded, while there are bounded conserved tensors that provide classical stability of the system for certain combinations of the parameters in the Lagrangian. We also demonstrate the examples of consistent interactions which are compatible with the requirement of stability.


Introduction
In this paper we consider a class of 1-form field A = A µ dx µ models on 3d Minkowski space with the action where m is a constant with dimension of mass, a 0 , a 1 , a 2 , a 3 , . . . are some real dimensionless coefficients, * is Hodge conjugation, and the signature is (+, −, −). The coefficient a 0 m 2 corresponds to the usual mass term, a 1 m is the Chern-Simons mass, a 2 is a coefficient at Maxwell's Lagrangian, a 3 corresponds to the extended Chern-Simons Lagrangian [1] and the fourth order term appears in the Podolsky's electrodynamics Lagrangian [2]. With appropriate choice of the coefficients a k , k = 0, 1, 2, ..., this action reproduces various known 3d models, including the Chern-Simons-Proca [3,4], Maxwell-Chern-Simons [5,6], Maxwell-Chern-Simons-Proca [7,8] and the other previously studied higher derivative models [9,10].
In any dimension, inclusion of the higher derivative terms results in the unbounded canonical energy, so classical stability becomes the issue. It is also known that the ghost poles can emerge in the propagator once higher derivatives are included in the action.
The specifics of higher order terms in three dimensions is that they can be viewed as derived from the Chern-Simons term by the repeated shift of the field by its strength: A → A + 2m −1 * dA. As a result, the operator of field equations is a polynomial in first order operator W = 2m −1 * dA. This special structure allows us to make some conclusions concerning conservation laws and stability. The observation is that the n-th order theory of the class (1) admits n parametric family of conserved second rank tensors whenever a 0 = 0. Once a 0 = 0 (the theory is gauge invariant in this case), there exists an n − 1 parametric family of conserved tensors.
The canonical energy-momentum is included in the family in every instance. We provide the general receipt for constructing these conservation laws, and related symmetries. The construction in fact applies to any system (of 1 any field A, not necessarily 1-form) with the operator of field equations being a polynomial in another operator where W can be any self-adjoint 1 differential operator, a k are real constants, and a n = 0. We term the models of the type (2) derived from the theory with equations W A = 0. For the case (1), when W = 2m −1 * d, we apply the general procedure to explicitly deduce the conserved tensors for the third order actions of this class. As we see, the bounded conserved tensors are contained in the family, once the polynomial M (2m −1 * d) has only simple roots, or at most one double zero root. In this generic case, the theory is classically stable even though the canonical energy-momentum is unbounded. As we explain, these models can admit certain interactions such that the stability survives at nonlinear level. The case of multiple roots is special. It also admits a family of n conserved tensors, including the canonical energy-momentum, though there are no bounded conserved quantities in this family. As we see, the corresponding representations of the Poincaré group are non-unitary, while in the generic case, the representation decomposes into unitary ones.
The article is organized in the following way. In the next section we describe the general structure of field equations in the higher derivative models that fall into the class of derived theories (2). For the generic derived system of order n we suggest a procedure of constructing n-parametric family of conserved tensors whose structure depends on the coefficients a k in the field equations (2). In Section 2, we explicitly construct the families of conserved tensors for the theory (1) involving terms up to third order. As we see, four different cases are possible from the viewpoint of existence the bounded representative in the family of conserved quantities.
These cases are distinguished by the structure of roots in the polynomial (2). Once the positive conserved quantity exists, the theory is stable at classical level, even though the canonical energy is unbounded. In Section 3, we demonstrate the example of the self-interaction such that the nonlinear theory remains stable. In conclusion, we summarize the results and comment on stability of the theory (1) at quantum level.

Derived theories, higher symmetries and conservation laws
In this section, we consider the field equations of general structure (2). We demonstrate that combining the space-time translations with the powers of operator W , one can construct non-trivial higher order symmetries and find related conserved tensors. The construction is quite general, it applies to any system of the form (2).
The explicit details for the extension Chern-Simons theory (1) are provided in Section 2.
1.1. Derived theories. Consider a set of fields A J (x) on d−dimensional Minkowski space with local coordinates x µ . The multi-index J accommodates all the tensor, spinor, isotopic indices labeling the field components.
Here, we suppose that the theory admits appropriate constant metrics that can be used to rise and lower the multi-indices. In this setting, any local linear system of field equations can be represented in the following form: where M IJ (∂) is a square matrix whose entries are polynomials in the formal variables ∂ µ . If ∂ µ are understood as the partial derivatives in Minkowski space coordinates x µ , (3) will be a linear PDE system. The formal adjoint to the operator M is defined by The field equations (3) are variational whenever M = M † , in which case the action reads Let us further suppose that the self-adjoint linear differential operator W IJ (∂) exists (Cf. (2)) such that the operator of field equations is polynomial in W : The real numbers λ i and complex conjugate numbers ω j , ω j are the roots of the polynomial M (W ) with multiplicities p i and q j , respectively. The multiplicity of roots is connected with the total degree of the polynomial If W is a differential operator of finite order n W , the order of PDE system (3) will not exceed n × n W .
Once the field equation operator M (∂) is a polynomial of another self-adjoint operator W (∂), we say that the theory is a derived model. In [11], the special case of the factorization (6) was studied, where M has two different simple real roots in W . This simple assumption has far-reaching consequences. In particular, each of the factors defines its own Lagrangian theory whose order is lower than that of the derived theory. Let us mention some of these consequences noticed in [11]. Once the two lower order theories are translation invariant, the derived higher derivative theory has a two-parameter family of independent conserved tensors. This family includes the canonical energy-momentum tensor of the derived theory. The canonical energy is unbounded in general, as it should be in the higher derivative system, while some other conserved quantities can be bounded in this family. The existence of the bounded conserved quantities guaranties the classical stability of dynamics.
As it was demonstrated in the paper [11], every conserved tensor in this family can be connected to translation invariance of the system by appropriate Lagrange anchor 2 . As we will see in this section, for any derived system (2), one can construct n-parameter family of conserved tensors, where n is the order of polynomial M (W ).
We consider two ways of constructing conserved tensors in the derived theories. At first, we make notice that the symmetry algebra of the derived theory includes higher order symmetries generated by the operator W , and translations, once W is translation invariant. Then, we derive the conserved tensors from these symmetries by the Noether theorem. Another option employs the procedure of reducing the order of higher derivative theory (2) by assigning a lower order system to every irreducible factor in decomposition of the polynomial (6). Then, making use of the canonical conserved tensors for the lower order systems, we get the family of the conserved tensors for the original theory (2). Although the Noether theorem provides a uniform way for deducing conservation laws from given symmetries, the conserved tensors obtained from the lower order equivalent system appear in a more convenient form in this case, and we will use them for further analysis of stability.

1.2.
Higher order symmetries and conservation laws. Provided the operator W is translation invariant, the action (5) admits the following symmetry transformations: The space-time translations correspond to k = 0. The higher order transformations with k = n, n + 1, . . .
are equivalent to the lower order ones with account of the equations of motion (3), while for k < n one has independent symmetries. By the Noether theorem one can link the symmetries (7) with the conserved tensors Here, k = 0 corresponds to the usual energy-momentum tensor. There are n independent tensors in the set (8).
1.3. Conservation laws by the reduction of order. Consider the polynomial (6). Denote the cofactors to the real roots λ i and complex roots ω j by Λ i and Ω j , respectively, By definition, the polynomials Λ i (W ) and Ω j (W ) are coprime. Obviously, The notion of the Lagrange anchor was introduced in the work [12] in relation to the path-integral quantization of not necessarily Lagrangian systems. Later it was shown that every Lagrange anchor admitted by the equations of motion maps the conserved quantity to the symmetry of equations [13]. In the paper [11] it was noticed that once the operator M decomposes into two self-adjoint independent factors, the equations admit a two-parameter family of Lagrange anchors such that map any representative of the family of conserved tensors to the space-time translation. In this sense, any of these tensors can be understood as energy-momentum of the theory.
For each cofactor, we introduce the new set of fields, called components. Once the original fields A are subject to the original field equations (3), the components satisfy the lower order derived equations where p i , q j are the multiplicities of the roots λ i , ω j in the operator of the original equations (6).
It is easy to see the one-to-one correspondence between solutions of these equations and the original system (3). The inverse transformation to (10) is established by the relations where the polynomials B i (W ) and C j (W ) can be found by the method of undetermined coefficients. The The last equality is just Bezout's identity for the coprime univariate polynomials Λ i (W ) and Ω j (W ).
Whenever the equivalent formulation (11) is known, the conserved tensors can be obtained by applying the relation (8) separately to every component and then summarizing the results. We denote the conserved tensors for the components by where the indices i, j label the corresponding components (10) while p i , q j are the multiplicities of corresponding roots (6). The conserved tensors of original derived theory are obtained by substitution (10): By construction, There are n conserved tensors (15). The relationship between "new" and "old" conserved tensors is established by comparing their divergences (8) and (16). In particular, the canonical energy-momentum tensor of the derived theory (2) has the following representation: with the coefficients of linear combination being defined by Rel. (13).
Notice that some combinations of the conserved tensors (8) or (15) may be trivial. A conserved tensor is said to be trivial if it is given by the divergence of an antisymmetric tensor modulo equations of motion, i.e., The trivial conserved tensors do not result in any conserved quantity and have to be systematically ignored.
However, we provide the expressions for the conserved tensors modulo divergence terms, but keep the contributions from the equations of motion. Consistency of the computations can then be verified by taking the divergence, see (8) and (16).
As the issue of stability is concerned, the positive conserved tensors are relevant. By positive tensor we mean the one whose 00-component is positive for any solution which is not a pure gauge. We consider the ansatz for the general conserved tensor of the derived theory (3) in the form The ansatz means that we consider the conserved tensors being additive in the contributions from bilinear combinations of Λ i A and Ω j A, where Λ i , Ω j are the cofactors (9) to the real roots λ i and complex roots ω j in the decomposition (6). The quadratic forms (T p i ) and (U q j ) are defined by Rels. (14), (15). In fact, they represent in terms of the original field A the conserved tensors (8) of the component fields ξ i , ζ j subject to equations (11). Here, (T 0 i ), (U 0 j ) are just the energy-momentum tensors for the component fields ξ i , ζ j expressed in terms of A by substitution (10), while p, q > 0 correspond to the higher order symmetries (7) of component fields.
As far as the components (10) are independent, the conserved tensor (17) is positive if and only if so are the In the other words, the derived theory (2) is stable if and only if all the components (10) are stable.
Below, we examine the third order extension of the Chern-Simons theory from the viewpoint of existence of bounded 00-components of conserved tensors we found above.

Conserved tensors in the third order extension of the Chern-Simons theory
The field equations of higher derivative extension of the Chern-Simons model (1) fall into the class of derived theories (2), with W being composition of the Hodge and de Rham operators: The n-th order theory (1)  The case of multiple roots is special because the set of conserved tensors (14) includes a number of terms corresponding to the multiplicity of root. One of the terms corresponds to the energy-momentum tensor of the component, while the others are connected to the higher order symmetries of the components where p i , q j are multiplicities of real and complex roots. Below we will observe that equations do not have positive conserved quantities in the family (19) once they involve tachyon or non-unitary representations (that corresponds to complex, double or higher multiplicity nonzero real or triple or higher multiplicity zero roots).
The models leading to the unitary representations (that corresponds to simple roots, or at most one double zero root in (6)) admit the conserved tensors with bounded 00-component even though the canonical energy is unbounded in all the instances.
The conserved tensors (8) and (15) of higher derivative extension of the Chern-Simons model are given by where the notation is used The expressions for the conserved tensors (19) can be simplified making use of the identity Applying this formula one can express all the conserved tensors in terms of W k A, k = 0, . . . , n − 1.
3 The irreducible massive vector corresponds to the massive representation of the proper Poincaré group. Being subject to the self-duality equation proposed in [3,4], it has one physical polarization. On the generalities of the Poincaré group unitary irreducible representations in d = 3 we refer to [14,15,16].
For a 0 = 0, only n − 1 of n conserved tensors (19) are non-trivial. The trivial conserved tensor reads with λ i = 0. The simplest example of that kind is provided by the energy-momentum tensor for the Chern-Simons theory, where n = 1 , λ 1 = 0 , p 1 = 1.
With account of (21), we consider the following ansatz for the general conserved tensor of the derived theory (3): where p i = p i if λ i = 0 and p i = p i − 1 otherwise. Here, its 00-component is given by the quadratic form in Let us turn to the case when a k = 0 for k > 3 and a 3 = 1. This is the most general case of the third order derived theory. The equations of motion (2) read Below we elaborate on each case separately.
The general solution to the theory (23) is decomposed into three components (10), that satisfy the Chern-Simons-Proca equations Each of the equations describes the massive vector field with the mass m|λ i |. Thus, the third-order theory describes a collection of three massive fields with different masses. At the level of propagator, the decomposition into irreducible components has been noticed in already in the original paper [1], where the third order extension was proposed for the Cher-Simons theory. In this paper we see the decomposition at the level of solutions to the equations of motion and elaborate on conserved tensors. In case of second-order theory, n = 2, the decomposition into components was noticed [17]. The solution (12) to the original theory (23) is reconstructed by the formula The conserved tensors (14) are labelled by the indices i = 1, 2, 3, p = 0 and have the form The sign of the corresponding 00-component coincides with the sign of −λ i , Here, the Euclidean scalar product is used, The conserved tensors (27) can be combined into the tensor In contrast, the 00-component of the canonical energy-momentum tensor with β 0 i = b 0 i is always unbounded because the different components contribute with different signs.
Finally, there is an option when one of the roots is zero. In this case, the corresponding conserved tensor becomes trivial and the positivity of the 00-component of the general conserved tensor (22) is ensured by imposing condition −β 0 i λ i > 0 for the nonzero roots. The 00-component of the canonical energy-momentum tensor is again unbounded.

Cases B and C.
We deduce the explicit expressions for the conserved quantities in the Case C. Corresponding expressions for Case B follow from the ones of the Case C by setting the imaginary part of complex root to zero.
The polynomial (6) has the simple real root λ 1 and the simple complex root ω 1 , i.e., Here, r = 1 and s = 1, so the indices i, j numerating real and complex roots take a single value i = j = 1. The parametrization for the coefficients a 0 , a 1 , a 2 of the polynomial (2) reads The general solution to the theory (23) decomposes into the pair of components (10) that satisfy the first-order and the second-order equations (11), respectively. The equations for the ξ-component correspond to the Chern-Simons-Proca theory [3,4] with mass m|λ 1 | . The ζ−field satisfies the (tachyon) Maxwell-Chern-Simons-Proca equations [7,8]. The solution (12) to the original theory (23) is reconstructed by the formula The conserved tensors (19) of the theory are parameterized by the indices p = 0 and q = 0, 1. The expressions for the tensors have the form The 00-components read The sign of (T 0 1 ) 0 0 coincides with the sign of −λ 1 , see (28). The linear combination of (U 0 1 ) and (U 1 1 ) does give rise to a positive conserved tensor unless ω 1 ω 1 = 0 (Case B, λ 2 = 0). Thus, Cases B and C of theory (23) are unstable unless the decomposition (6) has one simple nonzero root and double zero root. Degrees of freedom of stable theory include one massive and one massless vector mode.
2.3. Case D. The polynomial (6) has the simple real root λ 1 multiplicity 3, i.e., The comparison with (6) brings us to the identification r = 1 and s = 0. In this case, the index i can take a single value i = 1, and p 1 = 3. The parametrization for the coefficients a 0 , a 1 , a 2 of the polynomial (6) reads The general solution to the theory (23) consists of one component. The new variables (10) are not introduced.
The conserved tensors are constructed by the general rule (19) and parameterized by the indices i = 1 and p = 0, 1, 2. The expressions for the tensors have the form where the notation w = W − λ 1 is used.
The 00-components read One can check that the quantities (33) are not combined into a positive tensor. This result also applies to the case λ 1 = 0. The theory with root of multiplicity three has to be considered as unstable anyway.

An example of stable self-interactions
As we have seen, some of the higher derivative extensions of the Chern-Simons theory admit positive conserved tensors at free level. In this section, we provide an example of interaction in the Case A such that the theory still has positive conserved tensor and remains therefore classically stable. The equations of motion read where U (s) can be any scalar function, U ′ (s) = dU(s) ds and β 0 i are treated as the parameters of interactions. The interaction could be constructed by the factorization method of the papers [11,18] that ensures survival of selected conservation law of free theory at interacting level. Here, we do not elaborate on the procedure for constructing the interaction, we just examine consistency and stability of interacting model.
The theory admits the conserved tensor With account of equations of motion it can be rewritten as The conserved tensor is positive if β 0 i λ i < 0 and U > 0, U ′ < 0. The latter property is not satisfied by the polynomial interactions. The admissible choice can be U (s) = π/2 − arctg(s), for example.
The consistent inclusion of interactions should not change the degree of freedom number. The interaction (35) is consistent. This fact can be seen from decomposition of solution into components (24). The equations of motion for the components take the form At the free level, elimination of longitudinal degree of freedom is ensured by the transversality conditions ∂ α (ξ i ) α = 0 , i = 1, 2, 3. In non-linear theory, the transversality conditions are modified but still remain the first-order constraints, The degree of freedom number can be also covariantly computed without depressing the order, e.g. by bringing the original higher derivative equations into the involutive form as is it explained in [19]. Anyway, the equations (38) still describe three degrees of freedom, so the interaction (35) is stable (if U > 0, U ′ < 0) and consistent.
Rare examples are known of stable interactions in the higher derivative systems. The best known example is f (R)-gravity [20,21] where the canonical energy is bounded at linearized level. This exceptional phenomenon happens because the theory is strongly constrained. In the paper [22], the stability of some interactions is demonstrated for the Pais-Uhlenbeck oscillator (whose canonical energy is unbounded) by numerical simulations.
The stable interactions were recently proposed for the Podolsky electrodynamics [11] and for the higher order Pais-Uhlenbeck oscillator [18]. The example of this section extends the limited list of known stable interactions in higher derivative models.

Concluding remarks
Let us summarize the results. In this paper, we suggest a simple general procedure of constructing a family of higher order symmetries and related conservation laws for the derived theories whose equations are polynomial in certain operator (2). For the higher order extensions of the Chern-Simons theory (1), being an example of derived theory, we explicitly deduce the conserved tensors. In some cases, depending on the structure of roots in the polynomial (6), the positive tensors exist among the conserved quantities, while in the other cases, none of the conserved quantities is positive. Once a positive conserved tensor exists, the theory is classically stable, even though the canonical energy is unbounded. In the third order examples of the theory (1) we notice that the stable theories realize the irreducible unitary representations of the Poincaré group, while the models admitting only unbounded conserved tensors correspond to non-unitary representations. We also demonstrate that stable free theory can admit consistent interactions that does not break the stability.
Finally, we make remarks on stability at quantum level. Let us mention that derived theories (2) admit nontrivial Lagrange anchors that can be constructed as polynomials in W of order lower than n. The construction of the anchor for the case n = 2 demonstrated in [11]. This can allow one to quantize classically stable theory without loss of stability. As established in [23,24,25], every Lagrange anchor leads to a Poisson bracket and Hamiltonian in the first order formalism. The inequivalent Lagrange anchors lead to the canonically inequivalent Poisson brackets, so the theory will be multi-Hamiltonian in the first order formulation once it admits different Lagrange anchors. As demonstrated in [13], the Lagrange anchor maps conservation laws to symmetries. In the examples of classically stable higher derivative systems admitting the different Lagrange anchors [11,18], the anchor exists such that maps the positive conserved quantity to the time shift. This means that in the corresponding Hamiltonian formalism (which is not unique, once there exist inequivalent Lagrange anchors) the positive quantity can serve as Hamiltonian. As example, let us mention that for the Pais-Uhlenbeck oscillator positive Hamiltonians are known [26,27], also at interacting level [18]. As the Hamiltonian is bounded at the classical level, we can hope to have bounded spectrum of energy in quantum theory. In view of these reasons, we may expect that classically stable higher derivative extensions of the Chern-Simons model can remain stable at the quantum level once appropriate Lagrange anchor is applied to quantize the theory.