Stable interactions in higher derivative field theories of derived type

We consider the general higher derivative field theories of derived type. At free level, the wave operator of derived-type theory is a polynomial of the order $n\geq 2$ of another operator $W$ which is of the lower order. Every symmetry of $W$ gives rise to the series of independent higher order symmetries of the field equations of derived system. In its turn, these symmetries give rise to the series of independent conserved quantities. In particular, the translation invariance of operator $W$ results in the series of conserved tensors of the derived theory. The series involves $n$ independent conserved tensors including canonical energy-momentum. Even if the canonical energy is unbounded, the other conserved tensors in the series can be bounded, that will make the dynamics stable. The general procedure is worked out to switch on the interactions such that the stability persists beyond the free level. The stable interaction vertices are inevitably non-Lagrangian. The stable theory, however, can admit consistent quantization. The general construction is exemplified by the order $N$ extension of Chern-Simons coupled to the Pais-Uhlenbeck-type higher derivative complex scalar field.


Introduction
The higher derivative field theories are notorious for the stability problems at interacting level, and also in quantum theory, see [1][2][3][4] for discussions and further references. The most frequently discussed higher derivative field theories are the modified models of gravity where the stability is an issue [5][6][7]. Among the modified gravity theories, f (R) models (for review and further references see [8,9]) provide the best known example of stable nonlinear higher derivative field theory. Some other models with similar properties can be found in the references [10][11][12]. The stability of this exceptional class of higher derivative theories is related to the fact that the canonical energy is bounded because of strong second class constraints. For discussion of stability in various non-linear higher derivative mechanical models we refer to [13][14][15][16][17] and references therein.
In this paper, we consider a certain class of higher derivative models which we call derived theories. At free level, the field equations of the derived theory are defined by the higher derivative wave operator M being a polynomial of another skipping to explicitly employ the Lagrange anchor. The basic idea is that we have two subsets of fields such that the wave operator in each subset is a polynomial in certain primary operator. The primary operators are assumed to be Poincaré invariant. One of the primary wave operators is supposed to be gauge invariant. We also assume that the primary theories admit appropriate covariant interaction vertex consistent with the gauge symmetry. To construct the stable interactions between two derived theories, we identify the series of conserved tensors at free level such that every item is connected to the space-time translation symmetry. Some of the conserved tensors can have bounded 00-component, while the other ones are unbounded.
Then, we seek for the interactions in the derived theories such that generalize couplings between primary models and keep the appropriate bounded quantity conserved at interacting level. If the theory admits bounded conserved quantity, the stability will persist at interacting level once it is constructed by this method. The proposed construction can reproduce all the perviously known stable interactions in higher derivative systems of derived type [23][24][25] and we also add a new example in this paper, to illustrate the method.
As the illustration of the general method described in the paper, we construct the consistent and stable interaction between the order N extension of Chern-Simons and order 2n Pais-Uhlenbeck-type higher derivative complex scalar field. These models have been discussed earlier at free level in the papers [23][24][25][26] for some specific orders n and N . In this paper, the N + n-parameter series of conserved second-rank tensors is constructed for the free model being connected to the translation invariance. Also n-parameter series of conserved currents is identified to be connected to a single U (1)-symmetry of the scalar. There are bounded quantities among the conserved observables that make the theory stable at free level, while the canonical energy is unbounded. Non-minimal gauge invariant couplings are identified such that the higher derivative theory remains stable at interacting level. The stable interaction vertices are inevitably non-Lagrangian. That does not necessarily obstruct quantization as we explain in the conclusion.
The article is organized as follows. In the next section, we define the free derived theories and elaborate on the series of symmetries and conserved quantities connected to each symmetry of the primary wave operator W . In Section 3, we identify the gauge invariant couplings such that provide conservation of any fixed representative of the series of conserved quantities of the derived-type free system. In Section 4, we consider interactions between the higher derivative complex scalar field and higher derivative extension of Chern-Simons. Following general pattern of the previous section, we explicitly construct the conserved quantities and keep track of the stability making use of the bounded integrals of motion. In conclusion, we discuss the results.
2 Derived-type theories, higher symmetries, and conservation laws Consider a set of fields φ a in d-dimensional Minkowski space with local coordinates x µ , µ = 0, . . . , d − 1. We systematically use the DeWitt condensed notation [30] in this paper. The fields are labeled by the condensed indices a, b, . . .. Every condensed index accommodates all the vector, tensor, spinor, isotopic, etc. indices and the space-time coordinates. Summation in the condensed indices implies integration in x µ . In this notation, the linear differential operators are represented by matrices with condensed indices. We assume that the theory admits a constant metric which is used to raise and lower multiindices. In this way, every linear operator has the quadric form. These quadratic forms correspond to the local functionals which are bilinear in the fields. We also imply that the fields vanish at the infinity, so if the quadratic form vanishes in the condensed notation, this means that the corresponding local functional is the integral of total divergence.
In the condensed notation, any system of linear field equations reads: where M ab is the integral kernel of matrix differential operator. For the sake of simplicity, we assume that the matrix M ab is square, so that the number of equations in each space-time point coincides with the number of fields. In this class of theories, M ab is usually called the wave operator. The formal adjoint of the wave operator is defined by The field equations are variational whenever M † = M , in which case the action functional S[φ] reads where the brackets , denote the natural pairing between the fields, and summation is implied over the repeated multi-indices a when they stand at different levels. If M † * = −M * for some other operator M * , its diagonal elements are total divergencies, Once the expression M * φ vanishes on shell (1), the latter formula establishes the relation between conserved currents and anti-self-adjoint operators.
We say that the variational theory (1) is of the derived-type if the wave operator is a finite-order polynomial of another self-adjoint operator W a b , i.e.
where all the multi-indices are raised and lowered by the metric, and α p , p = 0, . . . , n , are some real constants. The order of the polynomial is assumed to be irreducible, so the coefficient at the highest order is nonzero, α n = 0. In accordance with the definition, each derived theory is defined by two ingredients: the self-adjoint operator W a b and finite-order polynomial, with z being formal complex-valued variable. We call M (α; z) the characteristic polynomial, while W a b is referred to as the primary wave operator. Being considered in itself, the primary operator defines the primary free field theory, As W ab is self-adjoint, the primary theory is variational. In our article, we mostly deal with the class of theories, where the primary wave operator does not involve higher derivatives. In this setting, the higher derivative derived model (1), (6) can be considered as originating from the lower order primary theory (8).
In this paper, we consider the linear operator X a b as a symmetry 1 of linear equations (1) if it is interchangeable with the wave operator of theory in the following sense: with Y a c being some other operator. Once this relation is valid, the operator X a b defines the linear transformation of the fields such that leaves the mass shell (1) intact, where ξ is the transformation parameter, being some constant. The sign ≈ means equality modulo equations (1). Each theory admits the trivial symmetries that read where X a b can be any operator. The corresponding transformations of fields vanish on-shell and do not contain any valuable information about the dynamics of model. We systematically ignore the trivial symmetries, considering any symmetry modulo trivial one.
The non-trivial symmetries of linear system are known to form an associative algebra. The multiplication operation is just a composition of operators. It is easy to verify the latter fact: for each pair of symmetries ( The elements of associative algebra that commute with the wave operator form a subalgebra. In this paper, we mostly consider this subalgebra as it is connected to conserved quantities. Now, consider the derived equations (1), (6). Assume that the primary model (8) has the symmetry X a b such that commutes with the primary wave operator, Even if the primary theory admits a single symmetry, the associative algebra of symmetries of derived theory have the two natural generators -X and W -that commute with each other, By composing the primary symmetry X with the degree of the primary wave operator W we get the symmetry of the derived equations (6): Only the terms with p = 0, . . . , n − 1 are relevant, because the higher powers of primary operator can be absorbed by the wave operator (6). In this way, for p ≥ n, the symmetry reduces on shell to the symmetry with p < n. The generators X p can be assembled into the n-parameter series of derived symmetries of derived model, with β p , p = 0, . . . , n − 1 , being real numbers. As is seen, all the representatives of the series originate from one and the same symmetry X of primary model.
The symmetry (9) preserves the action functional (3) if its operator is anti-self-adjoint and commutes with the wave operator, The corresponding conserved current j µ (φ), being quadratic in the fields, is defined by the condition This formula represents the Noetherian relationship between symmetries and conservation laws. In the class of derived theories, a single symmetry (13) of primary model (8) determines the series of derived symmetries (16). The associated n-parameter series of conserved currents reads where the quantities j p µ , p = 0, . . . , n − 1 , come from symmetries (15). In particular, j 0 µ represents the Noether conserved current for the symmetry (13) of primary model, while j p µ , p = 1, . . . , n − 1 , are other quantities.
The simplest possible symmetry of the free field theory is the translation invariance. The translation generators ∂ µ are automatically anti-self-adjoint and they commute with the primary wave operator (8), Once the primary wave operator is translation invariant, the derived theory enjoy n-parameter series (16) of derived symmetries originating from this invariance: Each of these symmetries preserves the action (3) of the derived model (1), (6). The corresponding conserved currents (19) constitute the series of second-rank energy-momentum tensors This series includes the canonical energy-momentum as T 0 µν (φ), while T p µν , p = 1, . . . , n − 1 , are different independent conserved tensors associated with space-time translation invariance of the model. In this way, the translation invariance of the derived-type field theory results in the series of the conserved tensors.
As the stability of higher derivative model is concerned, the 00-component of the conserved tensor is of interest. The

00-component of tensor (22) reads
This expression is given by the sum of canonical energy T 0 00 and the other contributions T p 00 , p = 1, . . . , n − 1. Even if the canonical energy is unbounded due to higher derivatives, the quantity (23) can define bounded conserved charge. The bounded conserved quantity, if it exists, stabilizes classical dynamics of derived model (1), (6) at free level.
The discussion above do not address the gauge symmetries. Now we explain how the gauge symmetry is accounted for in the field theories of derived type. The wave operator may have a null vector, In this case, the derived theory (1), (6) admits the gauge transformation such that preserves the mass shell, where ε α are the gauge parameters, being arbitrary functions of space-time coordinates. The operator R a α is the gauge symmetry generator. In the present work, we consider the class of derived models whose gauge symmetries come from null-vectors of primary model, i.e.
This property is not automatically satisfied for general derived theory, so this is an additional assumption. The Podolsky's electrodynamics and the extended Chern-Simons are consistent with this assumption, for example.

Consistent interactions of derived models
In this section, we construct the class of consistent interactions between two Poincaré invariant derived theories, with one of them being gauge. At free level, these theories admit series of conserved energy-momentum tensors. We construct the interaction such that provides conservation of the deformation of certain representative of energy-momentum tensor series.
In so doing, we admit not necessarily Lagrangian vertices 2 To solve the problem we proceed from the assumption that the two primary theories admit consistent interaction vertex. Then, we seek for the way to lift this vertex to the level of derived theory in such a way that does not spoil the stability of the free derived theory.
Consider two subsets of fields φ a , Φ i on d-dimensional Minkowski space, with the multi-indices a, i labeling the fields.
The primary operators are denoted by W ab and W ij , so the primary field equations read These equations are variational, with the action functional being the sum of actions (3) of the fields φ a and Φ i . The primary operators are assumed to be invariant under the space-time translations, The Poincaré symmetry implies the existence of conserved energy-momentum tensor where T µν (φ) and T µν (Φ) denote the contributions of free fields φ and Φ. The quantity Θ µν is determined by the formula The theory of the field Φ i is assumed to be gauge invariant. The gauge symmetry generator R i α of the field is the null-vector of the primary operator W ij , see Eq. (24). The corresponding gauge transformation reads where ε α are gauge parameters. We do not allow gauge freedom for the fields φ a .
We assume that there exists a variational interaction vertex between primary models such that (i) the gauge transformation (32) is preserved at the interacting level and (ii) the action functional is at most quadratic in φ. The most general action such that meets these requirements has the form where the operator of vertex Γ ab (βΦ) if a function of quantity βΦ, and β is the coupling constant, which can be arbitrary with where and the derivatives ∂ a , ∂ i are understood as variational in φ a , Φ i . As is seen, in the model with interaction, the operator of vertex Γ ab (βΦ) is added to free equations of originally non-gauge field φ a , while the gauge field theory get additional currentlike contribution J i (φ, βΦ). This structure of non-linear theory is typical for lower spin fields. For example, it includes the minimal coupling between the electromagnetic field and the charged matter fields.
The infinitesimal gauge transformation of the action functional (33) reads where R i α is the null-vector of primary operator W ij (31), and R a bα is some field-independent structure function. We assume that it is antisymmetric in the indices ab, The invariance of action (33) with respect to the gauge transformations (37) is equivalent to the condition As the gauge symmetries and gauge identities of Lagrangian theories are connected by the Noether theorem, conditions (38), (39) determine the selection rule for the structure functions Γ a b , R a bα . The meaning of these conditions is that the wave operator W ab (βΦ) (36) is gauge invariant in the following sense: The latter relation immediately ensures the gauge invariance of equations of motion of the field φ a .
The translation invariance of interacting theory (33) is understood in the following sense: Once this condition is met, the action functional (33) is preserved by the space-time translations. The energy-momentum tensor of the non-linear theory has the following structure: The expression T µν (φ, Φ) is the gauge invariant extension of the tensor T µν (φ) (29) of free theory, while T µν (Φ) is the energy-momentum of free gauge field. The defining condition for the conserved tensor reads The rhs of this expression is a total divergence because ∂ µ W (βΦ), ∂ µ W are anti-self-adjoint operators.
Let us explain the meaning of conditions (38), (39) and its impact on the structure of interactions. Assume that the gauge transformation of the gauge field Φ i has a null-mode ε = ε(x) such that In this case, the action functional (33) is preserved by the following transformation: with the parameter ξ being a constant. In the free limit, this transformation corresponds to certain internal symmetry of the free model of the fields φ a , which localizes at the interacting level. This symmetry does not follow from the relativistic invariance of equations (27), so we have additional prerequisite for constriction of interaction at free level. As we deal with internal symmetry, this gives a restriction on the multiplet of fields φ a that are involved into the interaction.
The current-like term J i (36) originates from the internal symmetry (45). As a consequence of relations (38), (39), it is gauge invariant and satisfies gauge identity, If the internal symmetry is the U (1)-transformation, and gauge generator R i α is gradient, the current-like term literally corresponds to the current of the U (1) charge, which meets condition (46). Once current-like term J i is added to the equations of motion of the gauge field, and the primary operator W is replaced by its gauge invariant extension W (βΦ) , the non-linear theory remains gauge invariant. This means that interactions (35) imitate the electromagnetic-like couplings between gauge and matter fields.
Let us now turn to the details of inclusion of interactions between derived theories. The most general ansatz for two derived theories with the primary operators W and W has the form The characteristic polynomials of derived models (48) are the most general of orders n and N , respectively, where z is a formal complex-valued variable. The real numbers α p , p = 0, . . . , n , and A q , q = 1, . . . , N , are model parameters that distinguish different theories in the considered class.
The derived theories (48) have the same space-time and gauge symmetries as primary models. In particular, models (48), (49) are invariant under gauge transformation (32). As for space-time translation, a single symmetry of primary theory induces the n + N -parameter series of derived symmetries such that where the real numbers β p , B q are parameters, and ξ is a constant vector. The corresponding set of the conserved quantities is given by the series of energy-momentum tensors where the quantities T p µν (φ), T q µν (Φ) are defined by the conditions By construction, the canonical energy-momentum tensor of the model is included in the series as Even though the canonical energy-momentum is almost always unbounded due to higher derivatives, the other bounded quantities can present in the series. Free higher derivative theory can be stabilized by these quantities.
By interaction between two gauge models (48) we mean a formal deformation of free theory with coupling parameter β such that the non-linear equations of motion read The structure functions Γ (k) are assumed to have the homogeneity degree k + 1 in the variables φ a , Φ i , so Γ (1) are quadratic in the fields, Γ (2) are cubic, and etc. No action principle is required for interacting theory, so non-linear equations (54) can be non-Lagragian. We consider interaction (54) consistent if the number of gauge symmetries and gauge identities is preserved by coupling and at least one representative of the free series of energy-momentum tensors (52) is still conserved at the non-linear level.
The n + N -parameter series of interaction vertices for free equations (48) can be presented in the following form where the notation is used: and the line above means that the gauge field Φ i is replaced by more general expression: (ii) there are the gauge identities between equations of motion, (iii) the second-rank conserved tensor Θ µν is defined by the condition The proof of relations (58), (59) uses identities which hold for p = 0, . . . , n − 1. We deduce these relations in the Appendix A. The conserved tensor Θ µν (60) is a deformation of a selected representative of free energy-momentum series (52), whose parameters are defined by coupling constants. In Appendix B, we ensure the rhs of Eq. (60) is a total divergence and establish the structure of conserved quantity.
Let us comment on the origin of interactions (55). The model (48) admits n-parameter series of gauge invariant currentlike terms Once the primary wave operator W is replaced by its gauge invariant extension W (βΦ), the inclusion of such current-like terms preserves gauge invariance. This means that theory (55) with the gauge-invariant primary operator (36) and current-like term (62) is consistent. At this ways, we conclude that there is n + 1-parameter series of interactions with coupling constants β, β 0 , . . . , β n−1 . To involve coupling constants B q into account, we perform the shift of the gauge field βΦ → Φ(β, B) (57). This step preserves consistency of interactions and inserts N − 1 coupling parameters B 1 , . . . , B N −1 into the field equations.
The non-linear theory (55) is stable if the bounded representative of free energy-momentum tensor series (51) is still conserved at the interacting level. The latter requirement can be interpreted as the selection rule for admissible couplings.
The derived theories, whose characteristic polynomial (7) has non-degenerate real roots, are usually stable at free level 3 . This gives a good chance for existence of stable interactions between such models. The theories with multiple real or/and complex roots of the characteristic polynomial usually have the degrees of freedom whose dynamics is unbounded by any conserved quantity. These degrees of freedom are the true ghosts unless they are suppressed by constraints or gauged out. The stability of interactions in the models having no bounded conserved quantity at free level is a subtle issue, which we do not consider in the present paper.
and all other constants vanish. The action functional reads The corresponding conserved quantity is the canonical energy-momentum tensor of the model. In the class of higher derivative derived equations, the variational interaction vertex usually suffers from the Ostrogradski instability. In contrast to Lagrangian interactions, the non-Lagrangian couplings can preserve bounded representative in the series of free conserved quantities (51).
These couplings define the stable non-linear theory.

Couplings between extended Chern-Simons and higher derivative scalar
Following the general pattern of the previous section, let us consider the interactions between the gauge vector field and the charged scalar. In this case, we have two subsets of fields on 3d Minkowski space, The primary operators read To make contact with the general scheme of previous section, we identify the d'Alembertian as the primary operator for the scalar, For the vector field, the Chern-Simons operator serves as the primary operator. The constant m > 0 has the dimension of mass. Here we use it to make the primary operators dimensionless. The primary equations of motion, have the gauge symmetry (32), with the gauge generator for the vector field being gradient, R i α ≡ ∂ µ . The primary models (67) admit obvious variational interaction vertex, where D µ is the covariant derivative, For the complex conjugate field φ * , the covariant derivative is given by the complex conjugation of (69). The action functional (68) has form (33) with the operator of vertex, being the second-order polynomial in the coupling constant β . The Lagrange equations for the action functional (68) read These equations have form (35) with As the gauge field is a vector, the current-like term holds world index. As we can see, the quantity J µ (φ, βΦ) is the charge current of complex scalar field. The infinitesimal gauge transformation (37) of the action functional (68) reads The canonical energy-momentum tensor (42) for the action (68) has the form It is obvious that its 00-component is bounded. So, the theory of interacting Chern-Simons and charged scalar is stable at the level of primary theories. Now we consider interactions between two derived theories, whose primary operators are given by (66). The free equations (48) are chosen in the form Here we assume that all the roots of characteristic polynomial (7) for scalar field are different and positive. We ignore all the other options because they do not result to the stable theory at the free or interacting level. The extended covariant derivative D µ is defined as follows: The gauge invariant extension of the free wave operator M (α; W ) (75) reads The series of current-like terms (62) can be represented in the form where the notation is used: 4 Let us notice that (J p ) µ are the linear combinations of (J p ) i in (62). We deal with objects (J p ) µ for reasons of convenience.
Equations (80) are obviously gauge invariant with respect to gauge symmetry (73). The gauge identity between equations (80) is seen in the form The series of the energy-momentum tensors (51) for non-linear theory (80) has the structure where the quantities (T p ) µν , p = 0, . . . , n − 1 , and (T q ) µν , q = 0, . . . , N − 1 , read The constants C q,q ′ r,s are defined as follows: As is seen from (82), the quantity Θ µν given by the deformation of selected representative of energy-momentum tensor series of free complex scalar and Chern-Simons, In this formula all the parameters of the conserved quantity are fixed by the interaction, so a single representative of free series of conserved quantities survives at the interacting level.
Let us discuss the stability of the theory (80) at the interacting level. The 00-component of (82) can be written in the form where the matrix C r,s (A, B) is defined by The expression (87)  In general, the interaction (80) is non-Lagrangian. The Lagrangian interaction vertex corresponds to the following values of coupling parameters: The conserved quantity for variational couplings is the canonical energy-momentum tensor. As the canonical energy (Θ can ) 00 is unbounded, so the Lagrangian interaction is unstable. The field equations (80) include stable interactions, so the corresponding coupling is non-Lagrangian. The coupling parameters that lead to stable interactions are determined by conditions β p > 0, p = 0, . . . , n − 1 and positive definiteness requirement for the form C r,s (A, B) (88).

Conclusion and discussion of results
At first, let us summarize the basic assumptions about their consequences for the class of higher derivative field theories we study in this paper. We consider inclusion of stable interactions in certain class of higher derivative field theories which we term as derived models. The derived higher derivative theory implies that the same field admits free field equation with the wave operator W without higher derivatives. We call this model a primary theory. The wave operator of the derived theory is a n-th order polynomial in W . We call it a characteristic polynomial of the derived theory. Every symmetry of primary theory results in the n-parametric series of symmetries of the derived theory. These symmetries are connected to the series of independent conserved quantities. In particular, the translation invariance of the primary model results in the n-parametric series of conserved tensors. Under certain assumptions about the roots of characteristic polynomial, the series can include bounded conserved quantities. The canonical energy-momentum is included into the series, though it is always unbounded.
Once the free derived theory admits bounded conserved quantities, it is considered stable.
We consider inclusion of interactions between two different derived field theories. One of these is supposed to be gauge invariant, and another one is non-gauge. We assume that the primary theories of these two models admit consistent and stable interactions, with the gauge symmetry remaining abelian at interacting level. We also impose certain technical restrictions on the interaction between primary theories as explained in the section 3. In particular, these restrictions exclude the "gauging" scenarios of inclusion of interactions 5 . In principle, these restrictions could be relaxed without breaking the general scheme of inclusion the of stable interactions in the derived-type theories, though this would make the consideration much more cumbersome. Once the primary theories admit consistent interactions, these can be lifted to the level of derived theory. The lift is not unique, we get n + N -parametric series of consistent interactions, where n, N are the orders of characteristic polynomials of the two models. At free level, the two derived theories admitted n + N parametric series of the conserved tensors. Upon inclusion of interaction, only one tensor conserves with any fixed set of the interaction parameters. The canonical energy momentum is conserved if the interaction is Lagrangian. As the canonical energy is unbounded, the Lagrangian interaction is unstable. Once the free derived theory admits bounded conserved quantity, its conservation can be always preserved with appropriate consistent and Poincaré invariant interaction, so the stability persists at interacting level, though with the non-Lagrangian vertex.
Also notice that even if the interaction is included in such a way that none of the bounded conserved quantities ( from the series identified in section 2) of the free theory remains conserving at interacting level, these non-conserved currents can be still informative. So they can be still relevant for Lagrangian interactions. If the dynamics is known of bounded non-conserved quantity, than we learn about the compact evolving phase-space surface where the dynamics is confined. This might be useful for identifying the isles of stability, and evaluating the velocity of running away solutions. For discussion of these ideas and further references see the recent work [34].
Even though the vertices of stable interactions are always non-Lagrangian in this class of higher derivative theories, they can still admit quantization. The matter is that these non-Lagrangian field equations can admit Lagrange anchor which allows one to quantize the non-Lagrangian field theory. The examples are provided in the paper [23] where the some special interactions are considered between the derived model with the quadratic characteristic polynomial, and the theory without higher derivatives. Another side of the existence of the Lagrange anchor that the theory should admit Hamiltonian formalism even though the higher derivative equations are non-Lagrangian. The explicit examples of Hamiltonian formalism for third order non-Lagrangian derived systems can be found in the recent papers [25,26]. The n-th order derived theory with the stable interactions constructed in section 4 should also admit constrained Hamiltonian formalism while the vertices are non-Lagrangian. The explicit construction is unknown at the moment, this will be considered elsewhere.
Now consider the issue of gauge invariance of current-like term (J p ) i . Computing the gauge variation of the expression (94) and using formula (61.a), we obtain Again, this expression vanishes identically as a result of identity (40). This proves formula (61.c).
Appendix B. Existence and structure of second-rank conserved tensor Θ µν (β, B) In this appendix, we prove that the rhs of equation (60) is the conserved tensor. Substituting the equations of motion (55) into this relation, using the definition of current-like term (56), we get The second term defines the series of energy-momentum tensors of free theory of gauge field (52), It remains to prove that the first term is a divergence of some tensor. Integrating by parts, we decompose the entries of the sum into three contributions where the square brackets [ , ] and braces { , } denote the commutator and anti-commutator of enclosed operators. The first term vanishes identically due to (41). The second term is the total divergence (91), as well as the third one. The consistency of formula (60) is proven.
The conserved tensor Θ µν (β, B) has the following structure: where T q µν (Φ) are the energy-momentum tensors of free gauge field, and Σ p+k µν (φ, Φ) is defined in (91). As in the free the quantity Θ µν (β, B) is a deformation of an exclusive representative in the series of energy-momentum tensors(52).