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A model is proposed for a classical bosonic string in $d=3$ Minkowski space with an action functional that includes Gauss and mean world-sheet curvature. The Lagrangian is invariant under $3d$ Poincar\'e transformations modulo total divergence. In addition to the diffeomorphism, the action enjoys the extra gauge symmetry with the second derivatives of the scalar gauge parameter. This symmetry gauges out all the local degrees of freedom (DoF's), while some global DoF's survive. The Hamiltonian constrained analysis confirms that the model does not have any local DoF. The world sheet of the string turns out to be a cylinder with time-like axis. The global DoF's of this string describe one single irreducible massive $3d$ particle with spin. The particle momentum is a conserved vector directed along the cylinder axis while the momentum square is a fixed constant determined by the parameters in the action. The total angular momentum is a conserved vector that defines the position of the axis of the cylinder whose specific value is defined by initial data, while the spin, being the product of momentum and angular momentum is fixed by the parameters in the string Lagrangian., Comment: 19 pages

The gauge symmetry is said unfree if the gauge transformation leaves the action functional unchanged provided for the gauge parameters are constrained by the system of partial differential equations. The best known example of this phenomenon is the volume preserving diffeomorphism being the gauge symmetry of unimodular gravity (UG). Various extensions are known of the UG, including the higher spin analogs - all with unfree gauge symmetry. Given the distinctions of the unfree gauge symmetry from the symmetry with unrestricted gauge parameters, the algebra of gauge transformations is essentially different. These distinctions have consequences for all the key constituents of general gauge theory, starting from the second Noether theorem, Hamiltonian constrained formalism, BRST complex, and quantization. In this review article, we summarise the modifications of general gauge theory worked out in recent years to cover the case of unfree gauge symmetry., Comment: 14 pages, prepared for the Proceedings of the XXXIV International Workshop on High Energy Physics "From Quarks to Galaxies: Elucidating Dark Sides", to be published in Physics of Particles and Nuclei

We consider the general free field theory such that system of equations of motion includes a subsystem with a special property. If the subsystem is considered by itself, it would be a topological field theory having no local degrees of freedom. Various well-known field theories admit such a subsystem, including Einstein's gravity. As the subsystem is a topological theory, the general solution is a pure gauge, modulo global degrees of freedom. Gauge symmetry of the subsystem of linear theory can be explicitly found, providing the general solution to the subsystem. Substituting this solution into entire system, we arrive at an equivalent field theory where the role of the field variables is played by the gauge parameters of the topological subsystem. These fields can be viewed as the potentials for the original ones. A general procedure is proposed for constructing a "parent action" which includes the fields of both dual formulations. At the level of this action, one can switch between dual formulations by imposing appropriate gauge fixing conditions. This general dualisation procedure is exemplified by massless and massive spin two fields. For the massless case, the hook type tensor serves as a potential for the metric, while for the massive case it is the fourth rank tensor with window Young diagram., Comment: 31 pages, minor corrections

We consider a class of theories with unfree gauge symmetry, whose gauge parameters are restricted by differential equations. We demonstrate that such theories admit global conserved quantities, whose on-shell values are defined by asymptotics of the fields rather than Cauchy data. The global conserved quantities can be deduced proceeding from the equations restricting gauge parameters, and they are treated differently by two BRST complexes corresponding to a system with unfree gauge symmetry., Comment: 6 pages, to appear in PEPAN Letters

The field equations are proposed for the third rank tensor field with the hook Young diagram. The equations describe the irreducible spin two massless representation in any $d\geq 3$. The starting point of the construction is the linearised system of Einstein equations which includes the Nordstr\"om equation. This equation, being considered irrespectively to the rest of the Einstein system, corresponds to the topological field theory. The general solution is a pure gauge, modulo topological modes which we neglect in this article. We find the sequence of the reducible gauge transformations for the linearised Nordstr\"om equation, with the hook tensor being the initial gauge symmetry parameter. By substituting the general solution of the Nordstr\"om equation into the rest of the Einstein's system, we arrive at the field equations for the hook tensor. The degree of freedom number count confirms, it is the spin two theory., Comment: 19 pages; minor corrections, reference added, to appear in European Physical Journal C

We work out the description of the gauge symmetry of unimodular gravity in the constrained Hamiltonian formalism. In particular, we demonstrate how the transversality conditions restricting the diffeomorphism parameters emerge from the algebra of the Hamiltonian constraints. The alternative form is long known as parametrizing the volume preserving diffeomorphisms by unrestricted two-forms instead of the transverse vector fields. This gauge symmetry is reducible. We work out the Hamiltonian description of this form of unimodular gravity (UG) gauge symmetry. Becchi-Rouet-Stora-Tyutin--Batalin-Fradkin-Vilkovisky (BFV-BRST) Hamiltonian formalism is constructed for both forms of the UG gauge symmetry. These two BRST complexes have a subtle inequivalence: Their BRST cohomology groups are not isomorphic. In particular, for the first complex, which is related to the restricted gauge parameters, the cosmological constant does not correspond to any nontrivial BRST cocycle, while for the alternative complex it does. In the wording of physics, this means $\Lambda$ is a fixed parameter defined by the field asymptotics rather than the physical observable from the standpoint of the first complex. The second formalism views $\Lambda$ as the observable with unrestricted initial data., Comment: 13 pages, minor corrections, referrence added, to appear in Phys. Rev. D

We consider the class of spinning particle theories, whose quantization corresponds to the continuous helicity representation of the Poincare group. The classical trajectories of the particle are shown to lie on the parabolic cylinder with a lightlike axis irrespectively to any specifics of the model. The space-time position of the cylinder is determined by the values of momentum and total angular momentum. The value of helicity determines the focal distance of parabolic cylinder. Assuming that all the world lines lying on one and the same cylinder are connected by gauge transformations, we derive the geometrical equations of motion for the particle. The timelike world paths are shown to be solutions to a single relation involving the invariants of trajectory up to fourth order in derivatives. Geometrical equation of motion is non-Lagragian, but it admits equivalent variational principle in the extended set of dynamical variables. The lightlike paths are also admissible on the cylinder, but they do not represent the classical trajectories of this spinning particle. The classical trajectories of massless particle (with zero helicity) are shown to lie on hyperplanes, whose spacetime position depends on momentum and total angular momentum., Comment: 26 pages

The unfree gauge symmetry implies that gauge variation of the action functional vanishes provided for the gauge parameters are restricted by the differential equations. The unfree gauge symmetry is shown to lead to the global conserved quantities whose on shell values are defined by the asymptotics of the fields or data on the lower dimension surface, or even at the point of the space-time, rather than Cauchy hyper-surface. The most known example of such quantity is the cosmological constant of unimodular gravity. More examples are provided in the article for the higher spin gravity analogues of the cosmological constant. Any action enjoying the unfree gauge symmetry is demonstrated to admit the alternative form of gauge symmetry with the higher order derivatives of unrestricted gauge parameters. The higher order gauge symmetry is reducible in general, even if the unfree symmetry is not. The relationship is detailed between these two forms of gauge symmetry in the constrained Hamiltonian formalism. The local map is shown to exist from the unfree gauge algebra to the reducible higher order one, while the inverse map is non-local, in general. The Hamiltonian BFV-BRST formalism is studied for both forms of the gauge symmetry. These two Hamiltonian formalisms are shown connected by canonical transformation involving the ghosts. The generating function is local for the transformation, though the transformation as such is not local, in general. Hence, these two local BRST complexes are not quasi-isomorphic in the sense that their local BRST-cohomology groups can be different. This difference in particular concerns the global conserved quantities. From the standpoint of the BRST complex for unfree gauge symmetry, these quantities are BRST-exact, while for the alternative complex, these quantities are the non-trivial co-cycles., Comment: 41 pages; minor typos are fixed, and relations (65)-(67), (148)-(149) are corrected

We propose a general procedure for iterative inclusion of Stueckelberg fields to convert the theory into gauge system being equivalent to the original one. In so doing, we admit reducibility of the Stueckelberg gauge symmetry. In this case, no pairing exists between Stueckelberg fields and gauge parameters, unlike the irreducible Stueckelberg symmetry. The general procedure is exemplified by the case of Proca model, with the third order involutive closure chosen as the starting point. In this case, the set of Stueckelberg fields includes, besides the scalar, also the second rank antisymmetric tensor. The reducible Stueckelberg gauge symmetry is shown to admit different gauge fixing conditions. One of the gauges reproduces the original Proca theory, while another one excludes the original vector and the Stueckelberg scalar. In this gauge, the irreducible massive spin one is represented by antisymmetric second rank tensor obeying the third order field equations. Similar dual formulations are expected to exist for the fields of various spins., Comment: 15 pages

We consider a gauge theory of vector fields in $3d$ Minkowski space. At the free level, the dynamical variables are subjected to the extended Chern-Simons (ECS) equations with higher derivatives. If the color index takes $n$ values, the third-order model admits a $2n$-parameter series of second-rank conserved tensors, which includes the canonical energy-momentum. Even though the canonical energy is unbounded, the other representatives in the series can have bounded from below $00$-component. The theory admits consistent self-interactions with the Yang-Mills gauge symmetry. The Lagrangian couplings preserve the unbounded from below energy-momentum tensor, and they do not lead to a stable non-linear theory. The non-Lagrangian couplings are consistent with the existence of conserved tensor with a bounded from below $00$-component. These models are stable at the non-linear level. The dynamics of interacting theory admits a constraint Hamiltonian form. The Hamiltonian density is given by the $00$-component of the conserved tensor. In the case of stable interactions, the Poisson bracket and Hamiltonian do not follow from the canonical Ostrogradski construction. The particular attention is paid to the "triply massless" ECS theory, which demonstrates instability already at the free level. It is shown that the introduction of extra scalar field, serving as Higgs, can stabilize dynamics in the vicinity of the local minimum of energy. The equations of motion of stable model are non-Lagrangian, but they admit the Hamiltonian form of dynamics with a bounded from below Hamiltonian., Comment: 34 pages

The BFV-BRST Hamiltonian quantization method is presented for the theories where the gauge parameters are restricted by differential equations. The general formalism is exemplified by the Maxwell-like theory of symmetric tensor field., Comment: 8 pages, submitted to AIP Conference Proceedings

A systematic procedure is proposed for inclusion of Stueckelberg fields. The procedure begins with the involutive closure when the original Lagrangian equations are complemented by all the lower order consequences. The involutive closure can be viewed as Lagrangian analogue of complementing constrained Hamiltonian system with secondary constraints. The involutively closed form of the field equations allows for explicitly covariant degree of freedom number count, which is stable with respect to deformations. If the original Lagrangian equations are not involutive, the involutive closure will be a non-Lagrangian system. The Stueckelberg fields are assigned to all the consequences included into the involutive closure of the Lagrangian system. The iterative procedure is proposed for constructing the gauge invariant action functional involving Stueckelberg fields such that Lagrangian equations are equivalent to the involutive closure of the original theory. The generators of the Stueckelberg gauge symmetry begin with the operators generating the closure of original Lagrangian system. These operators are not assumed to be a generators of gauge symmetry of any part of the original action, nor are they supposed to form an on shell integrable distribution. With the most general closure generators, the consistent Stueckelberg gauge invariant theory is iteratively constructed, without obstructions at any stage. The Batalin-Vilkovisky form of inclusion the Stueckelberg fields is worked out and existence theorem for the Stueckelberg action is proven., Comment: 30 pages, minor corrections in the abstract

We study Hamiltonian form of unfree gauge symmetry where the gauge parameters have to obey differential equations. We consider the general case such that the Dirac-Bergmann algorithm does not necessarily terminate at secondary constraints, and tertiary and higher order constraints may arise. Given the involution relations for the first-class constraints of all generations, we provide explicit formulas for unfree gauge transformations in the Hamiltonian form, including the differential equations constraining gauge parameters. All the field theories with unfree gauge symmetry share the common feature: they admit sort of "global constants of motion" such that do not depend on the local degrees of freedom. The simplest example is the cosmological constant in the unimodular gravity. We consider these constants as modular parameters rather than conserved quantities. We provide a systematic way of identifying all the modular parameters. We demonstrate that the modular parameters contribute to the Hamiltonian constraints, while they are not explicitly involved in the action. The Hamiltonian analysis of the unfree gauge symmetry is precessed by a brief exposition for the Lagrangian analogue, including explicitly covariant formula for degrees of freedom number count. We also adjust the BFV-BRST Hamiltonian quantization method for the case of unfree gauge symmetry. The main distinction is in the content of the non-minimal sector and gauge fixing procedure. The general formalism is exemplified by traceless tensor fields of irreducible spin $s$ with the gauge symmetry parameters obeying transversality equations., Comment: 30 pages

We consider the class of higher derivative field equations whose wave operator is a square of another self-adjoint operator of lower order. At the free level, the models of this class are shown to admit a two-parameter series of integrals of motion. The series includes the canonical energy. Every conserved quantity is unbounded in this series. The interactions are included into the equations of motion such that a selected representative in conserved quantity series is preserved at the non-linear level. The interactions are not necessarily Lagrangian, but they admit Hamiltonian form of dynamics. The theory is stable if the integral of motion is bounded from below due to the interaction. The motions are finite in the vicinity of the conserved quantity minimum. The equations of motion for fluctuations have the derived form with no resonance. The general constructions are exemplified by the models of the Pais-Uhlenbeck oscillator with multiple frequency and Podolsky electrodynamics. The example is also considered of stable non-abelian Yang-Mills theory with higher derivatives., Comment: 21 pages

The constrained Hamiltonian formalism is worked out for the theories where the gauge symmetry parameters are unfree, being restricted by differential equations. The Hamiltonian BFV-BRST embedding is elaborated for this class of gauge theories. The general formalism is exemplified by the linearized unimodular gravity., Comment: 14 pages

We consider inclusion of interactions between the higher derivative extended Chern-Simons and charged scalar field. We demonstrate that the order $N$ extended Chern-Simons and order $2n$ charged scalar admit the $(N+n)$-parameter series of interaction vertices. The interactions are in general non-Lagrangian, but they preserve a certain conserved second-rank tensor, whose parameters are determined by the coupling. The $00$-component of this tensor can be bounded even if the canonical energy of the model is unbounded before the inclusion of interaction. If the $00$-component of conserved tensor is bounded, the theory is stable., Comment: 7 pages, submitted to the AIP Conference Proceedings

The BV formalism is proposed for the theories where the gauge symmetry parameters are unfree, being constrained by differential equations., Comment: 39 pages

We proceed from the fact that the classical paths of irreducible massive spinning particle lie on a circular cylinder with the time-like axis in Minkowski space. Assuming that all the classical paths on the cylinder are gauge-equivalent, we derive the equations of motion for the cylindrical curves. These equations are non-Lagrangian, but they admit interpretation in terms of the conditional extremum problem for a certain length functional in the class of paths subjected to the constant separation conditions. The unconditional variational principle is obtained after inclusion of constant separation conditions with the Lagrange multipliers into the action. We explicitly verify that the states of the obtained model lie on the co-orbit of the Poincare group. The relationship with the previously known theory is demonstrated., Comment: Submitted to the AIP Conference Proceedings

We propose the model of massive spinning particle traveling in four-dimensional Minkowski space. The equations of motion of the particle follow from the fact that all the classical paths of the particle lie on a cylinder whose position in Minkowski space is determined by the particle's linear momentum and total angular momentum. All the paths on one and the same cylinder are gauge equivalent. The equations of motion are found in implicit form for general time-like paths, and they are non-Lagrangian. The explicit equations of motion are found for trajectories with small curvature and helices. The momentum and total angular momentum are expressed in terms of characteristics of the path in all the cases. The constructed model of the spinning particle has geometrical character, with no additional variables in the space of spin states being introduced., Comment: Submitted to Journal of Physics: Conference Series

The higher derivative field theories are notorious for the stability problems both at classical and quantum level. Classical instability is connected with unboundedness of the canonical energy, while the unbounded energy spectrum leads to the quantum instability. For a wide class of higher derivative theories, including the extended Chern-Simons, other bounded conserved quantities which provide the stability can exist. The most general gauge invariant extended Chern-Simons theory of arbitrary finite order $n$ admits $(n - 1)$-parameter series of conserved energy-momentum tensors. If the $00$-component of the most general representative of this series is bounded, the theory is stable. The stability condition requires from the free extended Chern-Simons theory to describe the unitary reducible representation of the Poincar\'e group. The unstable theory corresponds to nonunitary representation., Comment: 7 pages, submitted to Journal of Physics: Conference series

We consider constrained multi-Hamiltonian formulation for the extended Chern-Simons theory with higher derivatives of arbitrary finite order. The order $n$ extension of the theory admits $(n-1)$-parametric series of conserved tensors. The $00$-component of any representative of the series can be chosen as Hamiltonian. The theory admits a series of Hamiltonian formulations, including the canonical Ostrogradski formulation. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. Also, we demonstrate the inclusion of stable interactions with charged scalar field that preserves one specified Hamiltonian from the series., Comment: 10 pages, to appear in Russian Physics Journal

We study the general structure of field theories with the unfree gauge symmetry where the gauge parameters are restricted by differential equations. The examples of unfree gauge symmetries include volume preserving diffeomorphisms in the unimodular gravity and various higher spin field theories with transverse gauge symmetries. All the known examples of the models with unfree gauge symmetry share one common feature. They admit local quantities which vanish on shell, though they are not linear combinations of Lagrangian equations and their derivatives. We term these quantities as mass shell completion functions. In the case of usual gauge symmetry with unconstrained gauge parameters, the irreducible gauge algebra involves the two basic constituents: the action functional and gauge symmetry generators. For the case of unfree gauge symmetry, we identify two more basic constituents: operators of gauge parameter constraints and completion functions. These two extra constituents are involved in the algebra of unfree gauge symmetry on equal footing with action and gauge symmetry generators. Proceeding from the algebra, we adjust the Faddeev-Popov (FP) path integral quantization scheme to the case of unfree gauge symmetry. The modified FP action involves the operators of the constraints imposed on the gauge parameters, while the corresponding BRST transformation involves the completion functions. The BRST symmetry ensures gauge independence of the path integral. We provide two examples which admit the alternative unconstrained parametrization of gauge symmetry and demonstrate that they lead to the equivalent FP path integral., Comment: Minor corrections, references added

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., Comment: minor corrections, references added, to appear in Phys. Rev. D

We consider inclusion of interactions between 3d Einstein gravity and the third order extensions of Chern-Simons. Once the gravity is minimally included into the third order vector field equations, the theory is shown to admit a two-parameter series of symmetric tensors with on-shell vanishing covariant divergence. The canonical energy-momentum is included into the series. For a certain range of the model parameters, the series include the tensors that meet the weak energy condition, while the canonical energy is unbounded in all the instances. Because of the on-shell vanishing covariant divergence, any of these tensors can be considered as an appropriate candidate for the right hand side of Einstein's equations. If the source differs from the canonical energy momentum, the coupling is non-Lagrangian while the interaction remains consistent with any of the tensors. We reformulate these not necessarily Lagrangian third order equations in the first order formalism which is covariant in the sense of 1+2 decomposition. After that, we find the Poisson bracket such that the first order equations are Hamiltonian in all the instances, be the original third order equations Lagrangian or not. The brackets differ from canonical ones in the matter sector, while the gravity admits the usual PB's in terms of ADM variables. The Hamiltonian constraints generate lapse, shift and gauge transformations of the vector field with respect to these Poisson brackets. The Hamiltonian constraint, being the lapse generator, is interpreted as strongly conserved energy. The matter contribution to the Hamiltonian constraint corresponds to 00-component of the tensor included as a source in the right hand side of Einstein equations. Once the 00-component of the tensor is bounded, the theory meets the usual sufficient condition of classical stability, while the original field equations are of third order., Comment: 17 pages

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., Comment: 19 pages

The classical spinning particles are considered such that quantization of classical model leads to an irreducible massive representation of the Poincar\'e group. The class of gauge equivalent classical particle world lines is shown to form a $[(d+1)/2]$-dimensional world sheet in $d$-dimensional Minkowski space, irrespectively to any specifics of classical model. For massive spinning particles in $d=3,4$, the world sheets are shown to be cylinders. The radius of cylinder is fixed by representation. In higher dimensions, particle's world sheet turn out to be a toroidal cylinder $\mathbb{R}\times \mathbb{T}^D$, $D=[(d-1)/2]$. Proceeding from the fact that the world lines of irreducible classical spinning particles are cylindrical curves, while all the lines are gauge equivalent on the same world sheet, we suggest a method to deduce the classical equations of motion for particles and also to find their gauge symmetries. In $d=3$ Minkowski space, the spinning particle path is defined by a single fourth-order differential equation having two zero-order gauge symmetries. The equation defines particle's path in Minkowski space, and it does not involve auxiliary variables. A special case is also considered of cylindric null-curves, which are defined by a different system of equations. It is shown that the cylindric null-curves also correspond to irreducible massive spinning particles. For the higher-derivative equation of motion of the irreducible massive spinning particle, we deduce the equivalent second-order formulation involving an auxiliary variable. The second-order formulation agrees with a previously known spinning particle model., Comment: references added, formulas in Section III.B included

We detail the construction of a weak Poisson bracket over a submanifold of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle, interpreted as a weak Koszul bracket on differential forms on M. This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system.

We prove that the differential equation for the null-curves of pseudo-Euclidean space R^{2,n} defines a flat dynamical system in the sense of optimal control theory. The connection with general gauge theories is briefly discussed., Comment: 7 pages

A new method is proposed for switching on interactions that are compatible with global symmetries and conservation laws of the original free theory. The method is applied to the control of stability in Lagrangian and non-Lagrangian theories with higher derivatives. By way of illustration, a wide class of stable interactions is constructed for the Pais-Uhlenbeck oscillator., Comment: 12 pages, minor corrections

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., Comment: 14 pages, minor corrections

We study stability of higher-derivative dynamics from the viewpoint of more general correspondence between symmetries and conservation laws established by the Lagrange anchor. We show that classical and quantum stability may be provided if a higher-derivative model admits a bounded from below integral of motion and the Lagrange anchor that relates this integral to the time translation., Comment: A contribution to the Proceedings of the XXXIII Workshop on the Geometric Methods in Physics, 8 pages