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 ≥ 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 energymomentum. 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.

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 00-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.

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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.

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 00component 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. IntroductionThe Hamiltonian formulations for theories with higher derivatives have been discussed once and again for decades since the work of Ostrogradski [1]. The procedure for constructing such Hamiltonian formulation for degenerate theories was originally proposed in [2]. This procedure and its modifications can be applied to study different gauge theories, including gravity (see [3,4]). The canonical Ostrogradski Hamiltonian used in all these methods is not bounded. It leads to the wellknown stability problem and difficulties with constructing of quantum theory [5][6][7].The alternative Hamiltonian formulation was first introduced in the Pais-Uhlenbeck theory [8,9]. In the work [10], it was noticed that a wide class of theories with higher derivatives admits the series of Hamiltonians and the Poisson brackets that are not connected with each other by canonical transformations. The canonical Ostrogradski Hamiltonian is included in the series. The series of non-canonical Hamiltonian formulations was constructed explicitly in [11,12] for the extended Chern-Simons theory [13] of the third and fourth order.In the present work, we consider the Hamiltonian formulation for the extended Chern-Simons of arbitrary finite order. At free level, the theory of order n admits (n − 1)-parametric series of Hamiltonian formulations with Hamiltonian being the 00-component of any conserved tensor from the series [14]. We demonstrate that non-Lagrangian interaction vertices with * abakumova@phys.tsu.ru † dsc@phys.tsu.ru ‡ sll@phys.tsu.ru