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 the nth-order theory of this type, we provide a general recipe 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.

It is shown that curvature-squared gravity would be split into five essentially different variants at D)4. Every variant possesses its own number of physical degrees of freedom. The classification explicitly depends upon the dimension. The Dirac constrained canonical formalism is developed and its general structure is exhaustively studied. Explicit expressions of the first- and second-class constraints are specific for every variant of the theory.

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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 the third order. IntroductionVarious higher derivative field theories are studied once and again over many decades for several reasons. Among the most frequently mentioned advantages of the higher derivative systems are the better convergence properties at classical and quantum level comparing to the analogues without higher derivatives. For discussion of various types of higher derivative models we refer to the paper [1] and references therein. The higher derivative theories are also notorious for the instability problem. The simplest stability test -boundedness of energy -is usually failed by the models with higher order equations of * dsc@phys.tsu.ru † karin@phys.tsu.ru ‡ sll@phys.tsu.ru motion. The best known exception -f (R)-gravity [2] -is stable due to very strong second class constraints. Because of that, on the constrained surface, the Hamiltonian is bounded, so the theory meets the sufficient condition for stability.Even if the energy is unbounded for general higher-derivative dynamics, the theory is not necessarily unstable. If another bounded conserved quantity exists, it stabilizes dynamics at least at classical level. For example, the free fo...

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.