Third order extensions of 3d Chern–Simons interacting to gravity: Hamiltonian formalism and stability

Abstract

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...

Stable interactions in higher derivative field theories of derived type

“…Several known examples confirm this observation[27,29,28], though it is not a theorem at the moment.…”

“…Another side of the existence of the Lagrange anchor is 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 papers [28,29]. The constrained Hamiltonian formalism for n-th order derived theory with the stable interactions constructed in Sec.…”

“…The stability of interactions in various particular types of higher derivative theories of derived type has been previously studied in [26][27][28][29]. The proper deformation method has been suggested in [30] to systematically include stable interactions in the derived theory which is stable at free level.…”

“…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 [26][27][28][29] for some specific orders n and N. In this paper, the N þ 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 Uð1Þ-symmetry of the scalar.…”

Extended Chern-Simons model for a vector multiplet

“…If the Hamiltonian is bounded, it ensures the stability of the model at both classical and quantum levels. The model admits inclusion of stable non-Lagrangian interactions with scalar, fermionic and gravitational fields that preserve a selected representative in the series of conserved quantities of free model [38,43,44]. However, the gauge symmetry is abelian in the sector of vector field in all these examples.…”