Multi-Hamiltonian formulations and stability of higher-derivative extensions of 3d Chern–Simons

Abakumova, V. A.; Kaparulin, D. S.; Lyakhovich, S. L.

doi:10.1140/epjc/s10052-018-5601-y

Cited by 23 publications

(40 citation statements)

References 54 publications

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“…Equation (19) coincides with the definition (16) of the conjugate momenta p i . Equation (20) can be rewritten as…”

Section: Passage To Canonical Hamiltonian Formulationmentioning

confidence: 99%

“…To gain a better understanding of the constraints and the elimination of instabilities, we consider two simple examples of harmonic oscillator systems in more detail. Contrary to previous work [17][18][19][20], which exploits the existence of multiple, inequivalent Hamiltonian structures in linear systems (mainly Pais-Uhlenbeck oscillators) to switch to Hamiltonians bounded from below, we here keep unbounded Hamiltonians, like in the Ostrogradsky approach, but on a larger space.…”

Section: Examples: Harmonic Oscillatorsmentioning

confidence: 99%

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Natural Hamiltonian formulation of composite higher derivative theories

Öttinger

2019

J. Phys. Commun.

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If a higher derivative theory arises from a transformation of variables that involves time derivatives, a tailor-made Hamiltonian formulation is shown to exist. The details and advantages of this elegant Hamiltonian formulation, which differs from the usual Ostrogradsky approach to higher derivative theories, are elaborated for mechanical systems and illustrated for simple examples. Both a canonical space and a set of constraints emerge naturally from the transformation rule for the variables. In other words, the setting for quantization and the procedure for eliminating instabilities arise naturally.

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“…Equation (19) coincides with the definition (16) of the conjugate momenta p i . Equation (20) can be rewritten as…”

Section: Passage To Canonical Hamiltonian Formulationmentioning

confidence: 99%

Section: Examples: Harmonic Oscillatorsmentioning

confidence: 99%

Natural Hamiltonian formulation of composite higher derivative theories

Öttinger

2019

J. Phys. Commun.

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“…been recently found that the model is multi-Hamiltonian at free level [21]. To the best of our knowledge, it is the first known explicit example of higher derivative field theory that admits multi-Hamiltonian structure.…”

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confidence: 91%

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

Kaparulin¹,

Karataeva²,

Lyakhovich³

2018

Nuclear Physics B

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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...

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Section: Introductionmentioning

confidence: 99%

“…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].…”

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confidence: 99%

Stable Interactions Between the Extended Chern-Simons Theory and a Charged Scalar Field with Higher Derivatives: Hamiltonian Formalism

2019

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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

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Multi-Hamiltonian formulations and stability of higher-derivative extensions of 3d Chern–Simons

Cited by 23 publications

References 54 publications

Natural Hamiltonian formulation of composite higher derivative theories

Natural Hamiltonian formulation of composite higher derivative theories

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

Stable Interactions Between the Extended Chern-Simons Theory and a Charged Scalar Field with Higher Derivatives: Hamiltonian Formalism

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