Remarks on quantization of Pais–Uhlenbeck oscillators

Damaskinsky, E. V.; Sokolov, Mikhail A.

doi:10.1088/0305-4470/39/33/017

Cited by 23 publications

(40 citation statements)

References 7 publications

(16 reference statements)

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“…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%

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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Section: Examples: Harmonic Oscillatorsmentioning

confidence: 99%

Natural Hamiltonian formulation of composite higher derivative theories

Öttinger

2019

J. Phys. Commun.

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“…We term the symmetry variational if the transformation (11) preserves the action functional (4). The defining condition for the variational symmetry reads † ( ) ( ) ( ) ( ) 0, L M M L       (13) which is relation (10) for Q = L † . Trivial variational symmetries have the form var † triv ( )…”

Section: Symmetries Characteristics and Conserved Quantities Of Linmentioning

confidence: 99%

“…The recent research [3][4][5] demonstrates that the models with unbounded classical energy are not necessarily unstable. Various ideas were applied to the study of stability of higher-derivative theories, including the non-Hermitian quantum mechanics [6][7][8], alternative Hamiltonian formulations [9][10][11][12], adiabatic invariants [13], and special boundary conditions [14]. For constrained systems, the energy can be bounded on-shell due to constraints [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Conservation Laws and Stability of Field Theories of Derived Type

Kaparulin

2019

Symmetry

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We consider the issue of correspondence between symmetries and conserved quantities in the class of linear relativistic higher-derivative theories of derived type. In this class of models the wave operator is a polynomial in another formally self-adjoint operator, while each isometry of space-time gives rise to the series of symmetries of action functional. If the wave operator is given by n-th-order polynomial then this series includes n independent entries, which can be explicitly constructed. The Noether theorem is then used to construct an n-parameter set of second-rank conserved tensors. The canonical energy-momentum tensor is included in the series, while the other entries define independent integrals of motion. The Lagrange anchor concept is applied to connect the general conserved tensor in the series with the original space-time translation symmetry. This result is interpreted as existence of multiple energy-momentum tensors in the class of derived systems. To study stability we seek for bounded-conserved quantities that are connected with the time translations. We observe that the derived theory is stable if its wave operator is defined by a polynomial with simple and real roots. The general constructions are illustrated by the examples of the Pais-Uhlenbeck oscillator, higher-derivative scalar field, and extended Chern-Simons theory.

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“…In the case U = 0 theory (1) is poly-Hamiltonian. Each value of the parameters α i corresponds to some Hamiltonian (11) that can be bounded or unbounded from below depending on the signs of α i [21]. The statement of unboundedness from below (absence of a lower bound) pertains only to Ostrogradsky's Hamiltonian, which follows from action (3) and corresponds to α i = 1 [19].…”

Section: Nonlinear Oscillator With Higher Derivativesmentioning

confidence: 99%

“…The present work makes use of the idea that stability of a theory can be ensured by the existence of an alternative first-order formulation with a positive Hamiltonian [20][21][22] which is not the Ostrogradsky Hamiltonian. This approach allows us to overcome the prohibition established by the Ostrogradsky theorem.…”

Section: Introductionmentioning

confidence: 99%