Healthy degenerate theories with higher derivatives
Abstract
In the context of classical mechanics, we study the conditions under which higherorder derivative theories can evade the socalled Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time derivatives, of the form L(ϕ{sup ¨a}, ϕdot {sup a},ϕ{sup a}; qdot {sup i},q{sup i}) with a=1,⋯,n and i=1,⋯,m. For n=1, assuming that the q{sup i}’s form a nondegenerate subsystem, we confirm that the degeneracy of the kinetic matrix eliminates the Ostrogradsky instability. The degeneracy implies, in the Hamiltonian formulation of the theory, the existence of a primary constraint, which generates a secondary constraint, thus eliminating the Ostrogradsky ghost. For n>1, we show that, in addition to the degeneracy of the kinetic matrix, one needs to impose extra conditions to ensure the presence of a sufficient number of secondary constraints that can eliminate all the Ostrogradsky ghosts. When these conditions that ensure the disappearance of the Ostrogradsky instability are satisfied, we show that the EulerLagrange equations, which involve a priori higher order derivatives, can be reduced to a second order system.
 Authors:

 Kavli Institute for Cosmological Physics, The University of Chicago,Chicago, Illinois 60637 (United States)
 Laboratoire de Mathématiques et Physique Théorique,Université François Rabelais,Parc de Grandmont, 37200 Tours (France)
 Research Center for the Early Universe (RESCEU), Graduate School of Science,The University of Tokyo, Tokyo 1130033 (Japan)
 Department of Physics, Tokyo Institute of Technology,2121 Ookayama, Meguroku, Tokyo 1528551 (Japan)
 Laboratoire APC  Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris (France)
 Publication Date:
 Sponsoring Org.:
 SCOAP3, CERN, Geneva (Switzerland)
 OSTI Identifier:
 22572120
 Resource Type:
 Journal Article
 Journal Name:
 Journal of Cosmology and Astroparticle Physics
 Additional Journal Information:
 Journal Volume: 2016; Journal Issue: 07; Other Information: PUBLISHERID: JCAP07(2016)033; OAI: oai:repo.scoap3.org:16499; ccby Article funded by SCOAP3. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 License. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 14757516
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CLASSICAL MECHANICS; COSMOLOGICAL INFLATION; HAMILTONIANS; INFLATIONARY UNIVERSE; INSTABILITY; LAGRANGE EQUATIONS; LAGRANGIAN FUNCTION; LIMITING VALUES; MATRICES; NONLUMINOUS MATTER
Citation Formats
Motohashi, Hayato, Noui, Karim, Laboratoire APC  Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris, Suyama, Teruaki, Yamaguchi, Masahide, and Langlois, David. Healthy degenerate theories with higher derivatives. United States: N. p., 2016.
Web. doi:10.1088/14757516/2016/07/033.
Motohashi, Hayato, Noui, Karim, Laboratoire APC  Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris, Suyama, Teruaki, Yamaguchi, Masahide, & Langlois, David. Healthy degenerate theories with higher derivatives. United States. doi:10.1088/14757516/2016/07/033.
Motohashi, Hayato, Noui, Karim, Laboratoire APC  Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris, Suyama, Teruaki, Yamaguchi, Masahide, and Langlois, David. Wed .
"Healthy degenerate theories with higher derivatives". United States. doi:10.1088/14757516/2016/07/033.
@article{osti_22572120,
title = {Healthy degenerate theories with higher derivatives},
author = {Motohashi, Hayato and Noui, Karim and Laboratoire APC  Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris and Suyama, Teruaki and Yamaguchi, Masahide and Langlois, David},
abstractNote = {In the context of classical mechanics, we study the conditions under which higherorder derivative theories can evade the socalled Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time derivatives, of the form L(ϕ{sup ¨a}, ϕdot {sup a},ϕ{sup a}; qdot {sup i},q{sup i}) with a=1,⋯,n and i=1,⋯,m. For n=1, assuming that the q{sup i}’s form a nondegenerate subsystem, we confirm that the degeneracy of the kinetic matrix eliminates the Ostrogradsky instability. The degeneracy implies, in the Hamiltonian formulation of the theory, the existence of a primary constraint, which generates a secondary constraint, thus eliminating the Ostrogradsky ghost. For n>1, we show that, in addition to the degeneracy of the kinetic matrix, one needs to impose extra conditions to ensure the presence of a sufficient number of secondary constraints that can eliminate all the Ostrogradsky ghosts. When these conditions that ensure the disappearance of the Ostrogradsky instability are satisfied, we show that the EulerLagrange equations, which involve a priori higher order derivatives, can be reduced to a second order system.},
doi = {10.1088/14757516/2016/07/033},
journal = {Journal of Cosmology and Astroparticle Physics},
issn = {14757516},
number = 07,
volume = 2016,
place = {United States},
year = {2016},
month = {7}
}