skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Healthy degenerate theories with higher derivatives

Abstract

In the context of classical mechanics, we study the conditions under which higher-order derivative theories can evade the so-called Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time derivatives, of the form L(ϕ{sup ¨a}, ϕ-dot {sup a},ϕ{sup a}; q-dot {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 Euler-Lagrange equations, which involve a priori higher order derivatives, can be reduced to a second order system.

Authors:
 [1];  [2];  [3];  [4];  [5];  [6]
  1. Kavli Institute for Cosmological Physics, The University of Chicago,Chicago, Illinois 60637 (United States)
  2. Laboratoire de Mathématiques et Physique Théorique,Université François Rabelais,Parc de Grandmont, 37200 Tours (France)
  3. (France)
  4. Research Center for the Early Universe (RESCEU), Graduate School of Science,The University of Tokyo, Tokyo 113-0033 (Japan)
  5. Department of Physics, Tokyo Institute of Technology,2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551 (Japan)
  6. 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
Resource Relation:
Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2016; Journal Issue: 07; Other Information: PUBLISHER-ID: JCAP07(2016)033; OAI: oai:repo.scoap3.org:16499; cc-by 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)
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:10.1088/1475-7516/2016/07/033.
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:10.1088/1475-7516/2016/07/033.
Motohashi, Hayato, Noui, Karim, Laboratoire APC - Astroparticule et Cosmologie,Université Paris Diderot Paris 7,75013 Paris, Suyama, Teruaki, Yamaguchi, Masahide, and Langlois, David. 2016. "Healthy degenerate theories with higher derivatives". United States. doi:10.1088/1475-7516/2016/07/033.
@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 higher-order derivative theories can evade the so-called Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time derivatives, of the form L(ϕ{sup ¨a}, ϕ-dot {sup a},ϕ{sup a}; q-dot {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 Euler-Lagrange equations, which involve a priori higher order derivatives, can be reduced to a second order system.},
doi = {10.1088/1475-7516/2016/07/033},
journal = {Journal of Cosmology and Astroparticle Physics},
number = 07,
volume = 2016,
place = {United States},
year = 2016,
month = 7
}
  • The general second-order massive field equations for arbitrary positive integer spin in three spacetime dimensions, and their 'self-dual' limit to first-order equations, are shown to be equivalent to gauge-invariant higher-derivative field equations. We recover most known equivalences for spins 1 and 2, and find some new ones. In particular, we find a non-unitary massive 3D gravity theory with a 5th order term obtained by contraction of the Ricci and Cotton tensors; this term is part of an N=2 super-invariant that includes the 'extended Chern-Simons' term of 3D electrodynamics. We also find a new unitary 6th order gauge theory for 'self-dual'more » spin 3.« less
  • Theories with higher order time derivatives generically suffer from ghost-like instabilities, known as Ostrogradski instabilities. This fate can be avoided by considering ''degenerate'' Lagrangians, whose kinetic matrix cannot be inverted, thus leading to constraints between canonical variables and a reduced number of physical degrees of freedom. In this work, we derive in a systematic way the degeneracy conditions for scalar-tensor theories that depend quadratically on second order derivatives of a scalar field. We thus obtain a classification of all degenerate theories within this class of scalar-tensor theories. The quartic Horndeski Lagrangian and its extension beyond Horndeski belong to these degeneratemore » cases. We also identify new families of scalar-tensor theories with the property that they are degenerate despite the nondegeneracy of the purely scalar part of their Lagrangian.« less
  • Ostrogradskii's method for reducing theories with higher derivatives to Hamiltonian form is generalized to make it suitable for application to gauge field theories. A Hamiltonian formalism is constructed for the theory with the Lagrangian L-g(anti ..lambda.. - (1/x/sup 2/)R+aR/sub ..mu..v/R/sup ..mu..v/+bR/sup 2/). The structure of the constraints of this theory is investigated, and it is shown that, depending on the relationship between the parameters anti ..lambda.., x, a, b, five different variants of the theory are possible. In each of them, canonical quantization is performed and a local measure in the functional integral is found. The general form of localmore » measure for an arbitrary boson theory interacting with gravity is established.« less
  • This paper presents a method of constructing the Hamiltonian formulation for a singular Langrangian with higher derivatives. The method allows the specifics of the theory to be taken into account. It is shown that Hamiltonian formulations constructed from the same Lagrangian, but with different methods of introducing the additional generalized coordinates representing the higher derivatives, are related by a canonical point transformation.