Healthy degenerate theories with higher derivatives

Hayato Motohashi; Karim Noui; Teruaki Suyama; Masahide Yamaguchi; David Langlois

doi:10.1088/1475-7516/2016/07/033

paper

Healthy degenerate theories with higher derivatives

arXiv:1603.09355 · doi:10.1088/1475-7516/2016/07/033

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 $L (ϕ ¨^{a}, ϕ ˙^{a}, ϕ^{a}; q ˙^{i}, q^{i})$ with $a = 1, \dots, n$ and $i = 1, \dots, m$ . For $n = 1$ , assuming that the $q^{i}$ '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.

26 pages; Added footnotes 3 and 4, fixed typos in Appendix B 2, matches published version in JCAP

Healthy degenerate theories with higher derivatives

Abstract

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