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Higher derivative theories with constraints : Exorcising Ostrogradski's Ghost

Tai-jun Chen, Matteo Fasiello, Eugene A. Lim, Andrew J. Tolley

TL;DR

Non-degenerate higher-derivative theories generically suffer the Ostrogradski instability due to a Hamiltonian unbounded from below. The authors employ Dirac constraint analysis to show that simply adding auxiliary fields or constraints does not remove the ghost unless the effective phase-space dimensionality is reduced; they illustrate this with the Pais-Uhlenbeck oscillator and, crucially, provide a simple stable example where a proper reduction of phase-space yields a bounded Hamiltonian. They derive a general condition for stability in $N$-th order theories and construct a general two-auxiliary-Lagrangian form that can realize ghost-free dynamics, highlighting when constraints can meaningfully cure the instability. The work thus clarifies when higher-derivative modifications can be viable, outlining concrete pathways to ghost-free constructions with potential implications for modified gravity and related high-derivative field theories.

Abstract

We prove that the linear instability in a non-degenerate higher derivative theory, the Ostrogradski instability, can only be removed by the addition of constraints if the original theory's phase space is reduced.

Higher derivative theories with constraints : Exorcising Ostrogradski's Ghost

TL;DR

Non-degenerate higher-derivative theories generically suffer the Ostrogradski instability due to a Hamiltonian unbounded from below. The authors employ Dirac constraint analysis to show that simply adding auxiliary fields or constraints does not remove the ghost unless the effective phase-space dimensionality is reduced; they illustrate this with the Pais-Uhlenbeck oscillator and, crucially, provide a simple stable example where a proper reduction of phase-space yields a bounded Hamiltonian. They derive a general condition for stability in -th order theories and construct a general two-auxiliary-Lagrangian form that can realize ghost-free dynamics, highlighting when constraints can meaningfully cure the instability. The work thus clarifies when higher-derivative modifications can be viable, outlining concrete pathways to ghost-free constructions with potential implications for modified gravity and related high-derivative field theories.

Abstract

We prove that the linear instability in a non-degenerate higher derivative theory, the Ostrogradski instability, can only be removed by the addition of constraints if the original theory's phase space is reduced.

Paper Structure

This paper contains 16 sections, 59 equations.

Table of Contents

  1. Introduction
  2. Ostrogradski's theorem: an example
  3. Constraints do not cure Ostrogradski's instability if the dimensionality is not reduced
  4. First vs second class constraints
  5. General second order non-degenerate theory with second class constraint
  6. Pais-Uhlenbeck model with constraint
  7. Constraint: $\ddot{q}^2-\dot{q}^2=0$
  8. $N$-th order theory with $M$ auxiliary variables
  9. $M$ auxiliary variables with $2M$ constraints
  10. Exorcising Ostrogradski's ghost by reducing the dimensionality of phase space
  11. A simple example of stable non-degenerate higher derivative theory
  12. General condition of stable non-degenerate higher derivative theory
  13. $P_1$ entering in $\Phi_4$ does not lead to a stable theory
  14. $P_1$ entering at $\Phi_3$ and conditions for stability
  15. Summary
  16. ...and 1 more sections