Hamiltonian analysis of higher derivative scalar-tensor theories

TL;DR

This paper performs a Hamiltonian analysis of higher derivative scalar-tensor theories, clarifying the role of degeneracy in eliminating Ostrogradski ghosts and identifying the physical degrees of freedom.

Contribution

It provides a simplified Hamiltonian formulation for a broad class of scalar-tensor theories, including degenerate and nondegenerate cases, and discusses the implications for ghost elimination.

Findings

Degenerate theories have reduced phase space due to constraints.

Nondegenerate theories possess four degrees of freedom.

Hamiltonian analysis confirms ghost elimination in degenerate cases.

Abstract

We perform a Hamiltonian analysis of a large class of scalar-tensor Lagrangians which depend quadratically on the second derivatives of a scalar field. By resorting to a convenient choice of dynamical variables, we show that the Hamiltonian can be written in a very simple form, where the Hamiltonian and the momentum constraints are easily identified. In the case of degenerate Lagrangians, which include the Horndeski and beyond Horndeski quartic Lagrangians, our analysis confirms that the dimension of the physical phase space is reduced by the primary and secondary constraints due to the degeneracy, thus leading to the elimination of the dangerous Ostrogradski ghost. We also present the Hamiltonian formulation for nondegenerate theories and find that they contain four degrees of freedom, as expected. We finally discuss the status of the unitary gauge from the Hamiltonian perspective.