The Hamilton--Jacobi and Symplectic Analysis for Extended Hořava Gravity

Abstract

The Hamilton-Jacobi [HJ] analysis and the Faddeev-Jackiw [FJ] quantization for the linearized, non-projectable $λR$ model, extended by the lowest-order term of Blas, Pujolàs, and Sibiryakov and the higher-order term involving the square of the Cotton tensor, were carried out. The cases $λ\neq \frac{1}{3}$ and $λ= \frac{1}{3}$ are analyzed. Within the Hamilton-Jacobi framework, we obtain the complete set of Hamiltonians, generalized brackets, fundamental differential, characteristic equations, and gauge transformations of the theory. In the Faddeev-Jackiw framework, a symplectic tensor is constructed from which the fundamental brackets are identified. The HJ and FJ brackets coincide, and from the symplectic tensor, the measure for the quantum path integral is calculated. The differences between the cases $λ\neq \frac{1}{3}$ and $λ= \frac{1}{3}$ at the quantum path integral are discussed.