Abstract
We construct no-ghost theories of analytic mechanics involving arbitrary higher-order derivatives in Lagrangian. It has been known that for theories involving at most second-order time derivatives in the Lagrangian, eliminating linear dependence of canonical momenta in the Hamiltonian is necessary and sufficient condition to eliminate Ostrogradsky ghost. In the previous work we showed for the specific quadratic model involving third-order derivatives that the condition is necessary but not sufficient, and linear dependence of canonical coordinates corresponding to higher time-derivatives also need to be removed appropriately. In this paper, we generalize the previous analysis and establish how to eliminate all the ghost degrees of freedom for general theories involving arbitrary higher-order derivatives in the Lagrangian. We clarify a set of degeneracy conditions to eliminate all the ghost degrees of freedom, under which we also show that the Euler-Lagrange equations are reducible to a second-order system.
Highlights
Be recast into second-order system without introducing extra variables
In the previous work we showed for the specific quadratic model involving third-order derivatives that the condition is necessary but not sufficient, and linear dependence of canonical coordinates corresponding to higher time-derivatives need to be removed appropriately
Our finding is that elimination of the canonical momenta in the Hamiltonian by the constraints and degeneracy conditions does not kill all the ghost DOFs associated with the higher derivatives and the ghost DOFs still remain
Summary
The specific example of ghost-free theory of quadratic model involving third-order derivatives is presented in [21]. Which is clearly a second-order system for 2 variables q, φ and requires 4 initial conditions for {q, q, φ, φ}. We check the consistency condition 0 = Pd−1 = {Pd−1, H} + ν{Pd−1, Ψ} and obtain a tertiary constraint Pd−2 = 0. It is clear from the linear terms PiQi+1 in the Hamiltonian that we successively obtain the constraints. We expect the system possesses only healthy 2 DOFs. To count the number of DOFs, we classify all the constraints obtained above to first class and second class by checking the Poisson brackets between them, which form the Dirac matrix.
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