On Higher Time Derivative Theory

We review the canonical formulation for Lagrangians with higher time derivative and nonlocality of finite extent, and try to prove the relation between canonical quantization and path integral. The time-ordering is always mixed if we try to obtain the Poisson
bracket for a nonlocal theory in the path integral approach. We also developed the perturbative approach to deal with Lagrangians with arbitrary higher order time derivatives for both bosons and fermions. This approach enables us to find an effective Lagrangian with only first time derivatives order by order in the coupling. The
Hamiltonian is bounded from below whenever the potential is. Finally, we also got a nonlocal worldline action and consider its Virasoro algebra. It is equivalent to the worldsheet theory of a bosonic open string.