  The Euclidean action with acceleration has been analyzed in [1], hereafter
cited as reference I, for its Hamiltonian and path integral. In this paper, the
state space of the Hamiltonian is analyzed for the case when it is
pseudo-Hermitian (equivalent to a Hermitian Hamiltonian), as well as the case
when it is inequivalent. The propagator is computed using both
creation/destruction operators as well as the path integral. A state space
calculation of the propagator shows the crucial role played by the dual state
vectors that yields a result impossible to obtain from a Hermitian Hamiltonian
acting on a Hilbert space. When it is not pseudo-Hermitian, the Hamiltonian is
shown to be a direct sum of Jordan blocks.
