Classical and quantum mechanics of complex hamiltonian systems: An extended complex phase space approach

Certain aspects of classical and quantum mechanics of complex Hamiltonian systems in one dimension investigated within the framework of an extended complex phase space approach, characterized by the transformation x = x ₁ + ip ₂, p = p ₁ + ix ₂, are revisited. It is argued that Carl Bender inducted $ \mathcal{P}\mathcal{T} $ \mathcal{P}\mathcal{T} symmetry in the studies of complex power potentials as a particular case of the present general framework in which two additional degrees of freedom are produced by extending each coordinate and momentum into complex planes. With a view to account for the subjective component of physical reality inherent in the collected data, e.g., using a Chevreul (hand-held) pendulum, a generalization of the Hamilton’s principle of least action is suggested.