Time as a dynamical variable

Abstract

Clocks are dynamical systems, notwithstanding our usual formalism in which time is treated as an external parameter. A dynamical variable which may be identified with time is explicitly constructed for a variety of simple dynamical systems. This variable is canonically conjugate to the Hamiltonian. The complications brought about by the semiboundedness of the Hamiltonian are taken into account and the spectral properties of the time operator are examined. For relativistic systems boosts affect the time variable in a familiar manner. It is pointed out that all physical clock times are cyclic, contrary to Newton's view of a 'uniformly flowing time'.