Quantum clock and Newtonian time

An extension of standard quantum mechanics is proposed in which the Newtonian time appearing as a parameter in the unitary evolution operator is replaced with the time shown by a `quantum clock'.

Such a clock is defined by the following properties: (a) the time that the clock shows is nondecreasing, (b) the clock ticks at random Newtonian times with random tick sizes, and (c) on average the clock shows the Newtonian time.

We show that the leading term in the evolution equation for the density matrix associated with any quantum clock gives the von Neumann equation.

The leading correction to the von Neumann equation is given by the Lindblad equation generated by the Hamiltonian, but there are higher-order terms that generalize the von Neumann equation and the Lindblad terms.

Modifications to the von Neumann equation are worked out in detail in a parametric family of models for which the tick sizes are gamma distributed.

Lower bounds on the parameters of these quantum clock models are derived using the precision limit of an atomic clock.

An anomalous term in the Ehrenfest theorem for a free particle is derived, which in principle can be used as a basis such models.