Intrinsic operator time of stochastic systems

Abstract

Stochastic systems consisting of a very large number of independent elementary processes of the same kind, especially the radioactive decay, are considered as quantum clocks. By adapting the framework of the previously introduced concept of ideal quantum clocks, the time operator for these systems is derived and discussed. It is shown that the standard deviation of time measurement by such a stochastic device is bounded from below by the limitation of the number of elementary processes from physical reasons and by the Planck-time. As a result, any time dilatation, whether caused by extreme speed of the quantum clock or by gravitational fields, increases the standard deviation. This reduces the accuracy of time measurements especially in space navigation. In the vicinity of the Schwarzschild spherical shell of a black hole, time measurements are completely blurred and thus impossible.