Relational evolution with oscillating clocks

Abstract

A fundamental description of time can be consistent not only with the usual monotonic behavior but also with an oscillating physical clock variable, coupled to the degrees of freedom of a system evolving in time. Generically, one would in fact expect some kind of oscillating motion of a system that is dynamical and interacts with its surroundings, as required for a fundamental clock that can be noticed by any other system. Unitary evolution does not require a monotonic clock variable and can be achieved more generally by formally unwinding the oscillating clock movement, keeping track not only of the value of the clock variable but also of the number of cycles it has gone through at any moment. The specific treatment used here employs concatenations of evolution operators, alternating between forward and backward evolution of the oscillating clock. Because the clock and an evolving system have a common conserved energy, the clock is in different cycles for different energy eigenstates of the system state. As a result, the clock is generically in a quantum state with a superposition of different clock cycles, a key feature that distinguishes oscillating clocks from monotonic time. In addition, coherence could be lost faster than observed, for instance if a system that would be harmonic in isolation is made anharmonic by interactions with a fundamental clock, implying observational bounds on fundamental clocks. Numerical computations show that coherence is maintained over long timescales provided the clock period is much smaller than the system period. A small loss of coherence nevertheless remains and, measured in terms of the relative standard deviation of the system period, is proportional to the ratio of the system period and the clock period. Since the precision of atomic clocks could not be achieved if atomic frequencies would be subject to additional variations from coupling to a fundamental clock, an upper bound on the clock period can be obtained that turns out to be much smaller than currently available direct or indirect measurements of time.