Semiclassical limit problems with concurrent use of several clocks in quantum cosmology

Abstract

We revisit a recent proposal for a definition of time in quantum cosmology, to investigate the effects of having more than one possible type of clock "at the same time". We use as test tube an extension of Einstein gravity with a massless scalar field in which the gravitational coupling $G_N$ is only a constant on-shell, mimicking the procedure for $\Lambda$ in unimodular gravity. Hence we have two "simultaneous" clocks in the theory: a scalar field clock, and the conjugate of $G_N$. We find that attempts to use two coherent clocks concurrently are disastrous for recovering the classical limit. The Heisenberg relations, instead of being saturated, are always realized abundantly above their bound, with strong quantum effects expected at least in parts of the trajectory. Semi-classical states always result from situations where we effectively impose a single clock, either by making the other clock a failed clock (i.e. by choosing a state where its conjugate constant is infinitely sharp), or by choosing a basis of constants where all clocks but one are redundant, i.e. motion or change in phase space does not occur with the passing of their "times". We show how this conclusion generalizes to fluids with any equation of state. It also applies to systems where "sub-clocks" of the same type could be used, for example in mixtures of different fluids with the same equation of state.