Prerequisites for time emergence in quantum and classical mechanics

Is time fundamental or emergent? This is an old question but to date the answer remains elusive. Advocating for the latter, we put forth a set of basic prerequisites necessary for its emergence and examine their implications in quantum and classical mechanics.

In such timeless approaches, the unitary dynamics are derived from the description of a constrained global state of a bipartite system, comprised of a quantum `clock' and a generic quantum `system’. Guided by the fact that every measurement of change in time is made in relation to the reading of a clock, the central notion is that of a system state conditioned on the state of a clock. As a result, the concept of time or, more concretely, dynamics emerges from the inherent correlation between both subsystems.

In particular, based on our postulated prerequisites, we demonstrate the emergence of the Schrödinger, von Neumann and Liouville equation for pure quantum states, quantum density operators and classical phase space densities, respectively. For the classical setting, a Hilbert space formulation is employed and, thereby, shows the formal similarity to the quantum mechanical treatment. If a coupling between clock and system exists then the clock state must be a `quasi-eigenstate’ of the interaction in order to provide an effective time-dependent hermitian potential for the system.

Our work is independent of dimensionality or specific models and, therefore, generalizes and unifies already existing approaches.