The Nature of Time: from a Timeless Hamiltonian Framework to Clock Time of Metrology

Abstract

The problem of the Nature of Time is twofold: whether or not time is a fundamental quantity of Nature, and how does clock time of metrology emerge in the experimental description of dynamics. This work strongly supports the fundamental timelessness of Nature. However, the correct view that physics is described by relations between variables does not address the second problem of how time does emerge at the macroscopic scale on the ground of a timeless framework. In this work ordinary Hamiltonian dynamics is recast in a timeless formalism capable to provide a definition of parameter time on the basis of the only generalized coordinates, together with the Hamiltonian invariance on trajectories, and a variational principle. Next, by relaxing the assumption of periodicity of real clocks to the only cyclicity in the phase space, the second problem is addressed. Physical systems, if complex enough, can be separated in a subsystem whose dynamics is described, and another cyclic subsystem which behaves as a clock. The dynamics of the first is mapped in the states of the second cyclic subsystem which provides a discrete approximation of the parameter time. Useful clocks fulfill a stability prescription which guarantees that dynamics is expressed by simple laws also in terms of metric time as naturally happens for parameter time. The two ideas provide a unitary framework capable to account the fundamental timelessness of Nature, and the experimental evidence of time evolution in macroscopic systems experienced by the observers.