Time without time: A stochastic clock model

Abstract

We study a classical reparametrization-invariant system, in which "time" is not a priori defined. It consists of a nonrelativistic particle moving in five dimensions, two of which are compactified to form a torus. There, assuming a suitable potential, the internal motion is ergodic or more strongly irregular. We consider quasi-local observables which measure the system's "change" in a coarse-grained way. Based on this, we construct a statistical timelike parameter, particularly with the help of maximum entropy method and FisherRao information metric. The emergent reparametrization-invariant "time" does not run smoothly but is simply related to the proper time on the average. For sufficiently low energy, the external motion is then described by a unitary quantum mechanical evolution in accordance with the Schrödinger equation.

Emergent discrete time and quantization: relativistic particle with extra-dimensions

“…We do not discuss this further here, but refer to Ref. [1] for a detailed and analogous discussion. Instead, with an evolution obviously taking place in such systems, we conclude from these remarks that the space-time description of motion requires a gauge invariant construction of a suitable time, replacing the fictitious proper time τ .…”

“…In distinction, we pointed out in Ref. [1] that the emergent discrete time in our approach naturally leads to a H.-T. Elze "stroboscopic" quantization of the system. Further arguments for a deterministically induced quantization have recently been proposed, for example, in Refs.…”

“…Similarly as in the nonrelativistic example studied in Ref. [1], the lapse function introduces a gauge degree of freedom into the dynamics, which is related to the reparametrization of the evolution parameter s. In fact, the action, Eqs. (1)- (2), is invariant under the set of gauge transformations:…”

“…Recently we have shown that for a nonrelativistic particle with time-reparametrization invariant dynamics one can define quasi-local observables which characterize the evolution in a gauge invariant way [1].…”

Classical trajectories and quantum field theory

“…On the other hand, the opposite route, from classical to quantum, namely the problem of 'quantization' of a classical theory, is a central problem in many fields of research; paradigmatic examples are the ones of gravitation theories and of non-hamiltonian systems, such as dissipative systems. More recently, an alternative, novel perspective has been proposed [1] (see also [2][3][4]) for the route from classical to quantum, the one of the 'emergence' of the quantum-like behavior from a classical frame; namely, the possibility has been considered that classical deterministic systems with dissipation (information loss) may exhibit quantum behavior.…”