Non-inertial quantum clock frames lead to non-Hermitian dynamics

Abstract

The operational approach to time is a cornerstone of relativistic theories, as evidenced by the notion of proper time. In standard quantum mechanics, however, time is an external parameter. Recently, many attempts have been made to extend the notion of proper time to quantum mechanics within a relational framework. Here, we use similar ideas combined with the relativistic mass-energy equivalence to study an accelerating massive quantum particle with an internal clock system. We show that the ensuing evolution from the perspective of the particle’s internal clock is non-Hermitian. This result does not rely on specific implementations of the clock. As a particular consequence, we prove that the effective Hamiltonian of two gravitationally interacting particles is non-Hermitian from the perspective of the clock of either particle.

Quantum clock frames: Uncertainty relations, non-Hermitian dynamics and nonlocality in time

“…Given that, we started speculating whether the non-unitarity could result from the internal clock being a non-inertial frame of reference. This seems to be indeed the case according to our more recent analysis of non-inertial quantum clocks [18], which shows that when considering an accelerated or gravitating quantum particle with an internal clock degree of freedom, the evolution given by such an internal clock is non-unitary. We did so in a non-relativistic framework with a relativistic correction due to the mass-energy equivalence.…”

“…In this section, we connect the emergence of non-unitarity just studied, with the existence of acceleration of the clock frame [18]. For that, we need to understand how acceleration affects the Hamiltonian of a system.…”

“…Next is the case of gravitationally interacting clocks. We shall only present the gist of the argument, while the detailed analysis is available in [18]. In previous studies, it was shown that this type of interaction leads to a type of time dilation, although the unitarity of the evolution persists [5,48,49,6,11].…”

“…Time, then, becomes a dynamical quantum variable rather than a mere parameter. The Page-Wootters approach has attracted much interest in recent years leading to major development, generalization and various applications [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].…”

The Universe as a Telecommunication Network

“…It seems worth to investigate the relation between the description of fermions in our model and the connection between curvature, time dilation and non-Hermicity highlighted in Ref. [35] and [40].…”