A spacetime-covariant approach to inertial and accelerated quantum clocks in first-quantization

TL;DR

This work develops a spacetime-covariant, first-quantized framework for quantum clocks built on Hamiltonian constraints that decompose into positive- and negative-mass sectors, enabling unitary evolution of accelerated clocks when $H^{(-)}\approx0$ and $M>0$. Clock observables are treated with Temporal POVMs, and time evolution is described with respect to the clock's internal proper time via a Dirac-quantized physical Hilbert space; multiple clocks are handled by enforcing a common foliation through a global projector. The authors derive explicit worldfunctions for single clocks and construct the conditional density operator $\rho_{\tau_2|\tau_1}$ to quantify quantum time dilation, showing that classical time-dilation is recovered as peaks while quantum coherences persist, including Gaussian and periodic-modulation features in specific setups. Applications include inertial clocks at rest and in relative motion, and charged clocks in a uniform magnetic field, illustrating how relativistic time-dilation emerges alongside quantum fluctuations and potential observational signatures in low-energy quantum gravity contexts.

Abstract

It is expected that a quantum theory of gravity will radically alter our current notion of spacetime geometry. However, contrary to what was commonly assumed for many decades, quantum gravity effects could manifest in scales much larger than the Planck Scale, provided that there is enough coherence in the superposition of geometries. Quantum Clocks, i.e. quantum mechanical systems whose internal dynamics can keep track of proper-time lapses, are a very promising tool for probing such low-energy quantum gravity effects. In this work, we contribute to this subject by proposing a spacetime-covariant formalism to describe clocks in first quantization. In particular, we account for the possibility of dynamically accelerated clocks via suitable couplings with external fields. We find that a particular decomposition of the (quadratic) clock Hamiltonian into positive- and negative-mass sectors, when attainable, enables one to compute the evolution of the system directly in terms of the clock's proper-time while maintaining explicit covariance. When this decomposition is possible, the evolution obtained is always unitary, even with couplings with external fields used to, e.g., accelerate the clocks. We then apply this formulation to compute the joint time evolution of a pair of quantum clocks in two cases: (i) inertial clocks with relative motion and (ii) charged clocks accelerated by a uniform magnetic field. In both cases, when our clocks are prepared in coherent states, we find that the density matrix $ρ_{τ_2|τ_1}$ describing time-measurements of the two clocks yields not only conditional probabilities whose peaks match exactly the classical expected values for time dilation, but also yields coherent quantum fluctuations around that peak, with a profile which is either a pure Gaussian (i) or a Gaussian combined with a periodic modulation (ii).

Figures (4)

• Figure 1: Support of $\psi_0$ in momentum space, represented by the orange stripes in the future and past light cones. This region is delimited by $(M^-)^2<p^2<(M^+)^2$, where $M^\pm = m + \epsilon^\pm$, and $\epsilon^+$ ($\epsilon^-$) is the maximum (minimum) value in the internal energy spectrum $\sigma(H_C)$ of the clock.
• Figure 2: Classical time-dilation between clocks with relative 3-velocity $v$, evaluated for spacelike separated events $p$ and $q$, with respect to the simultaneity surface $\{t=\tau_1\}$ orthogonal to the 4-velocity of clock 1. Clocks 1 and 2 read proper-times $\tau_1$ and $\tau_2=\tau_1/\gamma$ at events $p$ and $q$, respectively. Our conditional density operator $\rho_{\tau_2|\tau_1}$ peaks exactly at the expected classical values, but also presents quantum coherences and fluctuations.
• Figure 3: Classical worldline of a charged particle in a uniform magnetic field. The particle's proper time $\tau$ as a function of the coordinate time $t$ is simply $\tau=t/\gamma$, where $\gamma$ can be expressed as a function of the energy, $\gamma=E/m$.
• Figure 4: Classical worldlines of two charged particles in a uniform magnetic field. If both particles carry a clock synchronized at $\tau_1=\tau_2=0$ at $t=0$, they will read $\tau_I=t^*/\gamma_I$ at $t=t^*$. Furthermore, if the particles' motions are arranged properly, their worldlines will intersect periodically at certain times, so that one can compare the proper time difference locally, independently of the choice of foliation.