Time-System Entanglement and Special Relativity

Abstract: We know that space and time are treated almost equally in classical physics, but we also know that this is not the case for quantum mechanics. A quantum description of both space and time is important to really understand the quantum nature of reality. The Page-Wootters mechanism of quantum time is a promising starting point, according to which the evolution of the quantum system is described by the entanglement between it and quantum temporal degrees of freedom. In this paper, we consider a qubit clock that is entangled with a quantum system due to the Wigner rotation induced by Lorentz transformation. We study how this time-system entanglement depends on the rapidity of the Lorentz boost. We consider the case of a spin-1/2 particle with Gaussian momentum distribution as a concrete example. We also compare the time-system entanglement entropy with the spin-momentum entanglement entropy and find that the former is smaller than the latter.

• The paper introduces a novel analysis of time-system entanglement using the Page-Wootters mechanism to model quantum time within special relativity.
• It quantifies entanglement using metrics like entropy and fidelity, showing how increasing Lorentz boost rapidity leads to greater state separation.
• The study differentiates time-system from spin-momentum entanglement and sets the stage for future work in multi-particle systems and quantum gravity contexts.

Time-System Entanglement and Special Relativity: An Analysis

The paper "Time-System Entanglement and Special Relativity" by Ngo Phuc Duc Loc explores a novel consideration of time-system entanglement, employing the framework of the Page-Wootters (PaW) mechanism for quantum time. This work aims to shed light on the entanglement between quantum systems and quantum time within the context of special relativity, focusing on the entanglement induced by the Wigner rotation resulting from a Lorentz transformation.

Theoretical Framework and Methodology

The foundation for this investigation is laid by addressing the dichotomy between space and time treatment in classical versus quantum physics. The PaW mechanism is a pivot in this discussion, offering a potential resolution by suggesting the evolution of a quantum system is faithfully described by its entanglement with quantum temporal degrees of freedom. In this approach, time is not merely a parameter but interwoven with the quantum state evolution.

Within this framework, the entangled state is expressed as $$\ket{\Psi}\rangle = \int dt \ket{t}_T \otimes \ket{\psi(t)}_S$$, indicating the entanglement between temporal and system states without conventional time-dependent interactions. The influence of a Lorentz transformation is then modeled as entanglement between a qubit clock and a quantum system, utilizing the Wigner rotation to understand this interaction's dependence on the rapidity of the Lorentz boost.

Entanglement Measures and Entropic Analysis

The study explores several entanglement metrics, including entanglement entropy, mutual information, quadratic entanglement entropy, Renyi entropy, and logarithmic negativity, applying them to assess the time-system entanglement. These measures hinge upon calculating fidelity, defined as the overlap between initial and final states post-boost, which decreases as rapidity increases, illustrating increased state separation in the Hilbert space.

Significantly, the entanglement entropy $E(T,S)$ is computed and shown to increase with rapidity. This is interpreted as a direct consequence of the quantum system traversing a greater distance in state space, becoming more entangled with quantum time as rapidity enhances. The entropic saturation observed mirrors a maximum attainable state separation, controlled by parameters such as the Gaussian momentum distribution's width relative to particle mass.

Comparative and Practical Implications

The work contrasts time-system entanglement with the more established spin-momentum entanglement, elucidating that while both follow similar saturation trends, the former maintains a lower entropic value. This distinction underscores the methodological novelty and practical implications of employing a quantum conceptualization of time, especially in relativistic contexts.

From a theoretical standpoint, this study provides initial steps toward integrating quantum time descriptions with relativistic transformations. Though not offering a complete relativistic extension of the PaW mechanism, it establishes groundwork pointing toward meaningful future explorations.

Future Directions

Considering the outlined methodologies and findings, the paper suggests a range of extensions and speculative directions: investigating time-system entanglement in multi-particle frameworks, applications to quantum field theory, and broader cosmological implications. The potential intersection with quantum gravity theories offers exciting possibilities, particularly regarding the interplay between entanglement, time, and the emergent classical phenomena.

In conclusion, the paper by Ngo Phuc Duc Loc holds significance as a tentative exploration into the quantum nature of time through the lens of special relativity, taking meaningful steps towards a deeper understanding of quantum systems' temporal dynamics. Its contributions lie both in theoretical insights and proposed future trajectories, essential for advancing the broader discourse on quantum time and entanglement.