Quantizing time: Interacting clocks and systems (1712.00081v3)

Abstract: This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time $t$. Doing so assigns a conditional state to the rest of the Universe $|\psi_S(t)\rangle$, referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state $|\psi_S(t)\rangle$ satisfies a time-nonlocal Schr\"{o}dinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schr\"{o}dinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of $G/c^4$ and inversely proportional to the distance between the clock and system.

• The paper derives a time-nonlocal Schrödinger equation by incorporating clock-system interactions within the Wheeler-DeWitt framework.
• It provides perturbative solutions and examples, including gravitational corrections at order G/c⁴ and two-level clock models, to illustrate modified system dynamics.
• The findings imply that clock-system entanglement introduces crucial corrections in quantum gravity models, potentially addressing decoherence and time evolution issues.

An Analysis of "Quantizing Time: Interacting Clocks and Systems"

The paper "Quantizing Time: Interacting Clocks and Systems" by Alexander R. H. Smith and Mehdi Ahmadi offers a sophisticated extension of the conditional probability interpretation of time in quantum mechanics, particularly addressing interactions between clocks and the systems they measure. This work is positioned within the broader context of quantum gravity, where one of the prominent challenges is the integration of time as a dynamic component, contrary to its treatment as a classical parameter in standard quantum mechanics.

The discourse is rooted in the canonical quantization of gravity, invoking the Wheeler-DeWitt equation, whereby the total Hamiltonian in the universe nullifies the temporal evolution of physical states, effectively presenting the "problem of time." The authors propose that by treating the clock as an interactive quantum entity, and not merely a classical parameter, a new time-nonlocal form of Schrödinger's equation can be derived. This modified equation accommodates clock-system interactions, leading to a non-trivial modification of system dynamics.

Key Contributions of the Paper

• Time-Nonlocal Schrödinger Equation: The authors derive a time-nonlocal Schrödinger equation wherein the system Hamiltonian is modified by a self-adjoint integral operator. This result arises when the clock-system interaction is integrated into the Wheeler-DeWitt framework.
• Perturbative Solutions and Examples: The paper provides a perturbative solution to the modified Schrödinger equation and explores several illustrative examples of clock-system interactions. For instance, it assesses Newtonian gravitational interactions and distinguishes the resultant dynamics at the order of $G/c^4$. Moreover, the treatment of a system employing two-level clocks elucidates how specific interactions can or cannot influence the relational dynamics.
• Implications on Quantum Gravity: These findings emphasize the necessity of factoring in gravitational interactions in quantum gravity models that utilize the conditional probability interpretation of time. The coupling between clocks and systems implies fundamental corrections to the Hamiltonian, which are negligible but crucial under specific conditions.

Implications and Future Directions

The authors establish that interactions intrinsically modify the perceived evolution of quantum systems over time. This insight bears significant implications for quantum gravity, suggesting that the quintessential Hamiltonian dynamics are modulated by intrinsic quantum correlations shared between clocks and systems, potentially inducing fundamental decoherence mechanisms.

Future research can leverage this groundwork to explore:

• Refinement of Quantum Reference Frames: Further explorations into quantum reference frames can optimize how systems interact with varied clock models, especially considering finite-dimensional clocks.
• Exploration in Alternative Time Interpretations: Integrating alternate interpretations of time within quantum mechanics could yield novel insights into longstanding paradoxes, such as time-energy uncertainty relations.
• Computational methods in Quantum Cosmology: Given the analytical nature of these corrections, computational models could simulate interactions over cosmologically significant timescales, paving the way for dynamic simulations within quantum cosmological frameworks.

In conclusion, Smith and Ahmadi's paper lays robust theoretical foundations for evolving quantum mechanics beyond classical temporality constraints. By postulating that the universe's clock and measured systems are inseparably interwoven, it provides novel perspectives for resolving key challenges in the gravity-quantum mechanics interface. Such advancements foster a deeper understanding of temporal aspects in quantum theory, inviting further exploration into the non-local nature of time.