Emergent Time and Time Travel in Quantum Physics

The notion of time travel presents significant challenges to fundamental physics, leading to numerous logical paradoxes across various established physical theories. While many arguments against time travel exist, ranging from classical to quantum perspectives, these arguments often rely on pre-conceived notions of time itself. Quantum gravity, notoriously grappling with the "problem of time," offers a potential framework to re-examine these arguments. The Page-Wootters (PW) formalism, providing an emergent notion of time within the canonical framework, is particularly relevant. This paper initiates a research program using toy models to investigate what emergent time implies about the possibility of time travel. The authors propose that the traditional objections to time travel, frequently rooted in our everyday experience of time and causality, might need revision in the context of quantum gravity and its potential resolution of the problem of time. The paper focuses on exploring simple models that can demonstrate potential phenomenology of time travel with emergent time, rather than directly attempting to derive a fully realistic model of time travel from a canonical quantization of gravity.

The paper reviews existing literature on the problem of time in quantum gravity, highlighting the challenges of defining time and energy operators and discussing no-go theorems related to canonical time operators. It examines different approaches to address this problem, including the use of positive operator-valued measures (POVMs) to define time observables and the Page-Wootters (PW) formalism, which describes time evolution in terms of conditional probabilities. The authors also discuss criticisms of the PW formalism and recent developments that address these criticisms, particularly the re-interpretation of the PW formalism within a gauge-theoretic framework. The introduction mentions various approaches to quantum gravity, including string theory, loop quantum gravity, and others, acknowledging the diversity of perspectives in the field. Finally, the paper briefly touches upon related research avenues, such as Deutsch's closed timelike curves (CTCs), analogue simulations of time travel, and the various concepts of time beyond the typical notions of Newtonian or even general relativistic time.

The core methodology centers around applying the Page-Wootters (PW) formalism to a simplified model system. This model consists of two non-interacting harmonic oscillators, a 'clock' system and a 'residual' system, subject to a Hamiltonian constraint analogous to the Wheeler-DeWitt equation in quantum cosmology. The PW formalism allows the authors to define the evolution of the 'residual' system with respect to the 'clock' system using conditional probabilities. To introduce periodic time and the notion of time travel, the authors incorporate the POVM approach to time operators for the harmonic oscillator, identifying phase and time. Time travel in this context is defined using Novikov's self-consistency conjecture: a system experiences time travel if it returns to the same state after a full clock period. The analysis involves solving the Hamiltonian constraint for the two-oscillator system and evaluating the conditional expectation value of the residual Hamiltonian at different clock times. The authors investigate different forms for the system's wave function, starting with a diagonal form for simplicity and then generalizing to a non-diagonal form. The mathematical tools employed heavily utilize ladder operators for the harmonic oscillator due to the simplicity of applying the algebra. The model, inspired by minisuperspace models, consciously avoids fully incorporating constraints derived from a specific theory of quantum gravity, prioritizing theory-agnosticism. Appendix C explains the construction of wave packets used in the illustration of the results, emphasizing that, despite the inspiration from quantum cosmology, the interpretation of the variables differs from the typical cosmological interpretations.

The analysis reveals that, within the chosen simplified model, the time evolution is trivial, regardless of the chosen clock states or the form of the wave function. The commensurability condition for square-integrable solutions to the Hamiltonian constraint plays a crucial role in this result. The authors demonstrate that even for non-diagonal wave functions (represented by a specific matrix decomposition), the conditional probabilities from the PW formalism result in constant time evolution. This means the system remains in the same state over time, according to the PW definition, a situation interpreted as a particularly uninteresting example of Novikov's self-consistency conjecture. The obtained trivial time evolution marks a significant difference from some related results in quantum cosmology, attributed to the restricted domain used in quantum cosmology, which does not allow for the usage of the same algebraic tools used here. The authors illustrate their findings with figures depicting wave packets for different dispersion parameters, comparing them with similar examples found in quantum cosmology studies.

The surprisingly trivial time evolution in the toy model, while initially disappointing, serves as a proof-of-concept for the feasibility of studying time travel in quantum systems with emergent time. The strong model-dependence of the PW formalism, stemming from its dependence on the choice of the 'universe' state, suggests that more complex systems might exhibit non-trivial time evolution. The authors argue that the very concept of emergent time challenges the traditional arguments against time travel, as these arguments rely on notions of time not applicable in emergent-time frameworks. The study highlights the need for further model building to explore the complexities of emergent time and time travel, moving beyond the oversimplification of the chosen two-oscillator model.

The paper presents a novel approach to investigating time travel within the context of emergent time in quantum physics. While the specific toy model considered yielded a trivial result, demonstrating constant time evolution and a self-consistent yet uneventful time travel scenario, the methodology laid the groundwork for more complex investigations. The authors propose several avenues for future research, focusing on incorporating more subsystems and interactions, studying entropic considerations, clarifying the distinctions between periodic clocks and time travel, and investigating time travel in the context of quantum reference frame transformations. The study concludes that emergent time necessitates a re-evaluation of conventional arguments against time travel and opens up new, rich possibilities for future research.

The primary limitation of the study is the significant simplification inherent in the two-harmonic-oscillator model. This simplicity, while advantageous for analytical tractability, restricts the generality of the results. The absence of interactions between the oscillators and the highly idealized nature of the system prevent the model from capturing the full complexity of potential time-travel scenarios in real-world quantum systems. Furthermore, the specific choice of wave function forms may also influence the outcome. Future research should aim at extending the model to more complex and realistic situations, addressing these limitations.