The Page-Wootters formalism is a proposal for reconciling the background-dependent, quantum-mechanical notion of time with the background independence of general relativity. However, the physical meaning of this framework remains debated. In this work, we compare two consistent approaches to the Page-Wootters formalism to clarify the operational meaning of evolution and measurements with respect to a temporal quantum reference frame. The so-called "twirled observable" approach implements measurements as operators that are invariant with respect to the Hamiltonian constraint. The "purified measurement" approach instead models measurements dynamically by modifying the constraint itself. While both approaches agree in the limit of ideal clocks, a natural generalization of the purified measurement approach to the case of non-ideal, finite-resource clocks yields a radically different picture. We discuss the physical origin of this discrepancy and argue that these approaches describe operationally distinct situations. Moreover, we show that, for non-ideal clocks, the purified measurement approach yields a time non-local evolution equation, which can lead to non-unitary evolution. Moreover, it implies a fundamental limitation to the operational definition of the temporal order of events. Nevertheless, unitarity and definite temporal order can be restored if we assume that time is discrete.
The Page-Wootters formalism is a proposal for reconciling the background-dependent, quantum-mechanical notion of time with the background independence of general relativity. However, the physical meaning of this framework remains debated. In this work, we compare two consistent approaches to the Page-Wootters formalism to clarify the operational meaning of evolution and measurements with respect to a temporal quantum reference frame. The so-called "twirled observable" approach implements measurements as operators that are invariant with respect to the Hamiltonian constraint. The "purified measurement" approach instead models measurements dynamically by modifying the constraint itself. While both approaches agree in the limit of ideal clocks, a natural generalization of the purified measurement approach to the case of non-ideal, finite-resource clocks yields a radically different picture. We discuss the physical origin of this discrepancy and argue that these approaches describe operationally distinct situations. Moreover, we show that, for non-ideal clocks, the purified measurement approach yields a time non-local evolution equation, which can lead to non-unitary evolution. Moreover, it implies a fundamental limitation to the operational definition of the temporal order of events. Nevertheless, unitarity and definite temporal order can be restored if we assume that time is discrete.
One of the most fundamental open problems in physics is the unification of general relativity and quantum theory to a theory of quantum gravity. An aspect that might become relevant in such a theory is that the dynamical nature of causal structure present in general relativity displays quantum uncertainty. This may lead to a phenomenon known as indefinite or quantum causal structure, as captured by the process matrix framework. Due to the generality of that framework, however, for many process matrices there is no clear physical interpretation. A popular approach towards a quantum theory of gravity is the Page-Wootters formalism, which associates to time a Hilbert space structure similar to spatial position. By explicitly introducing a quantum clock, it allows to describe time-evolution of systems via correlations between this clock and said systems encoded in history states. In this paper we combine the process matrix framework with a generalization of the Page-Wootters formalism in which one considers several observers, each with their own discrete quantum clock. We describe how to extract process matrices from scenarios involving such observers with quantum clocks, and analyze their properties. The description via a history state with multiple clocks imposes constraints on the physical implementation of process matrices and on the perspectives of the observers as described via causal reference frames. While it allows for describing scenarios where different definite causal orders are coherently controlled, we explain why certain non-causal processes might not be implementable within this setting.
The nature of time remains a central problem in fundamental physics, spanning quantum mechanics, thermodynamics, and relativity. In this work, we develop an observer-dependent framework in which time is not assumed as a fundamental background parameter, but emerges operationally from restricted access to a globally timeless quantum state. Building on and extending the Page–Wootters mechanism, we introduce a unified relational formula that shows how three traditionally distinct aspects of temporality arise from the same structure: quantum dynamics from conditioning on physical clock states, thermodynamic irreversibility from partial tracing over inaccessible degrees of freedom, and observer-relative time from the locality of the clock subsystem. No modification of unitary quantum mechanics, collapse postulates, or fundamental time flow is required. Analytical arguments and numerical simulations demonstrate that, in the limit of complete information access, all temporal ordering disappears, while effective Schrödinger dynamics and an arrow of time emerge naturally under informational and observational constraints. These results support a central claim: time is not a property of the universe itself, but a property of descriptions generated by physical observers embedded within it.
One of the most fundamental open problems in physics is the unification of general relativity and quantum theory to a theory of quantum gravity. An aspect that might become relevant in such a theory is that the dynamical nature of causal structure present in general relativity displays quantum uncertainty. This may lead to a phenomenon known as indefinite or quantum causal structure, as captured by the process matrix formalism. Due to the generality of that framework, however, for many process matrices there is no clear physical interpretation. A popular approach towards a quantum theory of gravity is the Page-Wootters formalism, which associates to time a Hilbert space structure similar to spatial position. By explicitly introducing a quantum clock, it allows to describe time-evolution of systems via correlations between this clock and said systems encoded in history states. In this paper we combine the process matrix framework with a generalization of the Page-Wootters formalism in which one considers several agents, each with their own discrete quantum clock. We describe how to extract process matrices from scenarios involving such agents with quantum clocks, and analyze their properties. The description via a history state with multiple clocks imposes constraints on the implementation of process matrices and on the perspectives of the agents as described via causal reference frames. While it allows for scenarios where different definite causal orders are coherently controlled, we explain why certain noncausal processes might not be implementable within this setting.
In timeless formulations of quantum theory, the Page-Wootters proposal (PW) recovers time and dynamics as relative clock-and-world states. By interpreting the total timeless system as a clock entangled with the world, the states having a definite clock reading appear to recover the temporal states of the world as relative states, and dynamics emerges. But an entangled state admits infinitely many decompositions as a superposition of product states, each decomposition corresponding to a different but equally valid time operator of the same clock system. Different valid choices of the time operator for the same clock system lead to different physical histories, where the world states in a history are superpositions of the world states from another history. Fixing one operator as "the" clock time would import the temporal meaning that the Page-Wootters proposal is meant to explain in the first place. Therefore, the interpretation of PW that the world state is passively conditioned on the clock state cannot hold. The timeless state is resolved into time-dependent states by the intrinsic pointer observables of the world, without having to outsource the role of the time operator to a separate clock. Fortunately, the formalism of PW remains valid, provided that it is interpreted in terms of intrinsic pointer observables of the world itself.
In canonical quantum gravity, the Hamiltonian constraint Ĥ|Ψ⟩=0 deprives the universe of any explicit temporal evolution: this is the "problem of time". The Page–Wootters (PaW) mechanism answers it by reconstructing a relational time: from a stationary global state, the evolution of a subsystem appears as a family of conditional states relative to an internal clock ("evolution without evolution"). The Time–Actualization (TA) theory answers the same question with an opposite primitive: time is the process of actualization of quantum potentialities into real events, with a primitive arrow — whose dynamical law remains an open problem — and objective events. We compare the two frameworks by carefully separating the operational target (the same conditional probabilities within a given protocol, as required of an admissible TA dynamics) from the ontological divergence, which bears on three axes: the status of events, the origin of irreversibility, and the dependence on the choice of clock. We argue that TA and PaW are not extensions of one another, but two distinct primitives — the conditional state and the ontic event — and we remain cautious about the metaphysical status of PaW, whose eternalist reading, common though it is, is itself debated.
The problem of describing the dynamics of quantum systems relative to quantum clocks with which they interact is relevant to the physics of quantum reference frames and quantum gravity alike. There have been different approaches that derive a relational time-dependent Schrödinger equation for the system relative to the clock from a time-independent Hamiltonian constraint. We show that some of these approaches, due to Smith and Ahmadi, Gemsheim and Rost, as well as the so-called Born-Oppenheimer approach, can be unified in a single framework for describing conditional quantum states. In particular, the Born-Oppenheimer approach to quantum cosmology is seen to correspond to a type of interacting Page-Wootters formalism. We discuss the issue of unitarity and linearity, or lack thereof, in the relational dynamics, and we conclude with a brief outlook to further developments.