In quantum theory, the concept of time rests on shaky ground. One way to address this problem is to remove the usual background time parameter as a primitive entity and explain its emergence via correlations between physical systems. This approach was adopted by Page and Wootters [Phys. Rev. D 27, 2885 (1983)], who showed how time can emerge in a stationary quantum universe from the correlations between two of its subsystems, one of them acting as a clock for the other. In this work, I study the robustness of the Page-Wootters construction across different pictures, states of the universe, and clock interactions, clarifying the role and the nature of the correlations between the subsystems of the universe. I start by showing how to formulate the Page-Wootters construction in the Heisenberg picture via a unitary change of basis. I consider both pure and mixed states of the universe and extend the analysis to include interactions between the clock and the other subsystem of the universe. The study reveals what kind of correlations are necessary for the construction to work. Interestingly, entanglement is not required as long as there are no interactions with the clock. The study also shows that these interactions can lead to a nonunitary evolution for some mixed states of the universe. In a simple two-level system, this aspect becomes relevant at scales where one would expect strong relativistic effects. At these scales, I also observe an inversion in the system's direction of time.
In quantum theory, the concept of time rests on shaky ground. One way to address this problem is to remove the usual background time parameter as a primitive entity and explain its emergence via correlations between physical systems. This approach was adopted by Page and Wootters [Phys. Rev. D 27, 2885 (1983)], who showed how time can emerge in a stationary quantum universe from the correlations between two of its subsystems, one of them acting as a clock for the other. In this work, I study the robustness of the Page-Wootters construction across different pictures, states of the universe, and clock interactions, clarifying the role and the nature of the correlations between the subsystems of the universe. I start by showing how to formulate the Page-Wootters construction in the Heisenberg picture via a unitary change of basis. I consider both pure and mixed states of the universe and extend the analysis to include interactions between the clock and the other subsystem of the universe. The study reveals what kind of correlations are necessary for the construction to work. Interestingly, entanglement is not required as long as there are no interactions with the clock. The study also shows that these interactions can lead to a nonunitary evolution for some mixed states of the universe. In a simple two-level system, this aspect becomes relevant at scales where one would expect strong relativistic effects. At these scales, I also observe an inversion in the system's direction of time.
Physical systems are usually assumed to evolve relative to an external time parameter, which is problematic because in quantum theory that parameter is not a physical observable. Page and Wootters (1984) solved this by proposing that the Universe is in a stationary state, eliminating the need for the external time parameter. Instead, their model contains an isolated subsystem, a clock, with which other subsystems are entangled, making the latter appear to evolve relative to different states of the clock. While this resolves the problem of the time parameter, the assumption that the clock is isolated prevents it from being measured, as this requires an interaction with another system. We prove that the clock can be measured while preserving the core features of the Page-Wootters construction. We also discuss clock synchronization.
Time, space and entanglement are the main characters in this work. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we propose a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.
Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, this mechanism explored a stationary pure bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge and an equation of motion can be obtained for a quantum particle conditioned to the measurement of the state of the clock. In this mechanism, time is seen as an inaccessible coordinate and the apparent passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that the internal coherence is the necessary ingredient for the emergence of time in the Page–Wootters model. Connecting it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence. In a second step we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the systems initial conditions.
In previous years several theoretical papers discussed if time can be an emergent property deriving from quantum correlations. Here, to provide insight into how this phenomenon can occur, we present an experiment that illustrates Page and Wootters' mechanism of ``static'' time [D. N. Page and W. K. Wootters, Phys. Rev. D 27, 2885 (1983)], and Gambini et al. for subsequent refinements [R. Gambini et al., Phys. Rev. D 79, 041501(R) (2009)]. A static, entangled state between a clock system and the rest of the universe is perceived as evolving by internal observers that test the correlations between the two subsystems. We implement this mechanism using an entangled state of the polarization of two photons, one of which is used as a clock to gauge the evolution of the second: An ``internal'' observer that becomes correlated with the clock photon sees the other system evolve, while an ``external'' observer that only observes global properties of the two photons can prove it is static.
Among the many proposals to approach the concept of time in quantum theory, the Page-Wootters mechanism has attracted much attention in the last few years. Originally, such a mechanism explored a stationary bipartite non-interacting global system, i.e., a system of interest together with an ancillary clock, to determine how the evolution in time can emerge as an equation of motion for a quantum particle conditioned to the measurement of the state of the clock. After the seminal proposal, many variations of it were considered, and different aspects of the mechanism were elucidated. Our contribution to these investigations is that we analyze such a timeless approach to quantum theory but deriving an equation of motion for a mixed state system that evolves according to its gravitationally induced interaction with a non-ideal quantum clock. The interaction considered is known to describe the gravitational decoherence mechanism, and the clock model is the recently proposed quasi-ideal clock, i.e., one constructed to approximate the time-energy canonical commutation relation. As a result of our considerations, we obtained an equation of motion that is non-linear in nature, dependent on the system's initial conditions.
I. Introduction The problem of time in quantum gravity is the following. General relativity is a diffeomorphisminvariant theory: there is no preferred time variable, and the Hamiltonian constraint HΨ = 0 impliesthat all physical states are ‘frozen’ in the sense that they carry no manifest time dependence [Dirac1964; DeWitt 1967]. Yet quantum mechanics requires time as a background parameter againstwhich evolution is measured. Reconciling these two facts — the timelessness of diffeomorphisminvariant quantum gravity and the temporal structure required for quantum measurement theory—is the problem of time [Isham 1993; Kuchăř 1992]. The problem is not merely philosophical: ithas concrete consequences for the interpretation of quantum cosmological perturbations, includingthose that seed the CMB anisotropies we observe today.One of the most elegant proposed resolutions is the Page–Wootters (PW) mechanism [Page & Wootters 1983]: time is not a background parameter but a relational quantity, encoded in correlationsbetween a ‘clock’ subsystem and the rest of the universe. Physical time is the reading of the clock,and quantum evolution is the conditional state of the system given a particular clock reading. Thisidea has been placed on rigorous mathematical footing by Höhn, Smith & Lock (HSL) [2021], whoproved the equivalence of three apparently distinct quantization procedures — the Dirac constraintquantization, the Page–Wootters mechanism, and the deparametrized Heisenberg picture — undera precise set of conditions. The HSL result reduces the problem of time, for any system satisfyingthose conditions, to the problem of identifying a clock with the right algebraic properties.In Loop Quantum Cosmology (LQC), relational time has been implemented using material degreesof freedom: a minimally-coupled massless scalar field [Ashtekar, Pawlowski & Singh 2006], a GFTcondensate [Gielen & Sindoni 2016], or dust fields [Brown & Kuchăř 1995; Pawłowski 2014]. Allof these are material clocks: dynamical degrees of freedom that must be added to, or identified1within, the matter content of the model. No worked example exists in which the clock arises fromthe algebraic structure of the kinematic Hilbert space itself — as an invariant of the theory’s gaugegroup rather than an additional field. Such an algebraic clock would be conceptually preferable: itrequires no extension of the Hilbert space, its back-reaction on geometry is controlled, and it is notsubject to the multiple-choice ambiguity [Kuchăř 1992] that afflicts material clock choices when thesystem has multiple degrees of freedom.This paper provides the first such example. The LQC bounce, treated via the Pöschl–Teller (PT)effective potential established in the companion series [Hillard 2026a–d], admits a spectral threshold = √2 that is fixed by the zero of the SU(2) Casimir polynomial W() = ³(²−2)/[C·sinh(/2)],where C = 3.60329… is the Hillard constant [Hillard 2026a, Paper 8]. This zero is a representationtheoretic invariant of the kinematic Hilbert space: it exists prior to any dynamical input, beforeany clock is chosen. We show that instantiates the full HSL relational clock construction. TheClock Decomposition Theorem (Section III) establishes the tensor product factorization H̃_kin H H and constructs the covariant POVM on H. The Temporal Trinity Formula (Section IV)is identified as the HSL clock reading in all three quantization pictures, with explicit clock-changemap T (Appendix A). The result holds exactly: is not a perturbative approximation to a clockbut a genuine algebraic invariant.A relational clock that is an algebraic invariant makes a concrete, falsifiable prediction. The signchange multipole * — the CMB multipole at which the LQC bounce imprint changes sign in thepower spectrum [Hillard 2026a] — is precisely the PW temporal origin in Fourier space: * =/_B. When GFT interactions of strength are included, the PT potential is deformed to U_ =−(2−)sech²(), which remains within the exactly solvable PT family (Section VI). The thresholdshifts to () = √(2−), giving the exact non-perturbative prediction:()/ = √(1 − /2) − 1 −/4 + O(²).* (1)*The sign is negative: GFT interactions push * to lower multipoles. The detection threshold forLiteBIRD is _crit 4/*, placing the prediction within reach of the near-term CMB survey (SectionV). This is the first quantitative prediction from a relational clock interpretation of the LQC bounce.A word on the relationship between this paper and the companion series [Hillard 2026a–d]. ThePWinterpretation of the bounce established here depends on W() only through its functional form:W() ³(²−2). Any function with that polynomial zero structure has a scattering threshold at =√2, and any such function in the Hamiltonian constraint generates the relational clock constructionof Sections III–IV. The validity of the LQG derivation of W in [Hillard 2026a] is therefore not aprerequisite for the results of this paper. The companion series and the present paper are mutuallyreinforcing rather than circularly dependent: [Hillard 2026a] provides the physical motivation andprecise form of W; this paper provides the relational interpretation that explains why has thevalue it has.The paper is organised as follows. Section II reviews the relevant background on LQC, the PTeffective potential, and the HSL formalism. Section III establishes the Clock Decomposition Theorem: spectral factorization (Proposition 1), covariant POVM, and temporal origin. Section IVidentifies the Temporal Trinity Formula in all three HSL pictures (Lemma 1, Proposition 2). Section V derives the physical inner product by group averaging and the full observational prediction(Table 2). Section VI proves that the intertwiner deformation is shape-invariant and promotes theprediction to an exact formula (Proposition 3). Appendix A constructs the clock-change map Texplicitly (Proposition A.1, Corollary A.1).22Main results at a glanceClock Decomposition Theorem (Section III): H̃_kin H H + O(). POVM E(d) =|W()|²/‖W‖² d is covariant. = √2 is the temporal origin.Trinity Formula (Section IV): N_total = N + ln(··k_piv/) is a Dirac observable, clockchange invariant. T acts as identity on (Appendix A).Exact prediction (Sections V–VI): ()/ = √(1−/2) − 1 −/4. LiteBIRD detection threshold:_crit 4/*.First algebraic PW clock in LQC (Section V.4): is an SU(2) Casimir invariant, not amaterial d.o.f. Back-reaction is O(); no Hilbert space extension needed.