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Problem of time

The problem of time is a foundational conceptual challenge in theoretical physics that emerges when attempting to reconcile quantum mechanics and general relativity in a theory of quantum gravity, primarily due to their fundamentally incompatible notions of time.^[1] In quantum mechanics, time functions as an absolute, external parameter that drives the evolution of wave functions via the Schrödinger equation, serving as a universal backdrop for all physical processes.^[2] By contrast, general relativity treats time as a dynamical component of spacetime geometry, where it can dilate, curve, or become relative depending on gravitational fields, motion, and the distribution of matter and energy.^[1] This discrepancy becomes acute in canonical quantum gravity formulations, where the Wheeler-DeWitt equation—a key constraint equation derived from quantizing the Hamiltonian of general relativity—imposes a "timeless" condition on the wave function of the universe, \hat{H} \Psi = 0, eliminating explicit time evolution and rendering traditional notions of dynamics ill-defined.^[3] Consequently, the problem manifests as the absence of a fixed background time, complicating interpretations of quantum gravity phenomena such as black hole evaporation, the early universe, or the arrow of time, and prompting diverse strategies to resolve it, including emergent time from quantum entanglement, internal clock variables, or timeless records theories.^[1] Pioneering work by physicists like Don Page and William Wootters in the 1980s proposed that time could emerge from the entanglement between a clock subsystem and the rest of a static quantum system, suggesting that apparent temporal evolution arises from correlations within a fundamentally timeless framework.^[4] Ongoing research, including in approaches like loop quantum gravity and the AdS/CFT correspondence, continues to explore these ideas, with implications for understanding whether time is a fundamental feature of reality or an illusory byproduct of deeper quantum structures.^[2]

Foundations of Time in Physics

Time in Classical Mechanics

In classical mechanics, time is treated as an absolute and universal parameter that flows independently of physical events or observers, providing a fixed framework for describing motion. Galileo Galilei pioneered this quantitative approach in the early 17th century through experiments on falling bodies and pendulums, where he measured durations to establish laws of motion, such as the distance fallen being proportional to the square of the time elapsed.^[5] Galileo's work emphasized time as a measure of duration, distinct from spatial or motional influences, enabling precise predictions of trajectories without reliance on qualitative observations.^[6] Isaac Newton built upon this foundation in his Philosophiæ Naturalis Principia Mathematica (1687), explicitly defining absolute time as a fundamental entity separate from relative measures like hours or days.^[7] He described it as "absolute, true, and mathematical time, of itself, and from its own nature, [flowing] equably without relation to anything external."^[8] This conception positions time as an immutable coordinate, unaffected by the positions, velocities, or interactions of bodies in space, ensuring a consistent temporal structure across the universe. Time's role in classical mechanics is primarily as a parameter that orders the evolution of systems in the equations of motion. In Newton's second law, F = ma, force equals mass times acceleration, with time t serving as the independent variable that traces the change in velocity and position along particle paths.^[9] This parameterization allows deterministic solutions, where initial conditions at a given t uniquely determine future and past states, treating time as a neutral backdrop rather than a dynamical quantity. Classical mechanics exhibits time reversibility at the fundamental level, as the equations of motion remain unchanged under the transformation t \to -t, implying no preferred direction for time from the laws themselves.^[10] This symmetry highlights time's neutrality, where reversing temporal flow yields equally valid trajectories, contrasting with emergent irreversibility in macroscopic phenomena.

Time in Quantum Mechanics

In non-relativistic quantum mechanics, time serves as an external, classical parameter that governs the evolution of the quantum state, rather than as a dynamical variable within the theory. The foundational equation describing this evolution is the time-dependent Schrödinger equation,

i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi,

where \psi is the wave function, \hat{H} is the Hamiltonian operator, \hbar is the reduced Planck's constant, and t is a real-valued c-number (ordinary number) parameter, not promoted to an operator.^[11] This formulation treats time as absolute and background-independent, external to the quantum system, enabling the wave function to propagate deterministically while incorporating the probabilistic nature of measurement outcomes, in contrast to the reversible trajectories of classical mechanics.^[12] The time evolution of the quantum state |\psi(t)\rangle is generated by the unitary operator U(t) = e^{-i \hat{H} t / \hbar}, which satisfies |\psi(t)\rangle = U(t) |\psi(0)\rangle for time-independent Hamiltonians, preserving the norm of the state and ensuring unitarity.^[13] In the Heisenberg picture, an alternative formulation, the states remain time-independent while operators evolve as \hat{A}(t) = U^\dagger(t) \hat{A}(0) U(t), yet time itself continues to function as an external label parametrizing this operator dynamics, without being quantized.^[14] This external role underscores the non-relativistic framework's reliance on a fixed temporal backdrop, distinct from the spatial coordinates treated as operators. Standard quantum mechanics lacks a fundamental self-adjoint time operator conjugate to the energy (Hamiltonian), as dictated by Pauli's theorem, which prohibits such an operator for Hamiltonians with spectra bounded from below, such as those ensuring positive energies in non-relativistic systems.^[15] Instead, the time-energy uncertainty relation \Delta E \Delta t \geq \hbar / 2 emerges as a heuristic tool, interpreting \Delta t as the timescale over which the system evolves appreciably and \Delta E as the energy spread, without deriving from a canonical commutator.^[16] Historically, Paul Dirac explored quantizing time as conjugate to energy in his early 1920s work on relativistic extensions, proposing a "quantum time" via Poisson brackets to mirror position-momentum duality, but this led to inconsistencies with the required positive energy spectrum for stable particles, prompting abandonment in favor of the parameter approach.^[17] These foundational aspects highlight the asymmetric treatment of time in quantum mechanics, setting the stage for tensions when unifying with theories where time is dynamical.

Time in General Relativity

In 1905, Albert Einstein introduced special relativity, which unified space and time into a four-dimensional Minkowski spacetime, where time is treated as a coordinate on equal footing with spatial dimensions. This framework was extended in 1915 with the development of general relativity, transforming gravity into the curvature of this spacetime, with the metric becoming dynamical and influenced by the distribution of mass and energy. The geometry of spacetime in general relativity is described by the line element ds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu, where g_{\mu\nu} is the metric tensor and the coordinates x^\mu include the time coordinate t. Unlike in special relativity's flat spacetime, the metric g_{\mu\nu} varies dynamically due to gravitational effects, making time an integral part of the curved geometry rather than a fixed background. Along timelike geodesics, which represent the paths of freely falling observers, the proper time \tau experienced by such an observer is given by d\tau^2 = -ds^2 (using the metric signature -+++), distinguishing it from the coordinate time t, which depends on the choice of reference frame. This relational nature of time highlights its dependence on the observer's path through the gravitational field. Gravitational effects manifest in time dilation, where clocks in stronger gravitational potentials tick more slowly relative to those in weaker fields, a phenomenon tied to the locality of time measurement. For instance, gravitational redshift occurs when light escaping a gravitational well loses energy, appearing redder to distant observers, thereby demonstrating time's relativity to position. This prediction was experimentally verified in the 1959 Pound-Rebka experiment, which measured the frequency shift of gamma rays traversing a 22.5-meter height in Harvard's Jefferson Laboratory tower, achieving agreement with general relativity to within 10% accuracy and later refined to 1%. Such effects underscore that time is not absolute but observer-dependent within curved spacetime. To formalize time evolution in general relativity, the ADM (Arnowitt-Deser-Misner) formalism decomposes spacetime into a 3+1 foliation of spatial hypersurfaces evolving along a time direction. This approach introduces the spatial metric q_{ab} on each hypersurface, along with the lapse function N, which relates the proper time interval to the coordinate time advance, and the shift vector N_i, which accounts for the spatial displacement between adjacent hypersurfaces. The dynamics are governed by the Hamiltonian constraint, ensuring the consistency of this evolution under diffeomorphism invariance, thus portraying time as a parameter driving geometric change rather than an independent entity. In the Newtonian limit of weak fields, this reduces to an approximate absolute time flow.

Origins of the Problem in Quantum Gravity

Conflict Between Theories

The problem of time in quantum gravity arises primarily from a fundamental incompatibility between the treatment of time in quantum mechanics (QM) and general relativity (GR). In QM, time serves as an external, absolute parameter that drives the evolution of the wave function via the Schrödinger equation, assuming a fixed spacetime background against which quantum states evolve.^[18] In contrast, GR describes time as a dynamical coordinate embedded within the geometry of spacetime, where the metric evolves according to Einstein's field equations and is subject to diffeomorphism invariance, meaning no preferred background exists and time is solved for as an internal degree of freedom through constraint equations like the Hamiltonian constraint.^[18] This mismatch becomes acute when attempting to unify the theories, as QM's reliance on an external time conflicts with GR's demand that time be emergent from the gravitational degrees of freedom themselves.^[19] The need for a consistent quantum gravity theory is most evident in regimes where both quantum and gravitational effects are strong, such as near the Planck scale (approximately $10^{-35} meters), including black hole horizons and the Big Bang singularity. At these scales, quantum fluctuations could significantly alter spacetime geometry, yet standard QM cannot accommodate GR's dynamical time without breaking unitarity or causality. For instance, black hole horizons require resolving quantum tunneling effects intertwined with gravitational collapse, while the Big Bang singularity demands a description of time's origin without presupposing an external clock.^[20] A central conceptual tension exacerbates this issue: GR's background independence, where physical predictions are invariant under arbitrary coordinate choices due to diffeomorphism invariance, clashes with QM's fixed background structure, which assumes a pre-existing spacetime metric for defining operators and states. This invariance in GR complicates canonical quantization, as promoting constraints to quantum operators leads to anomalies in enforcing diffeomorphism symmetry on Hilbert space.^[19]^[21] Historically, early attempts to merge QM and GR highlighted these temporal conflicts. For example, quantizing the Klein-Gordon equation on a curved spacetime background—treating gravity classically while applying quantum field theory (QFT) to matter—encountered severe ambiguities in defining time-ordered products and vacuum states, as the absence of a global time-like Killing vector prevents a unique Hamiltonian evolution.^[22] These issues arise because operator ordering in the covariant formalism lacks a canonical time parameter, leading to non-unique renormalization and inconsistencies in correlation functions.^[23] Such semiclassical approaches, while useful for approximations, ultimately fail to resolve the full quantum dynamics. A notable illustration of this hybrid inconsistency appears in the semiclassical calculation of Hawking radiation, where quantum fields are propagated on a fixed classical black hole metric to predict thermal emission. Here, time is treated externally for the quantum fields but dynamically for the background geometry, yielding a temperature T = \frac{[\hbar](/page/H-bar) c^3}{8\pi G M k_B} dependent on the black hole mass M. However, this approximation breaks down when quantum backreaction significantly alters the metric, as the evolving geometry undermines the fixed-time assumption, rendering the full quantum gravity treatment inconsistent without a unified temporal framework.^[24]

Wheeler-DeWitt Equation

The Wheeler-DeWitt equation arises from the canonical quantization of general relativity, building on John Archibald Wheeler's 1957 proposal for a quantum geometrodynamics that treats the geometry of three-dimensional hypersurfaces as the fundamental dynamical variables.^[25] Wheeler envisioned the evolution of spacetime as paths in a "superspace" of all possible three-metrics, setting the stage for a functional formulation of quantum gravity.^[26] Bryce DeWitt formalized this approach in 1967 by applying Dirac's quantization procedure to the constrained Hamiltonian formulation of general relativity, as developed in the ADM formalism.^[27] In this framework, the classical constraints of general relativity, which include the scalar (Hamiltonian) constraint and the vector (diffeomorphism) constraints, are first-class and generate gauge transformations preserving the theory's diffeomorphism invariance. The Hamiltonian constraint takes the form H = G^{ijkl} \pi_{ij} \pi_{kl} - \sqrt{g} \, R(g) \approx 0, where G^{ijkl} is the DeWitt supermetric on superspace, \pi_{ij} are the conjugate momenta to the three-metric g_{ij}, g = \det(g_{ij}), and R(g) is the three-dimensional Ricci scalar curvature.^[27] To quantize, the metric g_{ij}(x) and momenta \pi_{ij}(x) are promoted to operators \hat{g}_{ij}(x) and \hat{\pi}_{ij}(x) satisfying the canonical commutation relations [\hat{g}_{ij}(x), \hat{\pi}_{kl}(y)] = i \hbar \frac{1}{2} (\delta_i^k \delta_j^l + \delta_i^l \delta_j^k) \delta^{(3)}(x - y), acting on a wave functional \Psi[g_{ij}] of the three-metric.^[27] The first-class nature of the constraints implies that physical states must be annihilated by the quantized constraints, yielding the Wheeler-DeWitt equation \hat{H} |\Psi\rangle = 0 from the scalar constraint, alongside similar equations from the vector constraints.^[27] When matter fields, such as a scalar field \phi, are included to provide a possible internal clock, the wave functional becomes \Psi[g_{ij}, \phi], and the Wheeler-DeWitt equation takes the approximate functional form

\hat{H} \Psi[g_{ij}, \phi] = \left[ -16\pi G \hat{G}^{ijkl} \frac{\delta^2}{\delta g_{ij} \delta g_{kl}} + \frac{1}{16\pi G \sqrt{g}} \hat{R}(g) + \hat{H}_\text{matter} \right] \Psi[g_{ij}, \phi] = 0,

where \hat{H}_\text{matter} incorporates the quantized matter Hamiltonian, and the equation is defined on the infinite-dimensional superspace of three-metrics modulo diffeomorphisms.^[27] This equation resembles a time-independent diffusion equation in superspace, lacking any explicit time parameter, as the total Hamiltonian vanishes on physical states due to the constraint structure.^[27] Consequently, the wave function of the universe \Psi[g_{ij}, \phi] is stationary, with no Schrödinger-like unitary evolution in an external time, highlighting the timeless nature at the heart of the problem of time in quantum gravity.^[27] A key challenge in this quantization is the ambiguity in operator ordering within \hat{H}, arising because the classical Poisson brackets become non-commuting quantum operators. DeWitt proposed specific orderings, such as the Weyl ordering for the kinetic term (involving normal ordering with respect to the supermetric) or Laplacian-like orderings, to ensure the operator is Hermitian and covariant under superspace diffeomorphisms, but no unique choice is dictated by the theory.^[27] In formulations involving gauge fixing or additional constraints, second-class constraints may emerge, which are handled by transitioning to Dirac brackets to define a reduced phase space before quantization, preserving the physical content of the Wheeler-DeWitt equation.^[28]

Frozen Formalism

In the Wheeler-DeWitt framework, the frozen formalism arises because the wave function of the universe, denoted as \Psi, lacks an explicit time parameter, rendering it timeless such that any apparent change or dynamics must emerge from intrinsic correlations among degrees of freedom rather than evolution with respect to an external time t. This static nature implies that the quantum state does not evolve in the conventional sense, posing a fundamental interpretive challenge for recovering physical time and causality in quantum gravity. This situation is analogous to the standard time-dependent Schrödinger equation,

i \hbar \frac{\partial \Psi}{\partial t} = \hat{H} \Psi,

but with the partial derivative with respect to time absent, effectively halting unitary evolution and leaving a constraint equation \hat{H} \Psi = 0 that enforces timelessness. The absence of the time derivative term underscores how the Hamiltonian constraint in quantum gravity eliminates the parameter t, contrasting sharply with non-relativistic quantum mechanics where time serves as an external backdrop for dynamics.^[29] A key conceptual resolution involves interpreting time as an emergent illusion derived from conditional probabilities within the timeless wave function, particularly in deparameterized models where matter fields act as internal clocks to parameterize evolution.^[29] For instance, by selecting a monotonic matter degree of freedom as a clock variable, one can derive an effective Schrödinger-like equation conditioned on the clock's reading, thereby restoring a sense of progression through correlations in the quantum state.^[30] The term "frozen formalism" was coined by Karel Kuchař in his 1992 review of canonical quantum gravity, highlighting the pervasive issue across constrained Hamiltonian systems. These foundational discussions underscored the philosophical tension between the diffeomorphism invariance of general relativity and the time-parameterized evolution of quantum theory. As a specific example, in minisuperspace models that reduce quantum gravity to homogeneous cosmologies with finite degrees of freedom, the resulting Wheeler-DeWitt equation remains devoid of explicit time dependence unless additional matter fields are incorporated to serve as a dynamical clock.^[31] Without such fields, the wave function describes a static superposition of scale factors and other geometric variables, illustrating how the frozen problem persists even in simplified settings and necessitates emergent mechanisms for temporal structure.^[31]

Manifestations and Implications

Problem of Observables

In the context of timeless quantum gravity theories, such as those arising from the Wheeler-DeWitt equation, the problem of observables centers on identifying physical quantities that can be meaningfully measured without reference to an external time parameter. These observables must be gauge-invariant, meaning they remain unchanged under the diffeomorphism and Hamiltonian constraints that enforce the theory's timelessness. In general relativity (GR), physical observables are often described as "dirt" smeared on the spacetime manifold—such as the configuration of matter fields that break the symmetry of the bare geometry—requiring gauge-invariant combinations to yield predictable outcomes upon quantization. This challenge is exacerbated by the thin-sandwich problem, which arises in the canonical formulation of GR when attempting to specify initial data for evolution, as the freedom in choosing spacelike hypersurfaces prevents unique determination of gauge-invariant observables without additional relational structure. A key distinction in addressing this issue was introduced by Carlo Rovelli, who differentiated between partial observables and complete observables. Partial observables refer to gauge-dependent quantities, such as coordinate-based positions or momenta of fields (e.g., the value of a matter field at a specific spacetime point), which are useful for describing subsystems but require further specification to become physically measurable. Complete observables, in contrast, are fully gauge-invariant combinations of partial observables, such as the relational distance between two dust particles or the angle between matter configurations relative to a reference frame, ensuring they commute with the constraints and thus represent timeless physical facts. This framework, developed in the early 1990s, underscores that in quantum gravity, predictions must rely on correlations between systems rather than absolute locations, avoiding the pitfalls of background-dependent coordinates.^[32] Mathematically, in the classical limit, observables must satisfy the Poisson bracket algebra with the constraints, { \hat{H}, \hat{O} } = 0, where \hat{H} denotes the total Hamiltonian constraint and \hat{O} the observable; upon quantization, this promotes to the commutator [ \hat{H}, \hat{O} ] = 0, defining Dirac observables that are constants of motion across superspace. For instance, a relational observable might be constructed by smearing a matter field density along geodesics defined by another dynamical system acting as a "clock," such as the proper distance to a monotonically evolving particle, ensuring invariance under spacetime diffeomorphisms. These Dirac observables form a complete set only if they suffice to distinguish all physical states, but in practice, their construction in full quantum gravity remains incomplete due to the infinite-dimensional nature of the constraints. The frozen formalism, where the wave function of the universe appears non-evolving, further complicates measurements by implying that standard quantum observables evolve only conditionally relative to internal degrees of freedom.^[33]

Challenges in Canonical Quantization

In canonical quantization of general relativity, the primary constraints arise from the diffeomorphism invariance of the theory, where first-class constraints generate gauge transformations corresponding to spacetime diffeomorphisms. These constraints must be imposed on the physical states in the quantum theory via the Dirac procedure, ensuring that the algebra of constraints is preserved at the quantum level. However, anomalies can emerge if the quantum commutators fail to replicate the classical Poisson bracket algebra, leading to a breakdown in the structure of the constraints and violating the underlying diffeomorphism symmetry.^[34] A prominent example of such anomalies occurs in the Wheeler-DeWitt equation, the quantum version of the Hamiltonian constraint, where the classical algebra requires [\hat{H}, \hat{H}] = 0, but quantum effects from measure factors in the path integral or operator ordering ambiguities can induce non-zero commutators. This failure disrupts general covariance, as the quantum theory no longer respects the full symmetry group of the classical theory, rendering the quantization inconsistent without modifications. Such anomalies are particularly acute in models coupling gravity to matter fields, where ultraviolet divergences exacerbate the issue.^[35] The problem of time in quantum gravity is a specific manifestation of the broader "problem of constraints," encompassing difficulties in implementing these constraints quantum mechanically, including inconsistencies in semiclassical approximations where quantum backreaction on classical geometry leads to instabilities or violations of energy conditions. In semiclassical gravity, the expectation value of the quantum stress-energy tensor sources the Einstein equations, but fluctuations introduce backreaction effects that can destabilize the spacetime metric, highlighting the limitations of treating gravity classically while matter is quantum. Approaches like loop quantum gravity, building on Ashtekar variables introduced in 1986, address some ultraviolet issues by quantizing via holonomies of the connection, which provide a non-perturbative regularization and resolve singularities in cosmological models. However, time remains an emergent concept in this framework, arising relationally from matter degrees of freedom rather than as a fundamental parameter, underscoring persistent challenges in the canonical approach.^[36] To handle the gauge fixing in path integral formulations of general relativity, the Faddeev-Popov procedure introduces ghost fields to compensate for the infinite-dimensional diffeomorphism group, which generates unphysical degrees of freedom. Unlike finite-dimensional gauge theories, the infinite degrees of freedom in general relativity lead to additional complexities, such as potential anomalies in the ghost sector and difficulties in ensuring a well-defined measure on the space of metrics. This infinite-dimensional nature complicates the quantization, often requiring careful regularization to avoid inconsistencies.

Proposed Resolutions

Timeless Formalisms

Timeless formalisms in quantum gravity address the problem of time by treating the universe as a static, constraint-satisfying quantum state, where temporal evolution emerges relationally rather than parametrically. These approaches, motivated by the frozen formalism arising from the Wheeler-DeWitt equation, posit that the total Hilbert space of the universe is stationary, with no external time parameter governing dynamics. Instead, time arises through internal correlations within the quantum state, allowing for a consistent description of change without invoking a global clock. A seminal example is the Page-Wootters mechanism, proposed in 1983, which formalizes the universe as an entangled quantum state in a product Hilbert space factored into a clock subsystem C and the rest of the system S. The total state is given by

|\Psi\rangle = \int dt \, |t\rangle_C \otimes |\psi_t\rangle_S,

where the clock states |t\rangle_C entangle with the evolving system states |\psi_t\rangle_S, enabling the recovery of apparent time evolution through relational measurements. This factorization enforces a key constraint: the total Hamiltonian of the universe vanishes, H_C + H_S = 0, ensuring the global state remains timeless and stationary while local dynamics appear time-dependent. Within this framework, conditional probabilities for observables o given a clock reading t are computed by projecting the total state onto clock eigenstates, yielding P(o|t) = |\langle o_S, t_C | \Psi \rangle|^2, which recovers Schrödinger-like evolution for subsystem S without reference to an external time. This relational perspective has been extended to constrained systems, where timeless path integrals over all field configurations \phi provide the amplitude for the static wave function:

\Psi[\phi] = \int \mathcal{D}\phi' \, e^{i S[\phi'] / \hbar},

integrating over histories in configuration space without a time foliation, thus embodying the constraint structure directly. Historically, the Hartle-Hawking no-boundary proposal of 1983 complements these ideas by defining a timeless initial condition for the universe via a Euclidean path integral over compact geometries without a singularity or boundary in the early universe. This wave function \Psi[h, \phi] sums over metrics h and matter fields \phi that close off smoothly, providing a ground-state amplitude that avoids specifying an initial time hypersurface.

Emergent Time Approaches

Emergent time approaches in quantum gravity seek to resolve the problem of time by positing that time arises as an effective, approximate parameter from an underlying timeless quantum framework, particularly in regimes where classical general relativity is recovered. These strategies typically involve identifying a physical clock or relational variable that parameterizes evolution, allowing the Wheeler-DeWitt equation to be reformulated in a form resembling the Schrödinger equation. Such methods emphasize the semiclassical limit, where quantum fluctuations are suppressed, and classical spacetime geometry emerges dynamically. One prominent strategy is decoherence-induced time, where interactions with an environment select preferred foliations of spacetime, effectively defining a classical time direction. In this view, quantum superpositions of geometries decohere due to entanglement with matter or gravitational degrees of freedom, favoring hypersurfaces of constant extrinsic curvature, such as those parameterized by York time. York time, defined as the trace of the extrinsic curvature tensor K of spatial slices in the ADM formalism, serves as a monotonic parameter for cosmic expansion, enabling a deparametrized description that approximates classical evolution. This approach contrasts with fully timeless formulations by relying on environmental decoherence to suppress quantum interference and enforce a preferred slicing. Relational dynamics provide another key mechanism, where time emerges from correlations or ratios between changing physical quantities, without invoking an absolute background. A seminal example is the use of incoherent dust as a reference system, which introduces a privileged dynamical frame and foliation through the dust's four-velocity and proper time. In the Brown-Kuchař mechanism, the dust matter field defines relational observables, such as the expansion rate in cosmology, allowing time to parameterize changes relative to the dust distribution. This relational clock resolves the frozen formalism by expressing gravitational evolution in terms of dust-induced proper time, applicable in both classical and quantum regimes.^[37]^[38] In the semiclassical limit, emergent time is further justified via the Ehrenfest theorem, which ensures that expectation values of geometric operators follow classical general relativity trajectories. Applied to quantum gravity, the theorem implies that for states with small uncertainties in the metric and momenta, the average geometry evolves according to the Einstein equations, with time parameterized by a monotonic variable like the volume or scalar field. This correspondence holds approximately until Ehrenfest time scales, where quantum spreading becomes significant, but it establishes time as an effective parameter in the classical regime. Timeless entanglement may underlie this emergence, providing a quantum basis for relational correlations.^[39] Specific realizations appear in loop quantum cosmology, where discrete time steps emerge from the spin foam representation of dynamics. Spin foams encode the quantum evolution of spin networks, discretizing spacetime at the Planck scale, and in cosmological minisuperspaces, this leads to effective time evolution through holonomy-flux algebras. The discrete structure prevents singularities and yields emergent continuous time in the large-volume limit, consistent with semiclassical expectations.^[40] A mathematical cornerstone of these approaches is deparameterization, where a matter degree of freedom, such as a scalar field T, is chosen as the clock variable. The total Hamiltonian constraint H = P_T + H_g + H_m \approx 0 (with P_T the conjugate momentum, H_g the gravitational part, and H_m the matter part excluding the clock) is solved for P_T \approx - (H_g + H_m). Upon quantization, this yields a Schrödinger-like equation for the physical wave function \Psi:

i \frac{\partial}{\partial T} \Psi = \hat{H}_\text{rest} \Psi,

where \hat{H}_\text{rest} = \hat{H}_g + \hat{H}_m acts on the remaining degrees of freedom, restoring unitary time evolution relative to the scalar clock. This formulation is particularly effective in minisuperspace models, bridging the timeless Wheeler-DeWitt equation to dynamical quantum theory.

Thermal Time Hypothesis

The thermal time hypothesis, proposed by Alain Connes and Carlo Rovelli in 1994, posits that in generally covariant quantum theories, physical time emerges from the thermodynamic properties of equilibrium states without presupposing an external clock or time parameter.^[41] This approach draws on the Tomita-Takesaki theory of von Neumann algebras, where a given state determines a modular Hamiltonian flow that serves as the dynamical evolution, effectively recovering a notion of time from the algebra's internal structure.^[41] Central to the hypothesis is the concept of thermal time \tau, derived from the Kubo-Martin-Schwinger (KMS) condition, which characterizes thermal equilibrium states in quantum statistical mechanics.^[41] In this framework, an equilibrium state \omega on the algebra of observables defines a one-parameter group of automorphisms via the modular operator \Delta, associated with the state's density matrix, such that the flow is given by \alpha_t(A) = \Delta^{it} A \Delta^{-it} for any observable A.^[41] This modular flow \alpha_t acts as the time evolution, with the parameter t scaling with the inverse temperature \beta = 1/(k_B T), where k_B is Boltzmann's constant and T is the temperature; thus, equilibrium states inherently encode a relational time that aligns locally with the time of general relativity in stationary spacetimes.^[41] The hypothesis applies particularly to black hole horizons, where the modular flow for a thermal state at the Hawking temperature yields the geometric time evolution near the horizon, providing a relational perspective on black hole thermodynamics.^[41] This relational time helps address aspects of the black hole information paradox by framing information preservation in terms of state-dependent dynamics rather than absolute time, consistent with the unitary evolution in the full quantum theory.^[41] Furthermore, the framework extends to accelerated observers, interpreting the Unruh effect—where an observer with proper acceleration a perceives the Minkowski vacuum as a thermal bath at temperature T = a / (2\pi k_B)—as an instance of thermal time emerging from the modular flow in the Rindler wedge.^[42]

Weyl Time in Scale-Invariant Gravity

In scale-invariant gravity, local Weyl transformations—conformal rescalings of the metric g_{\mu\nu} \to e^{2\lambda} g_{\mu\nu} under arbitrary scalar functions \lambda(x)—extend general relativity by incorporating a dynamical scalar field that compensates for scale changes, rendering the theory invariant under these transformations. This framework, often formulated in the Jordan frame, includes a scalar field \phi coupled to the Ricci scalar, as in the action S = \int d^4x \sqrt{-g} \left[ \phi^2 R + \frac{6}{\phi^2} (\partial \phi)^2 + \mathcal{L}_m \right], where \mathcal{L}_m is the matter Lagrangian minimally coupled to \phi. The scale invariance leads to a conserved Noether current, known as the Weyl current J^\mu, derived from variations under \delta g_{\mu\nu} = 2\lambda g_{\mu\nu} and \delta \phi = \lambda \phi. This current satisfies \nabla_\mu J^\mu = 0 and provides a natural timelike vector field in cosmological contexts, enabling the definition of a preferred time direction.^[43] Weyl time emerges as a parameter \tau obtained by integrating the Weyl current along a timelike path, \tau = \int J^\mu dx_\mu, which acts as a harmonic time coordinate free from the reparameterization invariance that plagues canonical quantum gravity formulations. In loop quantum gravity adaptations of this theory, the discrete spectrum of the scale generator yields quantized values of \tau, interpreted as "quanta of time" with eigenvalues \tau_n = n / \omega, where \omega is a frequency related to the theory's fundamental scale and n is an integer labeling quantum states. Spin-network vertices in the quantum geometry carry these \tau labels, contributing phase factors e^{-i \omega \tau} to the wave function, thus restoring a unitary time evolution absent in the Wheeler-DeWitt equation. This construction resolves the frozen formalism of timeless quantum gravity by providing an intrinsic clock mechanism tied to the geometry's scale degrees of freedom.^[43] The approach addresses key aspects of the problem of time, such as the absence of a Hamiltonian constraint in the scale-invariant limit and the removal of the Immirzi ambiguity—a free parameter in standard loop quantum gravity—through absorption into the Weyl-rescaled Barbero-Immirzi parameter \gamma \to e^{\lambda} \gamma, which does not affect physical dynamics. In cosmological models, Weyl time parameterizes bounce solutions avoiding big bang singularities, with the universe emerging from \tau \to -\infty and expanding unitarily thereafter. For instance, in a flat FLRW background, the scale factor a(\tau) satisfies modified Friedmann equations where \tau drives expansion without invoking external matter clocks. This timeless yet dynamical structure aligns with shape dynamics interpretations, where Weyl time functions as a "pure shape time" decoupled from overall scale, facilitating semiclassical limits via WKB approximations that recover classical geodesics. Empirical implications include predictions for cosmic microwave background power spectra consistent with near-conformal invariance in the standard model, though testable deviations arise at high energies.^[43]

Recent Developments in Quantum Clocks and Entanglement

Recent advancements in quantum clocks have focused on deriving time from entanglement between a clock subsystem and the quantum system it measures, building on the foundational Page-Wootters mechanism where time emerges from conditional probabilities in entangled states. A 2022 review highlights how clock Hamiltonians, which govern the evolution of clock degrees of freedom, resolve key aspects of the problem of time by enabling precise measurements without introducing external time parameters, particularly in non-relativistic quantum mechanics. This approach treats the clock as an internal quantum entity entangled with the system, allowing time to manifest as correlations rather than a fundamental parameter.^[44] In 2025, researchers at Aalto University proposed a quantum theory of gravity formulated as a gauge theory, compatible with the Standard Model of particle physics, where gravitational effects arise from symmetries in an eight-dimensional spinor framework rather than spacetime curvature alone.^[45] Complementing this, a May 2025 study posits that quantum entanglement entropy directly influences spacetime curvature.^[46] A notable experimental proposal from July 2025 by physicists at the Stevens Institute of Technology demonstrates how quantum networks of entangled atomic clocks can probe spacetime curvature, testing the compatibility of quantum mechanics with general relativity on Earth-based setups. These networks leverage distributed entanglement to measure time dilation effects, revealing how gravity warps quantum time flows without requiring space travel, and paving the way for tabletop tests of quantum gravity.^[47]^[48] Further progress includes a June 2025 framework proposing a three-dimensional time structure as the primary fabric of the universe, with space emerging as a secondary effect from temporal symmetries, offering a novel resolution to the problem of time by prioritizing temporal primitives over spatial ones. In October 2025, a relational principle based on variety-coherence optimization was introduced, where spacetime emerges from background-independent informational primitives akin to quantum numbers, replacing traditional geometric coordinates with discrete relational degrees of freedom. These developments address gaps in earlier literature by incorporating unidirectional time flow through decoherence-induced irreversibility and forging interdisciplinary connections, such as linking quantum time emergence to neural decoherence models in neuroscience for understanding consciousness timelines.^[49]^[50]

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https://journals.aps.org/prd/abstract/10.1103/PhysRevD.27.2885
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The Feynman Lectures on Physics Vol. I Ch. 5: Time and Distance
Galileo decided that a given pendulum always swings back and forth in equal intervals of time so long as the size of the swing is kept small. A test comparing ...
[6]
Galileo Galilei - Stanford Encyclopedia of Philosophy
Jun 4, 2021 · He is renowned for his discoveries: he was the first to report telescopic observations of the mountains on the moon, the moons of Jupiter, the ...
[7]
Newton's views on space, time, and motion
Aug 12, 2004 · Absolute, true, and mathematical time, from its own nature, passes equably without relation to anything external, and thus without reference to ...Overview of the Scholium · Newton's Manuscript: De... · The Structure of Newton's...
[8]
Newton's Scholium on Time, Space, Place and Motion
Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called ...
[9]
[PDF] Time in fundamental physics
Oct 3, 2014 · Newtonian mechanics to conclude that there is no time dilation for particles in the CERN accelerator, or, using the flat space–time of.
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[PDF] Time Reversal - Bryan W. Roberts May 30, 2018 - PhilSci-Archive
May 30, 2018 · However, for more exotic forces, it is not generally guaranteed that classical mechanics is time reversal invariant, in spite of frequent ...
[11]
[PDF] Quantum mechanics with space-time noncommutativity
... time coordinates, in the sense that time is regarded as an c-number evolution parameter and not elevated to the level of operators, unlike the spatial ...
[12]
[PDF] 6. Time Evolution in Quantum Mechanics - MIT OpenCourseWare
of a system at a later time t is given by |ψ(t)> = U(t) |ψ(0)>, where U(t) is a unitary operator. An operator is unitary if its adjoint U† (obtained by ...<|separator|>
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[PDF] Time Evolution in Quantum Mechanics
These results are equivalent to expanding the time-evolution operator in energy eigenstates,. U(t) = e−iHt/¯h = Xn e−iEnt/¯h |n><n|.
[14]
Quantum Measurements of Time | Phys. Rev. Lett.
Mar 16, 2020 · Textbook quantum mechanics cannot describe measurements of time, since time is a parameter and not a quantum observable [1] . This is a clear ...
[15]
[quant-ph/9908033] Pauli's Theorem and Quantum Canonical Pairs
Apr 5, 2002 · Thus the conclusion that there exists no self-adjoint time operator conjugate to a semibounded or discrete Hamiltonian despite some well-known ...
[16]
[quant-ph/0105049] The Time-Energy Uncertainty Relation - arXiv
May 11, 2001 · The purpose of this chapter is to show that different types of time energy uncertainty relation can indeed be deduced in specific contexts.
[17]
[PDF] The genesis of dirac's relativistic theory of electrons - Research
PAUL DIRAC's relativistic quantum mechanics for electrons, published early in 1928, is one of the great landmarks in the history of science. According to.
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None
Below is a merged summary of the key sections on the conflict between time in quantum mechanics (QM) and general relativity (GR), combining all information from the provided segments into a single, comprehensive response. To retain maximum detail and clarity, I will use a structured format with tables where appropriate, followed by a narrative summary. The response includes all page references, historical attempts, concepts, and URLs mentioned across the segments.
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[hep-th/0507235] The case for background independence - arXiv
Jul 25, 2005 · The aim of this paper is to explain carefully the arguments behind the assertion that the correct quantum theory of gravity must be background independent.
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[PDF] arXiv:2505.04858v1 [gr-qc] 7 May 2025
May 7, 2025 · This clash between the external time in quantum theory and the dynamical time in GR leads to problems in combining the theories, and is ...
[21]
[PDF] BACKGROUND INDEPENDENCE - arXiv
Oct 5, 2013 · Background Independence is often (but not always) considered in attempts to combine General Relativity (GR) and Quantum. Theory. Standard ...
[22]
[PDF] Quantum Field Theory in Curved Spacetime - John Preskill
Prove that the only static (time-independent) solution to the Klein–Gordon equation in the r > 2M region of the Schwarzschild geometry that satisfies. (i) u ...
[23]
[1401.2026] Quantum fields in curved spacetime - arXiv
Jan 9, 2014 · Abstract:We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime.
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[PDF] arXiv:2504.00516v1 [gr-qc] 1 Apr 2025
Apr 1, 2025 · In our defini- tion of semiclassical approximation, quantum-gravitational effects can be present as long as the black-hole mass is bigger than ...
[25]
On the nature of quantum geometrodynamics - ScienceDirect.com
On the nature of quantum geometrodynamics. Author links open ... Misner, J.A. Wheeler. Annals of Physics, 2 (1957), p. 525. View PDFView articleView in Scopus.
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The Superspace of geometrodynamics | General Relativity and ...
Feb 14, 2009 · Wheeler's Superspace is the arena in which Geometrodynamics takes place. I review some aspects of its geometrical and topological structure.
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Quantum Theory of Gravity. I. The Canonical Theory | Phys. Rev.
Einstein's equations are revealed as geodesic equations in the manifold of 3-geometries, modified by the presence of a "force term." The classical phenomenon of ...
[28]
Derivation of the Wheeler-DeWitt equation from a path integral for ...
Oct 15, 1988 · We show how to derive the Wheeler-DeWitt equation from our path-integral expression. The relationship between the choice of measure in the path ...Missing: 1967 | Show results with:1967
[29]
[1206.2403] Problem of Time in Quantum Gravity - arXiv
Jun 11, 2012 · Strategizing in this Review is not just centred about the Frozen Formalism Problem facet, but rather about each of the eight facets.Missing: 1971 | Show results with:1971
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Summary of each segment:
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[PDF] Time and Structure in Canonical Gravity.1 - PhilSci-Archive
This, in capsule form, is the problem of the frozen formalism of the classical theory. Let me say a little more about the kinds of constraints that appear in.
[32]
[PDF] A Wheeler-DeWitt Equation with Time - arXiv
Nov 3, 2022 · One of these is- sues is, notoriously, the absence of an explicit time. ... Time, for the matter fields, is a multi-fingered (i.e., space ...
[33]
[gr-qc/0110035] Partial observables - arXiv
Oct 6, 2001 · Abstract: We discuss the distinction between the notion of partial observable and the notion of complete observable.Missing: 1991 | Show results with:1991
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Partial and complete observables for Hamiltonian constrained systems
Aug 22, 2007 · We will pick up the concepts of partial and complete observables introduced by Rovelli in Conceptional Problems in Quantum Gravity, ...<|control11|><|separator|>
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[0709.2540] Manifestly covariant canonical quantization of gravity ...
Sep 17, 2007 · Such diffeomorphism anomalies are invisible in field theory, because the relevant cocycles are functionals of the observer's trajectory in ...
[36]
[gr-qc/9506037] Quantal Modifications to the Wheeler DeWitt Equation
Jun 19, 1995 · Abstract: Commutator anomalies obstruct solving the Wheeler-DeWitt constraint equation in Dirac quantization of quantum gravity-matter theory.
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[PDF] QUANTUM NATURE OF THE BIG BANG IN LOOP ... - Abhay Ashtekar
Mar 6, 2006 · Loop quantum gravity is a non-perturbative approach to unifying general relativity with quantum physics while retaining this interplay between ...
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Dust as a Standard of Space and Time in Canonical Quantum Gravity
Sep 1, 1994 · Dust as a Standard of Space and Time in Canonical Quantum Gravity. Authors:J.D. Brown, K.V. Kuchar.Missing: relational | Show results with:relational
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Dust as a standard of space and time in canonical quantum gravity
May 15, 1995 · Abstract. The coupling of the metric to an incoherent dust introduces into spacetime a privileged dynamical reference frame and time foliation.Missing: relational | Show results with:relational
[40]
Phase time and Ehrenfest's theorem in relativistic quantum ... - arXiv
Aug 16, 2024 · Abstract:We revisit the concept of phase time, which has been previously proposed as a solution to the problem of time in quantum gravity.Missing: emergent semiclassical
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[0909.4221] Loop Quantum Cosmology and Spin Foams - arXiv
Sep 23, 2009 · Loop quantum cosmology (LQC) is used to provide concrete evidence in support of the general paradigm underlying spin foam models (SFMs).Missing: steps | Show results with:steps
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[gr-qc/9406019] Von Neumann Algebra Automorphisms and Time ...
Jun 14, 1994 · Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories. Authors:A. Connes, C. Rovelli.
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https://doi.org/10.1016/j.physletb.2019.135106
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https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.897305/full
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Time and Quantum Clocks: A Review of Recent Developments
In the standard formulation of quantum physics time is considered a parameter measured by an external clock and is independent of the observer and the system [7] ...
[46]
New theory of gravity brings long-sought Theory of Everything a ...
May 5, 2025 · Researchers at Aalto University have developed a new quantum theory of gravity which describes gravity in a way that's compatible with the Standard Model of ...Missing: emergent clock synchronization
[47]
Study Suggests Quantum Entanglement May Rewrite the Rules of ...
May 11, 2025 · A new study proposes that quantum information -- encoded in entanglement entropy -- directly influences the curvature of spacetime.
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Quantum Internet Meets Space-Time in This New Ingenious Idea
Jul 21, 2025 · The entangled clocks can test how quantum theory behaves in the presence of curved space-time as predicted by Einstein, or whether our current ...
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Testing quantum theory on curved spacetime with quantum networks
May 27, 2025 · Here we show that entangled clocks can test quantum theory on curved spacetime, and that such experiments are within reach of current technology.
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Time may emerge from quantum entanglement with clocks - Medium
Sep 18, 2025 · Recent research, published in Physical Review Letters, points to time being emergent from the quantum interaction between systems and their ...
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[PDF] arXiv:2411.13469v2 [quant-ph] 5 Mar 2025
Mar 5, 2025 · Alongside, the entanglement entropy shows a logarithmic growth over all time, reflecting the slow dynamics up to a thermal volume-law saturation ...
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New theory proposes time has three dimensions, with space as a ...
Jun 21, 2025 · "This theory demonstrates how viewing time as three-dimensional can naturally resolve multiple physics puzzles through a single coherent ...
[53]
Variety–Coherence Optimization: A Relational Principle for ...
Oct 10, 2025 · Figure 2 schematically illustrates the sequential emergence of spacetime from background-independent informational primitives: initially ...