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Problem of time
Problem of time
Problem of time
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In theoretical physics, the problem of time is a conceptual conflict between quantum mechanics and general relativity. Quantum mechanics regards the flow of time as universal and absolute, whereas general relativity regards the flow of time as malleable and relative.[1][2] This problem raises the question of what time really is in a physical sense and whether it is truly a real, distinct phenomenon. It also involves the related question of why time seems to flow in a single direction, despite the fact that no known physical laws at the microscopic level seem to require a single direction.[3]

Time in quantum mechanics

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In classical mechanics, a special status is assigned to time in the sense that it is treated as a classical background parameter, external to the system itself. This special role is seen in the standard Copenhagen interpretation of quantum mechanics: all measurements of observables are made at certain instants of time and probabilities are only assigned to such measurements. Furthermore, the Hilbert space used in quantum theory relies on a complete set of observables which commute at a specific time.[4]: 759 

Time in general relativity

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In general relativity time is no longer a unique background parameter, but a general coordinate. The field equations of general relativity are not parameterized by time but formulated in terms of spacetime. Many of the issues related to the problem of time exist within general relativity. At the cosmic scale, general relativity shows a closed universe with no external time. These two very different roles of time are incompatible.[4]

Impact on quantum gravity

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Quantum gravity describes theories that attempt to reconcile or unify quantum mechanics and general relativity, the current theory of gravity.[5] The problem of time is central to these theoretical attempts. It remains unclear how time is related to quantum probability, whether time is fundamental or a consequence of processes, and whether time is approximate, among other issues. Different theories try different answers to the questions but no clear solution has emerged.[6]

The frozen formalism problem

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The most commonly discussed aspect of the problem of time is the frozen formalism problem. The non-relativistic equation of quantum mechanics includes time evolution: where is an energy operator characterizing the system and the wave function over space evolves in time, t. In general relativity the energy operator becomes a constraint in the Wheeler–DeWitt equation: where the operator varies throughout space, but the wavefunction here, called the wavefunction of the universe, is constant. Consequently this cosmic universal wavefunction is frozen and does not evolve. Somehow, at a smaller scale, the laws of physics, including a concept of time, apply within the universe while the cosmic level is static.[4]: 762 

Proposed solutions to the problem of time

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Work started by Don Page and William Wootters[7][8][9] suggests that the universe appears to evolve for observers on the inside because of energy entanglement between an evolving system and a clock system, both within the universe.[10] In this way the overall system can remain timeless while parts experience time via entanglement. The issue remains an open question closely related to attempted theories of quantum gravity.[11][6] In other words, time is an entanglement phenomenon, which places all equal clock readings (of correctly prepared clocks – or any objects usable as clocks) into the same history.

In 2013, at the Istituto Nazionale di Ricerca Metrologica (INRIM) in Turin, Italy, Ekaterina Moreva, together with Giorgio Brida, Marco Gramegna, Vittorio Giovannetti, Lorenzo Maccone, and Marco Genovese performed the first experimental test of Page and Wootters' ideas. They confirmed for photons that time is an emergent phenomenon for internal observers of a quantum system but is absent for external observers, which is consistent with the predictions of the Wheeler–DeWitt equation.[10][12][13]

Consistent discretizations approach developed by Jorge Pullin and Rodolfo Gambini have no constraints. These are lattice approximation techniques for quantum gravity. In the canonical approach, if one discretizes the constraints and equations of motion, the resulting discrete equations are inconsistent: they cannot be solved simultaneously. To address this problem, one uses a technique based on discretizing the action of the theory and working with the discrete equations of motion. These are automatically guaranteed to be consistent. Most of the hard conceptual questions of quantum gravity are related to the presence of constraints in the theory. Consistent discretized theories are free of these conceptual problems and can be straightforwardly quantized, providing a solution to the problem of time. It is a bit more subtle than this. Although without constraints and having "general evolution", the latter is only in terms of a discrete parameter that isn't physically accessible. The way out is addressed in a way similar to the Page–Wootters approach. The idea is to pick one of the physical variables to be a clock and ask relational questions. These ideas, where the clock is also quantum mechanical, have actually led to a new interpretation of quantum mechanics — the Montevideo interpretation of quantum mechanics.[14][15] This new interpretation solves the problems of the use of environmental decoherence as a solution to the problem of measurement in quantum mechanics by invoking fundamental limitations, due to the quantum mechanical nature of clocks, in the process of measurement. These limitations are very natural in the context of generally covariant theories as quantum gravity where the clock must be taken as one of the degrees of freedom of the system itself. They have also put forward this fundamental decoherence as a way to resolve the black hole information paradox.[16][17] In certain circumstances, a matter field is used to de-parametrize the theory and introduce a physical Hamiltonian. This generates physical time evolution, not a constraint.

Reduced phase-space quantization constraints are solved first and then quantized. This approach was considered for some time to be impossible, as it seems to require first finding the general solution to Einstein's equations. However, with the use of ideas involved in Dittrich's approximation scheme (built on ideas of Carlo Rovelli) a way to explicitly implement, at least in principle, a reduced phase-space quantization was made viable.[18]

Avshalom Elitzur and Shahar Dolev argue that quantum-mechanical experiments such as the "quantum liar"[19] provide evidence of inconsistent histories, and that spacetime itself may therefore be subject to change affecting entire histories.[20] Elitzur and Dolev also believe that an objective passage of time and relativity can be reconciled and that it would resolve many of the issues with the block universe and the conflict between relativity and quantum mechanics.[21]

One solution to the problem of time proposed by Lee Smolin is that there exists a "thick present" of events, in which two events in the present can be causally related to each other, but in contrast to the block-universe view of time in which all time exists eternally.[22] Marina Cortês and Lee Smolin argue that certain classes of discrete dynamical systems demonstrate time asymmetry and irreversibility, which is consistent with an objective passage of time.[23]

Weyl time in scale-invariant quantum gravity

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Motivated by the Immirzi ambiguity in loop quantum gravity and the near-conformal invariance of the standard model of elementary particles,[24] Charles Wang and co-workers have argued that the problem of time may be related to an underlying scale invariance of gravity–matter systems.[25][26][27] Scale invariance has also been proposed to resolve the hierarchy problem of fundamental couplings.[28] As a global continuous symmetry, scale invariance generates a conserved Weyl current[25][26] according to Noether's theorem. In scale-invariant cosmological models, this Weyl current naturally gives rise to a harmonic time.[29] In the context of loop quantum gravity, Charles Wang et al. suggest that scale invariance may lead to the existence of a quantized time.[25]

Thermal time hypothesis

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The thermal time hypothesis is a possible solution to the problem of time in classical and quantum theory as has been put forward by Carlo Rovelli and Alain Connes. They develop a statistical mechanics model of gravity and characterize thermodynamic time as a vector flow of the statistical state. This thermodynamic time has the common characteristics of time concepts.[30][31]

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Problem of time

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from Grokipedia
The problem of time is a foundational conceptual challenge in theoretical physics that emerges when attempting to reconcile and in a theory of , primarily due to their fundamentally incompatible notions of time. In , time functions as an absolute, external parameter that drives the evolution of wave functions via the , serving as a universal backdrop for all physical processes. By contrast, treats time as a dynamical component of geometry, where it can dilate, curve, or become relative depending on gravitational fields, motion, and the distribution of matter and energy. This discrepancy becomes acute in canonical formulations, where the Wheeler-DeWitt equation—a key constraint equation derived from quantizing the Hamiltonian of —imposes a "timeless" condition on the wave function of the universe, H^Ψ=0\hat{H} \Psi = 0, eliminating explicit and rendering traditional notions of dynamics ill-defined. Consequently, the problem manifests as the absence of a fixed background time, complicating interpretations of phenomena such as evaporation, the early universe, or the , and prompting diverse strategies to resolve it, including emergent time from , internal clock variables, or timeless records theories. Pioneering work by physicists like Don Page and 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. Ongoing research, including in approaches like 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.

Foundations of Time in Physics

Time in Classical Mechanics

In , time is treated as an absolute and universal parameter that flows independently of physical events or observers, providing a fixed framework for describing motion. pioneered this quantitative approach in the early 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. 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. Isaac Newton built upon this foundation in his (1687), explicitly defining absolute time as a fundamental entity separate from relative measures like hours or days. He described it as "absolute, true, and mathematical time, of itself, and from its own nature, [flowing] equably without relation to anything external." 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 is primarily as a that orders the evolution of systems in the . In Newton's second law, F=maF = ma, equals times , with time tt serving as the independent variable that traces the change in and position along particle paths. This parameterization allows deterministic solutions, where initial conditions at a given tt uniquely determine and states, treating time as a neutral backdrop rather than a dynamical quantity. Classical mechanics exhibits at the fundamental level, as the remain unchanged under the transformation t→−tt \to -t, implying no preferred direction for time from the laws themselves. 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 , time serves as an external, classical that governs the evolution of the , rather than as a dynamical variable within the theory. The foundational equation describing this evolution is the time-dependent , iℏ∂ψ∂t=H^ψ,i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, where ψ\psi is the wave function, H^\hat{H} is the Hamiltonian operator, ℏ\hbar is the reduced Planck's constant, and tt is a real-valued c-number (ordinary number) , not promoted to an operator. 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 . The time evolution of the quantum state ∣ψ(t)⟩|\psi(t)\rangle is generated by the unitary operator U(t)=e−iH^t/ℏU(t) = e^{-i \hat{H} t / \hbar}, which satisfies ∣ψ(t)⟩=U(t)∣ψ(0)⟩|\psi(t)\rangle = U(t) |\psi(0)\rangle for time-independent Hamiltonians, preserving the norm of the state and ensuring unitarity. In the , an alternative formulation, the states remain time-independent while operators evolve as A^(t)=U†(t)A^(0)U(t)\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. This external role underscores the non-relativistic framework's reliance on a fixed temporal backdrop, distinct from the spatial coordinates treated as operators. Standard lacks a fundamental time operator conjugate to the (Hamiltonian), as dictated by Pauli's , which prohibits such an operator for Hamiltonians with spectra bounded from below, such as those ensuring positive in non-relativistic systems. Instead, the time- uncertainty relation ΔEΔt≥ℏ/2\Delta E \Delta t \geq \hbar / 2 emerges as a tool, interpreting Δt\Delta t as the timescale over which the system evolves appreciably and ΔE\Delta E as the spread, without deriving from a canonical . Historically, 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. These foundational aspects highlight the asymmetric treatment of time in , 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 ds2=gμν dxμ dxνds^2 = g_{\mu\nu} \, dx^\mu \, dx^\nu, where gμνg_{\mu\nu} is the metric tensor and the coordinates xμx^\mu include the time coordinate tt. Unlike in special relativity's flat spacetime, the metric gμνg_{\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 τ\tau experienced by such an observer is given by dτ2=−ds2d\tau^2 = -ds^2 (using the metric signature −+++-+++), distinguishing it from the coordinate time tt, 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 tied to the locality of time measurement. For instance, occurs when light escaping a gravitational well loses , 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 to within 10% accuracy and later refined to 1%. Such effects underscore that time is not absolute but observer-dependent within curved . To formalize time evolution in general relativity, the ADM (Arnowitt-Deser-Misner) formalism decomposes into a 3+1 of spatial evolving along a time direction. This approach introduces the spatial metric qabq_{ab} on each hypersurface, along with the lapse function NN, which relates the interval to the advance, and the shift vector NiN_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 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. 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. 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. 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 10−3510^{-35} meters), including horizons and the singularity. At these scales, quantum fluctuations could significantly alter geometry, yet standard QM cannot accommodate GR's dynamical time without breaking unitarity or causality. For instance, horizons require resolving quantum tunneling effects intertwined with gravitational collapse, while the singularity demands a description of time's origin without presupposing an external clock. 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 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. Historically, early attempts to merge QM and GR highlighted these temporal conflicts. For example, quantizing the Klein-Gordon equation on a curved background—treating classically while applying (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. These issues arise because operator ordering in the covariant formalism lacks a time parameter, leading to non-unique and inconsistencies in correlation functions. Such semiclassical approaches, while useful for approximations, ultimately fail to resolve the full . A notable illustration of this hybrid inconsistency appears in the semiclassical calculation of , where quantum fields are propagated on a fixed classical metric to predict thermal emission. Here, time is treated externally for the quantum fields but dynamically for the background , yielding a temperature T=ℏc38πGMkBT = \frac{\hbar c^3}{8\pi G M k_B} dependent on the black hole mass MM. However, this approximation breaks down when quantum backreaction significantly alters the metric, as the evolving undermines the fixed-time assumption, rendering the full treatment inconsistent without a unified temporal framework.

Wheeler-DeWitt Equation

The Wheeler-DeWitt equation arises from the of , building on John Archibald Wheeler's 1957 proposal for a quantum that treats the geometry of three-dimensional hypersurfaces as the fundamental dynamical variables. Wheeler envisioned the evolution of as paths in a "superspace" of all possible three-metrics, setting the stage for a functional formulation of . Bryce DeWitt formalized this approach in 1967 by applying Dirac's quantization procedure to the constrained Hamiltonian formulation of , as developed in the ADM formalism. In this framework, the classical constraints of , which include the scalar (Hamiltonian) constraint and the vector () constraints, are first-class and generate gauge transformations preserving the theory's diffeomorphism invariance. The Hamiltonian constraint takes the form H=Gijklπijπkl−g R(g)≈0H = G^{ijkl} \pi_{ij} \pi_{kl} - \sqrt{g} \, R(g) \approx 0
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