The clock ambiguity is back with a vengeance

Abstract: Page and Wootters (1983) showed how time and dynamics can emerge in a stationary system containing a clock. Albrecht (1995) later showed, for discrete time, that within this framework any dynamical evolution can be obtained simply by choosing a different clock. Marletto and Vedral (2017) claimed that this ambiguity disappears assuming that the clock and the rest of the world do not interact. I show that their proof relies on an incorrect mathematical assumption. Also, eliminating the ambiguity completely would obstruct spacetime symmetries. Whereas the original clock ambiguity concerns all possible histories of a discrete-time system evolving under arbitrary Hamiltonians, but not the Hamiltonians themselves, I prove a stronger version for continuous and discrete unbounded time: the ambiguity extends to both histories and Hamiltonians, including noninteracting ones. Only the dimension of the Hilbert space remains. One might hope to dismiss the ambiguity as merely perspectival, but I show that this would predict incorrect correlations between outcomes and their records, making even knowledge impossible. Purely relational approaches therefore face both the stronger and the original clock ambiguity problems. The ambiguity is removed by taking into account the physical meaning of the operators.

• The paper demonstrates that the clock ambiguity in the Page-Wootters framework is both unavoidable and maximized, particularly with infinite-dimensional ideal clocks.
• It refutes claims that noninteraction criteria can resolve the ambiguity by showing that unitary transformations can alter the clock-world split without affecting the noninteracting Hamiltonian.
• The findings imply that without assigning physical meaning to operators, emergent approaches in quantum gravity and relational time fail to yield unique, predictive histories.

The Return and Strengthening of the Clock Ambiguity in Relational Quantum Time

Introduction

The paper "The clock ambiguity is back with a vengeance" (2604.21805) offers a rigorous and systematic critique of attempts to resolve the so-called clock ambiguity in the Page-Wootters (PW) formalism for relational quantum time. The author extensively analyzes both historical and contemporary arguments regarding how time might emerge from a stationary universal quantum state and demonstrates—contrary to recent claims—that the ambiguity is both unavoidable and more severe than previously recognized. This essay summarizes the key technical results, clarifies the distinctions between different forms of the ambiguity, and discusses their implications for the foundations of quantum theory, quantum gravity, and approaches to emergent spacetime.

Theoretical Background and the Page-Wootters Formalism

In canonical quantum gravity, the Wheeler–DeWitt equation $H\Psi = 0$ formalizes the purported timelessness of the universe. Page and Wootters proposed a mechanism whereby time can emerge via correlations between a "clock" subsystem and the rest of the universe, with the universal state $\Psi$ taking the form of an entangled integral over clock and world states. Conditionalizing observables on the state of the clock subsystem reproduces Schrödinger-like dynamics, despite the overall stationary constraint.

This framework, motivated by superselection rules and relationalism, has been widely studied and even experimentally illustrated. However, it immediately raises the question of how, and whether, the choice of the clock affects the resulting dynamics—the core of the clock ambiguity problem.

The Clock Ambiguity: Historical and Technical Analysis

Albrecht-Iglesias Ambiguity

Albrecht and Iglesias formulated the "clock ambiguity" concretely: given a timeless PW state $\Psi$ in a Hilbert space with a chosen tensor product structure, one can—for any other possible sequence of world states—find a suitable change of clock so that the same stationary vector yields any desired history or even allows for arbitrary dynamics. This observation leads to a severe underdetermination: the formalism predicts everything, and therefore nothing, unless additional structure is imposed.

Marletto-Vedral's Noninteraction Criterion and its Failure

A more recent response by Marletto and Vedral [2017] argued that requiring the clock and world subsystems to be non-interacting removes the ambiguity, by uniquely specifying the decomposition. This paper provides a detailed mathematical refutation of that assertion:

• It demonstrates via explicit examples that nonlocal (but interaction-preserving) transformations exist, changing the clock-world split yet leaving the noninteracting form of the Hamiltonian intact.
• Even with noninteraction enforced, the decomposition into clock and world subsystems and the extraction of histories is non-unique.
• For infinite-dimensional clock Hilbert spaces (the so-called ideal clocks), the ambiguity is maximal: any pair of histories and evolution laws are connected by unitary transformations that preserve the noninteracting structure.

Thus, the central claim of Marletto and Vedral does not hold, and the clock ambiguity persists.

Strengthening: Distinction Between History and Law

The paper sharply distinguishes the "ambiguity for histories" (the inability to distinguish between different possible entire histories for the world) and the "ambiguity for histories and laws" (the inability even to identify which Hamiltonian or law generated a given history). For ideal clocks, even the latter ambiguity is unavoidable.

Formally, for separable Hilbert spaces, and particularly for infinite-dimensional clock Hilbert spaces, the author proves that

• For any two noninteracting PW systems with identical rest-of-world Hilbert space dimension, there exists a unitary transformation mapping one full history and law onto another.
• The eigenstructure of the total Hamiltonian becomes insensitive to the world Hamiltonian due to the presence of the clock's continuous spectrum.

This maximal ambiguity result generalizes and mathematically strengthens prior forms of the problem.

Implications for Quantum Theory and Emergent Structures

The author carefully addresses why maximal clock ambiguity is both theoretically unsatisfying and phenomenologically disastrous:

• Relationalism and Emergence: Some had hoped that such ambiguity allowed spacetime, locality, or even gauge structures to emerge relationally from raw quantum structure. However, the analysis shows that the ambiguity is so large (for ideal clocks) that, without external input, any physical law or history can be realized; thus, the "emergence" is unconstrained and uninformative.
• Physical Operators and Observables: The ambiguity is only controlled when physical meaning is assigned to operators—i.e., when position, momentum, and other observables are fixed by reference to physical properties rather than spectral data and algebraic relations alone. This is a reduction in symmetry: only those unitary transformations respecting this operator assignment are allowed, curtailing ambiguity to within spacetime (and perhaps gauge) symmetries.
• Observational Consequences: If one refuses to assign such operator meanings (to maintain a "fully relational" viewpoint), the consequences are extreme: there would be no correlation between records (memories, measurement devices) and external world states. In particular, empirical science, memory, and consistent histories would all be impossible, as there would be no fact of the matter connecting the state of a system to records about it.

Role of Symmetry and Reference Frames

The analysis clarifies that some ambiguity is physically meaningful and even required—namely, that arising from fundamental spacetime symmetries (translations, boosts, diffeomorphisms). However, the group of all unitary transformations is vastly larger, and the relational framework as formulated cannot differentiate between physically meaningful and unphysical forms of ambiguity.

Practical and Conceptual Effects

Given these technical results, any approach to emergent quantum gravity, spacetime, or "law without law" scenarios that presumes history can be relationally extracted from a stationary quantum state without additional physical input is mathematically unjustified. Rather, the assignment of physical operators is needed to extract consistent, predictive histories.

For practical quantum foundations, this suggests that the operational specification of observables and reference frames is not a dispensable addition but a requirement for physical meaning. In quantum gravity, approaches that rely on the PW mechanism or analogous devices must supplement their frameworks to uniquely identify observables or else face insurmountable ambiguity.

Conclusion

This paper demonstrates, with precise technical proofs, that the clock ambiguity in the relational quantum time program is not relieved—indeed, it is maximized—when only abstract Hilbert space and noninteracting decomposition data are given. Only the prescription of operators with physically specified meaning resolves the ambiguity to the level allowed by spacetime and gauge symmetries. Abandoning this practice leads not to interpretational flexibility but to a loss of empirical content and the collapse of scientific practice. These findings critically delimit the scope for relational and emergent approaches to quantum time and quantum gravity (2604.21805).