Clock Ambiguity: Internal Time & Operational Challenges

• Clock Ambiguity is the non-unique assignment of time when clocks are defined by internal physical subsystems, measurement channels, or synchronization conventions.
• It challenges predictivity in theories by allowing different effective dynamics and histories based on the choice of clock, impacting quantum gravity and cosmology.
• In metrology and signal processing, clock ambiguity manifests as phase, delay, and offset uncertainties, necessitating advanced methods like Kalman filtering for robust corrections.

Clock ambiguity appears in several technically distinct settings. In time-reparameterization-invariant and Page–Wootters-type frameworks, the choice of internal time or clock can change the effective dynamics associated with a single stationary state (0708.2743, Stoica, 23 Apr 2026). In gravitating superpositions, a coherent superposition of the large and small object introduces an ambiguity in the definition of a common time for both states (Oosterkamp et al., 2013). In synchronization theory, spacetime curvature implies obstacles to synchronization, while real symbol-handling requires a notion of “logical synchronization” distinct from Einstein synchronization (Myers et al., 2019). In metrology and sensing, ambiguity arises when phase readout leaves the inversion region of Ramsey interrogation (Kohlhaas et al., 2015) and when bistatic systems suffer timing and carrier-frequency offsets due to unsynchronized clocks (Wang et al., 12 Jul 2025). Taken together, these uses suggest a common theme: time becomes operationally non-unique when it is tied to physical subsystems, measurement channels, or reference conventions rather than supplied as a fixed external parameter.

1. Internal time and the original ambiguity in reparameterization-invariant theories

In time-reparameterization-invariant theories, the total Hamiltonian vanishes as a first-class constraint, and in Dirac quantization one imposes

$H\,|\Psi\rangle=0,\qquad H=0.$

Time is then recovered by splitting the full Hilbert space into a clock subsystem and the rest,

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$

choosing a clock basis $\{\ket{t_i}_C\}$, and conditioning the physical state on the clock reading:

$\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$

For sufficiently fine-grained clock labels, successive conditional states define unitary steps and an effective Hamiltonian through

$$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$

Albrecht and Iglesias show that a different choice of clock corresponds to a different tensor-factorization and basis, and that for any desired new set of coefficients $\beta_{ij}$ there exists a unitary $M$ on $\mathcal H_S$ such that the same physical state can be rewritten with those coefficients. Hence the same superspace state can be seen, under a different clock choice, as evolving under any desired Hamiltonian $H'_{\rm eff}(t')$ (0708.2743).

This construction gives the original meaning of clock ambiguity in the problem of time. The ambiguity is not merely a relabeling of a fixed dynamics: the apparent laws of physics extracted from the same constrained state depend on which subsystem is designated as clock. Albrecht and Iglesias argue that this threatens predictivity unless one supplements the formalism with additional selection criteria. Their proposed criterion is quasi-separability, defined by a Hamiltonian of the form

$H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$

because small interaction terms allow subsystems to maintain their identity. They further compare a local quantum field theory Hamiltonian with a completely random Hamiltonian and argue that any random Hamiltonian, constructed in a sufficiently large space, can yield a “good enough” approximation to a local field theory (0708.2743).

2. Stronger theorems, noninteracting clocks, and the status of the ambiguity

The Page–Wootters framework begins from a globally stationary state on

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$0

with Wheeler–DeWitt constraint

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$1

and, in the noninteracting case,

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$2

A time operator $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$3 on the clock subsystem defines generalized eigenvectors $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$4, and the conditional state

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$5

obeys an effective Schrödinger equation. In the ideal non-interacting case, this reduces to

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$6

Stoica argues that the ambiguity is stronger than the original Albrecht result: for continuous and discrete unbounded time, the ambiguity extends to both histories and Hamiltonians, including noninteracting ones, and only the dimension of the Hilbert space remains (Stoica, 23 Apr 2026).

A central controversy concerns whether the ambiguity disappears if the clock and the rest of the world do not interact. Marletto and Vedral claimed that uniqueness can be recovered by demanding that, after a change of tensor-product structure, the transformed Hamiltonian still has no interaction term. Stoica argues that their proof relies on an incorrect mathematical assumption. The counterexample is two qubits with

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$7

for which the swap operator $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$8 is nonlocal, satisfies $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$9, and leaves the Hamiltonian noninteracting:

$\{\ket{t_i}_C\}$0

Nonlocality of the transformation therefore does not force $\{\ket{t_i}_C\}$1 (Stoica, 23 Apr 2026).

Stoica also argues that the ambiguity cannot simply be eliminated by fiat. Spacetime symmetries require some relativity of clock versus system, while maximal ambiguity would predict incorrect correlations between outcomes and their records, making even knowledge impossible. The proposed way out is to invoke the physical meaning of operators: one specifies a complete set of generating observables whose physical interpretations are fixed, and these observables induce a unique tensor-product structure. This suggests that the mathematical freedom of clock choice must be constrained by physically labeled observables if a unique, physically meaningful clock–system split is to be recovered (Stoica, 23 Apr 2026).

3. Relational clock changes and internal clocks in quantum cosmology

A distinct response to the multiple-choice problem is to treat different clock choices as different perspectives on a single clock-choice-neutral Dirac quantized theory. In the parametrized free particle, one has canonical pairs $\{\ket{t_i}_C\}$2 and $\{\ket{t_i}_C\}$3 subject to the first-class constraint

$\{\ket{t_i}_C\}$4

Dirac quantization promotes this to

$\{\ket{t_i}_C\}$5

on the kinematical Hilbert space $\{\ket{t_i}_C\}$6, and the physical Hilbert space is the kernel of $\{\ket{t_i}_C\}$7. One may then deparametrize with respect to the $\{\ket{t_i}_C\}$8-clock, where the gauge-invariant observables are

$\{\ket{t_i}_C\}$9

or with respect to the $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$0-clock, where the time-of-arrival observable is

$\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$1

Because $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$2 is singular at $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$3, the $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$4-clock requires a split into left- and right-moving sectors and a regularization of the inverse-momentum operator. The switch between the two clock choices is implemented by first trivializing the constraint and then projecting onto the corresponding gauge-fixing surface; the concatenated map

$\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$5

acts as the clock-switch map between reduced theories (Hoehn et al., 2018).

Within this framework, the multiple-choice problem is reframed as a feature rather than an inconsistency. The Dirac Hilbert space carries all clock choices simultaneously, in analogy to a manifold carrying all coordinate charts. Yet the switch is not classically innocuous: quantum uncertainties lead to discontinuity in the relational dynamics when switching clocks, because the relevant relational observables do not commute. The mismatch between “when clock $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$6 reads $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$7, clock $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$8 reads $\ket{\psi(t_i)}_R\equiv\sum_j\alpha_{ij}\ket{j}_R.$9” and the inverse statement is therefore of order $$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$0 (Hoehn et al., 2018).

Quantum cosmology provides a second internal-clock setting. For flat, homogeneous minisuperspace models with a perfect fluid in Schutz formalism, the first-order action is

$$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$1

with

$$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$2

Because the fluid Hamiltonian enters linearly, the constraint $$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$3 becomes the Schrödinger equation

$$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$4

so the fluid variable $$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$5 acts as an internal time. However, operator-ordering ambiguity remains. For $$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$6, requiring manifest covariance under point-canonical transformations and invariance under arbitrary lapse rescalings fixes the curvature coupling to

$$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$7

and infinitely many reorderings collapse to a single physical quantum theory. For $$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$8, the lapse rescaling symmetry is lost in the quantum theory, and the description remains essentially ambiguous (Mondal et al., 20 Jan 2025).

4. Gravitational time-dilation ambiguity and Penrosian collapse

A different notion of clock ambiguity arises in the thought experiment of a clock containing a massive object in a superposition of radii. The object has the same mass $$U_i=\exp\!\bigl(-\tfrac{i}{\hbar}H_{\rm eff}(t_i)\,\Delta t\bigr).$$9 in both branches but different radii $\beta_{ij}$0 and $\beta_{ij}$1. In branch $\beta_{ij}$2, the weak-field metric component is

$\beta_{ij}$3

and the proper time over a common coordinate interval $\beta_{ij}$4 is

$\beta_{ij}$5

The proper-time difference is therefore

$\beta_{ij}$6

Since a quantum state carrying rest energy $\beta_{ij}$7 acquires a phase $\beta_{ij}$8, the branch-dependent proper times generate a relative phase

$\beta_{ij}$9

The paper asserts that a coherent superposition of the large and small object introduces an ambiguity in the definition of a common time for both states, and that the wave function collapse will occur when this phase difference becomes of order unity. Setting $M$0 gives

$M$1

equivalently $M$2, with gravitational self-energy $M$3 (Oosterkamp et al., 2013).

This argument is conceptually distinct from the problem-of-time literature. In ordinary quantum mechanics, a single global time parameter implies that only energy differences matter, while the mean energy is physically irrelevant. Here each branch carries its own proper time, so even the absolute rest-energy $M$4 enters the phase accumulation. The result is a concrete mechanism by which time-dilation differences are translated into a loss of coherent unitary evolution. The derivation depends on the weak-field, slow-motion limit, first-order expansion in $M$5, neglect of back-reaction or stress beyond the static gravitational potential, and the heuristic Penrosian criterion $M$6 (Oosterkamp et al., 2013).

5. Synchronization, curvature, and the operational construction of time

Einstein synchronization defines time through round-trip light signaling. If clock $M$7 emits at reading $M$8, clock $M$9 receives at $\mathcal H_S$0, and $\mathcal H_S$1 receives the echo at $\mathcal H_S$2, then Einstein synchronization requires

$\mathcal H_S$3

Logical synchronization, by contrast, is formulated for symbol-handling agents equipped with adjustable cyclic clocks and one-way clock tapes. If agent $\mathcal H_S$4 sends a symbol on square $\mathcal H_S$5 and agent $\mathcal H_S$6 records it on square $\mathcal H_S$7, the transmission relation is

$\mathcal H_S$8

Logical synchronization requires that arrivals occur during a receptive phase of the receiving clock:

$\mathcal H_S$9

The paper emphasizes that the timing of symbol arrival within the allowed phase cannot be registered by the process that recognizes distinct symbols, so phase steering requires an additional analog sensing mechanism. In generic curved spacetime, Einstein synchronization can be achieved, even allowing clock-rate adjustments, only for selected pairs of clocks; synchronizing some pairs necessarily precludes synchronizing other pairs (Myers et al., 2019).

The “horse race” example sharpens the point. Pairwise comparisons of three arrivals can yield

$H'_{\rm eff}(t')$0

which no single real-valued time function $H'_{\rm eff}(t')$1 can reproduce, because transitivity would require

$H'_{\rm eff}(t')$2

This does not merely limit accuracy; it blocks the existence of a single global time assignment adequate to all comparisons. A plausible implication is that clock ambiguity in this operational sense concerns not only synchronization conventions but also the admissibility of event orderings constructed from real devices (Myers et al., 2019).

A related but more dynamical issue appears in the exact analysis of Jefimenko’s longitudinal electromagnetic clock in uniform motion. The total period satisfies the textbook formula

$H'_{\rm eff}(t')$3

but the derivation also produces an extra term in the infinitesimal time element,

$H'_{\rm eff}(t')$4

which cancels only in the full period. Accordingly, one half-cycle is slightly shorter than $H'_{\rm eff}(t')$5 and the other half-cycle is slightly longer by the same amount. The overall retardation is exact, but the internal slowing is non-uniform because of Heaviside-field anisotropy and Lorentz contraction of the oscillation amplitude (Redzic, 2015).

6. Readout ambiguity in precision clocks and offset ambiguity in sensing systems

In a Ramsey-type atomic clock, the measurable quantity is the population difference or transition probability after two $H'_{\rm eff}(t')$6 pulses separated by interrogation time $H'_{\rm eff}(t')$7:

$H'_{\rm eff}(t')$8

Because $H'_{\rm eff}(t')$9 and $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$0 are two-to-one mappings on $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$1, they are only injective over an interval of length $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$2. In practice one chooses the inversion region

$H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$3

within which the measured population difference determines $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$4 unambiguously. If the local-oscillator phase drifts outside that window, the same measured $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$5 can correspond to two inequivalent phases, leading to fringe hops and a catastrophic loss of stability. The phase-lock protocol of Kohlhaas et al. repeatedly measures the evolving phase with coherence-preserving probes, applies immediate phase corrections with gain $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$6, and reconstructs the total phase drift through

$H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$7

In the reported demonstration, $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$8, $H=H_A\otimes \openone_B+\openone_A\otimes H_B+H_{\rm int}, \qquad \|H_{\rm int}\|\ll\|H_A\|,\|H_B\|,$9, $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$00, and $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$01; the Allan deviation improves from

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$02

to

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$03

a stability improvement of $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$04 (Kohlhaas et al., 2015).

In bistatic Integrated Sensing and Communication, clock ambiguity takes the form of clock asynchronism between far separated transmitters and receivers. At frame $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$05, the offsets are

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$06

and the Line-of-Sight observations are

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$07

With state vector

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$08

the Time-Varying Offset Estimation framework uses an Extended Kalman Filter to jointly estimate timing offset and carrier frequency offset from the geometrically predictable LoS path. The resulting estimates correct all Non-Line-of-Sight components through

$\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$09

The reported simulations show that the proposed TVOE method improves the estimation accuracy by $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$10, and that after offset removal, spatial-smoothing MUSIC attains near-Cramér-Rao performance for AoA, range and velocity; for example, AoA RMSE improves by $\mathcal H_S=\mathcal H_C\otimes\mathcal H_R,$11 over conventional MUSIC in multipath (Wang et al., 12 Jul 2025).

These metrological and signal-processing examples are not instances of the problem of time in quantum gravity, but they exhibit the same structural issue: a phase, delay, or ordering variable is not uniquely inferable unless the clock is kept within a controlled operating regime or tied to a physically justified reference. This suggests that “clock ambiguity” is best regarded as a family of operational non-uniqueness problems whose mathematical form depends on the domain—constraint quantization, gravitating superposition, synchronization theory, atomic interferometry, or asynchronous sensing—but whose common content is the absence of a unique time assignment without extra physical structure.