Time-reparameterisation invariant quantum evolution law: the lack of absolute time does not imply a stationary global state

AI summary

Overview

Research area: Quantum foundations and quantum gravity — specifically the "problem of time," the Page-Wootters formalism, and Dirac/Wheeler-DeWitt canonical quantisation of Hamiltonian constraints.

Technical level: Advanced. The paper is a purely theoretical and conceptual work using Hilbert-space operator mathematics, diffeomorphism invariance arguments, and density-matrix formalism. There are no experiments, datasets, benchmarks, or numerical simulations.

One-sentence scope: The paper constructs an underdetermined, time-reparameterisation-invariant quantum evolution law whose solutions are all time-reparameterised solutions of the Schrödinger equation, and argues that the absence of absolute time does not force the global quantum state to be stationary.

What This Paper Is About

Physicists often assume that if the universe has no absolute external time, then a quantum description of the universe without reference to an external clock must put the global state in a stationary state of its Hamiltonian, with all observed change arising from correlations with clock subsystems. The authors challenge this reasoning and ask whether one can instead write down a quantum evolution law that is invariant under reparameterisation of time yet still predicts the same nontrivial trajectory in state space as the Schrödinger equation, leaving only the speed along that trajectory undetermined. They construct such a law, show how ordinary Schrödinger evolution emerges relative to an internal clock, and re-examine the assumptions behind the stationary-state conclusions of Page-Wootters and Dirac quantisation.

Key Contributions

1. A new time-reparameterisation-invariant pure-state evolution law. Equations (10a) and (10b) replace the Schrödinger equation by a normalised form in which both sides are divided by the norms of the tangent vector and of Ĥ|ψ⟩, so that only the direction of motion in Hilbert space — not its speed — is fixed.

2. A proof that the solution set is exactly the time-reparameterised Schrödinger solutions. The authors show the equations are invariant under orientation-preserving diffeomorphisms τ(τ′) with ∂_{τ′}τ(τ′) > 0, that every Schrödinger solution satisfies them, and that every solution can be rewritten as a Schrödinger solution via a reparameterisation t(τ).

3. A density-matrix analogue and a relational recovery of the Schrödinger equation. Equations (13a) and (13b) give the trace-norm version for mixed states, and the authors show that with a Page-Wootters-type Hamiltonian Ĥ_CS = Ĥ_C ⊗ 1_S + 1_C ⊗ Ĥ_S, the system state relative to the clock reading obeys the ordinary Schrödinger equation — but with a non-stationary global state.

4. A critique of the assumptions behind stationary global states. The paper examines the energy-superselection argument used to motivate Page-Wootters and the treatment of the Hamiltonian constraint as a generator of gauge transformations in Dirac quantisation, arguing both need to be revisited when the time parameter itself is the gauge variable.

Main Findings

• Absence of absolute time does not imply a stationary state. The authors state explicitly that a system described without reference to an external clock need not be in a stationary state of its Hamiltonian, contradicting what they call a common belief underlying the Page-Wootters picture and Dirac's quantisation of the Hamiltonian constraint.

• Only the speed is lost, not the trajectory. For a given initial condition, the new law predicts the same trajectory in state space as the Schrödinger equation, except that for nontrivial trajectories it does not predict the rate at which the trajectory is traversed, and hence not its time parameterisation.

• Solution space is fully characterised. Every solution of Eqs. (10) is a time-reparameterised version of a solution of the Schrödinger equation; the converse also holds. The paper notes that, by invoking ancillary systems, one can conclude the equations describe unitary evolution at an undetermined rate.

• Hamiltonians differing by an overall factor are indistinguishable. According to Eqs. (10), Hamiltonian operators differing by an overall, possibly time-dependent, factor cannot be distinguished within this framework. The paper names the parameterisation that solves the Schrödinger equation for a specific Hamiltonian definition the "Schrödinger gauge."

• Relational emergence of standard evolution. Starting from an initial state in which the clock approximately shows a definite reading, |ψ(τ=0)⟩_CS ≈ |t=0⟩ ⊗ |ψ₀⟩, the gauge-free solution takes the form |t(τ)⟩_C ⊗ e^{−iĤ_S t(τ)}|ψ₀⟩_S with t(τ) an arbitrary monotonic diffeomorphism; when the clock reads approximately |t⟩_C, the system is approximately e^{−iĤt}|ψ(0)⟩.

• Global state can in principle be measured. Unlike the Page-Wootters model, the global state here is not stationary and could in principle be subjected to measurements under standard quantum rules — for example, an agent could measure the internal clock C in the |t⟩_C basis at an unknown instant τ and find a specific value t, with negligible disturbance to C.

• Sequential measurements distinguish the pictures. In the Appendix, the authors show that while for a single measurement the predictions are indistinguishable from those of the Page-Wootters picture, the situation changes radically for sequential measurements, which can be performed in a time-translation-invariant way; nontrivial global time evolution can then be observed through correlations between measurements.

• Averaged POVM construction. For a measurement at an unknown time, the averaged POVM element is Ē = Σ_i P̂_i Ê P̂_i, where {P̂_i} are projectors onto eigenspaces of Ĥ, obtained as the limit T → ∞ of the uniform average (1/T)∫_{−T/2}^{T/2} dt ê^{itĤ} Ê e^{−itĤ}; the off-diagonal blocks vanish because terms are proportional to (1/T)∫ dt e^{it(h_i − h_j)} → 0 for i ≠ j.

• Page-Wootters requires stronger assumptions than gauge freedom. The authors argue the Page-Wootters conclusion needs more than inability to observe an external time parameter; it may follow from inability to identify the order of instants or to recognise instants at all, which they say is strictly stronger than the gauge freedom known to be physically relevant from general relativity.

• Doubts about Dirac quantisation of the Hamiltonian constraint. The paper argues that viewing the Hamiltonian constraint as a generator of gauge transformations is justified only when the time parameter is assumed physically observable; when the time parameter itself is the gauge variable, imposing Ĥ|ψ⟩ = 0 (the Wheeler-DeWitt equation) raises doubts as to whether it is a correct quantisation procedure.

Methodology in Plain English

The authors begin from the standard Schrödinger equation and notice that its left-hand side is the velocity vector of the state as it moves through Hilbert space: its direction tells you where the state is heading, and its length tells you how fast. If you divide both sides by the length of the relevant vectors, you get an equation that fixes only the direction. That normalised equation is the proposed law. For states where the Hamiltonian annihilates the state — the trivial, constant trajectories — the law simply says the state does not change.

They then prove the law has the desired properties: it is unchanged under any smooth, order-preserving relabelling of the time parameter; every ordinary Schrödinger solution satisfies it; and every solution of it can be converted back into a Schrödinger solution by choosing a suitable reparameterisation, defined through the ratio of the norm of the state's velocity to the norm of Ĥ|ψ⟩. They repeat the construction for density matrices using the trace norm, and check that the pure-state and mixed-state versions agree on solutions.

To connect back to physics as actually practised, they model a clock and a system with the same additive Hamiltonian structure used by Page and Wootters, evolve it, and show that conditioning on the clock reading reproduces ordinary Schrödinger evolution of the system. Finally, they go through the argument that leads people to conclude the global state must be stationary — an energy superselection rule motivated by analogy with electric charge superselection and by the unobservability of global time shifts — and identify where an assumption is smuggled in: namely, that a solution of the equations of motion is described by a single family of states, rather than by an equivalence class of families related by the symmetry. The Appendix works out, via averaged POVM elements, why losing the reference frame is not operationally the same as having a trivial trajectory.

Why This Matters

Impact on research. The paper offers an alternative to the two dominant ways of handling the absence of absolute time in quantum theory — the Page-Wootters "evolution without evolution" proposal and Dirac's quantisation of the Hamiltonian constraint, which in gravity becomes the Wheeler-DeWitt equation. If the authors are right, canonical quantisation of time-reparameterisation-invariant theories may be quantising the wrong thing, treating as gauge a transformation that changes physically observable phase-space variables. This bears directly on the problem of time in quantum gravity.

Real-world applications. The paper reports no applications, experiments, or technological demonstrations. The following are domains the conceptual issues touch, not claims made by the authors:

• Quantum gravity and cosmology, where a background-free notion of time is required.
• Quantum reference frames, cited in the paper as the framework (Ref. [13]) for proving which observables remain accessible when a reference frame for a symmetry is lost.
• Quantum clock and timing models, where the relationship between an internal clock's reading and an external time parameter is the central object.
• Foundational analysis of sequential-measurement protocols, where the Appendix shows the new picture and the Page-Wootters picture diverge.

Industry relevance. None is reported. This is fundamental theoretical work with no stated commercial or near-term technological payoff.

Future Directions

• Heisenberg-picture formulation. The paper explicitly flags underdetermined evolution equations in the Heisenberg picture as "an interesting question for future investigation," noting that time translation acts on families of time-parameterised operators rather than on individual operators.

• Re-examining canonical quantisation. The authors suggest reconsidering the canonical methods of quantising classical theories with time-reparameterisation invariance, which they say may have nontrivial implications for quantum gravity.

• A group-theoretic / large-Hilbert-space formalism. The paper points to companion work (Ref. [18]) presenting a formalism describing quantum evolutions by operators on a suitable large Hilbert space, offering tools for studying these symmetries and their invariant objects.

• Testing the operational distinction. The Appendix's sequential-measurement analysis sets up a concrete discrimination between this framework and Page-Wootters; developing that into a sharp operational criterion, and extending the treatment from time-translation invariance to the full reparameterisation group, is a natural next step.

Target Audience

Advanced readers in quantum foundations, quantum gravity, and the philosophy of physics — particularly researchers working on the problem of time, Page-Wootters-type formalisms, quantum reference frames, and canonical quantisation of constrained systems. Graduate students with solid quantum mechanics and some differential-geometry background will follow the technical content; readers seeking experimental results or numerical comparisons will not find them here.

Authors’ abstract

We challenge the common belief that if there is no absolute time parameter in physics, a quantum system described without reference to an external clock can be assumed to be in a stationary state of its Hamiltonian. We present a time-reparameterisation invariant quantum evolution law, which for a given initial condition predicts the same trajectory in state space as the Schrödinger equation, except that for nontrivial trajectories it does not predict the speed at which the trajectory is traversed. The solutions of this evolution law are all time-reparameterised solutions of the Schrödinger equation. We show how the predictions of the Schrödinger equation are recovered relative to an internal clock in this framework. In contrast to the Page-Wootters formalism or Dirac's quantisation of the Hamiltonian constraint, here the global sate is not stationary. We discuss the assumptions leading to the common conclusion that the state can be taken stationary and suggest that they need to be revisited.