⚛️ quantum physics

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

This paper challenges the assumption that the absence of absolute time necessitates a stationary global quantum state by proposing a time-reparameterisation invariant evolution law that reproduces Schrödinger trajectories without fixing their speed, thereby recovering standard predictions via internal clocks while maintaining a non-stationary global state.

Original authors: Ognyan Oreshkov, Denis Bouvy

Published 2026-08-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ognyan Oreshkov, Denis Bouvy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a movie. In the real world, time is like the projector's motor: it ticks forward at a steady, unchangeable pace, and the film moves through it. But in the strange, microscopic world of quantum physics, things get weird. Usually, we treat time as a background stage that just exists, a clock ticking outside the system. However, when physicists try to describe the entire universe as a single quantum system, there is no "outside" and no external clock to tick for them. This creates a massive puzzle: if there is no master clock, does the universe just freeze in place? Many famous theories have suggested that without an external timekeeper, the universe must be stuck in a static, unchanging state, and that the "movement" we see is just an illusion created by how different parts of the universe relate to each other. This is a big deal because it challenges our most basic intuition that the universe is always evolving.

Now, enter a new idea from physicists Ognyan Oreshkov and Denis Bouvy. They are challenging that "frozen universe" belief. They propose a new way to write the rules of quantum evolution that doesn't need an absolute clock. Instead of saying the universe is frozen, they suggest the universe is like a runner on a track who is moving along a specific path, but nobody knows how fast they are running. The path itself—the sequence of states the universe goes through—is real and dynamic, just like the Schrödinger equation (the standard rule for quantum motion) predicts. But the "speed" at which the universe travels along that path is left undefined. It's as if the universe has a script that says, "Go from point A to point B," but the director forgot to say, "Do it in 5 seconds." The result is a universe that is definitely evolving, not frozen, but whose evolution is "underdetermined" regarding time.

Here is how their new law works, using a simple analogy. Imagine you are walking down a hallway. The standard Schrödinger equation is like a strict coach who tells you exactly where to step and exactly how fast to walk. "Step here, and take 1 second to get there." But Oreshkov and Bouvy suggest a different coach. This coach says, "Walk this exact path," but they don't care how fast you go. You could stroll, sprint, or pause for a moment; as long as you hit the right spots in the right order, you are following the rules. In their new mathematical law, the universe follows the same "track" (trajectory) as the standard theory, but the "speed" of the journey is a free variable. The math allows for any speed, meaning the universe isn't stuck in one spot; it's just that the "time" parameter is flexible, like a rubber band that can be stretched or squashed without changing the shape of the path.

This might sound like a small tweak, but it changes everything about how we view the universe's state. In the popular "Page-Wootters" model (a famous theory about time without clocks), the whole universe is described as a giant, stationary photo. The movement we see is just a correlation between a "clock" part of the universe and the "system" part, like two dancers moving in sync while the room stays still. Oreshkov and Bouvy argue that this isn't necessary. In their new framework, the global state of the universe is not stationary. It is actually moving! They show that if you look at the universe through the lens of an internal clock (like a physical clock built into the system), you recover the standard, fast-moving Schrödinger evolution. But unlike the frozen-photo theory, their universe is alive and changing globally, even if we can't pin down the exact "speed" of that change without a reference clock.

The authors are careful to point out that this isn't just a guess; it's a mathematically consistent alternative. They explicitly rule out the idea that a lack of absolute time forces the universe to be stationary. They argue that the common belief in a frozen universe comes from a misunderstanding of how "gauge" (flexible) time works. They suggest that the assumptions leading to the "frozen" conclusion—specifically, the idea that we must treat the universe's state as a single, unchanging snapshot—need to be revisited. Their work suggests that we can have a dynamic, evolving universe without an external clock, simply by accepting that the "speed" of time is a flexible part of the equation, not a fixed number.

So, what does this mean for us? It means that the universe might not be a still image where time is an illusion. Instead, it might be a movie where the projector is running, but the frame rate is variable. We can see the story unfolding, the characters moving, and the plot progressing, but the "clock" that tells us exactly how many seconds have passed is a mystery. The authors suggest that this "underdetermined" evolution is a perfectly valid way to describe reality, one that keeps the universe moving and changing, even in the absence of an external timekeeper. It's a playful yet rigorous way to say: the universe is definitely dancing, even if we don't know the exact tempo of the music.

Technical Summary: Time-Reparameterisation Invariant Quantum Evolution Law

Problem Statement
The paper addresses a foundational issue in quantum theory regarding systems described without reference to an external clock, such as the universe as a whole. A common belief posits that the absence of an absolute time parameter necessitates that the global quantum state be stationary (i.e., an eigenstate of the Hamiltonian with zero energy). This view underpins the Page-Wootters formalism ("evolution without evolution") and Dirac's quantisation of Hamiltonian constraints (e.g., the Wheeler-DeWitt equation), where physical states satisfy $\hat{H}|\psi\rangle = 0$ . In these frameworks, the observed non-trivial evolution of subsystems is recovered only through correlations with a "clock" subsystem, while the global state remains static. The authors challenge the assumption that the lack of an absolute time parameter logically implies a stationary global state.

Methodology
The authors construct a new quantum evolution law that is explicitly invariant under time reparameterisation. Starting from the standard Schrödinger equation, they identify that the equation determines both the trajectory in state space and the speed at which it is traversed. To remove the dependence on an absolute time parameter (the speed), they propose an underdetermined evolution law.

For pure states $|\psi(\tau)\rangle$ , where $\tau$ is an arbitrary parameter, the new law is formulated by normalizing the time derivative of the state vector against the norm of the Hamiltonian action:
$\frac{\partial_\tau |\psi(\tau)\rangle}{\|\partial_\tau |\psi(\tau)\rangle\|} = -i \frac{\hat{H}|\psi(\tau)\rangle}{\|\hat{H}|\psi(\tau)\rangle\|}$
(for $\hat{H}|\psi(\tau)\rangle \neq 0$ ), with a trivial solution $\partial_\tau |\psi(\tau)\rangle = 0$ when $\hat{H}|\psi(\tau)\rangle = 0$ . An analogous formulation is provided for density matrices using the trace norm.

This equation is shown to be invariant under orientation-preserving diffeomorphisms of $\tau$ . The authors demonstrate that the solutions to this new law are precisely the set of all possible time-reparameterised solutions of the standard Schrödinger equation. Consequently, the law predicts the same trajectory in state space as the Schrödinger equation but leaves the parameterisation (the "speed" of evolution) undetermined.

Key Results

Non-Stationary Global States: The authors show that a time-reparameterisation invariant evolution law can yield non-trivial, non-stationary global states. The global state evolves along a trajectory in Hilbert space, but the rate of traversal is gauge-dependent.
Recovery of Schrödinger Evolution: When a physical clock subsystem is introduced with a Hamiltonian of the form $\hat{H}_{CS} = \hat{H}_C \otimes \hat{1}_S + \hat{1}_C \otimes \hat{H}_S$ , the standard Schrödinger evolution for the system $S$ is recovered relative to the clock's reading. If the clock indicates time $t$ , the system state is $e^{-i\hat{H}_S t}|\psi_0\rangle$ . Crucially, unlike the Page-Wootters model, the global state $|\psi(\tau)\rangle_{CS}$ is not stationary; it evolves synchronously with the clock at an undetermined rate.
Operational Distinction: The paper argues that the inability to observe an external time parameter does not render the global trajectory trivial. While single measurements might appear indistinguishable from the stationary picture, sequential measurements can reveal non-trivial global evolution through correlations, provided the order of events is preserved.

Significance and Claims
The paper claims to provide an alternative framework for time-reparameterisation invariance in quantum theory that does not require the global state to be stationary. The authors suggest that the conclusion of a stationary state in the Page-Wootters formalism and Dirac's quantisation relies on specific assumptions that warrant reconsideration:

Regarding Page-Wootters: The assumption that the inability to observe a global time shift implies a superselection rule for energy (and thus a stationary state) may be too strong. The authors argue that without a reference frame for time translations, one should consider equivalence classes of solutions related by time translations, which can still possess non-trivial trajectories.
Regarding Dirac Quantisation: The authors question the application of Dirac's constraint quantisation ( $\hat{H}|\psi\rangle = 0$ ) when the time parameter itself is the gauge variable. They argue that treating the Hamiltonian constraint as a generator of gauge transformations that annihilate physical states is only justified if the time parameter is assumed to be physically observable (as in standard gauge theories like electromagnetism). If time is the gauge variable, the constraint equation may correspond to a theory where no physical evolution occurs, rather than a theory with reparameterisation-invariant evolution.

The authors conclude that their underdetermined evolution equations offer a way to realise time-reparameterisation invariance without sacrificing the concept of a dynamical global state, suggesting that canonical quantisation methods for theories with this symmetry may need to be revisited, with potential implications for quantum gravity.

Technical Summary: Time-Reparameterisation Invariant Quantum Evolution Law

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