A Stationary Action is Stable Information
[Updated 6 February 2026 -MFM]
There is a maximum amount of information that any observer can extract from a physical system. This limit emerges from the structure of phase space itself. A system's state occupies a region defined by its position and momentum, and quantum mechanics forbids this region from shrinking below a fundamental volume set by Planck's constant. No measurement, however precise, can resolve finer details.
We can thus view Planck's constant not as an arbitrary scale where quantum effects appear, but as a conversion factor between physical units and information-theoretic units. Planck's constant translates between the language of physics and the language of information. And if information is always relational — if it measures the divergence between one system's description and another's, as argued in It from Bit, Bit from It — then \(\hbar\) sets the minimum relative entropy that any observation must produce.
Action as Information
The principle of stationary action governs classical mechanics. A particle traveling from point A to point B follows the trajectory that makes the action \(S\) stationary. For centuries, physicists have treated this principle as fundamental but mysterious — why should nature care about stationarity?
Feynman's path integral formulation of quantum mechanics reframed the question. In his picture, a quantum system does not follow a single trajectory. It explores all possible paths simultaneously, each contributing an amplitude weighted by a phase factor:
$$\psi = \sum_{\text{paths}} e^{iS/\hbar}$$
The classical trajectory emerges from this sum through interference. Paths near the stationary point have similar actions, so their phases align and reinforce each other. Paths far from the stationary point have rapidly varying actions, so their phases point in random directions and cancel.
If \(\hbar\) converts between physical and informational units, then the phase \(S/\hbar\) measures the information content of a path. The amplitude \(e^{iS/\hbar}\) becomes \(e^{i \cdot (\text{information})}\), and the path integral sums over trajectories weighted by their informational phases.
This reframing transforms the principle of stationary action into a principle of information stability. At the stationary point, neighboring paths carry descriptions that barely diverge from each other — the relative entropy between their informational content vanishes to first order. Classical motion emerges not because nature optimizes anything, but because paths whose descriptions agree interfere constructively, while paths whose descriptions diverge cancel each other out.
Consensus and Cancellation
Consider two neighboring paths with nearly identical actions. Their phases \(e^{iS_1/\hbar}\) and \(e^{iS_2/\hbar}\) point in nearly the same direction on the complex plane. When summed, they reinforce each other. A family of such paths, clustered around the stationary point, builds a robust amplitude that survives the sum.
Now consider paths far from the classical trajectory. Small changes in the path produce large changes in the action, so the phases \(e^{iS/\hbar}\) rotate rapidly. One path contributes a phase pointing north; its neighbor points south. These contributions cancel, leaving no net amplitude.
The metaphor of consensus captures this dynamic. The observed path represents an informational consensus among neighboring trajectories. These paths carry compatible information—their phases agree on a common direction. Alternative paths carry excessive, conflicting information that fails to cohere. The noise cancels itself out.
We can quantify this cancellation. Near a stationary point where \( \delta S = 0 \), the action varies quadratically with small deviations \( \eta \) from the classical path:
$$S(\text{path} + \eta) \approx S(\text{classical}) + \frac{1}{2}\frac{\delta^2 S}{\delta \eta^ 2}\eta^ 2$$
In quantum mechanics, this Action determines the phase of the probability amplitude, \( \phi = S/\hbar \).
Because of this inverse relationship, variations in Action translate to rapid spinning of the phase. Paths within a neighborhood of size \( \eta \approx \sqrt{\hbar} \) change the Action by less than \( \hbar \) (one radian of phase), so they add coherently. Beyond this window, the phase spins wildly, averaging to zero.
The classical world thus emerges from a kind of informational election. Each path casts a vote weighted by its phase. Paths that agree on the information content (that is, those near \(\delta S = 0\)) form a coalition large enough to dominate the sum. Paths that disagree scatter their votes and elect nothing.
The width of the winning coalition — the neighborhood \(\eta \approx \sqrt{\hbar}\) around the classical path — measures the synchronization cost of establishing a classical fact. Within this window, paths synchronize their descriptions cheaply; the relative entropy between neighbors is small enough that consensus forms. Beyond it, the cost of reconciling divergent descriptions exceeds what constructive interference can sustain, and the paths decohere into noise. The classical world occupies the region where the synchronization tax is affordable.
The Silence of Isolation
This picture illuminates the puzzle of quantum coherence. When a system remains isolated from its environment, no external observer extracts information from it. The need to settle into consensus on a single path of uniform information vanishes.
In this isolation, phases across wildly divergent paths retain their coherence. A particle passing through two slits maintains amplitude for both trajectories because no information distinguishes them. The system physically explores multiple conflicting histories simultaneously, each contributing to the final amplitude.
Decoherence disrupts this exploration. When the environment interacts with the system — when air molecules scatter off the particle, or photons reflect from its surface — information leaks out. The environment effectively measures which path the particle took, even if no human observer records the result.
This environmental interrogation forces consensus. Paths that differ in ways the environment can detect acquire different environmental entanglements. Their phases no longer align when we trace over the environment's state. The interference terms wash out, and only the diagonal (classical) probabilities survive.
The transition from quantum to classical is therefore a relational effect. The classical world solidifies when the environment extracts enough information to distinguish between paths. Without this extraction, the web of quantum possibilities persists. Reality, on this view, is not a property that systems possess intrinsically. It condenses where the relative entropy between a system's self-description and its environment's description of it has been paid — where the synchronization tax has been collected. Without that payment, the system persists in superposition, and no classical fact emerges.
There is an inverse to this silence. As noted in It from Bit, Bit from It, systems measured too frequently freeze — the quantum Zeno effect pins them to their initial state. Too little synchronization and reality remains indefinite; too much and it cannot evolve. The classical world occupies the narrow regime between these extremes, where synchronization occurs often enough to establish facts but slowly enough to permit change.
Stationary Action as Stable Consensus
This interpretation inverts the usual hierarchy. We typically regard dynamics as fundamental and information as derivative — a system evolves according to physical law, and observers extract information about that evolution. The informational reading of the path integral suggests the opposite.
The action \(S\) encodes not what happens, but what can be known. The stationary action condition identifies trajectories where observers can extract stable, coherent information. The Lagrangian specifies which physical quantities can become correlated through interaction. Dynamics emerges as a consequence of informational structure, not the other way around.
Planck's constant, in this view, sets the fundamental exchange rate between physical evolution and extractable knowledge. It determines how finely observers can carve up phase space, how precisely they can distinguish one history from another. The quantum-classical boundary lies not at a particular scale of size or energy, but at the threshold where informational consensus becomes possible.
The principle of stationary action thus becomes a principle about the stability of information — a statement about where in the space of possible histories observers can find consensus.[1] The classical trajectory is not the path nature "chose." It is the history where descriptions agree — where the relative entropy between neighboring accounts of what happened vanishes to first order, and a stable, shared fact condenses from the noise of all possible histories.
Recent work suggests this principle operates far beyond classical mechanics. Ginestra Bianconi's "Gravity from Entropy" theory derives Einstein's equations by extremizing an action that is a quantum relative entropy — the divergence between the true spacetime metric and the metric induced by matter fields. Gravity, on this view, is stationary information: spacetime curves in the way that stabilizes the informational relationship between geometry and matter. Separately, Alain Connes' recent work on the Riemann Hypothesis shows that the zeros of the Riemann zeta function sit at the extrema of the Weil quadratic form, which Connes connects to the prolate spheroidal functions of communication theory — functions that optimally concentrate signals in both time and frequency. The distribution of primes, the curvature of spacetime, and the trajectory of a thrown ball may all be instances of the same principle: stable facts emerge where informational descriptions achieve stationary consensus. ↩︎