A Stationary Action is Stable Information
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.
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. The observed trajectory stabilizes informational cost — it requires the fewest bits to describe the system's evolution from start to end state. Classical motion emerges not because nature optimizes anything, but because paths of similar information complexity interfere constructively.
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 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 emerges with a consensus established by the mutual information exchanged between systems.
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. The classical trajectory is not the path nature "chose"; rather, it is the only history for which the internal phases of the system synchronize, allowing a stable, shared reality to condense out of the noisy background.