Information Theory of Action : Reconstructing Quantum Dynamics from Inference over Action Space

We develop an information-theoretic reconstruction of quantum dynamics based on inference over action space. The fundamental object is a density of action states encoding the multiplicity of dynamical alternatives between configurations. Maximum-entropy inference introduces a finite resolution scale in action, implying that sufficiently close action contributions are operationally indistinguishable. We show that this indistinguishability, together with probability normalization and action additivity, selects complex amplitudes and unitary evolution as the minimal continuous representation compatible with action additivity, probability normalization, and inference under finite resolution. Quantum interference and unitarity therefore emerge as consequences of these assumptions rather than independent postulates. From the resulting propagator, the Lagrangian, Hilbert-space structure, and Schrödinger equation follow as derived consequences. In the infinitesimal-time limit, action differences universally fall below the resolution scale, making coherent summation the minimal consistent description at every step. The numerical value of the action scale is fixed empirically and identified with $\hbar$.

💡 Research Summary

The paper proposes an “Information Theory of Action” (ITA) that reconstructs the formal structure of quantum mechanics from three minimal ingredients: (i) the additive nature of the classical action, (ii) a finite resolution in action space, and (iii) the standard rules of probabilistic inference (maximum‑entropy). The authors introduce a fundamental object, the density of action states g(A; b,T|a), which counts how many dynamical alternatives connect an initial configuration a to a final configuration b in a time T with total action A. Unlike a probability density, g is purely a multiplicity measure, defined only by five axioms: positivity and integrability, composition reflecting action additivity, time‑reversal symmetry, locality for short times, and finite variance. No reference is made to trajectories, wavefunctions, or path integrals at this stage.

From g, a joint probability over actions and endpoints is obtained by maximizing the relative entropy subject to a linear constraint on the average action. This yields the exponential family P(A,b|a) ∝ g(A; b,T|a) e^{‑ηA}, where η is the Lagrange multiplier associated with the action constraint. The Fisher information of this family leads to a Cramér–Rao bound ΔA_min = 1/η, interpreted as the smallest distinguishable action difference. The authors argue that when two contributions differ by less than ΔA_min they are operationally indistinguishable, forcing a coherent combination rule.

Using a Cox‑type functional argument, they show that the only continuous representation compatible with indistinguishability, action additivity, and probability normalization is a complex amplitude of the form ψ(A) = e^{iηA}. This reproduces the familiar phase factor e^{iS/ħ} of the Feynman path integral, provided one identifies η = 1/ħ. The identification is justified by matching the experimentally observed scale of quantum interference with the action resolution derived from the MaxEnt distribution.

The propagator is defined as K(b|a) = ∫ dA g(A; b,T|a) e^{iηA}. The composition law for g and the time‑reversal symmetry guarantee that K generates a linear, norm‑preserving semigroup on L²(Q). By invoking Stone’s theorem, the authors prove that this semigroup extends to a one‑parameter unitary group, implying the existence of a self‑adjoint Hamiltonian Ĥ with U(t) = e^{‑iĤt/ħ}. A short‑time expansion of K yields the Schrödinger equation iħ∂ₜψ = Ĥψ and the canonical commutation relation