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

TL;DR

The paper shows that quantum dynamics can be reconstructed from information-theoretic inference over additive action, beginning with a density of action states $g(A;b,T|a)$ and employing maximum-entropy methods. A finite action-resolution scale $\\Delta A_{ m min} \,= 1/\\eta$ emerges from a Cramér–Rao bound, forcing indistinguishable action contributions to combine coherently as complex amplitudes $e^{i\\eta A}$, with $\\eta$ identified empirically as $1/\\hbar$. The resulting propagator, together with composition laws and symmetry constraints, yields a derived Hilbert-space structure and unitary evolution, and, in the short-time limit, the standard Schrödinger equation and Hamiltonian mechanics as emergent, not postulated, constructs. The framework also shows how the Lagrangian and Euler–Lagrange equations arise from the short-time action distribution and composition, while the path integral appears as a derived representation when trajectories exist. Overall, ITA positions quantum mechanics as the minimal consistent theory for coherent inference over indistinguishable action alternatives, with $\\hbar$ acting as the universal action scale calibrated by experiment.

Abstract

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 $\hbar$.

Figures (1)

• Figure 1: Logical structure of Information Theory of Action. The derivation flows from the axiomatically defined density of action states $g(A;b, T|a)$ through maximum entropy inference, the Cramér--Rao bound establishing an action resolution scale, and indistinguishability combined with probability normalization forcing complex amplitudes and unitarity. The propagator $K(b|a)$ then yields Hilbert space structure via Stone's theorem, culminating in the Schrödinger equation and canonical commutation relations. Dashed arrows indicate external inputs: the axioms, Jaynes' inference principle, probability normalization (a definitional requirement, not a physical assumption), and Stone's theorem from functional analysis. Interference and unitarity emerge as consequences, not inputs.

Theorems & Definitions (71)

• Remark 1: Implicit definition and non-circularity
• Theorem 1: Complex amplitude representation
• Proposition 1: Gaussian short-time fluctuations
• Definition 1: Emergent Lagrangian
• Theorem 2: Stationary phase condition
• Theorem 3: Euler-Lagrange from composition
• Theorem 4: Classical action theorem
• Corollary 1: Classical limit of $g$
• proof
• Theorem 5: Existence and conditional uniqueness of $g$