Emergence of Planck's Constant Derived from State Space Theory——Quantization Conditions in Information Geometry

tractStarting from the information geometry of state space, we rigorously derive the dimension, magnitude, and physical origin of Planck’s constant $\hbar$. We prove: (1) The dimension $[\hbar] = [\text{energy} \times \text{time}]$ necessarily emerges from the symplectic geometry of phase space area; (2) The numerical value of $\hbar$ is determined by the universe’s information density $\rho_I$, the speed of light $c$, and the gravitational constant $G$: $\hbar \sim c^3/(G\rho_I^{2/3})$; (3) A discrete state space manifests as a continuous manifold at the observer’s scale, with the Planck scale $\ell_P = \sqrt{G\hbar/c^3}$ being the critical scale where discreteness becomes apparent; (4) The uncertainty principle $\Delta x \Delta p \geq \hbar/2$ is not a measurement "disturbance" but the minimal resolvable phase space area in information geometry. Our derivation demotes $\hbar$ from a "fundamental constant" to an "emergent constant," unifying quantum mechanics and information theory. Keywords: Planck’s constant, state space, information density, symplectic geometry, holographic principle, uncertainty principleClassification Code: 04.60.-m (Quantum Gravity); 03.65.-w (Quantum Mechanics); 89.70.-a (Information Theory)