Emergent Quantum Mechanics from Temporal Geometry
Mario Di Masi
We provide a mathematically rigorous derivation of the quantum mechanicalformalism — Hilbert space, Schrödinger equation, and Born rule — from thecondensate dynamics of Temporal Emergence Theory (TET), assuming Postulate P4of TET as the sole non-derived input at the level of stochastic dynamics. Starting from the Langevin equation for the condensate fluctuation δφ on thetemporal manifold H_τ = H^{3,1}, we apply stochastic differential geometry onRiemannian manifolds to extend Nelson's stochastic equivalence theorem to thecurved emergent spatial sections H^3 = SO(3,1)/SO(3). Three rigorous results are established: - Theorem A: the emergent Hilbert space H = L^2(H^3, dμ_{H^3}) is a separable complex Hilbert space with Plancherel orthonormal basis. - Theorem B: under Postulate P4 and a Nelson-type condition, the condensate diffusion on H^3 is unitarily equivalent to quantum mechanics with the identification ℏ = 2 m_φ D obtained as a definitional consequence of the Nelson formalism (an explicit dimensional analysis distinguishes what is derived from what is definitional); the Born rule follows from the polar decomposition of the diffusion density. - Theorem C (flat-space limit): the quantum evolution operator on H^3 converges to the standard QM evolution on R^3 with corrections bounded by C(l/R_τ)^2 t, reproducing all of standard quantum mechanics at laboratory scales with relative precision ≤ 4×10^{-53}. A conditional reduction of Postulate P4 to a single geometric assumption isalso presented. Under Assumption EQ — quantum ergodicity of the geodesic flowon H^{3,1}, proven for compact hyperbolic manifolds and conjectured for thenon-compact case — P4 follows from mode independence, a functional CLT onSO(3,1) (with explicit caveats on UV renormalization, since |c(s)|^{-2} growspolynomially at infinity), and the Lévy–Khintchine theorem on symmetric spaces.We do not claim that P4 is hereby derived; we claim that the problem is reducedto Assumption EQ. We do not claim a derivation of canonical commutation relations on H^3: this isexplicitly recorded as an open problem in the quantization of constrainedsystems on curved manifolds. Two consistency statements and one quantitative prediction follow from theframework:(i) an effective UV length cutoff l_UV = ℏc/μ_EW ≈ 8×10^{-19} m from the inter-sheet kink thickness of TET, presented as an internal consistency statement (degenerate with standard EW physics at present phenomenology);(ii) a pointwise suppression of the primordial scalar power spectrum of ~5.6% at k ≈ 2H_0 from the Plancherel spectral gap of H_τ, with the integrated effect on C_2^{TT} recorded as an open problem requiring a full Boltzmann calculation;(iii) a modified primordial spectral tilt δn_s ≈ -2.8×10^{-5} at the pivot scale, a geometric correction distinct from inflationary dynamics, potentially accessible to CMB-S4. This version (v4) supersedes earlier drafts (v1, v2, v3) and includes:- explicit retraction of the v1 "Theorem C" on canonical commutation relations (algebraically incorrect, foundationally ill-posed);- retraction of the v1 GUP framing of the inter-sheet UV scale;- corrected dimensional analysis of ℏ in TET vs laboratory units;- explicit UV/renormalization caveats in the P4 conditional reduction;- alignment of the abstract with the body (the v2/v3 numerical claim of 15–25% C_2^{TT} suppression is corrected to the pointwise ~5.6% at k ≈ 2H_0). Companion paper to: Temporal Emergence Theory (TET), Zenodo preprintdoi:10.5281/zenodo.19410988 Keywords: stochastic differential geometry, Nelson stochastic mechanics,emergent quantum mechanics, hyperbolic manifold, Plancherel decomposition,quantum ergodicity, CMB low-quadrupole anomaly. AI-assisted research: mathematical derivations and document elaboration weredeveloped with the assistance of online LLMs. The physical ideas andtheoretical framework are the author's original work.
•May 18, 2026•0 citations