Building on the evolutionary time field theory, this paper proves that the core mathematical structures of quantum mechanics can be derived from the constrained geometry of the evolutionary time field. Starting from the symplectic structure of the constrained geometry of the evolutionary time density and the order parameter, we establish effective canonical commutation relations and derive the time-energy uncertainty relation. The quantum excited states form a Fock space, whose linear structure yields the superposition principle, and the effective dynamics reduces to the standard Schrödinger equation in the flat-space limit. The measurement postulate corresponds to the coupling of the order parameter to an external dissipative channel—the contrapositive of the information bottleneck theorem applies directly, and the Born rule is reduced to an application of the Principle of Minimum Irreversible Cost. This paper further reveals the topological origin of the imaginary unit i, the geometric nonlocality mechanism of many-body entanglement, and the collapse threshold dissipation effect. The conclusion is that the core structures of quantum mechanics—Hilbert space, operator algebra, superposition principle, uncertainty relation, Schrödinger equation, measurement postulate, and Born rule—can all be derived from the constrained geometry of the evolutionary time field. This paper is the tenth in a series. The evolutionary time field theory is established in a companion paper (DOI: 10.5281/zenodo.21436542), the gauge field origin in a second (DOI: 10.5281/zenodo.21438149), the Hawking radiation quantum effects in a third (DOI: 10.5281/zenodo.21436715), and the axiomatic framework in a fourth (DOI: 10.5281/zenodo.21325460).
Building on the evolutionary time field theory, this paper proves that the core mathematical structures of quantum mechanics can be derived from the constrained geometry of the evolutionary time field. Starting from the symplectic structure of the constrained geometry of the evolutionary time density and the order parameter, we establish effective canonical commutation relations and derive the time-energy uncertainty relation. The quantum excited states form a Fock space, whose linear structure yields the superposition principle, and the effective dynamics reduces to the standard Schrödinger equation in the flat-space limit. The measurement postulate corresponds to the coupling of the order parameter to an external dissipative channel—the contrapositive of the information bottleneck theorem applies directly, and the Born rule is reduced to an application of the Principle of Minimum Irreversible Cost. This paper further reveals the topological origin of the imaginary unit i, the geometric nonlocality mechanism of many-body entanglement, and the collapse threshold dissipation effect. The conclusion is that the core structures of quantum mechanics—Hilbert space, operator algebra, superposition principle, uncertainty relation, Schrödinger equation, measurement postulate, and Born rule—can all be derived from the constrained geometry of the evolutionary time field. This paper is the tenth in a series. The evolutionary time field theory is established in a companion paper (DOI: 10.5281/zenodo.21436542), the gauge field origin in a second (DOI: 10.5281/zenodo.21438149), the Hawking radiation quantum effects in a third (DOI: 10.5281/zenodo.21436715), and the axiomatic framework in a fourth (DOI: 10.5281/zenodo.21325460).
We present complex-time (CT) theory, a framework in which the fundamental structures of quantum mechanics are derived from three physical postulates concerning the internal structure of time. The postulates are: (P1) physical time possesses two orthogonal components — past-fixed (pf), representing actualized, irreversible records, and future-pending (fp), representing physically real but undetermined possibilities; (P2) pf and fp are orthogonal (pf fp); and (P3) the total temporal magnitude is conserved (pf + fp = 1). From these three postulates alone, we show that: (i) the state space is the unit circle and the dynamical group is , so that the imaginary unit arises as the unique generator of temporal evolution — not as a mathematical convenience but as a consequence of temporal geometry; (ii) the Schrödinger equation follows from the identification of energy with the rate of phase rotation in the pf–fp plane, with the Planck constant playing the role of a natural constant intrinsic to the pf/fp structure; (iii) the de Broglie–Einstein relation holds universally — for massive and massless particles alike — because the wave structure belongs to time itself, not to any particular particle; (iv) the uncertainty principle is a geometric consequence of the impossibility of fully eliminating fp while preserving ; (v) the complex structure of Hilbert space, the inner product, and the Born rule all follow from the geometry of fp pf projection; and (vi) energy quantization emerges from a “real-axis sampling” rule without imposing discreteness on the underlying dynamics. We further show that the CT framework provides unified descriptions of double-slit interference, quantum tunneling, unstable-state decay widths, and the structural origin of the arrow of time. Preliminary experimental predictions for quantum-circuit observables are stated, and applications to cosmological dark-energy phenomenology, galaxy rotation curves, and Lorentzian path-integral contour selection are outlined with references to dedicated companion papers. The central thesis is that quantum mechanics is not a theory of particles with mysterious wave-like properties, but a consequence of the internal structure of time — time whose wave-like phase (fp) is projected, irreversibly and discretely, onto the real axis (pf) in every act of measurement. Keywords: complex time; temporal structure; imaginary unit; quantum mechanics derivation; past-fixed / future-pending decomposition; Born rule; energy quantization; measurement; irreversibility; arrow of time
The postulates of quantum mechanics — complex Hilbert space structure, unitary evolution, the Born rule, definite measurement outcomes, the canonical commutation relations, and the Pauli exclusion principle — have traditionally been taken as axioms. In this work we show that, within the framework of Modular Entropic Gravity (MEG), these structures admit a unified derivation from four geometric axioms governing the informational structure of the quantum vacuum.The framework is non-circular: its inputs are operator-algebraic structures of quantum field theory — including modular Hamiltonians, the JLMS identity, and the Bisognano–Wichmann theorem — which are defined independently of any probability interpretation. The probabilistic and dynamical rules of textbook quantum mechanics then arise as an infrared effective description of this underlying structure.The six results are obtained in logical order: (Q1) the emergence of complex structure from the entropion mass gap; (Q2) the L2 Hilbert space and Schrödinger equation from the long-wavelength limit of the vacuum propagator; (Q3) the Born rule from a relative entropy identity combined with the PLES–Petz correspondence; (Q4) definite measurement outcomes from PLES boundary conditions imposed by classical apparatus; (Q5) the canonical commutation relations from modular flow via the Bisognano–Wichmann and Araki reconstruction; and (Q6) the Pauli exclusion principle from the Z3 topological structure of the MEG vacuum.Three of these results (Q1, Q2, Q5) are established as complete theorems within the framework. The remaining three (Q3, Q4, Q6) are structurally complete derivations in which the underlying mechanism is fully identified, and any remaining conditions concern either formal infinite-dimensional extension or quantitative refinement.The paper therefore presents a coherent and non-circular programme in which the postulates of quantum mechanics are reconstructed from a smaller set of underlying principles, together with an explicit account of the remaining technical steps required for complete formalisation.
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.
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.
This paper presents a structural reconstruction of the formal core of quantum mechanics fromthe properties of a bounded, linear, distinguishability-reducing projection acting on a generalfunctional space. Effective quantum states are identified with equivalence classes under theprojection ΠQM : H∗ → HQM and the Hilbert-space structure of HQM is induced by a positiveoperator K on the underlying space.Within this framework, the characteristic elements of quantum theory emerge as direct conse-quences of projection geometry and operator structure. Linearity yields superposition, quadraticnorms on equivalence classes give rise to the Born rule, non-factorizability leads to entangle-ment and commuting local algebras ensure no-signalling. Schrödinger-type unitary dynamicsarises from projection of symmetric generators, Dirac-type operators follow from algebraic com-patibility with Clifford structures and violations of classical Bell inequalities together with theTsirelson bound are enforced by operator-norm constraints in the projected C∗-algebra.No hidden variables are introduced and no modification of standard empirical quantummechanics is proposed. Rather than postulating the quantum formalism as axiomatic, theanalysis shows that its defining structures form a highly constrained and stable outcome oflinear projection, equivalence-class geometry, and norm-preserving dynamics.Rather than introducing new physical dynamics, the purpose of this work is to clarify why theestablished quantum formalism admits so little structural freedom once a small set of minimalrequirements is imposed.
This paper derives the non-relativistic time-dependent Schrödinger equation as the logically unavoidable dynamical law for de Broglie matter waves, extending the reconstruction begun in the companion work “The Logical Origin of the de Broglie Matter Wave and its Physical Meaning (the Born Rule)” (Chan, 2026). Starting from the same minimal premises—continuous position in real space, Fourier conjugacy (and the logical uncertainty principle), translation invariance via Noether’s theorem, and probability conservation via statistical energy flux—the Schrödinger equation follows deductively once norm preservation and classical non-relativistic correspondence are enforced. The derivation proceeds in clear stages: Plane-wave solutions for free particles and constant potentials, Quadratic dispersion relation from classical kinematics, Exact local plane-wave structure (defined via phase gradients) that must hold at every point for any smooth wave function, The unique linear differential operator of order ≤ 2 that consistently “glues” these local structures across smooth external potentials V(x,t) while preserving the L²-norm (unitarity). No canonical quantization, operator promotion, commutation relations, or postulated wave equations are used. The Born rule is taken as already established from the prior paper and serves only to constrain unitary evolution. The Schrödinger equation thus emerges not as an empirical postulate or convenient ansatz, but as the mathematically and symmetrically forced description of indivisible particles propagating in continuous, translationally invariant (or smoothly perturbed) space. Domain of validity and natural limitations (non-relativistic kinematics, smooth potentials, single-particle/mean-field regime, closed-system unitarity) are explicitly delineated.