This record contains the manuscript Quantum Mechanics as the Unique Lorentzian Continuation of an Osterwalder–Schrader Positive Theory by Chast K. Wolfe. The paper proves that the full mathematical structure of quantum mechanics—complex Hilbert space, linear and unitary time evolution, the Born rule, canonical (anti-)commutation relations, Poincaré invariance, the spin–statistics correspondence, and the existence of a unique vacuum—arises uniquely from the standard Osterwalder–Schrader (OS) axioms: reflection positivity, Euclidean covariance, locality, regularity, and the Markov semigroup property. Starting from a reflection-positive Euclidean field theory, the work establishes a monotone chain of implications showing that OS positivity forces:(1) a unique complex structure on the reconstructed Hilbert space,(2) linear unitary Lorentzian time evolution generated by a self-adjoint Hamiltonian,(3) the Born rule via Gleason’s theorem,(4) CCR/CAR relations in the Gaussian case, and(5) full OS → Wightman reconstruction, including Poincaré covariance, TCP analyticity, spin–statistics, and vacuum uniqueness. The paper also proves that nonlinear, real, or quaternionic variants of quantum theory are incompatible with OS positivity together with Euclidean covariance and locality. Thus, within the class of OS-positive models, standard complex quantum mechanics is the unique Lorentzian continuation. This Zenodo deposit includes:• the full manuscript (PDF),• a license file (CC BY-NC-ND 4.0),• a bibliography file, and• a README describing the archive contents.
This record contains the manuscript Quantum Mechanics as the Unique Lorentzian Continuation of an Osterwalder–Schrader Positive Theory by Chast K. Wolfe. The paper proves that the full mathematical structure of quantum mechanics—complex Hilbert space, linear and unitary time evolution, the Born rule, canonical (anti-)commutation relations, Poincaré invariance, the spin–statistics correspondence, and the existence of a unique vacuum—arises uniquely from the standard Osterwalder–Schrader (OS) axioms: reflection positivity, Euclidean covariance, locality, regularity, and the Markov semigroup property. Starting from a reflection-positive Euclidean field theory, the work establishes a monotone chain of implications showing that OS positivity forces:(1) a unique complex structure on the reconstructed Hilbert space,(2) linear unitary Lorentzian time evolution generated by a self-adjoint Hamiltonian,(3) the Born rule via Gleason’s theorem,(4) CCR/CAR relations in the Gaussian case, and(5) full OS → Wightman reconstruction, including Poincaré covariance, TCP analyticity, spin–statistics, and vacuum uniqueness. The paper also proves that nonlinear, real, or quaternionic variants of quantum theory are incompatible with OS positivity together with Euclidean covariance and locality. Thus, within the class of OS-positive models, standard complex quantum mechanics is the unique Lorentzian continuation. This Zenodo deposit includes:• the full manuscript (PDF),• a license file (CC BY-NC-ND 4.0),• a bibliography file, and• a README describing the archive contents.
Quantum Theory as the Operational Shadow of the M--A Channel: a Substratum Reconstruction in Absolute Frame Theory
The reconstruction program of quantum foundations asks why nature is described by complex Hilbert spaces with the Born rule, and why its correlations stop at the Tsirelson bound rather than saturating the no-signaling limit. Absolute Frame Theory (AFT) embeds the observable four-dimensional manifold M in a continuous Euclidean substratum A carrying a pilot wave Ψ, with an information-bottleneck channel that projects A onto M and traces out an inaccessible fiber. We propose that finite-dimensional quantum theory is the operational shadow of that channel---the channel viewed from within M, restricted to its M-accessible, identifiable part---and we carry out the reconstruction in that reading. This is a program, not a completed theorem: we mark every claim as derived, as a structural correspondence, or as an open frontier, and we state plainly that AFT is not yet formulated as an operational-probabilistic theory in full rigor and that the complex Hilbert structure is here a mechanism (the analytic-signal representation of the channel's sampling), not a posit. Within those limits the reconstruction reduces the why-of-quantum-theory to a single physical fact---the M--A signature transition, the Euclidean-to-Lorentzian Wick rotation---of which the complex structure and the Born rule are the two faces. Concretely, a linear map carries both the complex generator and the conserved ∣Ψ∣2 density if and only if it is unitary. The same signature transition is, in AFT, the root of parity violation and of the arrow of time; quantum theory, parity violation, and time's arrow thus share one origin. The operator-level complex structure that makes the theory locally tomographic is supplied by the Osterwalder--Schrader reconstruction about that single Euclidean time---reflection positivity being a theorem for the free embedding sector and preserved, by a local Feynman--Kac argument, for static configurations---hence complex quantum theory is derived, not posited, in the static sector. The named open frontier is then the time-dependent and interacting reconstruction, while the Born measure ∣ψ∣2 is shown to be the equivariant density of the reconstructed unitary and generic macroscopic matter is derived to be classical, leaving only the strict Dürr--Goldstein--Zanghì uniqueness open. That the embedding yields the Lorentzian signature, and with it emergent local Lorentz invariance, is established in a companion paper on emergent Lorentz invariance.
As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.
Paper XXII of the Six-Throat Directed Geometry program derives the mathematical structure of one-particle quantum mechanics from the directed spinor dynamics established on the regular closed six-throat spacetime. Starting from the conserved curved-spacetime Dirac current, the construction produces a positive and slice-independent inner product and its Hilbert-space completion. The corresponding geometric propagator is unitary and satisfies composition, inverse, and adjoint closure. Real physical observables are represented by self-adjoint operators with real spectra and orthogonal spectral resolutions. The compact global geometry yields an exact periodic commutator algebra, while the standard canonical position-momentum relation is recovered on regular local charts. The Robertson and Schrödinger-Robertson uncertainty relations then follow from the same Hilbert-space structure. The tetrad Clifford algebra generates the intrinsic spin-one-half representation, including the expected spin magnitude, the spinor sign reversal under a full two-pi rotation, and the rotational multiplets obtained by coupling orbital and intrinsic angular momentum. Detector-surface current is promoted from the direct Born-flux construction of Paper XXI to a general positive measurement structure described by a non-projective POVM and the trace probability rule. Density operators, purity, von Neumann entropy, conditional access states, and Kraus dephasing channels arise within the same measurement framework. The positive-energy Dirac branch reduces at low energy to Pauli dynamics and, when the spin-covariant terms commute appropriately, to the Schrödinger Hamiltonian. The same unitary framework yields the Liouville-von Neumann equation, the Heisenberg operator equation, and the Ehrenfest relations in the local directed potential. A forty-three-condition numerical closure audit satisfies all 43 declared conditions across the derived one-particle structure. Representative numerical results include Hilbert-product and unitary residuals at approximately machine precision, exact spin-one-half algebra to numerical precision, POVM completeness at machine precision, low-energy wave-packet infidelity of 3.16 × 10⁻¹⁰ for the stated audit configuration, and Schrödinger-Heisenberg expectation agreement at 5.66 × 10⁻¹⁷. The result is a single geometric chain from conserved spacetime flux to one-particle state space, unitary dynamics, observables, commutators, uncertainty, spin, measurement operators, statistical states, and the low-energy quantum equations. Composite systems, entanglement, exchange symmetry, quantum statistics, and variable-particle-number field theory remain subsequent structural layers and are not claimed as part of the present one-particle closure.
Title Ontological Uniqueness of Quantum Mechanics: A Pre-Quantum Classification Theorem Description This work addresses one of the most fundamental questions in theoretical physics: Why does nature employ quantum mechanical structure rather than alternative mathematical frameworks? We prove a rigorous classification theorem showing that quantum mechanics is not one possible theory among many, but the unique mathematical structure compatible with ontological continuity—the principle that "being cannot cease to be." Unlike previous approaches that either interpret quantum mechanics within its existing formalism (many-worlds, Bohmian mechanics) or derive it from operational measurement axioms (Gleason, Hardy, Chiribella-D'Ariano-Perinotti), this work operates at a pre-quantum logical level: we classify which geometric structures are ontologically admissible before introducing any quantum formalism. Main Result: Any theory satisfying minimal ontological requirements—continuity of being, compositional independence of subsystems, and reversibility of closed-system evolution—must describe the space of identity-classes as complex projective space ℂℙⁿ equipped with the Fubini–Study metric. Complex Hilbert space, unitary dynamics, the Schrödinger equation, and the Born rule emerge as necessary geometric consequences, not as postulates. Alternative structures (real or quaternionic projective spaces, non-homogeneous manifolds) are proven ontologically inadmissible through rigorous exclusion arguments. Key Contributions: Rigorous derivation showing that ontological composition forces a symplectic structure, which combined with homogeneity and isotropy uniquely determines Kähler geometry, selecting ℂℙⁿ as the only admissible space. Complete defense against circularity concerns, with detailed appendices showing that the axioms assume no quantum structure—they are purely ontological requirements for coherent continuous being. Systematic distinction from Gleason/Kochen-Specker theorems and operational axiomatizations, clarifying that this work classifies geometries before quantum mechanics, not constraints within it. Testable predictions in cosmology (Hubble tension resolution, matter fluctuation scaling) and quantum foundations (decoherence rates in molecular interferometry), while rigorously separating structural necessity from empirical determination. Significance: This work transforms quantum mechanics from an empirically successful but conceptually mysterious framework into an ontologically inevitable structure—comparable to Einstein's geometrization of gravity. It shows that quantum structure has no alternative: it is the only mathematical form compatible with coherent, continuous, compositional being. The manuscript includes comprehensive technical appendices with full geometric proofs, axiom justifications, and comparative analysis with existing foundational approaches. Suitable for researchers in quantum foundations, mathematical physics, philosophy of physics, and differential geometry. Document Status: Preprint submitted to Foundations of Physics (January 2026). Related Work: This work is part of the Law of Continuity of Being (LCS) research program, which derives physical and ethical structures from fundamental ontological principles. See author's ORCID profile for related publications. Open Access: This preprint is made available under CC BY 4.0 license to facilitate discussion and feedback from the research community prior to peer review.
We develop a conditionally closed reconstruction of quantum mechanics from exactly three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime--vacuum substrate, and persistent causal source--response coupling. Wave functions, Hilbert spaces, complex amplitudes, density operators, the Born rule, commutation relations, Fock spaces, path integrals, completely positive maps, and Markovian dynamics are introduced only as derived mathematical or operational structures, not as additional physical axioms. The construction distinguishes the actual finite carrier and its causal history from the quantum state used to represent preparations, transformations, and measurement probabilities. It derives unitary dynamics from self-adjoint Hamiltonians, formulates spectral observables and probability currents, and states the explicit equilibrium, additivity, noncontextuality, and calibration conditions required for the Born representation. Measurement is described through instruments, pointer dynamics, stable records, weak values, continuous monitoring, diffusive and counting trajectories, and operational error--disturbance relations. Quantum trajectories remain record-conditioned predictive representations rather than automatically ontic paths. Composite systems, identical particles, spin, entanglement entropy, purification, steering, Bell correlations, dense coding, and no-broadcasting are developed on declared tensor-product and symmetry domains. Open-system dynamics are formulated through causal Volterra equations, memory kernels, initial-slip terms, auxiliary-mode realizations, and unitary dilations. Existence, positivity, complete positivity, trace preservation, no-signalling, and energy closure are treated as logically distinct properties. Semiclassical pseudodifferential calculus, controlled path-integral convergence, Ehrenfest-time bounds, semiclassical measures, decoherence, and macroscopic-record stability define the quantum--classical interface. Topological phases, defect-bound states, relativistic wave equations, infrared dressing, and effective quantum-field-theory domains are incorporated with explicit applicability and error conditions. Numerical spectral certification, statistical model comparison, twenty-five benchmark models, theorem-status ledgers, and a complete experimental statistical protocol provide falsification and replication criteria. The resulting framework preserves standard quantum mechanics on its validated domain while isolating finite-core and historical corrections as conditional, quantitatively bounded, and experimentally testable extensions. **Keywords** Quantum foundations; topological quantum mechanics; historical memory; three-axiom reconstruction; Born rule; quantum measurement; weak measurement; quantum trajectories; entanglement; spin and statistics; open quantum systems; Volterra equations; semiclassical limit; quantum topology; numerical certification; experimental falsification.
October 25th, 2026: updates following the development of the Absolute Frame Theory. Why is the physical action a phase, exp(iS) (oscillatory, norm-preserving, interference-bearing), rather than a real weight, exp(−S) (diffusive, dissipative)? This is the difference between a quantum and a merely statistical world. In Absolute Frame Theory (AFT) it is the difference between the Lorentzian observable manifold ℳ and the Euclidean substratum 𝒜. The imaginary unit is introduced by the ℳ–𝒜 signature transition, the Wick rotation. We show that this transition is the single deepest posit beneath the AFT account of quantum theory. It is the same root from which the complex structure, the Born rule, the arrow of time, and the Tsirelson bound were separately shown to follow. We then reduce it as far as a physical theory can reduce its own ground. The signature transition factorizes into three layers: the existence of a time direction (the sliding of the embedding interface), the orientation (the arrow, fixed by informational irreversibility), and the complex/unitary character (the i and the phase). For the last and only open layer we prove three statements. First, the complex structure is not smuggled in: it acts on each real mode as the real, antisymmetric generator J₀ of the first-order phase-space closure of the mode's oscillation, so that the imaginary unit is an operator of the real phase plane, not an imported ℂ. Real time and the imaginary unit enter together, through the reconstruction. Second, we prove a theorem (for the free sector with uniform sliding) that the reconstructed evolution on ℳ is unitary (the phase exp(iS), with a single complex structure), and that no dissipative exp(−S) sector is admissible. The comoving frame is static, and Osterwalder–Schrader reconstruction delivers a positive Hamiltonian and unitary continuation. Third, that single complex structure acts compatibly across subsystems, J_S ⊗ 𝟙 = 𝟙 ⊗ J_S′ (an operator identity, not a shared scalar sign). Hence the reconstructed theory is locally tomographic, complex rather than real (rebit). The phase is therefore forced by the fact that 𝒜 is Euclidean. That fact is not an arbitrary extra assumption. It is entailed by 𝒜's role as the atemporal, stable, deterministic ground. Atemporality forbids a timelike direction, and a bounded-below action requires positive-definiteness. Hence the two-signature architecture reduces to one posit. Asking why 𝒜 is Euclidean is asking the theory to derive its own ground. That is a foundational question of self-referential, not identifiability, character. The reduction is sharp: it does not derive the posit from its origins, but it shows the posit to be single, non-arbitrary, and necessary given the role. The open residual is thereby narrowed to two sectors, the interacting and the curved/horizon, the latter addressed by modular reflection positivity in the companion dynamical reconstruction.