Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac's `Quantum Mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving vectors in the domain of the Hamiltonian: The ``probability densities'' (hermitean forms) ψ^\dagger χfor ψ,χin this domain generate an algebra from which the classical configuration space with its topology (and with further refinements of the axiom, its C^K and C^infinity structures) can be reconstructed using Gel'fand - Naimark theory. Classical topology is an attribute of only certain quantum states for these axioms, the configuration space emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.
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.
Based on a number of experimentally verified physical observations, it is argued that the standard principles of quantum mechanics should be applied to the Universe as a whole. Thus, a paradigm is proposed in which the entire Universe is represented by a pure state wavefunction contained in a factorisable Hilbert space of enormous dimension, and where this statevector is developed by successive applications of operators that correspond to unitary rotations and Hermitian tests. Moreover, because by definition the Universe contains everything, it is argued that these operators must be chosen self-referentially; the overall dynamics of the system is envisaged to be analogous to a gigantic, self-governing, quantum computation. The issue of how the Universe could choose these operators without requiring or referring to a fictitious external observer is addressed. The processes by which conventional physics might be recovered from this fundamental, mathematical and global description of reality are particularly investigated. Specifically, it is demonstrated that by considering the changing properties, separabilities and factorisations of both the state and the operators as the Universe proceeds though a sequence of discrete computations, familiar notions such as classical distinguishability, particle physics, space, time, special relativity and endo-physical experiments can all begin to emerge from the proposed picture. A pregeometric vision of cosmology is therefore discussed, with all of physics ultimately arising from the relationships occurring between the elements of the underlying mathematical structure.
We reconstruct nonrelativistic 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. The single-run physical state is a finite topological core, its actual causal history, the substrate state, its sector, and the associated historical disturbance. A wave function is not identified with that ontic tuple. It is a ray in a Hilbert representation of admissible preparations and response amplitudes; a density operator is a statistical or reduced representation; and a record is a stable macroscopic event. The logical construction is acyclic. Finite-energy sector spaces and their continuous symmetry representations lead, under explicit representation conditions, to a complex Hilbert direct sum. A closed semibounded quadratic form defines the Hamiltonian and its self-adjoint realization. The standard Schrödinger action, unitary propagation, probability current, spectral observables, uncertainty relations, composite-system tensor products, spin representations, exchange sectors, instruments, and completely positive open dynamics then follow on their declared domains. Gleason-type reasoning fixes the Born form only after additivity, noncontextuality, continuity, and preparation–detector calibration are stated. Those operational conditions are not hidden physical axioms. The distribution of unresolved ontic initial data that realizes quantum equilibrium remains a realization certificate, not a theorem of topology alone. The Heisenberg representation is derived as a controlled transport of observables after Hilbert space and the self-adjoint generator have been constructed. Strongly continuous time translation, two-parameter nonautonomous propagation, common-domain Heisenberg equations, spectral matrix mechanics, regular Weyl relations, open-system adjoints, and instrument effects are connected by one certificate. Schrödinger and Heisenberg predictions are proved equivalent for one-time and finite instrument records, while finite-core, historical, truncation, domain, and detector errors remain separately bounded. Three frozen matrix benchmarks verify low-energy oscillator truncation and its state-weighted CCR defect, second-order unitary convergence for a driven doublet, Schrödinger–Heisenberg equality for a sequential instrument, finite auxiliary-history truncation, and quadratic leakage scaling for a conditional Dirac–Pauli interface. Independent reproduction, including the extended-precision audit, is closed for the frozen packet; physical coefficient calibration remains open. The next interface connects an owner-resolved Dirac–Pauli Hamiltonian to spin records through a Ward-compatible vertex, a causal passive history kernel, an independently audited doublet projection, SPAM-aware forward probabilities, nuisance-profiled identifiability, and a noncompensatory coefficient-to-record error theorem. The DPHC identifiability coordinate is executed on a frozen synthetic response family. A single-scale design is rank deficient, whereas an eighteen-setting frequency–gradient–order–time design has full profiled rank, controlled conditioning, near-nominal finite-sample coverage, and a calibrated held-out statistic. A robust twenty-setting design raises the worst registered information eigenvalue by
20.12%. A ten-parameter nonlinear audit exhibits an exact radiative–detector equivalence curve. An independent dark record breaks that symmetry and restores full local rank. Continuous-set robustness, validated global exclusion, and independent implementation are executed, while physical response identification and empirical calibration remain open. Measurement is treated as a dynamics of system, apparatus, environment, and actual history. Unitary premeasurement and decoherence select robust record sectors but do not by themselves select one record. The three-axiom ontology supplies one actual branch history, while a measurable response map from the complete initial state and instrument to the final record supplies the missing dynamical content. Bell correlations require contextual and nonfactorizable ontic response while parameter independence and operational no-signalling are retained; no controllable superluminal channel is introduced. Historical response produces causal non-Markov Hamiltonians and memory-kernel master equations. Their normalization, positivity, complete positivity, and energy ledger are derived from a unitary enlarged realization whenever such a dilation exists. Finite-core form factors and causal memory yield explicit departures from point-core Markov quantum mechanics. Standard quantum mechanics is recovered as the unresolved-core, sector-fixed, quantum-equilibrium, short-memory limit. Quantum field theory is retained as a successful local, spectral, and asymptotic calculus, but field operators, Fock vectors, and vacuum functionals remain effective representations until a noncircular map from finite carriers and their complete causal histories has been certified. The manuscript proves a sequence of previously open analytic, mathematical, measurement, topological, statistical, and computational obligations on explicitly declared domains. These include non-Markov diffusion, graph-norm stability, distorted-wave and pole variation, finite passive-memory realization, Fisher-information identifiability, detector pushforwards, unique-record basins, Born-representation bounds, measurement independence, topology–Dirac charge conjugation, Stern–Gerlach records, action-scale matching, rank-two spin projection, Finkelstein–Rubinstein topology, Clifford reconstruction, and parent-to-record realization maps. Eighteen groups of numerical obligations are executed on frozen reference models. They cover spectral and scattering calculations, oscillator and two-level dynamics, Schrödinger–Heisenberg equivalence, finite-history truncation, Dirac–Pauli and Foldy–Wouthuysen reduction, gauge covariance, interference, Bell and GHZ correlations, open-system memory, Stern–Gerlach records, identifiability, global searches, topological invariants, path products, semiclassical limits, partial-wave optical closure, the relativistic/QFT interface, and conservative-parent two-level dynamics. Every environment explicitly typed as a computational obligation now has an executed frozen-model certificate; empirical identification and separately typed experimental inputs remain open. The present manuscript differs so substantially from the earlier version in scope, logical architecture, proofs, and executable certificates that it is issued as a new upload rather than as a minor revision. Because the subject is both foundational and technically consequential, readers professionally engaged in quantum foundations, mathematical physics, quantum measurement, or related fields are encouraged to consult both versions: the earlier version preserves the developmental route, whereas the present version gives the substantially expanded and reorganized reconstruction. ## Keywords Axiomatic quantum mechanics; finite particle histories; retarded Green functions; Schrödinger equation; Dirac equation; Born rule; topological spin; quantum measurement; non-Markovian dynamics; classical limit.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
The mathematical framework of quantum mechanics is conventionally founded upon a set of abstract axioms governing a Hilbert space. This work investigates whether this formalism can be derived as the effective description of a deeper, deterministic physical reality. We propose a mechanical model wherein a fundamental particle is treated as a 1-dimensional elastic defect (a string-like oscillator) topologically anchored to a background medium. By analyzing the dynamics of this system in the linear regime, we demonstrate that the core tenets of quantum theory emerge as necessary theorems of classical wave mechanicssubject to geometric constraints. Specifically, we show that: (1) the quantization of energy levels is a direct consequence of boundary conditions and resonance; (2) the Hilbert space structure is the natural mathematical arena for describing the superposition of stable vibrational modes; and (3) the momentum and energy operators, along with the Schrödinger equation, arise uniquely from the definition of observables and the conservation of energy. This approach grounds the quantum formalism in the mechanics of a continuous medium, suggesting that quantum behavior is not a break from classical physics, but the natural manifestation of wave dynamics in a confined topology.