CONSTRUCTIBILITY IN QUANTUM MECHANICS D.J. BENDANIEL Cornell University (djb16@Cornell.edu) Ithaca NY, 14853, USA We pursue an approach in which space-time proves to be relational and its differential properties fulfill the strict requirements of Einstein-Weyl causality. Space-time emerges here from a set theoretical foundation for a constructible mathematics. In this theory the Schrödinger equation can be obtained by adjoining a physical postulate of action symmetry in generalized wave phenomena. This result now allows quantum mechanics to be considered conceptually cumulative with prior physics. A set theoretical foundation is proposed in which the axioms are the theory of Zermelo-Frankel (ZF) but without the power set axiom, with the axiom schema of subsets removed from the axioms of regularity and replacement and with an axiom of countable constructibility added. Four arithmetic axioms are also adjoined; these formulae are contained in ZF but must be added here as axioms. All sets of finite natural numbers in this theory are finite and hence definable. The real numbers are countable. We first show that this constructible theory gives polynomial functions of a real variable, which are of bounded variation and locally homeomorphic. Eigenfunctions governing physical fields can then be effectively obtained. By using an integral of the Lagrange density of a field over a compactified space, we produce a nonlinear sigma model. Finally, the Schrödinger equation follows directly from the postulate and a sui generis proof in this theory of the discreteness of the space-like and time-like terms of the model.
CONSTRUCTIBILITY IN QUANTUM MECHANICS D.J. BENDANIEL Cornell University (djb16@Cornell.edu) Ithaca NY, 14853, USA We pursue an approach in which space-time proves to be relational and its differential properties fulfill the strict requirements of Einstein-Weyl causality. Space-time emerges here from a set theoretical foundation for a constructible mathematics. In this theory the Schrödinger equation can be obtained by adjoining a physical postulate of action symmetry in generalized wave phenomena. This result now allows quantum mechanics to be considered conceptually cumulative with prior physics. A set theoretical foundation is proposed in which the axioms are the theory of Zermelo-Frankel (ZF) but without the power set axiom, with the axiom schema of subsets removed from the axioms of regularity and replacement and with an axiom of countable constructibility added. Four arithmetic axioms are also adjoined; these formulae are contained in ZF but must be added here as axioms. All sets of finite natural numbers in this theory are finite and hence definable. The real numbers are countable. We first show that this constructible theory gives polynomial functions of a real variable, which are of bounded variation and locally homeomorphic. Eigenfunctions governing physical fields can then be effectively obtained. By using an integral of the Lagrange density of a field over a compactified space, we produce a nonlinear sigma model. Finally, the Schrödinger equation follows directly from the postulate and a sui generis proof in this theory of the discreteness of the space-like and time-like terms of the model.
A physical principle that has consequences for the topology of a relativistic space-time is the principle of Einstein-Weyl causality. We show that these consequences also have quantum mechanical manifestations. Borchers and Sen have rigorously investigated the mathematical implications of Einstein-Weyl causality, regarded as a partial order, for the underlying spaces. This partial order was axiomatized by them and shown to admit Q^2 as an ordered space. They subsequently proved that, given certain topological assumptions, these spaces satisfying Einstein-Weyl causality are densely imbedded in spaces that have locally the structure of differentiable manifolds. They were left, however, with important philosophical paradoxes regarding the nature of the physical real line E, e.g., whether E = R, the real line of mathematics. We can view their original result, without further assumptions, as an insight into a denumerable space-time. This would suggest an investigation into a constructible foundation for relativistic physics. A set theoretical foundation is proposed including an axiom of countable constructibility. We first show that this constructible theory actually contains polynomial functions which are locally homeomorphic with a dense, denumerable metric space R*. Eigenfunctions governing fields can then be effectively obtained by iteration. Postulating a Lagrangian for fields in a compactified space-time, we get a general field description of which the Schrodinger equation is a special case. From these results we can then also show that this denumerable space-time is relational (in the sense that space is not infinitesimally small if and only if it contains a quantized field) and that, since Q^2 is imbedded in R*^2, it directly fulfills the strict topological requirements for Einstein-Weyl causality. Therefore, the theory predicts that E = R*.
AbstractStandard quantum mechanics is formulated using complex Hilbert spaces, real-valued probabilities, continuous one-parameter dynamics, and—where particles and fields have continuous degrees of freedom —function spaces constructed over continua such as R3 . In that precise mathematical sense, the real and complex continua are deeply embedded in the standard theory. It does not follow, however, that experiments have demonstrated the physical existence of an uncountable set of exact real numbers, still less that individual quantum systems physically “store” infinitely many digits. Every preparation, control operation, and measurement has finite precision, and many quantum predictions can be approximated arbitrarily closely using countable or finite descriptions.This distinction matters because the continuum is sometimes blamed for quantum theory's conceptual strangeness. That diagnosis is mostly unsupported. Superposition, interference, uncertainty, noncommutativity, entanglement, contextuality, Bell-inequality violations, and no-cloning already occur in finite-dimensional quantum systems and do not depend in any straightforward way on continuous position space or arbitrary irrational numbers. Restricting a Hilbert space to rational, algebraic, computable, or finitely specified states therefore does not automatically produce classical physics. Indeed, dense restricted subsets can approximate ordinary quantum predictions to arbitrary finite accuracy, although turning such a subset into a complete physical theory requires much more: closed dynamics, a composition rule, a measurement theory, robustness, and new empirical predictions.A particularly developed contemporary alternative is Tim Palmer's Invariant Set Theory and its recent formulation as Rational Quantum Mechanics (RaQM). RaQM is not accurately described as simply “quantum mechanics with irrational numbers banned.” It restricts the bases in which states are physically defined through number-theoretic rationality conditions, represents admissible states using finite bit strings, invokes a fractal invariant state space equipped with a non-Euclidean, p-adic-like metric, and violates Bell measurement independence at the level of exact counterfactual settings. Palmer's 2026 proposal further predicts a finite qubit-information capacity and an eventual saturation of quantum computational advantage. These are genuine, potentially falsifiable proposals, but they remain speculative and have not displaced standard quantum theory.The strongest conclusion is consequently asymmetric. There is a serious foundational question about whether the continuum is an ontological feature of nature or an extraordinarily efficient mathematical completion. Finite precision prevents existing experiments from proving a literal continuum. But there is presently little evidence that continuum mathematics causes quantum phenomena. Any discrete replacement must still account quantitatively for interference, contextuality, Bell correlations, entanglement, spectroscopy, atomic structure, and quantum computation. Discreteness can change quantum physics only when it is accompanied by additional physical postulates—and those postulates, rather than discreteness alone, must ultimately face experiment.
Understanding whether the mathematical structure of quantum mechanics is fundamental or emergent remains a central question in the foundations of physics. In particular, the special role played by time in quantum theory, appearing as an external evolution parameter rather than a dynamical observable, suggests that the formalism itself may arise from deeper structural considerations. In this work, we investigate the emergence of quantum mechanical formalism from classical wave dynamics by adopting a dimensional framework in which time is treated as a +1 evolution parameter relative to the dimensions through which physical phenomena (fields or disturbances of a field) propagate and interact. Within this perspective, different fields may evolve with respect to different effective dimensions, while remaining embedded in a common higher-dimensional space, allowing time to acquire a relational and context-dependent role. This means that in our proposed model, time is not a fixed dimension which is experienced the same way for every field or field interaction of any dimensionality. In that sense, time for one physical phenomenon can behave as space for a higher dimensional physical phenomenon, whose time is a different +1 dimension. The central objective of this paper is to determine how a higher-dimensional deterministic field can be consistently represented by a lower-dimensional description that lacks direct access to its full set of evolution parameters and evolves through a spatial (for the higher-dimensional field) dimension. To this end, we introduce a general projection framework in which a higher-dimensional field is mapped to a reduced field through an interaction-based recording process. Crucially, we do not assume the form of this mapping a priori. Instead, we impose the requirement that it preserve the maximum amount of physically accessible information. In particular, we demand the faithful encoding of phase relations, interference structure, and spectral composition, including the relative contributions of different Fourier modes and their superposition. We first demonstrate, within a purely classical 3+1-dimensional wave framework that these constraints severely restrict the admissible form of the reduced description and naturally lead to complex amplitudes, linear superposition, Hilbert space structure, and canonical operator relations. This analysis provides an intuitive and mathematically explicit route to quantum-like descriptions without assuming quantum postulates. We then generalize the construction to a 4+1-dimensional framework, introducing an additional evolution parameter and showing that under the same information-preserving constraints, the Schrödinger equation appears as an effective low-energy description of the reduced dynamics, while a relativistic dispersion relation emerges simultaneously through the encoding of the hidden evolution parameter as an invariant frequency scale. In this way, both quantum mechanical and relativistic structures arise from the same underlying requirement: the consistent and information-preserving representation of higher-dimensional wave propagation in a lower-dimensional observational framework. The results suggest that the formal structure of quantum mechanics need not be postulated a priori, but may instead be understood as the unique mathematical language required to encode the observable remnant of a higher-dimensional deterministic dynamics under strict constraints of symmetry, invariance, and information preservation.
The standard Copenhagen interpretation of quantum mechanics relies on a set of historically established, ad hoc postulates to describe particle-wave duality, measurement, and non-locality. In this paper, we propose a single foundational axiom that fully reproduces the standard mathematical formalism of quantum mechanics while eliminating the need for supplementary assumptions. From this singular postulate, we rigorously derive the fundamental origin of the wave function and the mathematical generator of the Schrödinger equation. Furthermore, we demonstrate via isomorfism that all seemingly inexplicable quantum phenomena—including wave-function collapse and quantum entanglement—are mathematically and conceptually reducible to the underlying dynamics of the double-slit experiment. By establishing a direct fundamental correlation between the double-slit interference mechanism and the Monty Hall phenomenon, we prove that these effects are distinct manifestations of the exact same principle. We framed the double-slit experiment as an interference of the space of possibilities affected by information and simetrys. Ultimately, by anchoring the existing quantum mathematical framework to this unique axiom, this work provides a coherent, paradox-free ontological foundation that resolves the interpretational crisis of quantum mechanics without altering its empirical predictive power.
Ever since quantum mechanics was first developed, it has been unclear what it really tells us about reality. A novel framework, based on 5 axioms, is presented here which offers an interpretation of quantum mechanics unlike any considered thus far: It is postulated that physical objects can exist in one of two distinct modes, based on whether they have an intrinsic actual spacetime history or not. If they do, their mode of existence is actual and they can be described by classical physics. If they do not, then their mode of existence is called actualizable and they must be described in terms of an equal-weight superposition of all possible actualizable (not actual) histories. The distinction is based on an axiom according to which there exists a limit in which spacetime reduces to a one-dimension reduced version, called areatime, and that objects which merely actualizably exist in spacetime actually exist in areatime. The operational comparison of the passage of time for such objects to the passage of time for a spacetime observer is postulated to be made possible by what is called an angular dual bilateral symmetry. This symmetry can be decomposed into the superposition of two imaginary phase angles of opposite sign. To mathematically describe the spacetime manifestation of objects which actually exist in areatime, each actualizable spacetime history is associated with an actualizable path, which in turn is associated with the imaginary phases. For a single free particle, the complex exponent is identified with a term proportional to its relativistic action, thus recovering the path integral formulation of quantum mechanics. Although based on some highly unfamiliar ideas, this framework appears to render at least some of the usual mysteries connected with quantum mechanics amenable to simple conceptual understanding. It also appears to connect the foundations of quantum theory to the foundation of the special theory of relativity while clarifying its relationship to the general theory of relativity and yields a testable prediction about a type of experiment, as yet unperformed, which under the current paradigm is utterly unexpected, namely, that the gravitational field of radiation is zero. The paper concludes with some speculations about how the theory may be extended to a metatheory of nature.(Note the paper was edited on 3-October-2011 to remove one sentence on page 7)