Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Reports

Emergence of Quantum Mechanical Formalism Through a Dimensional Redefinition of Time

Georgios I. Alamanos

Published Jul 30, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.3390/quantum8030072

Related papers

Time as a Relational Evolution Parameter: A Unified Framework for Composite Quantum Mechanics, Measurement, the Born Rule, Bell Correlations, and Internal Standard Model Symmetries

Georgios I. Alamanos

This work continues the framework developed in Ref. [1], in which time is treated relationally as the +1 evolution parameter associated with the dimensional configuration space of a physical phenomenon, and quantum-mechanical structure is proposed to emerge from the information-preserving dimensional recording of a higher-dimensional wave field. Here we extend that construction beyond the single-system description and investigate whether additional central elements of quantum mechanics can arise without being introduced as independent quantum postulates. We first clarify the recording process in terms of physically distinguishable equivalence classes of parent configurations and show how changes of the parent field can produce effective measurement-induced state updating in the reduced description. For independently preparable systems, preservation of local linear superposition leads to bilinear composition and, subject to completeness of the accessible composite state space, to Hilbert-space tensor products. Joint interactions then generate nonfactorizable reduced states, providing a structural origin for entanglement before any probabilistic interpretation is introduced. We subsequently derive the Born probability rule from the geometry of orthogonal reduced alternatives together with unitary invariance, noncontextual consistency, continuity, normalization, and additive coarse-graining. The resulting probability structure is compared with the independent robustness-based reconstruction of De Raedt, Katsnelson, and Michielsen [2]. Using the derived composite structure and probability rule, Bell–CHSH correlations are obtained from generic interaction-generated pure entangled states without assuming a singlet state or Pauli spin operators, while operational no-signalling is preserved. Finally, we explore a possible geometric extension toward internal gauge symmetries: aligned three-dimensional, tilted three-dimensional, and fully four-dimensional parent-field deformations lead respectively to one, two, and three component complex internal representations, suggesting U(1), SU(2), and SU(3) type symmetry structures. Unlike approaches that take the quantum formalism as their starting point, or reconstruction programs based primarily on operational or informational axioms, the present framework seeks a concrete physical mechanism from which these structures emerge together. The fact that a single modification in the dimensional role of time, combined with information-preserving reduction, consistently generates multiple otherwise independent features of quantum theory makes this possibility physically significant even before genuinely new empirical predictions are established.

•Oct 9, 2026•0 citations

Deriving the Quantum Formalism from the Primacy of Contextual Wholes in Phase Space

Viktoras Rimsha, Laimontas Risha

We present a unified, autonomous formulation of quantum mechanics derived strictly from the foundational principles of Contextual Probability Theory. Instead of initiating the formalism via complex wave functions in Hilbert space, we postulate that the primary physical reality is a realvalued joint distribution 𝑃(𝑢, 𝑣, 𝑡)in a hybrid phase space. Here, the variable 𝑢 is explicitly defined as the structural property (index) characterizing a contextual whole (equivalence class), while 𝑣 represents the internal spectral alternative within that context. We mathematically demonstrate that the traditional Hilbert space architecture, complex amplitudes, and non-commuting operators are not primary postulates of nature, but rather emergent information-compensating artifacts that inevitably arise when the unified contextual flow is projected onto lower-dimensional subspaces.To demonstrate the universal validity of this mechanism, we deliberately restrict our current derivation to the foundational, one-dimensional case. Proving that the entire quantum apparatus emerges seamlessly within this prototype guarantees that the underlying principle can be directly generalized to higher dimensions and more complex contextual configurations.“

•Jun 4, 2026•0 citations

The Emergence of the Quantum State Space from the Dynamics of a Constrained Continuum

S. Conde

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.

•Dec 29, 2025•0 citations

03 - Emergence of Quantum Mechanics and Born Measure from a Coherence-Driven TwoField Discrete Framework

Enrico Ferrero

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.

•May 13, 2026•0 citations

Derivation of the Schrödinger Equation from Ontological Field Theory

Guillermo C. Barraza

This article is the second part of a three-paper sequence developing Ontological Field Theory (OFT) as a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work complements explicit field-theoretic derivations by clarifying why quantum mechanics could not have been otherwise.

•Dec 25, 2025•0 citations

Quantum Mechanics and Cosmology Arising from the Unobservability of Scale

Paul Michel

We propose that the unobservability of absolute scale constitutes a foundational principle of physics. Starting from the requirement that physical predictions must be invariant under rescalings of the action, we show that only phase-based quantities remain well defined. This requirement uniquely selects a complex amplitude description, leading directly to the structure of quantum mechanics, including linear superposition, Hilbert space, and the Born rule. Within this framework, the classical notion of action loses absolute meaning, and physical observables arise from invariant phase relations. Quantum mechanics is therefore not postulated, but emerges as the minimal consistent description of systems in which scale cannot be directly observed. The formal structure of quantum theory is shown to follow from this principle without additional axioms. We further demonstrate that the same principle implies the existence of a dynamical scale field, which governs the effective quantum scale and connects naturally to cosmological models with a time- dependent Planck mass. In this way, the structure of quantum mechanics and the large-scale dynamics of the Universe are linked through a common origin. This framework provides a unified perspective in which quantum behavior and cosmological evolution arise from a single underlying principle, suggesting a new approach to the relationship between fundamental physics and cosmology.

•Apr 3, 2026•0 citations
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.
This work continues the framework developed in Ref. [1], in which time is treated relationally as the +1 evolution parameter associated with the dimensional configuration space of a physical phenomenon, and quantum-mechanical structure is proposed to emerge from the information-preserving dimensional recording of a higher-dimensional wave field. Here we extend that construction beyond the single-system description and investigate whether additional central elements of quantum mechanics can arise without being introduced as independent quantum postulates. We first clarify the recording process in terms of physically distinguishable equivalence classes of parent configurations and show how changes of the parent field can produce effective measurement-induced state updating in the reduced description. For independently preparable systems, preservation of local linear superposition leads to bilinear composition and, subject to completeness of the accessible composite state space, to Hilbert-space tensor products. Joint interactions then generate nonfactorizable reduced states, providing a structural origin for entanglement before any probabilistic interpretation is introduced. We subsequently derive the Born probability rule from the geometry of orthogonal reduced alternatives together with unitary invariance, noncontextual consistency, continuity, normalization, and additive coarse-graining. The resulting probability structure is compared with the independent robustness-based reconstruction of De Raedt, Katsnelson, and Michielsen [2]. Using the derived composite structure and probability rule, Bell–CHSH correlations are obtained from generic interaction-generated pure entangled states without assuming a singlet state or Pauli spin operators, while operational no-signalling is preserved. Finally, we explore a possible geometric extension toward internal gauge symmetries: aligned three-dimensional, tilted three-dimensional, and fully four-dimensional parent-field deformations lead respectively to one, two, and three component complex internal representations, suggesting U(1), SU(2), and SU(3) type symmetry structures. Unlike approaches that take the quantum formalism as their starting point, or reconstruction programs based primarily on operational or informational axioms, the present framework seeks a concrete physical mechanism from which these structures emerge together. The fact that a single modification in the dimensional role of time, combined with information-preserving reduction, consistently generates multiple otherwise independent features of quantum theory makes this possibility physically significant even before genuinely new empirical predictions are established.
We present a unified, autonomous formulation of quantum mechanics derived strictly from the foundational principles of Contextual Probability Theory. Instead of initiating the formalism via complex wave functions in Hilbert space, we postulate that the primary physical reality is a realvalued joint distribution 𝑃(𝑢, 𝑣, 𝑡)in a hybrid phase space. Here, the variable 𝑢 is explicitly defined as the structural property (index) characterizing a contextual whole (equivalence class), while 𝑣 represents the internal spectral alternative within that context. We mathematically demonstrate that the traditional Hilbert space architecture, complex amplitudes, and non-commuting operators are not primary postulates of nature, but rather emergent information-compensating artifacts that inevitably arise when the unified contextual flow is projected onto lower-dimensional subspaces.To demonstrate the universal validity of this mechanism, we deliberately restrict our current derivation to the foundational, one-dimensional case. Proving that the entire quantum apparatus emerges seamlessly within this prototype guarantees that the underlying principle can be directly generalized to higher dimensions and more complex contextual configurations.“
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.
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 article is the second part of a three-paper sequence developing Ontological Field Theory (OFT) as a unified framework grounding quantum mechanics in foundational ontological principles. The trilogy is structured as follows: Part I — Conceptual Necessity:Why Quantum Mechanics Is Inevitable: Ontological Continuity and the Necessity of Quantum Dynamics establishes that quantum mechanics follows necessarily from minimal requirements on coherent being, independently of specific physical postulates. Part II — Technical Realization:Derivation of the Schrödinger Equation from Ontological Field Theory provides the explicit field-theoretic derivation showing how the Schrödinger equation emerges as the non-relativistic limit of a covariant ontological field. Part III — Critical Context:Why Classical Physics Is an Approximation: Decoherence, Coarse-Graining, and the Limits of Determinism explains why classical determinism and trajectories arise only as emergent, approximate descriptions of fundamentally quantum and relational dynamics. Each article is logically self-contained, but the full argument is distributed across the three contributions. The conceptual analysis motivates the mathematics; the mathematics grounds the physics; and the critical examination situates classical intuition as a limiting case rather than a foundation. The papers are intended to be read as a coherent whole. We argue that quantum mechanics is not an arbitrary empirical framework but an inevitable consequence of minimal ontological requirements. Assuming only that being must exist necessarily, remain consistent, be relational, and admit self-reference, we show that classical deterministic physics is structurally insufficient. Quantum dynamics emerges as the unique low-energy realization of coherent ontological continuity. This work complements explicit field-theoretic derivations by clarifying why quantum mechanics could not have been otherwise.
We propose that the unobservability of absolute scale constitutes a foundational principle of physics. Starting from the requirement that physical predictions must be invariant under rescalings of the action, we show that only phase-based quantities remain well defined. This requirement uniquely selects a complex amplitude description, leading directly to the structure of quantum mechanics, including linear superposition, Hilbert space, and the Born rule. Within this framework, the classical notion of action loses absolute meaning, and physical observables arise from invariant phase relations. Quantum mechanics is therefore not postulated, but emerges as the minimal consistent description of systems in which scale cannot be directly observed. The formal structure of quantum theory is shown to follow from this principle without additional axioms. We further demonstrate that the same principle implies the existence of a dynamical scale field, which governs the effective quantum scale and connects naturally to cosmological models with a time- dependent Planck mass. In this way, the structure of quantum mechanics and the large-scale dynamics of the Universe are linked through a common origin. This framework provides a unified perspective in which quantum behavior and cosmological evolution arise from a single underlying principle, suggesting a new approach to the relationship between fundamental physics and cosmology.