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.“
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.“
Interpretational paradoxes of quantum mechanics may be associated with the application of the structure of classical probability theory to systems whose probabilistic organization is fundamentally contextual. In this work, we propose an outline of an alternative probability theory in which the primary object is not an elementary outcome but a contextual whole—a class of alternatives defined by a common structural property. In this approach, probability is organized on two levels. The first level describes the probabilities of contextual wholes, while the second level describes the conditional probabilities of alternatives within those wholes. Individual alternatives do not possess autonomous probabilities outside their context. Their probabilities arise only as contextual projections of a more fundamental probabilistic structure. Within this model, quantum mechanics appears as a special case of a more general theory in which contextual wholes are described by superpositions of probability amplitudes. Classical probability theory emerges as a limiting case in which all contexts are mutually compatible and can be combined into a single global description. The proposed framework provides a possible conceptual foundation for understanding quantum probability, contextuality, interference, and hybrid probability distributions within a unified structure.
Standard formulations of quantum mechanics successfully describe physical systems through isolated representations, most notably in position or momentum spaces. Attempts to unify these descriptions into a single joint phase space have historically faced severe mathematical dilemmas, such as the appearance of negative values in the Wigner quasiprobability distribution. This paper addresses this challenge by extending the recently developed Analytic Probability Framework and the Primacy of the Whole Principle into a geometric domain. We demonstrate that the separate, valid representations of standard quantum mechanics can be consistently unified into a joint hybrid representation without relying on classical phase space assumptions. By defining a strictly positive and normalized joint probability distribution, we construct a genuine quantum phase space where individual values of position and momentum do not exist as autonomous points, but rather function as interconnected structural elements of a single, indivisible quantum state. This approach provides a rigorous, non-paradoxical geometric foundation that naturally preserves quantum logic while extending the standard formalism.
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.
Quantum mechanics was initially conceptualized within the infinite-dimensional sequence space l2 by Born, Heisenberg, and Jordan, relying on the canonical commutation relation. However, as formalized by the Wintner-Wielandt theorem, this relation cannot hold universally using bounded, everywhere-defined operators. To prevent algebraic collapse, a structural restriction becomes inevitable, revealing a profound logical dichotomy. The traditional analytical approach, formalized by von Neumann, circumvents the Wintner prohibition to secure exact invariance of the relation, but abandons the Hilbert space framework for the non-normed Schwartz space and its dual distribution space. Conversely, the algebraic approach accepts the Wintner constraint as a structural law; it preserves the strict Hilbert space structure by accepting a boundary modification of the commutation relation within a finite, yet arbitrarily large dimension C(N+1) . This autonomous path directly yields a new physical observable, defined as the action difference D . While empirical correspondence holds perfectly within the interior matrix space, only this algebraic framework unlocks manageable eigenvectors for both position and momentum. Within this formulation, we rigorously prove that all position eigenvectors possess non-negligible components extending up to the highest energy levels. Confronted with this spectrum during a position measurement, a finite observer can resolve only a bounded energy spectrum, causing the accessible information to shrink to a set of measure zero. Consequently, quantum randomness emerges as intrinsic decoherence at the interface between the finite observation window and the space. Furthermore, we can demonstrate that the algebraic representation provides the foundational, contradiction-free justification for the continuous Cauchy-Hilbert core.
We present Unified Quantum Mechanics (UQM) by unifying real and standard complex quantum theories via real energy and momentum operators derived from the Hilbert Transform. This paper especially presents a fully real-valued formulation of quantum mechanics, showing that while complex numbers are commonly used in the standard theory, they are not fundamentally necessary for describing quantum phenomena. We develop Real Quantum Mechanics (RQM) by defining the real energy and momentum operators using the temporal Hilbert transform which shifts the phases of wave functions while preserving all essential quantum behavior. These real operators unify the proposed RQM and standard Complex Quantum Mechanics (CQM), as these are valid for both real and complex wavefunctions. We introduce a real-valued algebraic structure constructed from the identity operator and the Hilbert transform operator. This framework is mathematically equivalent to the system of complex numbers, preserving all fundamental algebraic properties while operating entirely within the realm of real operators. The isomorphism provides a novel representation of complex quantum mechanics using only real-valued components, offering potential advantages for both theoretical analysis and physical implementations. Importantly, we establish that the Hilbert transform operator is a generalization of the imaginary unit. The key insight is that the Hilbert transform generalizes the role of imaginary numbers in standard quantum mechanics, allowing us to construct real operators for both real and complex wave functions that produce the same physical predictions. We derive real versions of energy and momentum operators, leading to a modified wave equation that replaces the Schrödinger equation. Despite its non-local time evolution, this framework retains local spatial behavior and reproduces the usual energy-momentum relations. We rigorously test this approach on fundamental quantum systems, including the hydrogen atom and relativistic particles, confirming that it yields identical energy levels and wave functions as conventional quantum mechanics. Notably, we establish a consistent probabilistic framework by introducing real analogues of quantum probabilities via the analytic Born rule. We derive a unified uncertainty principle that coincides with the standard Heisenberg uncertainty bound for complex-valued wave functions, while admitting a vanishing lower bound in the case of real-valued wave functions. The implications of this work are significant: (1) it demonstrates that complex numbers are a mathematical convenience rather than a fundamental requirement in quantum theory, (2) it introduces new tools for analyzing quantum dynamics using signal-processing techniques, (3) it provides insights into non-local quantum evolution, (4) it connects quantum mechanics to real-field theories. Ultimately, RQM reproduces all known quantum predictions while offering a fresh perspective on operator structure, time evolution, and the mathematical foundations of the theory, and (5) it provides a real qubit representation on the modified Bloch sphere for quantum computing.
The persistent challenge of unifying quantum mechanics with general relativity is often traced to a series of apparent structural tensions: complex versus real ontology, probabilistic versus deterministic dynamics, linear versus nonlinear evolution, and spectral discreteness versus geometric continuity. We argue that these tensions do not signal a fundamental incompatibility between the theories, but rather originate from surplus structure in the conventional formulation of quantum mechanics. Beginning from the physically indispensable requirement of matter stability, we prove that the Hamiltonian spectrum can always be shifted to become non-negative. The consequent positivity of energy enforces one-sided temporal frequency support for all physical states, selecting the Hardy space of analytic signals as the admissible solution space. In this space, the real and imaginary parts of the wave function are not independent; they are linked uniquely via the temporal Hilbert transform. Hence, the complex structure of quantum mechanics emerges as a representational artifact of an underlying real operator algebra. Building on our earlier work on Unified Quantum Mechanics and the Unified Real Wave Equation, we establish a rigorous proof of exact equivalence between the real-field evolution and the Schrödinger equation on the physical subspace. Probabilistic quantities arise naturally as quadratic invariants of analytic signals, rather than as independent postulates. The resulting framework provides a realfield formulation of quantum mechanics and gravitation, with stability and causality implemented as foundational principles, and demonstrates that the Einstein field equations are compatible with a unified field-theoretic description of spacetime and matter. Extending the same analyticity principle to spacetime geometry, we propose an analytic formulation of the Einstein field equations, from which classical general relativity is recovered exactly as the real projection.