In this paper, we extend the framework introduced in [1], which proposed a two-dimensional model of time as a means to reconcile relativistic invariance with quantum discreteness. Building on this foundation, we derive key principles of quantum theory such as the uncertainty and exclusion principles and also introduce a fourth postulate: “The probability of an event is the same in all inertial frames of reference (IFR), independent of the observer's position in space.” This postulate enables us also to derive the probability amplitude interference from a purely relativistic time-phase geometry. We show that trajectories in the two-dimensional time plane, governed by relativistic constraints and discrete frequency-phase dynamics, naturally give rise to quantized measurements and path-based superposition. Using the Mach-Zehnder interferometer as an illustrative case, we demonstrate that the square of the proper time trajectory corresponds to quantum probability and interference phenomena emerge from the structure of phase evolution in time. Our formulation also aligns with Feynman’s path integral framework, showing that quantum mechanics can be interpreted as a direct geometric consequence of extended relativistic principles.
In this paper, we extend the framework introduced in [1], which proposed a two-dimensional model of time as a means to reconcile relativistic invariance with quantum discreteness. Building on this foundation, we derive key principles of quantum theory such as the uncertainty and exclusion principles and also introduce a fourth postulate: “The probability of an event is the same in all inertial frames of reference (IFR), independent of the observer's position in space.” This postulate enables us also to derive the probability amplitude interference from a purely relativistic time-phase geometry. We show that trajectories in the two-dimensional time plane, governed by relativistic constraints and discrete frequency-phase dynamics, naturally give rise to quantized measurements and path-based superposition. Using the Mach-Zehnder interferometer as an illustrative case, we demonstrate that the square of the proper time trajectory corresponds to quantum probability and interference phenomena emerge from the structure of phase evolution in time. Our formulation also aligns with Feynman’s path integral framework, showing that quantum mechanics can be interpreted as a direct geometric consequence of extended relativistic principles.
The Theory of Temporal Relativity proposes a unified theoretical framework that extends Special and General Relativity into the complex domain, integrating aspects of Quantum Mechanics within a single geometrical structure. The theory introduces a generalized Hilbert-space geometry and interprets quantum states as observer-dependent, arising from complex temporal relations between reference frames. Theory of Temporal Relativity is a theoretical framework proposing a unified extension of Special and General Relativity into the complex domain, with the goal of incorporating aspects of Quantum Mechanics within a consistent geometrical model. The theory generalizes Hilbert-space geometry and relativistic invariance to complex coordinates, introducing observer-dependent descriptions of physical quantities and a reinterpretation of quantum states as complex-phase relations between reference frames. Within this model, fundamental constants such as c, G, and h are not strictly universal but can exhibit apparent local variations arising from the observer’s position in the complex spacetime manifold. These variations preserve global covariance and CPT symmetry while reconciling quantum non-locality with relativistic kinematics. The framework predicts experimentally testable effects, including measurable phase shifts in interferometric experiments, possible frequency modulations in non-inertial systems, and small local deviations of physical constants detectable in condensed-matter and gravitational contexts. The supplementary work, “Quantum Uncertainty, Planck’s Constant, and Ontological Insights from Complex Spacetime Geometry and Local Metric Deformations”, develops the principle of Temporal Relativity as a foundational element of the theory. It formalizes the connection between quantum uncertainty, metric deformations, and complex temporal structures, within an alternative logical framework (AU). This extension provides both an ontological interpretation of the wavefunction and a basis for the emergence of quantum indeterminacy from geometric causes. Together, these works present a coherent attempt to unify the relativistic and quantum domains under a single complex-geometric view of spacetime, maintaining mathematical rigor while providing clear experimental pathways for empirical validation.
The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature (1,4), we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as Γ=L2/ℓ2 for states that saturate the uncertainty relations, where L and ℓ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit ℓ→0, the equation reduces to the de Sitter spacetime equation; in the limit L→∞, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both ℓ→0 and L→∞), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.
Quantum mechanics successfully predicts experimental outcomes, yet its geometric and causal interpretation remains an open area of foundational inquiry. Concepts such as superposition, interference, and decoherence are frequently described as intrinsically probabilistic or nonclassical, often leaving unresolved whether these behaviors reflect fundamental randomness or limitations of our representational framework. This paper proposes a geometric reinterpretation in which quantum phenomena arise from the projection of a deeper causal structure into observer-dependent frames. Through thought experiments only, this work explores the possible literal interpretation of an orthogonal space-causal frame of reference, associated with null propagation, within a physical tradition otherwise strongly rooted in time-causal description. In our familiar time-causal frame, time is treated as the ordered causal axis and space as a volumetric domain of alternatives. Introduced here is a complementary space-causal frame, in which space becomes the ordered causal axis while time forms a volumetric domain of alternatives. Within this framework, superposition, interference, and decoherence emerge as consequences of geometric constraint rather than physical collapse. This approach preserves relativistic causality, introduces no additional dimensions, and leaves experimental predictions unchanged while offering a unified conceptual account of quantum behavior.
This work introduces Time–Space Oscillations (TSO), a deterministic geometric framework that reinterprets fundamental quantum phenomena as natural consequences of oscillatory structures in complex spacetime. Unlike standard quantum mechanics, which is probabilistic and incomplete, TSO provides a complete underlying phase geometry from which uncertainty, quantisation, entanglement, zero-point energy, path integrals, and interference phenomena emerge naturally. This Version presents the framework as a conceptual and structural exploration intended to stimulate discussion and critical engagement. Readers are invited to explore how classical deterministic oscillations can underpin observed quantum behavior, bridging the gap between geometric intuition and the apparent probabilistic rules of the quantum world. This article with its preceding articles on TSO lays the foundation for a future monograph that will consolidate the full theory in a single work.This article, together with the preceding works — Time–Space Oscillations: A Geometric and Deterministic Approach to Relativistic and Quantum Phenomena (DOI: 10.5281/zenodo.17534735), Time Dilation and the Nature of Gravitational and Inertial Forces (DOI: 10.5281/zenodo.17543060), and Time–Space Oscillations and Electromagnetics (DOI: 10.5281/zenodo.17591326) — provides a comprehensive introduction to the TSO model and its deterministic approach to quantum mechanics. This introduction work on TSO is part of the following key works: Time–Space Oscillations: A Geometric and Deterministic Approach to Relativistic and Quantum PhenomenaZenodo DOI: 10.5281/zenodo.17534734The foundational article introducing the TSO framework and its applications to both relativistic and quantum phenomena. Time Dilation and the Nature of Gravitational and Inertial ForcesZenodo DOI: 10.5281/zenodo.17543059Explores the TSO perspective on time dilation and the interplay of inertial and gravitational forces. Time–Space Oscillations and ElectromagneticsZenodo DOI: 10.5281/zenodo.17591325Applies the TSO model to electromagnetic phenomena, revealing novel interpretations of wave propagation and field interactions. Time–Space Oscillations and Quantum Mechanics Zenodo DOI: 10.5281/zenodo.17670668Extends TSO to encompass quantum mechanical principles, including Heisenberg uncertainty, Schrödinger dynamics, path integrals, entanglement, zero-point energy, and the Quantum Zeno effect. For an overview of related work and publications, visit:https://ndl1971.github.io/time-space-oscillations/ Version 0.92 Corrected, mainly section 17. Incomplete normalisation descriptions have been corrected. V and H or E and B of the observer need normalisation through observer E and B. Which is a remark to the reader too. TSO oscillations may become nonoscillative images and patterns after normalisation/projection. Version 0.91 Corrected w_tso_base frequency in: 7.1 & 8 which scales linearly with mass and conform time dilation when accumulated particles arise.
We present complex-time (CT) theory, a framework in which the fundamental structures of quantum mechanics are derived from three physical postulates concerning the internal structure of time. The postulates are: (P1) physical time possesses two orthogonal components — past-fixed (pf), representing actualized, irreversible records, and future-pending (fp), representing physically real but undetermined possibilities; (P2) pf and fp are orthogonal (pf fp); and (P3) the total temporal magnitude is conserved (pf + fp = 1). From these three postulates alone, we show that: (i) the state space is the unit circle and the dynamical group is , so that the imaginary unit arises as the unique generator of temporal evolution — not as a mathematical convenience but as a consequence of temporal geometry; (ii) the Schrödinger equation follows from the identification of energy with the rate of phase rotation in the pf–fp plane, with the Planck constant playing the role of a natural constant intrinsic to the pf/fp structure; (iii) the de Broglie–Einstein relation holds universally — for massive and massless particles alike — because the wave structure belongs to time itself, not to any particular particle; (iv) the uncertainty principle is a geometric consequence of the impossibility of fully eliminating fp while preserving ; (v) the complex structure of Hilbert space, the inner product, and the Born rule all follow from the geometry of fp pf projection; and (vi) energy quantization emerges from a “real-axis sampling” rule without imposing discreteness on the underlying dynamics. We further show that the CT framework provides unified descriptions of double-slit interference, quantum tunneling, unstable-state decay widths, and the structural origin of the arrow of time. Preliminary experimental predictions for quantum-circuit observables are stated, and applications to cosmological dark-energy phenomenology, galaxy rotation curves, and Lorentzian path-integral contour selection are outlined with references to dedicated companion papers. The central thesis is that quantum mechanics is not a theory of particles with mysterious wave-like properties, but a consequence of the internal structure of time — time whose wave-like phase (fp) is projected, irreversibly and discretely, onto the real axis (pf) in every act of measurement. Keywords: complex time; temporal structure; imaginary unit; quantum mechanics derivation; past-fixed / future-pending decomposition; Born rule; energy quantization; measurement; irreversibility; arrow of time
Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schroedinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations.