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International Journal of Computing Science and Mathematics

The discrete topology of time operator in closed quantum dynamical systems: the case of a finite wave collapse

Yohannes Yebabe Tesfay

Published Jan 1, 2021• 0 citations
Publisher

Abstract

Sympathetic, the topological structure of time is one of the fundamental and the most sophisticated research themes of modern physics. All the calculations of Newtonian mechanics are derived based on the assumption of absolute and independent time. Though, Einstein, in the SR, had thrown away the absolute and independent time conception of the Newtonian mechanics. Later, Einstein, in his GT, showed that gravitation is the curvature of four-dimensional spacetime. In this paper, the author proposed to test whether the metric time is continuous or discrete in a closed quantum dynamical systems considering a finite wave collapse. By applying the energy-time uncertainty quantum formalisation, the author gives a complete proof that the metric time is discrete. The proof of the theorem, therefore, gives strong evidence of the scheme of quantum gravity.

DOI
10.1504/ijcsm.2021.10036749

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Sympathetic, the topological structure of time is one of the fundamental and the most sophisticated research themes of modern physics. All the calculations of Newtonian mechanics are derived based on the assumption of absolute and independent time. Though, Einstein, in the SR, had thrown away the absolute and independent time conception of the Newtonian mechanics. Later, Einstein, in his GT, showed that gravitation is the curvature of four-dimensional spacetime. In this paper, the author proposed to test whether the metric time is continuous or discrete in a closed quantum dynamical systems considering a finite wave collapse. By applying the energy-time uncertainty quantum formalisation, the author gives a complete proof that the metric time is discrete. The proof of the theorem, therefore, gives strong evidence of the scheme of quantum gravity.
We present an ontological and dynamic derivation of wave function collapse within the context of General Relativity formulated as a Quantum Theory (GRQT), a formulation that maintains ontological realism and conceptual continuity with General Relativity. Starting exclusively from the discrete geometric structure of spacetime and its fundamental elastic properties, we demonstrate that collapse constitutes neither an additional postulate of quantum mechanics nor an external phenomenological effect, but emerges as a necessary consequence of the requirement for geometric consistency in the temporal update of the causal lattice. We show that quantum superpositions of massive systems imply geometric bifurcations of spacetime, associated with incompatible metrics in the transition between successive simultaneity slices. The energy identified in Objective Reduction theories as gravitational self-interaction energy is reinterpreted, in GRQT, as the elastic deformation energy of the lattice, governed by the fundamental geometric impedance α\alphaα. We derive a quantitative criterion for the viability of temporal evolution and a closed-form expression for the characteristic collapse time, without introducing free parameters. Wave function collapse thus arises as the physical process by which spacetime eliminates geometric ambiguities incompatible with the unique definition of the subsequent slice, preserving the topological continuity of the causal manifold. The boundary between quantum and classical regimes becomes quantitative and derived, resolving the measurement problem in strictly geometric terms. The formulation maintains ontological realism, conceptual continuity with General Relativity, and provides explicit experimental predictions in mesoscopic regimes.
A novel model for the problem of Quantum Gravity. The formal topologies of continuous General Relativity and discrete Quantum Mechanics are fundamentally irreconcilable without an intervening boundary constraint. We propose the dual-manifold formalism: a deterministic framework that reclassifies wave-function collapse as a localised thermodynamic phase transition. By strictly bounding quantum state reduction at the Bekenstein limit, we demonstrate that macroscopic spacetime curvature emerges natively as the mandatory metric deformation tethered to localised mass-energy. Mapped via standard thermodynamic equations of state, this localised thermal flux mechanically recovers continuous gravity without invoking unobserved dimensions or novel gauge bosons.
Background:In quantum mechanics energy is an observable, whereas time is a parameter. Pauli's theorem and the problem of time in canonical quantum gravity reflect this asymmetry.Material and methods:We define the time operator t^=x^/c\hat{t}=\hat{x}/ct^=x^/c and form, with the relativistic energy E^R\hat{E}_RE^R​, Einstein's constant κ\kappaκ and the time t^0\hat{t}_0t^0​ in the rest system, operators of gravitational-field and curvature type. Commutation relations, wave equations and spectra are derived in the time and in the energy representation.Results:The commutator [t^,E^R]=iℏcv^x[\hat{t},\hat{E}_R]=\tfrac{i\hbar}{c}\hat{v}_x[t^,E^R​]=ciℏ​v^x​ is bounded with norm ℏ\hbarℏ and is not a multiple of the identity, so Pauli's theorem is not contradicted. The sum c t^+κE^Rc\,\hat{t}+\kappa\hat{E}_Rct^+κE^R​ is unitarily equivalent to the time operator and has a continuous spectrum with explicit eigenfunctions. A postulated field equation, G^0+Λ^=κE^R\hat{G}_0+\hat{\Lambda}=\kappa\hat{E}_RG^0​+Λ^=κE^R​, has the structure of Einstein's equation; the term Λ^\hat{\Lambda}Λ^ is an operator with continuous spectrum, the Ricci scalar operator contains the energy, and for small velocities the equation becomes the Schr\"odinger equation of a particle in a homogeneous field.Conclusion:Based on the postulated field equation, a quantization of the gravitational-field and curvature terms is derived: they become operators with explicit commutation relations and spectra. The metric tensor of Einstein's field equation has not been quantized.\par
The hypothesis that spacetime possesses a discrete structure at the Planck scale - potentially composed of quantized tetrahedral or simplicial building blocks - arises naturally in background-independent approaches to quantum gravity such as Loop Quantum Gravity (LQG) and Causal Dynamical Triangulations (CDT). In these frameworks, spatial geometry emerges from spin networks of quantum tetrahedra with discrete area and volume spectra, while spacetime evolution proceeds via spin foams or summed triangulations. Roger Penrose’s Objective Reduction (OR) mechanism further posits that quantum superpositions of massive states generate incompatible spacetime curvatures whose gravitational self-energy difference triggers an objective collapse when the associated energy uncertainty reaches ħ/τ, implicitly requiring a non-continuous spacetime geometry. This article presents an exhaustive theoretical foundation and proposes a hybrid tabletop experiment - the Planck Geometry Interferometer - designed to simultaneously measure gravitational phase shifts arising from proper-time differences and spontaneous collapse lifetimes. By placing mesoscopic masses in spatial superpositions and monitoring quantized deviations in phase or collapse time, the experiment can distinguish discrete tetrahedral geometries (LQG-style) from continuous spacetime or pure Penrose foam. Building directly on existing proposals, the setup is shown to be within reach of current optomechanical, cryogenic, and interferometric technologies. Detailed protocols, expected signatures of discreteness, feasibility assessments, and implications for falsifying or confirming Planck-scale discreteness are provided. A negative result would constrain or refute these models; a positive detection of quantization would constitute direct empirical evidence for discrete spacetime.
We propose a unified physical framework in which four-dimensional spacetime vacuum is the unique physical reality, and all quanta are its localized topological excitations. Matter particles correspond to stable knot-like configurations, while radiation quanta correspond to weak propagating ripples of the same substrate. From this single ontological commitment, we derive: (1) wave–particle duality is resolved—particle behavior belongs to the localized excitation, wave behavior to the disturbance field it generates in spacetime; (2) double-slit interference arises from coherent superposition between a quantum's newly sourced disturbance and its own historical disturbance field, not from ontological self-superposition; (3) the wave function is a statistical tool encoding ensemble distributions of trajectories modulated by disturbance feedback, and collapse is epistemic convergence rather than a physical process; (4) quantum entanglement corresponds to Planck-scale topological connectivity (microscopic Einstein–Rosen bridges), preserving relativistic causality; (5) gravity is the macroscopic accumulation of universal but individually weak quantum–vacuum couplings, explaining both its microscopic weakness and macroscopic dominance without requiring a graviton; (6) the four fundamental interactions are differentiated coupling modes of quanta to one spacetime vacuum field; and (7) apparent dark matter and dark energy are reinterpreted as collective topological curvature and large-scale vacuum self-relaxation, respectively. We formulate falsifiable predictions at both laboratory and cosmological scales, including disturbance–corpuscle separation interferometry, tunable decoherence threshold measurements, and galaxy rotation–topology correlations. The framework compresses the conventional axiomatic structure of quantum mechanics and cosmology into one master axiom and eliminates at least twenty-eight independent interpretive assumptions.
Modern Physics rests on two pillars: General Relativity (GR) and Quantum Mechanics (QM). The former describes gravitation as the continuous geometry of spacetime, based on differential fields. The latter, in turn, operates in a discretized regime, where energy levels, frequencies, and experimental results arise in quantized packets. The tension between both arises because GR assumes absolute continuity of time and space, while QM presents intrinsically discrete phenomena. Historically, the search for a unified theory started from the assumption that one of the two formulations must be modified — either GR should be discretized, or QM should emerge from an underlying continuous theory. In this work, we argue that the conflict is not fundamental: it stems from the traditional interpretation of time in Special Relativity (SR). When analyzed correctly, the proper time of distinct observers does not evolve synchronously, and this produces beating and discretization effects that explain, without additional postulates, many behaviors labeled as "quantum."