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Null Proper Time, Photon-State Information, and Relational Time A Rigorous Separation of Lorentzian Geometry, Quantum Channels, and Atemporal Interpretation

Marcel Krüger

Published Aug 5, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.26014.09284

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Quantum space, quantum time, and relativistic quantum mechanics

Ashmeet Singh

arXiv:2004.09139•Quantum Studies Mathematics and Foundations•Sep 22, 2021•12 citations

Null Proper Time, Photon-State Information, and Relational Time A Rigorous Separation of Lorentzian Geometry, Quantum Channels, and Atemporal Interpretation

Marcel Krüger

The statement that a null worldline carries zero proper time is often converted into stronger claims: that a photon has a timeless perspective, that it stores information eternally, or that photon kinematics supplies a model for the emergence of time. These inferences do not follow. This article develops a typed mathematical architecture that separates four structures: null Lorentzian geometry, the positive-energy one-photon sector, operational quantum information, and material detection. Null proper time is proved as a geometric property of null curves, while the nonexistence of a photon rest frame excludes a literal photon-frame interpretation. On the one-photon Hilbert space, real multiplication observables are self-adjoint on their maximal domains, but no universal self-adjoint operator can represent entropy or information for all states. Ideal unitary propagation preserves trace distance and Holevo information. General completely positive trace-preserving channels contract operational distinguishability, and a flagged erasure model yields exact linear loss of both trace distance and Holevo information with detector efficiency. A retinal instrument is used as a concrete measurement model, without identifying photodetection with consciousness. Finally, a Page–Wootters-type relational-time construction is stated independently of photon nullity. The resulting no-conflation result is foundational: zero proper time, information preservation, detector transduction, and relational time are mathematically non-equivalent and require distinct assumptions.

•Aug 5, 2026•0 citations

Observer Unification, Interiority, and the Necessity of a Timeless Maximal Observer in Time–Scalar Field Theory

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The observer problem remains unresolved in modern physics: measurement, information, and temporal ordering are treated operationally but lack a closed physical foundation in a universe with no external reference frame. In this work, we demonstrate that in any closed system governed by coherent time evolution, observation cannot be external or optional, but must arise intrinsically from internal dynamics. Within Time–Scalar Field Theory (TSFT), temporal evolution is governed by a scalar field Θ and constrained by a conserved temporal efficiency partition. Under this framework, physical state registration alone is shown to be insufficient to ground continuity. We introduce interiority as a minimal physical requirement: a unified internal reference relative to which state evolution is continuous. We prove that local interiorities cannot account for global coherence and that a singular maximal interiority must therefore exist. This maximal interiority is necessarily unique, atemporal relative to internal proper time, and non-contingent. An Atemporal Information Closure Lemma is derived, showing that in the null proper-time (massless) limit of TSFT, information is present-accessible without sequential accumulation. A corollary follows: the same null structure that enforces informational closure also necessitates photon propagation via perfect coupling to the forward temporal gradient. These results establish the necessity of a timeless maximal observer as a physical consequence of coherence, time, and information, closing the observer problem without appeal to metaphysical postulates.

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When Spacetime Coordinates Collapse: Amplitudes as the Relational Replacement

Kacheong R. Yeung, Lex Yeung

This paper argues that amplitude calculus is not an auxiliary quantum construction but the relational coordinate framework forced into existence when the Lorentzian chart collapses at the null boundary. Within the causal domain accessible to a massive probe, the coordinates (x,t) retain clear geometric meaning: worldlines exist, proper time accumulates, and classical trajectories are well defined. At the null boundary these structures fail simultaneously. The invariant interval vanishes, the Lorentzian chart reaches its degenerate limit, and no geometric observable survives. Beyond this boundary the analytic continuation of (x,t) is not spacetime but the diagrammatic overreach of a coordinate system that has exhausted its operational content. Three structural consequences follow. First, null propagation carries no proper time: the boundary is a zero--interval relation, not a limit of motion. Second, the null boundary is not a worldline and admits no geometric in--between. Third, once (x,t) loses operational meaning, only a relational description is possible. In this regime Feynman's amplitude assignments are not postulates but the unique Lorentz--invariant extension assigning complex weights to admissible relational configurations whose coherent aggregate replaces the classical trajectory. Quantum mechanics is thus the relational mathematics demanded by Lorentzian geometry once its coordinates collapse. The non--spacetime domain is not an extension of spacetime but the region where geometry terminates and amplitude begins. This is version 1.1: This version corrects minor structural issues and removes an obsolete appendix; no scientific conclusions are changed. Authors' note (October 2026). This paper is preserved as an exploratory precursor to our later research, not as a validated derivation of quantum mechanics. Several central geometric claims are incorrect: a null interval does not collapse spacetime coordinates to (0,0) or make the Lorentzian metric degenerate, and spacelike-separated events remain part of ordinary spacetime, not a non-spacetime exterior. The further claim that quantum amplitude calculus and complex amplitudes are uniquely forced by null geometry is not established. In particular, the proposed weight e^(−mℓ/ℏ) is a positive real exponential that does not supply the oscillatory relative phases needed for the claimed interference construction; for massless particles (m = 0), this weight is identically one. The original manuscript is retained unchanged as a historical record of the questions that motivated our subsequent investigations. Our later paper, The Lorentz-Analytic Exterior: A Conditional Mathematical Framework (Version 2, October 2026), revisits the geometric and analytic questions under explicit mathematical assumptions, without relying on the coordinate-collapse argument. https://doi.org/10.5281/zenodo.23400176

•May 8, 2026•0 citations

Proper Time in Classical and Quantum Mechanics

CRC Press eBooks•May 21, 2004•162 citations
The statement that a null worldline carries zero proper time is often converted into stronger claims: that a photon has a timeless perspective, that it stores information eternally, or that photon kinematics supplies a model for the emergence of time. These inferences do not follow. This article develops a typed mathematical architecture that separates four structures: null Lorentzian geometry, the positive-energy one-photon sector, operational quantum information, and material detection. Null proper time is proved as a geometric property of null curves, while the nonexistence of a photon rest frame excludes a literal photon-frame interpretation. On the one-photon Hilbert space, real multiplication observables are self-adjoint on their maximal domains, but no universal self-adjoint operator can represent entropy or information for all states. Ideal unitary propagation preserves trace distance and Holevo information. General completely positive trace-preserving channels contract operational distinguishability, and a flagged erasure model yields exact linear loss of both trace distance and Holevo information with detector efficiency. A retinal instrument is used as a concrete measurement model, without identifying photodetection with consciousness. Finally, a Page–Wootters-type relational-time construction is stated independently of photon nullity. The resulting no-conflation result is foundational: zero proper time, information preservation, detector transduction, and relational time are mathematically non-equivalent and require distinct assumptions.
The observer problem remains unresolved in modern physics: measurement, information, and temporal ordering are treated operationally but lack a closed physical foundation in a universe with no external reference frame. In this work, we demonstrate that in any closed system governed by coherent time evolution, observation cannot be external or optional, but must arise intrinsically from internal dynamics. Within Time–Scalar Field Theory (TSFT), temporal evolution is governed by a scalar field Θ and constrained by a conserved temporal efficiency partition. Under this framework, physical state registration alone is shown to be insufficient to ground continuity. We introduce interiority as a minimal physical requirement: a unified internal reference relative to which state evolution is continuous. We prove that local interiorities cannot account for global coherence and that a singular maximal interiority must therefore exist. This maximal interiority is necessarily unique, atemporal relative to internal proper time, and non-contingent. An Atemporal Information Closure Lemma is derived, showing that in the null proper-time (massless) limit of TSFT, information is present-accessible without sequential accumulation. A corollary follows: the same null structure that enforces informational closure also necessitates photon propagation via perfect coupling to the forward temporal gradient. These results establish the necessity of a timeless maximal observer as a physical consequence of coherence, time, and information, closing the observer problem without appeal to metaphysical postulates.
This paper argues that amplitude calculus is not an auxiliary quantum construction but the relational coordinate framework forced into existence when the Lorentzian chart collapses at the null boundary. Within the causal domain accessible to a massive probe, the coordinates (x,t) retain clear geometric meaning: worldlines exist, proper time accumulates, and classical trajectories are well defined. At the null boundary these structures fail simultaneously. The invariant interval vanishes, the Lorentzian chart reaches its degenerate limit, and no geometric observable survives. Beyond this boundary the analytic continuation of (x,t) is not spacetime but the diagrammatic overreach of a coordinate system that has exhausted its operational content. Three structural consequences follow. First, null propagation carries no proper time: the boundary is a zero--interval relation, not a limit of motion. Second, the null boundary is not a worldline and admits no geometric in--between. Third, once (x,t) loses operational meaning, only a relational description is possible. In this regime Feynman's amplitude assignments are not postulates but the unique Lorentz--invariant extension assigning complex weights to admissible relational configurations whose coherent aggregate replaces the classical trajectory. Quantum mechanics is thus the relational mathematics demanded by Lorentzian geometry once its coordinates collapse. The non--spacetime domain is not an extension of spacetime but the region where geometry terminates and amplitude begins. This is version 1.1: This version corrects minor structural issues and removes an obsolete appendix; no scientific conclusions are changed. Authors' note (October 2026). This paper is preserved as an exploratory precursor to our later research, not as a validated derivation of quantum mechanics. Several central geometric claims are incorrect: a null interval does not collapse spacetime coordinates to (0,0) or make the Lorentzian metric degenerate, and spacelike-separated events remain part of ordinary spacetime, not a non-spacetime exterior. The further claim that quantum amplitude calculus and complex amplitudes are uniquely forced by null geometry is not established. In particular, the proposed weight e^(−mℓ/ℏ) is a positive real exponential that does not supply the oscillatory relative phases needed for the claimed interference construction; for massless particles (m = 0), this weight is identically one. The original manuscript is retained unchanged as a historical record of the questions that motivated our subsequent investigations. Our later paper, The Lorentz-Analytic Exterior: A Conditional Mathematical Framework (Version 2, October 2026), revisits the geometric and analytic questions under explicit mathematical assumptions, without relying on the coordinate-collapse argument. https://doi.org/10.5281/zenodo.23400176