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The Probability Sector of PT-Symmetric Quaternionic Spacetime

Chien-Chih Chen

Published Apr 14, 2026• 1 citation
Publisher

Abstract

We investigate the probability sector of PT-symmetric quaternionic spacetime (PTQ) from a structural and geometric perspective. The aim of this work is not to claim a universal derivation of the Born rule or a solution of the measurement problem, but to identify a constrained and internally coherent route by which the Born-rule form becomes admissible within the projected pseudo-Hermitian sector of PTQ.The analysis proceeds in three steps. First, we show that the probability sector must be defined only after restriction to a real, PT-even, projectively invariant observable scalar sector. Second, we demonstrate that consistency of the projected dynamics selects a pseudo-Hermitian metric structure, leading to a physically distinguished G-inner product rather than a naive kinematic norm. Third, we show that the same G-metric induces a conserved probability current, whose density is j⁰ = ψ†Gψ and whose spatial integral is preserved under the projected evolution.These ingredients jointly imply a constrained route: projected observability → physical G-inner product → conserved probability current → Pᵢ = |⟨ψ | φᵢ⟩ᴳ|². Within this route, the Born-rule form is not introduced as an independent postulate, but appears as the probability assignment compatible with observability, metric compatibility, and norm-preserving dynamics in the PT-even sector. We clarify the scope of the result: the construction is restricted to the unbroken PT-even projected sector and does not address collapse, branch selection, or fully general measurement theory. Within these boundaries, however, PTQ provides a disciplined and auditable framework in which the probability sector is structurally constrained.

DOI
10.20944/preprints202604.0888.v1

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We investigate the probability sector of PT-symmetric quaternionic spacetime (PTQ) from a structural and geometric perspective. The aim of this work is not to claim a universal derivation of the Born rule or a solution of the measurement problem, but to identify a constrained and internally coherent route by which the Born-rule form becomes admissible within the projected pseudo-Hermitian sector of PTQ.The analysis proceeds in three steps. First, we show that the probability sector must be defined only after restriction to a real, PT-even, projectively invariant observable scalar sector. Second, we demonstrate that consistency of the projected dynamics selects a pseudo-Hermitian metric structure, leading to a physically distinguished G-inner product rather than a naive kinematic norm. Third, we show that the same G-metric induces a conserved probability current, whose density is j⁰ = ψ†Gψ and whose spatial integral is preserved under the projected evolution.These ingredients jointly imply a constrained route: projected observability → physical G-inner product → conserved probability current → Pᵢ = |⟨ψ | φᵢ⟩ᴳ|². Within this route, the Born-rule form is not introduced as an independent postulate, but appears as the probability assignment compatible with observability, metric compatibility, and norm-preserving dynamics in the PT-even sector. We clarify the scope of the result: the construction is restricted to the unbroken PT-even projected sector and does not address collapse, branch selection, or fully general measurement theory. Within these boundaries, however, PTQ provides a disciplined and auditable framework in which the probability sector is structurally constrained.
This work examines the quadratic probability rule (Born rule) from the perspective of structural admissibility constraints on state evolution. Rather than relying on interpretational or decision-theoretic arguments, the analysis proceeds from symmetry, geometric, and compositional principles. Assuming bounded norm-preserving symmetries, invariance under admissible isometries, tensor-product consistency for independent systems, and normalization, the quadratic rule emerges as the probability assignment compatible with inner-product geometry. Full unitary invariance implies the parallelogram identity and therefore inner-product structure via the Jordan–von Neumann theorem. Under these structural constraints, the quadratic rule is uniquely selected. The paper further analyzes controlled deviations from exact isometric evolution. Allowing bounded, contractive perturbations consistent with completely positive trace-preserving (CPTP) dynamics, the first-order structure of admissible departures from the quadratic rule is derived. Deviations are shown to correspond to contractive departures from strict isometry and must remain operationally realizable within standard quantum dynamical frameworks. This work situates the Born rule within a broader admissibility framework previously applied to dynamical and geometric sectors. The argument is structural rather than interpretational, and emphasizes compatibility with symmetry, composition, and operational consistency.
T99 provides a geometric interpretation of the Born rule within a real symplectic-metric framework. Rather than treating quantum probabilities as fundamental postulates, the theorem derives them as normalized projection weights onto an observable subspace of a larger real state space. The construction begins with a real vector space RRR equipped with a compatible metric-symplectic-complex structure (R,g,ω,J),(R,g,\omega,J),(R,g,ω,J), where \[J^2=-I,\qquadg(Ju,Jv)=g(u,v).\] The state space is assumed to be split into complementary components \[R=O\oplusO',\] where OOO is the observable subspace and O′=JOO'=JOO′=JO is its conjugate counterpart. The theorem introduces the observable projection operator ΠO:R→O\Pi_O:R\rightarrow OΠO​:R→O and an orthogonal decomposition of the observable sector into measurement directions O=⨁iOi,O=\bigoplus_i O_i,O=⨁i​Oi​, with associated projectors Πi\Pi_iΠi​. Every state v∈Rv\in Rv∈R admits a unique decomposition \[v=o+h,\qquado\in O,\qquadh\in O'.\] The observable component is o=ΠOv,o=\Pi_O v,o=ΠO​v, while the conjugate component lies outside the directly observed sector. A central result is that the conjugate sector is metrically equivalent to the observable sector. Because g(Jv,Jv)=g(v,v),g(Jv,Jv)=g(v,v),g(Jv,Jv)=g(v,v), The conjugate component possesses the same norm structure as the observable component. It is therefore not a lower-order correction or negligible remainder, but a geometrically equivalent part of the larger real state space. The theorem then derives the Born readout formula as a projection law. For each measurement direction OiO_iOi​, Pi(v)=∥Πiv∥2∥v∥2.P_i(v)=\frac{\|\Pi_i v\|^2}{\|v\|^2}.Pi​(v)=∥v∥2∥Πi​v∥2​. The probabilities satisfy 0≤Pi(v)≤1,0\le P_i(v)\le1,0≤Pi​(v)≤1, and ∑iPi(v)=PO(v)=∥ΠOv∥2∥v∥2.\sum_i P_i(v)=P_O(v)=\frac{\|\Pi_O v\|^2}{\|v\|^2}.∑i​Pi​(v)=PO​(v)=∥v∥2∥ΠO​v∥2​. Whenever the state is normalized entirely within the observable sector, ∥ΠOv∥=∥v∥,\|\Pi_O v\|=\|v\|,∥ΠO​v∥=∥v∥, The probabilities satisfy ∑iPi(v)=1.\sum_i P_i(v)=1.∑i​Pi​(v)=1. The Born rule, therefore, emerges as the normalized squared projection of the full state onto observable measurement directions. Choosing the complex structure JJJ recovers the familiar complex formulation. The real metric and symplectic structures combine into the Hermitian inner product g(⋅,⋅)+i ω(⋅,⋅),g(\cdot,\cdot)+i\,\omega(\cdot,\cdot),g(⋅,⋅)+iω(⋅,⋅), and the projection formula becomes Pi=∣⟨i∣ψ⟩∣2.P_i=|\langle i|\psi\rangle|^2.Pi​=∣⟨i∣ψ⟩∣2. The standard Born rule is thus recovered as the complex representation of the underlying projection geometry. Several consequences follow. Components lying entirely in the conjugate sector contribute no direct observable readout when measurements are restricted to the observable slice. The observable sector nevertheless remains a complete probability space, since normalized states within OOO satisfy the usual probability-sum rule. Standard quantum mechanics, therefore, appears as the special case in which measurements access only the observable sector and no probability weight leaks from the conjugate component into observable readout. T99 provides a geometric foundation for the Born rule within the reduced Q5 framework. The theorem interprets quantum probabilities as observable-slice projections of a larger real symplectic structure and identifies the standard complex Born rule as the closed-observable-sector limit of that broader geometric description.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
This paper is the third installment (Part III) of a research series dedicated to the geometric and algebraic foundations of higher-derivative scalar theories. Following the structural uniqueness of the □2\Box^2□2 operator (Part I: arXiv:2512.16955) and the emergence of spin-analogous dynamics (Part II), this work addresses the most fundamental axioms of quantum mechanics (QM). Key findings presented in this manuscript: Geometric Origin of Complex Numbers: We demonstrate that the standard complex field C\mathbb{C}C of quantum mechanics is not a fundamental postulation but an emergent infrared (IR) feature. In the Split-Quaternionic Krein Algebra (PQA), the decoupling of heavy indefinite-metric modes at the Lee-Wick scale (M≈11.3M \approx 11.3M≈11.3 TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to C\mathbb{C}C. Derivation of the Born Rule: We show that the Born rule (P=∣ψ∣2P = |\psi|^2P=∣ψ∣2) emerges as the unique, positive-definite inner product consistent with the internal Krein metric η\etaη after projecting out ghost-sector excitations. Algebraic Necessity of the Quadratic Form: The paper provides a rationale for the second-power dependence of probability, linking it to the conservation of the bilinear form under pseudo-unitary time evolution. By reframing standard QM as a low-energy manifestation of PQA geometry, this study provides a consistent "Why" for the mathematical structure of our quantum reality, suggesting that the "ghosts" of higher-derivative theories are the very keys to understanding the origin of quantum axioms.
Abstract The Born rule, which assigns probabilities as squared amplitudes, is traditionally introduced as an independent postulate of quantum mechanics. In this work, we demonstrate that the quadratic structure of probability is instead a geometric necessity. Within a multi-channel phase algebra admitting a spectral decomposition, measurement is identified as a projection onto a single stable sector of an underlying linear state space. The norm is not postulated independently; its quadratic form is forced by the requirement that measurement produce a positive, phase-insensitive scalar while discarding unobservable phase information. Probability thus emerges as the squared norm preserved under projection, rather than as an additional axiom. In this perspective, the Born rule is revealed as a direct consequence of phase geometry and measurement stability, grounded in the geometric structure of the state space itself.