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 $\Box^2$ 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 $\mathbb{C}$ 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 ( $M \approx 11.3$ TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to $\mathbb{C}$ . Derivation of the Born Rule: We show that the Born rule ( $P = |\psi|^2$ ) emerges as the unique, positive-definite inner product consistent with the internal Krein metric $\eta$ 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 $\Box^2$ 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 $\mathbb{C}$ 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 ($M \approx 11.3$ TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to $\mathbb{C}$. Derivation of the Born Rule: We show that the Born rule ($P = |\psi|^2$) emerges as the unique, positive-definite inner product consistent with the internal Krein metric $\eta$ 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.

Feb 1, 20260 citations

The Geometric Origin of Quantum Axioms: Emergence of Complex Numbers and the Born Rule from Split-Quaternionic Krein Algebra (Part III)

Masayuki Note

Jan 25, 20260 citations

Geometric Phase in Quaternionic Quantum Mechanics

M. D. Maia, V. B. Bezerra

arXiv:hep-th/0107107International Journal of Theoretical PhysicsJul 1, 20015 citations

Geometric Phase Associated with Spinor Rotations

Apoorva G. Wagh, Veer Chand Rakhecha

Springer eBooksJan 1, 19934 citations

\Box^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

\mathbb{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 \approx 11.3

TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to

\mathbb{C}

. Derivation of the Born Rule: We show that the Born rule (

P = |\psi|^2

) emerges as the unique, positive-definite inner product consistent with the internal Krein metric

\eta

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.

\Box^2

\mathbb{C}

M \approx 11.3

TeV) naturally reduces the internal degrees of freedom to a complex subalgebra isomorphic to

\mathbb{C}

. Derivation of the Born Rule: We show that the Born rule (

P = |\psi|^2

) emerges as the unique, positive-definite inner product consistent with the internal Krein metric

\eta