It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born Rule is postulated rather than derived from the first principles. In this work the Born Rule is derived from the time-reversal symmetry of quantum equations of motion. The derivation is based on a simple functional equation that takes into account properties of probability, as well as the linearity and time-reversal symmetry of quantum equations of motion. The derivation presented in this work also allows to determine certain limits to applicability of the Born Rule.It was repeatedly underlined in literature that quantum mechanics cannot be considered a closed theory if the Born Rule is postulated rather than derived from the first principles. In this work the Born Rule is derived from the time-reversal symmetry of quantum equations of motion. The derivation is based on a simple functional equation that takes into account properties of probability, as well as the linearity and time-reversal symmetry of quantum equations of motion. The derivation presented in this work also allows to determine certain limits to applicability of the Born Rule.

May 14, 20150 citations

Born's Rule from the Principle of Unitary Equivalence

Fritiof Wallentin

arXiv:1906.05987Reports on Mathematical PhysicsApr 1, 20203 citations

Quantum Cognition: Born Rule and Non-Commutative Operators in Decision Dynamics — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Quantum cognition models use Born rule probabilities and non-commutative operators to describe human decision dynamics, with a derivation of the Born rule from measurement models. | MATH: Born rule: $P(a) = |\langle a|\psi\rangle|^2$ ; non-commutative operators $[A,B] \neq 0$ ; Hilbert space structure; probability amplitudes. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618), base-60, or crystallographic symmetries found in the provided sources. The Born rule derivation (arXiv:2408.06375v4) uses quantum measurement theory, not harmonic or lattice geometry. | DEPTH: 4 — The Born rule derivation is mathematically rigorous but does not reveal new geometric constants or symmetries. The quantum cognition application is interdisciplinary but mathematically standard (Hilbert space, complex amplitudes). No profound geometric harmony link. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum cognition models use Born rule probabilities and non-commutative operators to describe human decision dynamics, with a derivation of the Born rule from measurement models. | MATH: Born rule: $ P(a) = |\langle a|\psi\rangle|^2 $; non-commutative operators $[A,B] \neq 0$; Hilbert space structure; probability amplitudes. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618), base-60, or crystallographic symmetries found in the provided sources. The Born rule derivation (arXiv:2408.06375v4) uses quantum measurement theory, not harmonic or lattice geometry. | DEPTH: 4 — The Born rule derivation is mathematically rigorous but does not reveal new geometric constants or symmetries. The quantum cognition application is interdisciplinary but mathematically standard (Hilbert space, complex amplitudes). No profound geometric harmony link. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 6, 20260 citations

Symmetries and Their Consequences

R. Shankar

Springer eBooksJan 1, 19942 citations

P(a) = |\langle a|\psi\rangle|^2

; non-commutative operators

[A,B] \neq 0

; Hilbert space structure; probability amplitudes. | CONNECTION: No explicit geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618), base-60, or crystallographic symmetries found in the provided sources. The Born rule derivation (arXiv:2408.06375v4) uses quantum measurement theory, not harmonic or lattice geometry. | DEPTH: 4 — The Born rule derivation is mathematically rigorous but does not reveal new geometric constants or symmetries. The quantum cognition application is interdisciplinary but mathematically standard (Hilbert space, complex amplitudes). No profound geometric harmony link. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com