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arXiv:1701.07746

Quantum Mechanics With Noncommutative Amplitudes

Gregory W. Moore

Published Jan 26, 2017• 3 citations
arXivHTMLPDF

Abstract

We formulate a Born rule for families of quantum systems parametrized by a noncommutative space of control parameters. The resulting formalism may be viewed as a generalization of quantum mechanics where overlaps take values in a noncommutative algebra, rather than in the complex numbers.
DOI
10.48550/arxiv.1701.07746

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News and views

AAPPS

The article gives an introductory summary of our recent results from the perspective of the new conceptual notion of noncommutative values of quantum observables, where basic quantum mechanics is seen a theory of particle dynamics that is intrinsically noncommutative.Real numbers as an algebra is really an abstract mathematical model for the description of values of physical quantities.Quantum mechanics illustrates the limitation of that perspective.The noncommutative values preserve the algebraic relations among the observables in their values.Such a value contains the complete information the theory presents for an observable on a given state.All physical quantities, including coordinates of the physical space for describing the location of a particle take definite noncommutative values.

AAPPS bulletin•Feb 8, 2022•0 citations
We prove that the parametric family x² − x + c (c = ±1) is uniquely forced by three axioms: conservation, minimality, and binary coupling. The two polynomials generate the simplest real and imaginary quadratic number fields — ℚ(√5) and ℚ(√−3) — and classify trace-1 dynamics in GL(2,ℤ) into hyperbolic and elliptic regimes. The elliptic regime determines a unique positive-definite inner product via Schur’s lemma. Under continuous-time propagation, the hyperbolic regime forces complex-valued amplitudes. The Born rule p(z) = |z|² is derived as the unique measurement map respecting the dynamically determined inner product and tensor product compositionality. Ten independently verified physical domains — spanning condensed matter, quantum foundations, dynamical systems, and statistical mechanics — are cataloged as empirical instances of this algebraic structure at critical boundaries. No free parameters appear in the algebraic structure itself.
real-valued calculations in artificial neural networks captures the amplitudes of neural signals but neglects their phases, which are critical parameters controlling the composition of cognitive waves in natural brains. To address this limitation, we present a complex-valued modification of singular matrix decomposition, a conceptual precursor to the tensor algebra underlying modern neural networks. In this approach, we generalize the diagonal matrix of singular values to a self-adjoint complex matrix, analogous to the density matrix in quantum theory. Within low-dimensional “semantic” spaces, the additional non-diagonal elements of the complex matrix account for the non-stationary logic of the cognitive systems that generate the data. As in the standard formulation, the density matrix is sandwiched between two orthogonal real-valued matrices, unfolding semantic regularities into the space of observable events. The squared modulus of the resulting set of complex-valued amplitudes then produces observable real-valued data, following the quantum-mechanical Born rule. By introducing a minor increase in the number of parameters, our method significantly enhances the precision of classical singular value decomposition. This improvement highlights the efficiency of wave-like and quantum-inspired principles in natural cognition, as expressed in the proposed algebra. The method provides new opportunities for semantic data analysis and offers pathways to advance modern neural network architectures.
The article gives an introductory summary of our recent results from the perspective of the new conceptual notion of noncommutative values of quantum observables, where basic quantum mechanics is seen a theory of particle dynamics that is intrinsically noncommutative.Real numbers as an algebra is really an abstract mathematical model for the description of values of physical quantities.Quantum mechanics illustrates the limitation of that perspective.The noncommutative values preserve the algebraic relations among the observables in their values.Such a value contains the complete information the theory presents for an observable on a given state.All physical quantities, including coordinates of the physical space for describing the location of a particle take definite noncommutative values.