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Galois and Class Field Theory for Quantum Chemists
Version 1
: Received: 30 June 2020 / Approved: 3 July 2020 / Online: 3 July 2020 (04:54:16 CEST)
How to cite: Kikuchi, I. Galois and Class Field Theory for Quantum Chemists. Preprints 2020, 2020070011. https://doi.org/10.20944/preprints202007.0011.v1 Kikuchi, I. Galois and Class Field Theory for Quantum Chemists. Preprints 2020, 2020070011. https://doi.org/10.20944/preprints202007.0011.v1
Abstract
Quantum mechanics could be studied through polynomial algebra, as has been demonstrated by a work (“An approach to first-principles electronic structure calculation by symbolic-numeric computation” by A. Kikuchi). We carry forward the algebraic method of quantum mechanics through algebraic number theory; the basic equations are represented by the multivariate polynomial ideals; the symbolic computations process the ideal and disentangle the eigenstates as the algebraic variety; upon which one canbuild the Galois extension of the number field, in analogy with the univariate polynomial case, to investigate the hierarchy of solutions; the Galois extension is accompanied with the group operations, which permute the eigenstates from one to another, and furnish the quantum system with a non-apparent symmetry. Besides, this sort of algebraic quantum mechanics is an embodiment of the class field theory; some of the important consequences of the latter emerge in quantum mechanics. We shall demonstrate these points through simple models; we will see the use of computational algebra facilitates such sort of analysis, which might often be complicated if we try to solve them manually.
Supplementary and Associated Material
https://github.com/kikuchiichio/GaloisClassFieldTheoryQ: Computer programs used in the study.
Keywords
quantum mechanics; algebraic geometry; algebraic number theory; commutative algebra; Gr¨onber basis; primary ideal decomposition, eigenvalue problem in quantum mechanics; molecular orbital theory; quantum chemistry; quantum chemistry in algebraic variety; symbolic computation; algebraic molecular orbital theory; Galois theory; class field theory
Subject
Physical Sciences, Quantum Science and Technology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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