Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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. Kikuchi, I. Galois and Class Field Theory for Quantum Chemists. Preprints 2020, 2020070011.


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 Computer programs used in the study.


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


Physical Sciences, Quantum Science and Technology

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