Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed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 (doi: 10.20944/preprints202007.0011.v1). Kikuchi, I. Galois and Class Field Theory for Quantum Chemists. Preprints 2020, 2020070011 (doi: 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.

Subject Areas

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

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