Quantum mechanics over Galois extensions of number fields

Matilde Marcolli

M. Kibler

Alain Connes A, Matilde Marcolli

Ichio Kikuchi, Akihito Kikuchi

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.

John Gardiner

Mamoru Sugamoto, Akio Sugamoto

Quantization is studied from the viewpoint of field extension. If the dynamical fields and their action have a periodicity, the space of the wave functions should be algebraically extended à la Galois, so that it may be consistent with the periodicity. This was pointed out by Nambu three decades ago. Having chosen quantum mechanics (1D field theory), this paper shows that a different Galois extension gives a different quantization scheme. A new scheme of quantization appears when the invariance under the Galois group is imposed as a physical state condition. Then, the normalization condition appears as a sum over the product of more than three wave functions, each of which is given for a different root adjoined by the field extension.