In the recent years, there has been a growing interest in analyzing physical theories defined over groundfields admitting an ultrametric structure. The most familiar fields of this kind are the fields \({\mathbb{Q}}_ p\) of p-adic numbers or their algebraic extensions. The relevance for using p-adic numbers has become particularly obvious in the recent development of string theories, which is basically due to the fact that the world sheet parameters are intrinsically non-observable.
Thus one is led to find a suitable formulation of a general theory of quantum mechanics over the p-adic number fields \({\mathbb{Q}}_ p\). During the past five years, various approaches to formulate a p-adic quantum mechanics have been suggested, in particular by V. S. Vladimirov, I. V. Volovich and their collaborators [cf., e.g., their paper Commun. Math. Phys. 123, 659-676 (1989; Zbl 0688.22004)].
In the present paper, the authors present another possible approach. Their formulation is basically a generalization to \({\mathbb{Q}}_ p\) of the classical Weyl formulation of quantum mechanics over the real number field \({\mathbb{R}}\). Thus the evolution matrices appearing here belong to abelian subgroups of \(SL(2,{\mathbb{Q}}_ p)\), and these are studied in detail. This is then used to determine, via the corresponding unitary characters of those subgroups, the eigenvalues and eigenfunctions (or the spectrum and the wave functions, respectively) of the evolution operators in some particular cases, namely for the p-adic free particle, the non- compact oscillator and the p-adic compact harmonic oscillator. Possible generalizations of this approach to more complicated physical p-adic systems are briefly discussed in the concluding section of the paper.
Altogether, this article is fairly self-contained and detailed. It contains, at the beginning, a section on the basic facts about the number fields \({\mathbb{Q}}_ p\) and, at the end, several appendices providing explicit calculations, computational procedures, and examples.
As for a closely related, however different approach to the same problem, one should compare this paper to the recent one by V. S. Vladimirov, I. V. Volovich and E. I. Zelenov [Izv. Akad. Nauk SSSR, Ser. Mat. 54, No.2, 275-302 (1990; see the preceding review Zbl 0709.22010)].