The p-adic valuations of rational numbers give rise to ultrametric properties, which may serve as alternative definitions of the notion of large and small in physical theories, especially in quantum field theories. This has led, in the very recent years, to various attempts to construct (and interpret) quantum field theories over p-adic number fields, for example by I. V. Volovich and his collaborators, P. Freund - M. Olson - E. Witten, P. Ruelle - E. Thiran - D. Verstegen - J. Weyers, and others.
In the present article, the author develops another approach to establish a p-adic quantum mechanics. His proposal is based upon the idea to take the Heisenberg group as the fundamental starting point for quantization, and to study its unitary representations and linear canonical transformations over a complex Hilbert space with dynamical variables over p-adic number fields. The main result is that the representations (up to sign) of the linear canonical transformations can be expressed as representations of the group \(SL(2,{\mathbb{Q}}_ p)\), as it is well-known in the classical case of real numbers, and that the spectra of the subgroups corresponding to the free particle and to the harmonic oscillator can be effectively computed. The results, involving the Dirichlet L-function, suggest the interesting possibility of an adelic interpretation of the whole approach, whereas the ultimate physical reason and interpretation of this amazing phenomenon is just as mysterious as exciting.