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p-adic quantum mechanics. (English) Zbl 0688.22004
We are now at the very start of p-adic quantum mechanics. The p-adic numbers could be useful, for instance, to describe the space-time geometry at very small distances, or order of the so called Planck length. The aim of the paper reviewed is to generalize the quantum mechanics formalism to the field of p-adic numbers. The authors start from the p-adic classical mechanics in which time, spatial coordinates, and momentum are defined over the p-adic field and further formulate p- adic quantum mechanics via the Weyl representation. As an example, the evolution operators for a free particle and harmonic oscillator are constructed. The existence of at least two generalized vacuum states for prime p of the \((4\ell +1)st\) form is proved. The authors also propose an approach for investigating spectral properties of p-adic quantum mechanical systems. Another version of the p-adic quantum mechanics originating from the Euclidean formalism is under study, too. It is demonstrated, particularly, that the asymptotics of propagator of a massive particle behave at infinity as \(| t|_ p^{-3}\), unlike the exponential one in the case of the real-number version of quantum mechanics.
Reviewer: E.Kryachko

MSC:
22E70 Applications of Lie groups to the sciences; explicit representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
22E35 Analysis on \(p\)-adic Lie groups
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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