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Journal of Mathematical Physics

Quantum mechanics on p-adic fields

Philippe Ruelle, E. Thiran, D. Verstegen, J. Weyers

Published Dec 1, 1989• 65 citations
Publisher

Abstract

A formulation of quantum mechanics on p-adic number fields is presented. Quantum amplitudes are taken as complex functions of p-adic variables and it is shown how the Weyl approach to quantum mechanics can be generalized to the p-adic case. The p-adic analogs of simple one-dimensional systems (free particle, compact and noncompact oscillators) are defined by a ‘‘group of motion,’’ which is an Abelian subgroup of SL (2,Qp). In each case the evolution operator is a unitary representation of the appropriate group. Its spectrum is given by characters and its eigenstates are calculated.

DOI
10.1063/1.528468

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A formulation of quantum mechanics on p-adic number fields is presented. Quantum amplitudes are taken as complex functions of p-adic variables and it is shown how the Weyl approach to quantum mechanics can be generalized to the p-adic case. The p-adic analogs of simple one-dimensional systems (free particle, compact and noncompact oscillators) are defined by a ‘‘group of motion,’’ which is an Abelian subgroup of SL (2,Qp). In each case the evolution operator is a unitary representation of the appropriate group. Its spectrum is given by characters and its eigenstates are calculated.
In this thesis we look at how it is possible to construct models in quantum mechanics by using p-adic numbers. First we look closely at different quantum mechanical models using the real numbers, as it is necessary to understand them well before moving on to p-adic numbers. The most promising model, where Weyl systems are used, is studied in detail. Here time translation is not generated by the Hamiltonian, but constructed directly as an operator possessing some fundamental structure in relation to the classical dynamics. Then we develop the relevant theory of the field of p-adic numbers Qp , with a focus on the properties of Qp as a locally compact abelian group. Here we present alternative proofs to those found in the literature. In particular, we give an independent proof of the selfduality of Qp. In the last chapters we look at some models using Qp . We generalize the idea of Weyl systems from real to p-adic numbers, and we discuss the physical implications. When using Weyl systems, time is p-adic. We also produce MatLab algorithms for numerical computations in connection with approximations of p-adic models by finite models.
In this thesis we look at how it is possible to construct models in quantum mechanics by using p-adic numbers. First we look closely at different quantum mechanical models using the real numbers, as it is necessary to understand them well before moving on to p-adic numbers. The most promising model, where Weyl systems are used, is studied in detail. Here time translation is not generated by the Hamiltonian, but constructed directly as an operator possessing some fundamental structure in relation to the classical dynamics. Then we develop the relevant theory of the field of p-adic numbers Qp , with a focus on the properties of Qp as a locally compact abelian group. Here we present alternative proofs to those found in the literature. In particular, we give an independent proof of the selfduality of Qp. In the last chapters we look at some models using Qp . We generalize the idea of Weyl systems from real to p-adic numbers, and we discuss the physical implications. When using Weyl systems, time is p-adic. We also produce MatLab algorithms for numerical computations in connection with approximations of p-adic models by finite models.
This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ℚ p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ℚ p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.
In this paper, we construct a p-adic path integral via p-adic multiple integrals. This integral describes the evolution of a wave function Ψ(x), defined as a map from a domain in Cp to Cp. Unlike the standard p-adic quantum mechanics formulated for functions Qp→C, our construction works directly on Cp-valued wave functions. Using p-adic Dirac delta measures and p-adic additive characters, we define the propagator as a finite-partition multiple integral, which allows explicit iterative integration. As an application, we compute the Feynman propagator for free particles and obtain a closed-form expression analogous to the classical counterpart. Our method provides a discrete, constructive alternative that complements the existing distribution-theoretic frameworks for p-adic functional integration.