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arXiv:physics/9702025

Path integrals for a class of p-adic Schr"odinger equations

Veeravalli S. Varadarajan

Published Feb 21, 1997• 14 citations
arXivHTMLPDF

Abstract

The theme of doing quantum mechanics on all abelian groups goes back to Schwinger and Weyl. If the group is a vector space of finite dimension over a non-archimedean locally compact division ring, it is of interest to examine the structure of dynamical systems defined by Hamiltonians analogous to those encountered over the field of real numbers. In this letter a path integral formula for the imaginary time propagators of these Hamiltonians is derived.

DOI
10.48550/arxiv.physics/9702025

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The theme of doing quantum mechanics on all abelian groups goes back to Schwinger and Weyl. If the group is a vector space of finite dimension over a non-archimedean locally compact division ring, it is of interest to examine the structure of dynamical systems defined by Hamiltonians analogous to those encountered over the field of real numbers. In this letter a path integral formula for the imaginary time propagators of these Hamiltonians is derived.
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.
We continue our study of quantum dynamics on a Lie group GGG, initiated in arXiv:2512.19840, by building the path integral that governs transition amplitudes in the Hilbert space L2(G)L^2(G)L2(G). This relies on the proper handling of both a noncommutative momentum space and the presence of compact directions in GGG. We show that compactness can be handled through a sum over winding numbers in maximal tori of GGG, generalizing the similar sum commonly encountered for path integrals on a circle. As applications, we compute semiclassical approximations of propagators and partition functions of Euler-Arnold systems, up to (and including) two-loop order.
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.
We review some recent results on the mathematical foundations of a quantum theory over a scalar field that is a quadratic extension of the non-Archimedean field of [Formula: see text]-adic numbers. In our approach, we are inspired by the idea — first postulated in [I. V. Volovich, [Formula: see text]-adic string, Class. Quantum Grav. 4 (1987) L83–L87] — that space, below a suitably small scale, does not behave as a continuum and, accordingly, should be modeled as a totally disconnected metrizable topological space, ruled by a metric satisfying the strong triangle inequality. The first step of our construction is a suitable definition of a [Formula: see text]-adic Hilbert space. Next, after introducing all necessary mathematical tools — in particular, various classes of linear operators in a [Formula: see text]-adic Hilbert space — we consider an algebraic definition of physical states in [Formula: see text]-adic quantum mechanics. The corresponding observables, whose definition completes the statistical interpretation of the theory, are introduced as SOVMs, a [Formula: see text]-adic counterpart of the POVMs associated with a standard quantum system over the complex numbers. Interestingly, it turns out that the typical convex geometry of the space of states of a standard quantum system is replaced, in the [Formula: see text]-adic setting, with an affine geometry; therefore, a symmetry transformation of a [Formula: see text]-adic quantum system may be defined as a map preserving this affine geometry. We argue that, as a consequence, the group of all symmetry transformations of a [Formula: see text]-adic quantum system has a richer structure with respect to the case of standard quantum mechanics over the complex numbers.