Infinitesimal and infinite numbers as an approach to quantum mechanics

Abstract

Non-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of ultrafunctions can be used as a richer framework for a description of a physical system in quantum mechanics. In this paper, we provide a discussion of the space of ultrafunctions and its advantages in the applications of quantum mechanics, particularly for the Schrödinger equation for a Hamiltonian with the delta function potential.Notice that, trivially, every superreal field is non-Archimedean. Infinitesimals allow introducing the following equivalence relation, which is fundamental in all non-Archimedean settings.Definition 2. We say that two numbers ξ, ζ ∈ K are infinitely close if ξ −ζ is infinitesimal. In this case we write ξ ∼ ζ.In the superreal case, ∼ can be used to introduce the fundamental notion of "standard part" 2 . Theorem 3. If K is a superreal field, every finite number ξ ∈ K is infinitely close to a unique real number r ∼ ξ, called the the standard part of ξ.Following the literature, we will always denote by st(ξ) the standard part of any finite number ξ. Moreover, with a small abuse of notation, we also put st(ξ) = +∞ (resp. st(ξ) = −∞) if ξ ∈ K is a positive (resp. negative) infinite number. Definition 4. Let K be a superreal field, and ξ ∈ K a number. The monad of ξ is the set of all numbers that are infinitely close to it,1 Without loss of generality, we assume that Q ⊆ K.2 For a proof of the following simple theorem, the interested reader can check, e.g., [21].

Halos and undecidability of tensor stable positive maps

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A characterization of singular Schrödinger operators on the half-line

“…Singular problems can be fruitfully studied by means of non-Archimedean methods, which provide rigorous ways to handle infinitesimal and infinite quantities; in particular, in our problem they give the possibility to formalize the idea of letting the potential be zero ranged and delta-like by taking infinitesimal and infinite in the expression (1.1). In this paper, we will focus mostly on non-Archimedean methods based on nonstandard analysis, on similar lines of those contained in the seminal paper [2] (see also the recent paper [7]), but there have also been other non-Archimedean approaches to the study of equations like (1.1), for example, those based on different versions of Colombeau algebras (see e.g. [16,23] and references therein).…”

Everettian Mechanics with Hyperfinitely Many Worlds

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