Infinitesimal and Infinite Numbers as an Approach to Quan- tum Mechanics Vieri Benci1, Lorenzo Luperi Baglini2, and Kyrylo Simonov2 1Dipartimento di Matematica, Universit`a degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy 2Fakult¨at fur¨ Mathematik, Universit¨at Wien, Oskar-Morgenstern Platz 1, 1090 Vienna, Austria April 29, 2019 Non-Archimedean mathematics is an ap- Non-Archimedean mathematics (particularly, proach based on fields which contain in- nonstandard analysis) is a framework that treats finitesimal and infinite elements. Within the infinitesimal and infinite quantities as num- this approach, we construct a space of a bers. Since the introduction of nonstandard anal- particular class of generalized functions, ysis by Robinson [2] non-Archimedean mathe- ultrafunctions. The space of ultrafunctions matics has found a plethora of applications in can be used as a richer framework for a physics [3,5,4,6,7], particularly in quantum me- description of a physical system in quan- chanical problems with singular potentials, such tum mechanics. In this paper, we pro- as δ-potentials [8,9, 10, 11, 12]. vide a discussion of the space of ultrafunc- In this paper, we build a non-Archimedean tions and its advantages in the applica- approach to quantum mechanics in a simpler tions of quantum mechanics, particularly way through a new space, which can be used for the Schr¨odinger equation for a Hamil- as a basic construction in the description of a tonian with the delta function potential. physical system, by analogy with the Hilbert space in the standard approach. This space 1 Introduction is called the space of ultrafunctions, a particu- lar class of non-Archimedean generalized func- Quantum mechanics is a highly successful physi- tions [13, 14, 15, 16, 17, 18, 19]. The ultrafunc- ∗ cal theory, which provides a counter-intuitive but tions are defined on the hyperreal field R , which accurate description of our world. During more extends the reals R by including infinitesimal and than 80 years of its history, there were devel- infinite elements into it. Such a construction al- oped various formalisms of quantum mechanics lows studying the problems which are difficult that use the mathematical notions of different to solve and formalize within the standard ap- complexity to derive its basic principles. The proach. For example, variational problems, that standard approach to quantum mechanics han- have no solutions in standard analysis, can be dles linear operators, representing the observ- solved in the space of ultrafunctions [18]. In this ables of the quantum system, that act on the way, non-Archimedean mathematics as a whole vectors of a Hilbert space representing the phys- and the ultrafunctions as a particular propose a ical states. However, the existing formalisms in- richer framework, which highlights the notions clude not only the standard approach but as well hidden in the standard approach and paves the arXiv:1901.10945v3 [math-ph] 26 Apr 2019 some more abstract approaches that go beyond way to a better understanding of quantum me- Hilbert space. A notable example of such an chanics. abstract approach is the algebraic quantum me- The paper is organized as follows. In Section1 chanics, which considers the observables of the we introduce the needed notations and the notion quantum system as a non-abelian C∗-algebra, of a non-Archimedean field. In Section2 we in- and the physical states as positive functionals on troduce a particular non-Archimedean field, the it [1]. field of Euclidean numbers E, through the notion of Λ-limit, which is a useful, straightforward ap- Vieri Benci: [email protected] proach to the nonstandard analysis. In Section3 Lorenzo Luperi Baglini: [email protected] we introduce the space of ultrafunctions, which Kyrylo Simonov: [email protected] are a particular class of generalized functions. In Accepted in Quantum 2019-04-25, click title to verify 1 Section4 we apply the ultrafunctions approach to 1.2 Non-Archimedean fields quantum mechanics and discuss its advantages in Our approach to quantum mechanics makes mul- contrast to the standard approach. In Section5 tiple uses of the notions of infinite and infinites- we provide a discussion of a quantum system with imal numbers. A natural framework to intro- a delta function potential for the standard and duce these numbers suitably is provided by non- ultrafunctions approaches. Last but not least, in Archimedean mathematics (see, e.g., [20]). This Section6 we provide the conclusions. framework operates with the infinite and in- finitesimal numbers as the elements of the new 1.1 Notations non-Archimedean field. N Let Ω be an open subset of , then 1 R Definition 1. Let K be an infinite ordered field . • C (Ω) denotes the set of continuous functions An element ξ ∈ K is: N defined on Ω ⊂ R , • infinitesimal if, for all positive n ∈ N, |ξ| < 1 • Cc (Ω) denotes the set of continuous func- n , tions in C (Ω) having compact support in Ω, • finite if there exists n ∈ N such that |ξ| < n, • Ck (Ω) denotes the set of functions defined on • infinite if, for all n ∈ , |ξ| > n (equiva- Ω ⊂ N which have continuous derivatives N R lently, if ξ is not finite). up to the order k, We say that is non-Archimedean if it contains • Ck (Ω) denotes the set of functions in K c an infinitesimal ξ 6= 0, and that is superreal if Ck (Ω) having compact support, K it properly extends R. • D (Ω) denotes the space of infinitely differ- Notice that, trivially, every superreal field is entiable functions with compact support de- non-Archimedean. Infinitesimals allow introduc- fined almost everywhere in Ω, ing the following equivalence relation, which is • L2 (Ω) denotes the space of square integrable fundamental in all non-Archimedean settings. functions defined almost everywhere in Ω, Definition 2. We say that two numbers ξ, ζ ∈ K 1 are infinitely close if ξ−ζ is infinitesimal. In this • Lloc (Ω) denotes the space of locally inte- grable functions defined almost everywhere case we write ξ ∼ ζ. in Ω, In the superreal case, ∼ can be used to intro- N duce the fundamental notion of \standard part"2. • mon(x) = {y ∈ E | x ∼ y} (see Def.4), Theorem 3. If K is a superreal field, every finite • given any set E ⊂ X, χE : X → R denotes the characteristic function of E, namely, number ξ ∈ K is infinitely close to a unique real number r ∼ ξ, called the the standard part of 1 if x ∈ E, ξ. χE(x) := 0 if x∈ / E, Following the literature, we will always denote by st(ξ) the standard part of any finite number • with some abuse of notation, we set χa(x) := ξ. Moreover, with a small abuse of notation, we also put st(ξ) = +∞ (resp. st(ξ) = −∞) if ξ ∈ χ{a}(x), K is a positive (resp. negative) infinite number. ∂ • ∂i = ∂x denotes the usual partial deriva- i Definition 4. Let K be a superreal field, and ξ ∈ tive, Di denotes the generalized derivative a number. The monad of ξ is the set of all (see Section 3.1), K numbers that are infinitely close to it, • R denotes the usual Lebesgue integral, H denotes the pointwise integral (see Section mon(ξ) = {ζ ∈ K : ξ ∼ ζ}. 3.1), 1 Without loss of generality, we assume that Q ⊆ K. • if E is any set, then |E| denotes its cardinal- 2For a proof of the following simple theorem, the inter- ity. ested reader can check, e.g., [21]. Accepted in Quantum 2019-04-25, click title to verify 2 Notice that, by definition, the set of infinitesi- Let Λ be an infinite set containing R and let L mals is mon(0) precisely. Finally, superreal fields be the family of finite subsets of Λ. A function can be easily complexified by considering ϕ : L → R will be called net (with values in R). The set of such nets is denoted by F (L, R) and K + iK, equipped with the natural operations namely, a field of numbers of the form (ϕ + ψ)(λ) = ϕ(λ) + ψ(λ), (ϕ · ψ)(λ) = ϕ(λ) · ψ(λ), a + ib, a, b ∈ K. and the partial order relation In this way, the complexification of non- Archimedean fields shows no particular difficulty ϕ ≥ ψ ⇐⇒ ∀λ ∈ L, ϕ(λ) ≥ ψ(λ). and is straightforward. In this way, F (L, R) is a partially ordered real algebra. 2 The field of Euclidean numbers Definition 5. We say that a superreal field E is a field of Euclidean numbers if there is a surjective Nonstandard analysis plays one of the most map prominent roles between various approaches to J : F (L, R) → E, non-Archimedean mathematics. One reason is that nonstandard analysis provides a handy tool which satisfies the following properties, to study and model the problems which come • J (ϕ + ψ) = J (ϕ) + J (ψ), from many different areas. However, the classical representations of nonstandard analysis can feel • J (ϕ · ψ) = J (ϕ) · J (ψ), overwhelming sometimes, as they require a good knowledge of the objects and methods of mathe- • if ϕ(λ) > r, then J (ϕ) > r. matical logic. This stands in contrast to the ac- J will be called the realization of . tual use of nonstandard objects in the mathemat- E ical practice, which is almost always extremely The proof of the existence of such a field is an close to the usual mathematical practice. easy consequence of the Krull { Zorn theorem. For these reasons, we believe that it is worth It can be found, e.g., in [24, 13, 14, 26]. In this to present nonstandard analysis avoiding most of paper, we also use the complexification of E, de- the usual logic machinery.