A non-archimedean algebra and the Schwartz impossibility theorem

Abstract

In the 1950s L. Schwartz proved his famous impossibility result: for every k ∈ N there does not exist a differential algebra (A, +, ⊗, D) in which the distributions can be embedded, where D is a linear operator that extends the distributional derivative and satisfies the Leibnitz rule (namely D(u ⊗ v) = Du ⊗ v + u ⊗ Dv) and ⊗ is an extension of the pointwise product on C 0 (R).In this paper we prove that, by changing the requests, it is possible to avoid the impossibility result of Schwartz. Namely we prove that it is possible to construct an algebra of functions (A,+, ⊗, D) such that (1) the distributions can be embedded in A in such a way that the restriction of the product to C 1 (R) functions agrees with the pointwise product, namely for every f, g ∈ C 1 (R)and (2) there exists a linear operator D : A → A that extends the distributional derivative and satisfies a weak form of the Leibnitz rule.The algebra that we construct is an algebra of restricted ultrafunctions, which are generalized functions defined on a subset Σ of a nonarchimedean field K (with R ⊂ Σ ⊂ K) and with values in K. To study the restricted ultrafunctions we will use some techniques of nonstandard analysis. Mathematics subject classification: 13N99, 26E30, 26E35, 46F30.

Generalized functions beyond distributions

“…Let us note that the second property presented in this desideratum is a weak form of the Leibniz rule. In fact our derivative will not satisfy Leibniz rule (in general); nevertheless once again we note that this weak form is sufficient for many applications (see [10] for a discussion on this point).…”

“…The relations between ultrafunctions, distributions and Schwartz impossibility result are precised in [10], where it is constructed an algebra of ultrafunctions in which all distributions can be embedded. Let us note that, in general, we do not require the space of ultrafunctions to be an algebra: in fact, the multiplication uv of two ultrafunctions u, v is defined, since ultrafunctions are internal functions, but in general we do not have (and for many applications, we do not need to require) that uv ∈ V (R).…”

“…In [10] we showed that, actually, it is possible to define an embedding of the whole space of distributions in a particular algebra of ultrafunctions. Our embedding preserves the pointwise moltiplication of C 1 functions; we do not contrast with Schwartz impossibility result because the derivative of ultrafunctions satisfies only a weak form of the Leibniz rule.…”

Infinitesimal and infinite numbers as an approach to quantum mechanics

“…An explicit technical construction of (several) ultrafunctions spaces has been provided in Ref. [13,16,17,14,15,19,18] by various reformulations of nonstandard analysis. However, we prefer to pursue an axiomatic approach to underscore the key properties of ultrafunctions needed for our aims since such technicalities are not important in the applications of the ultrafunctions spaces.…”

“…In this paper, we build a non-Archimedean approach to quantum mechanics in a simpler way through a new space, which can be used as a basic construction in the description of a physical system, by analogy with the Hilbert space in the standard approach. This space is called the space of ultrafunctions, a particular class of non-Archimedean generalized functions [13,14,15,16,17,18,19]. The ultrafunctions are defined on the hyperreal field R * , which extends the reals R by including infinitesimal and infinite elements into it.…”