Infinitesimal and Infinite Numbers as an Approach to Quantum Mechanics

Non-Archimedean mathematics is an approach based on the fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The ultrafunction space 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 ultrafunction space and its advantages in the applications of quantum mechanics, particularly for the Schr\"{o}dinger equation for a Hamiltonian with the delta function potential.


I. INTRODUCTION
Quantum mechanics is a highly successful physical theory, which provides a counter-intuitive but accurate description of our world. During more than 80 years of its history, there were developed various formalisms of quantum mechanics, which use the mathematical notions of different complexity, to derive its basic principles. The standard approach to quantum mechanics handles linear operators, representing the observables of the quantum system, that act on the vectors of a Hilbert space representing the physical states. However, the existing formalisms include not only the standard approach but as well some more abstract approaches that go beyond Hilbert space. A notable example of such an abstract approach is the algebraic quantum mechanics, which considers the observables of the quantum system as a nonabelian C * -algebra and the physical states as positive functionals on it [1].
Non-Archimedean mathematics (particularly, nonstandard analysis) is a framework that treats the infinitesimal and infinite quantities as numbers. Since the introduction of non-standard analysis by Robinson [2] non-Archimedean mathematics has found a plethora of applications in physics [3], particularly in quantum mechanical problems with singular potentials, such as δpotentials [4][5][6][7][8].
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 [9][10][11][12][13][14][15]. The ultrafunctions are defined on the hyperreal field R * , which extends the reals R by including infinitesimal and infinite elements into it. Such a construction allows studying the problems which are difficult to solve and formalize within the standard approach. For example, the variational problems, that have no solutions in the standard analysis, can be solved in the ultrafunction space [14]. In this way, non-Archimedean mathematics as a whole and the ultrafunctions as a particular propose a richer framework, which highlights the notions hidden in the standard approach and paves the way to better understanding of quantum mechanics.
The paper is organized as follows. In Section I we introduce the needed notations and the notion of a non-Archimedean field. In Section II we introduce a particular non-Archimedean field, the field of Euclidean numbers E, through the notion of Λ-limit, which is a useful, straightforward approach to the non-standard analysis. In Section III we introduce the space of ultrafunctions, which are a particular class of generalized functions. In Section IV we apply the ultrafunction approach to quantum mechanics and discuss its advantages in contrast to the standard approach. In Section V we provide a discussion of a quantum system with a delta function potential for the standard and ultrafunction approaches. Last but not least, in Section VI we provide the conclusions.

A. Notations
Let Ω be an open subset of R N , then • C (Ω) denotes the set of continuous functions defined on Ω ⊂ R N , • C c (Ω) denotes the set of continuous functions in C (Ω) having compact support in Ω, • C k (Ω) denotes the set of functions defined on Ω ⊂ R N which have continuous derivatives up to the order k, • C k c (Ω) denotes the set of functions in C k (Ω) having compact support, • D (Ω) denotes the space of infinitely differentiable functions defined almost everywhere in Ω, • L 2 (Ω) denotes the space of square integrable functions defined almost everywhere in Ω, • L 1 loc (Ω) denotes the space of locally integrable functions defined almost everywhere in Ω, • mon(x) = {y ∈ E N | x ∼ y} (see Def. 4), • given any set E ⊂ X, χ E : X → R denotes the characteristic function of E, namely, • with some abuse of notation we set χ a (x) := χ {a} (x), • ∂ i = ∂ ∂xi denotes the usual partial derivative, D i denotes the generalized derivative (see Section III A), • denotes the usual Lebesgue integral, denotes the pointwise integral (see Section III A), • if E is any set, then |E| denotes its cardinality.

B. Non-Archimedean Fields
Our approach to quantum mechanics makes multiple uses of the notions of infinite and infinitesimal numbers. A natural framework to introduce these numbers suitably is provided by the non-Archimedean mathematics (see, e.g., [16]). This framework operates with the infinite and infinitesimal numbers as the elements of the new non-Archimedean field. Definition 1. Let K be an infinite ordered field 1 . An element ξ ∈ K is: We say that K is non-Archimedean if it contains an infinitesimal ξ = 0, and that K is superreal if it properly extends R.
Notice that, trivially, every superreal field is non-Archimedean. Infinitesimals allow introducing the following equivalence relation, which is fundamental in all non-Archimedean settings. 1 Without loss of generality we assume that Q ⊆ K. 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: Notice that, by definition, the set of infinitesimals is mon(0) precisely. Finally, superreal fields can be easily complexified by considering namely, a field of numbers of the form a + ib, a, b ∈ K.
In this way, the complexification of non-Archimedean fields presents no particular difficulty and is straightforward.

II. THE FIELD OF EUCLIDEAN NUMBERS
The non-standard analysis plays one of the most prominent roles between various approaches to non-Archimedean mathematics. One reason is that the nonstandard analysis provides a handy tool to study and model the problems which come from many different areas. However, the classical representations of the nonstandard analysis can feel overwhelming sometimes, as they require a good knowledge of the objects and methods of mathematical logic. This stands in contrast to the actual use of non-standard objects in the mathematical practice, which is almost always extremely close to the usual mathematical practice.
For these reasons, we believe that it is worth to present the non-standard analysis avoiding most of the usual logic machinery. We will introduce the non-standard approach to the formalism of quantum mechanics via the notion of Λ-limit 3 and the Euclidean numbers. These numbers are the underlying object of Λ-theory, and can be introduced via a purely algebraic approach, as done in [20] The basic idea of the construction of Euclidean numbers is the following: as the real numbers can be constructed by completion of the rational numbers using the Cauchy notion of limit, the Euclidean numbers can be constructed by completion of the reals with respect to a new notion of limit, called Λ-limit, that we are now going to introduce.
Let Λ be an infinite set containing R and let L be the family of finite subsets of Λ. A function ϕ : L → E will be called net (with values in E). The set of such nets is denoted by F (L, R) and equipped with the natural operations and the partial order relation In this way, F (L, R) is a partially ordered real algebra.

Definition 5. We say that a superreal field E is a field of Euclidean numbers if there is a surjective map
which satisfies the following properties, • if ϕ(λ) ≥ r, then J (ϕ) ≥ r.
J will be called the realization of E.
The proof of the existence of such a field is an easy consequence of the Krull-Zorn theorem. It can be found, e.g., in [9,10,20,22]. In this paper, we also use the complexification of E, denoted 5 by The name "limit" has been chosen because the operation ϕ → lim λ↑Λ ϕ(λ) satisfies the following properties, • (Λ-1) Existence. Every net ϕ : L → R has a unique limit L ∈ E, • (Λ-3) Sum and product. For all ϕ, ψ : The Cauchy limit of a sequence (as formalized by Weierstrass) satisfies the second and the third 6 condition above. The main difference between the Cauchy and the Λ-limit is given by the property (Λ − 1), namely, by the fact that the Λ-limit always exists. Notice that it implies that E must be larger than R since it must contain the limit of diverging nets as well.
For those nets that have a Cauchy limit, the relationship between the Cauchy and the Λ-limit is expressed by the following identity, lim λ→Λ ϕ(λ) = st lim λ↑Λ ϕ(λ) . ( With the introduction of the Euclidean numbers there arises a natural question: What do they look like? Let us give a few examples. Then ε > 0, as ϕ(λ) > 0 for every λ ∈ L. However, ε < r for every positive r ∈ R as ϕ(λ) < r eventually in λ. Therefore, ε is a nonzero infinitesimal in E.
2. Analogously, let ϕ(λ) := |λ| for every λ ∈ L. Let σ = lim λ↑Λ ϕ(λ). Straightforwardly, σ is a positive infinite element in E, and σ · ε = 1. where |F | denotes the number of elements of a finite set F . As λ is a finite set, |E ∩ λ| ∈ N for every λ. Now, if E is finite then E belongs to L, and the net |E ∩ λ| is eventually constant (in λ) and equal to |E|, and so n (E) = |E|. If E is an infinite set, the above limit gives an infinite number (which, in the literature, is called the numerosity of E. The reader interested in the details and the developments of the theory of numerosities is referred to [23][24][25][26]).

Definition 7. A mathematical entity is called internal
if it is a Λ-limit of some other entities.
A. Extension of functions, hyperfinite sets and grid functions The Λ-limit allows extending the field of real numbers to the field of Euclidean numbers. Similarly, we can use it to extend sets and functions in arbitrary dimensions, we let the natural extension of f to A * to be a function such that, for every x = lim • the above case can be generalized, in order to define directly the functions between subsets of E N : if is a net of functions, its Λ-limit is a function such that, for any x = lim λ↑Λ x λ ∈ E N , • finally, if V is a function space, we let its natural extension to be by (2), it follows that Moreover, it is clear that E is the natural extension of R, namely, E = R * . Notice that this choice justifies also our notation (1) for the complexification of E, Now we introduce a fundamental notion for all our applications, the "hyperfinite" set.
Hyperfinite sets play the role of finite sets in the non-Archimedean framework since they share many properties with finite sets. We will often use the following feature provided by the hyperfinite sets: it is possible to "add" the elements of a hyperfinite set of numbers. If F is a hyperfinite set of numbers, the hyperfinite sum of the elements of F is defined in the following way, Let us illustrate the hyperfinite sets with a very simple example. Let F λ = {n ∈ N | n ≤ λ} for every λ ∈ L. Further, let F = {lim λ↑Λ x λ | x λ ∈ F λ }. Then F is hyperfinite and F ⊆ N * . Moreover, n ∈ F , for every n ∈ N, as n ∈ F λ , for every λ, so that |λ| ≥ n. Therefore, we have constructed a hyperfinite subset of N * which contains the infinite set 7 N.
The kind of hyperfinite sets that we will use are the so-called "hyperfinite grids".
If {Γ λ } Λ∈L is a family of finite subsets of R, which satisfies the property then it is not difficult to prove that the set Below Γ denotes a hyperfinite grid extending R, which is fixed once and forever.
namely, f • is a restriction to Γ of the natural extension f * which is defined on the whole E N . It is easy to check that, for every a ∈ Γ, the characteristic function χ a (x) of {a} is a grid function. In perfect analogy with the classical finite case, every grid function can be represented by the following hyperfinite sum, namely, {χ a (x)} a∈Γ is a set of generators for G(R N ). Moreover, from Definition 10 it follows that actually {χ a (x)} a∈Γ is a basis for the space of grid functions on Γ.
In general, if E is a subset of R N and f is defined only on E, we set namely, we set it to be 0 on every grid point that does not belong to the natural extension of its domain. For

III. ULTRAFUNCTIONS
In this section, we introduce the key ingredient needed for the description of a quantum system, the ultrafunction space. An explicit technical construction of (several) ultrafunction spaces has been provided in Ref. [9][10][11][12][13][14][15] by various reformulations of the non-standard 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 ultrafunction spaces. For an explicit construction, we refer to [27]. The basic idea one has to keep in mind is that in order to build an ultrafunction space we start with a classical space of functions V (Ω) and extend it to a space of grid functions V • (Ω) with several ad hoc properties.

A. Axiomatic definition of ultrafunctions
In order to build a space of ultrafunctions that suites for an adequate description of a quantum system, we have to fix an appropriate function space V (Ω). In that way, we choose V (Ω) ⊇ C 0 (Ω) to be a function space that includes infinitely differentiable real functions and is a subspace of the space of locally integrable functions (which includes real p-integrable functions with p ≥ 1), In the further step we build a family of all finitedimensional subspaces of the chosen function space V (Ω). We label this family by {V λ } λ∈L so that it shares the following property, Such a family provides a net {V λ } λ∈L of the finite subspaces of V (Ω). Hence it allows us to perform a Λ-limit of this net resulting in the required hyperfinite function space, the space of ultrafunctions, This leads to a clear definition of the ultrafunction space, which we equip with the axioms providing some properties needed in quantum mechanics.
(called generalized partial derivative), which satisfy the axioms below.
there exists a net u λ , such that ∀λ ∈ L u λ ∈ V λ , and where , then its generalized partial derivative at the point x is defined as Axiom 6. If we set the support of an ultrafunction u as Let us notice that, in most previous papers, ultrafunctions spaces where assumed to witness only some of the above axioms. In fact, constructing a space of ultrafunctions that witnesses all these axioms presents several technical challenges. However, these properties are fundamental to develop many applications better, as we are going to show. For more details on the construction of such a space, as well as for the relevance of these axioms, we refer to [27].
B. Discussion of the axioms of the ultrafunction space Axiom 1 means that every ultrafunction is a grid function (based on V (Ω)), which shares the following peculiar property: at every step λ the corresponding finitedimensional vector space which contains all u λ is V λ .
Hence, every ultrafunction is a Λ-limit of functions in V λ .
Axiom 2 is a definition of the pointwise integral. This integral extends the usual Riemann integral from functions in C 0 (R N ) to ultrafunction in V • (Ω). In fact, by definition, ∀f ∈ C 0 (R N ), since for the functions of a compact support the net lim as the quantity becomes arbitrarily small for large λ. Axiom 3 shows that the above inequality cannot hold for all arbitrary Riemann integrable functions. This is the reason why the new notation has been introduced.
A key example (being very important when dealing with singular potentials) is the characteristic function of a singleton. In fact, if a ∈ R ∩ Γ, then with ε > 0, so that it represents the "weight" of the point a. At the first sight, this Axiom might seem unnatural. However, we work in a non-Archimedean approach, thus, the infinitesimals cannot be forgotten as is done in the Riemann integration. Axiom 4 shows that the generalized derivative extends the usual derivative. In fact, if f ∈ C 1 (R N ) and x ∈ R N , then However, the less intuitive fact is that the operator D i is defined on all ultrafunctions. The last three axioms have been introduced to highlight that all the most useful properties of the usual derivative are also satisfied by D i . In fact, Axiom 5 says that the ultrafunctions behave as compactly supported C 1 functions.
Axiom 6 states that the derivative is a local operator 8 , which is a fundamental fact in the applications of ultrafunctions to quantum mechanics.
Axiom 7 provides a weak form of Leibniz rule, which is of primary importance in the theory of weak derivatives, distribution, calculus of variations, etc. We express this rule in its weak form since the Leibniz rule D(f g) = Df g + f Dg cannot be satisfied by every ultrafunction, because of the Schwartz impossibility theorem (see [11,28]). In fact, if Leibniz rule held for all ultrafunctions, we would construct a differential algebra extending C 0 , that is not possible. In some sense, ultrafunctions provide a "solution" of Schwartz impossibility theorem at the cost of using the weak formulation for Leibniz rule instead of the full one 9 .
C. The structure of the space of ultrafunctions Since the space of ultrafunctions is generated by {χ a } a∈Γ , it follows from Axiom 3 that the ultrafunction integral is actually a hyperfinite sum. In fact, for every u ∈ V • its integral can be expanded as In order to analyze the applications of the ultrafunction space to quantum mechanics we consider the complexvalued ultrafunctions, so that we provide a complexification of the ultrafunction space, Coherently with this notation, we let is all we need to develop the applications we have in mind. That said, using some basic tools of non-standard analysis (underspill and saturation), it would be simple to prove that, actually, there exists an infinitesimal number η such that, for every a ∈ Γ, the expansion of the derivative Dχa(x) in the base {χ b } b∈Γ would involve only the points b ∈ [a − η, a + η]. Therefore, the second derivative of χa(x) would involve only the points b ∈ [a − 2η, a + 2η] and, more in general, the n-th derivative would involve the points in [a − nη, a + nη]. This shows that for every n ∈ N * such that nη ∼ 0, the operator D n would still be local. This includes some infinite n, but not all infinite n, e.g., let n > 1 η . In particular, the infinite matrix M D that corresponds to the operator D in the base {χ b } b∈Γ is close to be diagonal, in the sense that if we let N = max |{[a − η] ∩ Γ|, |[a + η] ∩ Γ| | a ∈ Γ}, then in every row of M D the only nonzero elements are the Mn,m with m ∈ [n − N, n + N ]. This is similar to the computational approximations of the derivative. 9 The reader interested in a deeper discussion of this fact is referred to [28]. Let us also mention that Colombeau functions provide an alternative "solution" of the Schwartz theorem, see [29]. In the Colombeau approach, Leibniz's rule is preserved. However, we have to take C ∞ functions instead of C k functions. and The pointwise integral allows to define the following sesquilinear form on H • , where z is a complex conjugation of z. Due to Axiom 3 this form is a scalar product.
with ∼ substituted by equality when u, v, u · v ∈ L 2 ∩ C 0 R N . This means that this sesquilinear form extends the usual L 2 scalar product. The norm of an ultrafunction is given by In the theory of ultrafunctions, the Dirac delta function has a simpler interpretation due to the pointwise integral. In fact, we can define the delta ultrafunction (called also the Dirac ultrafunction) as follows. For every a ∈ Γ, Our choice can be easily motivated, since for every u ∈ In particular, if u = f * | Γ for some f ∈ D(Ω), this shows that the scalar product between δ a and u equals f * (a); in particular, if a ∈ R then one recovers the classical expected property of a delta function.
Moreover, as in the ultrafunction framework delta functions are actual functions (and not functionals, like in distributions theory), we can perform on them all the classical operations that do not have sense in the standard analysis like, for example, δ 2 (x).
As D ⊆ V , this shows that the delta ultrafunction behaves as the delta distribution when tested against functions in D. Moreover, delta functions are mutually orthogonal with respect to the scalar product (10).
Hence, being normalized they provide an orthonormal basis, called delta-basis, given by Hence, every ultrafunction can also be expanded in the following way, The scalar product allows the following ultrafunction version of the Riesz representation theorem.
and for every f ∈ V, is a linear functional over H λ and hence, since there exists u λ ∈ H λ such that ∀v ∈ H λ , If we set u Φ = lim λ↑Λ u λ , the conclusion follows.
As already stated, our goal is to apply the theory of ultrafunctions to quantum mechanics. In this way, we are interested in understanding the relationship between ultrafunctions and L 2 -functions. Even though the scalar product in V • can be seen as an extension of the L 2scalar product, it is still not clear whether L 2 functions can be embedded into V • . The basic idea is to use Eq. (3) to do such an association. However, this does not work since the L 2 -functions are not defined pointwise. For this reason, we introduce the following definition, which uses a weak form of association.
Definition 13. Given a function ψ ∈ L 2 (Ω) we denote by ψ • the unique ultrafunction such that, for every v = lim Proposition 12 ensures that the above definition is well posed, as the map Φ : v → ψv dx is a functional on the space H • .

D. Ultrafunctions and distributions
Distributions can be easily embedded into ultrafunction spaces by identifying them with equivalence classes [14].
Definition 14. The space of generalized distributions on Ω is defined as follows, The equivalence class of an ultrafunction u ∈ V • (Ω) is denoted by [u] D(Ω) . It contains all ultrafunctions, whose action on the functions in D(Ω) differs from the action of u by at most an infinitesimal quantity. An obvious idea is to identify this action with the action of a distribution. However, it is not directly possible since there are ultrafunctions, whose action does not correspond to the action of any distribution. For example, if u = τ δ 0 , where τ is an infinite number, then ∀ϕ ∈ D(Ω) uϕ * dx = τ ϕ(0), which is an infinite quantity, whenever ϕ(0) is different from 0.
This issue can be overcome by considering the so-called "bounded" ultrafunctions. In turn, the spaces of generalized distributions and bounded generalized distributions can be identified by an isomorphism, as shows the following theorem. Proof. For the proof see, e.g., [11].

E. Self-adjoint operators on ultrafunctions
If an operator is an internal linear operator, then it can be regarded as an infinite matrix since, by the Λ-limit characterization, we have can be represented by a finite matrix (as every space V λ has a finite dimension). In particular, if L is a self-adjoint operator, Luv dx = uLv dx, then the matrices L λ are Hermitian. Hence, L can be regarded as an infinite-dimensional Hermitian matrix. Therefore, the spectrum σ(L) of L consists of eigenvalues only, more precisely and its corresponding normalized eigenfunctions form an orthonormal basis of H • . In that way, the ultrafunction approach resembles the finite-dimensional vector spaces approach, in the sense that the distinction between selfadjoint operators and Hermitian operators is not needed. We have proven the following Theorem 17. Every Hermitian operator L : In [27] it is possible to find a detailed analysis of the ultrafunction formalization of the position and the momentum operators. For our applications to the Schrödinger equation, let us notice that the Laplacian operator ∆ : C 2 R N → C 0 R N has the following expression as its ultrafunction formulation, For applications to quantum mechanics, the following Hamiltonian operator is fundamental, where we assume that the mass of the i-th particle m i = 1 and = 1.
There is a deep and important difference between the standard and the ultrafunction approach to the study of H. In the classical L 2 -theory, a fundamental problem is a choice of an appropriate potential V such that (15) makes sense. Moreover, it is fundamental to define an appropriate self-adjoint realization ofĤ. On the other hand, in the theory of ultrafunctions any internal function V : Γ → E provides a self-adjoint operator on H • , given byĤ which always has a discrete (in the R * sense) spectrum that consists of eigenvalues only. Of course, this spectrum will be hyperfinite, of cardinality equal to the cardinality of Γ (once the multiplicities of the eigenvalues are taken into account). For these reasons, we believe that the ultrafunction approach allows a much simpler study of "very singular potentials" such as where τ and Ω might be infinite numbers. The use of non-Archimedean methods is fundamental to give a reasonable model of these potentials, which have an interesting physical meaning. Particularly, the delta function potential (17), which represents a very short-ranged interaction, appears in many physical problems. For example, it can serve as a reasonable model for the interactions between atoms and electromagnetic fields (particularly, dipole-dipole interactions) and interatomic potential in a many-body system (particularly, Bose-Einstein condensate). In this way, in the following, we will take a look a quantum system with a delta potential within the ultrafunction framework. Before to start, we will reformulate the basis of quantum mechanics -its system of axioms -through the ultrafunctions.

A. Axioms of QM based on ultrafunctions
In the following table, we provide a set of axioms of quantum mechanics formulated within the ultrafunction approach and compare it with the standard Dirac -von Neumann axioms.

Axiom 1
A physical state of a quantum system is completely described by a unit vector in a complex Hilbert space |ψ ∈ H. Particularly, it can be described by a complex-valued quadratically integrable wavefunction ψ ∈ L 2 (Ω).
A physical state of a quantum system is described by a unit complex-valued ultrafunction ψ ∈ H • .

Axiom 2 A physical observable A is represented by a linear Hermitian operatorÂ acting in H.
A physical observable A is represented by a linear Hermitian operatorÂ acting in H • .

Axiom 3
The only possible outcomes of a measurement of an observable A form a set {µj }, where µj are the eigenvalues of the operatorÂ.
The only possible outcomes of a measurement of an observable A form a set {st(µj)}, where µj are the eigenvalues of the operatorÂ.

Axiom 4
An outcome µj of a measurement of an observable A can be obtained with a probability where ψj is the eigenstate associated with the observed eigenvalue µj . After the measurement the quantum system is left in the state ψj .
An outcome st(µj ) of a measurement of an observable A can be obtained with a probability where ψj is the eigenstate associated with the observed eigenvalue µj . After the measurement the quantum system is left in the state ψj.

Axiom 5
The time evolution of the state of the quantum system is described by the Schrödinger equation whereĤ is the Hamiltonian, which represents the energy of the system.
The time evolution of the state of the quantum system is described by the Schrödinger equation whereĤ • is the Hamiltonian, which represents the energy of the system. In the Dirac -von Neumann formalism, a physical system is described by a vector in Hilbert space. The ultrafunction formulation of Axiom 1 guarantees that we use ultrafunction space H • (hence, non-Archimedean mathematics) instead of the Hilbert one H. In particular, working within wave functions, the state |ψ can be represented by a function ψ ∈ L 2 (Ω), Ω ⊂ R N . Since L 2 can be embedded in V • , there exists a canonical embedding due to the Def. 13, given by Since H • is a space much richer than H, the ultrafunction framework recovers all standard states and provides more possible states, particularly, the ideal states of Axiom 6.

Axiom 2: Observables
The standard and ultrafunction formulations of Axiom 2 highlight many similarities as well as some differences. The main difference is the fact that the ultrafunction formalism needs the von Neumann's notion of the self-adjoint operator no more. In fact, the observables of a quantum system can be represented by internal Hermitian operators, which are trivially self-adjoint due to the Theorem 17.

Axioms 3, 4: Measurement
By self-adjointness of internal Hermitian operators, it follows that any observable has exactly κ = dim * (H • ) = |Γ| eigenvalues (taking into account their multiplicity). In this way, no more essential distinction between eigenvalues and a continuous spectrum is needed since a continuous spectrum can be considered as a discrete spectrum containing eigenvalues infinitely close to each other.
For example, if we consider the eigenvalues of the position operator q of a free particle, then the eigenfunction relative to the eigenvalue q ∈ R is the Dirac ultrafunction δ q . Therefore, it can be trivially seen that its spectrum is Γ, which is not a continuous spectrum in R * . However, the standard continuous spectrum can be recovered since the eigenvalues of an internal Hermitian operatorÂ are Euclidean numbers. Hence, assuming that measurement gives a real number, we have imposed in the Axiom 2 that its outcome is st(µ). In the case of the spectrum of the position operator, we can show that because every real number lies in Γ.
Working in a non-Archimedean framework, we have set in a natural way that the transition probabilities should be non-Archimedean. In fact, we assume that the probability is better described by the Euclidean number |(ψ, ψ j )| 2 , rather than the real number st(|(ψ, ψ j )| 2 ). For example, let ψ ∈ H • be the state of a system. The probability of an observation of the particle in a position q is given by which is an infinitesimal number. The standard probability can be recovered by the means of the standard part, in the case of the position operator it would be zero (as is expected). We refer to [30,31] for a presentation and discussion of the non-Archimedean probability.

Axiom 5: Evolution
The ultrafunction version of this axiom is very similar to the standard one. SinceĤ • is an internal operator defined on a hyperfinite-dimensional vector space H • , it can be represented by an Hermitian hyperfinite matrix due to the Theorem 17. Hence, the evolution operator of the quantum system is described by the exponential matrixÛ • (t) = e −iĤ • t .

Axiom 6: Physical and ideal states
This is the most peculiar axiom in the ultrafunction approach. In the ultrafunction theory, the mathematical distinction between the physical eigenstates and the ideal eigenstates is intrinsic. It does not correspond to anything in the standard formalism. Hence, it opens a very interesting problem of the physical relevance of such states. Basically, we can intuitively say that the physical states correspond to the states which can be prepared and measured in a laboratory, while the ideal states represent "extreme" states useful in the foundations of quantum mechanics, thought experiments (Gedankenexperiment) and computations.
For example, the Dirac ultrafunction is not a physical state but an ideal state. It represents a state in which the position of the particle is perfectly determined, which is precisely what one has in mind considering the Dirac delta distribution. Clearly, this state cannot be produced in a laboratory since it requires infinite energy. However, it is useful in our description of the physical world since such a state makes more explicit the standard approach. Therefore, Axiom 6 highlights a concept that is already present (but somehow hidden) in the standard approach.
For example, in the Schrödinger representation of a free particle in R 3 , consider the state We see that ψ(x) ∈ L 2 (R 3 ) but this state cannot be produced in a laboratory, since the expected value of its energy is infinite (even if the result of a single experiment is a finite number).

V. EXAMPLE: HAMILTONIAN WITH A δ-POTENTIAL
To compare the standard and the ultrafunction approach, in this section, we want to study the standard HamiltonianĤ which corresponds to the problem of a particle in the delta potential of transparency τ at the point x = 0.
As we are going to show, there are two main differences between the standard and the ultrafunction approaches, 1) the standard approach changes if we change the dimensionality of the space (as in the space of dimension D > 1 one of the main problems is to find a self-adjoint representation ofĤ), while in the ultrafunction approach we always have a self-adjoint representation ofĤ, independently of D, due to Theorem 17,2) in the ultrafunction approach the constant τ can be an infinitesimal or an infinite number, which allows us to construct the models which cannot be considered within the standard framework (for example, a potential equal to the characteristic function of a point 10 ).

A. The standard approach
Let us review the solution of the problem within the standard approach (namely, τ ∈ R) [32,33]. At first, it is assumed that the particle is in a box of length 2L. In turn, the delta potential is approximated as a square wall (well) of a finite length 2ε and a finite height V 0 , where the transparency of the potential is related to the parameters of the approximation as τ = 2εV 0 .
Potential barrier (τ > 0) We start with the case of a potential barrier with τ > 0, which is illustrated on the Fig. 1.
FIG. 1. Delta potential as a limit of a finite barrier.
Using the abbreviations k 2 = 2E and κ 2 = 2(V 0 − E) we obtain the following solution of the Schrödinger equation, where A ± and B ± are the normalization constants, f + (x) = cosh(κx), f − (x) = sinh(κx), "+" corresponds to the even function and "−" corresponds to the odd function. We seek the energies of the eigenstates, which can be found with use of the continuity condition, In turn, the original problem is recovered by taking the limit (20), which leads to the following solution, with the normalization |A ± n | −2 = 2k ± n L−sin(2k ± n L) 2k ± n and the eigenvalues k ± n , which are defined by the following equations due to Eq. (22), At the singularity point one obtains Potential well (τ < 0) Let us now consider the case of a potential well, namely, the negative transparency τ < 0.
In this case, we have two different possibilities for the definition of energy, E > 0 (scattering states) and E < 0 (bound states). Positive energies (E > 0): scattering states. Using the abbreviations k 2 = 2E and κ 2 = 2(V 0 + E) we obtain the following solution of the Schrödinger equation, where A ± and B ± are the normalization constants, f + (x) = cos(κx), f − (x) = sin(κx), "+" corresponds to the even function and "−" corresponds to the odd function. Performing the limits as done above one obtains the following solution, with the normalization |A ± n | −2 = 2k ± n L−sin(2k ± n L) 2k ± n and eigenvalues k ± n , which are defined by the equations k + n cot(k + n L) = τ, At the singularity point one obtains Negative energies (E < 0): bound states. Using the abbreviations k 2 = 2|E| and κ 2 = 2(V 0 − |E|) we obtain the following solution of the Schrödinger equation, where A ± and B ± are the normalization constants, f + (x) = cos(κx), f − (x) = sin(κx), "+" corresponds to the even function and "−" corresponds to the odd function. Performing the limits as done above one obtains the following solution, with the normalization |A ± n | −2 = 2k ± n L−sinh(2k ± n L) 2k ± n and eigenvalues k ± n , which are defined by the equations k + n coth(k + n L) = τ, In this way, we find that, for E < 0, there exists a unique eigenvalue k + defined by Eq. (36). With expanding the box by taking the limit L → ∞, it corresponds to the energy of the unique bound state At the singularity point one obtains Multidimensional δ-potentials In the analysis of the delta potential, we have focused on the 1-dimensional case. In this case, one can easily prove that there exists a self-adjoint realization of the HamiltonianĤ bounded below. As shown above, there exists a unique bound state for a Hamiltonian with a one-dimensional delta potential well.
The delta potentials of a higher dimensionality D provide not only a pedagogical model but finds its applications in nuclear and condensed matter physics [34][35][36][37]. However, solving a Schrödinger equation with these potentials is a more cumbersome problem since there is no rigorous construction of a self-adjoint realization of the Hamiltonian (19) as long as it is not extended to a non-interacting Hamiltonian on a space with a removed point [7,[38][39][40][41][42]. Let us illustrate it by the case of a twodimensional and three-dimensional delta potential well.
As in the one-dimensional case, the delta potential well of dimensionality D can be approximated by a finite square well described by the Heaviside function θ(r), where its depth V 0 and length ε are related to the transparency of the delta potential well as τ 2D = πε 2 V 0 in the two-dimensional case and τ 3D = 4π 3 ε 3 V 0 in the threedimensional case, respectively [35,36].
The most interesting part of the problem with a delta potential well is the estimation of the bound states. After solving the corresponding Schrödinger equation, the bound states can be found from the continuity equation. In the two-dimensional case, it is more difficult to do because the wave functions are Bessel functions in this case. However, since the Bessel function J 0 (r) with an azimuthal symmetry asymptotically behaves as a cosine, the number of the bound states can be approximately estimated as N = εκ max /π, where κ 2 max = 2V 0 . Plugging in the transparency, we find which is a finite number. The solution for the threedimensional potential well can be found in the form of trigonometric functions, that simplifies the calculations. The number of bound states is given by [35,36] which is already an infinite number. The energy of a bound state for the delta potential can be found by solving the following equation [43,44], where D is the number of dimensions. The integral in this equation is divergent for D ≥ 2, that results in infinite energies of the bound states. In particular, in the twodimensional case, it can be shown that [35] which is infinite. In this way, we conclude that multidimensional delta potential wells provide, generally speaking, an infinite number of bound states with the energies diverging to −∞.
In order to avoid the problems with the infinite quantities (diverging energies) within the standard approach, one usually applies the so-called regularization and renormalization procedures to the calculations. The regularization provides a cut-off 11 σ for the divergent integral, and the renormalization re-defines the transparency τ as τ R (where R stands for "renormalized"), which absorbs the dependence on the cut-off in order to absorb the divergence of the integral.
In particular, in the two-dimensional case, the renormalized transparency is defined by the following equation [43,44], where µ is an arbitrary parameter, which represents the renormalization scale. Then, one takes a limit σ → ∞ and varies the "bare" transparency τ in such a way that the renormalized transparency τ R remains finite [43,44]. Such a renormalized transparency leads to a unique bound state with the binding energy Finally, this procedure means that one introduces an additional scale in order to "forget" about the unboundedness of the Hamiltonian from below.

B. The ultrafunction approach
Solution in the singularity point In the ultrafunction approach, the Schrödinger equation reads where τ, E ∈ R * . We use the linearity of V For every a ∈ Γ, let D 2 χ a (x) := b∈Γ D a,b χ b (x). Notice that the locality of the ultrafunction derivative entails that, actually, we can write 12 Therefore, considering that a∈Γ u(a)δ(a)χ a (x) = u(0) χ0(x) d(0) , we can rewrite Eq. (47) as Let us explicitly note that the above discussion does not depend on the dimensionality of the space. In fact, for any number of dimensions, Eq. (47) reveals the selfadjoint representation of the Hamiltonian (19) in the ultrafunction setting.
Outside the monad of 0, the equation actually reads which is formally the same that one gets for the equation On the other hand, at the point 0 ∈ Γ we obtain the equation which corresponds to . (52) Notice that for every other point γ ∈ mon(0) ∩ Γ we have In particular, it shows that, in the basis {χ a } a∈Γ , the matrix representation M δ of Eq. (46) and the matrix representation M ∆ of Eq. (50) satisfy the relation Particularly, in the case of a one-dimensional potential well with an infinitesimal transparency, there will still exist a unique bound state with infinitesimal negative energy.

Connection with the standard solutions
As usual, in order to introduce a new solution concept for a standard problem, it is important to discuss the relationship between the standard and new solution. Our first result shows that the approximation procedure used in most standard approaches to the Schrödinger equation with a delta potential can be reproduced within the ultrafunction approach, in the following sense.
Theorem 18. Let 0 ∼ a Λ = lim λ↑Λ a λ and τ Λ = lim λ↑Λ τ λ . Then there exists an ultrafunction space V • that contains a solution u Λ of the equation which is a Λ-limit of a net of solutions u λ ∈ V (Ω) of the standard Schrödinger equations where A a λ is an approximation of the delta distribution by a square well of length a λ ∈ R, and for every λ E λ is an eigenvalue of the energy of the solution of the standard Schrödinger equation with the approximated delta potential.
Proof. For every λ ∈ Λ let u λ be a solution of Eq. (55). Let {V λ } λ be an increasing net of finite dimensional subspaces of V (Ω) as in Definition 11. If ∀λ the solution u λ ∈ V λ , then a Λ-limit u Λ = lim λ↑Λ u λ is contained in the ultrafunction space V • generated by the net {V λ } λ , and theorem is proven. If u λ / ∈ V λ , then we extend V λ by V ′ λ := span(V λ ∪ {u λ }), and build the Λ-limit with the net {V ′ } λ . In this way, the Λ-limit of the net {u λ } gives an ultrafunction in V • by construction.
The second result we want to show precises the relationship between the ultrafunction and standard solution in the case of finite energy and transparency.
where ψ is such that for any f ∈ D(Ω), the Hamiltonian (19) that could be constructed in V • (Ω). This gives us a degree of freedom, which is not present in the standard approach.

Comparison of the approaches
We summarize the discussion of the interpretation of the Schrödinger equation with a delta potential within the ultrafunction approach in the following table.

Standard approach Ultrafunction approach
Analysis of the solutions of the Schrödinger equation (19) needs an approximation of the delta function by a finite potential (for example, a square barrier/well) and performing a limit of its zero width.
The solutions can be analyzed directly with the use of the delta ultrafunction, which describes an infinite jump concentrated in 0. However, the approximation procedure can be used as well due to Theorem 18.
Multidimensional delta potentials are difficult to interpret because of divergence of the corresponding integrals.
There is a unique framework for the delta potentials of any dimension based on hyperfinite spaces and, thus, hyperfinite matrices. Infinite or infinitesimal numbers can be used as parameters of the equation. In this way a model comes out which better describes the physical phenomenon.
The number of the eigenstates depends on the number of physical dimensions D. In particular, the number of bound states is infinite for D ≥ 3.
There always exists the number of eigenstates equal to the hyperfinite dimensionality of the ultrafunction space.
The finite energies of the eigenstates can be calculated directly only for a one-dimensional delta potential.
Natural bounds for the energies of the eigenstates can be estimated independently of the dimension of the system. Moreover, it is possible to obtain the bound states with finite energies by fixing a suitable transparency.
-The splitting the corresponding ultrafunction can recover the standard content of the solution.

VI. CONCLUSIONS
In this paper, we have introduced a new approach to quantum mechanics using the advantages of non-Archimedean mathematics. It is based on ultrafunctions, non-Archimedean generalized functions defined on a hyperreal field of Euclidean numbers E = R * , which is a natural extension of the field of real numbers. Euclidean numbers have been chosen because they allow constructing a non-Archimedean framework through a simple notion of Λ-limit, which gives an advantage concerning other non-standard approaches to quantum me-chanics [4][5][6][7][8].
The ultrafunction approach proposes a new set of the axioms of quantum mechanics. Even though the new axioms are obviously related with the standard Diracvon Neumann formulation of quantum mechanics, ultrafunctions makes it possible to build a framework which is closer to the matrix approach. Moreover, the presence of infinite and infinitesimal elements allows constructing new simplified models of physical problems. In particular, this framework is based on Hermitian operators, so that unbounded self-adjoint operators are not more needed. Furthermore, the difference between continuous and discrete spectra is not more present. These aspects were illustrated by a comparison between standard and ultrafunction solutions of a Schrödinger equation with a delta potential. We have shown that the ultrafunction approach provides a more straightforward way to solve the Schrödinger equation by unifying the cases of a potential well and a potential barrier and providing the extreme cases of infinite and infinitesimal transparency of the potential. Moreover, it gives a connection between energy and transparency, providing a bound for the allowed energies of the quantum system.