infinitesimal and infinite numbers as an approach to
TRANSCRIPT
Infinitesimal and Infinite Numbers as an Approach to Quan-tum MechanicsVieri Benci1, Lorenzo Luperi Baglini2, and Kyrylo Simonov2
1Dipartimento di Matematica, Universita degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy2Fakultat fur Mathematik, Universitat Wien, Oskar-Morgenstern Platz 1, 1090 Vienna, AustriaApril 29, 2019
Non-Archimedean mathematics is an ap-proach based on fields which contain in-finitesimal and infinite elements. Withinthis approach, we construct a space of aparticular class of generalized functions,ultrafunctions. The space of ultrafunctionscan be used as a richer framework for adescription of a physical system in quan-tum mechanics. In this paper, we pro-vide a discussion of the space of ultrafunc-tions and its advantages in the applica-tions of quantum mechanics, particularlyfor the Schrodinger equation for a Hamil-tonian with the delta function potential.
1 IntroductionQuantum mechanics is a highly successful physi-cal theory, which provides a counter-intuitive butaccurate description of our world. During morethan 80 years of its history, there were devel-oped various formalisms of quantum mechanicsthat use the mathematical notions of differentcomplexity to derive its basic principles. Thestandard approach to quantum mechanics han-dles linear operators, representing the observ-ables of the quantum system, that act on thevectors of a Hilbert space representing the phys-ical states. However, the existing formalisms in-clude not only the standard approach but as wellsome more abstract approaches that go beyondHilbert space. A notable example of such anabstract approach is the algebraic quantum me-chanics, which considers the observables of thequantum system as a non-abelian C∗-algebra,and the physical states as positive functionals onit [1].
Vieri Benci: [email protected] Luperi Baglini: [email protected] Simonov: [email protected]
Non-Archimedean mathematics (particularly,nonstandard analysis) is a framework that treatsthe infinitesimal and infinite quantities as num-bers. Since the introduction of nonstandard anal-ysis by Robinson [2] non-Archimedean mathe-matics has found a plethora of applications inphysics [3, 5, 4, 6, 7], particularly in quantum me-chanical problems with singular potentials, suchas δ-potentials [8, 9, 10, 11, 12].
In this paper, we build a non-Archimedeanapproach to quantum mechanics in a simplerway through a new space, which can be usedas a basic construction in the description of aphysical system, by analogy with the Hilbertspace in the standard approach. This spaceis called the space of ultrafunctions, a particu-lar class of non-Archimedean generalized func-tions [13, 14, 15, 16, 17, 18, 19]. The ultrafunc-tions are defined on the hyperreal field R∗, whichextends the reals R by including infinitesimal andinfinite elements into it. Such a construction al-lows studying the problems which are difficultto solve and formalize within the standard ap-proach. For example, variational problems, thathave no solutions in standard analysis, can besolved in the space of ultrafunctions [18]. In thisway, non-Archimedean mathematics as a wholeand the ultrafunctions as a particular propose aricher framework, which highlights the notionshidden in the standard approach and paves theway to a better understanding of quantum me-chanics.
The paper is organized as follows. In Section 1we introduce the needed notations and the notionof a non-Archimedean field. In Section 2 we in-troduce a particular non-Archimedean field, thefield of Euclidean numbers E, through the notionof Λ-limit, which is a useful, straightforward ap-proach to the nonstandard analysis. In Section 3we introduce the space of ultrafunctions, whichare a particular class of generalized functions. In
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Section 4 we apply the ultrafunctions approach toquantum mechanics and discuss its advantages incontrast to the standard approach. In Section 5we provide a discussion of a quantum system witha delta function potential for the standard andultrafunctions approaches. Last but not least, inSection 6 we provide the conclusions.
1.1 NotationsLet Ω be an open subset of RN , then
• C (Ω) denotes the set of continuous functionsdefined on Ω ⊂ RN ,
• Cc (Ω) denotes the set of continuous func-tions in C (Ω) having compact support in Ω,
• Ck (Ω) denotes the set of functions defined onΩ ⊂ RN which have continuous derivativesup to the order k,
• Ckc (Ω) denotes the set of functions inCk (Ω) having compact support,
• D (Ω) denotes the space of infinitely differ-entiable functions with compact support de-fined almost everywhere in Ω,
• L2 (Ω) denotes the space of square integrablefunctions defined almost everywhere in Ω,
• L1loc (Ω) denotes the space of locally inte-
grable functions defined almost everywherein Ω,
• mon(x) = y ∈ EN | x ∼ y (see Def. 4),
• given any set E ⊂ X, χE : X → R denotesthe characteristic function of E, namely,
χE(x) :=
1 if x ∈ E,
0 if x /∈ E,
• with some abuse of notation, we set χa(x) :=χa(x),
• ∂i = ∂∂xi
denotes the usual partial deriva-tive, Di denotes the generalized derivative(see Section 3.1),
•∫
denotes the usual Lebesgue integral,∮
denotes the pointwise integral (see Section3.1),
• if E is any set, then |E| denotes its cardinal-ity.
1.2 Non-Archimedean fieldsOur approach to quantum mechanics makes mul-tiple uses of the notions of infinite and infinites-imal numbers. A natural framework to intro-duce these numbers suitably is provided by non-Archimedean mathematics (see, e.g., [20]). Thisframework operates with the infinite and in-finitesimal numbers as the elements of the newnon-Archimedean field.
Definition 1. Let K be an infinite ordered field1.An element ξ ∈ K is:
• infinitesimal if, for all positive n ∈ N, |ξ| <1n ,
• finite if there exists n ∈ N such that |ξ| < n,
• infinite if, for all n ∈ N, |ξ| > n (equiva-lently, if ξ is not finite).
We say that K is non-Archimedean if it containsan infinitesimal ξ 6= 0, and that K is superreal ifit properly extends R.
Notice that, trivially, every superreal field isnon-Archimedean. Infinitesimals allow introduc-ing the following equivalence relation, which isfundamental in all non-Archimedean settings.
Definition 2. We say that two numbers ξ, ζ ∈ Kare infinitely close if ξ−ζ is infinitesimal. In thiscase we write ξ ∼ ζ.
In the superreal case, ∼ can be used to intro-duce the fundamental notion of “standard part”2.
Theorem 3. If K is a superreal field, every finitenumber ξ ∈ K is infinitely close to a unique realnumber r ∼ ξ, called the the standard part ofξ.
Following the literature, we will always denoteby st(ξ) the standard part of any finite numberξ. Moreover, with a small abuse of notation, wealso put st(ξ) = +∞ (resp. st(ξ) = −∞) if ξ ∈ Kis 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 allnumbers that are infinitely close to it,
mon(ξ) = ζ ∈ K : ξ ∼ ζ.1Without loss of generality, we assume that Q ⊆ K.2For a proof of the following simple theorem, the inter-
ested reader can check, e.g., [21].
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Notice that, by definition, the set of infinitesi-mals is mon(0) precisely. Finally, superreal fieldscan be easily complexified by considering
K + iK,
namely, a field of numbers of the form
a+ ib, a, b ∈ K.
In this way, the complexification of non-Archimedean fields shows no particular difficultyand is straightforward.
2 The field of Euclidean numbersNonstandard analysis plays one of the mostprominent roles between various approaches tonon-Archimedean mathematics. One reason isthat nonstandard analysis provides a handy toolto study and model the problems which comefrom many different areas. However, the classicalrepresentations of nonstandard analysis can feeloverwhelming sometimes, as they require a goodknowledge of the objects and methods of mathe-matical logic. This stands in contrast to the ac-tual use of nonstandard objects in the mathemat-ical practice, which is almost always extremelyclose to the usual mathematical practice.
For these reasons, we believe that it is worthto present nonstandard analysis avoiding most ofthe usual logic machinery. We will introduce thenonstandard approach to the formalism of quan-tum mechanics via the notion of Λ-limit3 andthe Euclidean numbers. These numbers are theunderlying object of Λ-theory, and can be intro-duced via a purely algebraic approach, as donein [24]4.
The basic idea of the construction of Euclideannumbers is the following: as the real numberscan be constructed by completion of the rationalnumbers using the Cauchy notion of limit, theEuclidean numbers can be constructed by com-pletion of the reals with respect to a new notionof limit, called Λ-limit, that we are now going tointroduce.
3In [22] and [23], the reader can find several other ap-proaches to nonstandard analysis and an analysis of them.
4An elementary presentation of (part of) this theorycan be found in [25] and [26].
Let Λ be an infinite set containing R and let Lbe the family of finite subsets of Λ. A functionϕ : L → R will be called net (with values in R).The set of such nets is denoted by F (L,R) andequipped with the natural operations
(ϕ+ ψ) (λ) = ϕ(λ) + ψ(λ),(ϕ · ψ) (λ) = ϕ(λ) · ψ(λ),
and the partial order relation
ϕ ≥ ψ ⇐⇒ ∀λ ∈ L, ϕ(λ) ≥ ψ(λ).
In this way, F (L,R) is a partially ordered realalgebra.
Definition 5. We say that a superreal field E is afield of Euclidean numbers if there is a surjectivemap
J : F (L,R)→ E,
which satisfies the following properties,
• J (ϕ+ ψ) = J (ϕ) + J (ψ),
• J (ϕ · ψ) = J (ϕ) · J (ψ),
• 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 aneasy consequence of the Krull – Zorn theorem.It can be found, e.g., in [24, 13, 14, 26]. In thispaper, we also use the complexification of E, de-noted5 by
C∗ = E + iE. (1)
Definition 6. Let E be a field of Euclidean num-bers, let J be its realization and let ϕ ∈ F (L,R).We say that J (ϕ) is the Λ-limit of the net ϕ.We will also denote it by
J (ϕ) := limλ↑Λ
ϕ(λ).
The name “limit” has been chosen because theoperation
ϕ 7→ limλ↑Λ
ϕ(λ)
satisfies the following properties,
• (Λ-1) Existence. Every net ϕ : L → R hasa unique limit L ∈ E.
5The choice of the notation will become clear later on.
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• (Λ-2) Constant. If ϕ(λ) is eventually con-stant, namely, ∃λ0 ∈ L, r ∈ R such that∀λ ⊃ λ0, ϕ(λ) = r, then
limλ↑Λ
ϕ(λ) = r.
• (Λ-3) Sum and product. ∀ϕ,ψ : L→ R
limλ↑Λ
ϕ(λ) + limλ↑Λ
ψ(λ) = limλ↑Λ
(ϕ(λ)+ψ(λ)),
limλ↑Λ
ϕ(λ) · limλ↑Λ
ψ(λ) = limλ↑Λ
(ϕ(λ) · ψ(λ)).
The Cauchy limit of a sequence (as formalizedby Weierstraß) satisfies the second and the third6
condition above. The main difference betweenthe Cauchy and the Λ-limit is given by the prop-erty (Λ− 1), namely, by the fact that the Λ-limitalways exists. Notice that it implies that E mustbe larger than R since it must contain the limitof diverging nets as well.
For those nets that have a Cauchy limit, therelationship between the Cauchy and the Λ-limitis expressed by the following identity,
limλ→Λ
ϕ(λ) = st(
limλ↑Λ
ϕ(λ)). (2)
With the introduction of the Euclidean num-bers there arises a natural question: What dothey look like? Let us give a few examples.
1. Let ϕ(λ) := 1|λ| for every λ 6= ∅, and ε =
limλ↑Λ ϕ(λ). Then ε > 0, because ϕ(λ) > 0for every λ ∈ L. However, ε < r for everypositive r ∈ R since ϕ(λ) < r eventually inλ. Therefore, ε is a non-zero infinitesimal inE.
2. Analogously, let ϕ(λ) := |λ| for every λ ∈ L.Let σ = limλ↑Λ ϕ(λ). Straightforwardly, σ isa positive infinite element in E, and σ ·ε = 1.
3. Let us generalize the previous example. IfE ⊂ R ⊂ Λ, we set
n (E) = limλ↑Λ|E ∩ λ|,
where |F | denotes the number of elementsof a finite set F . Since λ is a finite set,|E∩λ| ∈ N for every λ. If E is finite, then Ebelongs to L, and the net |E ∩ λ| is eventu-ally constant (in λ) and equal to |E|, so that
6When both sequences have a limit.
n (E) = |E|. On the other hand, if E is aninfinite set, the above limit gives an infinitenumber (called the numerosity of E7).
Definition 7. A mathematical entity is calledinternal if it is a Λ-limit of some other entities.
2.1 Extension of functions, hyperfinite sets andgrid functionsThe Λ-limit allows extending the field of realnumbers to the field of Euclidean numbers. Sim-ilarly, we can use it to extend sets and functionsin arbitrary dimensions,
• if N ∈ N, then, in order to extend RN toEN , we proceed in a trivial way: if ϕ(λ) :=(ϕ11(λ), ..., ϕN (λ)) ∈ RN , we set
limλ↑Λ
ϕ(λ) =(
limλ↑Λ
ϕ1(λ), ..., limλ↑Λ
ϕN (λ))∈ EN ,
• if A ⊂ RN , we let the natural extensionof A to be
A∗ =
limλ↑Λ
ϕ(λ) | ∀λ, ϕ(λ) ∈ A,
• iff : A→ R, A ⊂ RN ,
we let the natural extension of f to A∗
to be a function such that, for every x =limλ↑Λ xλ ∈ A∗,
f∗(
limλ↑Λ
xλ
)= lim
λ↑Λf∗ (xλ) ,
• the above case can be generalized, in order todefine directly the functions between subsetsof EN , if
uλ : A→ R, A ⊂ RN
is a net of functions, its Λ-limit
u = limλ↑Λ
uλ : A∗ → E
is a function such that, for any x =limλ↑Λ xλ ∈ EN ,
u(x) = limλ↑Λ
uλ (xλ) ,
7The reader interested in the details and the de-velopments of the theory of numerosities is referred to[27, 28, 29, 30].
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• finally, if V is a function space, we let itsnatural extension to be
V ∗ :=
limλ↑Λ
uλ | uλ ∈ V.
Notice that if, for all x ∈ RN ,
v(x) = limλ→Λ
uλ (x)
by Eq. (2), it follows that
∀x ∈ RN , v(x) = st [u(x)] .
Moreover, it is clear that E is the natural ex-tension of R, namely, E = R∗. Notice that thischoice justifies also our notation (1) for the com-plexification of E,
C∗ = E + iE = R∗ + iR∗.
Now we introduce a fundamental notion for allour applications, the “hyperfinite” set.
Definition 8. We say that a set F ⊂ E is hy-perfinite if there is a net Fλλ∈Λ of finite setssuch that
F =
limλ↑Λ
xλ | xλ ∈ Fλ.
Hyperfinite sets play the role of finite sets inthe non-Archimedean framework since they sharemany properties with finite sets. We will oftenuse the following feature provided by the hyper-finite sets: it is possible to “add” the elements ofa hyperfinite set of numbers. If F is a hyperfi-nite set of numbers, the hyperfinite sum of theelements of F is defined in the following way,∑
x∈Fx = lim
λ↑Λ
∑x∈Fλ
x.
Let us illustrate the hyperfinite sets with avery simple example. Let Fλ = n ∈ N |n ≤ |λ| for every λ ∈ L. Further, let F =limλ↑Λ xλ | xλ ∈ Fλ
. Then F is hyperfinite
and F ⊆ N∗. Moreover, n ∈ F , for every n ∈ N,as n ∈ Fλ, for every λ, so that |λ| ≥ n. There-fore, we have constructed a hyperfinite subset ofN∗ which contains the infinite set8 N.
The kind of hyperfinite sets that we will useare the so-called “hyperfinite grids”.
8At first sight, it might seem absurd to have a set thatextends N but still behaves like a finite set. This is one ofthe peculiar and key properties of hyperfinite sets.
Definition 9. A hyperfinite set Γ such thatRN ⊂ Γ ⊂ EN is called hyperfinite grid.
If Γλλ∈L is a family of finite subsets of RN ,which satisfies the property
RN ∩ λ ⊂ Γλ,
then it is not difficult to prove that the set
Γ =
limλ↑Λ
xλ | xλ ∈ Γλ
is a hyperfinite grid.Below Γ denotes a hyperfinite grid extending
RN , which is fixed once and forever.
Definition 10. The space of grid functions on Γis the family G(Γ) of functions
u : Γ→ R
such that, for every x = limλ↑Λ xλ ∈ Γ,
u(x) = limλ↑Λ
uλ(xλ).
If f ∈ F(RN ), and x = limλ↑Λ xλ ∈ Γ, we set
f(x) := limλ↑Λ
f(xλ), (3)
namely, f is a restriction to Γ of the naturalextension f∗ which is defined on the whole EN .
It is easy to check that, for every a ∈ Γ, thecharacteristic function χa(x) of a is a grid func-tion. In perfect analogy with the classical finitecase, every grid function can be represented bythe following hyperfinite sum,
u(x) =∑a∈Γ
u(a)χa(x), (4)
namely, χa(x)a∈Γ is a set of generators forG(RN ).
Moreover, from Definition 10 it follows thatactually χa(x)a∈Γ is a basis for the space ofgrid functions on Γ.
In general, if E is a subset of RN and f isdefined only on E, we set
f(x) =∑
a∈Γ∩E∗f∗(a)χa(x),
namely, we set it to be 0 on every grid point thatdoes not belong to the natural extension of itsdomain. For example, if f(x) = 1
|x| , then f(0) =0.
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3 UltrafunctionsIn this section, we introduce the key ingredientneeded for the description of a quantum system,the space of ultrafunctions. An explicit techni-cal construction of (several) ultrafunctions spaceshas been provided in Ref. [13, 16, 17, 14, 15,19, 18] by various reformulations of nonstandardanalysis. However, we prefer to pursue an ax-iomatic approach to underscore the key proper-ties of ultrafunctions needed for our aims sincesuch technicalities are not important in the ap-plications of the ultrafunctions spaces. For anexplicit construction, we refer to [31]. The basicidea one has to keep in mind is that, in order tobuild a space of ultrafunctions, we start with aclassical space of functions V (Ω) and extend it toa space of grid functions V (Ω) with several adhoc properties.
3.1 Axiomatic definition of ultrafunctions
In order to build a space of ultrafunctions thatsuites for an adequate description of a quantumsystem, we have to fix an appropriate functionspace V (Ω). In that way, we choose V (Ω) ⊇C0(Ω) to be a function space that includes in-finitely differentiable real functions and is a sub-space of the space of locally integrable functions(which includes real p-integrable functions withp ≥ 1),
C0(Ω) ⊂ D(Ω) ⊂ V (Ω) ⊂ L1loc(Ω).
In the further step, we build a family of all finite-dimensional subspaces of the chosen functionspace V (Ω). We label this family by Vλλ∈L,so that it has the following property,
∀λ1, λ2 ∈ L λ1 ⊆ λ2 ⇒ Vλ1 ⊆ Vλ2 .
Such a family provides a net Vλλ∈L of the fi-nite subspaces of V (Ω). Hence it allows us toperform a Λ-limit of this net resulting in the re-quired hyperfinite function space, the space ofultrafunctions,
V (Ω) := limλ↑Λ
Vλ. (5)
This leads to a clear definition of the space ofultrafunctions, which we equip with the axiomsproviding some properties needed in quantummechanics.
Definition 11. A space of ultrafunctionsV (Ω) generated by V (Ω) and modelled on thefamily of its finite subspaces Vλλ∈L is a spaceof grid functions
u : Γ→ E
equipped with an internal functional∮: V (Ω)→ E
(called pointwise integral) and N internal op-erators
Di : V (Ω)→ V (Ω)
(called generalized partial derivative), whichsatisfy the axioms below.
Axiom 1. ∀u ∈ V (Ω), there exists a net uλsuch that ∀λ ∈ L uλ ∈ Vλ, and
u = limλ↑Λ
uλ.
Axiom 2. ∀u = limλ↑Λ uλ, uλ ∈ Vλ, its point-wise integral is defined as∮
u(x) dx = limλ↑Λ
∫Qλ
uλ dx, (6)
where
Qλ =x ∈ RN
∣∣∣ |xi| ≤ |λ| .Axiom 3. ∀a ∈ Γ, there exists a positive in-finitesimal number d(a) such that
d(a) =∮χa(x)dx > 0.
Axiom 4. If x = limλ↑Λ xλ ∈ Γ and u =limλ↑Λ uλ such that ∀λ ∈ L uλ ∈ Vλ ∩ C1
c (RN ),then its generalized partial derivative at the pointx is defined as
Diu(x) = limλ↑Λ
∂iuλ(xλ). (7)
Axiom 5. If D : V (Ω)→ (V (Ω))N is the gen-eralized gradient, i.e.,
Du := (D1u ... DNu),
thenDu = 0 ⇔ u = const .
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Axiom 6. If we set the support of an ultrafunc-tion u as
supp (u) = x ∈ Γ | u(x) 6= 0 ,
thensupp (Diχa(x)) ⊂ mon(a).
Axiom 7. For every u, v ∈ V (Ω),∮(Diu(x))v(x)dx = −
∮u(x)(Div(x))dx. (8)
Let us notice that, in most previous papers,ultrafunctions spaces were assumed to witnessonly some of the above axioms. In fact, con-structing a space of ultrafunctions that witnessesall these axioms presents several technical chal-lenges. However, these properties are fundamen-tal to develop many applications better, as we aregoing to show. For more details on the construc-tion of such a space, as well as for the relevanceof these axioms, we refer to [31].
3.2 Discussion of the axioms of the space ofultrafunctionsAxiom 1 means that every ultrafunction is agrid function (based on V (Ω)), which has the fol-lowing peculiar property: at every step λ the cor-responding finite-dimensional vector space whichcontains all uλ is Vλ. Hence, every ultrafunctionis a Λ-limit of functions in Vλ.
Axiom 2 is a definition of the pointwise inte-gral. This integral extends the usual Riemann in-tegral from functions in C0
c (RN ) to ultrafunctionsin V (Ω). In fact, by definition, ∀f ∈ C0
c (RN ),∮f(x) dx =
∫f(x) dx,
since for the functions of a compact support thenet limλ↑Λ
∫Qλu dx becomes constantly equal to∫
u(x)dx. Moreover, if f ∈ L1(RN ) ∩ V , then∮f(x) dx ∼
∫f(x) dx,
as the quantity∣∣∣∣∣∫Qλ
f(x) dx −∫RN
f(x) dx∣∣∣∣∣
becomes arbitrarily small for large λ.Axiom 3 shows that the equality
∮f(x) dx =∫
f(x) dx cannot hold for all arbitrary Riemann
integrable functions. This is the reason why thenew notation
∮has been introduced. A key ex-
ample (being very important when dealing withsingular potentials) is the characteristic functionof a singleton. In fact, if a ∈ RN ∩ Γ, then∫
χa(x)dx = 0
6=∮χa(x)dx = ε ∼ 0, (9)
with ε > 0, so that it represents the “weight” ofthe point a. At the first sight, this Axiom mightseem unnatural. However, we work in a non-Archimedean approach, thus, the infinitesimalscannot be forgotten as is done in the Riemannintegration.
Axiom 4 shows that the generalized deriva-tive extends the usual derivative. In fact, iff ∈ C1(RN ) and x ∈ RN , then
Dif(x) = lim
λ↑Λ∂if(x) = ∂if(x).
However, the less intuitive fact is that the op-erator Di is defined on all ultrafunctions. Thelast three axioms have been introduced to high-light that all the most useful properties of theusual derivative are also satisfied by Di. In fact,Axiom 5 says that the ultrafunctions behave ascompactly supported C1 functions.
Axiom 6 states that the derivative is a localoperator9, which is a fundamental fact in the ap-plications of ultrafunctions to quantum mechan-ics.
Axiom 7 provides a weak form of Leibniz rule,which is of primary importance in the theory of
9We do not want to enter too much into details here,as “locality” is all we need to develop the applications wehave in mind. That said, using some basic tools of non-standard analysis (underspill and saturation), it would besimple to prove that, actually, there exists an infinitesi-mal number η such that, for every a ∈ Γ, the expansionof the derivative Dχa(x) in the base χbb∈Γ would in-volve only the points b ∈ [a − η, a + η]. Therefore, thesecond derivative of χa(x) would involve only the pointsb ∈ [a−2η, a+2η] and, more in general, the n-th derivativewould involve the points in [a − nη, a + nη]. This showsthat, for every n ∈ N∗ such that nη ∼ 0, the operator Dn
would still be local. This includes some infinite n, but notall infinite n, e.g., let n > 1
η. In particular, the infinite
matrix MD that corresponds to the operator D in the baseχbb∈Γ is close to be diagonal, in the sense that if we letN = max |[a− η]∩Γ|, |[a+ η]∩Γ| | a ∈ Γ, then in everyrow of MD the only non-zero elements are the Mn,m withm ∈ [n−N,n+N ]. This is similar to the computationalapproximations of the derivative.
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weak derivatives, distribution, calculus of varia-tions, etc. We express this rule in its weak formsince the Leibniz rule
D(fg) = (Df)g + fDg
cannot be satisfied by every ultrafunction be-cause of the Schwartz impossibility theorem (see[32, 15]). In fact, if Leibniz rule held for all ul-trafunctions, we would construct a differentialalgebra extending C0, that is not possible. Insome sense, ultrafunctions provide a “solution”of Schwartz impossibility theorem at the cost ofusing the weak formulation for Leibniz rule in-stead of the full one10.
3.3 The structure of the space of ultrafunc-tions
Since the space of ultrafunctions is generated byχaa∈Γ, it follows from Axiom 3 that the ul-trafunctions integral
∮is actually a hyperfinite
sum. In fact, for every u ∈ V , its integral canbe expanded as∮
u(x) dx =∑a∈Γ
u(a)d(a).
In order to analyze the applications of the spaceof ultrafunctions to quantum mechanics we con-sider the complex-valued ultrafunctions, so thatwe provide a complexification of the space of ul-trafunctions,
H := V ⊕ iV .
Coherently with this notation, we let
H := V ⊕ iV
and
Hλ := Vλ ⊕ iVλ.
The pointwise integral allows to define the follow-ing sesquilinear form with values in C∗ on H,∮
u(x)v(x) dx =∑x∈Γ
u(x)v(x)d(x), (10)
10The reader interested in a deeper discussion of thisfact is referred to [32]. Let us also mention thatColombeau functions provide an alternative “solution” ofthe Schwartz theorem, see [33]. In the Colombeau ap-proach, Leibniz rule is preserved. However, we have totake C∞ functions instead of Ck functions.
where z is a complex conjugation of z. Due toAxiom 3 this form is a scalar product. More-over, if u, v, u · v ∈ V ∩ L2 ∩ C0, then∫
u(x) · v(x) dx ∼∮u(x)v(x) dx
=∑x∈Γ
u(x)v(x)d(x), (11)
with ∼ substituted by equality when u, v, u · v ∈L2 ∩ C0
c
(RN
). This means that this sesquilinear
form extends the usual L2 scalar product.The norm of an ultrafunction is given by
‖u‖ =(∮|u(x)|2 dx
) 12
=
∑a∈Γ|u(a)|2d(a)
.In the theory of ultrafunctions, the Dirac delta
function has a simpler interpretation due to thepointwise integral. In fact, we can define thedelta ultrafunction (called also the Dirac ul-trafunction) as follows. For every a ∈ Γ,
δa(x) = χa(x)d(a) .
Our choice can be easily motivated, since, for ev-ery u ∈ V ,∮
δa(x)u(x) dx =∑x∈Γ
u(x)δa(x)d(x)
=∑x∈Γ
u(x)χa(x)d(a) d(x)
= u(a). (12)In particular, if u = f∗|Γ for some f ∈ D(Ω),
this shows that the scalar product between δaand u equals f∗(a). In particular, if a ∈ R thenone recovers the classical expected property of adelta function.
Moreover, as in the ultrafunctions frameworkdelta functions are actual functions (and notfunctionals, like in distributions theory), we canperform on them all the classical operations thatdo not have sense in the standard analysis like,for example, δ2(x).
As D ⊆ V , this shows that the delta ultra-function behaves as the delta distribution whentested against functions in D. Moreover, deltafunctions are mutually orthogonal with respectto the scalar product (10). Hence, being normal-ized they provide an orthonormal basis, calleddelta-basis, given by√
δaa∈Γ
=
χa√d(a)
a∈Γ
. (13)
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Hence, every ultrafunction can also be expandedin the following way,
u(x) =∑a∈Γ
(∮u(ξ)δa(ξ)dξ
)χa(x). (14)
The scalar product allows the following ultra-functions version of the Riesz representation the-orem.
Proposition 12. If
Φ : H → C
is a linear internal functional, then there existsuΦ such that, for all v = limλ↑Λ vλ ∈ H,∮
uΦv dx = limλ↑Λ
Φ(vλ),
and, for every f ∈ V ,∮uΦf
dx = Φ(f).
Proof. If v ∈ Hλ, then the map
v 7→ Φ(v)
is a linear functional over Hλ and hence, sincethere exists uλ ∈ Hλ such that ∀v ∈ Hλ,∫
uλv dx = Φ (v) .
If we setuΦ = lim
λ↑Λuλ,
the conclusion follows.
As already stated, our goal is to apply thetheory of ultrafunctions to quantum mechanics.In this way, we are interested in understandingthe relationship between ultrafunctions and L2-functions. Even though the scalar product in V
can be seen as an extension of the L2-scalar prod-uct, it is still not clear whether L2 functions canbe embedded into V . The basic idea is to useEq. (3) to do such an association. However, thisdoes not work since the L2-functions are not de-fined pointwise. For this reason, we introducethe following definition, which uses a weak formof association.
Definition 13. Given a function ψ ∈ L2(Ω), wedenote by ψ the unique ultrafunction such that,for every v = limλ↑Λ vλ(x) ∈ H,∮
ψv dx = limλ↑Λ
∫Ωψvλ dx.
Proposition 12 ensures that the above defini-tion is well posed, as the map
Φ : v 7→∫ψv dx
is a functional on the space H.
3.4 Ultrafunctions and distributionsDistributions can be easily embedded into ultra-functions spaces by identifying them with equiv-alence classes [18].
Definition 14. The space of generalized dis-tributions on Ω is defined as follows,
D′(Ω) = V (Ω)/N,
where
N =τ ∈ V (Ω) | ∀ϕ ∈ D(Ω),
∮τϕ dx ∼ 0
.
The equivalence class of an ultrafunction u ∈V (Ω) is denoted by [u]D(Ω). It contains all ultra-functions, whose action on the functions in D(Ω)differs from the action of u by at most an in-finitesimal quantity. An obvious idea is to iden-tify this action with the action of a distribution.However, it is not directly possible since there areultrafunctions, whose action does not correspondto the action of any distribution. For example,if u = τδ0, where τ is an infinite number, then∀ϕ ∈ D(Ω)
∮uϕ∗ dx = τϕ(0), which is an in-
finite quantity, whenever ϕ(0) is different from0.
This issue can be overcome by considering theso-called “bounded” ultrafunctions.
Definition 15. Let [u]D(Ω) be a generalized dis-tribution. We say that [u]D(Ω) is a bounded gen-eralized distribution if ∀ϕ ∈ D(Ω) the integral∮uϕ∗ dx is finite. We will denote by D′B(Ω) the
set of bounded generalized distributions.
In turn, the spaces of generalized distributionsand bounded generalized distributions can beidentified by an isomorphism, as shows the fol-lowing theorem.
Theorem 16. There is a linear isomorphism
Φ : D′B(Ω)→ D′(Ω)
such that, for every [u]D(Ω) ∈ D′B(Ω) and for ev-ery ϕ ∈ D(Ω),⟨
Φ([u]D(Ω)
), ϕ⟩D(Ω)
= st(∮
uϕ∗ dx
).
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Proof. For the proof see, e.g., [15].
In this way, any equivalence class [u]D(Ω) ∈D′B(Ω) can be substituted by Φ ([u]D) ∈ D′(Ω),so that we can define its action on the functionsin D(Ω),⟨
[u]D(Ω) , ϕ⟩D(Ω)
:=⟨
Φ([u]D(Ω)), ϕ⟩D(Ω)
= st(∮
uϕ∗ dx
).
In particular, if f ∈ C0c (Ω) and f∗ ∈ [u]D(Ω),
then, for all ϕ ∈ D(Ω),⟨[u]D(Ω) , ϕ
⟩D(Ω)
= st(∮
uϕ∗ dx
)= st
(∮f∗ϕ∗ dx
)=∫fϕ dx.
3.5 Self-adjoint operators on ultrafunctionsLet an operator
L : H → H
be an internal linear operator on the space of ul-trafunctions, which is a hyperfinite-dimensionalspace by definition. In this way, L can be repre-sented by a hyperfinite matrix (viewed as an in-finite matrix by the standard analysis), becausewe can build a Λ-limit
Lu := limλ↑Λ
Lλuλ,
where the operators
Lλ : Hλ → Hλ
can be represented by finite matrices (as everyspace Vλ is finite-dimensional). In particular, ifL is a self-adjoint operator,∮
Luv dx =∮uLv dx,
then the matrices Lλ are Hermitian. Hence, Lcan be represented by a hyperfinite-dimensionalHermitian matrix. Therefore, the spectrum σ(L)of L consists of eigenvalues only, more precisely,
σ(L) =
limλ↑Λ
µλ ∈ E | ∀λ, µλ ∈ σ(Lλ),
and its corresponding normalized eigenfunctions(being Λ-limits of the corresponding eigenfunc-tions of Lλ) form an orthonormal basis of H. In
that way, the ultrafunctions approach resemblesthe finite-dimensional vector spaces approach, inthe sense that the distinction between self-adjointoperators and Hermitian operators is not needed.We have proven the following
Theorem 17. Every Hermitian operator L :H → H is self-adjoint.
In [31], it is possible to find a detailed analysisof the ultrafunctions formalization of the positionand the momentum operators. For our applica-tions to the Schrodinger equation, let us noticethat the Laplacian operator
∆ : C2(RN
)→ C0
(RN
)has the following expression as its ultrafunctionsformulation,
D2 =N∑j=1
D2j : V → V .
For applications to quantum mechanics, thefollowing Hamiltonian operator is fundamental,
Hu(x) = −12∆u(x) + V(x)u(x), (15)
where we assume that the mass of the i-th parti-cle mi = 1 and ~ = 1.
There is a deep and important difference be-tween the standard and the ultrafunctions ap-proach to the study of H. In the classical L2-theory, a fundamental problem is a choice of anappropriate potential V such that (15) makessense. Moreover, it is fundamental to define anappropriate self-adjoint realization of H. On theother hand, in the theory of ultrafunctions anyinternal function V : Γ → E provides a self-adjoint operator on H, given by
Hu(x) := −12D
2u(x) + V(x)u(x), (16)
which always has a discrete (in the R∗ sense)spectrum that consists of eigenvalues only. Ofcourse, this spectrum will be hyperfinite, of car-dinality equal to the cardinality of Γ (once themultiplicities of the eigenvalues are taken into ac-count). For these reasons, we believe that theultrafunctions approach allows a much simplerstudy of “very singular potentials” such as
V(x) = τδa(x), τ ∈ E, (17)V(x) = ΩχE(x), Ω ∈ E, (18)
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where τ and Ω might be infinite numbers. Theuse of non-Archimedean methods is fundamentalto give a reasonable model of these potentials,which have an interesting physical meaning. Par-ticularly, the delta function potential (17), whichrepresents a very short-ranged interaction, ap-pears in many physical problems. For example,it can serve as a reasonable model for the inter-actions between atoms and electromagnetic fields(particularly, dipole-dipole interactions) and in-teratomic potential in a many-body system (par-ticularly, Bose – Einstein condensate). In thisway, in the following, we will take a look at thequantum system with a delta potential withinthe ultrafunctions framework. Before to start, we
will reformulate the basis of quantum mechanics— its system of axioms — through the ultrafunc-tions.
4 Ultrafunctions and quantum me-chanics
4.1 Axioms of QM based on ultrafunctions
In the following table, we provide a set of axiomsof quantum mechanics formulated within the ul-trafunctions approach and compare it with thestandard axioms11.
Standard QM Ultrafunctions QM
Axiom 1 A physical state of a quantum systemis described by a unit vector |ψ〉 in acomplex Hilbert space H.
A physical state of a quantum system isdescribed by a unit complex-valuedultrafunction ψ ∈ H.
Axiom 2 A physical observable A is representedby a linear self-adjoint operator A act-ing in H.
A physical observable A is representedby a linear Hermitian operator A actingin H.
Axiom 3 The only possible outcomes of a mea-surement of an observable A form a setµj, where µj ∈ σ(A) are the (gener-alized) eigenvalues of A.
The only possible outcomes of a mea-surement of an observable A form aset st(µj), where µj ∈ σ(A) are theeigenvalues of the operator A.
Axiom 4 An outcome µj of a measurement of anobservable A can be obtained with aprobability
Pj = |〈ψj |ψ〉|2,
where |ψj〉 is the (generalized) eigen-state associated with the observed(generalized) eigenvalue µj . After themeasurement, the quantum system isleft in the state |ψj〉.
An outcome st(µj) of a measurement ofan observable A can be obtained witha probability
Pj = |〈ψ,ψj〉|2,
where ψj is the eigenstate associatedwith the observed eigenvalue µj . Afterthe measurement, the quantum systemis left in the state ψj .
11For the sake of simplicity, we focus on the system ofaxioms based on pure states of the quantum system. How-ever, it can be straightforwardly generalized to the case ofmixed states, which describe a statistical mixture of thepure states.
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Axiom 5 The time evolution of the state of thequantum system is described by theSchrodinger equation
i∂|ψ〉∂t
= H|ψ〉,
where H is the Hamiltonian of thesystem.
The time evolution of the state of thequantum system is described by theSchrodinger equation
i∂ψ
∂t= Hψ,
where H is the Hamiltonian of thesystem.
Axiom 6 — In a laboratory only the states associ-ated to a finite expectation value of thephysically relevant quantities can be re-alized. These states are called phys-ical states, the rest of the states iscalled ideal states.
4.2 Discussion of the axioms
Axiom 1: States
In the standard formalism, a physical system isdescribed by a unit vector in Hilbert space. Theultrafunctions formulation of Axiom 1 guar-antees that we use the ultrafunctions space H(hence, non-Archimedean mathematics) insteadof the Hilbert one H. In particular, workingwithin wave functions, the state |ψ〉 can be repre-sented by a normalized function ψ ∈ L2(Ω), Ω ⊂RN . Since L2 can be embedded in V , there ex-ists a canonical embedding due to the Def. 13,
: H → H,
given by
ψ 7→ ψ.
SinceH is a space much richer thanH, the ultra-functions framework recovers all standard statesand provides more possible states, particularly,the ideal states of Axiom 6.
Axiom 2: Observables
The standard and ultrafunctions formulations ofAxiom 2 highlight many similarities as well assome differences. The main difference is thefact that the ultrafunctions formalism needs vonNeumann’s notion of the self-adjoint operator nomore. In the standard formalism, it is not enoughfor an operator A to be Hermitian – it has to be
self-adjoint (A = A†) so that it can represent aphysical observable12.
In the ultrafunctions formalism, the observ-ables of a quantum system can be representedby internal Hermitian operators, which are triv-ially self-adjoint due to the Theorem 17. Hence,for observables, “Hermitian” and “self-adjoint”become equivalent.
Axioms 3, 4: Measurement
In the standard formalism, the possible measure-ment outcomes of an observable A belong to thespectrum of the corresponding self-adjoint oper-ator A, which, generally speaking, is decomposedinto two sets, discrete spectrum, and continuousspectrum. While the discrete spectrum containsthe eigenvalues of A, the continuous one is a set ofthe so-called generalized eigenvalues, which cor-respond to the generalized eigenstates, the eigen-states which do not belong to H. A typical ex-ample is given by the position operator q on theHilbert space L2(R), whose spectrum is purely
12For example, if we consider the system of a parti-cle in an infinite potential well, the momentum operatorP = −i d
dxis Hermitian but not self-adjoint: the domain
of P is only a subset of the domain of its adjoint. Thisfact leads to problems with a physical interpretation ofthe spectrum of an Hermitian but not self-adjoint opera-tor, which can turn out to be the so-called residual spec-trum (being not interpretable physically). The reader in-terested in a deeper discussion of the difference betweenHermitian and self-adjoint operators, and its consequencesin the standard quantum mechanics is referred to [36].
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continuous (the whole real line R), and the cor-responding eigenfunctions ψq(x) = δ(x − q) donot belong to L2(R). In that way, the mannerthe standard formalism treats the discrete spec-trum in does not fit for the continuous spectrumleading to numerous misunderstandings: an in-troduction of some additional constructions suchas rigged Hilbert spaces is needed [36].
In the ultrafunctions formalism, due to theself-adjointness of internal Hermitian operators,it follows that any observable has exactly κ =dim∗(H) = |Γ| eigenvalues (taking into accounttheir multiplicity). In this way, no more essentialdistinction between eigenvalues and a continuousspectrum is needed since a continuous spectrumcan be considered as a discrete spectrum contain-ing eigenvalues infinitely close to each other.
For example, if we consider the eigenvalues ofthe position operator q of a free particle, then theeigenfunction relative to the eigenvalue q ∈ R isthe Dirac ultrafunction δq. Therefore, it can betrivially seen that its spectrum is Γ, which is nota continuous spectrum in R∗. However, the stan-dard continuous spectrum can be recovered sincethe eigenvalues of an internal Hermitian operatorA are Euclidean numbers. Hence, assuming thatmeasurement gives a real number, we have im-posed in the Axiom 3 that its outcome is st(µ).In the case of the spectrum of the position oper-ator, we can show that
st(µ) | µ ∈ σ(q) = R,
because every real number lies in Γ.Working in a non-Archimedean framework, we
have set in a natural way that the transitionprobabilities should be non-Archimedean. Infact, we assume that the probability is betterdescribed by the Euclidean number |〈ψ,ψj〉|2,rather than the real number st(|〈ψ,ψj〉|2). Forexample, let ψ ∈ H be the state of a system.The probability of an observation of the particlein a position q is given by∣∣∣∣∮ ψ(x)δq(x)
√d(q) dx
∣∣∣∣2 = |ψ(q)|2 d(q),
which is an infinitesimal number. The standardprobability can be recovered by the means of thestandard part, in the case of the position operatorit would be zero (as is expected). We refer to[34, 35] for a presentation and discussion of thenon-Archimedean probability.
Axiom 5: Evolution
The ultrafunctions version of this axiom is verysimilar to the standard one. Since H is an inter-nal operator defined on a hyperfinite-dimensionalvector spaceH, it can be represented by an Her-mitian hyperfinite matrix due to the Theorem 17.Hence, the evolution operator of the quantumsystem is described by the exponential matrix
U(t) = e−iHt.
Axiom 6: Physical and ideal states
This is the most peculiar axiom in the ultra-functions approach. In the ultrafunctions theory,the mathematical distinction between the phys-ical eigenstates and the ideal eigenstates is in-trinsic. It does not correspond to anything inthe standard formalism. Hence, it opens a veryinteresting problem of the physical relevance ofsuch states. Basically, we can intuitively saythat the physical states correspond to the stateswhich can be prepared and measured in a labora-tory, while the ideal states represent “extreme”states useful in the foundations of quantum me-chanics, thought experiments (Gedankenexperi-mente) and computations.
For example, a Dirac ultrafunction is not aphysical state but an ideal state. It representsa state in which the position of the particle isperfectly determined, which is precisely what onehas in mind considering the Dirac delta distribu-tion. Clearly, this state cannot be produced in alaboratory since it requires infinite energy. How-ever, it is useful in our description of the physi-cal world since such a state makes more explicitthe standard approach. Therefore, Axiom 6highlights a concept that is already present (butsomehow hidden) in the standard approach.
For example, in the Schrodinger representationof a free particle in R3, consider the state
ψ(x) = ϕ(x)|x|
, ϕ ∈ D(R3), ϕ(0) > 0.
We see that ψ(x) ∈ L2(R3) but this state cannotbe produced in a laboratory, since the expectedvalue of its energy
〈Hψ, ψ〉 = 12
∫|∇ψ|2 dx
is infinite (even if the result of a single experimentis a finite number).
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5 Example: Hamiltonian with a δ-potentialTo compare the standard and the ultrafunctionsapproaches, in this section, we want to study theSchrodinger equation
Hψ ≡ −12∆ψ + τδ0(x)ψ = Eψ, (19)
which corresponds to the problem of a particlemoving in the Dirac delta potential of trans-parency τ at the point x = 0. As we are goingto show, there are two main differences betweenthe standard and the ultrafunctions approaches,
1) the standard approach changes if we changethe dimension of the space (as in the spaceof dimension D > 1 one of the main prob-lems is to find a self-adjoint representationof H), while in the ultrafunctions approachwe always have a self-adjoint representationof H, independently of D , due to Theorem17,
2) in the ultrafunctions approach the constantτ can be an infinitesimal or an infinite num-ber, which allows us to construct the mod-els which cannot be considered within thestandard framework (for example, a poten-tial equal to the characteristic function of apoint13).
5.1 The standard approachLet us review the solution of the problem withinthe standard approach (namely, τ ∈ R) [37, 38].At first, it is assumed that the particle moves ina box of length 2L. In turn, the delta potentialis approximated as a square wall (well) of a finitelength 2ε and a finite height V0,
Vε(x) = V0θ(|ε− x|)V0→∞, ε→0−−−−−−−−→ τδ0(x),(20)
where the transparency of the potential is re-lated to the parameters of the approximation asτ = 2εV0. In that way, within the standard ap-proach, one solves the Schrodinger equation withthe approximated potential,
− 12∆ψ + Vε(x)ψ = Eψ, (21)
13In fact, within standard approach such a potentialwould be indistinguishable from 0.
and, in the obtained solution, takes the limit ε→0 and expands the box by the limit L → ∞, inorder to recover the original problem.
Potential barrier (τ > 0)
We start with the case of a potential barrier withτ > 0, which is illustrated on the Fig. 1.
−L 0 L −L −ε ε L
V0
Figure 1: Delta potential as a limit of a finite barrier.
Using the abbreviations k2 = 2E and κ2 =2(V0−E) we obtain the following solution of theSchrodinger equation,
ψ±(x) =
∓A± sin(k(x+ L)), x ∈ [−L,−ε],
B±f±(x), x ∈ [−ε, ε],
A± sin(k(x− L)), x ∈ [ε, L],
where A± and B± are the normalization con-stants, f+(x) = cosh(κx), f−(x) = sinh(κx),“+” corresponds to the even function and “−”corresponds to the odd function. We seek theenergies of the eigenstates, which can be foundwith use of the continuity condition,
ψ′(x)ψ(x)
∣∣∣∣∣x→ε−0
= ψ′(x)ψ(x)
∣∣∣∣∣x→ε+0
. (22)
In turn, the original problem is recovered bytaking the limit (20), which leads to the followingsolution,
ψ±n (x) =
∓A±n sin(k±n (x+ L)), x ∈ [−L, 0],
A±n sin(k±n (x− L)), x ∈ [0, L],
with the normalization |A±n |−2 = 2k±n L−sin(2k±n L)2k±n
and the quantities k±n , which are defined by the
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following equations due to Eq. (22),
k+n cot(k+
n L) = −τ, (23)
k−n cot(k−n L) = −∞ ⇒ k−n = π + 2πnL
,(24)
and, in turn, define the energies En = k2n2 of the
eigenstates. At the singularity point, we obtain
ψ+n (0) = −A+
n sin(k+n L), (25)
ψ−n (0) = 0. (26)
Potential well (τ < 0)
Let us now consider the case of a potential well(illustrated on the Fig. 2), namely, the negativetransparency τ < 0. In this case, we have twodifferent situations with respect to energy, E > 0(scattering states) and E < 0 (bound states).
−L0
L −L−ε ε
L
−V0
Figure 2: Delta potential as a limit of a finite well.
Positive energies (E > 0): scatteringstates. Using the abbreviations k2 = 2E andκ2 = 2(V0 + E) we obtain the following solutionof the Schrodinger equation,
ψ±(x) =
∓A± sin(k(x+ L)), x ∈ [−L,−ε],
B±f±(x), x ∈ [−ε, ε],
A± sin(k(x− L)), x ∈ [ε, L],
where A± and B± are the normalization con-stants, f+(x) = cos(κx), f−(x) = sin(κx), “+”corresponds to the even function and “−” corre-sponds to the odd function. By performing the
limits as done above one obtains the followingsolution,
ψ±n (x) =
∓A±n sin(k±n (x+ L)), x ∈ [−L, 0],
A±n sin(k±n (x− L)), x ∈ [0, L],
with the normalization |A±n |−2 = 2k±n L−sin(2k±n L)2k±n
and the quantities k±n , which are defined by theequations
k+n cot(k+
n L) = τ, (27)
k−n cot(k−n L) =∞ ⇒ k−n = 2πnL
, (28)
and, in turn, define the energies En = k2n2 of the
eigenstates. At the singularity point, we obtain
ψ+n (0) = −A+
n sin(k+n L), (29)
ψ−n (0) = 0. (30)
Negative energies (E < 0): bound states.Using the abbreviations k2 = 2|E| and κ2 =2(V0 − |E|) we obtain the following solution ofthe Schrodinger equation,
ψ±(x) =
∓A± sinh(k(x+ L)), x ∈ [−L,−ε],
B±f±(x), x ∈ [−ε, ε],
A± sinh(k(x− L)), x ∈ [ε, L],
where A± and B± are the normalization con-stants, f+(x) = cos(κx), f−(x) = sin(κx), “+”corresponds to the even function and “−” corre-sponds to the odd function. By performing thelimits as done above one obtains the followingsolution,
ψ±n (x) =
∓A±n sinh(k±n (x+ L)), x ∈ [−L, 0],
A±n sinh(k±n (x− L)), x ∈ [0, L],
with the normalization |A±n |−2 = 2k±n L−sinh(2k±n L)2k±n
and the quantities k±n , which are defined by theequations
k+n coth(k+
n L) = τ, (31)k−n coth(k−n L) =∞, (32)
and, in turn, define the energies En = −k2n2 of
the eigenstates. In this way, we find that, for
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E < 0, there exists a unique value k+ defined byEq. (31). With expanding the box by taking thelimit L→∞, it corresponds to the energy of theunique bound state
E1D = −τ2
2 . (33)
At the singularity point, we obtain
ψ+(0) = −A+ sin(k+L). (34)
Multidimensional δ-potentials
In the analysis of the delta potential, we have fo-cused on the 1-dimensional case. In this case, onecan easily prove that there exists a self-adjoint re-alization of the Hamiltonian H bounded below.As shown above, there exists a unique boundstate for a Hamiltonian with a one-dimensionaldelta potential well.
The delta potentials of a higher dimensional-ity D provide not only a pedagogical model butfind their applications in nuclear and condensedmatter physics [39, 40, 41, 42]. However, solvinga Schrodinger equation with these potentials is amore cumbersome problem since there is no rig-orous construction of a self-adjoint realization ofthe Hamiltonian in Eq. (19) as long as it is notextended to a non-interacting Hamiltonian on aspace with a removed point [11, 43, 44, 45, 46,47]. Let us illustrate it by the case of a two-dimensional and three-dimensional delta poten-tial well.
As in the one-dimensional case, the delta po-tential well of dimensionality D can be approx-imated by a finite square well described by theHeaviside function θ(r),
Vε(r) = −V0θ(ε− r)V0→∞, ε→0−−−−−−−−→ −τδ0(r), (35)
where its depth V0 and length ε are related tothe transparency of the delta potential well asτ2D = πε2V0 in the two-dimensional case andτ3D = 4π
3 ε3V0 in the three-dimensional case, re-
spectively [40, 41].
The most interesting part of the problem witha delta potential well is the estimation of thebound states. After solving the correspondingSchrodinger equation, the bound states can befound from the continuity equation. In the two-dimensional case, it is more difficult to do, be-cause the wave functions are Bessel functions
in this case. However, since the Bessel func-tion J0(r) with an azimuthal symmetry asymp-totically behaves as a cosine, the number of thebound states can be approximately estimated asN = εκmax/π, where κ2
max = 2V0. Plugging inthe transparency, we find
N2D '1π
√2τπ, (36)
which is a finite number. The solution for thethree-dimensional potential well can be found inthe form of trigonometric functions, that simpli-fies the calculations. The number of bound statesis given by [40, 41]
N3D '1π
√3τ2πε, (37)
which is already an infinite number.The energy of a bound state for the delta po-
tential can be found by solving the followingequation [48, 49],
1τ
+ 1(2π)D
∫dDk
k2 − E= 0, (38)
where D is the number of dimensions. The inte-gral in this equation is divergent for D ≥ 2, thatresults in infinite energies of the bound states. Inparticular, in the two-dimensional case, it can beshown that [40]
E2D ' −1
2ε2 e− 2π
τ ,
which is infinite. In this way, we conclude thatmultidimensional delta potential wells provide,generally speaking, an infinite number of boundstates with the energies diverging to −∞.
In order to avoid the problems with the in-finite quantities (diverging energies) within thestandard approach, one usually applies the so-called regularization and renormalization proce-dures to the calculations. The regularization pro-vides a cut-off14σ for the divergent integral, andthe renormalization re-defines the transparencyτ via τR (where R stands for “renormalized”),which absorbs the dependence on the cut-off inorder to absorb the divergence of the integral.
14In the renormalization theory, as well as in the ref-erences, the cut-off is denoted by Λ. However, to avoidconfusion with the Λ-limit we prefer to change the nota-tion here, as we will not discuss renormalization in detailin this paper.
Accepted in Quantum 2019-04-25, click title to verify 16
In particular, in the two-dimensional case, therenormalized transparency is defined by the fol-lowing equation [48, 49],
1τR
= 1τ
+ 14π ln
(σ2
ω2
), (39)
where ω is an arbitrary parameter, which repre-sents the renormalization scale. Then, one takesa limit σ → ∞ and varies the “bare” trans-parency τ in such a way that the renormalizedtransparency τR remains finite [48, 49]. Sucha renormalized transparency leads to a uniquebound state with the binding energy
E2D = −ω2e− 4πτR . (40)
Finally, this procedure means that one introducesan additional scale in order to “forget” about theunboundedness of the Hamiltonian from below.
5.2 The ultrafunctions approachSolution in the singularity point
In the ultrafunctions approach, the Schrodingerequation reads
−D2u+ τδ0u = Eu, (41)
where τ, E ∈ R∗. We use the linearity of V towrite the solution u of Eq. (41) as
u(x) =∑a∈Γ
u(a)χa(x).
Therefore, we can rewrite Eq. (41) as
−∑a∈Γ
u(a)D2χa(x) + τ∑a∈Γ
u(a)χa(x)δ0(x)
= E∑a∈Γ
u(a)χa(x). (42)
For every a ∈ Γ, let D2χa(x) :=∑b∈ΓDa,bχb(x).
Notice that the locality of the ultrafunctionsderivative entails that, actually, we can write15
D2χa(x) =∑
b∈Γ∩mon(a)Da,bχb(x).
15Readers familiar with nonstandard analysis will noticethat our notation
∑b∈mon(a) is not proper, as mon(a) ∩
Γ is not an internal set (hence, in particular, it is nothyperfinite). However, by underspill it is trivial to provethat Da,b 6= 0 only for a hyperfinite amount of pointsb ∈ mon(a)∩Γ, as we have already noticed in Section 3.2,hence ours is just a small abuse of notations.
Therefore, since∑a∈Γ u(a)δ(a)χa(x) = u(0)χ0(x)
d(0)we can rewrite Eq. (42) as
−∑a∈Γ
u(a)∑
b∈Γ∩mon(a)Da,bχb(x)
+ τu(0)χ0(x)d(0)
= E∑a∈Γ
u(a)χa(x). (43)
Let us explicitly note that the above discus-sion does not depend on the dimensionality ofthe space. In fact, for any number of dimensions,Eq. (42) reveals the self-adjoint representation ofthe Hamiltonian in Eq. (19) in the ultrafunctionssetting.
Outside the monad of 0, the equation actuallyreads
−∑
a∈mon(x)∩Γu(a)D2
a(x) = Eu(x), (44)
which is formally the same that one gets for theequation
−D2u = Eu. (45)On the other hand, at the point 0 ∈ Γ, we obtainthe equation
−∑
a∈mon(0)∩Γu(a)D2
a(0) + τ
d(0)u(0) = Eu(0),
which corresponds to
u(0) =
∑a∈mon(0)∩Γ
(u(a)
∑b∈Γ∩mon(0)
Da,bχb(x))
E − τd(0)
.
Notice that, for every other point γ ∈ mon(0)∩Γ, we obtain
u(γ) =
∑a∈mon(0)∩Γ
(u(a)
∑b∈Γ∩mon(0)
Da,bχb(x))
E.
In particular, it shows that, in the ba-sis χaa∈Γ, the matrix representation Mδ ofEq. (41) and the matrix representation M∆ ofEq. (45) satisfy the relation Mδ = M∆+H, whereH is a matrix with
Ha,b =
τd(0) , if a = b = 0,0, otherwise.
Particularly, in the case of a one-dimensional po-tential well with an infinitesimal transparency,there will still exist a unique bound state withinfinitesimal negative energy.
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Connection with the standard solutions
As usual, in order to introduce a new solutionconcept for a standard problem, it is importantto discuss the relationship between the standardand new solution. Our first result shows that theapproximation procedure used in most standardapproaches to the Schrodinger equation with adelta potential can be reproduced within the ul-trafunctions approach, in the following sense.
Theorem 18. Let 0 ∼ aΛ = limλ↑Λ aλ andτΛ = limλ↑Λ τλ. Then there exists a space of ul-trafunctions V that contains a solution uΛ of theequation
−D2uΛ + τΛδ0uΛ = EΛuΛ, (46)
which is a Λ-limit of a net of solutions uλ ∈ V (Ω)of the standard Schrodinger equations
−∆uλ + τλAaλuλ = Eλuλ, (47)
where Aaλ is an approximation of the delta dis-tribution by a square well of length aλ ∈ R, and,for every λ, Eλ is an eigenvalue of the energyof the solution of the standard Schrodinger equa-tion with the approximated delta potential suchthat EΛ = limλ↑ΛEλ.
Proof. For every λ ∈ Λ, let uλ be a solution ofEq. (47). Let Vλλ be an increasing net of finitedimensional subspaces of V (Ω) as in Definition11. If ∀λ the solution uλ ∈ Vλ, then a Λ-limituΛ = limλ↑Λ uλ is contained in the space of ul-trafunctions V generated by the net Vλλ, andtheorem is proven.
If uλ /∈ Vλ, then we extend Vλ by
V ′λ := span(Vλ ∪ uλ),
and build the Λ-limit with the net V ′λ. Inthis way, the Λ-limit of the net uλ gives anultrafunction in V by construction.
The second result we want to show precisesthe relationship between the ultrafunction andstandard solution in the case of finite energy andtransparency.
Theorem 19. Let u ∈ V be an ultrafunctionsolution of the Eq. (41), where the transparency
τ and the energy E are finite numbers in R∗. Letw ∈ D′(Ω) be such that16
1. [u]D(Ω) = w,
2. [δ · u]D(Ω) = δ · w.
Then w is a solution of the standard Schrodingerequation
−∆w + 2 st(τ)δ0w = st(E)w. (48)
Proof. By the definition of [u]D(Ω), we can write
u = w + ψ, (49)
where ψ is such that for any f ∈ D(Ω),
〈ψ, f〉 ∼ 0. (50)
If f ∈ D(Ω), we have
〈w, f〉 = 〈w, f〉 = 〈u, f〉 − 〈ψ, f〉 ∼ 〈u, f〉.
Using Eq. (41) we can write
〈u, f〉 = 1E
(〈D2u, f〉+ τ〈δ0u, f
〉).
The first term can be rewritten in the followingway,
1E〈D2u, f〉 = 1
E〈u,D2f〉
∼ 1st(E)〈w,∆f〉 = 1
st(E)〈∆w, f〉.
Using the hypothesis on [δ · u]D(Ω), the secondterm can be rewritten in the following way,τ
E〈δ0u, f
〉 ∼ 2 st(τ)st(E) 〈δ0u, f
〉 ∼ 2 st(τ)st(E) 〈δ0w, f〉.
This means that
〈w, f〉 = 1st(E)〈∆w, f〉+ 2 st(τ)
st(E) 〈δ0w, f〉,
or, equivalently,
∆w + 2 st(τ)δ0w = st(E)w,
which is the corresponding standard Schrodingerequation.
16In condition 2, we have a distributional product δ ·w,which is not defined, in general. We tacitly assume thatw is such that this product makes sense as a distribution.This is the case, e.g., when w is smooth. Notice that,in contrast, the product of ultrafunctions is always welldefined.
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Bounds for the energy
Some easy bounds on the energy of the solutionof Eq. (41) can be found using the scalar productof ultrafunctions. By Axiom 7 of the space ofultrafunctions,
〈∆u(x), u(x)〉 = −〈Du(x), Du(x)〉 ≤ 0. (51)
Using the Schrodinger equation we can rewritethe previous equation, in order to get
〈∆u(x), u(x)〉 = −2⟨(E−τδ0(x)
)u(x), u(x)
⟩≤ 0.
Calculating the scalar product and applying thenormalization of the wave function ||u(x)||2 =∑a∈Γ u
2(a)d(a) = 1 we obtain
E − τu2(0) ≥ 0. (52)
Taking into account that the transparency cantake both positive (potential barrier) and nega-tive (potential well) values we obtain inequalitiesfor allowed energies
E
τ≥ u2(0), τ ≥ 0, (53)
E
τ≤ u2(0), τ < 0. (54)
The first inequality implies
E ≥ 0, τ ≥ 0, (55)
which means that the negative energies are notallowed in the case of a potential barrier.
In order to understand better the second in-equality, let us take a look to the following in-equality,
u2(0)d(0) ≥∑a∈Γ
u2(a)d(a) = ||u(x)||2 = 1. (56)
Implying this inequality results in
|E| ≤ 2|τ |d(0) , τ < 0. (57)
In this way, we see that the value of transparencyof the potential well establishes an upper boundfor the allowed negative energies. This bound is
valid not for the one-dimensional potential only,but for the multidimensional ones as well.
Let us notice that this bound reveals a possibil-ity to fix always an (infinitesimal) transparency τclose to d(0), so that there exists a solution of thecorresponding Schrodinger equation with finiteenergy. For example, we can fix τ = d(0). Noticethat, in this case, the potential τδ0 is equal to theultrafunction indicator χ0, that in the standardapproach would not differ from 0.
Splitting of the ultrafunction solution
Let us discuss a particular aspect of non-Archimedean mathematics in general, and ul-trafunctions in particular briefly. As we havehighlighted above, the ultrafunction solution ofEq. (19) has the form of the sum of the “ex-tended” standard solution ψ and the non-Archimedean contribution ϕ. Now, one mightask whether it is possible for ϕ to be concen-trated in the monad of zero or have some similaradditional property.
In principle, it is possible if the delta distri-bution δ is represented within the ultrafunctionsframework differently. In fact, we have usedthe delta ultrafunction δ0 = χ0
d(0) to model thedelta distribution, which appears in the standardHamiltonian in Eq. (19), because we believe thisis a natural approach to follow in our setting.However, there are several ultrafunctions u inV (Ω) which correspond to δ0 as distributions,so that the scalar product 〈u, f〉 = f(0) for ev-ery f ∈ D(Ω)17. Of course, by changing the setof test functions, we obtain different conditions.For example, if we use V instead of D(Ω), thenthe unique “delta” is precisely the delta ultra-function. In this sense, there are several differentmodels of the Hamiltonian in Eq. (19) that couldbe constructed in V (Ω). This gives us a degreeof freedom which is not present in the standardapproach.
Comparison of the approaches
We summarize the discussion of the interpreta-tion of the Schrodinger equation with a delta po-tential within the ultrafunctions approach in thefollowing table.
17This is the notion of “association”, which is often con-sidered in non-Archimedean mathematics when dealingwith distributions, see, e.g., [33].
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Standard approach Ultrafunction approach
Analysis of the solutions of theSchrodinger equation (19) needs an ap-proximation of the delta function by afinite potential (for example, a squarebarrier/well) and performing a limit ofits zero width.
The solutions can be analyzed directlywith the use of the delta ultrafunction,which describes an infinite jump con-centrated in 0. However, the approxi-mation procedure can be used as welldue to Theorem 18.
Multidimensional delta potentials aredifficult to interpret because of the di-vergence of the corresponding integrals.
There is a unique framework for thedelta potentials of any dimension basedon hyperfinite spaces and, thus, hyper-finite matrices. Infinite or infinitesimalnumbers can be used as parameters ofthe equation. In this way, a modelcomes out which better describes thephysical phenomenon.
The number of the eigenstates dependson the number of physical dimensionsD . In particular, the number of boundstates is infinite for D ≥ 3.
There always exists the number ofeigenstates equal to the hyperfi-nite dimensionality of the space ofultrafunctions.
The energies of the eigenstates are fi-nite only for a one-dimensional deltapotential.
Natural bounds for the energies of theeigenstates can be estimated indepen-dently of the dimension of the system.Moreover, it is possible to obtain thebound states with finite energies by fix-ing suitable transparency.
— The splitting of the corresponding ul-trafunction can recover the standardcontent of the solution.
6 Conclusions
In this paper, we have introduced a new approachto quantum mechanics using the advantages ofnon-Archimedean mathematics. It is based on ul-trafunctions, non-Archimedean generalized func-tions defined on a hyperreal field of Euclideannumbers E = R∗, which is a natural extensionof the field of real numbers. Euclidean numbershave been chosen because they allow constructinga non-Archimedean framework through a simplenotion of Λ-limit, which gives an advantage con-cerning other nonstandard approaches to quan-tum mechanics [8, 9, 10, 11, 12].
The ultrafunctions approach proposes a new
set of the axioms of quantum mechanics. Eventhough the new axioms are obviously related withthe standard Dirac – von Neumann formulationof quantum mechanics, ultrafunctions make itpossible to build a framework which is closerto the matrix approach. Moreover, the pres-ence of infinite and infinitesimal elements allowsconstructing new simplified models of physicalproblems. In particular, this framework is basedon Hermitian operators, so that unbounded self-adjoint operators are not more needed. Further-more, the difference between continuous and dis-crete spectra is not more present.
These aspects were illustrated by a compari-son between standard and ultrafunction solutions
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of a Schrodinger equation with a delta potential.We have shown that the ultrafunctions approachprovides a more straightforward way to solve theSchrodinger equation by unifying the cases of apotential well and a potential barrier and provid-ing the extreme cases of infinite and infinitesimaltransparency of the potential. Moreover, it givesa connection between energy and transparency,providing a bound for the allowed energies of thequantum system.
AcknowledgementsL. Luperi Baglini and K. Simonov acknowledgefinancial support by the Austrian Science Fund(FWF): P30821.
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