Towards a Usable Ontology for the Quantum World

Marcin Skulimowski

Faculty of Physics and Applied Informatics, University of Lodz, Pomorska 149/153, 90-236 Lodz, Poland

Keywords:

Ontology, Quantum Mechanics, Semantic Enhancement.

Abstract:

We present and discuss selected issues related to the problem of representing in a machine-readable way data

and knowledge about the quantum level of reality. In particular, we propose a method of creating an ontology

for the quantum world. The method uses a mathematical structure of quantum mechanics. We apply the

method to obtain a toy ontology corresponding to the Hilbert space formulation of quantum mechanics. We

use the terms from the ontology to describe a simple quantum system. We also show how we can use the

ontology to create semantic enhancements of scientiﬁc publications on quantum mechanics.

1 INTRODUCTION

The term Ontology (with uppercase initial) is usu-

ally linked to a philosophy where it means a disci-

pline which deals with the nature and structure of

reality. Another reading of the term (with lower-

case initial) corresponds to computer science, where

it means a formal, explicit speciﬁcation of a shared

conceptualization (Guarino, 1998). In recent years

many ontologies corresponding to various domains

have been proposed. They play essential roles in

various ﬁelds, e.g. knowledge engineering, knowl-

edge representation, database design, information re-

trieval and extraction. Ontologies (with lowercase ini-

tial) can be divided into different kinds according to

their level of generality. The most general ontolo-

gies are the so-called top-level ontologies, describing

concepts which are independent of a particular prob-

lem or domain, e.g. space, time, events and macro-

scopic objects. An exceptional concept among them

is space. All physical (macroscopic) objects are lo-

cated in space, which is an ”arena”, a ”support” for

them. Consequently, the problem of representing

knowledge about space and macroscopic objects has

been widely discussed in the literature (Kutz et al.,

2003; Borgo et al., 1997; Casati and Varzi, 1996).

Recently, an ontology of space, time, and physical en-

tities in classical mechanics has been proposed (Bit-

tner, 2018). The crucial point is that considerations

about space and macroscopic objects are based on our

observations. Indeed, we can observe objects, their

positions in space and spatial relations among them.

Based on our observations and in general, our sen-

sual perception, we can build an ontology represent-

ing given ”part of reality”. According to the ”realism-

based” approach, good ontologies in the support of

the natural sciences (and thus also physics) have to

be reality representations (Smith, 2004). Some au-

thors have expressed doubts about this requirement

(Dumontier and Hoehndorf, 2010; Merrill, 2010). For

example, Dumontier pointed out that ontologies satis-

fying this condition face the problem of representing

issues which are very important in scientiﬁc commu-

nication (e.g. hypotheses) and objects which cannot

(yet) be shown to exist (e.g. hypothetical elementary

particles) (Dumontier and Hoehndorf, 2010). The

problem with the ”realism-based” approach is partic-

ularly evident when the domain of a considered on-

tology is not accessible through our sensual percep-

tion. This problem happens on the most fundamental

level of our world, i.e. on the quantum level

. We

are not able to observe this level directly through our

senses. However, despite this, we have successfully

explored the quantum world since the theory called

quantum mechanics (QM) was created. However, the

notion of quantum reality is still not clear. Surpris-

ingly, it is difﬁcult to say what really (actually) ex-

ists on the quantum level. Indeed, in the mathemati-

cal structure of QM, nothing is corresponding to the

concept of a quantum object (Heller, 1994a). In con-

sequence, there are severe problems with the Ontol-

ogy and interpretations of QM (Bohm and Kaloyerou,

Throughout this paper, we use the terms ’the quantum

level’ and ’the quantum world’ interchangeably. The second

term frequently appears in the literature to emphasize the

difference between the quantum and macroscopic level.

Skulimowski, M.

Towards a Usable Ontology for the Quantum World.

DOI: 10.5220/0008365904870494

In Proceedings of the 11th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2019), pages 487-494

ISBN: 978-989-758-382-7

487

1987; da Costa and Lombardi, 2014; Busch, 2002;

Rudolph, 2006). In this paper, we are going to dis-

cuss selected issues related to an ontology (with low-

ercase initial) of the quantum world. Our motivation

is twofold. Firstly, taking into account the multitude

of ontologies corresponding to the macroscopic, ob-

servable level of reality it seems interesting how to

build an ontology for the quantum level. Our sec-

ond motivation is a more practical one. We are in-

terested in the representation of data (and knowledge)

about the quantum world in a way enabling easy in-

tegration, sharing and linking the data (Skulimowski,

2014; Skulimowski, 2015). To this end, we need an

appropriate ontology for this world. A preliminary

and highly incomplete version of a quantum ontol-

ogy represented in OWL (Web Ontology Langauge)

was shortly described in one of our previous papers

(Skulimowski, 2010). In this paper, we are going to

carry out more detailed research.

The paper is organized as follows. First, to intro-

duce the reader to the amazing quantum world, we

present in Section 2 two properties of the quantum

world which are contrary to our macroscopic experi-

ence. In Section 3, we propose a method for creating

a quantum ontology. The method is used in Section 4,

where we present a toy quantum ontology and apply

it to describe a simple quantum system. We also use

terms from the ontology to create RDF links between

entities from publication on QM. Finally, concluding

remarks are given in Section 5.

2 AMAZING QUANTUM WORLD

The macroscopic world differs signiﬁcantly from the

quantum world. In order to familiarize the reader with

the differences, we present below two interesting ex-

amples related to localizability and individuality of

entities. These two notions are essential in the onto-

logical analysis and rather evident on the macroscopic

level. Let us see how the situation looks on the quan-

tum level.

Localizability. All macroscopic physical objects

are located in space. The localization is an essen-

tial property of macroscopic objects. Consequently,

(Casati and Varzi, 1996) formulated the following ax-

iom: ∀x∃yL(x, y) where L(x,y) means that entity x is

exactly located at region y of space. In his Formal on-

tology of space, time, and physical entities in Classi-

cal Mechanics Bittner assumes that from the classical

point of view every particle is located as a unique re-

gion of space at every time of its existence (Bittner,

2018). It turns out that in the quantum world, space

is no longer support for any ”quantum objects”. Ac-

cording to QM, the physical space appears as the set

of possible results of a position measurement (which

belong to our macroscopic world). That is why Heller

proposed that space should be regarded as the macro-

scopic entity (Heller, 1994a). However, from the

macroscopic point of view, the behaviour of micro-

objects looks very strange. Namely, it turns out that if

at t = 0 a microscopic particle is strictly localized in

a bounded region y

then unless it remains in y

for

all times, it cannot be strictly localized in a bounded

region y, however large, for any t > 0 (Hegerfeldt,

1998). In other words, a particle which is initially

strictly localized after that becomes non-localizable

because it is spread over all space. This result contra-

dicts our macroscopic experience.

Individuality. We can describe macroscopic ob-

jects. The description of an object can be seen as a set

of properties that apply to this object. Based on the

properties, we can distinguish objects by comparing

their properties. Consequently, using the properties

of objects, we can also ascribe individuality to them.

This statement is evident on the macroscopic level. At

the quantum level, the situation is more complicated.

Namely, when two quantum systems, each of which

could originally be considered as having a complete

set of deﬁnite properties, have once interacted, it is

generally no longer possible to think of either of them

as having a complete set of deﬁnite properties of its

own (D’Espagnat, 1999). However, we can assign

the set of speciﬁc properties to a compound quantum

system consisting of these two systems. In quantum

world the best possible knowledge of a whole does not

necessarily include the best possible knowledge of all

its parts, even though they may be entirely separate

and therefore virtually capable of being ’the best pos-

sibly known’, i.e., of possessing, each of them, a rep-

resentative of its own (Schr

odinger, 1936). Heller re-

calls a well-known example: if two particles interact

with each other and then go apart, it is not possible

to consider them as two different quantum objects, or

even as two parts (subobjects) of the same object each

of which would be found in a state independent of the

other (Heller, 1994a) . The two (entirely separated)

particles do not have deﬁnite properties of its own.

We can say that these particles lack the individuality,

which is a very characteristic property of macroscopic

objects.

The above two examples suggest that our experience

obtained in the macroscopic world may not be useful

at the quantum level. Consequently, the tools of for-

mal ontology created based on our macroscopic expe-

KEOD 2019 - 11th International Conference on Knowledge Engineering and Ontology Development

488

riences (observations) may be useless in the ontolog-

ical analysis of the quantum world. So how to create

an ontology for the quantum level?

3 QUANTUM ONTOLOGY

The Hidden Structure. We have no direct access

to the quantum world. All, we have is the theory

called quantum mechanics which models the world

very precisely, i.e. its predictions agree with the re-

sults of numerous experiments. However, despite this

agreement, we have severe problems with the under-

standing of physical reality to which the theory refers.

This misunderstanding leads to interpretative prob-

lems with QM and, in consequence, with the Ontol-

ogy of quantum world (Bohm and Kaloyerou, 1987;

da Costa and Lombardi, 2014; Busch, 2002; Rudolph,

2006). Heller (Heller, 1994b) proposed an interest-

ing method for overcoming these problems. The ap-

proach uses an observation that everything we know

about the structure of the quantum world we owe to

mathematical models of it (Heller, 1994b). The ex-

traordinary success of QM legitimizes the statement

that the mathematical structure of QM strongly in-

teracts with the hidden quantum structure. Conse-

quently, in order to explore the hidden quantum struc-

ture, we have to analyze the mathematical structure of

QM. The analysis of this structure will give us knowl-

edge about entities and relations it requires to exist.

Heller proposes: let us not bother about the ”world”

to which quantum mechanics supposedly refers, and

let us consider only univers de discours of this theory.

By the univers de discours of a physical theory I mean

the collection and only the collection of objects, rela-

tionships, sets, and so on, which is presupposed by the

mathematical structure of this theory (Heller, 1994a).

Thus, we can treat the Heller’s approach as the mod-

iﬁed Quine’s program: we postulate the existence of

only those objects that are presupposed by the math-

ematical structure of the theory (Heller, 1994b). It is

worth to stress that the reconstruction of the univers

de discours for the case of empirical theories using

complex mathematical structures may not be a simple

task. However, this effort can pay off. According to

Heller a rigorously reconstructed univers de discours

of QM can be useful in solving interpretative prob-

lems of QM. Moreover, we believe that the univers de

discours can also be used to obtain an ontology (with

lowercase initial) for the quantum world, which can

be useful in the creation of semantic enhancements of

scientiﬁc papers.

Conceptualizations and Ontologies. An ontology

corresponds to a conceptualization, which is a simpli-

ﬁed view of the world we want to represent for some

purpose (Guarino, 1998). In order to obtain an ex-

plicit speciﬁcation of a conceptualization, we have to

ﬁx a language we want to use to talk about the concep-

tualization and constrain the interpretations of such

a language using suitable axioms which usually have

the form of a ﬁrst-order logical (FOL) theory (Guar-

ino et al., 2009). A set of such axioms is called an

ontology. In order to create an ontology for the quan-

tum world, we propose the following 3-steps strategy.

• First, we consider the univers de discours U of

QM as deﬁned by Heller, i.e. the collection of

objects, sets, relationships which are presupposed

by some mathematical structure of QM, e.g. the

Hilbert space formulation of QM.

• Then we choose from U essential elements to ob-

tain a conceptualization.

• Finally, we create a set of axioms for the concep-

tualization.

One can object that an ontology obtained in this way

will be an ontology of QM rather than the ontology

of the ”real” quantum world. It is true. Notice, how-

ever, that in this particular case, the theory is the only

way to explore the quantum world. We can say that

the theory gives us a unique ”access” to the world.

Moreover, it is worth to note that the ontology ob-

tained in this way will not be unique. This non-

uniqueness is because we can formulate QM by em-

ploying several mathematical structures, e.g. Hilbert

space formalism, C*-algebraic approach, Feynmann’s

approach. Each of these structures contains different

collections of objects, relationships, sets and so on

(Heller, 1994a). Therefore, we will obtain different

conceptualizations and different ontologies. At ﬁrst

sight, this multitude of quantum ontologies may seem

confusing. However, this is not the case taking into

account our primary motivation, i.e. usability. In-

deed, these various ontologies can be used to enrich

with machine-readable data publications using vari-

ous mathematical formalisms (structures).

Finally, it is worth to mention that if we are in-

terested in the Ontology (with uppercase initial), we

have to go one step further. Namely, we have to con-

sider representation invariants which are preserved

if we change from one mathematical structure to an-

other. According to Heller, the collection of these in-

variants constitutes the structuralist ontology of the

quantum world (Heller, 2006).

Axioms for the Quantum World? An ontology is a

set of axioms in the form of FOL theory for a concep-

Towards a Usable Ontology for the Quantum World

489

tualization of some domain, which in our case is the

quantum world. One can doubt whether it is possible

to create such axioms because QM and, in general,

physical theories are not axiomatic systems. Some

terms (names) from logic are indeed used by physi-

cists, e.g. axioms, postulates, consequence, equiv-

alence and contradiction. However, they cannot be

treated in a strictly logical way. Physical theories

are rather models of reality (Wole

nski, 1991). De-

spite this, there have been attempts to formulate strict

logical axioms for physical theories (Suppes, 1974;

Madarasz et al., 2006). According to the advocates of

axiomatization, we can better understand the physi-

cal theory by proving a basis of explicit postulates for

the theory (Madarasz et al., 2006). Moreover, hav-

ing axioms for physical theory, one could ask what

happens if we change one or more axioms. All ax-

iomatic approaches to QM were sharply criticized by

Mielnik, who noticed that axioms in physics can be

very deceptive, even if they look obvious (Mielnik,

1980). According to Mielnik, strict axiomatic foun-

dations of QM can be an obstacle in the further de-

velopment of the theory. Putting aside the discussion

Figure 1: Axioms for ontology, not for quantum mechanics.

about axiomatization of QM let us clarify that we are

not going to formulate axioms for QM. Our goal is

much more modest. We intend to specify a conceptu-

alization for QM and not QM using suitable axioms.

We believe that it is possible because a conceptual-

ization is a simpliﬁed ”model” of the theory (see Fig.

1), i.e. it contains only selected entities and relations

from the univers de discours of QM.

4 A TOY ONTOLOGY FOR

QUANTUM WORLD

In this section, we are going to propose a toy ontology

for the Hilbert space formulation of QM. The detailed

description of this formulation falls outside the scope

of this paper. It can be found in any textbook on QM

(see, e.g. (Isham, 1995)). For simplicity, we limit

ourselves to the following basic facts:

1. A state space of a quantum system is given by a

Hilbert space.

2. Observables (physical quantities) are self-adjoint

operators deﬁned on the Hilbert space

3. Vectors in the Hilbert space represent states of a

quantum system.

4. Real eigenvalues of a self-adjoint operator corre-

spond to results of a measurement of observable

represented by this operator.

5. The Schr

odinger equation describes the time evo-

lution of quantum states.

Let us now consider the univers de discours for this

formulation of QM. According to Heller it contains

objects, relationships and sets which are required by

the mathematical structure (Heller, 1994a). Listing

all elements of this collection is not an easy task.

Below, we limit ourselves to the following set of

relations, which are the backbone of the Hilbert

space formulation of QM and are usually present in

publications on this formulation:

1-ary relations:

• StateSpace(x) - x is a state space.

• HilberSpace(x) - x is a Hilbert space.

• VectorHS(x) - x is a vector is a Hilbert space.

• Observable(x) - x is an observable.

• SAOperator(x) - x is a self-adjoint operator

• State(x) - x is a state.

• SchroedingerEq(x) - x is a Schr

odinger equation.

• Hamiltonian(x) - x is a Hamiltonian operator.

• Real(x) - x is a real number.

Binary relations:

• isElementO f (x,y) - a state x belongs to a state

space y.

• x ⊥ y - a state x is orthogonal to a state y.

• hasEigenVal(x,y) - an operator x has an eigen-

value y.

• hasEigenV (x, y) - an operator x has an eigenvector

• hasMeasurementR(x,y) - a result of a measure-

ment of an observable x may be y.

• correspondsToEigenV (x,y) - an eigenvalue x cor-

responds to eigenvector y

• solO f Schroedinger(x, y) - a state x is the solution

of a Schr

odinger equation y.

Tenary relations:

• commutator(c,x, y) - c is the result of the commu-

tator of x and y.

For brevity, we consider only self-adjoint operators.

KEOD 2019 - 11th International Conference on Knowledge Engineering and Ontology Development

490

• meanValue(x,y,z) - an observable x in a state y

has the expected value z.

• uncertainty(d,o,x) - d is the dispersion of an ob-

servable o in a state x.

• timeEvolution(x,y,h) - x is a state which evolved

from a state y according to Hamiltonian h.

4-ary relation:

• probabilityO f R(p, r,y,o) - p is the probability of

obtaining result r in a state y, in the measurement

of o.

In order to specify our conceptualization more

precisely, we propose the following simple set of

axioms specifying the quantum domain

Taxonomic information

StateSpace(x) → HilberSpace(x)

Observable(x) → SAOperator(x)

State(x) → VectorHS(x)

Hamiltonian(x) → Observable(x)

hasMeasurementR(x, y) → hasEigenVal(x,y)

Domains and ranges:

x ⊥ y → State(x) ∧ State(y)

hasEigenVal(x,y) → SAOperator(x) ∧ Real(y)

hasEigenV (x,y) → SAOperator(x) ∧VectorHS(y)

isElementO f (x,y) → State(x) ∧ StateSpace(y)

hasMeasurementR(x, y) → Observable(x) ∧ Real(y)

solO f Schroedinger(x,y) →

State(x) ∧ SchroedingerEq(y)

We can specify the properties of the relations:

• Symmetry: ⊥.

• Irreﬂexivity: isElementO f , hasMeasurementR,

hasEigenVal, hasEigenV , correspondToEigenV ,

solO f Schroedinger, ⊥.

• Asymmetry: isElementO f , hasMeasurementR,

hasEigenVal, hasEigenV , correspondToEigenV ,

solO f Schroedinger.

Using the above basic set of relations, we can formu-

late more axioms, e.g.:

hasEigenV (x,y

) ∧ hasEigenV (x,y

) ∧ y

6= y

⇒

⊥ y

hasMeasurementR(x, y) ⇒ hasEigenVal(x,y)

correspondsToEigenV (x, y) ⇒

∃z SAOperator(z)∧

hasEigenVal(z,x) ∧ hasEigenV (z,y)

For brevity, we present only selected axioms.

In the formulas below, we omit the quantiﬁers ∀x∀y ...

The proposed conceptualization contains only the

most essential entities from the Hilbert space formu-

lation of QM. Note that, it consists of physical entities

(e.g. Observable) and purely mathematical (e.g. ha-

sEigenVal) entities. It is quite reasonable taking into

account that the conceptualization we obtained from

the mathematical structure of QM. In the future, all

mathematical relations and axioms should be taken

from the appropriate mathematical ontology. We can

also observe that the ontology contains nine classes

(1-ary relations), but there are only ﬁve taxonomic re-

lations. Moreover, note that for almost all binary re-

lations domains differ from ranges. The reason for

this is that the structure of QM contains relations be-

tween entities of different types. Consequently, there

is only one symmetric relation (⊥). All other relations

are irreﬂexive and asymmetric. It is interesting, that

the proposed conceptualization does not contain any

transitive relation (the transitivity is usually required

in the case of mereological relations). Note also that,

apart from binary relations there are also ternary rela-

tions (e.g. commutatorOf, meanValue) and even one

4-ary relation (probabilityOfR). These relations are

fundamental in the formalism of QM.

It is worth to mention that in our ontology, there is

no quantum counterpart of the predicate L related to

the location in space mentioned in Section 2. The re-

sult obtained by Hegerfeldt tells us something about

the expected value of a particular self-adjoint opera-

tor N (corresponding the probability to ﬁnd a particle

or system inside some region V ) (Hegerfeldt, 1998).

Consequently, formalization of the result would re-

quire the use of the predicate meanValue, the operator

N and some quantum state x.

Example. Let us now try to use the terms from our

ontology to represent knowledge about some quan-

tum system. To this end we consider the electron

described by the Hamiltonian operator

H =

1 + ασ

where α ∈ R,



0 −i

i 0



(1)

is the Pauli spin-matrix and

1 is the unit 2 × 2 matrix.

Hilbert space for this system is C

. The spin opera-

tors for this system are given by:



0 −i

i 0





0 −i

i 0





0 −i

i 0



(2)

After some calculations one can show that

(Isham,

1995):

The detailed explanation of this quantum system falls

outside the scope of this paper. All we want to show is

that the knowledge about this quantum system can be rep-

resented using terms from the proposed ontology.

Towards a Usable Ontology for the Quantum World

491

• The possible results of measurement of the ob-

servable

H are 1 ± α.

• The Schr

odinger equation:

d|ψ

H|ψ

i (3)

• If the state at some time t = 0 is |ψ

i =





then

at some later time t the state will be:

|ψ

i = e

−it/~



cos(αt/~)

sin(αt/~



(4)

(solution of equation (3))

• The probability of obtaining result ~/2 (corre-

sponding to state | ↑i) in state (4) is given by:

Prob(S

;|ψ

i) =

h↑ |ψ

= cos

αt

(5)

Similarly:

Prob(S

= −

;|ψ

i) =

h↓ |ψ

= sin

αt

(6)

• The expected result of S

in the state (4):

i =

sin

2αt

(7)

• The uncertainty in state (4):



cos

2αt



(8)

• The commutator of

and







i 0

0 −i



= i~

(9)

We can see that the above description of the quantum

system contains formulas of two kinds:

A. expression = numerical value (depending on a

certain parameter)

B. expression 1 = expression 2

Examples of type A formulas are: (5), (6), (7), (8).

Examples of type B formulas are: (1), (2), (3), (4) (9).

Below we accept the following convention: for for-

mulas of type A, the use of a formula number #n

means the reference to a numerical value on the right

side of the formula. For formulas of type B, the use of

a formula number #n means the reference to the whole

formula. For example, by #6 we refer to the value

sin

αt

, by #1 we refer to the formula σ



0 −i

i 0



The following symbols appear in the description

of our quantum system:

H,C

,1 + α, 1 − α,







−i



,|ψ

i,|ψ

,−

Below we use the following simpliﬁed (text) versions

of these symbols:

#H,#Cˆ2,#S_x,#S_y,#S_z,#1+alpha,#1-alpha,

#(1_i),#(1_-i),#psi_0,#psi_t,

#hbar_div_2,#-hbar_div_2

Using the vocabulary from our simple ontology, we

can describe the quantum system presented above as

follows:

Hamiltonian(#H)

Observable(#S_x),Observable(#S_y)

Observable(#S_z)

hasMeasurementR(#H,#1+alpha)

hasMeasurementR(#H,#1-alpha)

correspondsToEigenV(#1+alpha,#(1_i))

correspondsToEigenV(#1-alpha,#(1_-i))

SchroedingerEq(#3)

solOfSchroedinger(#4,#3)

timeEvolution(#psi_t,#psi_0,#H)

meanValue(#S_x,#psi_t,#7)

commutator(#S_z,#S_x,#S_y)

isElementOf(#psi_0,#Cˆ2)

isElementOf(#psi_t,#Cˆ2)

probabilityOfR(#5,#hbar_div_2,#4,#S_z)

probabilityOfR(#6,#-hbar_div_2,#4,#S_z)

uncertainty(#8,#S_x,#psi_t)

Exemplar inferences:

StateSpace(#Cˆ2),HilbertSpace(#Cˆ2)

Observable(#H),SAOperator(#H)

State(#psi_0),State(#psi_t)

Real(#1+alpha),Real(#1-alpha)

State(#4), SchroedingerEq(#3)

SAOperator(#S_x)

SAOperator(#S_y)

SAOperator(#S_z)

hasEigenVal(#H,#1+alpha)

hasEigenVal(#H,#1-alpha)

The ontology proposed in this paper is very sim-

ple and require further work. However, the ontology

is enough to show that terms from it can be used to

create semantic enhancements for scientiﬁc publica-

tions on QM. There is a vast amount of literature on

semantic publishing and semantic enhancements of

scientiﬁc publications (Shotton et al., 2009; Shotton,

2009). In general, the enhancements facilitate the in-

tegration of data and knowledge between articles. Be-

low, we show that the terms from the proposed ontol-

ogy can be used to create RDF (Resource Description

Framework)

statements. For this purpose, the on-

tology should be represented in OWL (Web Ontology

Language)

language. For reasons of space, we do not

address the issue in this paper. It was preliminarily

discussed in our previous paper (Skulimowski, 2010).

https://www.w3.org/RDF/

https://www.w3.org/OWL/

KEOD 2019 - 11th International Conference on Knowledge Engineering and Ontology Development

492

Moreover, we need a method of assigning URIs (Uni-

form Resource Identiﬁers) to entities from scientiﬁc

papers. For this purpose, we can use the URL of a

paper and a local name of an entity from the paper:

articleURL#LocalName (Skulimowski, 2014). Be-

cause all the entities considered in the above exam-

ple come from the same publication, we below omit

articleURL. For simplicity, we also omit the preﬁx

related to the namespace of the proposed ontology.

Below, we present examples of RDF statements using

relations from the proposed ontology.

• Unary relations allow describing types of entities

from scientiﬁc papers on quantum mechanics. For

example

<#S_x> a :Observable .

<#H> a :Hamiltonian .

<#3> a :SchroedingerEq .

• Binary relations allow representing relations be-

tween entities from scientiﬁc publications. For

example:

<#psi_0> :isElementOf <#Cˆ2> .

<#H> :hasMeasurementR <#1-alpha> .

<#4> :solOfSchroedinger <#3> .

• N-ary relations, where N > 2 - it turns out

that many important concepts of quantum me-

chanics correspond to relations which link

more than two entities e.g.: meanValue(x,y, z),

commutator(x,y,z). In OWL n-ary relation can be

represented as classes with n properties. Instances

of such class correspond to instances of the rela-

tion (W3C, 2006). For example, the commutator

relation we can represent as follows:

_:c a :Commutator;

:element1 <#S_x> ; :element2 <#S_y> ;

:value <#S_z> .

The mean value relation:

_:m a :MeanValue; :ofObservable <#S_x> ;

:inState <#psi_t> ;

:value <#7> .

Please note that we need several new relations to

implement the above two relations formally.

5 FINAL REMARKS

In this short paper, we have proposed a method for

obtaining a semantic ontology for the quantum world.

The method uses the fact that the full knowledge we

Throughout this paper, we use Turtle syntax for RDF

(https://www.w3.org/TR/turtle/).

have about the quantum level we owe to quantum

mechanics (Heller, 1994a). Consequently, we pro-

pose to create an ontology from the univers de dis-

cours of this theory deﬁned as the collection of ob-

jects, relationships, sets and so on which are presup-

posed by the mathematical structure used in the the-

ory (Heller, 1994a). An ontology obtained by using

the proposed method is not unique because quantum

mechanics can be formulated using various mathe-

matical structures. Ontologies of QM corresponding

to these structures can be used to create semantic en-

hancements and RDF links between entities from sci-

entiﬁc papers using various formalisms. Moreover, it

is worth to mention that the ontologies obtained us-

ing the proposed method do not face the problem of

hypothetical entities pointed out by critics of realism-

based approaches in natural sciences (Dumontier and

Hoehndorf, 2010). Such entities, even if only hypo-

thetical, can be usually represented somehow in the

mathematical formalism of the theory.

We have applied the proposed approach to the Hilbert

space formulation of QM and presented a toy ontol-

ogy based on this formalism. The reason for choos-

ing this formulation was that most of the articles on

QM use it. However, taking into account the goal we

want to achieve (create an ontology as a set of axioms

in a FOL language), the more natural candidate for

creating an ontology is another ”formulation” of QM

namely called yes-no measurements (Mielnik, 1968).

The point is that yes-no measurements possess spe-

ciﬁc properties analogous to those of logical systems.

That is why their set is called quantum logic.

Future studies should focus on the development

of a more mature ontology for Hilbert space formu-

lation of QM and other formalisms of QM (e.g. C*-

algebraic approach, Feynmann’s approach). The toy

ontology presented in this paper is very simple and

contains only selected terms from the Hilbert space

formulation of QM. If we want to create more precise

semantic enhancements for advanced publications on

QM, we have to broaden the set of relations and ax-

ioms in the ontology. We believe that to create a use-

ful ontology of quantum mechanics (used, for exam-

ple, in creating semantic enhancements), the explicit

formalization of this theory is not required. Never-

theless, the axiomatization of quantum mechanics re-

mains an interesting research problem.

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