An OWL ontology for quantum mechanics is presented in short. The motivation for its development, structure and characteristic features are depicted. In particular, some essential concepts from the ontology are described. Finally, some problems encountered during the development of the ontology are discussed.
The main aim of the Semantic Web is to make the information on the World Wide
Web more accessible to machines [
Our aim is to build an OWL (Ontology Web Language)[
After building such a preliminary ontology we may try to take into account, omitted previously, mathematical details of the theory. It turns out, however, that in many cases the details cannot be modeled in OWL (see Section 4). As a consequence an ontology obtained in this way will model quantum mechanics only approximately. Nevertheless, we think that it will be useful in applications. 3
quONTOm A preliminary version of an OWL ontology for quantum mechanics named quON
ogy is written using Protege 4.1 ontology editor.
Actually the ontology is contained in one OWL document. Aside from quantum mechanical concepts, the ontology contains concepts which are not parts of quantum mechanics, i.e. they are, for example, purely mathematical objects and should be contained in an ontology for mathematics. Unfortunately, to our best knowledge, an appropriate ontology for mathematics, which could be imported and used in quONTOm does not exist. The situation is similar for physical concepts. In the future the concepts will be separated from the ontology and become parts of auxiliary ontologies.
The ontology is at the moment incomplete and will be gradually developed towards a more complete form. In spite of this we may try to say something about the structure and the characteristic features of the ontology. 3.1
According to the de nition, an ontology provides shared vocabulary used to
describe entities in some domain of interest [
There are various kinds of objects in quantum mechanics. Some of them are purely mathematical objects (e.g. linear operators), some objects really exist (e.g. particles). The objects can be grouped in classes. It turns out however that it is very often impossible to establish subsumption relation between classes. As a consequence the class hierarchy of the ontology will be rather at. However, the ontology will be rich in other relations. 3.3
In OWL, a property is a binary relation. Instances of properties link two
individuals. However, in quantum mechanics we meet not only binary relations
but also n-ary relations. It turns out that many important concepts of quantum
mechanics correspond to n-ary relations where n > 2. Consider, for example,
the following statement which can be found in any book on quantum mechanics:
commutator of operators O1 and O2 is equal to 0. This statement falls under
the category of 3-ary relations. In general an n-ary relation can be represented
in OWL as a class with n properties [
In the case of quantum evolution a 5-ary relation is needed. The evolution of a state (initalState) is generated by (generatedBy) a Hamiltonian H, starts at t1 (startTime) and ends at t1 (endTime). The result of the evolution is a state ( nalState). Manchester syntax:
finalState exactly 1 State initialState exactly 1 State isGeneratedBy exactly 1 Hamiltonian endTime exactly 1 xsd:float startTime exactly 1 xsd:float
Other fundamental concepts are also represented by n-ary relations (where n > 2), for example quantum measurement. It seems that as the ontology will be developed the number of such relations will increase. 3.4
Relations between concepts in quantum mechanics determine characteristics of corresponding properties in the ontology. We observe that there is a numerous set of functional properties in the ontology e.g. hasAdjoint - a property relating an operator to its adjoint, isRepresentedByOperator - a property relating an observable to the corresponding self-adjoint operator, startTime - a property relating some initial time to the evolution. There are also numerous classes of objects that have exactly 1 value for some property (e.g. SelfAdjointOperator, Measurement ). By contrast transitive and symmetric properties are rather rare. This is mainly because, in quantum mechanics binary relations between objects of the same kind are rare (e.g. a commutation relation between two operators). It seems that the development of the ontology will not signi cantly change this situation. We also note that the ontology does not contain a merological relation a partOf. This is due to the speci c domain of the ontology. In the case of objects existing (and extended) in space a partOf relation seems to be very natural. For the case of mathematical objects which are not placed in space it is di cult to talk about such a relation. Admittedly, in the ontology a relation isElementOf is de ned. However, the relation is asymmetric and irre exive but not transitive: a state isElementOf a Hilbert space, the Hilbert space isElementOf a family of Hilbert spaces but it is not true that the state isElementOf the family of Hilbert spaces. Finally, it is worth to notice that the majority of relations in the ontology are those which are asymmetric and irre exive.
During the development of the ontology it turned out that some relationships between concepts in quantum mechanics cannot be represented in OWL 1. To represent them we have to use OWL 2. For example, we can de ne properties as a composition of other properties. Thanks to this we are able to de ne a result of a measurement as an eigenvalue of a self-adjoint operator representing some observable.
In a similar way, we de ne an object property corresponding to a quantum mechanical state reduction. And these are not the only bene ts of using OWL 2.
It happens very often in mathematics that a property of objects belonging to some set is used to de ne some subset of objects. For example, in a set of matrices we may consider a property transpose which assign the transpose matrix to a matrix. Using the property we can de ne a symmetric matrix as a matrix that is equal to its transpose. For linear operators on a Hilbert space we consider a property adjoint assigning the adjoint to an operator. An operator is self-adjoint if it is equal to its own adjoint. In order to manage such a de nition in quONTOm we use a self restriction o ered by OWL 2. First, we introduce a hasAdjoint object property and then we de ne a class of self-adjoint operators (SelfAdjointOperator ) as linear operators that are related to themselves via the hasAdjoint property. Manchester syntax:
In quantum mechanics there are also objects which can be uniquely identi ed by some set of attributes. For example, a complex number is uniquely identi ed by its real and imaginary parts. In such cases we can use keys available in OWL 2.
OWL 2 supports restrictions of datatypes by facets, as in XML Schema. Thanks to this we can de ne new datatypes. In quONTOm we de ne a datatype probValue by constraining the datatype oat to values between (inclusively) 0 and 1. The datatype is very important in quONTOm taking into account that quantum mechanics is probabilistic in nature. 4
In this paper we have brie y described an OWL ontology for quantum mechanics. It seems that OWL provides means to model the main conceptual issues that arise in the theory. Problems arise when we try to express in OWL (and even in OWL 2) mathematical details of the theory. Let us give some examples.
Starting to build the ontology we have encountered a problem with complex numbers which are ubiquitous in standard formulation of quantum mechanics (e.g. probability amplitudes are complex numbers, they also occur in the timedependent Schrodinger equation). One may try to de ne complex numbers as a new datatype. However, from the point of view of quantum mechanics it is important that real numbers ( oat) are subsumed in complex numbers. In order to express the subsumption we consider complex numbers not as a datatype but as a class (ComplexNumber ). Each individual from the class has two properties realPart and imaginaryPart.
Another problem is related to the representation of an n-ary relations as
classes [
There are many places in the ontology where classes (and relations between them) are de ned mathematically. Let us consider a class of linear operators (LinearOperator ). The de nition of a linear operator is very simple. However, OWL does not provide means to specify it. In these cases mathematical de nitions are included as comments.
In this paper, we have presented an OWL ontology for quantum mechanics. The current version of the ontology covers only a small part of actual paradigm of quantum mechanics. We plan to develop the ontology to a more complete form which will be useful for applications.