 
Summary: A Theory of Concepts and Their Combinations II:
A Hilbert Space Representation
Diederik Aerts
Center Leo Apostel for Interdisciplinary Studies
Department of Mathematics and Department of Psychology
Vrije Universiteit Brussel, 1160 Brussels, Belgium
EMail: diraerts@vub.ac.be
Liane Gabora
Center Leo Apostel for Interdisciplinary Studies
Vrije Universiteit Brussel and Department of Psychology
University of California at Berkeley, Berkeley, CA 947201650, USA
EMail: liane@berkeley.edu
Abstract
The sets of contexts and properties of a concept are embedded in the complex Hilbert space of quantum
mechanics. States are unit vectors or density operators, and contexts and properties are orthogonal pro
jections. The way calculations are done in Hilbert space makes it possible to model how context influences
the state of a concept. Moreover, a solution to the combination of concepts is proposed. Using the tensor
product, a procedure for describing combined concepts is elaborated, providing a natural solution to the
`pet fish' problem. This procedure allows the modeling of an arbitrary number of combined concepts. By
way of example, a model for a simple sentence containing a subject, a predicate and an object, is presented.
