 
Summary: Helvetica Physica Acta, Vol. 51 (1978), Birkhäuser Verlag, Basel
About the structure  preserving maps
of a quantum mechanical
propositional system
by Dirk Aerts1) and Ingrid Daubechies1)
Theoretische Natuurkunde, Vrije Universiteit Brüssel, Pleinlaan 2, B 1050 Brussels
(2. II. 1978; rev. 26. VII. 1978)
Abstract. We study cmorphisms from one Hilbert space lattice (with dimension at least three) to
another one; we show that for a cmorphism conserving modular pairs, there exists a linear structure
underlying such a morphism, which enables us to construct explicitly a family of linear maps generating this
morphism. As a special case we prove that a unitary cmorphism which preserves the atoms (i.e. maps one
dimensional subspaces into onedimensional subspaces) is necessarily an isomorphism. Counterexamples
are given when the Hilbert space has dimension 2.
1. Definition of a propositional system and Piron's representation theorem
According to Piron's axiomatic description of quantum mechanics [1], the
structure of the set of the propositions corresponding to 'yesno' experiments on a
physical system is that of a complete, orthocomplemented, weakly modular and
atomic lattice which satisfies the covering law. Such a lattice is called a propositional
system. If the physical system has no superselection rules, the propositional system is
irreducible. We will first give some definitions concerning propositional systems. For
