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Helvetica Physica Acta, Vol. 51 (1978), Birkhuser Verlag, Basel About the structure -preserving maps

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 c-morphisms from one Hilbert space lattice (with dimension at least three) to
another one; we show that for a c-morphism 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 c-morphism which preserves the atoms (i.e. maps one-
dimensional subspaces into one-dimensional 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 'yes-no' 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 super-selection rules, the propositional system is
irreducible. We will first give some definitions concerning propositional systems. For


Source: Aerts, Diederik - Leo Apostel Centre, Vrije Universiteit Brussel


Collections: Multidisciplinary Databases and Resources; Physics