A Mathematical Condition for a Sublattice of a Propositional System to Represent a Physical Subsystem, with a Physical Interpretation

We display three equivalent conditions for a sublattice, isomorphic to a ~ ( ~ ) , of the propositional system ~(JC) of a quantum system to be the representation of a physical subsystem (see [1 ] ). These conditions are valid for dim ~ ~> 3. We prove that one of them is still necessary and sufficient if dim ~ < 3. A physical interpretation of this condition is given. I N T R O D U C T I O N Working in the propositional formalism, we studied in [2] the simultaneous description of two physical systems. On this base we gave in [1 ] a characterization of the propositional system (PROP) of a physical subsystem, when considered as a sublattice of the PROP of the big system. All this was done for both classical and quantum systems. For quantum systems in particular, we showed that a connection exists with results proven in [5]. Using again results proven in [5], we give here three different necessary and sufficient conditions for a sub-PROP of a PROP ,~(~) to represent a physical subsystem. The theorems of [5] we shall use here are only valid, however, when the sub-PROP contains at least three orthogonal atoms; hence our conditions are necessary and sufficient only under this restriction. Discussing the case where the restriction does not hold, we find that one of the three conditions is still necessary and sufficient. We have thus a generally valid criterion to decide whether a sub-PROP represents a physical subsystem. Moreover, a close scrutiny of this criterion leads to a physical interpretation which is completely consistent with our search for physical subsystems. We review some definitions and previous results in Section 1. In Section 2, we give the three equivalent conditions; Section 3 contains the discussion of the case where the sub-PROP does not contain three orthogonal atoms. Finally, a physical interpretation of the one condition valid in the two cases is given in Section 4. *Wetenschappelijke medewerkers bij het Interuniversitair Instituut Poor Kernwetenschappen (in het kader van navorsingsprogramma 21 EN). Letters in Mathematical Physics 3 (1979) 19 27. 0377 9017/79/0031-0019 $00.90. Copyright 9 1979 by D. Reidel Publishing Company, Dordrech t, Holland, and Boston, U.S.A. 19 1. DEFINITIONS AND PREVIOUS RESULTS In the propositional formalism, a physical system is described by the propositional system (PROP) of its properties, i.e. by a complete, orthocomplemented, weakly modular, atomic lattice satisfying the covering law [3]. In particular, a pure quantum system is described by an irreducible PROP : no superselection rules exist [3]. Such an irreducible PROP can be shown to be isomorphic to the lattice of all dosed subspaces of a complex Hilbert space [3] 9. (Notation: ~(~C)). To study the joint description of two systems in one PROP, or to characterize sublattices as representing physical subsystems, we need a tool to compare different PROP's. This tool are the c-morphisms: maps from one orthocomplemented, complete lattice to another, which preserve the structure [3]. In the case where c-morphisms from one ~ ( ~ ) to a ~ Q f ' ) are studied, one can consider a special class of c-morphisms, namely the m-morphisms (see [4], [6]). An m-morphism f from ~(~f ) to ~ ( ~ ( ' ) is a c-morphism for which f(2) + f(~) is a dosed subspace of ~C' for any x, y in 3s (we use the symbol ~ to denote the closed subspace Irx). One can show that any such m-morphism f i s generated by a family of isometric maps (r with orthogonal images: W E J : r 3s -* Jr ' isometric