Failure of standard quantum mechanics for the description of compound quantum entities (Q703008)

The paper starts with the State Property Systems formalism for quantum or classical mechanics. A physical entity is associated with a set \(\Sigma\) of states and set \(\mathcal Q\) of ``yes/no experiments''. A relation on \(\Sigma\times \mathcal Q\) is defined such that \((p,\alpha)\) holds when state \(p\) assigns unit probability to experiment \(\alpha\), i.e. \(p\) finds \(\alpha\) certainly true. Equivalence classes of states are generated by this relation -- states equivalent to \(p\) assign certain truth to the same experiments that \(p\) does. Properties correspond to experiments in the sense that a property is considered ``actualised'' in a physical entity described by state \(p\), if the corresponding experiment is certain in \(p\). A mapping \(k:\mathcal L \to \mathcal P(\mathcal Q)\) takes each property in \(\mathcal L\) to the subset of states in which it is ``actualised''. This induces a partial ordering of properties, \(a < b \Leftrightarrow k(a) \subset k(b)\). Lastly the triplet \((\Sigma,\mathcal L , k)\) is defined as a State Property System, for \(\Sigma\) a set, \(\mathcal L\) a complete lattice, and \(k\) a structure preserving mapping that preserves the 1 and 0. In the case of classical mechanics, \(\Sigma\) is phase space \(\Omega\), \(\mathcal L\) the set \(\mathcal P(\Omega)\) of subsets of phase space and \(k\) the identity mapping. For standard quantum mechanics, \(\Sigma\) is the set of rays of Hilbert space \(\mathcal H\), \(\mathcal L\) the set of closed subspaces \(\mathcal P(\mathcal H)\), and \(k\) the mapping that takes each closed subspace to the set of rays contained in it. Two axioms ensure the structures are appropriate for classical and quantum mechanics. Axiom 1 ensures each state is determined by the properties ``actualised'' in the state, Axiom 2 requires states correspond to atoms in the lattice of properties. Three more axioms are added ``because they are satisfied in the lattice of closed subspaces of a complex Hilbert Space'' (p. 254). Axiom 3 asserts orthocomplementation \('\), Axiom 4 introduces the covering law, and axiom 5, orthomodularity on the lattice of properties. Both classical and quantum entities satisfy all five axioms. Classical properties are distinguished as those \(a\) in \(\mathcal L\) for which \(p\in k(a)\) or \(p\in k(a)'\) for all \(p\) in \(\Sigma\). It is noted that quantum systems have no non-trivial classical properties. Section 3 discusses direct unions of State Property Systems, with Representation Theorems. Separated physical entities are defined in terms of joint experiments that determine outcomes for separated systems. Finally the title issue of this paper is addressed. The major theorem here establishes that a product of two separated systems meets the requirements of all five axioms only if one of the constituent State Property Systems is classical. The author finds: ``A classical entity that is separated from a quantum entity, and two separated classical entities do not cause any problem, but two separated quantum entities need a structure where neither the covering law nor weak modularity are satisfied'' (p. 262). The implications are briefly discussed, with argument for more general structures that do not satisfy covering and orthomodularity axioms. The authors end with the comment: ``All this convinces us that the shortcoming of standard quantum mechanics to be able to describe separated quantum entities is really a shortcoming of the mathematical formalism used by standard quantum mechanics, and more notably of the vector space structure of the Hilbert space used in standard quantum mechanics.'' Many earlier papers by these authors are cited in the References.