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International Journal of Theoretical Physics

Models of the axiomatic theory associated with the lattice of subspaces of a finite-dimensional Hilbert space

Othman Qasim Malhas

Published Aug 1, 1995• 0 citations
Publisher
DOI
10.1007/bf00676266

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A set of statements about the properties of a quantum system is looked at as at a partially ordered set of subspaces of finite or infinite dimensional Hilbert space. The operation of ordering is performed on a set of propositions comparing the truth values of these propositions and on the set of subspaces as the opera- tion of inclusion. Based on the required properties these structures are translated into operations on the lattice. The correspondence with Heisenberg uncertainty principle is shown there. Furthermore, it is shown that the lattices correspond- ing to the subspaces of infinite dimensional Hilbert space are not modular. This property is replaced with weaker property of orthomodularity, when operation of the negation is added. Following the work of G. Birkhoff and J. von Neumann, the structure of quantum logic is looked for in projective spaces, which are in- troduced either arithmetically or axiomatically. The examples of quantum logic, their physical implementation and eventual implementation in projective spaces are analysed. 1
Von Neumann algebras associated with the normal representation of canonical commutation relations are studied. Corresponding to each subspace of a real Hilbert space (test function space), a von Neumann algebra on another complex Hilbert space (the Fock space) is defined. This correspondence is proved to be an isomorphism between a certain complemented lattice of subspaces and that of the von Neumann algebras. This result has an application to the duality theorem in the theory of a free scalar field, which is to be discussed in a separate paper. A necessary and sufficient condition on a subspace, in order that the corresponding von Neumann algebra is of type I, is obtained.
Given a complete orthocomplemented lattice L and a set S of nonnegative real functions on L, sufficient conditions are established that S should fulfill in order that L be atomic. The conditions are investigated under which L may be represented by the lattice of all closed subspaces of a separable Hilbert space. (As is well known, the atomicity of L plays an important role here.) Some unsolved problems are pointed out. In axiomatic quantum mechanics, the lattice L may represent the set of propositions whereas the set of functions S represents the set of physical states. The conditions imposed on the pair (L,S) then have a simple and plausible physical interpretation; an important condition imposed on (L,S) is the existence of the ``maximal'' (i.e., maximally determined) states which appear in the theory as limit constructions.