Abstract
The investigation of transition amplitude spaces (tas’s) introduced by Gudder and Pulmannová [J. Math. Phys. 28, 376 (1987)] is continued. In particular, ordered structures related to tas’s are considered. Under some conditions, which are analogous to the conditions obtained for transition probability spaces by Pulmannová [J. Math. Phys. 27, 1791 (1986)], the ordered structure related to a tas can be represented by the orthocomplemented lattice of all f-closed subspaces of a generalized Hilbert space (𝒱,𝒟,θ, f ). It is shown that, provided that the above representation takes place for a total tas, the division ring 𝒟 must be isomorphic with a subfield C1 of the field of complex numbers C. Sufficient conditions are also given under which the ordered structure of a tas can be represented by the lattice of all closed subspaces of a complex Hilbert space.
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