Authors: J.A.J. van Leunen
Hilbert spaces can store discrete members of division rings and continuums that are formed by members of division rings in the eigenspaces of operators that reside in these Hilbert spaces. Only three suitable division rings exist. These are the real numbers, the complex numbers and the quaternions. The reverse bra-ket method is an extension of the bra-ket notation that was introduced by P.M. Dirac. The reverse bra-ket method can create natural parameter spaces from number systems that are division rings and can relate the combinations of functions and their parameter spaces with eigenspaces and eigenvectors of corresponding operators that reside in non-separable Hilbert spaces. This also works for separable Hilbert spaces. The defining functions relate the separable Hilbert space with its non-separable companion. In this way, the method links Hilbert space technology with function technology, differential technology and integral technology. Quaternionic number systems exist in several versions that differ in the way that they are ordered. This is applied by defining multiple types of parameter spaces in the same Hilbert space. The set of closed subspaces of a separable Hilbert space has the relational structure of an orthomodular lattice. This fact makes the Hilbert space suitable for modelling quantum physical systems. The reverse bra-ket method is a powerful tool for generating quaternionic models that help investigating quantum physical models.
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