By Blackburn Ch. M.
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Extra info for An Application of the Quantum Theory of Band Spectra to the First Negative Deslandres Group of Carbon
6. G. Mackey, The Mathematical Foundations of Quantum Mechanics, Benjamin, New York (1963). 7. C. Piron, "Axiomatique quantique", Helv. Phys. Acta 37, 439 (1964). 8. C. , W. A. Benjamin (1976). 9. D. Aerts, "Foundations of quantum physics: a general realistic and operational approach", Int. J. Theor. Phys. 38, 289-358 (1999), lanl archive ref: quant-ph/0105109. 10. D. Aerts, E. Colebunders, A. Van der Voorde and B. Van Steirteghem, "State property systems and closure spaces: a study of categorical equivalence", Int.
The two examples that we propose here are the archetypical examples for quantum axiomatics. The first example represents (part of) the mathematical structure related to a classical physical entity, where Q corresponds with its state space, and the second example the one of a quantum physical entity, where Ji is the complex Hilbert space representing the states of the quantum entity. A lattice is a mathematical object that has some more structure than just this partial order relation. Suppose we consider two elements a, 6 6 £ of the lattice, then we demand that there exists an infimum and a supremum for a and b for the partial order relation < in £.
The triple (E, £, K), where E is a set, £ a complete lattice, and K a map satisfying (9), (10), and (11) has been called a state-property system 9 ' 10 , and it is this mathematical structure that can be derived from the operational aspects of the axiomatic approach. Let us consider our two examples and see how this structure appears there. For the quantum case, E is the set of rays of the Hilbert space and £ the set of closed subspaces. The Cartan map maps each closed subspace on the set of rays that are contained in this closed subspace, and indeed (9), (10), and (11) 27 are satisfied.