Quantum logics derived from asymmetric Mielnik forms (Q798653)

Usual axiomatizations of quantum mechanics use the set of all states. The author presents a new axiomatics using only the set M of pure states and a function \(\to:M\times M\to [0,1]: (1)\quad p\to p=1, (2)\quad if\quad p\to q=0\) then \(q\to p=0\) (such p, q are called orthogonal), \((3)\quad if\quad (p_ i)\) is a maximal set of orthogonal elements, then \(\sum_{i}(p\to p_ i)=1\) for all \(p\in M\). He shows that such (M,\(\to)\) structure allows to reconstruct a logic for which M is a set of pure states, i.e., such a subset of the set of all states that \((1)\quad if\quad pa=1\) implies \(pb=1\) for all \(p\in M\), then \(a\leq b\), \((2)\quad for\) every p, \(L_ p=\inf\{a| pa=1\}\) exists, and \(p\to q\) coincides with the Mielnik form \(p(L_ q)\). Vice versa, for every quantum logic, the Mielnik form satisfies the author's axioms.