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.

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The physical interpretation of (A) and (C) is clear: (A) mneans that for any proposition aεL there exists a state αεS̄ such that a is true in the state α. (C) means that if a implies b, then if a is true in the state αεS̄, b is true in the state α.
31.
Since, in the set of admissible states, a maximal state m is uniquely determined by the maximal set E1(m), the physical situation it describes cannot be further specified by any information. It seems worthwhile to remark that hypothesis (H) of Theorems 2 and 3 is a weakening of the postulate that, to any aεL, there exists a maximal pure state m with m(a) = 1. The latter postulate might be physically interpreted as follows: To any proposition aεL, there exists a set of propositions E1(m) (including a), which represents a maximal specification of a physical situation m of the system that cannot be further refined by any propositions concerning the system [the state m being uniquely determined31 by E1(m) and vice versa].
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(a) Lemma 1, 11; (b) Lemma 1, 12.
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© 1972 The American Institute of Physics.
1972
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