Some Results on BZ Structures from. Hilbertian Unsharp Quantum Physics. Gianpiero Cattaneo ~ and Roberto Giuntini 2. Received March 7, 1994; revised April ...
Foundations o f Physics, Vol. 25, No. 8, 1995
Some Results on BZ Structures from Hilbertian Unsharp Quantum Physics Gianpiero Cattaneo ~ and Roberto Giuntini 2 Received March 7, 1994; revised April 24, 1995 Some algebraic structures determined by the class ,~(oct~) o f all effects o f a Hilbert space ,,~ and by some subclasses o f 27(~) are investigated, in particular de Morgan-Brouwer-Zadeh posets [it is proved that Z ( ~ " ) (n < co) has such a structure], Brouwer-Zadeh* posets (a quite trivial example consistbzg of suitable effects is given), and Brouwer-Zadeh 3 posets which are both de Morgan and *. It is shown that a nontrivial class o f effects of a Hilbert space exists which is a BZ 3 poset. An e-preclusivity relation on the set o f all vectors of ogg is introduced, and it is shown that it satisfies the regularity condition also for e e [ 1/2, 1 ].
1. F R O M S H A R P TO U N S H A R P Q U A N T U M M E C H A N I C S O N HILBERT SPACES
The conventional model of the axiomatic approach to the foundations of quantum mechanics (QM) is based on a complex Hilbert space Jr the "logic" of all quantum simple propositions is realized by the set jg(~ug,) of all subspaces (i.e., closed linear manifolds) of ~t~. J r is an orthocomplemented orthomodular atomic complete lattice with respect to the usual set theoretic inclusion _ , bounded by the trivial subspaces {_0} and ~ ; the g.l.b, of any family {Mj} of subspaces, denoted by A Mj, is just the set theoretic intersection (i.e., A Mj = 0 Mj), whereas the 1.u.b. is the subspace, denoted by V Mj, generated by the set theoretic union (i.e., V Mj= (U M j ) " ) ; the orthocomplementation z corresponds to the passage from ~ Dipartimento di Scienze dell'Informazione, Universitfi. di Milano, Via Comelico 39-41, 20135, Milano, Italy. 2 Dipartimento di Filosofia, Universit/t di Firenze, via Bolognese 52, 50139 Firenze, Italy. 1147 0015-9018/95/0800-1147507.50/09 1995 Plenum PublishingCorporation
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Cattaneo and Giuntini
a subspace M to its orthogonal subspace, or annihilator, M • All this can be summarized by the structure P e g , = (dg(.g,), A, V, • {_0}, .Xf)
(1.1)
The collection of all quantum events (tested by suitable yes-no macroscopic apparatuses) is realized by the set//(~vt~) of all orthogonal projections on Jr. The structure (/-/(~), O, ~ such that
(BZ)-poset
(resp., lattice) is a
(a)
for co(x) # 0 otherwise
Theorem 4.1. The sequence of effect operators VnEN, (FA,~o)I/2": ~Vf~ ~vf, implicitly defined by
V~ke~,~,
(~l(FA.oA'/2"~k)=ll~Pll2 IRco~/Vd(pV, oeA)
(4.11)
is monotone not decreasing, i.e.,
| ...-1-eandP(~p2,F)/1 - e and P(0 2, F) ~ is called the fuzzy-intuitionisticpropositionalalgebra on the frame (X, # ). The restrictions to Lc(X, # ) of the two orthocomplementations defined on LflX, # ) coalesce and define a unique orthocomplementation; moreover, the following is a one-to-one correspondence between exact propositions and simple propositions
Le(X, # ) - . / , ( X , #)
(A,A#)
