Mathematical Foundations of Quantum Mechanics 133Proof. L 0 includes both 0 andIbecauseL 0 is maximally commutative. Having
(i) and (ii), due to (iii) in proposition 2.3.9, the sup and the inf of a sequence
of projectors ofL 0 commute with the elements ofL 0 , maximality implies that
they belong toL 0. Finally (i) and (ii) prove by direct inspection that∨and
∧are mutually distributive. Let us prove (ii) and (i) to conclude. IfPQ=
QP,PQis an orthogonal projector andPQ(H)=QP(H)⊂ P(H)∩Q(H).
On the other hand, ifx∈P(H)∩Q(H)thenPx=xandx=Qxso that
PQx=xand thusP(H)∩Q(H)⊂PQ(H) and (ii) holds. To prove (i) observe
that<P(H),Q(H)>
⊥
=P(H)⊥∩Q(H)⊥. Using (ii), this can be rephrased as
I−P∨Q=(I−P)(I−Q) which entails (i) immediately.
Remark 2.3.12.
(a)Every set of pairwise commuting orthogonal projectors can be completed to
a maximal set as an elementary application of Zorn’s lemma. However, since the
commutativity property isnottransitive, there are many possible maximal subsets
of pairwise commuting elements inL(H)with non-empty intersection.
(b)As a consequence of the stated proposition, the symbols∨,∧and¬have
thesamepropertiesinL 0 as the corresponding symbols of classical logicOR, AND
andNOT. MoreoverP≥Qcan be interpreted as “QIMPLIESP”.
(c)There has been many attempts to interpret∨and∧as connectives of a new
non-distributive logic when dealing with the wholeL(H):aquantum logic.The
first noticeable proposal was due to Birkhoff and von Neumann[ 14 ]. Nowadays
there are lots of quantum logics[15; 16]all regarded with suspicion by physicists.
Indeed, the most difficult issue is the physical operational interpretation of these
connectives taking into account the fact that they put together incompatible propo-
sitions, which cannot be measured simultaneously. An interesting interpretative
attempt, due to Jauch, relies upon a result by von Neumann (e.g.,[ 5 ])
(P∧Q)x= limn→+∞(PQ)nx for everyP,Q∈L(H)andx∈H.Notice that the result holds in particular ifPandQdo not commute, so they are
incompatible elementary observables. The right hand side of the identity above
can be interpreted as the consecutive and alternated measurement of an infinite
sequence of elementary observablesP andQ.As
||(P∧Q)x||^2 = limn→+∞||(PQ)nx||^2 for everyP,Q∈L(H)andx∈H,the probabilty thatP∧Qis true for a state represented by the unit vectorx∈H
is the probabilty that the infinite sequence of consecutive alternated measurements
ofP andQproduce is true at each step.