136 From Classical Mechanics to Quantum Field Theory. A Tutorial
P∧Qfor everyP,Q. There are several proposals, very different in nature, to
introduce this lattice structure (see[ 15 ]and[ 16 ]for a general treatise) and make
the physical meaning explicit in terms of measurement outcome. See Aerts in[ 16 ]
for an abstract but operational viewpoint and Sect. 21.1 of[ 15 ]for a summary on
several possible ways to introduce the lattice structure on the partially ordered set
of abstract elementary propositionsL.
If we accept the lattice structure on elementary propositions of a quantum
system, then we may define the operation of orthocomplementation by the familiar
logical/physical negation. Compatible propositions can then be defined in terms of
commuting propositions, i.e. commutingelements of a orthocomplemented lattice
as follows.
Definition 2.3.14. Let(L,≥,¬)an orthocomplemented lattice. Two elements
a, b∈Lare said to be:
(i)orthogonalwrittena⊥b,if¬a≥b(or equivalently¬b≥a);
(ii)commuting,ifa=c 1 ∨c 3 andb=c 2 ∨c 3 withci⊥cjifi=j.These notions of orthogonality and compatibility make sense beacuse,a posteriori,
they turn out to be the usual ones when propositions are interpreted via projectors.
As the reader may easilyprove, two elementsP,Q∈L(H) are orthogonal in
accordance with Definition 2.3.14 if and only ifPQ=QP = 0 (in other words
they project onto mutually orthogonal subspaces), and commute in accordance
with Definition 2.3.14 if and only ifPQ=QP.(IfP=P 1 +P 3 andQ=P 2 +P 3
where the orthogonal projectors satisfyPi⊥Pj=0fori=j, we trivially have
PQ=QP.Ifconversely,PQ=QP, the said decomposition arises forP 3 :=PQ,
P 1 :=P(I−Q),P 2 :=Q(I−P).)
Now fully-fledged with an orthocomplemented lattice and the notion of com-
patible propositions, we can attach a physical meaning (an interpretation backed
by experimental evidence) to the requests that the lattice be orthocomplemented,
complete, atomistic, irreducible and that it have the covering property[ 15 ]. Under
these hypotheses and assuming there exist at least 4 pairwise-orthogonal atoms,
Piron ([18; 19], Sect. 21 of[ 15 ],Aertsin[ 16 ]) used projective geometry techniques
to show that the lattice of quantum propositions can be canonically identified
with the closed (in a generalized sense)subsets of a generalized Hilbert space of
sorts. In the latter: (a) the field is replaced by a division ring (usually not com-
mutative) equipped with an involution, and (b) there exists a certain non-singular
Hermitian form associated with the involution. It has been conjectured by many
people (see[ 15 ]) that if the lattice is also orthomodular and separable, the divi-
sion ring can only be picked amongR,CorH(quaternion algebra). More recently