Mathematical Foundations of Quantum Mechanics 1312.3.2 The non-Boolean logic of QM, the reason why observables
are selfadjoint operators
It is evident that the classical like picture illustrated in Sect. 2.3.1 is untenable if
referring to quantum systems. The deep reason is that there is pair of elementary
propertiesE,Fof quantum systems which are incompatible. Here an elementary
property is an observable which, if measured by means of a corresponding experi-
mental apparatus, can only attain two values: 0 if it is false or 1 if it is true. For
instance,E=“thecomponentSxof the electron is/2” andF=“thecomponent
Syof the electron is/2”. There is no physical instrument capable to establish if
EANDFis true or false. We conclude that some of elementary observables of
quantum systems cannot be logically combined by the standard operation of the
logic. The model of Borelσ-algebra seems not to be appropriate for quantum sys-
tems. However one could try to use some form of lattice structure different form
the classical one. The fundamental ideas by von Neumann were the following pair.
(vN1)Given a quantum system, there is acomplex separable Hilbert space
Hsuch that theelementary observables– the ones which only assume values
in{ 0 , 1 }– are one-to-one represented by all the elements ofL(H), the orthogonal
projectors inB(H).
(vN2)Two elementary observablesP,Qare compatible if and only if they
commute as projectors.
Remark 2.3.7.
(a)As we shall see later (vN1) has to be changed for those quantum systems
which admitsuperselection rules. For the moment we stick to the above version of
(vN1).
(b)The technical requirement of separability will play a crucial role in several
places.
Let us analyze the reasons for von Neumann’s postulates. First of all we observe
thatL(H) is in fact a lattice if one remembers the relation between orthogonal
projectors and closed subspaces stated in Proposition 2.2.46.
Notation 2.3.8.Referring to Proposition 2.2.46, ifP,Q∈L(H), we writeP≥Q
if and only ifP(H)⊃Q(H).
P(H)⊃Q(H) is equivalent toPQ=Q. Indeed, ifP(H)⊃Q(H) then there is
a Hilbert basis ofP(H)NP =NQ∪NQ′ whereNQia a Hilbert basis ofQ(H)
andNQ′ ofQ(H)⊥P, the notion of orthogonal being referred to the Hilbert space
P(H). FromQ=
∑
z∈NQ〈z,·〉zandP =Q+∑
z∈NQ′〈z,·〉zwe havePQ=Q.