From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 135

P∈L(H)\{ 0 }there exists an atomAwithA≤P (L(H)is then called

atomic); For everyP ∈L(H)\{ 0 },P is thesupof the set of atoms

A≤P (L(H)is then calledatomistic);

(iii)orthomodularity:P≤QimpliesQ=P∨((¬P)∧Q);

(iv)covering property:ifA, P∈L(H),withAan atom, satisfyA∧P=0,

then (1)P ≤A∨PwithP=A∨P,and(2)P ≤Q≤A∨P implies

Q=PorQ=A∨P;

(v)irreducibility:only0andIcommute with every element ofL(H).

The orthogonal projectors onto one-dimensional spaces are the only atoms ofL(H).

Irreducibility can easily be proven observing that ifP∈L(H) commutes with all
projectors along one-dimensional subspaces,Px=λxxfor everyx∈H.Thus
P(x+y)=λx+y(x+y) but alsoPx+Py=λxx+λyyand thus (λx−λx+y)x=
(λx+y−λy)y, which entailsλx=λyifx⊥y.IfN ⊂His a Hilbert basis,
Pz=

∑

x∈N〈x, z〉λx =λzfor some fixedλ∈C.SinceP =P†=PP,we
conclude that eitherλ=0orλ= 1, i.e., eitherP =0orP =I,aswanted.
Orthomodularity is a weaker version of distributivity of∨with respect to∧that
we know to be untenable inL(H).
Actually each of the listed properties admits a physical operational interpre-
tation (e.g. see[ 15 ].) So, based on the experimental evidence of quantum sys-
tems, we could try to prove, in the absence of any Hilbert space, that elementary
propositions with experimental outcome in{ 0 , 1 }form a poset. More precisely,
we could attempt to find a bounded, orthocomplementedσ-complete lattice that
verifies conditions (i)–(v) above, and then prove this lattice is described by the
orthogonal projectors of a Hilbert space.
The partial order relation of elementary propositions can be defined in various
ways. But it will always correspond to the logical implication, in some way or
another. Starting from[ 17 ]a number of approaches (either of essentially physical
nature, or of formal character) have been developed to this end: in particular,
those making use of the notion of (quantum)state, which we will see in a short
while for the concrete case of propositions represented by orthogonal projectors.
The object of the theory is now[ 17 ]the pair (O,S), whereOis the class of
observables andSthe one of states. The elementary propositions form a subclass
LofOequipped with a natural poset structure (L,≤) (also satisfying a weaker
version of some of the conditions (i)–(v)). A states∈S, in particular, defines
the probabilityms(P)thatP is true for everyP ∈L[ 17 ]. As a matter of fact,
ifP,Q∈L,P≤Qmeans by definition that the probabilityms(P)≤ms(Q)for
every states∈S. More difficult is to justify that the poset thus obtained is a
lattice, i.e. that it admits a greatest lower boundP∨Qand a least upper bound