From Classical Mechanics to Quantum Field Theory

132 From Classical Mechanics to Quantum Field Theory. A Tutorial

The converse implication is obvious.
As preannounced, it turns out that (L(H),≥) is a lattice and, in particular, it
enjoys the following properties (e.g., see[ 5 ]) whose proof is direct.

Proposition 2.3.9.LetHbe a complex separable Hilbert space and, ifP∈L(H),
define¬P:=I−P (the orthogonal projector ontoP(H)⊥). With this definition,
(L(H),≥,¬)turns out to be bounded, orthocomplemented,σ-complete lattice which
is not distributive ifdim(H)≥ 2.
More precisely,

(i)P∨Qis the orthogonal projector ontoP(H)+Q(H).

The analogue holds for a countable set{Pn}n∈N⊂P(H),∨n∈NPnis the

orthogonal projector onto+n∈NPn(H).

(ii)P∧Qis the orthogonal projector onP(H)∩Q(H).

The analogue holds for a countable set{Pn}n∈N⊂P(H),∧n∈NPnis the

orthogonal projector onto∩n∈NPn(H).

(iii)The bottom and the top are respectively 0 andI.

Referring to (i) and (ii), it turns out that

∨n∈NPn= limk→+∞∨n≤kPn and ∧n∈NPn= limk→+∞∧n≤kPn

with respect to the strong operator topology.

Remark 2.3.10.The fact that the distributive property does not hold is evident
from the following elementary counterexample inC^2 (so that it is valid forevery
dimension> 1 ). Let{e 1 ,e 2 }be the standard basis ofC^2 and define the subspaces
H 1 :=span(e 1 ),H 2 := span(e 2 ), H 3 :=span(e 1 +e 2 ).FinallyP 1 , P 2 ,P 3
respectively denote the orthogonal projectors onto these spaces. By direct inspection
one sees thatP 1 ∧(P 2 ∨P 3 )=P 1 ∧I=P 1 and(P 1 ∧P 2 )∨(P 1 ∧P 3 )=0∨0=0,
so thatP 1 ∧(P 2 ∨P 3 )=(P 1 ∧P 2 )∨(P 1 ∧P 3 ).

The crucial observation is that, nevertheless (L(H),≥,¬) includes lots of Boolean
σalgebras, and precisely the maximal sets of pairwise compatible projectors[ 5 ].

Proposition 2.3.11.LetHbe a complex separable Hilbert space and consider the
lattice(L(H),≥,¬).IfL 0 ⊂L(H)is a maximal subset of pairwise commuting
elements, thenL 0 contains 0 ,Iis¬-closed and, if equipped with the restriction of
the lattice structure of(L(H),≥,¬), turns out to be a Booleanσ-algebra.
In particular, ifP,Q∈L 0 ,

(i)P∨Q=P+Q−PQ;

(ii)P∧Q=PQ.