From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 129

For future convenience, we observe that our model ofclassicalelementary prop-
erties can be also viewed as another mathematical structure, when referring to the
notion oflattice.

Definition 2.3.2. A partially ordered set(X,≥)is alatticewhen, fora, b∈X,

(a)sup{a, b}exists, denoteda∨b(sometimes called “join”);

(b)inf{a, b}exists, writtena∧b(sometimes “meet”).

(The partially ordered set is not required to be totally ordered.)

Remark 2.3.3.
(a)In our considered concrete case,X=B(R)and≥is nothing but⊃and
thus∨means∪and∧has the meaning of∩.
(b)In the general case,∨and∧turn out to be separatelyassociative, therefore
it make sense to writea 1 ∨···∨ananda 1 ∧···∧anin a lattice. Moreover they
are also separatelycommutativeso

a 1 ∨···∨an=aπ(1)∨···∨aπ(n) and a 1 ∧···∧an=aπ(1)∧···∧aπ(n)

for every permutationπ:{ 1 ,...,n}→{ 1 ,...,n}.

Let us pass to some relevant definitions.

Definition 2.3.4. A lattice(X,≥)is said to be:

(a) distributiveif∨and∧distribute over one another: for anya, b, c∈X,

a∨(b∧c)=(a∨b)∧(a∨c),a∧(b∨c)=(a∧b)∨(a∧c);

(b) boundedif it admits a minimum 0 and a maximum 1 (sometimes called

“bottom” and “top”);

(c) orthocomplementedif bounded and equipped with a mappingXa→

¬a,where¬ais theorthogonal complementofa, such that:

(i)a∨¬a= 1 for anya∈X;

(ii)a∧¬a= 0 for anya∈X;

(iii)¬(¬a)=afor anya∈X;

(iv)a≥bimplies¬b≥¬afor anya, b∈X;

(d)σ-complete, if every countable set{an}n∈N⊂Xadmits least upper bound

∨n∈Nan.

A lattice with properties (a), (b) and (c) is called aBoolean algebra.ABoolean
algebra satisfying (d) is aBooleanσ-algebra.