From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 109

Remark 2.2.48.
(a)(i) and (iii) withN={ 1 , 2 }imply thatP∅=0usingE 1 =XandE 2 =∅.
Next (ii) entails thatPEPF=0ifE∩F=∅. An important consequence is that for
Ninfinite, the vector given by the sum on the left hand side of (iii)is independent
from the chosen order because that vector is a sum of pairwise orthogonal vectors
PEjx.

(b)Ifx, y∈H,Σ(X)E→〈x, PEy〉=:μ(xyP)(E)is acomplex measurewhose

(finite)total variation[ 8 ]will be denoted by|μ(xyP)|. From the definition ofμxy,we
immediately have:

(i)μ(xyP)(X)=〈x, y〉;

(ii)μxx(P)is always positive and finite andμ(xxP)(X)=||x||^2 ;

(iii)ifs=

∑n

k=1skχEkis asimple function[^8 ],

∫

Xsdμxy=〈x,

∑n

k=1skPEky〉.

Example 2.2.49.
(1)The simplest example of PVM is related to a countable Hilbertian basisNin
a separable Hilbert spaceH. We can define Σ(N) as the class of all subsets ofN
itself. Next, forE∈Σ(N)andz∈Hwe define

PEz:=

∑

x∈E

〈x, z〉x

andP∅:= 0. It is easy to prove that the class of allPEdefined this way form
aPVMonN. (This definition can be also given ifHis non-separable andNis
uncountable, since for everyy∈Honly an at most countable subset of elements
x∈Esatisfy〈x, y〉= 0). In particularμxy(E)=〈x, PEy〉=

∑

z∈E〈x|z〉〈z|y〉and
μxx(E)=

∑

z∈E|〈x, z〉|

(^2).
(2)A more complicated version of (1) consists of a PVM constructed out of a
orthogonal Hilbertian decomposition of a separable Hilbert space,H=⊕n∈NHn,
whereHn⊂His a closed subspace andHn⊥Hmifn=m. Again defining Σ(N)
as the set of subsets ofN,forE∈Σ(N)andz∈Hwe define
PEz:=

∑

x∈E

Qnz

whereQnis the orthogonal projector ontoHn(the reader can easily check that the
sum always converges using Bessel’s inequality). It is easy to prove that the class
of∑PEs defined this way form a PVM onN. In particularμxy(E)=〈x, PEy〉=

n∈E〈x, Qny〉andμxx(E)=

∑

n∈E||Qnx||^2.

(3)InL^2 (R,dx) a simple PVM, not related with a Hilbertian basis, is made
as follows. To everyE∈B(R), the Borelσ-algebra, associate the orthonormal