From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 119

and, more generally, for any measurablef:σ(A)→C,

f(A)(Ha∩D(f(A)))⊂Ha

(ii)for anya∈S, there exist a unique finite positive Borel measureμaon

σ(A)⊂R, and a surjective isometric operatorUa:Ha→L^2 (σ(A),μa),

such that:

Uaf(A)|HaUa−^1 =f·

for any measurablef:σ(A)→C,wheref·is the point-wise multiplication

byfonL^2 (σ(A),μa).

(b)Ifsupp{μa}a∈Sis the complementary set to the numbersλ∈Rfor which

there exists an open setOλ⊂RwithOλλ,μa(Oλ)=0for anya∈S,

then

σ(A)=supp{μa}a∈S.

Notice that the theorem encompasses the case of an operatorAinHwithσ(A)=
σp(A). Suppose in particular that every eigenspace is one-dimensional and the
whole Hilbert space is separable. Letσ(A)=σp(A)={λn|n∈N}.Inthiscase

A=

∑

n∈N

λn〈xn, 〉xn,

wherexλis a unit eigenvector with eigenvalueλn. Consider theσ-algebra onσ(A)
made of all subsets and defineμ(E) := number of elements ofE⊂σ(E). In this
case,His isomorphic toL^2 (σ(A),μ) and the isomorphism isU:Hx→ψx∈
L^2 (σ(A),μ) withψx(n):=〈xn|x〉ifn∈N. With this surjective isometry, trivially

Uf(A)U−^1 =U

∫

σ(A)

f(λ)dP(A)(λ)U−^1 =U

∑

n∈N

f(λn)〈xn, 〉xnU−^1 =f·.

If all eigenspaces have dimension 2, exactly two copies ofL^2 (σ(A),μ)aresufficient
to improve the construction. If the dimension depends on the eigenspace, the
construction can be rebuilt exploiting many copies of differentL^2 (Sk,μk), where
theSkare suitable (not necessarily disjoint) subsets ofσ(A)andμkthe measure
which counts the elements ofSk.
The last tool we introduce is the notion ofjoint spectral measure. Everything
is stated in the following theorem[5; 6].

Theorem 2.2.63 (Joint spectral measure). Consider selfadjoint operators
A 1 ,A 2 ,...,An in the complex Hilbert spaceH. Suppose that the spectral mea-
sures of those operators pairwise commute:

PE(Akk)PE(Ahh)=PE(Ahh)PE(Akk) ∀k,h∈{ 1 ,...,n},∀Ek,Eh∈B(R).