From Classical Mechanics to Quantum Field Theory

Mathematical Foundations of Quantum Mechanics 179

[ 5 ], proving that the CCRs always give rise to the standard representation in
L^2 (Rn,dnx). We state and prove this version of the theorem, adding a last state-
ment which is the selfadjoint version of Mackey completion to Stone von Neumann
statement[5; 6].

Theorem 2.3.78 (Stone-von Neumann-Mackey theorem). LetHbe a complex
Hilbert space and suppose that there are 2 nselfadjoint operators inHwe indicate
withQ 1 ,...,QnandM 1 ,...,Mnsuch the following requirements are valid where
h, k=1,...,n.

(1)There is a common dense invariant subspaceD⊂Hwhere the CCRs hold

[Qh,Mk]ψ=iδhkψ, [Qh,Qk]ψ=0, [Mh,Mk]ψ=0 ψ∈D.

(2.116)

(2)The representation is irreducible, in the sense that there is no closed sub-

spaceK⊂Hsuch thatQk(K∩D(Qk))⊂KandMk(K∩D(Mk))⊂K

such thatQkandMkare selfadjoint as operators overK.

(3)The operator

∑n

k=1Q^2 k|D+Mk^2 |Dis essentially selfadjoint.

Under these conditions, there is a Hilbert space isomorphism, that is a

surjective isometric map,U:H→L^2 (Rn,dnx)such that

UQkU−^1 =Xk and UMkU−^1 =Pk k=1,...,n (2.117)

whereXkandPkrespectively are the standard position (2.35) and momen-

tum (2.36) selfadjoint operators inL^2 (Rn,dnx).InparticularHresults

to be separable.

If (1), (2) and (3) are valid with the exception that the representation is not re-
ducible, thenHdecomposes into an orthogonal Hilbertian sumH=⊕r∈RHkwhere
Ris finite or countable ifHis separable, theHr⊂Hare closed subspaces with

Qk(Hr∩D(Qk))⊂Hr and Mk(Hr∩D(Mk))⊂Hr

for allr∈R,k=1,...,nand the restrictions of all theQkandMkto eachHr
satisfy (2.117) for suitable surjective isometric mapsUr:Hr→L^2 (Rn,dnx).

Proof. If (1) holds, the restrictions toDof the selfadjoint operatorsQk,Mkde-
fine symmetric operators (since they are selfadjoint andDis dense and included
in their domains), also their powers are symmetric sinceDis invariant. If also
(2) is valid, in view of Nelson theorem (since evidently the symmetric operator
I|^2 D+

∑n

k=1Q^2 k|D+Mk^2 |Dis essentially selfadjoint if

∑n
k=1Q^2 k|D+Mk^2 |Dis), there
is a strongly continuous unitary representationW g→Vg ∈B(H)ofthe
simply connected 2n+ 1-dimensional Lie groupW whose Lie algebra is defined
by (2.116) (correspondingly re-stated for the operators−iI,−iQk,−iMk) together