180 From Classical Mechanics to Quantum Field Theory. A Tutorial
with [−iQh,−iI]=[−iMk,−iI] = 0, where the operator−iIrestricted toDis the
remaining Lie generator.Wis theHeisenberg-Weylgroup[ 5 ]. The selfadjoint gen-
erators of this representation are just the operatorsQkandPk(andI), since they
coincide with the closure of their restrictions toD, because they are selfadjoint
(so they admit unique selfadjoint extensions) andDis a core. If furthermore the
Lie algebra representation is irreducible, the unitary representation is irreducible,
too: IfKwere an invariant subspace for the unitary operators, Stone theorem
would imply thatKbe also invariant under the selfadjoint generators of the one
parameter Lie subgroups associated to eachQkandPk. This is impossible if the
Lie algebra representation is irreducible as we are assuming. The standard version
of Stone-von Neumann theorem[ 5 ]implies that there is isometric surgective oper-
atorU:H→L^2 (Rn,dnx) such thatWg→UVgU−^1 ∈B(L^2 (Rn,dnx)) is the
standard unitary representation of the groupWinL^2 (Rn,dnx) genernated byXk
andPk(andI)[ 5 ]. Again, Stone theorem immediately yields (2.117). The last
statement easily follows from the standard form of Mackey’s theorem completing
Stone-von Neumann result[ 5 ].
Remark 2.3.79.
(a) The resulta posteriorigives, in particular, a strong justification of the
requirement that the Hilbert space of an elementary quantum system, like a particle,
must be separable.
(b)Physical Hamiltonian operators have spectrum bounded from below to avoid
thermodynamical instability. This fact prevents the definition of a “time operator”
canonically conjugated withHfollowing the standard way. This result is some-
time quoted asPauli theorem. As a consequence, the meaning of Heisenberg
relations ΔEΔT≥/ 2 is different from the meaning of the analogous relations
for position and momentum. It is however possible to define a sort of time os-
servable just extending the notion of PVM to the notion of POVM i.e.,positive
valued operator measure(see the first part and[ 5 ]). POVMs are exploited to de-
scribe concrete physical phenomena related to measurement procedures, especially
in quantum information theory[30; 31].
Corollary 2.3.80. If the Hamiltonian operatorσ(H)of a quantum system is
bounded below, there is no selfadjoint operator (time operator) T satisfying the
standard CCR withHand the hypotheses (1), (2), (3) of Theorem 2.3.78.
Proof. The coupleH,T should be mapped to a corresponding coupleX,P in
L^2 (R,dx), or a direct sum of such spaces, by means of a Hilbert space isomorphism.
In all cases the spectrum ofHshould therefore be identical to the one ofX,namely
isR. This fact is false by hypotheses.