142 From Classical Mechanics to Quantum Field Theory. A Tutorial
Proposition 2.3.23.LetHbe a complex Hilbert space and letT∈B 1 (H)satisfy
T≥ 0 andTrT=1, then the map
ρT:L(H)P→tr(TP)is well defined andρT∈S(H).
The very remarkable fact is that these operators exhaustS(H)ifHis separable
with dimension= 2. As established by Gleason in a celebrated theorem, we restate
re-adapting it to these lecture notes (see[ 5 ]for a the original statement and[ 21 ]
for a general treatise on the subject).
Theorem 2.3.24(Gleason’s theorem).LetHbe a complex Hilbert space of finite
dimension=2, or infinite dimensional and separable. Ifρ∈S(H)there exists a
unique operatorT∈B 1 (H)withT≥ 0 andtr T=1such thattr(TP)=ρ(P)for
everyP∈L(H).
Concerning the existence ofT, Gleason’s proof works for real Hilbert spaces too.
If the Hilbert space is complex, the operatorTassociated toρis unique for the
following reason. Any otherT′of trace class such thatρ(P)=tr(T′P) for any
P ∈L(H)mustalsosatisfy〈x,(T−T′)x〉=0foranyx∈H.Ifx=0this
is clear, while ifx= 0 we may complete the vectorx/||x||toabasis,inwhich
tr((T−T′)Px)=0reads||x||−^2 〈x,(T−T′)x〉=0,wherePxis the projector onto
span(x). By (3) in exercise 2.2.31, we obtainT−T′=0.^17
Remark 2.3.25.
(a)ImposingdimH=2is mandatory, a well-known counterexample can be
found, e.g. in[ 5 ].
(b)Particles with spin 1 / 2 , like electrons, admit a Hilbert space – in which the
observable spin is defined – of dimension 2. The same occurs to the Hilbert space
in which the polarization of light is described (cf. helicity of photons). When these
systems are described in full, however, for instance including degrees of freedom
relative to position or momentum, they are representable on a separable Hilbert
space of infinite dimension.
Gleason’s characterization of states has an important consequence known as
theBell-Kochen-Specker theorem. It proves in particular (see[ 6 ]for an extended
discussion) that in QM there are no states assigning probability 1 to some ele-
mentary observables and 0 to the remaining ones, differently to what happens in
CM.
(^17) In a real Hilbert space〈x, Ax〉=0forallxdoes not implyA= 0. Think of real antisymmetric
matrices inRnequipped with the standard scalar product.