FOUNDATIONS OF QUANTUM MECHANICS

186 APPENDIX A. GLEASON’S THEOREM
Now we have the following statements,
(a) P (E) ⊂ P (H),
(b) the restriction of µ 0 to P (E) is a measure on P (E),
(c) the restriction of µ to P (E) is a measure on P (E),
(d) the measures µ 0 and µ differ on P (E).
Statement (a) follows immediately from E ⊂ H. The statements (b) and (c) follow from the
fact that both µ 0 and µ, being concentrated on P 0 , assign the value 1 to E, thereby assigning
the value 0 to all subspaces of H perpendicular to E. Statement (d) follows from our assumption
(A. 8).
Next, we have to show that the Hilbert space E can be real. A Hilbert space is real if scalar
multiplication and linear combinations of vectors are only carried out with real coefficients and
the inner products are real. Choosing the vectors |e 0 ⟩, |ẽ 1 ⟩ and |e 2 ⟩, we have the freedom to
absorb an arbitrary phase factor, which means that we can also take them real. Furthermore, we
can exploit that freedom to bring about that the vector |e 1 ⟩, lying in the plane spanned by |e 0 ⟩
and |ẽ 1 ⟩, becomes a linear combination with real coefficients, i.e.,
|e 1 ⟩ = a |e 0 ⟩ + b |ẽ 1 ⟩ with a, b ∈ R. (A. 10)
All inner products of the four vectors |e 0 ⟩, |ẽ 1 ⟩, |e 2 ⟩ and |e 1 ⟩ now have a real value. The required
real Hilbert space is obtained by taking all linear combinations of |e 0 ⟩, |ẽ 1 ⟩ and |e 2 ⟩ with real
coefficients. Because both |e 0 ⟩ and |e 1 ⟩ are elements of this Hilbert space, (a) through (d) remain
valid.
We see that, if Gleason’s theorem for pure states is not true for a complex Hilbert space with
dim > 3, it is also not true for a real 3 - dimensional Hilbert space. Now assume that the theorem
is proven to be true for a real Hilbert space with dim = 3. At the same time supposing that it is
not true for a Hilbert space with dim > 3, so that it would, as we showed, also not be true for a
real H with dim = 3, yields a contradiction. Therefore, theorem 1 is true. □
A. 3 FORMULATION OF THE PROBLEM ON THE SURFACE OF A SPHERE
While by proving theorem 1 we showed that, if Gleason’s theorem for pure states is true for a
real, 3 - dimensional Hilbert space, it is also true for a complex Hilbert space with dim > 2, we did
not prove that µ = µ 0 . In this section we will take the next steps towards proving that indeed µ = µ 0
for all P ∈ P (H) in a real, 3 - dimensional Hilbert space.
Conversion of an arbitrary complex Hilbert space to a 3 - dimensional real Hilbert space is convenient
because this space is isomorphic with the usual 3 - dimensional Euclidean space R 3 . Here,
the 1 - dimensional projectors correspond to lines through the origin, and we can identify them with
points on the surface of a unit sphere, or actually, with half of the unit sphere because |e⟩ and −|e⟩ represent
the same state. Those points will be designated by means of their spherical coordinates (θ, ϕ),
or as points, or directions, on the surface of the unit sphere p, q, r, s, t, . . . , ∈ S 2 , where S 2 is the
standard notation for this surface, and the index 2 refers to the fact that it is 2 - dimensional.