FOUNDATIONS OF QUANTUM MECHANICS

188 APPENDIX A. GLEASON’S THEOREM
A. 3. 1 STEP 2
THEOREM 2:
If the function µ (s) or, equivalently, µ (θ s , ϕ s ), satisfies the requirements (A. 11)
to (A. 15), then µ(s) is a nonincreasing function in θ s .
We will prove this theorem using two lemmas.
A. 3. 1. 1 LEMMA 1
A LITTLE LEMMA:
Let {s ∈ S 2 | s ⊥ r} be the great circle with axis r ≠ p 0 . Furthermore, let s 0 represent
the most northern point of this circle. Then for all points s of this great circle it holds
that
µ(s 0 ) µ(s), (A. 18)
i.e., if we let s travel along a great circle, µ(s) will have its maximum value in the most
northern point s 0 .
Proof
Choose a set of three orthogonal directions r, s, t, with s ∈ S 2 an arbitrary point on the great
circle around axis r. From (A. 13) we have
µ(r) + µ(s) + µ(t) = 1. (A. 19)
Now carry out a rotation of the orthogonal pair s and t around the axis r until s arrives at the most
northern point s 0 of the great circle. Under this rotation t arrives at a point t ′ at the equator as
can be seen in figure A. 2.
r
p 0
s 0
θ
t ′
t
∆ϕ
s
equator
Figure A. 2: Rotation of s to s 0 and t to t ′ along a great circle around axis r