FOUNDATIONS OF QUANTUM MECHANICS

A. 3. FORMULATION OF THE PROBLEM ON THE SURFACE OF A SPHERE 195
Proof
For every pair of points (s 1 , s 2 ) ∈ C 1 , where θ s1 < θ q and θ s2 > θ q , we can always find a
pair (t 1 , t 2 ) ∈ C 2 , such that θ s1 < θ t1 < θ q and θ s2 > θ t2 > θ q , see figure A. 10.
s 1
t 1
q
s 2 t 2
C 1
C 2
θ q
Figure A. 10: Two continuous curves on S 2 , intersecting in q
Using (A. 24), (A. 36) and (A. 37), we have for t ∈ C 2
θ s < θ t < θ q ⇒ µ(s) µ(t) a, (A. 38)
and
θ s > θ t > θ q ⇒ µ(s) µ(t) a − ε. (A. 39)
This holds no matter how close to q the points s and t are chosen, which proves the lemma. □
Proof, second part
Now we continue the proof of theorem 3. In the first part of the theorem we proved for the pair
(s, s ⊥ ) that if µ jumps with ε in p, it also jumps with ε in q. The same rigidity holding for any
pair (s i , s i⊥ ) ∈ C i , we concluded that µ jumps in every point q, q ′ , q ′′ , . . . , q n with at least ε.
With the accessory lemma, we proved that, if µ makes a jump of at least ε at some point on one
curve C, it does so on any curve C i through that point.
Since we chose the directions q, q ′ , q ′′ , . . . , q n perpendicular to p, see figure A. 9, they all lie
on C p , a great circle with axis p. Starting in its most northern point q, upon descending along this
great circle C p towards the equator, µ(s) remains constant or decreases, as we showed by proving
theorem 2.
But according to the first part of this proof and the accessory lemma, upon descending along
this great circle C p towards the equator, in each of the points q, q ′ , q ′′ , . . . , q n , µ jumps with
at least −ε while passing their various latitudes. Since we can choose n arbitrary large, we can
choose n to be larger than n > ε 1 , making the total jump nε > 1. This leads to µ acquiring values
smaller than 0, which is contradictory to the requirement that 0 µ 1. We have to conclude
that ε = 0, which yields M (θ 0 ) = m(θ 0 ).
We proved that if on the surface of the unit sphere a horizontal circle B exists for which µ is not
constant, then µ /∈ [0, 1], hence µ is constant on constant latitude and does not depend on ϕ,
which proves theorem 3. □