FOUNDATIONS OF QUANTUM MECHANICS

A
GLEASON’S THEOREM
Proofs really aren’t there to convince you that something is true - they’re there to show
you why it is true.
— Andrew Gleason
Of course mathematics works in physics! It is designed to discuss exactly the situation
that physics confronts; namely, that there seems to be some order out there - let’s find
out what it is.
— Andrew Gleason
In section III. 2 we mentioned that Von Neumann suggested for a quantum mechanical probability
measure the trace formula Tr P W , with P a projector. Gleason’s theorem shows that
this probability measure in fact characterizes all probability measures on P (H), the set of all
projectors on H. Since Gleason’s original proof is very difficult, in this appendix we will give a
simplified version by proving the theorem for pure states only.
A. 1 INTRODUCTION
Let H be a real or complex Hilbert space with dim H > 2, and P (H) the set of all projectors
on H. Let µ be a mapping µ : P (H) → [0, 1]. This µ is called a measure on H if it is additive,
satisfying
P i ⊥ P j =⇒ µ(P i + P j ) = µ(P i ) + µ(P j ) ∀ P i , P j ∈ P (H) (A. 1)
µ(0 ) = 0 and µ(11) = 1. (A. 2)
Combination of (A. 1) and the last requirement of (A. 2) implicates that µ attributes the value 1 to any
orthogonal decomposition of unity.
In section III. 2, p. 46, we saw that pure states are represented by the extreme elements of a convex
set, and by proving the theorem on p. 49 we showed that the extreme elements of the convex set S(H)
of state operators on H are the 1 - dimensional projectors in P (H). Consequently, the measure µ is
called extreme if there exists a 1 - dimensional projector P such that
µ(P ) = 1. (A. 3)
This is also expressed by saying that µ is concentrated on P . We can now formulate Gleason’s
theorem for pure states.