(B) If H is a Hilbert space of denumerable dimension, and if the
Pi are operators projecting on
qi, where the set
{qi} forms an orthonormal basis of H,
then
i
Pi=I (where I is the identity operator) (the
Pi form ‘a resolution of the
identity’).
| This is a file in the archives of the Stanford Encyclopedia of Philosophy. |
(A) Pi2 = Pi (the Pi are ‘idempotent’);Consider now an arbitrary state |(B) If H is a Hilbert space of denumerable dimension, and if the Pi are operators projecting on qi, where the set {qi} forms an orthonormal basis of H, then
> and an arbitrary nondegenerate
operator Q on H3, its eigenvectors |q1>,
|q2>, |q3>, and
projection operators P1, P2, P3
whose ranges are the rays spanned by these vectors. The eigenvectors
form an orthonormal basis, thus, by (B):
P1 + P2 + P3 = INow, P1, P2, P3 are compatible, so from assumption KS2 (a) (Sum Rule):
v(P1) + v(P2) + v(P3) = v(I)Now, from KS2 (b) (Product Rule) and (A):
Now, assume an observable R such that v(R)
v(Pi)2 = v(Pi2) = v(Pi) v(Pi) = 1 or 0
0 in state
|>. From this assumption and KS2
(b) (Product Rule):
Hence:
v(R) = v(I R) = v(I) v(R) v(I) = 1
(VC1) v(P1) + v(P2) + v(P3) = 1where v(Pi) = 1 or 0, for i = 1, 2, 3.
Return to The Kochen-Specker Theorem
| Carsten Held carsten.held@uni-erfurt.de |
