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Supplement to The Kochen-Specker Theorem

(A) PConsider now an arbitrary state |> and an arbitrary nondegenerate operator Q on H3, its eigenvectors |_{i}^{2}= P_{i}(the P_{i}are ‘idempotent’);(B) If H is a Hilbert space of denumerable dimension, and if the P

_{i}are operators projecting onq_{i}, where the set {q_{i}} forms an orthonormal basis of H, then_{i}P_{i}=I (where I is the identity operator) (the P_{i}form ‘a resolution of the identity’).

PNow, P_{1}+ P_{2}+ P_{3}= I

Now, from KS2 (b) (Product Rule) and (A):v(P_{1}) +v(P_{2}) +v(P_{3}) =v(I)

Now, assume an observable R such that

v(P_{i})^{2}=v(P_{i}^{2}) =v(P_{i})v(P_{i}) = 1 or 0

Hence:

v(R) =v(I R) =v(I)v(R)v(I) = 1

(VC1)wherev(P_{1}) +v(P_{2}) +v(P_{3}) = 1

Carsten Held

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*First posted: September 10, 2000*

*Last modified: September 10, 2000*