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Now let s=1. Then it follows (see e.g. Kochen and Specker 1967: 308, Redhead 1987: 37-38) that S_{x}^{2}, S_{y}^{2}, S_{z}^{2} are all mutually commuting and that:
S_{x}^{2} + S_{y}^{2} + S_{z}^{2} = 2I,where I is the identity operator. Now, from KS2 (a) (Sum Rule):
v(S_{x}^{2}) + v(S_{y}^{2}) + v(S_{z}^{2}) = 2v(I)Now, assume an observable R such that v(R) 0 in state |>. From this assumption and KS2 (b) (Product Rule):
Hence:
v(R) = v(I R) = v(I) v(R) v(I) = 1
(VC2) v(S_{x}^{2}) + v(S_{y}^{2}) + v(S_{z}^{2}) = 2where v(S_{i}^{2}) = 1 or 0, for i = x, y, z.
Return to The Kochen-Specker Theorem
Carsten Held cheld@ruf.uni-freiburg.de |