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Supplement to The Kochen-Specker Theorem

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In order to derive Sum Rule and Product Rule from FUNC, we use the following mathematical fact: LetFUNC:Letbe a self-adjoint operator associated with observable A, let f:ARRbe an arbitrary function, such thatis self-adjoint operator, and let | f > be an arbitrary state; thenf(A)is associated uniquely with an observable f(A) such that:f(A)v(f(A))^{}= f(v(A))^{}

Sum Rule:IfandAare commuting self-adjoint operators corresponding to observables A and B, respectively, then A + B is the unique observable corresponding to the self-adjoint operatorBandA + Bv(A + B)^{}=v(A)^{}+v(B)^{}

Product Rule:IfandAare commuting self-adjoint operators corresponding to observables A and B, respectively, then if A B is the unique observable corresponding to the self-adjoint operatorBAandBv(AB)^{}=v(A)^{}v(B)^{}

So, for two commuting operators __ A__,

SinceTherefore:= f(A) andC= g(B), there is a function h = f+g, such thatC= h(A + B).C

Similarly:

v(A + B)^{}= h( v(C)^{})(by FUNC) = f( v(C)^{}) + g(v(C)^{})= v(f(C))^{}+v(g(C))^{}(by FUNC) = v(A)^{}+v(B)^{}(Sum Rule)

SinceTherefore:= f(A) andC= g(B), there is a function k = fg, such thatCA= k(B).C

v(A B)^{}= k( v(C)^{})(by FUNC) = f( v(C)^{}) g(v(C)^{})= v(f(C))^{}v(g(C))^{}(by FUNC) = v(A)^{}v(B)^{}(Product Rule)

Return to The Kochen-Specker Theorem

Carsten Held carsten.held@uni-erfurt.de |

Stanford Encyclopedia of Philosophy