self-adjoint

Each bounded simple random variable f gives rise to a bounded self-adjoint operator A =

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It is not difficult to show that a self-adjoint operator P with spectrum contained in the two-point set {0,1} must be a projection; i.e.,

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There remains to recover, e.g., the representation of “observables” by self-adjoint operators, and the dynamics (unitary evolution).

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Studying the properties off these observations one sees that they satisfy the necessary axioms to be linear operators, and, in fact, self-adjoint operators.

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Pauli pointed out that a (self-adjoint) time operator is incompatible with a Hamiltonian spectrum bounded below.

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The observable quantities are represented by self-adjoint operators B on the Hilbert space.

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This amounts to rejecting that for every self-adjoint operator, there is a well-defined observable.

The Kochen-Specker Theorem Held, Carsten 2006

If we further assume that to every self-adjoint operator there corresponds a QM observable, then the principle can be formulated thus:

The Kochen-Specker Theorem Held, Carsten 2006

The principle trades on the mathematical fact that for a self-adjoint operator A operating on a Hilbert space, and an arbitrary function f:

The Kochen-Specker Theorem Held, Carsten 2006

If the non-intrinsic, state-dependent properties are identified with all the monadic or relational properties which can be expressed in terms of physical magnitudes associated with self-adjoint operators that can be defined for the particles, then it can be shown that two bosons or two fermions in a joint symmetric or anti-symmetric state respectively have the same monadic properties and the same relational properties one to another (French and Redhead 1988; see also Butterfield 1993).

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