Abstract
In this chapter we will analyse one of the main theorems (another one would be Bell’s inequality) which states the impossibility of quantum theory, as it is canonically expressed, to be a realist theory. This theorem is know as the Kochen-Specker theorem and will be explained in details in Sect. 3.3. However, in order to fully understand this theorem, we first of all have to introduce the concept of a valuation function. We will do so first for classical theory and then quantum theory.
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Notes
- 1.
In Chap. 13 we will explain, in detail, what a spectral decomposition is, but for now we will simply state that each self-adjoint operator \(\hat{A}\) can be written as \(\hat{A}=\int_{\sigma(A)}\lambda d\hat{E}^{\hat{A}}_{\lambda}\). Such an expression is called the spectral decomposition of \(\hat{A}\). Here \(\sigma(A)\subseteq \mathbb{R}\) represents the spectrum of the operator \(\hat{A}\) and \(\{\hat{E}^{\hat{A}}_{\lambda}|\lambda\in\sigma (\hat{A})\}\) is the spectral family of \(\hat{A}\). In the discrete case we would have \(\hat{A}=\sum_{\sigma(A)}\lambda\hat{P}^{\hat {A}}_{\lambda}\), where the projection operators \(\hat{P}^{\hat {A}}_{\lambda}\) project on subspaces of the Hilbert space for which the states ψ have value λ for A.
- 2.
Note that in the following we will use probabilistic valuation functions of the form \(V:\mathcal{O}\rightarrow P([\mathbb{R}])\) where \(P([\mathbb{R}])\) represents the space of probability distributions.
- 3.
Values are said to be mutually exclusive and collectively exhaustive if one and only one gets assigned the value 1 (true), while the rest gets assigned the value 0 (false).
- 4.
Mutually exclusive means that only one value of an observable can be realised at a given time, while collectively exhaustive means that at least one of the values has to be realised at a given time.
- 5.
Here \(\hat{P}_{1}\vee\hat{P}_{2}=\hat{P}_{1}+\hat{P}_{2}-\hat{P}_{1}\hat{P}_{2}\).
- 6.
As we will explain in detail later on, in standard quantum theory, propositions are represented by projection operators. In particular the proposition ``A=a 1”, indicating that the value of the physical quantity is a 1, will be represented by the projection operator \(\hat {P}\) which projects on the subspace of eigenvectors which have eigenvalue precisely a 1.
- 7.
It should be noted that the probabilistic predictions of quantum theory, are not affected by the notion of contextuality. In fact, the result of measuring a property A of a system does not depend on what else is measured at the same time, since the probability of obtaining a m as the value of A will always be \(\langle\psi|\hat{P}_{a_{m}}|\psi\rangle\).
References
C.J. Isham, Lectures on Quantum Theory, Mathematical and Structural Foundations (Imperial College Press, London, 1995)
J. Bub, Interpreting the Quantum World (Cambridge University Press, Cambridge, 1997)
M. Kernaghan, Bell-Kochen-Specker theorem for 20 vectors. J. Phys. A 27 (1994)
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Flori, C. (2013). Kochen-Specker Theorem. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_3
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