Abstract
We now arrive at the very important concept of truth values in a topos and how they are assigned to quantum propositions. The important feature of these new truth values is that, given any set of quantum propositions, even incompatible ones, it is always possible to assess their truthfulness simultaneously. Moreover the set of truth values forms a Heyting algebra thus, we obtain an intuitionistic logic. What this implies is that a more realist picture of quantum theory emerges. In fact, it now makes sense to say that quantities possess values since, at each moment in time, we can assess whether any statement regarding properties of a physical quantity is true or not. However, since the set of truth values is bigger than the classical set {0,1} we do not obtain a strictly realist interpretation, but rather a broader definition of realism, which is called neo-realism.
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Notes
- 1.
Note that \(\underline{0}_{\underline{\varOmega}_{V}}=\underline{0}_{\underline{\varOmega}_{V_{\hat{P}_{i},\hat{P}_{j}}}} =\underline{0}_{\underline{\varOmega}_{V_{\hat{P}_{i}}}}=\{ \emptyset\}\).
- 2.
Note that \(\hat{E}[A\in \varDelta ]=\hat{P}_{A\in \varDelta }\).
- 3.
Recall that \(\mathfrak {w}^{\,|\psi\rangle }_{V}:=\delta ^{o}(|\psi\rangle\langle\psi|)_{V}\) represents the projection operator while \(\underline {\mathfrak {w}}^{\,|\psi\rangle }_{V}\) indicates the subset of \(\underline{\varSigma}_{\,V}\). Although ultimately they are equivalent, it is always worth highlighting what specific role \(\mathfrak {w}^{\,|\psi\rangle }\) has.
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© 2013 Springer-Verlag Berlin Heidelberg
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Flori, C. (2013). Truth Values. 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_12
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DOI: https://doi.org/10.1007/978-3-642-35713-8_12
Publisher Name: Springer, Berlin, Heidelberg
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