Abstract
In classical physics a pure state, s, is a point in the state space. It is the smallest subset of the state space which has measure one with respect to the Dirac measure δ s .
Recall that a Dirac measure δ s on some set S is defined by
for any s∈S and any measurable subset A⊆S.
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Notes
- 1.
Recall that in a topos τ, a ‘point’ (or ‘global element’ or just ‘element’) of an object O is defined to be a morphism from the terminal object, 1 τ , to O.
- 2.
This was shown when we defined the sub-object classifier (see Definition 8.1).
- 3.
Here \(\mathit{Hom}_{\mathit{cl}}(\underline{\mathcal{O}}\times \underline{\varSigma}, \underline{\varOmega})\) indicates \(\mathit{Hom}(\underline{\mathcal{O}}\times \underline{\varSigma}, \underline{\varOmega})\) but restricted to clopen sub-objects of \(\underline{\mathcal{O}}\times \underline{\varSigma}\).
- 4.
Recall that in the topos framework, propositions are identified with clopen sub-objects of the state space.
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Flori, C. (2013). Topos Analogues of States. 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_11
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DOI: https://doi.org/10.1007/978-3-642-35713-8_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35712-1
Online ISBN: 978-3-642-35713-8
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