Abstract
From Chap. 16 we know that self-adjoint operators are represented as arrows, from the state space \(\underline{\varSigma}\) to the quantity value object \(\underline {\mathbb {R}}^{\leftrightarrow}\). The natural question to ask is whether such a representation can be extended to all normal operators. To this end one needs to, first of all, define the topos analogue of the complex numbers. Of course there exists the trivial object \(\underline {\mathbb {C}}\) but this, as we will see, can not be identified with the complex number object since (a) it does not reduce to \(\underline {\mathbb {R}}^{\leftrightarrow}\), and (b) the presheaf maps in \(\underline {\mathbb {C}}\) are the identity maps which do not respect the ordering induced by the daseinisation of normal operators, yet to be defined.
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- 1.
The presheaf \(\underline {\mathbb {R}}\) is defined on each \(V\in \mathcal{V}(\mathcal{H})\) as \(\underline {\mathbb {R}}_{V}:=\mathbb {R}\) and given the maps i:V′⊆V then the respective presheaf morphisms are simply the identity maps. See Definition 20.9.
- 2.
Recall that a local is essentially a frame (see Definition 17.1) but a morphisms between two locales is a morphisms between the frames which goes in the opposite direction.
- 3.
Note that in this internal approach one is working with co-presheaves instead of presheaves.
- 4.
A set P is directed if for any x,y∈P there exists a z∈P such that x,y⊑z.
- 5.
Here ⨆↑ indicates the supremum of a directed set and the relation x≪y indicates that x approximates y. In particular x≪y if, for any directed set S with a supremum, then y⊑⨆↑ S⇒∃s∈S:x⊑s.
- 6.
Here l([α,β] n ) represents the length of the largest side of the rectangle and thus is defined as l([α,β] n )=Max{|a n −c n |,|b n −d n |} where α=a+ib and β=c+id.
- 7.
We recall that, given a poset 〈P,≤〉 a subset G is said to be Scott-open if (i) x∈G∧x≤y⇒y∈G; (ii) for any directed set S with supremum then ⨆↑ S∈G⇒∃s∈S|s∈G. In other words all supremums in G have a non-empty intersection with G.
- 8.
Given this topology then it is clear that, as topological spaces \(\mathbb {C}\simeq max\mathbf{I}\mathbb {C}\) where \(\mathbb {C}\) is equipped with the (general) open rectangles topology and \(max\mathbf{I}\mathbb {C}\) has the topology inherited by \(\mathbf{I}\mathbb {C}\). The homeomorphisms can be see by the fact that
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Flori, C. (2013). Normal Operators. 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_18
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DOI: https://doi.org/10.1007/978-3-642-35713-8_18
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