Abstract
In this chapter we will describe how topos quantum theory can be seen as a contextual quantum theory, in the sense that each element is defined as a collection of ‘context dependent’ descriptions. Such context dependent descriptions will turn out to be classical snapshots.
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Notes
- 1.
\(\mathcal{B}(\mathcal{H})\) indicates the algebra of bounded operators on the Hilbert space \(\mathcal{H}\).
- 2.
Note that in order to understand the general idea of topos quantum theory it is not necessary to understand in details what a von Neumann algebra is, all that is needed is to roughly understand what they are, how they can be formed and the philosophical implications of their usage in topos quantum theory.
- 3.
A module over a ring is, roughly, the same thing as a vector space over a filed but now, instead of multiplication by a scalar, one has multiplication by an element of the ring.
- 4.
For completeness of exposition we will report both the corollary and the proof.
Corollary 9.1  Let the topological group G act on a Hausdorff topological space X in a continuous way, i.e., the G-action map G×X→X is continuous. Then the stabiliser, G x , of any x∈X is a closed subgroup of G.
Proof For any given x∈X, consider the maps f x ,h x :G→X defined by f x (g):=gx and h x (g):=x for all g∈G. The first is continuous since the G-action on X is continuous, and the second is continuous because constant maps are always continuous. Now
$$ E(f_x, h_x) = \{g \in G | f_x(g) = h_x(g)\} \\ = \{g\in G | gx = x\} = G_x. $$(9.14)We then need to show that E(f x ,h x ) is closed. This follows from the fact that, given any two continuous maps f,h:X→Y between topological spaces X,Y, such that Y is Hausdorff, then E(f,h):={x∈X|f(x)=h(x)} is closed. In fact consider E c(f,h):={x∈X|f(x)≠h(x)}. Let x∈E c(f,h). Then since Y is Hausdorff, there exists open neighbourhoods N x,f of f(x) and N x,h of h(x) such that N x,f ∩N x,h =∅. Since f,h are continuous f −1(N x,f ) and h −1(N x,h ) are open. Thus f −1(N x,f )∩h −1(N x,h ) is open and non-empty (since x∈f −1(N x,f )∩h −1(N x,h )). In fact, for all y∈f −1(N x,f )∩h −1(N x,h ) we have that f(y)≠h(y). It follows that E c(f,h) is open, and hence that E(f,h) is closed. This result implies that G x is closed.  □
- 5.
Recall that a neighbourhood filter of a point x∈X of a topological space X, is simply the collection of all neighbourhoods of x.
- 6.
Definition does not really matter here and we will simply call it isomorphisms.
References
J.B. Geloun, C. Flori, Topos analogues of the KMS state. arXiv:1207.0227v1 [math-ph]
C. Flori, Group action in topos quantum physics. arXiv:1110.1650 [quant-ph]
J. Dugundji, Topology (Allyn and Bacon, Needham Heights, 1976)
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Flori, C. (2013). Topos Analogue of the State Space. 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_9
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