An Introduction to Noncommutative Spaces and their by Giovanni Landi (auth.)

By Giovanni Landi (auth.)

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Our ‘detectors’ will be taken to be (possibly overlapping) open subsets of S 1 with some mechanism which switches on the detector when the particle is in the corresponding open set. The number of detectors must be clearly limited and we take them to consist of the following three open subsets whose union covers S 1 , G. Landi: LNPm 51, pp. 21–58, 2002. 1) U3 = {π < ϕ < 2π}. Now, if two detectors, U1 and U2 say, are on, we will know that the particle is in the intersection U1 ∩ U2 although we will be unable to distinguish any two points in this intersection.

We refer to [149] for an introduction to Jordan algebras. 41) 3 Projective Systems of Noncommutative Lattices The idea of a ‘discrete substratum’ underpinning the ‘continuum’ is somewhat spread among physicists. With particular emphasis this idea has been pushed by R. Sorkin who, in [141], assumes that the substratum be a finitary (see later) topological space which maintains some of the topological information of the continuum. It turns out that the finitary topology can be equivalently described in terms of a partial order.

3 How to Recover the Space Being Approximated We shall now briefly describe how the topological space being approximated can be recovered ‘in the limit’ by considering a sequence of finer and finer coverings, the appropriate framework being that of projective (or inverse) systems of topological spaces [141]. 4 Here we are relaxing the harmless assumption made in Sect. 1 that each U is already a subtopology, namely that U = τ (U). In Sect. 1 we have associated with each covering Ui a T0 -topological space Pi and a continuous surjection π i : M → Pi .

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