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Let us consider as an example a two dimensional spin system.

The operator representing the spin in the z direction is represented by the following matrix

We know that the elements and of the coarse grainig presheaf G are represented by the spectral algebras and so we need to identify these algebras.

Specifically, we can decompose, through the spectral theorem, the operator in terms of its spectral projector as follows

where

On the other hand the spectral decomposition of the operator is as follows

Thus the spectral algebras is

Thus it follows that

In fact we can define the morphisms , where the function indicates the operation of taking the square, as

In terms of propositions it is easy to see that the proposition is a coarse graining of the proposition since in terms of the projection operators we have

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**Previous:**Example of Coarse Graining Cecilia Flori 2008-05-02