Worked example of the Coarse Graining Presheaf

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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

$\displaystyle \hat{S}_z=\frac{1}{2}\left( \begin{array}{cc}
 \end{array} \right)$

If we take the square of this operator then the matrix representing it would be

$\displaystyle \hat{S}_z^2=\frac{1}{4}\left( \begin{array}{cc}
 \end{array} \right)$

Our aim is to define both the object $ G(S_z)=W_{S_z}$ and $ G(S_z^2)=W_{S^2_z}$ of the coarse graining presheaf G associated to the operators $ S_z$ and $ S_z^2$ respectively, and the morphism $ G(f):W_{S_z}\rightarrow W_{S^2_z}$.
We know that the elements $ G(S_z)=W_{S_z}$ and $ G(S_z^2)=W_{S^2_z}$ of the coarse grainig presheaf G are represented by the spectral algebras $ W_{S_z}$ and $ W_{S^2_z}$ so we need to identify these algebras.
Specifically, we can decompose, through the spectral theorem, the operator $ \hat{S}_z$ in terms of its spectral projector as follows

$\displaystyle \hat{S}_z=\frac{1}{4}\hat{P}_1-\frac{1}{4}\hat{P}_2$    


$\displaystyle \hat{P}_1=\left( \begin{array}{cc}
 \end{array} \right)$


$\displaystyle \hat{P}_2=\left( \begin{array}{cc}
 \end{array} \right)$

Thus it follows that the spectral algebra associated to $ \hat{S}_z$ is

$\displaystyle W_{\hat{S}_z}=\{\hat{0},\hat{1},\hat{P}_1,\hat{P}_2\}$    

On the other hand the spectral decomposition of the operator $ \hat{S}_z^2$ is as follows

$\displaystyle \hat{S}_z^2$ $\displaystyle =\frac{1}{8}\hat{P}_1+\frac{1}{4}\hat{P}_2$    
  $\displaystyle =\frac{1}{4}(\hat{P}_1\vee\hat{P}_2)$    
  $\displaystyle =\frac{1}{4}(\hat{P}_1+\hat{P}_2+\hat{P}_1\cdot\hat{P}_2)$    

Thus the spectral algebras is

$\displaystyle W_{\hat{S}_z^2}=\{\hat{0},\hat{1},(\hat{P}_1\vee\hat{P}_2)=\hat{1}\}$    

Thus it follows that $ W_{\hat{S}_z^2}\subseteq W_{\hat{S}_z}$
In fact we can define the morphisms $ G(f):W_{S_z}\rightarrow W_{S^2_z}$, where the function $ f$ indicates the operation of taking the square, as

$\displaystyle G(f)$ $\displaystyle :W_{S_z}\rightarrow W_{S^2_z}$    
  $\displaystyle \hat{0}\mapsto\hat{0}$    
  $\displaystyle \hat{1}\mapsto\hat{1}$    
  $\displaystyle \hat{P}_1\mapsto(\hat{P}_1\vee\hat{P}_2)$    
  $\displaystyle \hat{P}_2\mapsto(\hat{P}_1\vee\hat{P}_2)$    

In terms of propositions it is easy to see that the proposition $ \hat{S}^2_z=\frac{1}{4}$ is a coarse graining of the proposition $ \hat{S}_z=\frac{1}{2}$ since in terms of the projection operators we have

$\displaystyle \hat{S}_z=\frac{1}{2}:=\hat{P}_1\leq\hat{S}^2_z=\frac{1}{4}:=\hat{P}_2$    

Next: Appendix Up: topos Previous: Example of Coarse Graining
Cecilia Flori 2008-05-02