## Example of Coarse Graining Presheaf

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# Example of Coarse Graining Presheaf

In order to give a more rigorous explanation of the process of Coarse Graining I will consider an example concerning propositions in Quantum Mechanics, and the way in which these propositions are said to be true or false. The reason why I have chosen this example is given by the fact that the process of Coarse Graining will be essential in the definition of a Quantum Logic using Topos Theory (see section Topos and logic Topos and Logic . ).

In order to carry out this example we need to accept a few things which will not be proved in this context, the reader should refer to , ,  , My Master Dissertation 2005 for proves and explanations.
So what we need to accept is the following:
1. In Quantum Mechanics propositions of the form $(A\in\Delta)$2.3 are represented by spectral projectors $\hat{E}[A\in\Delta]=\hat{P}_{A\in\Delta}$
2. Spectral projectors related to different propositions belong to different spectral algebras (see section 3.7)

3. In Quantum Mechanics valuations of propositions of the form $A\in\Delta$ can be given in terms of their coarse granings $f(A)\in f(\Delta)$.
Statement 3) implies that in some way we can compare proposition belonging to different spectral algebras. In fact, the projector $P_{A\in\Delta}$, representing proposition $A\in\Delta$, belongs to a different spectral algebra as does the projector $P_{f(A)\in f(\Delta)}$ which represents the proposition $f(A)\in f(\Delta)$. Therefore, in order to assign valuations to propositions in terms of their coarse grainings, one requires to compare propositions which are related to different spectral algebras. This is precisely what the coarse graining presheaf does. In fact the coarse graining presheaf defines a logical ordering between propositions, where the logical ordering is given in terms of the scope of the implications of a proposition. For example: the proposition $A\in\Delta$ implies its coarse graning $f(A)\in f(\Delta)$ since, intuitively, if $A\in\Delta$ is true, then $f(A)\in f(\Delta)$ is also necessarily true. Therefore, $f(A)\in f(\Delta)$ is considered weaker than $A\in\Delta$ since the former is contained within the scope of implications of the latter. It follows that the coarse granings of a proposition X belong to the group of implications of X. To clarify, let us consider a proposition of the form $A\in\Delta$ whose associated projection operator is $\hat{E}[A\in\Delta]=\hat{P}_{A\in\Delta}$. Given a Borel function $f:\sigma(\hat{A})\rightarrow{\mathchoice {\setbox0=\hbox{\displaystyle\rm R}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{\textstyle\rm R}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{\scriptstyle\rm R}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}} {\setbox0=\hbox{\scriptscriptstyle\rm R}\hbox{\hbox to0pt {\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}$ such that $\hat{B}=f(\hat{A})$ then, from the Spectral Theorem2.4, it follows that $\displaystyle \hat{E}[A\in\Delta]\leq\hat{E}[f(A)\in f(\Delta)]\equiv\hat{P}_{A\in\Delta}\leq \hat{P}_{f(A)\in f(\Delta)}$ (2.1)

However, since $\hat{P}_{A\in\Delta}$ belongs to spectral algebra $W_A$, whereas $\hat{P}_{f(A)\in f(\Delta)}$ belongs to spectral algebra $W_{f(A)}$, then equation 2.1 implies that we "projected" the operator $\hat{P}_{f(A)\in f(\Delta)}$ from its spectral algebra $W_{f(A)}$ to $W_A$, so that the partial order ( $<$) in equation 2.1 is defined with respect to $W_A$. This is mathematically translated as $\displaystyle \hat{P}_{A\in\Delta}\leq i_{W_{f(A)}W_A}\hat{P}_{f(A)\in f(\Delta)}$

The map $i_{W_{f(A)}W_A}:W_{f(A)}\rightarrow W_A$ in the above equation represents an inclusion map i.e. $W_{f(A)}\subseteq W_A$. Therefore "implication ordering" of propositions, such as $A\in\Delta$, is given in terms of the partial ordering of their respective spectral algebras where, partial ordering is identified as subset inclusion. We can then reiterate that the conceptual motivation behind the definition of the coarse graning presheaf G, is the need to be able to compare propositions which are related to different spectral algebras, so as to define "implication ordering" between them.

Next: Worked example of the Coarse Graining Presheaf Up: Contents Previous: Presheaves in Quantum Mechanics
Cecilia Flori 2007-03-17