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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 [1], [6], [2] , My Master Dissertation 2005 for proves and explanations.
So what we need to accept is the following:
- In Quantum Mechanics propositions of the form
2.3 are represented by spectral
projectors
- Spectral projectors related to different propositions belong to different spectral algebras (see section 3.7)
- In Quantum Mechanics valuations of propositions of the form can be given in terms of their coarse granings .
However, since belongs to spectral algebra , whereas belongs to spectral algebra , then equation 2.1 implies that we "projected" the operator from its spectral algebra to , so that the partial order () in equation 2.1 is defined with respect to . This is mathematically translated as
The map in the above equation represents an inclusion map i.e. . Therefore "implication ordering" of propositions, such as , 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