The Yoga of Linearization

This chapter gives the partitional treatment of the concept of a quantum observable. This is done using a bit of mathematical folklore that we call the “Yoga of Linearization.” It is a method to transform set concepts into the corresponding vector space concepts. Starting with a real-valued numerical attributeNumerical attribute  \(f:U\rightarrow \mathbb {R} \) on a set U (like weight, height, age, etc.), we apply the Yoga to obtain the notion of a real-valued operator on a vector space over a field containing \( \mathbb {R}\) . If U is taken as a basis set for a vector space over \( \mathbb {C}\) , then this procedure obtains a Hermitian operator, i.e., a quantum observable. But the Yoga also has an interesting intermediate step to transform set concepts into vector space concepts over the two element field \( \mathbb {Z}_{2}\) . This intermediate step gives a simplified pedagogical (or toy) model of QM over sets. Hence we develop that model to illustrate results in standard QM over \( \mathbb {C} \) such as the double-slit experiment and Bell’s Theorem.