Equation (8) defines the conditional wave function of a microscopic physical system in terms of the wave function of the universe and the true positions of the particles in the environment (= the universe except the microscopic system). Such a definition is essential for the interpretation of quantum mechanical systems which are not usually analyzed together with the whole universe.
It is well-known that electrons, protons and neutrons, the constituents of ordinary matter, are fermions and hence have spin. Thus the wave function of the universe must account for particles with spin 1/2. But equation (8) ignores the spin. What is the correct generalization to spinning particles?
I admit that I lack understanding. But textbooks usually give at least explicit computations the first time something is encountered, so that readers have a template to do the exercises for themselves. Please provide the template so that I can improve my understanding. After all, a triviality should be easy to explain with all details.the generalization is trivial. [...] But of course, to see that something is trivial requires understanding. That's why textbooks have exercises, to test your understanding.