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To link the two approaches, it can be useful to understand the wave function as an object that takes values in a tensor space of C^n. In particular, if we had two particles, both spin-1/2 (C^2), then at the position configuration point {x,y}, the wave function takes a value in C^2 tensor C^2 where the tensoring is not done over the particle numbering 1 and 2 (which does not exist), but done over the set {x,y}. This is how one naturally leads into the choices of bosons vs. fermions.

To say it another way, the configuration of the system remains rooted in position in physical space and it is the value space of the wave function where spin resides. This is made most clear in Bohmian mechanics, where the position of the particle is always defined in the theory, but whether a particle will end up in the spin-z up or down regions does depend on the experimental setup. One can arrange experiments where the particle, due to symmetry for example, will always go up given the same starting point even if one flips the magnetic field in the Stern-Gerlach device so that the experimental conclusion would be spin up in one scenario and spin down in the other. Same exact path happens in both cases, but the spin value conclusion is the opposite. That is to say, while position is determined ahead of time, spin is not. Spin is not a real property of the particle; it is a property of the wave function.




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