## Why isn't spin a 5th dimension?

Since spin is a separate variable in a wave function, independent from its location in spacetime, why isn't it considered a dimension beyond the 3+1 of spacetime?
 The spin vector (if we allow ourselves to use the semi-classical picture, in which spin forms a vector) has a fixed lenght for any particle (that is, the electron spin 1/2 doesn't change), so it's not really a dimension in that (infinite) sense. You could add a number of "spin dimensions" equal to the number of components of the spin vector and restrict it to a sphere with radius S, but this product space would only describe one particle so it's not very useful. This is why one, for spin chains, sometimes form local product spaces (fibre bundles) where, for each site, there is a space for the spin to rotate in. Such geometrical constructions don't necessarily make the physics easier to learn though.
 Thank you, Hypersphere. Excellent, complete answer.

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