Why isn't spin a 5th dimension?

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SUMMARY

The discussion centers on the classification of spin in quantum mechanics, specifically why it is not regarded as a fifth dimension beyond the established 3+1 dimensions of spacetime. Spin is treated as a separate variable in a wave function, maintaining a fixed length for particles like electrons with spin 1/2. While one could theoretically create a "spin dimension" by considering the components of the spin vector, this approach is limited and primarily applicable to individual particles. The use of local product spaces or fibre bundles for spin chains is mentioned, but these geometrical constructs complicate rather than simplify the understanding of the underlying physics.

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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?
 
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The spin vector (if we allow ourselves to use the semi-classical picture, in which spin forms a vector) has a fixed length 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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