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This question comes from seeing that the triangle addition rule for quantum mechanics comes out of groups/representation theory; I thought this was odd as we haven't used any group ideas in QM up to this point. After thinking about it I realised/belive that it comes from the use of a tensor product.

In quantum mechanics if we have 2 systems (2 particles, a particle with spin and orbital wavefunctions etc) why do we join them as a tensor product and not a direct sum or cartesian product or a new 'quantum product'?

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