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I read that we need scalars, spinors, vectors and rank two tensors to describe spin-0, spin-1/2, spin-1 and spin-2 particles, respectively.

But then I reacall from quantum mechanics courses that the intrinsic spin of a particle is described by different finite representations of so(3), (2j+1)(2j+1) matrices acting on (2j+1) vectors, where j is half-integer or integer.

Why then tensors to describe spin-2 particles? Why not a 5-vector that gets transformed by 5x5 matrices that belong to SO(3)?

(EDIT: Or do all half-integer-j reps of so(3) behave like spinors (particles have spin 1/2) and all integer-j reps of so(3) behave like vectors (particles have spin 1)?

Or has it something to do with when we go from SO(3) to SO(3,1)?

hope my question makes sense

thank you

But then I reacall from quantum mechanics courses that the intrinsic spin of a particle is described by different finite representations of so(3), (2j+1)(2j+1) matrices acting on (2j+1) vectors, where j is half-integer or integer.

Why then tensors to describe spin-2 particles? Why not a 5-vector that gets transformed by 5x5 matrices that belong to SO(3)?

(EDIT: Or do all half-integer-j reps of so(3) behave like spinors (particles have spin 1/2) and all integer-j reps of so(3) behave like vectors (particles have spin 1)?

Or has it something to do with when we go from SO(3) to SO(3,1)?

hope my question makes sense

thank you

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