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Anyone who has studied quantum mechanics know the following relation: $ 2 \otimes 2 = 3 \oplus 1 $

But how did a man woke up and said "Hell yeah, I'll use tensor product of two spin $1/2$ to simulate the interaction of two particles with spin $1/2$" ? Why didn't he start with the direct sum ?

(And then, group theory made the magic leading to the relation above)

In fact, i'm wondering this because i don't fully understand why we use the tensor product to unite the two Hilbert's space.

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marked as duplicate by Jon Custer, ZeroTheHero, Kyle Kanos, Cosmas Zachos, Aaron Stevens Aug 28 at 17:45

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  • $\begingroup$ Are you referring to Clebsch–Gordan expansion? $\endgroup$ – Sunyam Nov 4 '17 at 22:07
  • $\begingroup$ If two particles each have $n$ states, the total number of possible states the two particles can be in is reasonably $n^2$, not $2n$. So that favors tensor product over direct sum. $\endgroup$ – knzhou Nov 4 '17 at 22:12
  • $\begingroup$ Not really, I just wonder how did Pauli (I assume it's him) knew he had to do a tensor product (and then the CG expansion) and not something else (like a direct sum) $\endgroup$ – user65854 Nov 4 '17 at 22:12
  • $\begingroup$ Total spin of two spin-1/2 particles $\endgroup$ – Frobenius Nov 4 '17 at 23:03
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    $\begingroup$ Possible duplicate of Should it be obvious that independent quantum states are composed by taking the tensor product? (and several other related questions). $\endgroup$ – Emilio Pisanty Aug 19 at 8:31
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The decomposition $\frac{1}{2}\otimes \frac{1}{2}\to 0\oplus 1$ is useful when the Hamiltonian (or at least its dominant part) is $SU(2)$-invariant. The part that mixes $SU(2)$ irreps can then be treated perturbatively.

Group theory notwithstanding, it’s a pretty natural thing to do from the perspective of finding eigenstates of commuting operators, since the eigenstates of the total angular momentum $\hat J^2$ naturally break into a (degenerate) triplet and a singlet.

The idea of irreducible representations and decomposing tensor products of representations (for finite groups) was well known from crystallography. But then came Wigner, who considerably expanded the use of representation theory in physics.

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