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Consider the direct product space of two angular momentum eigenfunctions:

$$|j_1, j_2; m_1, m_2⟩ = |j_1, m_1⟩|j_2, m_2⟩$$

for the simple case when

$$j_1 = j_2 = 1/2.$$

How can i construct the direct product space explicitly and make the connection with group representation theory?

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The direct product space is spanned by the 4 states $\vert{s_1m_1}\rangle\vert{s_2,m_2}\rangle$. This Hilbert space carries a representations of $\frac{1}{2}\otimes\frac{1}{2}$, and group theory tells you this representation is reducible as $1\oplus 0$.

Moreover, group theory explicitly helps in constructing states which reduce this representation since the search for irreducible subspaces amounts to a search for highest weight states, i.e. states killed by the action of the (here unique) raising operator so that $J_+\vert{j,j}\rangle=0$.

Once you have the highest weight you can generate the rest of the irreducible piece by the action of $J_-$.

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  • $\begingroup$ Thank you very much for your help.I have one more question: which are the invariant subspaces and why? $\endgroup$ – user243882 Oct 5 '19 at 14:34
  • $\begingroup$ The invariant subspaces are generates by lowering from the highest weight state, and they are invariant by definition. Thus the 3-dim subspace generated by lowering from the highest weight $\vert 1/2,1/2\rangle\vert 1/2,1/2\rangle$ is invariant. $\endgroup$ – ZeroTheHero Oct 5 '19 at 14:41

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