I know it is convention to taken Clebsch-Gordan $\color{blue}{<j_1,j_2;m_1,m_2|j_1,j_2;j,m>}$coefficients are real.If I want to make proof it's reality from any physical defination in backward then how can I do that?
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Mmmm... I don't think there is a proof as such, since it's simply a choice of phase for the individual highest weight states $|j,j\rangle$ states in the decomposition . It would be like asking for a proof that the coefficients $\sqrt{j(j+1)-m(m-1)}$ in the action of the ladder operator $J_-$ are real. This latter reality is merely a choice of phase for the states $|j,m\rangle$ relative to that of $|j,j\rangle$.
answered Oct 15, 2019 at 12:44
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$\begingroup$ In this case, the relevant proof would be to show that it is always possible to choose the phase so that the Clebsch-Gordan coefficients are real. $\endgroup$ Commented Nov 4, 2023 at 17:33