TITLE: Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients
QUESTION [5 upvotes]: An identity exists for CG coefficients:
$$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$
But why is there a phase factor $(-1)^{j_1+j_2-J}$?
It seems to me that 
$$|JM\rangle =\sum_{m_{1},m_{2}}|j_{1}m_{1}\rangle\otimes|j_{2}m_{2}\rangle\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle
 =\sum_{m_{1},m_{2}}|j_{2}m_{1}\rangle\otimes|j_{1}m_{2}\rangle\langle j_{2}m_{2}j_{1}m_{1}|JM\rangle
 $$
And since $|j_{1}m_{1}\rangle\otimes|j_{2}m_{2}\rangle$ and $|j_{2}m_{1}\rangle\otimes|j_{1}m_{2}\rangle$ are the same physical state, there should be no difference between $\langle j_1 m_1 j_2 m_2 |J M \rangle$ and $\langle j_2 m_2 j_1 m_1|J M\rangle$. What do I get wrong?

REPLY [3 votes]: The actual overall phase of a Clebsch-Gordan is a matter of convention.  This is easy to see in the simplest example: the coupling of two spin-1/2 particles.  The state with angular momentum L=0 and M=0 is antisymmetric and can be written (up to a normalization) as 
$$|\textstyle\frac{1}{2},\frac{1}{2}\rangle_1|\frac{1}{2},-\frac{1}{2}\rangle_2 - |\frac{1}{2},-\frac{1}{2}\rangle_1|\frac{1}{2},\frac{1}{2}\rangle_2$$
The relative phase is essential if this state is to be orthogonal to the L=1,M=0 state, but we could also have taken 
$$-|\textstyle\frac{1}{2},\frac{1}{2}\rangle_1|\frac{1}{2},-\frac{1}{2}\rangle_2 + |\frac{1}{2},-\frac{1}{2}\rangle_1|\frac{1}{2},\frac{1}{2}\rangle_2$$
as the $L=0, M=0$  state: this second choice is still annihilated by $L_+$ and still an eigenstate of $L_z$ with eigenvalue $0$, thus is an equally good choice of $L=0,M=0$ state.
In more general cases, whenever $L$ is not $L_1+L_2$, the Clebsch's must contain some negative signs to enforce orthogonality between states of different $L$'s but same values of $M$. The relative position of these signs is fully determined by orthogonality with other states, but the overall sign is a matter of convenience: what we call the first or the second "system" is purely a matter of convention.
There are several conventions used to choose the overall sign.  In angular momentum theory, the most common is the Condon-Shortley (but it is not the only one).  The encyclopedic text by Varshalovich et al gives a summary of the conventions used by several authors.
One of the features of the CS phase convention is precisely that it produces phase relations of the type given above when we interchange two labels.  This choice is therefore convenient for tabulation purposes: basically one only needs to tabulate the coefficients when $L_1\ge L_2$ since the cases where $L_2>L_1$ are obtained by permutation and related to the original case by a phase change.
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