# What are phase conventions in angular momentum and rotation calculations?

I work with complicated angular momentum calculations related to atomic physics; nevertheless, I never need to use anything related to a phase convention (apparently because it's taken care of in a consistent way everywhere with me not noticing it), but I keep reading this term within subjects related to Clebsch-Gordan coefficients, Wigner D-matrices, 3j and 6j symbols, and Racah W-Coefficients, and more.

Can someone please explain what a phase convention is and how to use/utilize it?

Or guide me to some good book that would explain that?

Since time reversal $T$ anticommutes with the total angular momentum, it is convenient to combine $T$ with a rotation $R$ through the angle $\pi$ about an axis perpendicular to the $z$-axis (the axis of space quantization). Such a rotation also inverts [angular momentum] $I_z$ and thus \begin{align*} [ RT, I_z ] &= 0 \\ [ RT, (\mathbf I)^2] &= 0. \end{align*} It is therefore possible to construct a set of basis states with quantum numbers $IM$, which are also eigenvectors of $RT$. By suitably choosing the phases of these states, the eignvalues of $RT$ may be set equal to unity, $$RT \left| \alpha IM\right> = \left| \alpha IM \right>,$$ where $\alpha$ represents a set of additional quantum numbers specifying the internal structure of the states. The conventional phasing corresponds to choosing the rotation axis of $R$ to be the $y$ axis.
In the representation in which $j_z$ is diagonal, the nonvanishing matrix elements of the angular momentum operators are \begin{align*} \left< jm \middle| j_z \middle| jm\right> &= m \\ \left< jm\pm1 \middle| j_x\pm ij_y \middle| jm\right> &= \sqrt{(j\mp m)(j\pm m+1)} \end{align*} The nondiagonal matrix elements of $j_x\pm ij_y$ involve arbitrary phase factors associated with the choice of relative phases for the states with different $m$. The phase convention [above] implies that the matrix elements of $j_x$ are real while those of $j_y$ are purely imaginary, since $j_x$ commutes with $RT$ while $j_y$ anticommutes with $RT$. We are thus left with arbitrary real phase factors (±1), which are conventionally fixed by the requirement that the matrix elements of $j_x ± i j_y$ be positive (Condon and Shortley, 1935).
Nowadays these phase "choices" are swallowed up in the standard definitions of the spherical harmonics (particularly the relationship between phase and $m$) and in the typical representation for the three spin axes (one diagonal, one purely real and one purely imaginary).