momentum operators:

[rCl]{} \[Jp\_orbital\] \[J\_+, \] && \_+() = e\^[i]{}(\_+i\_) ,\
\[Jm\_orbital\] \[J\_-,\] && \_[-]{}() = -e\^[-i]{}(\_-i\_) ,\
\[Jz\_orbital\] \[J\_z, \] && \_[z]{}() = -i .

Takahashi-Shibata-Schwinger representation
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Working with spin coherent states can eventually be cumbersome as they do not have the simplicity of standard coherent states. Here we shall also use a different approach put forth by Takahashi and Shibata [@Takahashi1975], which consists of first using the Schwinger operators to map the spin operators into two bosonic modes and then defining standard phase-space measures using bosonic coherent states. We shall thus refer to this as the Takahashi-Shibata-Schwinger (TSS) approach. This method gives the same result that would be obtained without resorting to the mapping, but considerably simplifies the formal approach to the problem.

We thus proceed by implementing Schwinger’s map that transforms the spin operators into two bosonic operators $a$ and $b$ according to $$\begin{aligned}
J_z &=& \frac{1}{2}(a^\dagger a - b^\dagger b),\qquad J_+ = (J_-)^\dag= a^\dagger b.    
\end{aligned}$$ To fix the total spin $J$, we impose to work on the restricted subspace where $n_a + n_b = 2J$, with $n_{a,b}$ the expectation value of the number operators for the two Schwinger modes. We now introduce standard bosonic 