Show that the coherent state corresponding to the single-fermion problem worked out in Problems 21.3 through 21.6 is

where \(\theta\) and \(\phi\) are angular variables parameterizing a sphere \(S^{2}\). The coset representative \(\Omega(\xi)\) is the same as for the example worked out in Section 21.4 .

Data from Eq.  21.4

Construct the Hilbert space corresponding to the Lie algebra for a single fermion in Problem 21.3 . Remember that for a fermion the Pauli principle must be obeyed, which greatly restricts allowed states in the Hilbert space.

Data from Eq.  21.5

For the single-fermion example worked out in Problems 21.3 and 21.4 , take as a reference state the minimal weight \(\mathrm{SU}(2)\) state \(\left|\frac{1}{2}-\frac{1}{2}\rightangle\) corresponding to the fermion vacuum. Find the stability subgroup and the coset space.

Data from Eq.  21.6

Show that the coset representative is \[\Omega(\xi)=e^{\xi J_{+}-\xi^{*} J_{-}}=e^{\xi a^{\dagger}-\xi^{*} a}\]
for the generalized coherent state approximation corresponding to the single-fermion example worked out in Problems 21.3 , 21.4 , and 21.5 .

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