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 .

Data from Eq.  21.3

Show that the fermion operator set \(\left\{a, a^{\dagger}, a^{\dagger} a-\frac{1}{2}\right\}\) obeys

and that this is equivalent to the \(\mathrm{SU}(2)\) Lie algebra of Eq. (3.18).

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.

Transcribed Image Text:

[a,a] = 2(a*a-) [a'a-,a] = ==a [a'a-, a'] = a,