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 Problem  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).

Transcribed Image Text:

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