For the single-fermion example worked out in Problems 21.3 and 21.4 , take as a reference state
Question:
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 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).
Data from Problem 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.
Step by Step Answer:
Symmetry Broken Symmetry And Topology In Modern Physics A First Course
ISBN: 9781316518618
1st Edition
Authors: Mike Guidry, Yang Sun