  In this article, we shall focus on the collective dynamics of the fermions in
a $\nu = 1$ quantum Hall droplet. Specifically, we propose to look at the
quantum Hall ferromagnet. In this system, the electron spins are ordered in the
ground state due to the exchange part of the Coulomb interaction and the Pauli
exclusion principle. The low energy excitations are ferromagnetic magnons. In
order to obtain an effective Lagrangian for these magnons, we shall introduce
bosonic collective coordinates in the Hilbert space of many-fermion systems.
These collective coordinates describe a part of the fermionic Hilbert space.
Using this technique, we shall interpret the magnons as bosonic collective
excitations in the Hilbert space of the many-electron Hall system. Furthermore,
by considering a Hall droplet of finite extent, we shall also obtain the
effective Lagrangian governing the spin collective excitations at the edge of
the sample.
