  Problems of strongly interacting electrons can be greatly simplified by
reducing them to effective quantum spin models. The initial step is
renormalization of the Hamiltonian into a lower energy subspace. The positive
and negative U Hubbard models are explicitely transformed into the Heisenberg
and -x-xz models respectively. Basic tools of quantum magnetism are introduced
and used: spin coherent states path integral, spin wave theory, and continuum
theory of rotators.The last lecture concerns pseudospin approaches to
superconductivity and superfluidity. The SO(3) rotator theory for the -x-xz
model describes a charge density wave to superconductor transition.
Analogously, Zhang's SO(5) rotator theory describes the antiferromagnet to
d-wave superconductor transition in high Tc cuprates. Finally the Magnus force
on the two dimensional vortices and their momentum, are derived from the Berry
phase of the spin path integral.
