  Identifying the time reversal symmetry of spins as a symplectic symmetry, we
develop a large N approximation for quantum magnetism that embraces both
antiferromagnetism and ferromagnetism. In SU(N), N>2, not all spins invert
under time reversal, so we have introduced a new large N treatment which builds
interactions exclusively out of the symplectic subgroup [SP(N)] of time
reversing spins, a more stringent condition than the symplectic symmetry of
previous SP(N) large N treatments. As a result, we obtain a mean field theory
that incorporates the energy cost of frustrated bonds. When applied to the
frustrated square lattice, the ferromagnetic bonds restore the frustration
dependence of the critical spin in the Neel phase, and recover the correct
frustration dependence of the finite temperature Ising transition.
