Wick's Theorem for Spin-½ Operators, with an Application to Spin Waves in Antiferromagnets

An analog of Wick's theorem is developed for spin-½ operators, and a linked diagram expansion for spin Green's functions is derived. As an application we derive the familiar Anderson approximation for spin waves in antiferromagnets, and we then obtain the leading dynamical and kinematical corrections to that approximation. The Oguchi form of correction, usually obtained by a formal expansion in $\frac{1}{S}$ extrapolated to $S=\frac{1}{2}$, is found here as the leading term of an expansion in powers of $\frac{1}{z}$, where $z$ is the number of nearest neighbors. However, the Oguchi result is here found to be valid only for spin waves with wavelengths greater than two or three interspin distances.