Maier-Saupe Theory of Nematics in 4D

We extend the Maier-Saupe theory of nematics to 4 dimensions. We consider the interaction of cylindrically symmetric particles, and derive an effective single particle potential. Using this, we obtain the free energy and the self-consistent equation for the order parameter -- a second rank traceless tensor. In 4D, the order parameter has three independent eigenvalues. We solve the self-consistent equation, and study the solutions as function of temperature. Our results give insight into the relation between orientational order parameters in different dimensions.