Statics and dynamics of spin and electric dipoles in three, four, and other dimensions

Properly, spin is an antisymmetric tensor, and therefore in n-dimensional spaces where polar vectors have n components, spin has n(n-1)/2 components. Moreover, although a rotation can make an arbitrary polar vector have only one nonzero component, the same is not true for spin (and magnetic field). In particular, for n=4 an experimentalist can generate two independent field components (e.g., $H_{12}$ and $H_{34}$) and, further, systems can develop two types of spontaneous symmetry-breaking internal fields. To illustrate some dynamical implications of the additional field component we have derived the equation of motion for spin in n dimensions, and for n=4 we apply it to free Larmor precession, where we find two modes [at γ($H_{12}$±$H_{34}$)]. Simple ferromagnets and spin glasses are also discussed for n=4. Since no true two-component spin can exist in any dimension, we consider XY spin dynamics for n=3 spins S→ subject to strong uniaxial anisotropy. The behavior of electric and magnetic dipoles is contrasted, for the usual (n=3) case. It is also shown that normal modes for XY electric dipoles p→ have only an in-plane polarization, contrasting to XY spins, which have a normal mode with an out-of-plane magnetization. The hypothesis, for n=3, of dipoles due to magnetic charge and a ‘‘gyroelectric effect’’ (p→∝S→) is briefly discussed. It is noted that the usual concept of a scalar magnetic source (magnetic charge) is appropriate only to n=3.