We present a generalized rate equation model for the array and polarization dynamics of general one- or two-dimensional arrays of vertical-cavity surface-emitting lasers (VCSELs). It is demonstrated that our model includes both the previous theory for edge-emitting laser arrays and the theory of polarization dynamics in quantum-well VCSELs in a single unified description. The model is based on the physical assumption of separated carrier density pools individually coupled to different light field modes. These modes, defined by the separate gain profile of each pool, interact through the coherent dynamics of the light field derived from Maxwell's equations and represented by the coefficients for index and loss guiding. The special case of two densities and two light field modes is solved and the implications of the results for large VCSEL arrays are discussed. Our analytic results show that typical solutions of the split-density model range from phase locking to chaos, depending on the magnitude of the frequency coefficients. For weak coupling, the stable supermode is always the mode of highest frequency. This indicates that anti-phase locking is the only stable phase locking possible in arrays.