The gravitational field about a spinning particle

We consider the action for N = I and N = 2 spinning particles in the presence of a background gravitational field. The action for the gravitational field induced by one-loop effects is examined to lowest order in the metric. This is the one-dimensional analogue of calculations performed in two-dimensional nonlinear sigma models. The inherent infrared divergences are quite severe, and it is found that the effective action depends crucially on how they are treated, as is the case in two-dimensional nonlinear sigma models. Using one approach, an intractable infrared divergence arises, while with another technique, the effective action vanishes. A calculational technique introduced by Onofri is employed. A novel N = 4 spinning particle is considered briefly.