Particles and Strings in Degenerate Metric Spaces

We consider relativistic and non-relativistic particles and strings in spaces (or space-times) with a degenerate metric. We show that the resulting dynamics is described by a rich structure of constraints. All constraints are classified and the dynamics depends strongly on the parity of the difference between the dimension of the space (or space-time) and the rank of the degenerate metric. For a particular class of degenerate metrics we can identify the null eigenvectors of the metric with its Killing vectors. We also give the first steps towards the quantization of the non-relativistic particle using the Senjanovic path integral quantization and the Batalin-Fradkin-Tyutin conversion method.