  We define and investigate the properties of the jaggedness of path integral
trajectories. The new quantity is shown to be scale invariant and to satisfy a
self-averaging property. Jaggedness allows for a classification of path
integral trajectories according to their relevance. We show that in the
continuum limit the only paths that are not of measure zero are those with
jaggedness 1/2, i.e. belonging to the same equivalence class as random walks.
The set of relevant trajectories is thus narrowed down to a specific subset of
non-differentiable paths. For numerical calculations, we show that jaggedness
represents an important practical criterion for assessing the quality of
trajectory generating algorithms. We illustrate the obtained results with Monte
Carlo simulations of several different models.
