A novel approach to the study of fermionic systems in d-dimensional Euclidean spacetime is presented according to which an original, field-theoretical form of description is converted into a particle-based language. An important aspect of the advocated procedure is that it employs a spacetime resolution scale which does not have to serve, at the same time, as an ultraviolet cutoff for matter field fluctuations. At the particle level of description, such fluctuations are independently regularized by a scale associated with a ``proper-time'' parameter. A key feature of our representation for fermionic systems is its purely geometrical content. In particular, Polyakov's spin factor, which enters the path integral description of spin-1/2 entities, emerges very naturally in the course of passing from the field-theoretical to the particle-based language. The applications considered in this paper pertain to evaluations of the Dirac determinant. In the presence of a coupling to an external gauge field, such computations lead to effective-action terms. Both Maxwell and topological terms are retrieved in two, three, and four spacetime dimensions.