  In this article we are concerned with finite dimensional Fermions, by which
we mean vectors in a finite dimensional complex space embedded in the exterior
algebra over itself. These Fermions are spinless but possess the characterizing
anticommutativity property. We associate invariant complex vector fields on the
Lie group $\mathrm{Spin}(2n+1)$ to the Fermionic creation and annihilation
operators. These vector fields are elements of the complexification of the
regular representation of the Lie algebra $\mathfrak{so}(2n+1)$. As such, they
do not satisfy the canonical anticommutation relations, however, once they have
been projected onto an appropriate subspace of $L^2(\mathrm{Spin}(2n+1))$,
these relations are satisfied. We define a free time evolution of this system
of Fermions in terms of a symmetric positive-definite quadratic form in the
creation-annihilation operators. The realization of Fermionic creation and
annihilation operators brought by the (invariant) vector fields allows us to
interpret this time evolution in terms of a positive selfadjoint operator which
is the sum of a second order operator, which generates a stochastic diffusion
process, and a first order complex operator, which strongly commutes with the
second order operator. A probabilistic interpretation is given in terms of a
Feynman-Kac like formula with respect to the diffusion process associated with
the second order operator.
