Finite dimensional systems of free Fermions and diffusion processes on Spin groups

Abstract: 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.