  Let $\Cl$ denote the Clifford algebra over $\R^n$, which is the von Neumann
algebra generated by $n$ self-adjoint operators $Q_j$, $j=1,...,n$ satisfying
the canonical anticommutation relations, $Q_iQ_j+Q_jQ_i = 2\delta_{ij}I$, and
let $\tau$ denote the normalized trace on $\Cl$. This algebra arises in quantum
mechanics as the algebra of observables generated by $n$ Fermionic degrees of
freedom. Let $\Dens$ denote the set of all positive operators $\rho\in\Cl$ such
that $\tau(\rho) =1$; these are the non-commutative analogs of probability
densities in the non-commutative probability space $(\Cl,\tau)$. The Fermionic
Fokker-Planck equation is a quantum-mechanical analog of the classical
Fokker-Planck equation with which it has much in common, such as the same
optimal hypercontractivity properties. In this paper we construct a Riemannian
metric on $\Dens$ that we show to be a natural analog of the classical
2-Wasserstein metric, and we show that, in analogy with the classical case, the
Fermionic Fokker-Planck equation is gradient flow in this metric for the
relative entropy with respect to the ground state. We derive a number of
consequences of this, such as a sharp Talagrand inequality for this metric, and
we prove a number of results pertaining to this metric. Several open problems
are raised.
