Relative Entropy for Fermionic Quantum Field Theory

We study the relative entropy, in the sense of Araki, for the representation of a self-dual CAR algebra ${\mathfrak{A}}_{SDC}(\mathcal{H},\mathrm{\Gamma })$. We notice, for a specific choice of $f\in \mathcal{H}$, that the associated element in ${\mathfrak{A}}_{SDC}(\mathcal{H},\mathrm{\Gamma })$ is unitary. As a consequence, we explicitly compute the relative entropy between a quasifree state over ${\mathfrak{A}}_{SDC}(\mathcal{H},\mathrm{\Gamma })$ and an excitation of it with respect to the abovely mentioned unitary element. The generality of the approach, allows us to consider $\mathcal{H}$ as the Hilbert space of solutions of the classical Dirac equation over globally hyperbolic spacetimes, making our result, a computation of relative entropy for a Fermionic Quantum Field Theory. Our result extends those of Longo and Casini et al. for the relative entropy between a quasifree state and a coherent excitation for a free Scalar Quantum Field Theory, to the case of fermions. As a first application, we computed such a relative entropy for a Majorana field on an ultrastatic spacetime.