Relative Entropy for Fermionic Quantum Field Theory

TL;DR

This paper computes the relative entropy between quasifree states and their excitations in Fermionic Quantum Field Theory, extending previous scalar field results to fermions and applying it to Majorana fields in curved spacetime.

Contribution

It provides an explicit calculation of relative entropy for fermionic fields in quantum field theory, generalizing prior scalar field results and applying to curved spacetime scenarios.

Findings

Explicit formula for relative entropy between quasifree and excited states.

Extension of scalar QFT results to fermionic fields.

Application to Majorana fields on ultrastatic spacetimes.

Abstract

We study the relative entropy, in the sense of Araki, for the representation of a self-dual CAR algebra $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$. We notice, for a specific choice of $f \in \mathcal{H}$, that the associated element in $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ is unitary. As a consequence, we explicitly compute the relative entropy between a quasifree state over $\mathfrak{A}_{SDC}(\mathcal{H},\Gamma)$ 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.