Dynamical C*-Algebras and Quadratic Perturbations for Fermions

We study interacting Fermionic fields on globally hyperbolic spacetimes in the context of dynamical C*-algebras. Our focus is on a particular class of interactions, that of kinetic perturbations; we provide their definition and we show that these induce, in the fermionic dynamical C*-algebra, symbols that obey the canonical anticommutation relation in a locally perturbed spacetime. We also study the adjoint action on the CAR algebra, a C*-subalgebra of the fermionic dynamical C*-algebra, induced by symbols associated with functionals that are compactly supported, self-adjoint, gauge-invariant, quadratic, and which induce normally hyperbolic operators (these are objects which can be interpreted as quadratic perturbations of the free Dirac Lagrangian); we show that this amounts to a Bogoliubov transformation of the CAR algebra. Said Bogoliubov transformation is shown to satisfy the Shale-Stinespring criterion for every gauge-invariant, quasifree, Hadamard state on the CAR algebra in a globally hyperbolic spacetime with compact Cauchy surfaces, and for the Minkowski vacuum; therefore the Bogoliubov transformation is implemented via unitary operators on the associated representation Fock space. In the case of the Minkowski vacuum, we also show that the phases of the unitary implementers can be chosen so that these satisfy the Bogoliubov causal factorization, provided that the representation of the perturbed CAR algebra satisfies twisted Haag duality.