Linear response of a one-dimensional conductor coupled to a dynamical impurity with a Fermi edge singularity

I study the dynamical correlations that a quantum impurity induces in the Fermi sea to which it is coupled. I consider a quantum transport setup in which the impurity can be realized in a double quantum dot. The same Hamiltonian describes tunneling states in metallic glasses, and can be mapped onto the Ohmic spin-boson model. It exhibits a Fermi edge singularity, i.e., many fermion correlations result in an impurity decay rate with a nontrivial power-law energy dependence. I show that there is a simple relation between temporal impurity correlations on the one hand and the linear response of the Fermi sea to external perturbations on the other. This results in a power-law singularity in the space and time dependence of the nonlocal polarizability of the Fermi sea, which can be detected in transport experiments.