Renormalization of bosonic quadratic Hamiltonians involving rank-one perturbations
Thomas Gamet
TL;DR
This work analyzes ultraviolet divergences in bosonic quadratic Hamiltonians with rank-one perturbations and provides a rigorous Bogoliubov-diagonalization framework. Depending on the regularity of the form factor $f$ relative to the one-particle operator $\omega$, the authors prove existence (and explicit form) of a renormalized, self-adjoint Hamiltonian either without renormalization, or with renormalization of energy and/or charge, via a diagonalization $\mathbb{U}_{\lambda}^* \mathbb{H}_{\lambda} \mathbb{U}_{\lambda} = \mathrm{d}\Gamma(\xi_{\lambda}) + \text{const}$. They establish precise trace-class conditions ensuring the Bogoliubov implementation and derive convergence results for regularized models, including norm-resolvent convergence of $\mathrm{d}\Gamma(\xi_{\lambda,n})$ to $\mathrm{d}\Gamma(\xi_{\lambda})$. In irregular regimes, they show that while a Bogoliubov transform may fail to exist, the diagonalized Hamiltonians can still converge, shedding light on when charge renormalization is necessary and highlighting cases of non-equivalence between formal and renormalized theories. The findings connect to Pauli-Fierz-type models and offer a controlled, explicit treatment of energy and coupling constant renormalization in simple quadratic bosonic systems.
Abstract
We study the renormalization of a bosonic quadratic Hamiltonian with an ultraviolet divergence. The Hamiltonian is composed of the sum of a free part and the square of the smeared field operator. We explicitly diagonalize the Hamiltonian via Bogoliubov transformations, thus simplifying its definition as a self-adjoint operator. Depending on the field operator's smearing, we discuss different renormalizations, either of the energy alone, or the energy and coupling constant together.