Renormalization of bosonic quadratic Hamiltonians involving rank-one perturbations

TL;DR

This paper addresses the renormalization of bosonic quadratic Hamiltonians with ultraviolet divergences, explicitly diagonalizing them via Bogoliubov transformations and exploring different renormalization schemes based on the field operator's smearing.

Contribution

It provides an explicit diagonalization method for bosonic quadratic Hamiltonians with divergences and analyzes various renormalization approaches depending on the smearing of the field operator.

Findings

Explicit diagonalization of the Hamiltonian using Bogoliubov transformations

Different renormalization schemes depending on field smearing

Clarification of self-adjointness and divergence handling

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.