Elementary example of exact effective-Hamiltonian computation

TL;DR

This paper demonstrates an exact method for computing effective Hamiltonians in a simplified Yukawa model using the RGPEP, illustrating how perturbative and approximation methods improve with evolution.

Contribution

It provides an exact solution for effective Hamiltonians in a simplified Yukawa model, showcasing the RGPEP method's potential for realistic theories.

Findings

Effective Hamiltonians can be computed exactly in the model.

Perturbative expansion accuracy improves with RGPEP evolution.

Tamm-Dancoff approximation becomes more accurate along the RGPEP flow.

Abstract

We present an exact computation of effective Hamiltonians for an elementary model obtained from the Yukawa theory by going to the limit of bare fermions being infinitely heavy and bare bosons being at rest with respect to the fermions that emit or absorb them. The coupling constant can be arbitrarily large. The Hamiltonians are computed by solving the differential equation of the renormalization group procedure for effective particles (RGPEP). Physical fermions, defined in the model as eigenstates of the effective Hamiltonians, are obtained in the form of an effective fermion dressed with a coherent state of effective bosons. The model computation illustrates the method that can be used in perturbative computations of effective Hamiltonians for realistic theories. It shows the mechanism by which the perturbative expansion and Tamm-Dancoff approximation increase in accuracy along the RGPEP evolution.