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Scaling Limit of aDirac Particle Interacting with the Quantum Radiation Field

Asao Arai

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Aquantum system of aDirac particle -arelativistic charged particle with spin 1/2 -interacting with the quantum radiation field is considered and an eective particle Hamiltonian is derived as ascaling limit of the total Hamiltonian of the system.

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The Hamiltonian of a system of charged particles interacting with the electromagnetic field is investigated. For an arbitrary system the multipole expansion of the interaction between the system and the field is derived by means of a suitable canonical transformation. The transformed Hamiltonian is obtained from the Hamiltonian of the system by replacing the momenta by the transformed kinetic momenta and by adding to the Hamiltonian a term representing the interaction of the system with the electric component of the field. By expanding this interaction term, as well as the transformed momenta, in powers of the dimension of the system over the wavelength, the multipole expansion of the Hamiltonian is obtained. For a system interacting with a classical field the multipole form of the Hamiltonian is exactly equivalent to the original Hamiltonian. For a quantized field this is not true, and the multipole form of the transformed Hamiltonian is shown to be equivalent to the original Hamiltonian only for first-order radiation processes.
We consider a scaling limit of the Hamiltonian of the generalized spin-boson (GSB) model which is an abstract quantum field theoretical model of particles interacting with a Bose field. Applying it to a Hamiltonian of the field of the nuclear force with isospin, we obtain an effective potential of the interaction between nucleons. Also, we discuss an application to a Hamiltonian of a lattice spin system interacting with a Bose field and obtain a spin–spin interaction in the vacuum of the Bose field. An interaction model between a Fermi field and a Bose field yields an interaction in the vacuum of the Bose field.
In this article, we consider fixed spin-1/2 particles interacting through the quantized electromagnetic field in a constant magnetic field. We give approximate evolutions of coherent states. This uses spins-photon classical Hamiltonian mechanics. These approximations enable to derive that the approximate average fields and spins follow Maxwell-Bloch equations with a current density coming from spins. In addition, we obain a law concerning the evolution of the approximate average number of photons. Next, we provide stationary points of the spins-photon Hamiltonian when the spin particles belong to an orthogonal plane to the constant magnetic field. This allows for the construction of quasimodes with spins colinear to the constant magnetic field. Finally, a quasimode with an arbitrary high accuracy is built up and its first order radiative correction is computed.
There is no relativistic Hamiltonian for many particles systems except for free particles and this has been accepted since the 1960s from the work of Currie, Jordan and Sudarshan, Cannon and Jordan, and Leutwyler. This is the problem we will address in this paper by coupling relativistic particles with a scalar field. The Hamiltonian we derive is explicitly relativistic and in the non-relativistic limit gives a Hamiltonian with two body interaction. However, we find that the Poincare algebra is still not satisfied because of the inherent time delayed interaction in relativistic particle systems.
The classical limit and the infrared divergence problem for a non-relativistic charged quantum particle interacting with the quantized electromagnetic field are analized. Overall three momentum conservation is taken into account. A unitary transformation associated to the coherent state corresponding to a particle surrounded by a cloud of soft photons is performed upon the hamiltonian and the particle-field states. The transformed state represnting a moving dressed quantum particle and its energy are given by the Brillouin-Wigner perturbation theory. It is shown formally that the quantum energy approaches the classical electromagnetic field at the Plank constant h goes to zero. Moreover,all Feynman diagrams contributing to this quantum energy are infrared finite, without needing add diagrams of the same order in the electric charge to obtain the infrared finiteness. Those properties justify the usefulness of the unitary transformation. The Compton effect in the forward direction is studied using dressed charged particle states after the unitary transformation has been performed. The quantum cross section approaches the classical limit (Thomson´s formula) as h ͢͢͢͢͢ 0, and the Feynman diagrams are free of infrared divergences.