Renormalization of the regularized relativistic electron-positron field

Contents

1 Motivation

2 Basics of QED

2.1 Why Fields?

2.2 Fields

2.3 Canonical Quantization

2.4 Renormalization

2.4.1 Bare and renormalized parameters

2.4.2 First-order radiative corrections in QED

2.4.3 Normal ordering

3 Mathematical description

4 Calculations

4.1 Fully normal ordered interaction

4.1.1 Bare and normal ordered Hamiltonian

4.1.2 Interim result for connection between bare and normal ordered Hamiltonian

4.1.3 Cutoff dependent constant

4.1.4 Terms with one particle operators

4.2 Dressed electron

4.2.1 Zero bare mass

4.2.2 Positive bare mass

4.3 Properties of the renormalized Hamiltonian

5 Interpretation

Objectives and Topics

The primary objective of this thesis is to provide a rigorous mathematical renormalization of the regularized relativistic electron-positron field, incorporating Coulomb interaction, by migrating the electrostatic energy into a renormalized Dirac-type operator using a non-perturbative redefinition of states.

• Mathematical foundations of Quantum Electrodynamics (QED)
• Renormalization and normal ordering procedures
• Calculation of the dressed electron and renormalized Hamiltonian
• Analysis of mass, charge, and field renormalization implications

Excerpt from the Book

Summary of Chapters

Motivation: Introduces the field of Mathematical Quantum Electrodynamics and the thesis objective of performing a rigorous renormalization of the electron-positron field.

Basics of QED: Establishes the physical and mathematical foundations, including field theory, canonical quantization, and the conceptual need for renormalization.

Mathematical description: Provides the formal setup of the Hilbert space, Fock space, annihilation/creation operators, and the Dirac operator used for the calculations.

Calculations: The core chapter where the transition from the bare to the renormalized Hamiltonian is computed, including the derivation of the "dressed electron" and proof of convergence via the Banach fixed point theorem.

Interpretation: Discusses the physical implications of the mathematical findings, specifically analyzing mass, field, and charge renormalization.

Keywords

Quantum Electrodynamics, QED, Renormalization, Hamiltonian, Dirac operator, Normal ordering, Coulomb interaction, Dressed electron, Banach fixed point theorem, Field theory, Vacuum polarization, Self-energy, Vertex correction, Bare parameters, Radiative corrections

Frequently Asked Questions

What is the core subject of this thesis?

The work focuses on the mathematical renormalization of the regularized relativistic electron-positron field within the framework of Quantum Electrodynamics (QED).

What are the primary areas of research?

The research covers canonical quantization, normal ordering as a renormalization technique, the derivation of a renormalized Hamiltonian, and the theoretical interpretation of mass and charge renormalization.

What is the main goal or research question?

The goal is to determine if the original (bare) Hamiltonian can be rigorously related to a renormalized Hamiltonian by redefining one-electron states in a non-perturbative manner.

Which scientific methods are applied?

The author uses operator theory, field quantization, and specifically the Banach fixed point theorem to demonstrate the existence of solutions for the coupled equations describing the system.

What topics are discussed in the main section?

The main section details the full normal ordering of the interaction energy, the mathematical derivation of the "dressed electron" operator, and the proof that the transformation exists for non-zero bare masses.

Which keywords best characterize this work?

Key terms include QED, renormalization, normal ordering, dressed electron, and the Hamiltonian of the relativistic electron-positron field.

What is the role of the Coulomb interaction in this study?

The Coulomb interaction is treated as an electrostatic potential that is migrated into the renormalized Dirac operator during the renormalization process.

How does the author define the "dressed electron"?

The dressed electron is identified by the non-perturbative choice of one-electron states, allowing the Hamiltonian to be expressed in a form similar to a Dirac operator.

What is the significance of the Banach fixed point theorem here?

It is used to prove that the system of equations derived for the mass and field operators is solvable, confirming the existence of a unique fixed point under certain conditions.