This thesis is motivated by questions arising in the eld of Mathematical Quantum Electrodynamics, the attempt of a proper mathematical description of Quantum Electrodynamics (QED). QED is the relativistic quantum eld theory of electrodynamics, which uni es Quantum Mechanics and Special Relativity in a consistent manner. From the mathematical point of view, QED is an abelian gauge theory with the symmetry group U(1) (phase factors) and the gauge eld mediating the interaction between the charged spin-1/2 elds is the electromagnetic eld. Aim of this thesis is the renormalization of the regularized relativistic electron positron eld together with a Coulomb interaction. The idea of the renormalization procedure is to compare the normal-ordered Hamiltonian with the original one. Choosing a conventional normal ordering, the change in Hamiltonian is given by a quadratic term. The choice of a suitable normal ordering amounts in a non-perturbative rede nition of the electron/positron states. This allows for the interpretation of change in the Hamiltonian as a certain renormalization. The proper interpretation of this renormalization and the various implications are discussed at the end.
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
4.1.1 Bare and normal ordered Hamiltonian
First we recall the most important facts from section 3 concerning the field operators and the non vanishing anticommutators between the creation & annihilation operators, which have to be used for normal ordering.
The bare interaction energy is only partially normal ordered, its renormalization will be determined here by carrying out the full normal ordering. This entails determining the splitting of h which is denoted by subscript α.
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.
- Quote paper
- Vanessa Grabelt (Author), 2011, Renormalization of the regularized relativistic electron-positron field, Munich, GRIN Verlag, https://www.grin.com/document/175581
