Renormalization of the regularized relativistic electron-positron field

Inhaltsverzeichnis (Table of Contents)

• Motivation
• Basics of QED
  • Why Fields?
  • Fields
  • Canonical Quantization.
  • Renormalization
    • Bare and renormalized parameters
    • First-order radiative corrections in QED
    • Normal ordering
• Mathematical description
• Calculations
  • Fully normal ordered interaction.
    • Bare and normal ordered Hamiltonian
    • Interim result for connection between bare and normal ordered Hamiltonian. . .
    • Cutoff dependent constant
    • Terms with one particle operators
  • Dressed electron.
    • Zero bare mass
    • Positive bare mass
  • Properties of the renormalized Hamiltonian .
• Interpretation

Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)

This thesis delves into the realm of Mathematical Quantum Electrodynamics, aiming to provide a rigorous mathematical framework for the physical theory of Quantum Electrodynamics (QED). QED, as a relativistic quantum field theory of electrodynamics, integrates Quantum Mechanics and Special Relativity coherently. Mathematically, QED is defined as an abelian gauge theory with the symmetry group U(1), where the gauge field mediating the interaction between charged spin-1/2 fields is the electromagnetic field. The thesis focuses on the renormalization of the regularized relativistic electron-positron field, incorporating a Coulomb interaction. This renormalization procedure involves comparing the normal-ordered Hamiltonian with the original one. Choosing a standard normal ordering, the change in Hamiltonian manifests as a quadratic term. The selection of a suitable normal ordering corresponds to a non-perturbative redefinition of the electron/positron states, leading to the interpretation of the change in Hamiltonian as a form of renormalization. The thesis concludes with an examination of the various interpretations and implications of this renormalization.

• Renormalization of the relativistic electron-positron field
• Mathematical description of Quantum Electrodynamics (QED)
• Coulomb interaction and normal ordering
• Interpretation of the renormalized Hamiltonian
• Implications of renormalization in QED

Zusammenfassung der Kapitel (Chapter Summaries)

• Motivation: This chapter introduces the research area of Mathematical Quantum Electrodynamics and its objective of providing a rigorous mathematical framework for Quantum Electrodynamics (QED). It outlines QED's core features as a relativistic quantum field theory and its mathematical representation as an abelian gauge theory with the symmetry group U(1).
• Basics of QED: This chapter delves into the foundational concepts of QED, explaining the use of fields in describing particle interactions. It introduces the concepts of Canonical Quantization, renormalization, and normal ordering.
• Mathematical description: This chapter provides a detailed mathematical description of the system under investigation, laying the groundwork for the calculations presented in the subsequent chapter.
• Calculations: This chapter focuses on the detailed calculations involved in the renormalization procedure. It examines the fully normal ordered interaction, including the relationship between bare and normal ordered Hamiltonians, the cutoff dependent constant, and terms with one particle operators. The chapter also explores the properties of the dressed electron in the context of both zero and positive bare masses.

Schlüsselwörter (Keywords)

This thesis explores the fundamental concepts of Quantum Electrodynamics (QED), emphasizing the mathematical framework and its application in understanding the renormalization of the relativistic electron-positron field. Key topics include: QED, relativistic quantum field theory, abelian gauge theory, U(1) symmetry group, electromagnetic field, Coulomb interaction, normal ordering, Hamiltonian, renormalization, electron-positron field, bare and renormalized parameters, radiative corrections, dressed electron, and interpretation of renormalization.