Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Relativistic Quantum Mechanics
- 2 Fock Space, the Scalar Field, and Canonical Quantization
- 3 Symmetries and Conservation Laws
- 4 From Dyson's Formula to Feynman Rules
- 5 Differential Transition Probabilities and Predictions
- 6 Representations of the Lorentz Group
- 7 Two-Component Spinor Fields
- 8 Four-Component Spinor Fields
- 9 Vector Fields and Gauge Invariance
- 10 Reformulating Scattering Theory
- 11 Functional Integral Quantization
- 12 Quantization of Gauge Theories
- 13 Anomalies and Vacua in Gauge Theories
- 14 SU(3) Representation Theory
- 15 The Structure of the Standard Model
- 16 Hadrons, Flavor Symmetry, and Nucleon-Pion Interactions
- 17 Tree-Level Applications of the Standard Model
- 18 Regularization and Renormalization
- 19 Renormalization of QED: Three Primitive Divergences
- 20 Renormalization and Preservation of Symmetries
- 21 The Renormalization Group Equations
- Appendix
- References
- Index
6 - Representations of the Lorentz Group
Published online by Cambridge University Press: 31 October 2009
- Frontmatter
- Contents
- Preface
- Introduction
- 1 Relativistic Quantum Mechanics
- 2 Fock Space, the Scalar Field, and Canonical Quantization
- 3 Symmetries and Conservation Laws
- 4 From Dyson's Formula to Feynman Rules
- 5 Differential Transition Probabilities and Predictions
- 6 Representations of the Lorentz Group
- 7 Two-Component Spinor Fields
- 8 Four-Component Spinor Fields
- 9 Vector Fields and Gauge Invariance
- 10 Reformulating Scattering Theory
- 11 Functional Integral Quantization
- 12 Quantization of Gauge Theories
- 13 Anomalies and Vacua in Gauge Theories
- 14 SU(3) Representation Theory
- 15 The Structure of the Standard Model
- 16 Hadrons, Flavor Symmetry, and Nucleon-Pion Interactions
- 17 Tree-Level Applications of the Standard Model
- 18 Regularization and Renormalization
- 19 Renormalization of QED: Three Primitive Divergences
- 20 Renormalization and Preservation of Symmetries
- 21 The Renormalization Group Equations
- Appendix
- References
- Index
Summary
A mathematical introduction to representation theory of Lie groups and Lie algebras providing details of the irreducible representations of the Lorentz groups in preparation for theories of spinor fields, vector fields, and their interactions.
Introduction
This Chapter begins the second part of the text, which comprises Chapters 6 to 9. Here, we develop the representation theory of the Lorentz group, thereby providing the spinor and vector fields which the following chapters quantize. Part 2 introduces all the types of quantum field used in the Standard Model.
At this point, we have quite a well-developed view of scalar field theory. Scalar fields, however, are necessarily bosonic. If we tried to change their statistics, we would find that the Hamiltonian is no longer bounded below. Furthermore, scalar fields do not have a polarization and so cannot represent the electromagnetic field.
The problem is the Lorentz transformation law of Axiom 3, Section 2.4, which only permits scalar fields. Now we want to change this axiom and allow the Lorentz group to mix field components. We will see just how many possibilities there are and find candidates for fermion and photon fields among these possibilities.
This chapter is a fairly mathematical one. The reason for this is that representation theory is mainly linear algebra, and so it is convenient to clarify the aspects of representation theory used in physics by plunging into the appropriate mathematical terminology.
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- Quantum Field Theory for Mathematicians , pp. 136 - 165Publisher: Cambridge University PressPrint publication year: 1999