Chapter I SPINORS AND TENSORS

1. The Lorentz Group

The Minkowski metric

Lorentz transformations

2. The Eestricted Lorentz Group and the Infinitesimal Lorentz Trans

formations

A theorem

Special Lorentz group

Restricted transformations

Infinitesimal transformations

3. Vector Spaces [Fields] and Representations

Representations of the Lorentz group

Equivalent representations

Infinitesimal transformations

4. The TJnimodular Group c2 and its Representations

Unimodular transformations

Representations of the group e2

6. The Connexion between the Group c2 and the Lorentz Group

Representations of the restricted Lorentz group

Representations of the full Lorentz group

Tensors

6. The Connexion between Tensors and Spinors [or Bispinors]

The connexion between four-vectors and spinors

The connexion between tensors and spinors

Invariant relations between tensors and spinors

7. Spinor and Tensor Analysis

Differential operations

Invariant differential equations

Invariant integration processes

World lines

8. Inversions of Space and Time and their Two-component Represen

tations

Various types of tensors

Various types of spinors

Two-component representations

Invariant equations and the determination of a

Connexions between the various types of y-matrices

Bilinear forms

Two-component equations

Chapter II VARIATIONAL PRINCIPLES

1. The General Variational Principle for Fields

The action functional

Variational formalism

The principle of stationary action

2. Integral Conservation Laws

Gauge transformations of the first kind

Translations and rotations

The functional F[o]

Integral conservation laws

3. Verification of the Integral Conservation Laws

Necessary and sufficient conditions for the conservation laws

Discussion

The local case

4. Differential Conservation Laws

The non-local case

Interpretation

The local case

Canonical transformations

5. The General Variational Principle for one Independent Variable

The action functional

The first variation of the action functional

The equations of motion

Invariance with respect to parametrization

Integral conservation laws

Differential conservation laws

Chapter III EXAMPLES

1. Non-Interacting Fields. Neutral and Complex Scalar Fields

Neutral scalar field, variational principle, field equations and conservation laws

Solutions of the field equations

The. initial value problem

Complex scalar field

2. Non-Interacting Fields. The First Rank Bispinor Field

Variational principle, field equations and conservation laws

Solutions of the field equations

The initial value problem

3. Non-Interacting Fields. Vector, Pseudo-Vector and Pseudo-Scalar

Fields

Vector field

Pseudo-scalar and pseudo-vector field

4. External Interactions. Scalar [or Pseudo-Scalar] Field

Interaction with external sources

Solutions of the inhomogeneous equations

Interaction with external fields

Solutions of the perturbed equations

An equivalence theorem

6. External Interactions. Bispinor Field

Interaction with external sources

Solutions of the inhomogeneous equations

Interaction with external fields

Solutions of tho perturbed equations

An equivalence theorem

6. Mutually Interacting Fields. Electromagnetic Field in Interaction

with the Electron Field

The action functional

Gauge invariance of the second kind

Energy-momentum

Solutions of the field equations

7. Fokker's Variational Principle

Action functional and equations of motion

Field equations

The subsidiary condition

The potentials of Lienard and Wiechert

The procedure of Wheeler and Feynman

Modified interactions

Self-action

Radiation damping

Conservation laws

Solutions of the equations of motion

8. Non-Local Interactions

Action functional

The local limit

Field equations and conservation laws

Solutions of the field equations

Mass eigenvalue problem

9. Example of a Non-Linear Theory

The action functional and the field equations

Static solutions

Energy-momentum

Chapter rv

THE BASIC SOLUTIONS OP THE FIELD EQUATIONS

1. Introduction of Various Types of delta-Functions

Homogeneous Klein-Gordon equation. The functions delta and delta(1)

The functions delta(+) and delta(-)

Integral relations

Inhomogeneous equation. The function delta

The functions deltaret, deltaadv and deltaF

Further integral relations

The homogeneous and inhomogeneous Dirac equations. The S-functions

2. Fourier Transforms of the delta- and S-Functions

Three-dimensional Fourier transforms

Four-dimensional Fourier transforms

3. Explicit Calculation of the delta-Functions

The function delta

The function delta(1)

Absolute integrability

4. Representation of the delta-Functions by Integrals in the Complex

k-Plane