##
**The Dirac equation.**
*(English)*
Zbl 0765.47023

Texts and Monographs in Physics. Berlin: Springer-Verlag. xvii, 357 p. (1991).

This is a textbook based on a qualified knowledge on the physical theory of the Dirac particle. On the other hand it is an research book because it contains several recent results, in particular with respect to the spectral theory of Dirac operators. The reader should be familiar with the fundamentals in spectral theory of selfadjoint operators, in the theory of Lie groups and their representations, in quantum mechanics and electrodynamics.

The main objective of the monograph is to give a mathematically rigorous foundation of the spectral theory for Dirac operators. It is useful as a guide to the analysis of the Dirac equation. It contains short descriptions of several mathematical theories which are necessary for understanding this analysis and its related mathematical theories.

The content of this book is divided into two parts. The first three chapters are used to establish the free Dirac operator, which is an ambitious first main task. After introducing the possible external fields the other main part is devoted to supersymmetric Dirac operators. The supersymmetric quantum mechanics is a powerful tool for the rigorous spectral theoretical treatment in the second half of the book.

The content of every chapter is well explained in a separate preface. In the notes one can find historical remarks and useful hints for additional literature concerning the mathematical and physical background.

The Dirac equation describes a freely moving relativistic electron or positron. It is derived heuristically, but then investigated as evolution equation in a Hilbert space. The selfadjointness of the Dirac operator is proved, spectral properties are given, negative energies are interpreted, the Zitterbewegung, localization and acausality are explained. In order to show the relativistic invariance of the Dirac theory Lorentz and Poincaré groups are introduced, projective representations of the Poincaré covering group are given. The 4-dimensional representation of the Poincaré covering group in the space of Dirac spinors is studied in detail. The role of the Dirac equation in the general theory of unitary group representations is explained. A Dirac equation arises in a construction of irreducible representations of the Poincaré covering group using Mackey’s theory of induced representations.

External fields are given by matrix-valued functions. They are classified according to Poincaré transformations. The selfadjointness of the perturbed Dirac operators is ensured, Weyl’s theorem gives information about the essential spectrum. Time-dependent potentials are treated and Klein’s paradox is demonstrated for the one-dimensional Dirac equation. Spherically symmetric potentials and their spectral properties are studied in more detail.

For the further mathematical rigorous treatment the supersymmetric theory plays the fundamental role. The structure of the Dirac operator fits well to the supersymmetric theory. An abstract Dirac operator is a generalization of a supercharge. An important example is a Dirac operator with supersymmetry. The supersymmetric Dirac operator can be diagonalized (or off-diagonalized). Therefore the spectral analysis reduces to the diagonal (or off-diagonal) operators. That implies the possibility to study for instance the Fredholm theory, Witten index theory or spectral shift function for super-symmetric Dirac operators.

In this supersymmetric framework the rest of the book consists of a collection of results mainly in spectral theory. The non-relativistic limit holds in norm resolvent sense. The analyticity regions of the resolvents depend on the light velocity. Expansions of eigenvalues and eigenfunctions are possible. For different external fields (homogeneous magnetic fields, cylindrically symmetric fields, Coulomb potentials) eigenvalues and essential spectra are considered. The time-dependent Enss theory yields the proof of the asymptotic completeness for short range and Coulomb like potentials. The supersymmetric scattering theory is used to study the scattering in magnetic fields. Also the scattering time- dipendent electric field is mentioned.

Solitons of the modified Korteweg-de Vries (mKdV) equation are related to the Dirac equation with a time-dependent scalar potential. The link between mKdV and KdV (Korteweg-de Vries) with Dirac and Schrödinger operators is described. The supersymmetry is exploited to describe the solutions of the mKdV equations in terms of KdV solutions. KdV solutions can be obtained by the inverse scattering method.

The second quantization of the Dirac field is sketched. Time-evolution, number, and charge operators in Fock spaces are introduced. Certain external fields can be implemented. For weak and time-independent external fields the quantum field theory reproduces the one-particle theory. Some scattering problems in QED are discussed.

Further possible topics like resonance theory, magnetic monopoles or gravitational fields are omitted.

The main objective of the monograph is to give a mathematically rigorous foundation of the spectral theory for Dirac operators. It is useful as a guide to the analysis of the Dirac equation. It contains short descriptions of several mathematical theories which are necessary for understanding this analysis and its related mathematical theories.

The content of this book is divided into two parts. The first three chapters are used to establish the free Dirac operator, which is an ambitious first main task. After introducing the possible external fields the other main part is devoted to supersymmetric Dirac operators. The supersymmetric quantum mechanics is a powerful tool for the rigorous spectral theoretical treatment in the second half of the book.

The content of every chapter is well explained in a separate preface. In the notes one can find historical remarks and useful hints for additional literature concerning the mathematical and physical background.

The Dirac equation describes a freely moving relativistic electron or positron. It is derived heuristically, but then investigated as evolution equation in a Hilbert space. The selfadjointness of the Dirac operator is proved, spectral properties are given, negative energies are interpreted, the Zitterbewegung, localization and acausality are explained. In order to show the relativistic invariance of the Dirac theory Lorentz and Poincaré groups are introduced, projective representations of the Poincaré covering group are given. The 4-dimensional representation of the Poincaré covering group in the space of Dirac spinors is studied in detail. The role of the Dirac equation in the general theory of unitary group representations is explained. A Dirac equation arises in a construction of irreducible representations of the Poincaré covering group using Mackey’s theory of induced representations.

External fields are given by matrix-valued functions. They are classified according to Poincaré transformations. The selfadjointness of the perturbed Dirac operators is ensured, Weyl’s theorem gives information about the essential spectrum. Time-dependent potentials are treated and Klein’s paradox is demonstrated for the one-dimensional Dirac equation. Spherically symmetric potentials and their spectral properties are studied in more detail.

For the further mathematical rigorous treatment the supersymmetric theory plays the fundamental role. The structure of the Dirac operator fits well to the supersymmetric theory. An abstract Dirac operator is a generalization of a supercharge. An important example is a Dirac operator with supersymmetry. The supersymmetric Dirac operator can be diagonalized (or off-diagonalized). Therefore the spectral analysis reduces to the diagonal (or off-diagonal) operators. That implies the possibility to study for instance the Fredholm theory, Witten index theory or spectral shift function for super-symmetric Dirac operators.

In this supersymmetric framework the rest of the book consists of a collection of results mainly in spectral theory. The non-relativistic limit holds in norm resolvent sense. The analyticity regions of the resolvents depend on the light velocity. Expansions of eigenvalues and eigenfunctions are possible. For different external fields (homogeneous magnetic fields, cylindrically symmetric fields, Coulomb potentials) eigenvalues and essential spectra are considered. The time-dependent Enss theory yields the proof of the asymptotic completeness for short range and Coulomb like potentials. The supersymmetric scattering theory is used to study the scattering in magnetic fields. Also the scattering time- dipendent electric field is mentioned.

Solitons of the modified Korteweg-de Vries (mKdV) equation are related to the Dirac equation with a time-dependent scalar potential. The link between mKdV and KdV (Korteweg-de Vries) with Dirac and Schrödinger operators is described. The supersymmetry is exploited to describe the solutions of the mKdV equations in terms of KdV solutions. KdV solutions can be obtained by the inverse scattering method.

The second quantization of the Dirac field is sketched. Time-evolution, number, and charge operators in Fock spaces are introduced. Certain external fields can be implemented. For weak and time-independent external fields the quantum field theory reproduces the one-particle theory. Some scattering problems in QED are discussed.

Further possible topics like resonance theory, magnetic monopoles or gravitational fields are omitted.

Reviewer: M.Demuth (Potsdam)

### MSC:

47N50 | Applications of operator theory in the physical sciences |

47A40 | Scattering theory of linear operators |

81T60 | Supersymmetric field theories in quantum mechanics |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

47B25 | Linear symmetric and selfadjoint operators (unbounded) |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |