##
**Finite quantum electrodynamics. The causal approach.
2nd ed.**
*(English)*
Zbl 0844.53052

Texts and Monographs in Physics. Berlin: Springer-Verlag. x, 409 p. (1995).

The book is the second edition of author’s book with the same title. But it differs considerably from its first edition. The present edition is more expository and explicit, sometimes with necessary detail calculations. The setup of the book is essentially guided by the observation (author’s conviction?) that the nature of causality is highly non-trivial. It shows how the scattering matrix can be constructed using causality, instead of dynamical equation, leading to finite physical observable quantities. The main programmes are the construction of the \(S\)-matrix by means of causality and the inductive construction of causal perturbation theory.

The book starts with a preliminary (0-th) chapter, giving a concise historical development of second quantization, the Lorentz group, vectors and tensors in Minkowski space, and some elementary notions about scattering theory. The \(S\)-matrix is introduced as a functional equation and power series of suitable switching functions (for external fields). The first chapter on relativistic quantum mechanics is concerned with the one-particle Dirac theory. Section headings are: Spinor Representation of the Lorentz Group, Invariant Field Equation, Algebraic Properties of the Dirac Equation, Gauge Invariance and Electromagnetic field, and the chapter ends with the application to the hydrogen atom. The second chapter is on field quantization. The method of second quantization in Fock space which transforms a one-particle theory into a many-particle theory, is applied to quantization of the Dirac field in an external electromagnetic field and a discussion of the commutation functions. The phase of the \(S\)-matrix in Fock space is fixed by the requirement of causality which is free from divergence and leads to finite results. The construction is extended to non-perturbative cases. The chapter includes applications to electron scattering, pair production, vacuum polarization and quantization of radiation fields. The next chapter is devoted to the construction of the \(S\)-matrix with both the electron positron field and the radiation field quantized. The most important point in the construction is the decomposition of the operator (distribution) with causal support into retarded and advanced parts. This is done first by multiplying with a \(C^\infty\) function and then performing the appropriate limits to the step function. It seems that this distribution splitting is the crux of the matter, leading to finite results. It further contains an illuminative discussion on adiabatic limits (switching process) and their consequences. In the penultimate chapter, the author analyses vacuum graphs and shows that the perturbative \(S\)-matrix is a well-defined operator in Fock space. The inductive construction of causal perturbation theory is then applied to proofs of normalizability, various symmetries, gauge invariance and unitarity. It also contains some more sophisticated techniques, namely the renormalization group and interacting fields. Finally, in the last chapter the author takes a sojourn beyond the orthodox QED to discuss scalar QED with derivative coupling – its basic properties and gauge invariance, axial anomalies and \(2 + 1\) QED. In addition, it is supplemented with an epilogue on non-abelian gauge theories and instructive appendices of particular interest such as regularly varying functions and Spence functions.

Each chapter of the book is preceded by a very concise summary of the matter to be followed and ends with a collection of problems which is, in fact, supplementing the preceding topics. The book ends with a section Bibliographical Notes, consisting of collections of some important references which are strongly related to the various chapters in the book. These references to the literature are expected to be extremely helpful to serious readers.

It is more than half a century that QED was initiated by Dirac to explain ‘spontaneous emission’, nevertheless it is not yet closed. There are many questions, which have not yet been answered with full satisfaction of those who are still working in the field. Notwithstanding, QED is still a source of ideas and methods for current research in quantum field theory. The book is a welcome addition to the literature in the field of quantum field theory. It should be very useful not only to those who are working in this field but also to the graduate students and advanced scholars.

The book starts with a preliminary (0-th) chapter, giving a concise historical development of second quantization, the Lorentz group, vectors and tensors in Minkowski space, and some elementary notions about scattering theory. The \(S\)-matrix is introduced as a functional equation and power series of suitable switching functions (for external fields). The first chapter on relativistic quantum mechanics is concerned with the one-particle Dirac theory. Section headings are: Spinor Representation of the Lorentz Group, Invariant Field Equation, Algebraic Properties of the Dirac Equation, Gauge Invariance and Electromagnetic field, and the chapter ends with the application to the hydrogen atom. The second chapter is on field quantization. The method of second quantization in Fock space which transforms a one-particle theory into a many-particle theory, is applied to quantization of the Dirac field in an external electromagnetic field and a discussion of the commutation functions. The phase of the \(S\)-matrix in Fock space is fixed by the requirement of causality which is free from divergence and leads to finite results. The construction is extended to non-perturbative cases. The chapter includes applications to electron scattering, pair production, vacuum polarization and quantization of radiation fields. The next chapter is devoted to the construction of the \(S\)-matrix with both the electron positron field and the radiation field quantized. The most important point in the construction is the decomposition of the operator (distribution) with causal support into retarded and advanced parts. This is done first by multiplying with a \(C^\infty\) function and then performing the appropriate limits to the step function. It seems that this distribution splitting is the crux of the matter, leading to finite results. It further contains an illuminative discussion on adiabatic limits (switching process) and their consequences. In the penultimate chapter, the author analyses vacuum graphs and shows that the perturbative \(S\)-matrix is a well-defined operator in Fock space. The inductive construction of causal perturbation theory is then applied to proofs of normalizability, various symmetries, gauge invariance and unitarity. It also contains some more sophisticated techniques, namely the renormalization group and interacting fields. Finally, in the last chapter the author takes a sojourn beyond the orthodox QED to discuss scalar QED with derivative coupling – its basic properties and gauge invariance, axial anomalies and \(2 + 1\) QED. In addition, it is supplemented with an epilogue on non-abelian gauge theories and instructive appendices of particular interest such as regularly varying functions and Spence functions.

Each chapter of the book is preceded by a very concise summary of the matter to be followed and ends with a collection of problems which is, in fact, supplementing the preceding topics. The book ends with a section Bibliographical Notes, consisting of collections of some important references which are strongly related to the various chapters in the book. These references to the literature are expected to be extremely helpful to serious readers.

It is more than half a century that QED was initiated by Dirac to explain ‘spontaneous emission’, nevertheless it is not yet closed. There are many questions, which have not yet been answered with full satisfaction of those who are still working in the field. Notwithstanding, QED is still a source of ideas and methods for current research in quantum field theory. The book is a welcome addition to the literature in the field of quantum field theory. It should be very useful not only to those who are working in this field but also to the graduate students and advanced scholars.

Reviewer: N.D.Sengupta (Bombay)

### MSC:

53Z05 | Applications of differential geometry to physics |

81V10 | Electromagnetic interaction; quantum electrodynamics |

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

81U20 | \(S\)-matrix theory, etc. in quantum theory |