Foundations of Quantum Field Theory. Klaus D Rothe

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       World Scientific Lecture Notes in Physics — Vol. 84

       FOUNDATIONS OF QUANTUM FIELD THEORY

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      Printed in Singapore

      To my wife, Neusa Maria, and my son, Thomas

PREFACE

      Quantum Field Theory (QFT) emerged in the 1930’s as a natural extension of Quantum Mechanics to include Special Relativity and particle creation in its second quantized formulation.

      Over the years QFT has gone through an extensive evolution with regard to the role of the quantum fields involved, the range of applicability (elementary particles, phase transitions in solid state physics), the treatment of infinities (renormalization) resulting from its local structure (microcausality), the asymptotic behaviour of Green functions (Callan–Symanzik equation, asymptotic freedom), and the analyticity property of transition amplitudes (S-matrix theory).

      In particular one has learned why Quantum Electrodynamics (QED) is so successful at low energies, whereas perturbative Quantum Chromodynamics (QCD), the theory of the strong interactions, is successful at high energies (asymptotic freedom, deep inelastic scattering). One has further learned that phenomena such as spontaneous symmetry breaking observed in solid state physics (ferromagnet) also plays a role in the theory of weak interactions in particle physics, where it is referred to as “Higgs mechanism”.

      The present lectures essentially represent the content of a two-semester course held by the author at the University of Heidelberg, and thus provides an adequate time-frame for the lecturer and student. As such it was intended to be a compact book providing a bird’s eye view of the very basic foundations of QFT, including the traditional operator, as well as the more modern path integral approach, and should serve as a good basis for post-graduate students, and as orientation for lecturers. Very extensive treatises of the subject can be found in the still excellent book of Bjorken and Drell, as well as in more up-to-date books, such as by C. Itzykson and J.-B. Zuber, E. Peskin and D.V. Schroeder, Lewis H. Ryder and S. Weinberg, which have also served as a basis for these lectures.¹ Aside from the author’s point of view in presenting, choosing and arranging the material, most of it can be found in some or other way in the existing literature. We have tried to present the material in reasonable detail, with emphasis on transparency and repeated cross references, at the expense of being sometimes pedantic. We therefore believe that the reader will be able to follow the material without engaging in detailed calculations, which are cumbersome at times. Though we have exemplified various regularization procedures (Pauli–Villars, Dimensional, Taylor-subtraction), we have dominantly used the traditional Pauli–Villars regularization as being the most intuitive one.

      We paid much attention in Chapter 2 to the Lorentz group and its representations in Hilbert-space, since they play a fundamental role in Chapters 3 and 5, where some knowledge of Group Theory on the part of the student is assumed. Much of these particular chapters is based on a series of remarkable articles by Steven Weinberg in Physical Review 1964, which underline the fundamental ideas behind the construction of a Quantum Field Theory from the operator point of view.

      Many interesting topics could not be covered in these lectures. Thus subjects such as the conformal group and analyticity of scattering amplitudes have only been marginally touched. In particular, Quantum Chromodynamics and the Weak Interactions were not included. They would have exceeded the intended size of this book, and have been left to other more extensive