Elementary Particle Theory. Volume 2, Quantum Electrodynamics

In a successful theory of elementary particles, at least three important conditions must be fulfilled: (1) relativistic invariance in the instant form of dynamics; (2) cluster separability of the interaction; (3) description of processes involving creation and destruction of particles. In the first volume of our book we discussed interacting quantum theories in Hilbert spaces with a fixed set of particles. We showed how it is possible to satisfy the first two requirements (relativistic invariance and cluster separability).1 However, these theories were fundamentally incomplete, due to their inability to describe physical processes that change the types and/or number of particles in the system. Thus, condition 3 from our list was not fulfilled. Familiar examples of the creation and annihilation processes are emission and absorption of light (photons), decays, neutrino oscillations, etc. Particles are produced especially intensively at high energies. This is due to the famous Einstein formula E = mc2, which says, in particular, that if the system has sufficient energy E of relative motion, then this energy can be transformed into the mass m of newly created particles. Even in the simplest two-particle case, the energy of the relative motion of these reactants is unlimited. Therefore, there is no limit to the number of new particles that can be created in a collision. To advance in the study of such processes, the first thing to do is to build a Hilbert space of states H , which is capable of describing particle creation and annihilation. Such a space must include states with arbitrary numbers (from zero to infinity) of particles of all types. It is called the Fock space. This construction is rather simple. However, the next step – the definition of realistic interaction operators in the Fock space – is highly nontrivial. A big part of our third volume will be devoted to the solution of this problem. Here we will prepare ourselves to this task by starting with a more traditional approach, which is known as the renormalized relativistic quantum field theory (QFT). Our discussions in this book are limited to electromagnetic phenomena, so we will be interested in the simplest and most successful type of QFT – quantum electrodynamics (QED). In Chapter 1, Fock space, we will describe the basic mathematical machinery of Fock spaces, including creation and annihilation operators, normal ordering and classification of interaction potentials. A simple toy model with variable number of particles will be presented in Chapter 2, Scattering in Fock space. In this example, we will discuss such important ingredients of QFT as the S-matrix formalism, renormalization, diagram technique and cluster separability. Our first two chapters have a mostly technical character. They define our terminology and notation and prepare us for a more in-depth study of QED in the two following chapters. In Chapter 3, Quantum electrodynamics, we introduce the important concept of the quantum field. This idea will be applied to systems of charged particles and photons in the formalism of QED. Here we will obtain an interacting theory, which satisfies the principles of relativistic invariance and cluster separability, where the number of particles is not fixed. However, the “naïve” version of QED presented here is unsatisfactory, since it cannot calculate scattering amplitudes beyond the lowest orders of perturbation theory. Chapter 4, Renormalization, completes the second volume of the book. We will discuss the plague of ultraviolet divergences in the “naïve” QED and explain how they can be eliminated by adding counterterms to the Hamiltonian. As a result, we will get the traditional “renormalized” QED, which has proven itself in precision calculations of scattering cross sections and energy levels in systems of charged particles. However, this theory failed to provide a well-defined interacting Hamiltonian and the interacting time evolution (= dynamics). We will address these issues in the third volume of our book. As in the first volume, here we refrain from criticism and unconventional interpretations, trying to keep in line with generally accepted approaches. The main purpose of this volume is to explain the basic concepts and terminology of QFT. For the most part, we will adhere to the logic of QFT formulated by Weinberg in the series of articles and in the excellent textbook. A critical discussion of the traditional approaches and a new look at the theory of relativity will be presented in Volume 3. References to Volume 1 of this book will be prefixed with “1-”. For example, (1-7.14) is formula (7.14) from Volume 1.