In this chapter, we will quantize the Dirac field using canonical quantization, i.e. with a similiar method as was used for the Klein–Gordon field (cf. Chapter 6). However, we will observe that we need to replace the canonical commutation relations with canonical anticommutation relations in order to obtain positive energy for the quantized Dirac field and to obey the Pauli exclusion principle. In addition, we will study the transformations of parity, time reversal, and charge conjugation as well as the CPT symmetry for this field. Especially, we will investigate the Majorana field, which is a special case of the Dirac field. We will also derive Green's functions and propagators and briefly investigate interactions.

The Dirac equation (cf. Chapter 3) establishes a single-particle theory (usually known as the Dirac theory), since it cannot take into account creation and annihilation of particles. Therefore, the Dirac equation for wave functions has to be replaced by the Dirac equation for quantum fields, which means that the problems of Dirac theory are circumvented by introducing a quantum field theory reformulation of this theory; thus abandoning wave functions in favour of quantum fields. In fact, adding to the theory of the Dirac equation for quantum fields the quantized version of the electromagnetic field (see Chapter 8), we will end up with the theory of QED, which will be investigated in detail in Chapters 11–13.

In ordinary non-relativistic quantum mechanics, quantizing a point particle, we interpret the generalized coordinates qn (normally the three Cartesian spatial coordinates x, y, and z) and their corresponding 3-momenta pn as operators, which act on the wave function Ψ that is a representation of the vector of states, and they fulfil the commutation relations [qn, pm] = iħδnm. However, in field theory, the position coordinates xi have a different meaning and are used to label the infinitely many ‘coordinates’ φ(x) and their corresponding conjugate momenta π(x). In order to quantize a field theory, we impose on the fields φ(x) and π(x) the commutation relations [φ(t, x), π(t, y)] = iħδ(x - y), which are the natural generalizations of the canonical commutation relations for ordinary quantum mechanics. Note that, in what follows, we will again set ħ = 1. In addition, the discussion in this chapter will be performed for a free neutral Klein–Gordon field. However, in the last two sections, we will present two natural extensions. First, in Section 6.5, we will extend the discussion with an example of interactions in the form of a classical external source, and second, in Section 6.6, we will describe how a free charged Klein–Gordon field can be treated.

Canonical quantization

Next, we want to develop the canonical quantum field theory for the neutral Klein–Gordon field. This theory was invented in the 1930s and it is indeed a very successful theory.

In the previous chapter, we investigated elementary processes in QED and calculated cross-sections to lowest order in perturbation theory. Taking higher orders into account, one will obtain corrections of the order of the Sommerfeld fine-structure constant α to the lowest-order results. However, performing such calculations, one will encounter divergent integrals. Nevertheless, we will try to remedy this situation in three steps. First, we want to regularize the theory, which means that we modify the theory in order to keep it finite and well-defined to all orders in perturbation theory. This step is called regularization for which there are several methods. Specific methods of regularization include: Pauli–Villars regularization, dimensional regularization, lattice regularization, Riemann's ξ-function regularization, etc. Here, we will mainly investigate Pauli–Villars regularization and dimensional regularization. Second, we recognize that the non-interacting particles (i.e. the free asymptotic fields) are not identical to the real physical particles that interact. Thus, the interactions modify the properties of the particles such as the masses and the charges. Of course, all the relevant predictions of the theory must be expressed in terms of the properties of the physical particles and not the noninteracting (or bare) particles. This step is called renormalization. Third, we have to revert from the regularized theory back to QED, and thus, the infinities of QED will appear in the relations between the bare and physical particles. Of course, these relations as well as the bare particles are completely unobservable.

We start this chapter with a brief history of quantum field theory. In the 1920s, Dirac attempted to quantize the electromagnetic field (see Chapter 8). This attempt was the beginning of the history of quantum field theory. Then, in 1926, M. Born, W. Heisenberg, and P. Jordan invented canonical quantization, which we will use in Chapters 6 and 7 in order to quantize the Klein–Gordon field and the Dirac field, respectively, that will bring us from relativistic quantum mechanics to relativistic quantum field theory. Next, in 1927, Dirac created and presented the first reasonably complete theory of quantum electrodynamics (QED), and in the following year, 1928, he formulated the Dirac equation (cf. Chapter 3). In addition, in 1928, E.Wigner found that the quantum field describing electrons, or other (spin-1/2) fermions, had to be expanded using anticommuting creation and annihilation operators due to the Pauli exclusion principle. After World War II, in the 1940s, H. Bethe, F. Dyson, R. Feynman, J. Schwinger, and S.-I. Tomonaga solved the socalled ‘divergence problem’ through renormalization (see Chapter 13). This was the start of the modern theory of QED. In the 1950s, C.-N. Yang and R. Mills generalized QED to gauge theories – known as Yang–Mills theories – which we will discuss classically (i.e. without quantization) in Chapter 9.

This book is based on my lectures in the course ‘Relativistic Quantum Physics’ at the Royal Institute of Technology (KTH) in Stockholm, Sweden. These lectures have been given four times during the academic years 2006–2007, 2007–2008, 2008–2009, and 2009–2010. The main sources of inspiration for the lectures were the books A. Z. Capri, Relativistic Quantum Mechanics and Introduction to Quantum Field Theory, World Scientific (2002) and M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley (1995), and indeed, this book serves as a textbook for relativistic quantum mechanics with continuation to basic quantum field theory. The book is mainly intended for final-year undergraduate students in physics or first-year graduate students in physics and/or theoretical physics, who want to learn relativistic quantum mechanics, the basics of quantum field theory, and the techniques of calculating cross-sections for elementary reactions in quantum electrodynamics. Thus, the book should be suitable for any course on relativistic quantum mechanics as well as it might be suitable for a beginners' course on quantum field theory. In summary, the book is a self-contained technical treatment on relativistic quantum mechanics, introductory quantum field theory, and the step in between, i.e. it should fill the gap between advanced quantum mechanics and quantum field theory, which I have called relativistic quantum physics. It contains a thorough and detailed mathematical treatment of the subject with smaller exercises throughout the whole text and larger problems at the end of most chapters.

In general, perturbation theory is a mathematical method that is used to find approximative solutions to problems that cannot be solved exactly. Therefore, the starting point of perturbation theory is the exact solution to a related problem. Perturbation theory can be applied if the original problem can be reformulated by adding a ‘small’ term to the exactly solvable problem. Thus, perturbation theory gives rise to an expression for the original solution in terms of a power series in some ‘small’ parameter, which measures the deviation from the exactly solvable problem. The leading term in this power series is the solution to the exactly solvable problem, whereas the higher-order terms may be found by some iterative procedure. In order for the perturbation theory to work properly, the higher-order terms in the ‘small’ parameter need to become successively smaller, i.e. the power series should converge. However, normally in quantum field theory, the terms will not anymore become successively smaller at some specific order, since the number of possible Feynman diagrams will grow so fast that the terms will instead become successively larger. Thus, perturbation theory will only be successful up to this order, and then, it will break down.

This chapter is devoted to the study of perturbation theory, which is, nevertheless, a very powerful tool in quantum field theory. Note that the discussion in Sections 11.1–11.6 will mainly be performed for a real scalar field, i.e. a neutral Klein–Gordon field, but it can, of course, be naturally extended to other types of fields, such as a Dirac field, if the appropriate changes are made.

In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that

• had a positive-definite conserved probability density and

• was first order both in time and space.

One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E2 = m2 + p2 via the correspondence principle that includes a term E2. Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.

In 1861, J. C. Maxwell published a theoretical and important work on electromagnetism. Indeed, this work contains essentially the equations that today are known as Maxwell's equations. These equations, together with the Lorentz force law F = q(E + v × B), constitute the complete set of laws for classical electromagnetism. An important property of Maxwell's equations is that they are Lorentz covariant. However, classical electromagnetism is, of course, non-quantized, as the prefix ‘classical’ suggests. Later on, in 1900, M. Planck introduced the concept of quantization of radiation that can be studied by using emission of radiation from heated bodies, which has become known as black-body radiation. Thus, the idea of quantization was born by Planck's discovery. Then, in 1905, Einstein introduced the concept of ‘quanta of light’, since the quantum nature of light was revealed by the photoelectric effect that was claimed by himself. In addition, the quanta of light were dubbed photons.

In this chapter, we will first present Maxwell's equations, we will then discuss different gauges and quantization of the electromagnetic field, we will next consider the Casimir effect that arises when quantizing the electromagnetic field, and finally, we will try to obtain a covariant quantization of the electromagnetic field.

In this chapter, the Lehmann–Symanzik–Zimmermann (LSZ) formalism is presented. This was developed in 1955 and is named after its inventors, namely, the three German physicists H. Lehmann, K. Symanzik, and W. Zimmermann. Especially, the LSZ formalism includes the LSZ reduction formula that provides an explicit way of expressing physical S matrix elements (i.e. scattering amplitudes) in terms of T-ordered correlation functions of an interacting field within a quantum field theory to all orders in perturbation theory. Thus, the goal is to express the S matrix elements in terms of asymptotic free fields instead of the unknown interacting field. Therefore, given the Lagrangian of some quantum field theory, it leads to predictions of measurable quantities. Note that the LSZ reduction formula cannot treat massless particles, bound states, and so-called topological defects. For example, in quantum chromodynamics (QCD), the asymptotic states are bound, which means that they are not free states. However, the original formula can be generalized in order to include bound states, which are states described by non-local composite fields. In addition, in statistical physics, a general formulation of the fluctuation–dissipation theorem, which is a theorem that can be used to predict the non-equilibrium behaviour of a system, can be obtained using the LSZ formalism.

Schrödinger’s equation and the “wave function” that satisfies it are the first bold brushstrokes of a new portrait of Nature that physicists gradually filled in during the middle half of the twentieth century. The emerging pattern was strange, but not entirely alien to science. It was drawn, to paraphrase Galileo, in a mathematical idiom “without some knowledge of which it is humanly impossible to understand.” While nonmathematical accounts are conceivable, I would rather at this point say more about the mathematics because it will deepen our view of the whole enterprise of modern physics. Most people never get beyond the “math is numbers and numbers are boring” barrier that hides behind it a limitless universe of beautiful things. So let us delve briefly into the nonboring side of mathematics.