Introductory Lectures on Quantum Field Theory a b L. Álvarez-Gaumé ∗ and M.A. Vázquez-Mozo † a CERN, Geneva, Switzerland b Universidad de Salamanca, Salamanca, Spain Abstract In these lectures we present a few topics in quantum field theory in detail. Some of them are conceptual and some more practical. They have been se- lected because they appear frequently in current applications to particle physics and string theory. 1 Introduction These notes are based on lectures delivered by L.A.-G. at the 3rd CERN–Latin-American School of High- Energy Physics, Malargüe, Argentina, 27 February–12 March 2005, at the 5th CERN–Latin-American School of High-Energy Physics, Medellín, Colombia, 15–28 March 2009, and at the 6th CERN–Latin- American School of High-Energy Physics, Natal, Brazil, 23 March–5 April 2011. The audience on all three occasions was composed to a large extent of students in experimental high-energy physics with an important minority of theorists. In nearly ten hours it is quite difficult to give a reasonable introduction to a subject as vast as quantum field theory. For this reason the lectures were intended to provide a review of those parts of the subject to be used later by other lecturers. Although a cursory acquaintance with the subject of quantum field theory is helpful, the only requirement to follow the lectures is a working knowledge of quantum mechanics and special relativity. The guiding principle in choosing the topics presented (apart from serving as introductions to later courses) was to present some basic aspects of the theory that present conceptual subtleties. These are the topics with which one is often uncomfortable after a first introduction to the subject. Among them we have selected the following: – The need to introduce quantum fields, with the great complexity this implies. – Quantization of gauge theories and the role of topology in quantum phenomena. We have included a brief study of the Aharonov–Bohm effect and Dirac’s explanation of the quantization of electric charge in terms of magnetic monopoles. – Quantum aspects of global and gauge symmetries and their breaking. – Anomalies. – The physical idea behind the process of renormalization of quantum field theories. – Some more specialized topics, like the creation of particles by classical fields and the very basics of supersymmetry. These notes have been written following closely the original presentation, with numerous clarifi- cations. Sometimes the treatment given to some subjects has been extended, in particular the discussion of the Casimir effect and particle creation by classical backgrounds. Since no group theory was assumed, we have included an Appendix with a review of the basics concepts. Through lack of space, and on purpose, few proofs have been included. Instead, very often we illustrate a concept or property by describing a physical situation where it arises. An expanded version of these lectures, following the same philosophy but including many other topics, has been published in [1]. ∗[email protected] †[email protected], [email protected] 1 L. ÁLVAREZ-GAUMÉ AND M.A. VÁZQUEZ-MOZO For full details and proofs, we refer the reader to the many textbooks on the subject, and in particular to those provided in the bibliography [2–11]. Especially modern presentations, very much in the spirit of these lectures, can be found in references [5, 6, 10, 11]. We should nevertheless warn the reader that we have been a bit cavalier about references. Our aim has been to provide mostly a (not exhaustive) list of reference for further reading. We apologize to any authors who feel misrepresented. 1.1 A note about notation Before starting, it is convenient to review the notation used. Throughout these notes we will be using the metric η = diag (1, 1, 1, 1). Derivatives with respect to the four-vector xµ = (ct, ~x) will be µν − − − denoted by the shorthand ∂ 1 ∂ ∂ = , ~ . (1.1) µ ≡ ∂xµ c ∂t ∇ As usual, space-time indices will be labelled by Greek letters (µ,ν,... = 0, 1, 2, 3), while Latin indices will be used for spatial directions (i, j, . = 1, 2, 3). In many expressions we will use the notation σµ = (1,σi), where σi are the Pauli matrices 0 1 0 i 1 0 σ1 = , σ2 = − , σ3 = . (1.2) 1 0 i 0 0 1 − µ Sometimes we make use of Feynman’s slash notation /a = γ aµ. Finally, unless stated otherwise, we work in natural units ~ = c = 1. 2 Why do we need quantum field theory after all? In spite of the impressive success of quantum mechanics in describing atomic physics, it was immediately clear after its formulation that its relativistic extension was not free of difficulties. These problems were clear already to Schrödinger, whose first guess for a wave equation of a free relativistic particle was the Klein–Gordon equation ∂2 2 + m2 ψ(t, ~x) = 0. (2.1) ∂t2 −∇ This equation follows directly from the relativistic “mass-shell” identity E2 = ~p 2 + m2 using the corre- spondence principle ∂ E i , → ∂t ~p i~ . (2.2) → − ∇ Plane-wave solutions to the wave equation (2.1) are readily obtained µ ipµx iEt+i~p ~x 2 2 ψ(t, ~x) = e− = e− · with E = ω ~p + m . (2.3) ± p ≡± p In order to have a complete basis of functions, one must include plane waves with both E > 0 and E < 0. This implies that, given the conserved current i j = (ψ∗∂ ψ ∂ ψ∗ψ), (2.4) µ 2 µ − µ its time component is j0 = E and therefore does not define a positive-definite probability density. 2 2 INTRODUCTORY LECTURES ON QUANTUM FIELD THEORY Energy m 0 −m Fig. 1: Spectrum of the Klein–Gordon wave equation. A complete, properly normalized, continuous basis of solutions of the Klein–Gordon equation (2.1) labelled by the momentum ~p can be defined as 1 f (t, ~x) = e iωpt+i~p ~x, p 3/2 − · (2π) 2ωp 1 f (t, ~x) = p eiωpt i~p ~x. p 3/2 − · (2.5) − (2π) 2ωp Given the inner product p 3 ψ ψ = i d x (ψ∗∂ ψ ∂ ψ∗ψ ), h 1| 2i 1 0 2 − 0 1 2 Z the states (2.5) form an orthonormal basis f f = δ(~p ~p ′), h p| p′ i − f p f p = δ(~p ~p ′), (2.6) h − | − ′ i − − fp f p = 0. h | − ′ i 2 2 The wave functions fp(t,x) describe states with momentum ~p and energy given by ωp = ~p + m . On the other hand, the states f p not only have a negative scalar product but they actually correspond | − i p to negative energy states, 2 2 i∂0f p(t, ~x)= ~p + m f p(t, ~x). (2.7) − − − Therefore, the energy spectrum of the theory satisfiesp E > m and is unbounded from below (see Fig. | | 1). Although in the case of a free theory the absence of a ground state is not necessarily a fatal problem, once the theory is coupled to the electromagnetic field this is the source of all kinds of disasters, since nothing can prevent the decay of any state by emission of electromagnetic radiation. 3 3 L. ÁLVAREZ-GAUMÉ AND M.A. VÁZQUEZ-MOZO Energy photon particle m −m antiparticle (hole) Dirac Sea Fig. 2: Creation of a particle–antiparticle pair in the Dirac sea picture. The problem of the instability of the “first-quantized” relativistic wave equation can be heuristi- 1 cally tackled in the case of spin- 2 particles, described by the Dirac equation ∂ iβ~ + ~α ~ m ψ(t, ~x) = 0, (2.8) − ∂t · ∇− where ~α and β~ are 4 4 matrices × 0 iσi 0 1 ~αi = , β~ = , (2.9) iσi 0 1 0 − with σi the Pauli matrices, and the wave function ψ(t, ~x) has four components. The wave equation (2.8) can be thought of as a kind of “square root” of the Klein–Gordon equation (2.1), since the latter can be obtained as 2 ∂ † ∂ ∂ iβ~ + ~α ~ m iβ~ + ~α ~ m ψ(t, ~x)= 2 + m2 ψ(t, ~x). (2.10) − ∂t · ∇− − ∂t · ∇− ∂t2 −∇ An analysis of Eq. (2.8) along the lines of that presented above for the Klein–Gordon equation leads again to the existence of negative energy states and a spectrum unbounded from below as in Fig. 1. Dirac, however, solved the instability problem by pointing out that now the particles are fermions and therefore they are subject to Pauli’s exclusion principle. Hence, each state in the spectrum can be occupied by at most one particle, so the states with E = m can be made stable if we assume that all the negative energy states are filled. If Dirac’s idea restores the stability of the spectrum by introducing a stable vacuum where all negative energy states are occupied, the so-called Dirac sea, it also leads directly to the conclusion that a single-particle interpretation of the Dirac equation is not possible. Indeed, a photon with enough energy (E > 2m) can excite one of the electrons filling the negative energy states, leaving behind a “hole” in the Dirac sea (see Fig. 2). This hole behaves as a particle with equal mass and opposite charge that is interpreted as a positron, so there is no escape from the conclusion that interactions will produce particle–antiparticle pairs out of the vacuum. 4 4 INTRODUCTORY LECTURES ON QUANTUM FIELD THEORY Fig. 3: Illustration of the Klein paradox. In spite of the success of the heuristic interpretation of negative energy states in the Dirac equation, this is not the end of the story. In 1929 Oskar Klein stumbled into an apparent paradox when trying to describe the scattering of a relativistic electron by a square potential using Dirac’s wave equation [12] (for pedagogical reviews see [13, 14]).