Chapter 3
Quantum field theory
Quantum mechanics describes nature as absurd from the point of view of common sense. And yet it fully agrees with experiment. So I hope you can accept nature as She is — absurd. — Richard Feynman
The standard model is a quantum field theory (QFT). It describes the universe as a collection of fields associated with the various elementary particles. At each point in spacetime, there is a random probability for these fields to interact and create or destroy their respective particles.
This means we have an electron field, a photon field, a Higgs field, etc. spread across the universe, and all electrons, photons, and Higgs bosons are identical quantum excitations of these. The interactions of the electron and photon fields, for example, are what we experience as electromagnetism.
As Feynman says, this may all sound absurd. Fields are highly unintuitive, seemingly “unphysical” concepts. It can be hard to imagine that particles, matter, and, hence, all of us, are simply a collection of quanta probabilistically popping out and dropping back into an abstract cosmic sea.
Not only that, historically, QFT often appeared intractable and even nonsensical, yielding results such as negative energy and infinite mass particles. Indeed its development underwent multiple periods of stagnation and ardent opposition, including by Richard Feynman who suggested in 1945 that field theory be abandoned altogether [25] before changing his mind and making seminal contributions to quantum electrodynamics.
Yet, through the collective efforts of generations of physicists, QFT can now explain nearly every observed phenomenon in particle physics, up to the highest experimental energies. Not only that, it has made some of the most staggering and precise predictions in the history of physics, all of which proved to be in complete agreement with experiment. These range from the calculation of the electron’s magnetic moment up to 12 significant digits, to the prediction of the Higgs boson 50 years before its discovery. Its unprecedented experimental success is why we believe “it is the language in which the laws of Nature are written” (Tong SM [5]).
In this chapter, we start first with a brief review of classical field theory (Section 3.1), primarily discussing free scalar fields, before outlining the process of quantization and the emergence of particles (Section 3.2). We then introduce interactions to these free theories and Feynman diagrams for calculating physical observables, i.e. the -matrix elements, of QFTs (Section 3.3). Section 3.4 extends the theory to spin- spinor fields, which describe fermions and have a much richer group-theoretic structure, and Section 3.5 then discusses spin- gauge fields, describing fundamental forces and gauge bosons. Finally, we conclude with the famous Higgs mechanism in Section 3.6.
3.1.1 Lagrangian mechanics
3.1.2 Free scalar field theory
3.1.3 Symmetries and Noether’s theorem
3.1.4 Hamiltonian mechanics
3.2 Quantization
3.2.1 Canonical quantization
3.2.2 The Hamiltonian and the vacuum catastrophe
3.2.3 Particles
3.2.4 The complex scalar field and antiparticles
3.2.5 Propagators and Green functions
3.3 Interactions
3.3.1 Interactions in the Lagrangian
3.3.2 S-matrix elements
3.3.3 Feynman diagrams
3.3.4 Decay rates and cross sections
3.4 Spinor field theory
3.4.1 The Dirac equation
3.4.2 Spinors
3.4.3 The Dirac Lagrangian
3.4.4 Quantizing the Dirac field
3.4.5 Interactions and Feynman rules
3.4.6 CPT Symmetries
3.5 Gauge theories
3.5.1 Why gauge invariance?
3.5.2 Maxwell Theory
3.5.3 Quantum electrodynamics
3.5.4 Yang-Mills Theory
3.5.5 Quantized Yang-Mills Theory
3.5.6 Running coupling and asymptotic freedom
3.6 The ABEGHHK (Higgs) mechanism
3.6.1 The abelian Higgs mechanism
3.6.2 The non-abelian Higgs mechanism