Quantum Field Theory

Harald Fritzsch

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789813141742
• Topic: Physical Sciences
• Subtopic: Quantum Theory

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The matter in our universe is composed of electrons and quarks. The dynamics of electrons and quarks is described by the Standard Model of particle physics, which is based on quantum field theories. The general framework of quantum field theories is described in this book. After the classical mechanics and the relativistic mechanics the details of classical scalar fields, of electrodynamics and of quantum mechanics are discussed. Then the quantization of scalar fields, of spinor fields and of vector fields is described.

The basic interactions are described by gauge theories. These theories are discussed in detail, in particular the gauge theories of quantum electrodynamics (QED) and of quantum chromodynamics (QCD), based on the gauge group SU(3). In both theories the gauge bosons, the photon and the gluons, have no mass. The gauge theory of the electroweak interactions, based on the gauge group SU(2) x U(1), describes both the electromagnetic and the weak interactions. The weak force is generated by the exchange of the weak bosons. They have a large mass, and one believes that these masses are generated by a spontaneous breaking of the gauge symmetry.

It might be that the strong and the electroweak interactions are unified at very high energies ("Grand Unification"). The gauge groups SU(3) and SU(2) x U(1) must be subgroups of a big gauge group, describing the Grand Unification. Two such theories are discussed, based on the gauge groups SU(5) and SO(10).

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Chapter 1

Introduction

From Classical Physics to Modern Physics

The classical mechanics is valid for systems where all velocities are much less than the velocity of light. If the velocities are close to the velocity of light, the classical mechanics must be replaced by the relativistic mechanics, introduced by Albert Einstein in 1905.

In the relativity theory the speed of light is a fundamental constant:

This implies that the flow of time depends on the reference system. Energy and mass are related to each other — mass is “frozen” energy. For a mass at rest the corresponding energy is given by the equation of Einstein: E = mc².

The Lorentz transformations are transformations between the space-time coordinates of one system of reference and another one. Space and time are unified to the four-dimensional space-time. The Lorentz transformations can be regarded as rotations in the four-dimensional space-time.

Electrodynamics is another important classical field of physics, which describes the behavior of electric charges and of electromagnetic fields. The theory of classical electrodynamics was introduced by James Clerk Maxwell in 1864. The basic equations of electrodynamics are the Maxwell equations.

Quantum Mechanics

Besides relativistic mechanics, quantum mechanics is another important branch of modern physics. Quantum physics started in 1900, when Max Planck wrote a paper on the quantization of the energy in electromagnetic processes. He introduced the new fundamental constant h.

In 1905 Albert Einstein proposed that the energy of light is quantized. Light is a collection of particles, which are called photons. The energy of a photon is given by the product of Planck’s constant h and the frequency of the light.

In 1924 Louis de Broglie published a theory of matter waves. A particle, e.g. an electron, is at the same time also a wave, just like the case of a photon. The wave length is given by the ratio of Planck’s constant and the momentum of the electron. Thus electrons and photons are similar, but photons move at the speed of light, while electrons can have any speed smaller than the speed of light.

Werner Heisenberg discovered in 1927 that in quantum physics observables cannot be precisely determined; they have a fundamental uncertainty, which is related to Planck’s constant h. For example, the product of the uncertainty of the position of a particle and the uncertainty of the momentum cannot be smaller than Planck’s constant:

In 1926 Erwin Schrödinger introduced the wave mechanics. He interpreted the matter waves as the wave functions of particles. The time evolution of these wave functions is described by a differential equation, the Schrödinger equation. He showed that his wave mechanics is equivalent to the matrix mechanics. Following this school of thought, Max Born interpreted the square of the wave function as the probability density.

In 1914 James Chadwick discovered that energy and momentum were not conserved in the beta decay of atomic nuclei. For many years this phenomenon was not understood. In 1930 Wolfgang Pauli suggested that in the beta decay not only an electron was emitted, but also a neutral particle, which could not be observed. The energy and the momentum of this particle, later called the neutrino, would be the observed missing energy and momentum. Pauli assumed that neutrinos could never be observed directly. But in 1956 they were discovered by Clyde Cowan and Frederick Reines while investigating the neutrino emission of a big reactor in South Carolina.

Quantum Electrodynamics

In 1929 Werner Heisenberg and Wolfgang Pauli went one step further from quantum physics and quantized the electromagnetic field, which started the development of quantum electrodynamics (QED). In this theory the electromagnetic forces are generated by the exchange of photons.

The Schrödinger equation cannot be used to describe relativistic effects, since it contains the second derivatives of the three-dimensional space, but only the first derivative of time. A relativistic field equation was introduced by Paul Dirac in 1928. In the Dirac equation only the first derivatives with respect to space and time appear. Dirac noticed that his equation was only consistent if the electron has an antiparticle, later called the positron. The positron was discovered in 1932 in the cosmic rays.

In quantum electrodynamics there is a serious problem. An electron can emit virtual photons, but can also absorb them afterwards. If one calculates this process, one obtains a divergent result — the charge and also the mass of an electron are infinite. This problem was solved by Julian Schwinger, Richard Feynman and Freeman Dyson.

An electron without the electromagnetic interaction has a “naked mass” and a “naked charge.” If the electromagnetic interaction is introduced, the contributions to the mass and to the charge are infinite. If one assumes that the naked mass and the naked charge are also infinite, but with a negative sign, the sum of the naked mass plus the corrections should be equal to the observed mass of the electron. The same can be done for the charge. Then all infinities have disappeared, due to the renormalization of the mass and the charge.

In quantum electrodynamics many quantities can be calculated using this renormalization technique, for example the magnetic moment of the electron. The calculated values agree with the observed values — the difference between the observed values and the calculated values is less than 10⁻⁸!

Gauge Theories

In quantum electrodynamics the electron interacts with the photon. Two fields are present, the Dirac field for the electron and the vector field for the photon. If the phase of the Dirac field is changed in such a way that the change depends on space and time and if the vector field is changed by adding the space-time derivative of the phase of the Dirac field, then nothing changes. This symmetry is called a gauge symmetry. It was discovered in 1918 by Hermann Weyl.

Theories of this type are called gauge theories. The associated gauge group is the group of phase transformations, the group U(1). Thus quantum electrodynamics is a gauge theory with the gauge group U(1) — the photon is a gauge boson.

Wolfgang Pauli studied in 1953 a gauge theory with the gauge group SU(2). In this theory the gauge bosons are a triplet of the gauge group, thus there would be three gauge bosons without a mass. But in nature such gauge bosons do not exist, unless they have a very large mass. Pauli did not know how to introduce a mass for the gauge bosons. Thus he did not publish his idea.

But in 1954 the SU(2) gauge theory was published by Chen Ning Yang and Robert Mills, who worked in Princeton at the Institute for Advanced Study. However Yang and Mills also did not know how to introduce a mass for the gauge bosons.

After 1960 Sheldon Glashow, Abdus Salam and Steven Weinberg unified quantum electrodynamics and the theory of the weak interactions. They constructed a theory of the electroweak interactions based on the gauge group SU(2)×U(1). In this theory the weak forces are generated by the exchange of very massive gauge bosons. The theory has four gauge bosons: three massive bosons, which mediate the weak interactions, and the photon.

The masses of these bosons were generated by a spontaneous symmetry breaking. This mechanism was introduced in 1964 by Robert Brout, Francois Englert and Peter Higgs. In 1971 it was shown by Gerard ’t Hooft and Martinus Veltman, that a gauge theory is renormalizable, if the masses of the gauge bosons are introduced by a spontaneous symmetry breaking.

Quantum Chromodynamics

The atomic nuclei are bound states of protons and neutrons. But these nucleons are not elementary, but consist of the quarks. A proton is a bound state of three quarks. Two different quarks are needed to describe all atomic nuclei, the up quarks and the down quarks. The electric charge of the up quark is (+2/3)e, the charge of the down quark is (–1/3)e — the electric charge of the proton is (+e).

The interactions among the quarks are described by the theory of quantum chromodynamics (QCD), introduced by Murray Gell-Mann and the author in 1972. The forces among the quarks are generated by the exchange of the gauge bosons of QCD, the gluons.

The quarks and gluons do not exist as free particles — they are permanently confined in the hadrons. The quarks can be observed indirectly in the scattering of electrons and atomic nuclei. Such experiments were started in 1967 at the Stanford Linear Accelerator Center in California.

Standard Model of Particle Physics

In nature there exist not only the up and down quarks, but four other quarks as well, the strange quark, the charm quark, the bottom quark and the top quark. The stable matter in the universe consists only of the up and down quarks. The other four quarks are inside unstable hadrons.

Besides the electron and its neutrino, there are also four other similar particles, the muon and the tau-lepton (or tauon), plus the two associated neutrinos. These six particles are called leptons.

Since there are six quarks and six leptons, it is possible to arrange them into three families. Each family consists of two quarks and two leptons. The electron, the electron neutrino, the up quark and the down quark form the first family; the muon, the muon neutrino, the charm quark and the strange quark define the second family. Lastly the third family consists of the tauon, the tau neutrino, the top quark and the bottom quark. With these three families of fundamental particles and the fundamental interactions (electroweak theory and quantum chromodynamics) put together, we have the Standard Model of particle physics.

It is not known, how the masses of the quarks and the leptons are generated and why the masses are quite different. The mass of the electron is only 0.5 MeV, but the mass of the top quark is very large, about 174 000 MeV.

Also we do not know, whether the Standard Model is an exact description of nature or only a good approximation. Most physicists think that at very high energies new interactions and new particles will play an important role. There are indications that there is physics beyond the Standard Model. For example, now it is known that neutrinos must have a small mass — in the Standard Model the neutrinos do not have any mass.

Beyond the Standard Model

About 80% of the matter in the universe is not the nuclear matter inside the stars, but dark matter. However, it remains unknown whether the dark matter consists of yet unknown neutral stable particles. Since 2012 one has been searching for physics beyond the Standard Model with the Large Hadron Collider (LHC) at CERN, thus far without any success.

Many physicists assume that at very high energies the electroweak theory and quantum chromodynamics are unified. In a Grand Unified Theory the electroweak gauge group and the color group of QCD are subgroups of a larger group. An interesting possibility is the group SO(10), introduced in 1975 by Peter Minkowski and the author (see Chapter 16).

In a Grand Unified Theory both leptons and quarks are described together in certain representations of the gauge group. Thus the proton is not stable, but can decay, e.g. into a positron and a neutral pion. One has searched for the proton decay with large detectors, e.g. with the detector Kamiokande near Kamioka in Japan, thus far without success. According to these experiments the lifetime of a proton must be larger than about 10³² years.

It is possible that the leptons and quarks are not elementary point-like particles, but small one-dimensional objects, the strings. A unification of all interactions, including gravity, might be achieved in a string theory.

Chapter 2

Mechanics

Classical Mechanics

The movement of a point mass is determined by the principle of least action. The Lagrange function of a point mass is given by the difference of the kinetic energy ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Contents
• Chapter 1. Introduction
• Chapter 2. Mechanics
• Chapter 3. Classical Fields
• Chapter 4. Quantum Theory
• Chapter 5. Group Theory
• Chapter 6. Free Scalar Fields
• Chapter 7. Free Spinor Fields
• Chapter 8. Free Vector Fields
• Chapter 9. Perturbation Theory
• Chapter 10. Quantum Electrodynamics
• Chapter 11. Symmetry
• Chapter 12. Gauge Theories
• Chapter 13. Quantum Chromodynamics
• Chapter 14. Spontaneous Symmetry Breaking
• Chapter 15. Electroweak Interactions
• Chapter 16. Grand Unification
• Appendix
• Bibliography