From Photons to Higgs

A Story of Light

Moo-Young Han

• Publisher: WSPC
• Year: 2014
• ISBN:
  9789814579971
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This book presents a brief introduction to the quantum field theory of the Standard Model for quarks and leptons. With minimal use of mathematics, it covers the basics of quantum field theory, local gauge field theory, spontaneous symmetry breaking mechanism, the Higgs mechanism and quantum chromodynamics.

From the time when the first edition was published until today, the field of particle physics has seen some major break-through with the possible discovery of Higgs particle, also known as the Higgs boson. In the second edition, the famous Higgs mechanism is included to explain the symmetry breaking in the Standard Model and the origin of mass, and all of this is explained in high-school level algebra.

Aimed at both scientists and non-specialists, it requires only some rudimentary knowledge of the Lagrangian and Hamiltonian formulation of Newtonian mechanics as well as a basic understanding of the special theory of relativity and quantum mechanics to enjoy this book.

Contents:

• Particles and Fields I: Dichotomy
• Lagrangian and Hamiltonian Dynamics
• Canonical Quantization
• Particles and Fields II: Duality
• Equations for Duality
• Electromagnetic Field
• Emulation of Light I: Matter Fields
• Road Map for Field Quantization
• Particles and Fields III: Particles as Quanta of Fields
• Emulation of Light II: Interactions
• Triumph and Wane
• Leptons and Quarks
• What is Gauge Field Theory?
• The Weak Gauge Fields
• The Higgs Mechanism and the Electroweak Gauge Fields
• The Higgs Particle
• Evolution of the Strong Force
• The History of Color SU(3) Symmetry
• Quantum Chromodynamics, QCD
• Appendices:
  • The Natural Unit System
  • Notation
  • Velocity-Dependent Potential
  • Fourier Decomposition of Field
  • Mass Units for Particles
  • Mass-Range Relation

Readership: Students, researchers, academics and non-specialists interested in quantum field theory.

Particles and Fields I:
Dichotomy

One may have wondered when first learning Newtonian mechanics, also called the classical mechanics, why the concept of a field, the force field of gravity in this case, is hardly mentioned. One usually starts out with the description of motion under constant acceleration — the downward pull of gravity with the value of 9.81 m/s². Even when the universal law of gravity is discussed, for example, to explain the Kepler’s laws, we do not really get into any detailed analyses of the force field of gravity.

In classical mechanics the primary definition of matter is the point mass, and the emphasis is on the laws of motion for point masses under the influence of force. The focus is on the laws of motion rather than the nature of force field, which is not really surprising when we consider the simplicity of the terrestrial gravitational force field — uniform and in one parallel direction, straight down toward the ground. A point mass is an abstraction of matter that carries mass and occupies one position at one moment of time and this notion of a point mass is diagonally opposite from the notion of a field, which, by definition, is an extended concept, spread out over a region of space.

As we proceed from the study of classical mechanics to that of classical electromagnetism, we immediately notice a big change; from day one it is all about fields. First the electric field, then the magnetic field, and then the single combined entity, the electromagnetic field. No sooner than the Coulomb’s law is written down, one defines the electric field and its spatial dependence is determined by Gauss’ Law. Likewise, Ampere’s Law determines the magnetic field and finally the laws of Faraday and Maxwell lead to the spatial as well as temporal dependence of electromagnetic field.

This dichotomy of the concept of point particle and that of field is in fact as old as the history of physics. From the very beginning, back in the 17^th century, there were two distinct views of the physical nature of light. Newton advocated the particle picture — the corpuscular theory of light — whereas Christian Huygens advanced the wave theory of light. For some time — for almost a century and half — these two opposing views remained compatible with what was then known about light — refraction, reflection, lenses, etc. Only when in 1801 Thomas Young demonstrated the wave nature of light by the classic double-slit interference experiment, with alternating constructive and destructive interference patterns, the wave theory triumphed over the particle theory of light.

One might have wondered why the notion of field did not play a prominent role in the initial formulation of Newtonian mechanics, especially since both the gravitational force law and the Coulomb’s law obey the identical inverse square force law:

    for gravity

and

    for Coulomb’s law

where G and k are the respective force constants, m is mass and q is the electric charge.

The disparity is simply a practical matter of scale. At the terrestrial level, in our everyday world, the inverse square law really does not come into play; the curvature of the surface of the earth is approximated by a flat ground and the gravitational force lines directed toward the center of the earth become, in this approximation, parallel lines pointing downward. In this scale of things, the field aspect of gravity is just too simple to be taken into account. There is no need to bring in any analyses of the gravitational field in the flat surface approximation.

On the contrary, with electric and magnetic forces, we notice and measure in the scale of tabletop experiments the spatial and temporal variations of these fields. The gradients, divergences and curls, to use the language of differential vector calculus, of the electric and magnetic fields come into play in the scale of the human-sized world and this is why the study of electromagnetism always starts off with the definition of electric and magnetic fields.

This well-defined dichotomy of particles and fields, diagonally opposite concepts in classical physics, would evolve through many twists and turns in the twentieth century physics of relativity and quantum mechanics, ending up eventually with the primacy of the concept of field over that of particle in the framework of quantum field theory.

The process of evolution of the concepts of particles and fields has taken a quite disparate path. The Newtonian mechanics has evolved through several steps, some quite drastic. First, there was the Lagrangian and Hamiltonian formulation of mechanics. One of the most important outcomes of this formalism is the definition of what is called the canonically conjugate momentum and this would pave the way for the transition from classical mechanics to quantum mechanics. Quantum field theory could not have developed had it not been the idea of canonically conjugate momentum defined within the Lagrangian and Hamiltonian formalism. As quantum mechanics is merged with special theory of relativity, the culmination of the particle view was reached in the form of relativistic quantum mechanical wave equations, such as the Klein–Gordon and Dirac equations, wherein the wavefunction solutions of these equations provide the relativistic quantum mechanical description of a particle. (More on these equations in later chapters.)

In contradistinction to this development of particle theory, the field view of classical electromagnetism remained almost totally unmodified. The equation of motion for charged particles in an electromagnetic field is naturally accommodated in the Lagrangian and Hamiltonian formalism. In the Lagrangian formulation of classical mechanics, Maxwell’s equations find a natural place by being one of the few examples of what is called the velocity-dependent potentials (more on this in the next chapter). The very definition of the canonically conjugate momentum for charged particles to be the sum of mechanical momentum and the vector potential of the electromagnetic field, discovered back in the 19^th century, is in fact the foundation for quantum electrodynamics of the 20^th century.

The contrast between the mechanics of particles and the field theory of electromagnetic fields becomes sharper when dealing with the special theory of relativity. The errors of Newtonian mechanics at speeds approaching the speed of light are quite dramatic, and of course, the very foundation of mechanics had to be drastically modified by the relativity of Einstein. Maxwell’s equations for the electromagnetic field, on the other hand, required no modifications whatsoever at high speeds; the equations are valid for all ranges of speeds involved, from zero to all the way up to the speed of light. At first, this may strike as quite surprising, but the fact of the matter is that Maxwell’s equations lead directly to the wave equations for propagating electromagnetic radiation — light itself. Maxwell’s theory of the electromagnetic field is already fully relativistic and hence need no modifications at all.

The development of relativistic quantum mechanics demonstrates quite dramatically the primacy of the classical field concept over that of particles. To cite an important example, in relativistic quantum mechanics, the first and foremost wave equation obeyed by particles of any spin, both fermions of half-integer spin and bosons of integer spin, is the Klein–Gordon equation. Fermions must also satisfy the Dirac equation in addition to the Klein–Gordon equation (more on this in later chapters).

For a vector field ϕ_μ(x) [μ = 0, 1, 2, 3] for spin one particles with mass m, the Klein–Gordon equation is¹

(∂_λ∂^λ + m²)ϕ_μ(x) = 0

where

For the special case of mass zero particles, of spin one, the Klein–Gordon equation reduces to

∂_λ∂^λϕ_μ(x) = 0

The classical wave equation for the electromagnetic four-vector potential A_μ(x), on the other hand, in the source-free region is

∂_λ∂^λ A_μ(x) = 0

An equation for a zero-mass particle of spin one (photon) in relativistic quantum mechanics turns out to be none other than the classical wave equation for the electromagnetic field of the 19^th century that predates both relativity and quantum physics!

---

¹Notations and the natural unit system are given in Appendices 1 and 2.

Lagrangian and Hamiltonian
Dynamics

Lagrange’s equations were formulated by the 18^th century mathematician Joseph Louis Lagrange (1736–1813) in his book Mathematique Analytique published in 1788. In its original form Lagrange’s equations made it possible to set up Newton’s equations of motion, F = dp/dt, easily in terms of any set of generalized coordinates, that is, any set of variables capable of specifying the positions of all particles in the system. The generalized coordinates subsume the rectangular Cartesian coordinates, of course, but also include angular coordinates such as those in the plane polar or spherical polar coordinates. The generalized coordinates also allow us to deal easily with constraints of motion, such as a ball constrained to move always in contact with the interior surface of a hemisphere; the forces of constraints do not enter into the description of dynamics. As originally proposed, the Lagrange’s equations provided a convenient way of implementing Newton’s equations of motion.

Lagrange’s equations became much more than just a powerful addition to the mathematical t...