A Story of Light

The following sections are included:

• Acknowledgments

• Contents

The relativistic quantum field theory, or quantum field theory (QFT) for short, is the theoretical edifice of the standard model of elementary particle physics. One might go so far as to say that the standard model is the quantum field theory. Having said that as the opening statement of this book, we must be mindful that both quantum field theory and the standard model of elementary particle physics are topics that are not necessarily familiar to many individuals. They are subject areas that are certainly not familiar to those outside the specialty of elementary particle physics, and in some cases not too well grasped even by those in the specialty…

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…

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…

Transition from classical to quantum physics, together with the discovery of relativity of space and time, represents the beginning of an epoch in the history of physics, signaling the birth of modern physics of the 20^th century. Quantum physics consists, broadly, of three main theories — non-relativistic quantum mechanics, relativistic quantum mechanics, and the quantum theory of fields. In each case, the principle of quantization itself is the same and it is rooted in the canonical formalism of the Lagrangian and Hamiltonian formulation of classical mechanics. In the Hamiltonian formulation, the coordinates and momenta are accorded an equal status as independent variables to describe a dynamical system, and this is the point of departure for quantum physics…

The departure of quantum mechanics from classical mechanics is quite drastic, rather extreme in contemporary parlance. Ordinary physical quantities are replaced by quantum mechanical operators that do not necessarily commute with each other and the Heisenberg's uncertainty principles between the canonically conjugate pairs of variables, between coordinates and momenta and between time and energy, deny the complete determinability of classical physics…

The following sections are included:

• The Schrödinger's Equation

• The Klein–Gordon Equation

• The Dirac Equation

The classical theory of electromagnetism, as mentioned in Chapter 1, developed along an entirely different path from that of Newton's classical mechanics. From day one, electromagnetism was based on properties of force fields — the electric and magnetic fields that are extended in space. An electric field due to a point charge, for example, is defined over the entire three-dimensional space surrounding the point charge. The works of Coulomb, Gauss, Biot–Savart, Ampèere, and Faraday led Maxwell to the great unification of electricity and magnetism into a single theory of an electromagnetic field. Together with Einstein's theory of gravitational field, Maxwell's theory of electromagnetic field is one of the most elegant of classical field theories…

We are now at the point, after the first six chapters, to look back and compare where the equations of motion for the field and particles stand with respect to each other. As far as electromagnetic fields are concerned, the equations remain intact in its original form, as Maxwell had written down. As discussed in the last chapter, Maxwell's equations for the radiation field, that is, in the source-free region, are of very compact expression

which also coincided with the Klein–Gordon equation for mass-zero case. The equations for particles, on the other hand, evolved through several phases — from Newton to Lagrange and Hamilton and through relativity and quantum mechanics — and ended up in the form of wave equations for non-interacting one-particle within the framework of relativistic quantum mechanics, the Klein–Gordon and Dirac equations being prime examples…

We are now ready to proceed with the quantization of classical fields — the classical electromagnetic, Klein–Gordon and Dirac fields — that were discussed in the last chapter. The quantization of these fields is to be carried out following the rules of canonical quantization, as discussed in Chapter 3. Before imposing canonical quantization onto the classical fields, however, we need to extend the Lagrangian formalism from that of point mechanics to one more suitable for continuous classical field variables…

Quantum field theory presents the third and the latest stage in the evolution of the concept of particles. This concept has evolved from that of a localized point mass in classical physics to that of wave–particle duality in quantum mechanics and, as shown in this chapter, to that of a quantum of quantized field. As classical fields are quantized, following the road map outlined in the last chapter, we will see that the concept of particles has become secondary to that of quantized fields. The quantal structure of fields, or more precisely the quantal structure of energy and momentum of fields, defines particles as discrete units of the field carrying the energy and momentum characteristics of each particle. In this sense, fields play the primary physical role and particles only the secondary role as units of discrete energy of a given field…

The quantization of fields and the emergence of particles as quanta of quantized fields discussed in Chapter 9 represent the very essence of quantum field theory. The fields mentioned so far — Klein–Gordon, electromagnetic as well as Dirac fields — are, however, only for the non-interacting cases, that is, for free fields devoid of any interactions, the forces. The theory of free fields by itself is devoid of any physical content: there is no such thing in the real world as a free, non-interacting electron that exerts no force on an adjacent electron. The theory of free fields provides the foundation upon which one can build the framework for introducing real physics, namely, the interaction among particles…

The success of quantum electrodynamics in agreeing with and predicting some of the most exact measurements is nothing less than spectacular. The quantitative agreements between calculations of QED and experimental data for such atomic phenomena as the Lamb shift, the hyperfine structure of hydrogen, and the line shape of emitted radiation in atomic transitions are truly impeccable and has helped to establish QED as the most successful theory of interacting particles. As stated previously, this is what made QED the shining example to emulate for other interactions…

As mentioned in the last chapter, the heyday of quantum electrodynamics was over by the early 1950s and in the next two decades, the 1950s and 1960s, the canonical Lagrangian field theory was rarely spoken of. The 50s and 60s were primarily occupied by the search for patterns of symmetries in the world of elementary particles — such discoveries as strangeness, charm, unitary symmetry, the eightfold way, the introduction of quarks, and many others — and the pursuit of quantum field theory was carried out by those investigating the formal framework of the theory, generally called the axiomatic field theory, starting from scratch seeking new ways to deal with weak and strong nuclear forces. During this period that may be called the second phase of quantum field theory, the Lagrangian field theory was almost completely sidelined and the emphasis was on the formal and analytic properties of scattering matrix, the so-called S-matrix theories and the axiomatic approaches to field theory. These new axiomatic approaches, however, did not bring solutions to quantum field theories any closer than the Lagrangian field theories. Entering the 1970s, there was a powerful revival of the Lagrangian field theory that continues to this day. This is what is called the (Lagrangian) gauge field theory, and it starts — yes, once again — from electrodynamics! The gauge field theory represents the third and current phase in the development of quantum field theory…

The elite group of particles that constitute elementary particles — the basic building blocks of all known matter in the universe — are divided into two distinct camps, a group of heavier particles called hadrons and a group of relatively lighter ones called leptons. The premier member of hadrons, proton, for example, is about 1,874 times more massive than electron, the premier member of leptons. The names “hadrons” and “leptons” originate from Greek words meaning “strong” and “small,” respectively, although this distinction becomes blurred as the heaviest “lepton” turns out to be about twice as massive as proton. What really separates hadrons from leptons is more dynamical in nature than the gaps in their masses: hadrons interact via the strong nuclear force whereas leptons have nothing to do with the strong nuclear force. All particles, both hadrons and leptons interact via the weak nuclear force and electrically charged ones via the electromagnetic force…

The non-Abelian gauge symmetry described in the last chapter is, historically speaking, a combination of new and old. The weak SU(2) and the color SU(3) symmetries of quarks and leptons are certainly “new” ideas, having been developed in the 1960s and 1970s, but the idea of a non-Abelian gauge field theory itself is an “old” one, having been proposed in 1954 by C. N. Yang and R. L. Mills. The Yang–Mills theory, as it is called, actually predates the idea of quarks by about ten years. The gauge fields of the original Yang–Mills theory had to be massless and the only known massless gauge field at that time was the electromagnetic field. The force particles then known for non-electromagnetic interactions — pions for the strong nuclear force between protons and neutrons as well as W-bosons (sometimes called the intermediate vector bosons, the IVBs) that mediated the weak nuclear force — all had mass and Yang-Mills theory remained an interesting but unrealistic idea for almost two decades. Then came the weak SU(2) and the color SU(3) symmetries of quarks and leptons and Yang–Mills formalism was accorded a powerful revival…

The following sections are included:

• Epilogue: Leaps of Faith

• Appendix 1: The Natural Unit System

• Appendix 2: Notation

• Appendix 3: Velocity-Dependent Potential

• Appendix 4: Decomposition of Field

• Appendix 5: Evolution of Color Charges

• Index