An Elementary Primer for Gauge Theory

K Moriyasu

• Publisher: WSPC
• Year: 1983
• ISBN:
  9789814338011
• Topic: Physical Sciences
• Subtopic: Nuclear Physics

Gauge theory is now recognized as one of the most revolutionary discoveries in physics since the development of quantum mechanics. This primer explains how and why gauge theory has dramatically changed our view of the fundamental forces of nature. The text is designed for the non-specialist. A new, intuitive approach is used to make the ideas of gauge theory accessible to both scientists and students with only a background in quantum mechanics. Emphasis is placed on the physics rather than the formalism. Contents:

• Introduction
• The Einstein Connection
• Weyl's Gauge Theory
• The Aharonov-Bohm Effect
• Isospin and the New Gauge Theory
• Yang-Mills Gauge Theory
• Maxwell Equations in Gauge Theory
• Leptons and Quarks
• The Weak Interaction
• The Dark Age of Field Theory
• Symmetry Breaking in Gauge Theory
• The Superconductor Analogy
• Weinberg-Salam Unified Theory
• Unification and Renormalization
• Color Gauge Theory
• Aymptotic Freedom and the Running Coupling Constant
• Topology in Gauge Theory
• Flux Trapping and Vortices
• Dirac Magnetic Monopole
• Appendix on Key Concepts in Group Theory

Readership: Students and researchers in high energy physics and mathematical physics.

CHAPTER II

THE REDISCOVERY OF GAUGE SYMMETRY

. . gauge invariance has no physical meaning, but must be satisfied for all observable quantities in order to ensure that the arbitrariness of A and ϕ does not affect the field strength.

Röhrlich, 1965¹

2.1 Introduction

Gauge invariance was recognized only recently as the physical principle governing the fundamental forces between the elementary particles. Yet the idea of gauge invariance was first proposed by Hermann Weyl² in 1919 when the only known elementary particles were the electron and proton. It required nearly 50 years for gauge invariance to be “rediscovered” and its significance to be understood. The reason for this long hiatus was that Weyl’s physical interpretation of gauge invariance was shown to be incorrect soon after he had proposed the theory. Gauge invariance only managed to survive because it was known to be a symmetry of Maxwell’s equations and thus became a useful mathematical device for simplifying many calculations in electrodynamics. In view of the present success of gauge theory, we can say that gauge invariance was a classical case of a good idea which was discovered long before its time.

In this chapter, we present a brief historical introduction to the discovery and evolution of gauge theory. The early history of gauge theory can be divided naturally into old and new periods where the dividing line occurs in the 1950’s. In the old period, we will return to Weyl’s original gauge theory to gain insight into several key questions. The most important question is what motivated Weyl to propose the idea of gauge invariance as a physical symmetry? How did he manage to express it in a mathematical form that has remained almost the same today although the physical interpretation has altered radically? And, how did the development of quantum mechanics lead Weyl himself to a rebirth of gauge theory?

The new period of gauge theory begins with the pioneering attempt of Yang and Mills³ to extend gauge symmetry beyond the narrow limits of electromagnetism. Here we will review the radically new interpretation of gauge invariance required by the Yang-Mills theory and the reasons for the failure of the original theory. By comparing the new theory with that of Weyl, we can see that many of the original ideas of Weyl have been rediscovered and incorporated into the modern theory.

2.2 The Einstein Connection

In 1919, only two fundamental forces of nature were thought to exist – electromagnetism and gravitation. In that same year, a group of scientists also made the first experimental observation of starlight bending in the gravitational field of the sun during a total eclipse⁴. The brilliant confirmation of Einstein’s General Theory of Relativity inspired Hermann Weyl to propose his own revolutionary idea of gauge invariance in 1919. To see how this came about, let us first briefly recall some basic ideas involved in relativity.

The fundamental concept underlying both special and general relativity is that there are no absolute frames of reference in the universe. The physical motion of any system must be described relative to some arbitrary coordinate frame specified by an observer, and the laws of physics must be independent of the choice of frame.

In special relativity, one usually defines convenient reference frames which are called “inertial”, i.e. moving with uniform velocity. For example, consider a particle which is moving with constant velocity v with respect to an observer. Let S be the rest frame of of the observer and S′ be an inertial frame which is moving at the same velocity as the particle. The observer can either state that the particle is moving with velocity v in S or that it is at rest in S′. The important point to be noted from this trivial example is that the inertial frame S′ can always be related by a simple Lorentz transformation to the the observer’s frame S. The transformation depends only on relative velocity between particle and observer, not on their positions in space-time. The particle and observer can be infinitesimally close together or at opposite ends of the universe; the Lorentz transformation is still the same, Thus the Lorentz transformation, or rather the Lorentz symmetry group of special relativity, is an example of “global” symmetry.

In general relativity, the description of relative motion is much more complicated because one is dealing with the motion of a system in a gravitational field. For the sake of illustration, let us consider the following “gedanken” exercise for measuring the motion of a test particle which is moving through a gravitational field. The measurement is to be performed by a physicist in an elevator. The elevator cable has broken so that the elevator and physicist are falling freely⁵. As the particle moves through the fields the physicist determines its motion with respect to the elevator. Since both particle and elevator are falling in the same field, the physicist can describe the particle’s motion as if there were no gravitational field. The acceleration of the elevator cancels out the acceleration of the particle due to gravity. This is a simple example of the principle of equivalence, which follows from the well-known fact that all bodies accelerate at the same rate in a given gravitational field (e.g. 9.8 m/sec² on the surface of the earth).

Let us now compare the physicist in the falling elevator with the observer in the inertial frame in special relativity. It might appear that the elevator corresponds to an accelerating or “non-inertial” frame that is analogous to the frame S′ in which the particle appeared to be at rest, However, this is not true because a real gravitational field does not produce the same acceleration at every point in space. As one moves infinitely far away from the source, the gravitational field will eventually vanish. Thus, the falling elevator can only be used to define a reference frame within an infinitesimally small region where the gravitational field can be considered to be uniform. Over a finite region, the variation of the field may be sufficiently large for the acceleration of the particle not to be completely cancelled.^a

We see that an essential difference between special and general relativity is that a reference frame can only be defined “locally” or at a single point in a gravitational field. This creates a fundamental problem. To illustrate the difficulty, let us now suppose that there are many more physicists in nearby falling elevators. Each physicist makes an independent measurement so that the path of the particle in the gravitational field can be determined. How are the individual measurements to be related to each other? The measurements were made in separate elevators at different locations in the field. Clearly, one cannot perform an ordinary Lorentz transformation between the elevators. If the different elevators were related only by a Lorentz transformation, the acceleration would have to be independent of position and the gravitational field could not decrease with distance from the source.

Einstein solved the problem of relating nearby falling frames by defining a new mathematical relation known as a “connection”. To understand the meaning of a connection, let us consider a 4-vector A_μ which represents some physically measured quantity. Now suppose that the physicist in the elevator located at x observes that A_μ changes by an amount dA_μ and a second physicist in a different elevator at x′ observes a change

How do we relate the changes and dA_μ and

In special relativity, the differential dA_μ is also a vector like A_μ itself. Thus, the differential

in the the elevator at x′ is given by the familiar relation

where, according to the usual convention^b, the repeated index μ is summed over the values = 0, 1, 2, 3. The simple relation (II - 1) follows from the fact that the Lorentz transformation betwee...

Table of contents

• Cover
• Half title
• Title
• Copyright
• Dedication
• Contents
• Preface
• I. Introduction
• II. Rediscovery of Gauge Symmetry
• III. Gauges, Potentials and All That
• IV. Yang-Mills Gauge Theories
• V. The Maxwell Equations
• VI. The Difficult Birth of Modern Gauge Theory
• VII. The Breaking of Gauge Symmetry
• VIII. The Weinberg-Salam Unified Theory
• IX. Color Gauge Theory
• X. Topology and Gauge Symmetry
• Appendix: Some Key Group Theory Terms
• Index