Gauge Theories

Type of physical science: Elementary particle (high-energy) physics

Field of study: Systematics (particle physics)

Gauge theories are that class of quantum theories which describes the basic forces of nature in terms of the restoration of symmetry and the exchanges of elementary particles.

Overview

The central tenets of gauge theory were first explored by the German mathematician Hermann Weyl in 1921, when he attempted to find a geometrical basis to unify James Clerk Maxwell's theory of electromagnetism and electrodynamics, and Albert Einstein's theory of relativity, which describes the operation of the force caused by gravity. "Gauge theory" is not one particular theory; rather, it is a group of theories that exhibits particular mathematical and phenomenological properties. The mathematical properties, gauge symmetry and invariance, have made it possible to develop very powerful theories that describe the elementary structure of matter--the behavior of those particles that constitute atomic structure and the forces that intervene between them.

Gauge theories have made possible the science of "high-energy physics," so named because the particles under study are examined with the aid of particle accelerators, giant scientific apparatus which accelerate particles such as protons and electrons to very high velocities, and thus energies. These high-energy particles are then forced to collide with detector/targets or other accelerating particles in order to create other particles. This is the phenomenological level of gauge theories. Gauge theories postulate the existence of a set of subatomic particles, the bosons, which transmit the four forces of nature: electromagnetism, strong and weak nuclear forces, and gravitation. Photons, the massless particles that transmit the electromagnetic force, are also the constituents of electromagnetic radiation, of which visible light is but one part. Gluons are responsible for the strong nuclear force, which is the force that holds the hadrons and quarks together in atomic nuclei. The intermediate vector bosons, first detected in 1983, are very massive particles that transmit the weak nuclear force, the force behind radioactive α and β decay processes. Gravitons are postulated to carry the gravitational force. The existence of these particles, and other particles such as quarks, which make up protons and neutrons (the hadrons), can often be confirmed in high-energy experiments.

Gauge symmetry is the most important notion in the study of gauge theories.

Something is symmetrical if it remains unchanged under a certain operation. For example, a human face is symmetrical, for if a line is drawn down the middle, the face is the same on both sides. A sphere is symmetrical, for it looks the same from whatever angle one looks at it or whatever axis one draws through it. Gauge symmetries involve the idea of "regauging" the scale with which physical properties are measured, such as charge and magnetic field strength, in space and time. The central idea behind gauge theories is that physicists believe that all forces in nature exist to maintain symmetries in the world. These forces act as fields--namely, quantized gauge fields--that preserve the symmetry in the universe. What stays the same here, in other words, what is symmetric, are the laws of physics themselves--those regularities that govern the behavior of the universe. The most simple illustration of gauge symmetry and gauge invariance is Maxwell's electromagnetic theory, the simplest of all gauge theories, although it is not a quantum gauge theory in the twentieth century sense.

When the laws of physics remain unchanged under particular transformations through all of space and time, one says that these laws are globally symmetric. For example, suppose that a group of static, electrically charged particles exists in space. Maxwell's theory states that a certain electrical field is generated by their presence. In addition, the forces between the particles are determined by the differences in their charges, known as the potential difference or voltage. If the magnitude of the potential surrounding these particles were to be raised to any arbitrary value, the forces between the particles would not change, for it is only the differences between the charges on the particles, not their absolute values, that determine the forces. There are, however, local symmetries as well. These come into effect when the particles are allowed to move and thus generate a magnetic field in addition to the electric field. Arbitrary changes can be applied to this system on a particle by particle basis--that is, locally--and still leave the forces governing the particle motions unchanged, for a change in local electric potential is compensated by the subsequent change in local magnetic field. As a result, the electrical and magnetic field values for the system are said to be "gauge invariant."

In general, the four forces in nature exist to preserve local gauge symmetry and to compensate for any local gauge changes. Another example is that of the gravitational force. The pilot of a spaceship moving in a circular path at constant velocity experiences the sensation of being forced to the outside walls of the ship by centripetal acceleration. Yet, if the ship is moving in the same circular path, but around a planet, which introduces a gravitational field, the pilot has the sensation of moving in a straight line, for the gravitational field of the planet--a gauge field--has compensated for the motion of the ship and restored the local gauge symmetry.

Physicists have been astounded with the mathematical power of gauge theories. One such theory, known as "quantum electrodynamics," or QED, predicts the existence of the electrical and magnetic fields explained by Maxwell's theories of electrodynamics. Without ever conducting an empirical experiment, such as those of Maxwell's predecessor, the great experimentalist Michael Faraday, the mathematics requires that such fields exist. In all gauge theories, such theoretical predictions spur future experimental work.

One advantage of gauge theories is that they exhibit a mathematical property known as "renormalization." When a quantum field theory is renormalizable, the mathematics can be reformulated to eliminate the appearance of "infinite terms." Particles in quantum theory, such as electrons, are not described as the traditional "billiard ball" styled particles. Instead, they are seen as structureless and dimensionless points in space which emanate energy fields.

Unfortunately, when other particles, with their associated quantum fields, approach this field at arbitrarily small distances, the mathematics shows that the force--in this case, an electromagnetic force--rises to infinity. Infinite forces, and their subsequent infinite energies, are not observed in nature. Also, Einstein's special theory of relativity illustrates the fact that a particle's mass is equivalent to its energy, so that the electron, in this case, has an infinite mass.

Such a prediction does not agree with known experimental results.

To compensate, the theorist adjusts the scale by which mass is measured by a compensating infinite amount, effectively adjusting the "zero point" on the scale so that it yields a finite reading, one that is consistent with the workings of the physical world. This is the process of renormalization. While this process may seem to be an example of ad hoc adjustment in science, it is simply the realization that measuring scales are arbitrary, and that there is no predetermined place to fix a zero point on a scale. Adjusting it by a factor of ten or infinity makes no difference. The realization that gauge theories were amenable to such a mathematical operation made the rise of gauge theories in modern physics possible.

Applications

Gauge theories are expected to give physicists the most precise and complete understanding of the underlying structure of the universe. While Maxwell's electrodynamical theory of the nineteenth century was shown, in retrospect, to be a gauge theory, it is not consistent with the twentieth century quantum description of the world. The quantum version is known as quantum electrodynamics (QED).

QED describes the effects associated with electromagnetic phenomena, including light of all wavelengths, electricity, and magnetism, as quantum field phenomena. Freeman J. Dyson, Richard P. Feynman, and others developed the theory in the late 1940's. It combines Maxwell's theory, known as "classical electrodynamics," with the two most powerful and broadly explanatory physical theories of the twentieth century--quantum mechanics, which details the wave-particle duality of all matter, describes energy in terms of discrete packets or "quanta," and places limits upon the ability to observe particle properties; and Einstein's special theory of relativity, which equates a particle's mass with its energy and describes the interrelationship of space and time.

As an example, if one imagines two electrons, or other leptons, approaching each other, they eventually approach each other to a point where their like charges repel. The electrical force between the two electrons is transmitted by photons. All electrons, and other charged particles, are surrounded by a swarm of so-called virtual photons, photons that do not emit visible quanta of radiation, but which nevertheless generate the quantum electromagnetic field around all solitary electrons. When two electrons approach each other, their two swarms do as well.

Repulsion occurs by the mutual exchange of the virtual photons, the carriers of the electromagnetic force.

QED is an incredibly successful theory and has had wide empirical success. QED predicts accurately the energy levels of the simplest of atomic systems, the hydrogen atom, as well as the magnetic field carried by an electron. Such accurate predictions led the physics community to hypothesize analogous gauge theories for the explanation of the strong and weak forces.

During the 1970's, two new gauge theories were developed that are analogous to QED and that explain the strong and weak nuclear forces. The first of these, the so-called electroweak theory developed by Steven Weinberg, Abdus Salam, and Sheldon L. Glashow, is doubly significant, for it not only provides a gauge theoretical description of the weak interaction but also unifies the weak nuclear force and the electromagnetic force in one broadly explanatory theory.

The weak force is more complicated than the electromagnetic force, for it involves many different particles, such as the muon, neutron, proton, electron, and neutrino, which, under the influence of the weak force, have the ability to transmute among themselves. This transmutation, say, from electron to neutrino and vice versa, disturbs the local gauge symmetry, unless a force, in this case, the weak nuclear force, is introduced. The complication of the weak interaction and its associated transmutations is reflected in the fact that the theory requires three particles for the transmission of the weak force, as opposed to only one in the electromagnetic interaction. These three particles are the W⁺, W^-, and Z⁰ particles, which are known as the intermediate vector bosons and, unlike the massless photon, are very massive. The predicted existence of the neutral Z was a bold step for the electroweak theory, for prior accounts of the weak interaction precluded the existence of such particles. In 1973, experiments at the CERN (European Laboratory for Particle Physics) particle accelerator laboratory in Geneva concluded that forces caused by the neutral boson--so-called weak neutral currents--do indeed exist.

Despite such obvious benefits, the electroweak theory was, in its initial stages, beset with two problems: that of renormalization and another more vexing problem, that of the masses of the intermediate vector bosons at the heart of the theory. Weinberg and Salam resolved the latter problem in 1967, with the notion of "spontaneous symmetry breaking." Simply put, the weak force is a short distance force, contrary to the electromagnetic force, which acts over large distances in space. Such long distances require bosons to be massless, such as the photon.

Nevertheless, the short distances of the weak interaction require the intermediate vector bosons to be quite massive. Massive particles, such as the W⁺, W^-, and Z⁰, would break the gauge symmetry provided by the weak nuclear force. In spontaneous symmetry breaking, the intermediate vector bosons do not exist normally in a state that reflects the gauge symmetry because it is an unstable state. The particles have the ability to enter a stable state and preserve symmetry resulting from the presence of another massive particle, the so-called Higgs boson, named for the English physicist Peter Higgs. The fields generated by the Higgs boson and the intermediate vector bosons interact to give mass to the W⁺, W^-, and Z⁰ particles. This process explains the "weakness" of the weak nuclear force, which is a function of the intermediate vector bosons' masses.

Upon devising spontaneous symmetry breaking and unifying the gauge theoretic description of the weak interaction with QED to form the electroweak theory, physicists needed to demonstrate the renormalization of the theory. Gerard 't Hooft accomplished this task in 1971.

Weinberg, Salam, and Glashow shared the 1979 Nobel Prize in Physics for their achievement. In 1984, Carlo Rubbia and Simon van der Meer received the Nobel Prize in Physics for demonstrating the existence of the W⁺, W^-, and Z⁰ particles in high-energy experiments at CERN, thus confirming the electroweak theory.

Similar to the electroweak theory, a gauge theory for the strong interaction has been developed. The strong force is that which binds the atomic nucleus and the quarks that compose its constituent hadrons. The strong force is carried by a set of eight particles known as gluons.

Quarks carry an analogue to electrical charge which is known (not literally) as "color." Quarks come in three colors--blue, green, and red--and all particles that are composed of quarks (for example, neutrons, protons, and the unstable mesons) are "color neutral," with quark colors adding to "white," just as the visible colors of the spectrum. This notion gave rise to the name of the theory, quantum chromodynamics, or QCD.

The strong force, transmitted by the eight varieties of gluons, preserves an intricate local gauge symmetry under color changes within the hadrons, for quarks continually change colors in this theory. Unfortunately, isolated quarks have yet to be confirmed to exist, which would seem to give nearly decisive confirmation to QCD. Instead, it appears that quarks are the victims of a phenomenon known as "confinement," which does not permit them to exist outside of color combinations which add to white. In spite of this fact, QCD is a powerful gauge theory that has yielded great explanatory and predictive success in the understanding of nuclear phenomena such as hadron collisions.

Context

The drive to unify physics is an old undertaking that reflects the faith that, given a universe that seems to exist as a unified whole, a theory or small set of theories should be able to explain it. Gauge theories are yet another step in this tradition, which traces its roots to the birth of modern science, in the seventeenth century work of Sir Isaac Newton. Newton's theories of classical mechanics and universal gravitation can be seen as the mathematical unification of the terrestrial physics of Galileo and the celestial mechanics of Johannes Kepler.

Unification was a driving force in the nineteenth century as well. In 1820, Hans Christian Orsted first demonstrated the relationship between electricity and magnetism. Faraday refined rsted's work and developed the concept of the magnetic field and, by the end of the nineteenth century, Maxwell succeeded in mathematizing the idea and developed a theory that described and explained the phenomena of electricity and magnetism.

This spirit of unification persists in twentieth century physics. The development of the electroweak gauge theory is one manifestation. The most ambitious example, however, is the attempt to develop so-called grand unified theories, or GUTs. Where

Weinberg, Salam, and Glashow unified the electromagnetic and weak nuclear forces, attempts began in the late 1970's to develop unifications of the electroweak theory with QCD and to unite these with Einstein's theory of general relativity, the widely accepted explanation of gravitational phenomena, space and time. Such a unified theory promises many insights into the origins of the universe and the ideas of "big bang" cosmology.

Many obstacles stand in the way of such a theory. One is the fact that, while gravitation is a gauge field, it appears that gauge theories of gravity may be nonrenormalizable. The quantum description of gravity requires virtual particles known as gravitons. Since all particles are affected by gravity, they would be surrounded by gravitons. This would include gravitons themselves, and their gravitons, and theirs, and so on, until an infinite gravitational field emerges. A gauge theory of gravity is impossible without the ability to eliminate this infinity by renormalization. Finally, general relativity is mathematically inconsistent with quantum mechanics, which poses yet another theoretical problem for those wishing a grand unification.

Nevertheless, the drive to unify a theoretical understanding of the world has been successful throughout the history of modern science and, as in the case of the electroweak and QCD theories, building upon past success has been shown to be a profitable undertaking.

Principal terms

ALPHA PARTICLES: energetic helium nuclei, a product of radioactive decay

BETA PARTICLES: energetic electrons, a product of radioactive decay

BOSONS: the group of elementary particles that transmit the basic forces of nature, including the photon, gluon, and intermediate vector bosons

FIELD: a physical parameter defined at all points in space and time in a particular region, such as an electromagnetic field

HADRONS: the group of particles that experience the strong nuclear force, such as protons and neutrons; believed to be composed of three quarks

LEPTONS: the group of elementary particles that experience the weak nuclear force, such as electrons, muons, and neutrinos

QUANTUM FIELD THEORY: a type of physical theory in which the strength of a field, for example, an electromagnetic field, can take on only certain discrete values

QUANTUM MECHANICS: a branch of physics that details the workings of the basic constituents of matter and is characterized by the use of mathematical probabilities and the notion that matter behaves as both particles and waves

QUARK: the fundamental constituent of the hadrons; these particles have a fractional electrical charge

SYMMETRY: the property of a system to remain unchanged in one or more properties under changing conditions; for example, a sphere is geometrically symmetrical under changes in position

Bibliography

Cline, David B., Carlo Rubbia, and Simon van der Meer. "The Search for Intermediate Vector Bosons." SCIENTIFIC AMERICAN 246 (March, 1982): 48-59. An excellent introduction to the successful experimental confirmation of one of the most important of all gauge theories--the Weinberg-Salam electroweak theory.

Davies, Paul. THE FORCES OF NATURE. New York: Cambridge University Press, 1986. This beginner's introduction takes the reader through all the fundamental concepts that underlie gauge theories, from forces, fields, and symmetry to elementary particles.

Davies, Paul. SUPERFORCE. New York: Simon & Schuster, 1984. This nontechnical introduction to high-energy physics and the program of unifying the four forces in nature devotes ample detail to the important concept of the gauge symmetry, without which an understanding of gauge theories is virtually impossible. The book suffers from a lack of illustrations and diagrams that could help to explicate some concepts in the absence of mathematical formalism.

Duff, Brian G. FUNDAMENTAL PARTICLES: AN INTRODUCTION TO QUARKS AND LEPTONS. Philadelphia: Taylor and Francis, 1986. This book outlines the formulation of gauge theories, the particles and forces that they describe, and some of their consequences. Contains some sophisticated mathematics, although the interested reader will find chapters 5 and 6, which describe the electroweak and quantum chromodynamic gauge theories, most enlightening.

Hawking, Stephen W. A BRIEF HISTORY OF TIME: FROM THE BIG BANG TO BLACK HOLES. New York: Bantam Books, 1988. This is how popular science should be written. It is clear and wonderfully illustrated, and the ample diagrams take the place of the mathematical formalism that often obscures the salient features of the subject for the lay reader. While the book is not about gauge theories specifically, two chapters discuss their consequences for the scientific understanding of the four fundamental forces in nature and the attempt to develop one unified theory to explain them all.

Nambu, Yoichiro. QUARKS: FRONTIERS IN ELEMENTARY PARTICLE PHYSICS. Philadelphia: World Scientific, 1981. An introduction to the electroweak and quantum chromodynamic gauge theories. Chapters 16 and 19 are especially relevant. Unfortunately, a background in higher-level calculus is assumed, and the illustrations are more along the line of amusing cartoons than informative heuristics.

Pickering, Andrew. CONSTRUCTING QUARKS: A SOCIOLOGICAL HISTORY OF PARTICLE PHYSICS. Chicago: The University of Chicago Press, 1984. Pickering's constructivist epistemology tends to be misleading, but the book contains one of the best accounts of the development of twentieth century gauge theories written for a nonscientific audience. Part 2 is recommended.

Quigg, Chris. GAUGE THEORIES OF THE STRONG, WEAK, AND ELECTROMAGNETIC INTERACTIONS. Reading, Mass.: Benjamin/Cummings, 1983. Entirely devoted to the principles behind gauge theory, this book is based upon Quigg's lecture series and is primarily geared to a scientific audience. The technicalities, however, are explained in clear illustrations.

The Structure of the Atomic Nucleus

Grand Unification Theories and Supersymmetry

Leptons and the Weak Interaction

Quarks and the Strong Interaction

The Unification of the Weak and Electromagnetic Interactions