50 Years of Yang-Mills Theory

The following sections are included:

• PREFACE

• CONTENTS

Yang and Mills' paper, originally published in The Physical Review [1] has been reproduced several times, such as in C. N. Yang's collection of Selected Papers [2]. Let us here briefly summarize the idea…

Gerardus 't Hooft wants me to write something about the early origin of non-Abelian gauge theory. I searched through my notes and found a few pages which I now contribute to this volume that he is editing…

We take the Royal Path in this book, by first presenting the formal arguments and attempts to quantize the Yang–Mills theory. After that, in Chapters 4–17, we turn to its physical aspects, the question of mass for the Yang–Mills quanta, and the many other questions posed by elementary particle physicists. The answers to all these questions came in bits and pieces while research continued, and now they exhibit the mathematical richness of these equations with impressive clarity…

The following sections are included:

• Introduction
  • Notation
  • Gauge invariance
  • Gauge transformations. The gauge group
  • The structure of Φ
  • A natural metric on Φ
  • A family of natural connections on Φ
  • Fibre-adapted coordinate patches
  • A metric for . Vilkovisky's connection
  • The functional integral for “in-out” amplitudes
  • Properties of the Jacobian J[φ]
  • A special choice for Ω[φ]
  • The ghost
  • One-loop approximation for the measure
  • Geodesic normal fields

• References

Even though the Yang–Mills theory was quoted every now and then, a vast majority of theoreticians continued to dismiss the notion of a quantized field. But, on both sides of the Iron Curtain, there were a few stubborn exceptions. Ludwig D. Faddeev is known for his powerful insights in Mathematical Physics, while he understood the language of the more pragmatic physicists who try to interpret experimental data. He, and his co-author V. N. Popov, were also attracted by the challenging questions posed by Feynman, and their way of formulating their results was more concise and accessible to their colleagues, even though a closer study reveals that they did arrive at the same conclusions as DeWitt. The important message of all was that the ghosts discovered by Feynman are due to a Jacobian factor in the functional integral. The Feynman rules are nothing but a perturbative expansion of an integral, and if the variables in an expression for an integral are being transformed, then one has to take Jacobian factors into account…

A method is developed for the manifestly covariant quantization of gaugeinvariant fields by means of a functional integration. It is shown that for the fields with non-Abelian gauge groups (the Yang–Mills and gravitational fields) fictitious particles appear naturally in the diagram technique, which are not present in the initial Lagrangian. An appearance of these particles restores the transversality of scattering amplitudes and the unitarity of the S-matrix.

Yang and Mills had not written their paper as just an esoteric curiosity to serve as a simplified model for Quantum Gravity. Their idea was that this force law had to play a role in the regime of the sub-atomic particles, but, from the very beginning, there had been the question of mass. Yang recalls that Pauli bugged him about it [1]. It was clear that

i) the theory, as it stood, predicted massless vector particles, mutually interacting as if they were charged photons, while

ii) such particles, regardless of whether they refer to ‘ordinary’ electric charge, or to some localized version of isospin, do not exist in Nature, and

iii) the mass is forbidden by gauge-invariance. The very basis of the theory would be destroyed if we simply would add mass terms to the Yang–Mills equations…

From its inception in statistical physics to its role in the construction and in the development of the asymmetric Yang–Mills phase in quantum field theory, the notion of spontaneous broken symmetry permeates contemporary physics. This is reviewed with particular emphasis on the conceptual issues.

After the dust settled, and renormalizability of these theories had been established, the “final word” was given by C. H. Llewellyn Smith [1] and Cornwall, Levin and Tiktopoulos [2]. They showed that renormalizability requires a sufficiently convergent high-energy behaviour of the on-shell amplitudes, in order to obey unitarity bounds, and found that the only theories obeying these conditions are Yang–Mills theories with any number of scalar and spinor fields with minimal couplings…

I have been asked to review the history of the formation of the Standard Model. It is natural to tell this story as a sequence of brilliant ideas and experiments, but here I will also talk about some of the misunderstandings and false starts that went along with this progress, and why some steps were not taken until long after they became possible. The study of what was not understood by scientists, or was understood wrongly, seems to me often the most interesting part of the history of science. Anyway, it is an aspect of the standard model with which I am very familiar, for as you will see in this talk, I shared in many of these misunderstandings…

The following sections are included:

• An Introduction
  • The largest time equation
  • Dressed propagators
  • Wave functions for in- and outgoing particles
  • Dispersion relations
  • Renormalization
  • Regularization schemes
  • Pauli–Villars regularization
  • Dimensional regularization
  • Equivalence of regularization schemes
  • Renormalization of gauge theories

• References

A reconstruction of perturbative gauge theories based on causality requires a suitable characterization of asymptotic fields. The algebraic machinery constructed by J. L. Koszul in the fifties, when applied to the free Maxwell and Proca fields via Wightman's reconstruction theorem, is shown to produce the usual collection of Faddeev–Popov ghosts and antighosts, Nakanishi–Lautrup–Lagrange multipliers, and Stueckelberg's scalar fields, as well as the corresponding antifields.

We show, in the simplest non-trivial case, the SU(2) pure Yang–Mills model, how the renormalization conditions implementing the whole set of Slavnov–Taylor identities can be reduced to a finite number of prescriptions for few renormalized one-particle irreducible Green functions. The result is strictly analogous to the famous QED Ward identity relating the electron vertex to the momentum derivative of its propagator.

The Slavnov–Taylor identities provide for rigorous constraints on the subtraction constants when on- or off-shell amplitudes are renormalized. A difficult question, however, is whether the theory can be renormalized at all in accordance with these constraints. We have much more identities than constants that can be adjusted, so the question can be raised whether this procedure does not lead to internal conflicts…

I give an account of my involvement with the chiral anomaly, and with the non-renormalization theorem for the chiral anomaly and the all orders calculation of the trace anomaly, as well as related work by others. I then briefly discuss implications of these results for more recent developments in anomalies in supersymmetric theories.

On the fiftieth anniversary of Yang–Mills theory, I review the contribution to its understanding by my collaborators and me.

Contents: 1. Gauge Theories and Quantum Anomalies; 2. Mathematical Connections; 3. Gauge Field Dynamics other than Yang–Mills; 4. Gauge Formalism for General Relativity Variables; A) Christoffel connection as a gauge potential, B) Gravitational Chern–Simons term from gauge theory Chern–Simons term, C) Coordinate transformations in general relativity and gauge theory, (i) Response to changes in coordinates; (ii) Invariant fields and constants of motion. References.

The fact that the entire electro-weak sector of the interactions between elementary particles could be described by a remarkably simple Yang–Mills theory came as a pleasant surprise. For the first time, a theory had become available that allowed us to compute higher-order corrections to these interactions. Furthermore, we learned how to impose the condition on a theory that such calculations can be done unambiguously, and indeed it was found that only Yang–Mills theories (with specially chosen sets of scalar and Dirac spinor fields added to them) comply. This severe restriction on the structure of preferred theories provided us with a tool to make detailed predictions. It may sound odd that we assume Nature to follow a scheme of predictable and computable interactions, but what this really amounts to is that there is a limited class of ‘simple’ theories, with as few as possible independent dynamical degrees of freedom. We first compare observed phenomena with the predictions of these simple schemes. If there is anything that is not foreseen in our ‘Standard Model’, then the most efficient way to proceed is to add an ‘epicycle’ of more fields and parameters. Such epicycles must be due to deeper lying layers and structures, and apparently, such layers are scarce. It is not obvious why this should be so, but we can argue that, if there are many more epicycles of interactions, there should be a ‘reason’ for that. Apparently, Nature can do without too many epicycles…

The primary interactions of Yang–Mills theory [1] are visibly embodied in hard processes, most directly in jets. The character of jets also reflects the deep structure of effective charge, which is dominated by the influence of intrinsically non-Abelian gauge dynamics. These proven insights into fundamental physics ramify in many directions, and are far from being exhausted. I will discuss three rewarding explorations from my own experience, whose point of departure is the hard Yang–Mills interaction, and whose end is not yet in sight. Given an insight so profound and fruitful as Yang and Mills brought us, it is in order to try to consider its broadest implications, which I attempt at the end.

Before 1974, speculations concerning the existence of pure magnetic charges had been diverse. Many experimental searches had been carried out, and up to today, no single magnetic charge has ever been isolated, apart from some indirect evidence [1] that slowly evaporated when it appeared to be impossible to reproduce it. The theoretical situation was also somewhat confused. Dirac [2] had written a brilliant paper on the subject, showing the Dirac quantization rule. But then Julian Schwinger came with an argument that a factor 2 should be added to this quantization rule — this would be falsified by our later results; presently, we know that if the Dirac quantum is odd, there will be a violation of the spin-statistics addition theorem: fermions can be made out of bosons. Many researchers tried to devise a perturbative scheme to handle monopoles in field theory — in vain, because, if the electric charge unit e is small enough for perturbation theory to make sense, then the magnetic charge unit g = 2πn/e will be far too big. In particular, the use of a separate ‘dual vector potential’ for magnetic charges is doomed to lead to inconsistencies if also electric charges occur. Either g or e is small, but never both…

Magnetic monopoles form an inspiring chapter of theoretical physics, covering a variety of surprising subjects. We review their role in non-Abelian gauge theories. An exposé of exquisite physics derived from a hypothetical particle species, because the fact remains that in spite of ever more tempting arguments from theory, monopoles have never reared their head in experiment. For many relevant particulars, references to the original literature are provided.

QCD was proposed as a theory for the strong interactions long before we had any idea as to how it could be that its fundamental constituents, the quarks, are never seen as physical particles. Massless gluons also do not exist as free particles. How can this be explained? The first indication that this question had to be considered in connection with the topological structure of a gauge theory came when Nielsen and Olesen observed the occurrence of stable magnetic vortex structures [1] in the Abelian Higgs model. Expanding on such ideas, the magnetic monopole solution was found [2]. Other roundabout attempts to understand confinement involve instantons. Today, we have better interpretations of these topological structures, including a general picture of the way they do lead to unbound potentials confining quarks. It is clear that these unbound potentials can be ascribed to a string-like structure of the vortices formed by the QCD field lines. Can string theory be used to analyze QCD? Many researchers think so. The leading expert on this is Sacha Polyakov. In his instructive account he adds how he experienced the course of events in Gauge Theory, emphasizing the fact that quite a few discoveries often ascribed to researchers from the West, actually were made independently by scientists from the Soviet Union…

This is a review of topics which haunted me for the last 40 years, starting with spontaneous symmetry breaking and ending with gauge/string/space-time correspondence. While the first part of this article is mostly historical, the second contains some comments, opinions and conjectures which are new.

Once absolutely confining forces have been accepted as a natural feature of gauge theories, more systematic approaches were searched for in order to be able to do more accurate calculations. We wish to compute the masses of the hadronic states in QCD, their decay properties as well as other details of the confining forces. It is understood that the perturbation expansion becomes reasonably accurate at small distance scales. This motivates some investigators to subject Yang–Mills systems confined to a box with periodic boundary conditions, to a detailed study…

We review in this contribution to celebrate fifty years of Yang–Mills theory the intricate non-perturbative aspects of gauge fixing and its relation to the topologically non-trivial gauge invariant configuration space.

All of our quantum field theories have been designed to be valid at some particular distance and time scale, and to apply to particles in a mass-energy scale that corresponds to that. If we want to study their effects on much larger distance scales, we have to solve the field equations, include the quantum corrections, and hope that complications due to infra-red divergences, or more general forms of complexity, like quark confinement, can be handled technically…

The Yang–Mills theory lies at the heart of our understanding of elementary particle interactions. For the strong nuclear forces, we must understand this theory in the strong coupling regime. The primary technique for this is the lattice. While basically an ultraviolet regulator, the lattice avoids the use of a perturbative expansion. I discuss some of the historical circumstances that drove us to this approach, which has had immense success, convincingly demonstrating quark confinement and obtaining crucial properties of the strong interactions from first principles.

As noted by Creutz, there is a problem with Yang–Mills theory in the lattice. The fact that it breaks Lorentz-invariance is something one can live with, since Lorentz symmetry is expected to be restored automatically in the continuum limit — there exist no low-dimensional effective interactions that break Lorentz-invariance so that no special measures need to be taken to restore it in the continuum limit. However, the situation is quite different in the case of chiral symmetry. Just like Pauli–Villars regulators and the dimensional regulator, also the lattice breaks chiral symmetry. The symmetry is not automatically restored in the continuum limit, since effective mass terms will be generated that survive. The reason why all known regulators have this ‘disease’ in common is the anomaly: even in the continuum theory chiral symmetry is broken down…

As a non-perturbative and gauge invariant regularization the lattice provides a tool for deeper understanding of the celebrated Yang–Mills theory, QCD and chiral gauge theories. For illustration, I discuss some analytic developments on the lattice related to chiral symmetry, chiral fermions and improvement programs. Chiral symmetry on the lattice has an amazing history, and it might influence our perception of a symmetry beyond this example.

Glancing through many of the previous chapters, one might be tempted to overlook an important fact: that Physics is an experimental science. Indeed, Quantum Field Theory in general, and Yang–Mills Theory in particular, would not have made any imprint in the history of subatomic particle physics if it hadn't been for the numerous meticulous experimental tests and searches. Most of our knowledge concerning the sub-atomic world comes from large particle accelerators and ingeniously designed detectors. To bring order to these enormous piles of observed data is the task of the theoretical physicists. It is here that Yang–Mills theory is claiming numerous successes. As Alvaro De Rújula recalls, experimentalists see it as their task to disprove all theories and all theoreticians, or at least discover things that were not foreseen. Here is his account…

On the occasion of the celebration of the first half-century of Yang–Mills theories, I am contributing a personal recollection of how the subject, in its early times, confronted physical reality, that is, its “phenomenology”. There is nothing original in this work, except, perhaps, my own points of view. But I hope that the older practitioners of the field will find here grounds form nostalgia, or good reasons to disagree with me. Younger addicts may learn that history does not resemble at all what is reflected in current textbooks: it was orders of magnitude more fascinating.

Elementary Particle Physics did not stop at the ‘Standard Model’. As explained in Chapter 15, it can only be valid at a specific range of distance scale, or at energy scales below some limit. Presently, we take that limit somewhere near 1 TeV. Beyond that scale, we do not believe the present Model can be relied upon. Already in the early 1970's, new symmetry principles were investigated. A theorem was derived stating that internal symmetries, such as the local gauge symmetry of Yang–Mills systems, cannot be combined in a non-trivial manner with space-time symmetries. But this was before the introduction of anti-commutators as possible replacements for commutators in a symmetry algebra. They are the exception: supersymmetry is the one counterexample disproving the theorem…

We give a simple introduction to ordinary and conformal supergravity, and write their actions as squares of curvatures.

When Superstring Theory entered the arena of sub-atomic particle physics, with its first new milestone in 1984, its enthusiastic supporters proclaimed that ‘this would be the physics of the 21st century!’. That statement, of course, was in error by an amount of 16 years. However, today's physics does still have some of its roots in superstring theory and related topological constructions. The undisputed leader in the field is Edward Witten. He has introduced an impressive amount of mathematics into new theories of physics, and vice versa. Some of those theories are highly speculative and difficult to value; they may or may not survive this century. But, results from algebraic topology and other well-established corners of pure mathematics, are finding their way in physics, and they are likely to stay…

We review some of the unexpected simplifications of perturbative Yang–Mills theory, and motivate the idea of explaining them by using a string theory in which the target space is twistor space. Perturbative Yang–Mills scattering amplitudes arise from an instanton expansion in this string theory.