The Higgs Boson: Searching for the God Particle (6 page)

It was not immediately apparent that quantum electrodynamics could qualify as a physically acceptable theory. One problem arose repeatedly in any attempt to calculate the result of even the simplest electromagnetic interactions,
such as the interaction between two electrons. The likeliest sequence of events in such an encounter is that one electron emits a single virtual photon and the other electron absorbs it. Many more complicated exchanges are also possible, however; indeed, their number is infinite. For example, the electrons could interact by exchanging two photons,
or three, and so on. The total probability of the interaction is determined by the sum of the contributions of all the possible events.

Feynman introduced a systematic procedure for tabulating these contributions by drawing diagrams of the events in one spatial dimension and one time dimension. A notably troublesome class of diagrams are those that include
"loops, " such as the loop in space-time that is formed when a virtual photon is emitted and later reabsorbed by the same electron. As was shown above, the maximum energy of a virtual particle is limited only by the time needed for it to reach its destination. When a virtual photon is emitted and reabsorbed by the same particle, the distance covered and the time required can be reduced to zero, and so the maxim um energy can be infinite. For this reason some diagrams with loops make an infinite contribution to the strength of the interaction.

The infinities encountered in quantum electrodynamics led initially to predictions that have no reasonable interpretation as physical quantities. Every interaction of electrons and photons was assigned an infinite probability. The infinities spoiled even the description of an isolated electron: because the electron can emit and reabsorb virtual particles it has infinite mass and infinite charge.

The cure for this plague of infinities is the proced ure called renormalization.
Roughly speaking, it works by finding one negative infinity for each positive infinity, so that in the sum of all the possible contributions the infinities cancel.
The achievement of Schwinger and of the other physicists who worked on the problem was to show that a finite residue could be obtained by this method.
The finite residue is the theory's prediction.
It is uniquely determined by the requirement that all interaction probabilities come out finite and positive.

The rationale of this procedure can be explained as follows. When a measurement is made on an electron, what is actually measured is not the mass or the charge of the pointlike particle with which the theory begins but the properties of the electron together with its enveloping cloud of virtual particles. Only the net mass and charge, the measurable quantities, are required to be finite at all stages of the calculation. The properties of the pointlike object, which are called the "bare" mass and the "bare" charge,
are not well defined.

Initially it appeared that the bare mass would have to be assigned a value of negative infinity, an absurdity that made many physicists suspicious of the renormalized theory. A more careful analysis, however, has shown that if the bare mass is to have any definite value, it tends to zero. In any case all quantities with implausible values are unobservable,
even in principle. Another objection to the theory is more profound: mathematically quantum electrodynamics is not perfect. Because of the methods that must be used for making predictions in the theory the predictions are limited to a finite accuracy of some hundreds of decimal places.

Clearly the logic and the internal consistency of the renormalization method leave something to be desired.
Perhaps the best defense of the theory is simply that it works very well. It has yielded results that are in agreement with experiments to an accuracy of about one part in a billion, which makes quantum electrodynamics the most accurate physical theory ever devised.
It is the model for theories of the other fundamental forces and the standard by which such theories are judged.

At the time quantum electrodynamics was completed another theory based on a local gauge symmetry had already been known for some 30 years. It is Einstein's general theory of relativity.
The symmetry in question pertains not to a field distributed through space and time but to the structure of space-time itself.

Every point in space-time can be labeled by four numbers, which give its position in the three spatial dimensions and its sequence in the one time dimension.
These numbers are the coordinates of the event, and the procedure for assigning such numbers to each point in space-time is a coordinate system. On the earth, for example, the three spatial coordinates are commonly given as longitude,
latitude and altitude; the time coordinate can be given in hours past noon. The origin in this coordinate system,
the point where all four coordinates have values of zero, lies at noon at sea level where the prime meridian crosses the Equator.

The choice of such a coordinate system is clearly a matter of convention.
Ships at sea could navigate just as successfully if the origin of the coordinate system were shifted to Utrecht in the Netherlands. Every point on the earth and every event in its history would have to be assigned new coordinates, but calculations made with those coordinates would invariably give the same results as calculations made in the old system.
In particular any calculation of the distance between two points would give the same answer.

The freedom to move the origin of a coordinate system constitutes a symmetry of nature. Actually there are three related symmetries: all the laws of nature remain invariant when the coordinate system is transformed by translation,
by rotation or by mirror reflection.
It is vital to note, however, that the symmetries are only global ones. Each symmetry transformation can be defined as a formula for finding the new coordinates of a point from the old coordinates.
Those formulas must be applied simultaneously in the same way to all the points.

The general theory of relativity stems
from the fundamental observation that the structure of space-time is not necessarily consistent with a coordinate system made up entirely of straight lines meeting at right angles; instead a curvilinear coordinate system may be needed. The lines of longitude and latitude employed on the earth constitute such a system, since they follow the curvature of the earth.

In such a system a local coordinate transformation can readily be imagined. Suppose height is defined as vertical distance from the ground rather than from mean sea level. The digging of a pit would then alter the coordinate system, but only at those points directly over the pit. The digging itself represents the local coordinate transformation. It would appear that the laws of physics (or the rules of navigation) do not remain invariant after such a transformation, and in a universe without gravitational forces that would be the case. An airplane set to fly at a constant height would dip suddenly when it flew over the excavation, and the accelerations needed to follow the new profile of the terrain could readily be detected.

As in electrodynamics, local symmetry ean be restored only by adding a new field to the theory; in general relativity the field is of course that of gravitation.
The presence of this field offers an alternative explanation of the accelerations detected in the airplane: they could result not from a local change in the coordinate grid but from an anomaly in the gravitational field. The source of the anomaly is of no concern: it could be a concentration of mass in the earth or a distant object in space. The point is that any local transformation of the coordinate system could be reproduced by an appropriate set of gravitational fields.
The pilot of the airplane could not distinguish one effect from the other.

Both Maxwell's theory of electromagnetism and Einstein's theory of gravitation owe much of their beauty to a local gauge symmetry; their success has long been an inspiration to theoretical physicists. Until recently theoretical acco unts of the other two forces in nature have been less satisfactory. A theory of the weak force formulated in the 1930's by Enrico Fermi accounted for some basic features of the weak interaction,
but the theory lacked local symmetry.
The strong interactions seemed to be a jungle of mysterious fields and resonating particles. It is now clear why it took so long to make sense of these forces: the necessary local gauge theories were not understood.

The first step was taken in 1954 in a theory devised by C. N. Yang and Robert L. Mills, who were then at the Brookhaven National Laboratory. A similar idea was proposed independently at about the same time by R. Shaw of the University of Cambridge. Inspired by the success of the other gauge theories,
these theories begin with an established global symmetry and ask what the consequences would be if it were made a local symmetry.

The symmetry at issue in the Yang-Mills theory is isotopic-spin symmetry,
the rule stating that the strong interactions of matter remain invariant (or nearly so) when the identities of protons and neutrons are interchanged. In the global symmetry any rotation of the internal arrows that indicate the isotopic-spin state must be made simultaneously everywhere. Postulating a local symmetry allows the orientation of the arrows to vary independently from place to place and from moment to moment. Rotations of the arrows can depend on any arbitrary function of position and time.
This freedom to choose different conventions for the identity of a nuclear particle in different places constitutes a local gauge symmetry.

As in other instances where a global symmetry is converted into a local one,
the invariance can be maintained only if something more is added to the theory.
Because the Yang-Mills theory is more complicated than earlier gauge theories it turns out that quite a lot more must be added. When isotopic-spin rotations are made arbitrarily from place to place, the laws of physics remain invariant only if six new fields are introduced. They are all vector fields, and they all have infinite range.

The Yang-Mills fields are constructed on the model of electromagnetism, and indeed two of them can be identified with the ordinary electric and magnetic fields. In other words, they describe the field of the photon. The remaining Yang-Mills fields can also be taken in pairs and interpreted as electric and magnetic fields, but the photons they describe differ in a crucial respect from the known properties of the photon: they are still massless spin-one particles, but they carry an electric charge. One photon is negative and one is positive.

The imposition of an electric charge on a photon has remarkable consequences.
The photon is defined as the field quantum that conveys electromagnetic forces from one charged particle to another. If the photon itself has a charge, there can be direct electromagnetic interactions among the photons.
To cite just one example, two photons with opposite charges might bind together to form an "atom" of light. The familiar neutral photon never interacts with itself in this way.

The surprising effects of charged photons become most apparent when a local symmetry transformation is applied more than once to the same particle. In quantum electrodynamics, as was pointed out above, the symmetry operation is a local change in the phase of the electron field, each such phase shift being accompanied by an interaction with the electromagnetic field. It is easy to imagine an electron undergoing two phase shifts in succession, say by emitting a
photon and later absorbing one. Intuition suggests that if the sequence of the phase shifts were reversed, so that first a photon was absorbed and later one was emitted, the end result would be the same. This is indeed the case. An unlimited series of phase shifts can be made,
and the final result will be simply the algebraic sum of all the shifts no matter what their sequence.

In the Yang-Mills theory, where the symmetry operation is a local rotation of the isotopic-spin arrow, the result of m ultiple transformations can be quite different. Suppose a hadron is subjected to a gauge transformation, A, followed soon after by a second transformation,
B; at the end of this sequence the isotopic-spin arrow is found in the orientation that corresponds to a proton. N ow suppose the same transformations were applied to the same hadron but in the reverse sequence: B followed by A. In general the final state will not be the same;
the particle may be a neutron instead of a proton. The net effect of the two transformations depends explicitly on the sequence in which they are applied.

Because of this distinction quantum electrodynamics is called an Abelian theory and the Yang-Mills theory is called a non-Abelian one. The terms are borrowed from the mathematical theory of groups and honor Niels Henrik Abel, a Norwegian mathematician who lived in the early years of the 19th century.
Abelian groups are made up of transformations that, when they are applied one after another, have the commutative property; non-Abelian groups are not commutative.