In the absence of electromagnetic fields the frequency of the oscillations in the electron field is proportional to the energy of the electron, and the wavelength of the oscillations is proportional to the momentum. In order to define the oscillations completely one additional quantity must be known: the phase. The phase measures the displacement of the wave from some arbitrary reference point and is usually expressed as an angle. If at some point the real part of the oscillation, say, has its maximum positive amplitude, the phase at that point might be assigned the value zero degrees. Where the real part next falls to zero the phase is 90 degrees and where it reaches its negative maximum the phase is 180 degrees. In general the imaginary part of the amplitude is 90 degrees out of phase with the real part, so that whenever one part has a maximal value the other part is zero.
It is apparent that the only way to determine the phase of an electron field is to disentangle the contributions of the real and the imaginary parts of the amplitude. That turns out to be impossible, even in principle. The sum of the squares of the real and the imaginary parts can be known, but there is no way of telling at any given point or at any moment how m uch of the total derives from the real part and how much from the imaginary part. Indeed, an exact symmetry of the theory implies that the two contributions are indistinguishable. Differences in the phase of the field at two points or at two moments can be measured, but not the absolute phase.
The finding that the phase of an electron wave is inaccessible to measurement has a corollary: the phase cannot have an influence on the outcome of any possible experiment. If it did, that experiment could be used to determine the phase. Hence the electron field exhibits a symmetry with respect to arbitrary changes of phase. Any phase angle can be added to or subtracted from the electron field and the res ults of all experiments will remain invariant.
This principle can be made clearer by considering an example: the two-slit diffraction experiment with electrons, which is the best-known demonstration of the wavelike nature of matter. In the experiment a beam of electrons passes through two narrow slits in a screen and the number of electrons reaching a second screen is counted. The distribution of electrons across the surface of the second screen forms a diffraction pattern of alternating peaks and valleys.
The quantum-mechanical interpretation of this experiment is that the electron wave splits into two segments on str iking the first screen and the two diffracted waves then interfere w ith each other. Where the waves are in phase the interference is constructive and many electrons are counted at the second screen; where the waves are out of phase destructive interference reduces the count. Clearly it is only the difference in phase that determines the pattern formed. If the phases of both waves were shifted by the same amount, the phase difference at each point would be unaffected and the same pattern of constructive and destructive interference would be observed. It is symmetries of this kind, where the phase of a quantum field can be adjusted at will, that are called gauge symmetries. Although the absolute value of the phase is irrelevant to the outcome of experiments, in constructing a theory of electrons it is still necessary to specify the phase. The choice of a particular value is called a gauge convention.
Gauge symmetry is not a very descriptive term for such an invariance, but the term has a long history and cannot now be dislodged. It was introduced in about 1920 by Hermann Weyl, who was then attempting to formulate a theory that would com bine electromagnetism and the general theory of relativity. Weyl was led to propose a theory that remained invariant with respect to arbitrary dilatations or contractions of space. In the theory a separate standard of length and time had to be adopted at every point in space-time. He compared the choice of a scale convention to a choice of gage blocks, the polished steel blocks employed by machinists as a standard of length. The theory was nearly correct, the necessary emendation being to replace "length scales" by "phase angles." Writing in German, Weyl had referred to "Eich Invarianz," which was initially translated as "calibration invariance," but the alternative translation "gauge" has since become standard.
The symmetry of the electron matter field described above is a global symmetry: the phase of the field must be shifted in the same way everywhere at once. It can easily be demonstrated that a theory of electron fields alone, with no other forms of matter or radiation, is not invariant with respect to a corresponding local gauge transformation. Consider again the two-slit diffraction experiment with electrons. An initial experiment is carried out as before and the electron-diffraction pattern is recorded. Then the experiment is repeated, but one slit is fitted with the electron-optical equivalent of a half-wave plate, a device that shifts the phase of a wave by 180 degrees. When the waves emanating from the two slits now interfere, the phase difference between them will be altered by 180 degrees. As a result wherever the interference was constructive in the first experiment it will now be destructive, and vice versa. The observed diffraction pattern will not be unchanged; on the contrary, the positions of all the peaks and depressions will be interchanged.
Suppose one wanted to make the theory consistent with a local gauge symmetry. Perhaps it co uld be fixed in some way; in particular, perhaps another field could be added that would compensate for the changes in electron phase. The new field would of course have to do more than mend the defects in this one experiment. It would have to preserve the invariance of all observable quantities when the phase of the electron field was altered in any way from place to place and from moment to moment. Mathematically the phase shift must be allowed to vary as an arbitrary function of position and time.
Although it may seem improbable, a field can be constructed that meets these specifications. It turns out that the required field is a vector one, corresponding to a field quantum with a spin of one unit. Moreover, the field must have infinite range, since there is no limit to the distance over which the phases of the electron fields might have to be reconciled. The need for infinite range implies that the field quantum must be massless. These are the properties of a field that is already familiar: the electromagnetic field, whose quantum is the photon.
How does the electromagnetic field ensure the gauge invariance of the electron field? It should be remembered that the effect of the electromagnetic field is to transmit forces between charged particles. These forces can alter the state of motion of the particles; what is most important in this context, they can alter the phase. When an electron absorbs or emits a photon, the phase of the electron field is shifted. It was shown above that the electromagnetic field itself exhibits an exact local symmetry; by describing the two fields together the local symmetry can be extended to both of them.
The connection between the two fields lies in the interaction of the electron's charge with the electromagnetic field. Because of this interaction the propagation of an electron matter wave in an electric field can be described properly only if the electric potential is specified. Similarly, to describe an electron in a magnetic field the magnetic vector potential must be specified. Once these two potentials are assigned definite values the phase of the electron wave is fixed everywhere. The local symmetry of electromagnetism, however, allows the electric potential to be given any arbitrary value, which can be chosen independently at every point and at every moment. For this reason the phase of the electron matter field can also take on any value at any point, but the phase will always be consistent with the convention adopted for the electric and the magnetic potentials.
What this means in the two-slit diffraction experiment is that the effects of an arbitrary shift in the phase of the electron wave can be mimicked by applying an electromagnetic field. For example, the change in the observed interference pattern caused by interposing a half-wave plate in front of one slit could be ca used instead by placing the slits between the poles of a magnet. From the resulting pattern it would be impossible to tell which procedure had been followed. Since the gauge conventions for the electric and the magnetic potentials can be chosen locally, so can the phase of the
electron field.
The theory that results from combining electron matter fields with electromagnetic fields is called quantum electrodynamics. Formulating the theory and proving its consistency was a labor of some 20 years, begun in the 1920's by P. A. M. Dirac and essentially completed in about 1948 by Richard P . Feynman, Julian Schwinger, Sin-itiro Tomonaga and others.
The symmetry properties of quantum electrodynamics are unquestionably appealing, but the theory can be invested with physical significance only if it agrees with the results of experiments. Indeed, before sensible experimental predictions can even be made the theory must pass certain tests of internal consistency. For example, quantum-mechanical theories predict the probabilities of events: the probabilities must not be negative, and all the probabilities taken together must add up to 1. In addition energies must be assigned positive values but should not be infinite.
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 ea
rth 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.
The Higgs Boson: Searching for the God Particle Page 5
