The Higgs Boson: Searching for the God Particle (5 page)

The first gauge theory with local symmetry was the theory of electric and magnetic fields introd uced in 1868 by James Clerk Maxwell. The foundation of Maxwell's theory is the proposition that an electric charge is surrounded by an electric field stretching to infinity,
and that the movement of an electric charge gives rise to a magnetic field also of infinite extent. Both fields are vector quantities, being defined at each point in space by a magnitude and a direction.

In Maxwell's theory the value of the electric field at any point is determined ultimately by the distributron of charges around the point. It is often convenient,
however, to define a poteptial, or voltage,
that is also determined by the charge distribution: the greater the density of charges in a region, the higher its potential. The electric field between two points is then given by the voltage difference between them.

The character of the symmetry that makes Maxwell's theory a gauge theory can be illustrated by considering an imaginary experiment. Suppose a system of electric charges is set up in a laboratory and the electromagnetic field generated by the charges is measured and its properties are recorded. If the charges are stationary, there can be no magnetic field (since the magnetic field arises from movement of an electric charge); hence the field is purely an electric one. In this experimental situation a global symmetry is readily perceived.
The symmetry transformation consists in raising the entire laboratory to a high voltage, or in other words to a high electric potential. If the measurements are then repeated, no change in the electric field will be observed. The reason is that the field, as Maxwell defined it, is determined only by differences in electric potential,
not by the absolute value of the potential. It is for the same reason that a squirrel can walk without injury on an uninsulated power line.

This property of Maxwell's theory amounts to a symmetry: the electric field is invariant with respect to the addition or subtraction of an arbitrary overall potential. As noted above, however,
the symmetry is a global one, because the result of the experiment remains constant only if the potential is changed everywhere at once. If the potential were raised in one region and not in another,
any experiment that crossed the boundary would be affected by the potential difference, just as a squirrel is affected if it touches both a power line and a grounded conductor. .

A complete theory of electromagnetic fields must embrace not only static arrays of charges but also moving charges.
In order to do that the global symmetry of the theory must be converted into a local symmetry. If the electric field were the only one acting between charged particles, it would not have a local symmetry.
Actually when the charges are in motion (but only then), the electric field is not the only one present: the movement itself gives rise to a second field,
namely the magnetic field. It is the effects of the magnetic field that restore the local symmetry.

Just as the electric field depends ultimately on the distribution of charges but can conveniently be derived from an electric potential, so the magnetic field is generated by the motion of the charges but is more easily described as resulting from a magnetic potential. It is in this system of potential fields that local transformations can be carried out leaving all the original electric and magnetic fields unaltered. The system of dual, interconnected fields has an exact local symmetry even though the electric field alone does not. Any local change in the electric potential can be combined with a compensating change in the magnetic potential in such a way that the electric and magnetic fields are invariant.

Maxwell's theory of electromagnetism is a classical or non-quantum-mechanical one, but a related symmetry can be demonstrated in the quantum theory of electromagnetic interactions.
It is necessary in that theory to describe the electron as a wave or a field, a convention that in quantum mechanics can be adopted for any material particle. It turns out that in the quantum theory of electrons a change in the electric potential entails a change in the phase of the electron wave.

The electron has a spin of one-half unit and so has two spin states (parallel and antiparallel). It follows that the associated field must have two components.
Each of the components must be represented by a complex number, that is, a number that has both a real, or ordinary,
part and an imaginary part, which includes as a factor the square root of
–1. The electron field is a moving packet of waves, which are oscillations in the amplitudes of the real and the imaginary components of the field. It is important to emphasize that this field is not the electric field of the electron but instead is a matter field. It would exist even if the electron had no electric charge.
What the field defines is the probability of finding an electron in a specified spin state at a given point and at a given moment.
The probability is given by the sum of the squares of the real and the imaginary parts of the field.

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.