The Higgs Boson: Searching for the God Particle Read online

  --Originally published: Scientific American 252(4), 84-95. (April 1985)

  Gauge Theories of the Forces between Elementary Particles

  by Gerard 't Hooft

  An understanding of how the world is put together requires a theory of how the elementary particles of matter interact with one another. Equivalently, it requires a theory of the basic forces of nature. Four such forces have been identified, and until recently a different kind of theory was needed for each of them. Two of the forces, gravitation and electromagnetism, have an unlimited range; largely for this reason they are familiar to everyone. They can be felt directly as agencies that push or pull. The remaining forces, which are called simply the weak force and the strong force, cannot be perceived directly because their influence extends only over a short range, no larger than the radius of an atomic nucleus. The strong force binds together the protons and the neutrons in the nucleus, and in another context it binds together the particles called quarks that are thought to be the constituents of protons and neutrons. The weak force is mainly responsible for the decay of certain particles.

  A long-standing ambition of physicists is to construct a single master theory that would incorporate all the known forces. One imagines that such a theory would reveal some deep connection between the various forces while accounting for their apparent diversity. Such a unification has not yet been attained, but in recent years some progress may have been made. The weak force and electromagnetism can now be understood in the context of a single theory. Although the two forces remain distinct, in the theory they become mathematically intertwined. What may ultimately prove more important, all four forces are now described by means of theories that have the same general form. Thus if physicists have yet to find a single key that fits all the known locks, at least all the needed keys can be cut from the same blank. The theories in this single favored class are formally designated non-Abelian gauge theories with local symmetry. What is meant by this forbidding label is the main topic of this article. For now, it will suffice to note that the theories relate the properties of the forces to symmetries of nature.

  Symmetries and apparent symmetries in the laws of nature have played a part in the construction of physical theories since the time of Galileo and Newton. The most familiar symmetries are spatial or geometric ones. In a snowflake, for example, the presence of a symmetrical pattern can be detected at a glance. The symmetry can be defined as an invariance in the pattern that is observed when some transformation is applied to it. In the case of the snowflake the transformation is a rotation by 60 degrees, or one-sixth of a circle. If the initial position is noted and the snowflake is then turned by 60 degrees (or by any integer multiple of 60 degrees), no change will be perceived. The snowflake is invariant with respect to 60-degree rotations. According to the same principle, a square is invariant with respect to 90-degree rotations and a circle is said to have continuous symmetry because rotation by any angle leaves it unchanged.

  Although the concept of symmetry had its origin in geometry, it is general enough to embrace invariance with respect to transformations of other kinds. An example of a nongeometric symmetry is the charge symmetry of electromagnetism. Suppose a number of electrically charged particles have been set out in some definite configuration and all the forces acting between pairs of particles have been measured. If the polarity of all the charges is then reversed, the forces remain unchanged.

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  SYMMETRIES OF NATURE determines the properties of forces in guage theories. The famiiar symmetry of a snowflake can be characterized by noting that the pattern is unchanged when it is rotated 60 degrees; the snowflake is said to be invariant with respect to such rotations. In physics nongeometric symmetries are introduced. Charge symmetry, for example, is the invariance of the forces acting among a set of charged particles when the polarities of all the charges are reveresed. Isotopic-spin symmetry is based on the observation that little would be changed in the strong interactions of matter if the identities of all protons and neutrons were interchanged. Hence proton and neutron become merely the alternative states of a single particle, the nucleon, and transitions between the states can be made (or imagined) by adjusting the orientation of an indicator in an internal space. It is symmetries of this kind, where the transformation is an internal rotation or a phase shift, that are referred to as guage symmetries.

  Illustration by Allen Beechel

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  Another symmetry of the nongeometric kind concerns isotopic spin, a property of protons and neutrons and of the many related particles called hadrons, which are the only particles responsive to the strong force. The basis of the symmetry lies in the observation that the proton and the neutron are remarkably similar particles. They differ in mass by only about a tenth of a percent, and except for their electric charge they are identical in all other properties. It therefore seems that all protons and neutrons could be interchanged and the strong interactions would hardly be altered. If the electromagnetic forces (which depend on electric charge) could somehow be turned off, the isotopic-spin symmetry would be exact; in reality it is only approximate.

  Although the proton and the neutron seem to be distinct particles and it is hard to imagine a state of matter intermediate between them, it turns out that symmetry with respect to isotopic spin is a continuous symmetry, like the symmetry of a sphere rather than like that of a snowflake. I shall give a simplified explanation of why that is so. Imagine that inside each particle are a pair of crossed arrows, one representing the proton component of the particle and the other representing the neutron component. If the proton arrow is pointing up (it makes no difference what direction is defined as up), the particle is a proton; if the neutron arrow is up, the particle is a neutron. Intermediate positions correspond to quantum-mechanical superpositions of the two states, and the particle then looks sometimes like a proton and sometimes like a neutron. The symmetry transformation associated with isotopic spin rotates the internal indicators of all protons and ne utrons everywhere in the universe by the same amount and at the same time. If the rotation is by exactly 90 degrees, every proton becomes a neutron and every neutron becomes a proton. Symmetry with respect to isotopic spin, to the extent it is exact, states that no effects of this transformation can be detected.

  All the symmetries I have discussed so far can be characterized as global symmetries; in this context the word global means "happening everywhere at once." In the description of isotopic-spin symmetry this constraint was made explicit: the internal rotation that transforms protons into neutrons and neutrons into protons is to be carried out everywhere in the universe at the same time. In addition to global symmetries, which are almost always present in a physical theory, it is possible to have a "local" symmetry, in which the convention can be decided independently at every point in space and at every moment in time. Although "local" may suggest something of more modest scope than a global symmetry, in fact the requirement of local symmetry places a far more stringent constraint on the construction of a theory. A global symmetry states that some law of physics remains invariant when the same transformation is applied everywhere at once. For a local symmetry to be observed the law of physics must retain its validity even when a different transformation takes place at each point in space and time.

  Gauge theories can be constructed with either a global or a local symmetry (or both), but it is the theories with local symmetry that hold the greatest interest today. In order to make a theory invariant with respect to a local transformation something new must be added: a force. Before showing how this comes about, however, it will be necessary to discuss in somewhat greater detail how forces are described in modern theories of elementary-particle interactions.

  The basic ingredients of particle theory today include not only particles and forces but also fields. A field is simply a quantity defined at every point throughout some region of space and time. For example, the quantity might be temperature and the region might be the surface of a frying
pan. The field then consists of temperature values for every point on the surface.

  Temperature is called a scalar quantity, because it can be represented by position along a line, or scale. The corresponding temperature field is a scalar field, in which each point has associated with it a single number, or magnitude. There are other kinds of field as well, the most important for present p urposes being the vector field, where at each point a vector, or arrow, is drawn. A vector has both a magnitude, which is represented by the length of the arrow, and a direction, which in three-dimensional space can be specified by two angles; hence three numbers are needed in order to specify the value of the vector. An example of a vector field is the velocity field of a fluid; at each point throughout the volume of the fluid an arrow can be drawn to show the speed and direction of flow.

  In the physics of electrically charged objects a field is a convenient device for expressing how the force of electromagnetism is conveyed from one place to another. All charged particles are supposed to emanate an electromagnetic field; each particle then interacts with the sum of all the fields rather than directly with the other particles.

  In quantum mechanics the particles themselves can be represented as fields. An electron, for example, can be considered a packet of waves with some finite extension in space. Conversely, it is often convenient to represent a quantum-mechanical field as if it were a particle. The interaction of two particles through their interpenetrating fields can then be summed up by saying the two particles exchange a third particle, which is called the quantum of the field. For example, when two electrons, each surrounded by an electromagnetic field, approach each other and bounce apart, they are said to exchange a photon, the quantum of the electromagnetic field.

  The exchanged quantum has only an ephemeral existence. Once it has been emitted it must be reabsorbed, either by the same particle or by another one, within a finite period. It cannot keep going indefinitely, and it cannot be detected in an experiment. Entities of this kind are called virtual particles. The larger their energy, the briefer their existence. In effect a virtual particle borrows or embezzles a quantity of energy, but it must repay the debt before the shortage can be noticed.

  The range of an interaction is related to the mass of the exchanged quantum. If the field quantum has a large mass, more energy must be borrowed in order to support its existence, and the debt must be repaid sooner lest the discrepancy be discovered. The distance the particle can travel before it must be reabsorbed is thereby reduced and so the corresponding force has a short range. In the special case where the exchanged quantum is massless the range is infinite.

  The number of components in a field corresponds to the number of quantum-mechanical states of the field quantum. The number of possible states is in turn related to the intrinsic spin angular momentum of the particle. The spin angular momentum can take on only discrete values; when the magnitude of the spin is measured in fundamental units, it is always an integer or a half integer. Moreover, it is not only the magnitude of the spin that is quantized but also its direction or orientation. (To be more precise, the spin can be defined by a vector parallel to the spin axis, and the projections, or components, of this vector along any direction in space must have values that are integers or half integers.) The number of possible orientations, or spin states, is equal to twice the magnitude of the spin, plus one. Thus a particle with a spin of one-half, such as the e lectron, has two spin states : the spin can point parallel to the particle's direction of motion or antiparallel to it. A spin-one particle has three orientations, namely parallel, antiparallel and transverse. A spin-zero particle has no spin axis; since all orientations are equivalent, it is said to have just one spin state.

  A scalar field, which has just one component (a magnitude), must be represented by a field q uantum that also has one component, or in other words by a spin-zero particle. Such particles are therefore called scalar particles. Similarly, a three-component vector field req uires a spin-one field quantum with three spin states: a vector particle. The electromagnetic field is a vector field, and the photon, in conformity with these specifications, has a spin of one unit. The gravitational field is a more complicated structure called a tensor and has 10 components; not all of them are independent, however, and the quantum of the field, the graviton, has a spin of two units, which ordinarily corresponds to five spin states.

  In the cases of electromagnetism and gravitation one further complication must be taken into account. Since the photon and the graviton are massless, they must always move with the speed of light. Because of their velocity they have a property not shared by particles with a finite mass: the transverse spin states do not exist. Although in some formal sense the photon has three spin states and the graviton has five, in practice only two of the spin states can be detected.

  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 f
ields 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.