shorter distances. At the so- called Planck length, which is about 10−35 m, or some 100 billion billion times smaller than a single proton, not just particles but space and time themselves are expected to join in the quan-tum dance of waves and probability (see Question 93). Quanquan-tum foam is the name given to this roiling broth of spacetime by the American physicist John Wheeler.* And it is at this dimension that strings, if they exist, writhe about to produce what we see as particles.
16. How far can one particle “reach out” to infl uence another one? Different forces have different ranges. Let me start with the weak-est of all forces, gravity. It pulls each of us to Earth but reaches out far beyond the human domain, keeping the Moon in orbit around Earth, keeping Earth in orbit about the Sun, and holding galaxies together as giant rotating wheels. Gravity reaches from one end of the universe to the other. From a quantum perspective it does so because the force car-rier, or “exchange particle” of gravity, the graviton, has zero mass. This enables it to reach out without limit, although with diminishing strength as the distance increases.
Electromagnetism is also mediated by a massless particle, the photon.
And, like gravity, it reaches out without limit, also decreasing in strength as the inverse square of the distance. An important difference in the uni-verse at large is that astronomical bodies have no appreciable net electric charge, so they don’t attract or repel one another electrically, even though they continue to attract one another gravitationally. Although electro-magnetism is intrinsically a much stronger force than gravity, its effects in the large are wiped out by the cancellation of positive and negative charge.
Within an atom, on the other hand, electromagnetism far outcompetes gravity— so much so that gravity can be entirely ignored in the atomic realm. And how does the electrical attraction between a proton and an electron manifest itself within an atom? By the emission and absorption of photons— literally by their creation and annihilation, trillions of
* Wheeler loved to come up with memorable phrases. He is responsible also for the terms Planck length and black hole.
events every second— making the electromagnetic force, like every other force, an exchange force.
There is one other force, besides gravity and electromagnetism, that involves the exchange of massless particles. Those particles are gluons, and that force is the strong, or nuclear, force. Oddly enough, however, the strong force doesn’t reach out to great distance like gravity or electro-magnetism. It is confi ned to a distance of about 1 fm. It is not by chance that this distance is the same as the size of a proton. A proton (or neutron) has that size because gluons won’t let quarks get much farther apart than that. The strong force, as it happens, gets stronger, not weaker, with in-creasing distance. It keeps a very tight rein on quarks, so if the three quarks within a proton started to drift farther apart, they would be pulled back with an even more powerful attractive force.*
*There is a subtlety here. If enough energy is poured into a proton— say in a collision in an accelerator— the quarks can be separated, but only if some of the
Sheldon Glashow (b. 1932), shown here lecturing on the universe at large in the early 1980s. The 2 × 1010 LY on the board refers to the then- estimated 20- light- year radius of the universe.
Glashow and his fellow Nobelist Steven Weinberg, both sons of Jewish immigrants, were members of the same class of 1950 at New York’s Bronx High School of Science and the same class of 1954 at Cornell University.
It is to Glashow that we owe the quirky name of the fourth quark, charm, a name that reappears in his essay collection The Charm of Physics. Photo courtesy of AIP Emilio Segrè Visual Archives, Physics Today Collection.
The fourth and last in the pantheon of forces is the weak force—
weak, as it turns out, relative to the electromagnetic and strong forces, but still much stronger than gravity. Its exchange particles, the Ws and Z, do have mass, a lot of mass. They weigh in at more than eighty times the mass of a proton. The weak force, which is responsible for the radio-active decay of many nuclei (and for the decay of a lone neutron) has a very short range, just a fraction of a fermi.
Physicists are always looking for unifi cations that will simplify the de-scription of nature. One very satisfying unifi cation— achieved by the the-orists Abdus Salam, Steven Weinberg, and Sheldon Glashow in the 1960s— was between the weak and electromagnetic forces. In their “elec-troweak” theory, the crucial difference between the weak and electromag-netic interactions* is in the masses of the exchange particles. The massless photon as an exchange particle reaches out to human- sized dimensions or
energy goes into making new quark- antiquark pairs, so that new composite particles are formed and fl y apart. Individual quarks are not released.
* I will be using forces and interactions more or less synonymously.
Abdus Salam (1926– 1996), shown here at a physics conference in Rochester, New York, at age 29. After earning a doctorate in En gland, Salam returned to his native Pakistan for three years as a professor of mathematics. From his later base at Imperial College in London, where I remember the warmth of his personality and the chalk- stained gown he wore while lecturing, he reached out to promote physics throughout the developing world.
Salam is the only Pakistani to have received a Nobel Prize. Photo courtesy of AIP Emilio Segrè Visual Archives, Marshak Collection.
beyond (think of cloud- to- ground lightning). The massive W and Z ex-change particles* reach out to a distance less than the size of a single pro-ton. In Section IX, where I introduce Feynman diagrams, you will see the similarity of these two interactions— much alike at the level of particle creation, annihilation, and exchange. Yet because they do differ in signifi -cant ways, they are still often described as distinct forces. The weak inter-action is not only much weaker and of far less range than the electromag-netic interaction, it is also more universal. Neutral particles that experience no electromagnetic force still experience a weak force.
17. How fast do particles move? There is one particle that has no choice. The photon, being massless, has to move as fast as it is possible to move: at the speed of light (for which we use the symbol c). Other
parti-* These particles were discovered only after their existence was predicted by Salam, Weinberg, and Glashow.
Steven Weinberg (b. 1933).
Weinberg’s stature among physicists is refl ected in the pop u lar but unfortunately untrue story that when he moved from Harvard to the University of Texas, he asked, outrageously, to be paid as much as the football coach. Weinberg is not only a deep theorist but also a gifted writer, as evidenced in The First Three Minutes and other books.
Photo courtesy of AIP Emilio Segrè Visual Archives, Physics Today Collection.
cles can, in principle, be slowed to a crawl or brought to rest, but in practice they are more likely to be seen scooting around at not much less than the photon’s speed. Cosmic- ray particles arrive from outer space at very close to the speed of light. Electrons ejected from nuclei in radioac-tive beta decay are nearly as fast. In the Stanford Linear Accelerator, electrons are pushed to within 0.02 meters per second (m/s) of the 299,792,458 m/s speed of light. Within atoms, electrons move at from 1 percent to more than 10 percent of the speed of light.
Bulkier entities do move more slowly. At ordinary temperature, mol-ecules of air laze along at an average speed of around 500 m/s. We hu-mans normally move much more slowly than that, although fi ghter pi-lots can slice through air at about that speed. Astronauts in orbit move at about 7,000 m/s, some forty thousand times slower than light.
The mind- stretching effects of relativity theory become evident only for speeds near the speed of light. In everyday life, we and the objects we deal with move much more slowly than light, so we are not normally aware of relativistic effects, just as we are not normally aware of quantum effects. Yet the speeds we regularly encounter don’t differ as much from the speed of light as, say, our everyday distances and times differ from those in the subatomic domain. You probably learned in school that it takes eight minutes for light to reach us from the Sun. So, despite the ap-parent instantaneous illumination that results when you turn on a light, you know that light can take a while to get from one place to another in the astronomical realm. From the nearest star other than the Sun, it takes four years to reach us.
On Earth, light (in an optical fi ber, for example) can get from any place to any other place in less than a tenth of a second— not a duration you are likely to notice in surfi ng the Web. But in the summer of 1969, we Earthlings did directly sense the speed of light (or, what is the same thing, the speed of radio waves). Those of us who listened in on the con-versation between NASA mission controllers on Earth and astronauts on the Moon noticed an appreciable time lag between a controller’s question and an astronaut’s answer. Beyond the normal human reaction time of less than a second, there was an extra two- and- a-half- second delay as the radio waves made their way to the Moon and back.
Is the speed of light really nature’s speed limit? There is strong reason to believe that it is, but nothing in science is absolute. Physicists have dared to speculate about particles called tachyons that move faster than light (and can only move faster than light). They have looked for them, and found none. So far the speed of light is a barrier uncrossed.
18. What are some quantum scales of time? In our everyday world, we think of the blink of an eye (perhaps a tenth of a second) as a very short time. In the subatomic domain it is an eternity. In a tenth of a second an electron can circle around the nucleus of its atom 10 million billion (1016) times. For particle interactions, a useful scale of time is the time it takes light to travel across a proton, a distance of about 1 fm. This time— you can think of it as one tick of the particle’s clock— is about three trillionths of a trillionth of a second (3 × 10−24 second). It is about how long a gluon lasts as it is exchanged between two quarks. Needless to say, that is not the shortest time that physicists have conceived of. Just as there is a “Planck length,” there is also a “Planck time.” This is the time it takes light to travel one Planck length. It is an unimaginably short time, about 10−43 second. It is the time scale where gravity and the quantum intermingle and quantum foam takes over.
Not all times associated with particles are as short as the times men-tioned above. Most particles are unstable (radioactive) and some of them live long enough to move an appreciable distance in the laboratory, or even to make it from high in the atmosphere to Earth’s surface. The unstable particle with the longest mean life* is the neutron, which lives, on average, fi fteen minutes.† Even the muon’s mean life of two micro-seconds (two millionths of a second) is much longer than that of most particles. Lifetimes around 10−10 second are common, but even so short a lifetime allows a particle to move a few centimeters and to leave a track in a detector before it expires.
* At Question 27 I discuss the meaning of mean life and its relation to half- life.
† Stabilized within a nucleus, a neutron can live forever.
19. What is the meaning of E = mc2? In 1905, the same year in which he explained Brownian motion, introduced the theory of relativity, and proposed the photon, Einstein wrote down what has surely become the world’s most famous equation, E = mc2. It states that energy (E) is equiva-lent to mass (m)— thus the equal sign. This means that mass can be converted to energy (as it is in nuclear fi ssion and fusion) and energy can be converted to mass (as it is when particles collide at high energy in an accelerator). It means even more: that mass and energy are really the same thing. Mass is just congealed energy, and energy has inertia (the defi ning feature of mass). Energy is mass. Mass is energy.
But how did the square of the speed of light (c2) get into the act? A chunk of matter need not be moving at all, much less moving at the speed of light, in order to have mass and energy. To explain the appear-ance of c2 in Einstein’s equation, let me take a brief detour to a fabric store. If you buy three yards of a material priced at fi ve dollars per yard, you know that your cost will be fi fteen dollars. In equation form, C = NP:
Your cost C is equal to the number of yards N times the price per yard P.
The cost is proportional to the number of yards. If you change your mind and decide to buy twice as many yards, your cost will be twice as much.
What will not change (for this par tic u lar material) is P, its price per yard.
P is called a proportionality constant. It converts the number of yards N to a cost C. The “essence” of the equation is the proportionality of C to N, written C ~ N. The constant P does the job of converting a number of yards to a number of dollars.
In a similar way, c2 in Einstein’s equation is a proportionality con-stant. The essence of the equation is the proportionality of E to m: E ~ m. The constant c2 does the job of converting a number of kilograms (a mass unit) to a number of joules (an energy unit). And c2 is, in everyday units, a very large number, 9 × 1016 joules per kilogram (kg). A 1- kg rock thrown at a speed of 10 m/s has a kinetic energy of 50 joules, enough to do a lot of damage if it hits you in the head. Yet locked within that rock is an energy— a mass energy— of 9 × 1016 joules, more than a million bil-lion times greater than its kinetic energy. Let’s put that in perspective:
The energy locked in 1 kg of mass is 1,500 times greater than the energy
released in the Hiroshima atomic bomb. Less than 1 gram of mass was converted to energy in that explosion.
Does this mean that mass- to- energy conversion takes place only in the nuclear domain? No, it occurs also when you light a match or put another log on the fi re or rev up your automobile engine. In those cases, however, the loss of mass is so totally minuscule that it can’t be mea sured.
This isn’t exactly the way it happened. Courtesy of ScienceCartoonsPlus .com.
Even now, the conservation of mass (no gains or losses) is a solid princi-ple of chemistry. Yet, at the deepest level, it is incorrect! Every time en-ergy is released (in an “exothermic” reaction), there is some decrease in mass. Every time energy is gained (in an “endothermic” reaction), there is some increase in mass.
Even as you grasp the idea that c2 is a constant of proportionality linking energy and mass, you may still rightly wonder what the speed of light has to do with the mass- energy equivalence. The answer starts from the fact that in relativity theory, the speed of light links space and time. (Note that speed is distance per unit of time.) Einstein brought the formerly quite disparate idea of space (mea sured, say, in meters) and time (mea sured, say, in seconds) into the single concept of four- dimensional spacetime. Had ancient scientists appreciated this unifi ca-tion, perhaps they would have adopted the second as a unit of distance (about 300 million meters) or the meter as a unit of time (about three nanoseconds). But they didn’t, and we are stuck with space and time mea sured in different units. To join them into a single four- dimensional whole, we need to multiply time by the speed of light. Then ct, mea-sured in meters, can join distance x as a full partner. As it has turned out, the other unifi cations in relativity theory— mass with energy, mo-mentum with energy, and magnetic fi eld with electric fi eld— also need constants of proportionality, all of which include the speed of light. As to why, for mass and energy, the constant is c2, not c, a rough- and- ready answer is that it takes the combination (meter)2/(second)2 to convert kilograms to joules.
In our everyday world, the joule is a con ve nient unit. In the quantum world, a more common unit is the much smaller electron volt, or eV, defi ned in the footnote on page 22. 1 eV is equal to 1.6 × 10−19 joule. In a cathode- ray TV tube, electrons strike the screen with an energy of about 1,500 eV (1.5 keV). Electrons in the Stanford Linear Accelerator reach an energy of 50 GeV. Protons in the Fermilab Tevatron reach 1 TeV. The current record- holder accelerator is CERN’s (Centre Européen pour la Recherche Nucléaire) Large Hadron Collider in Geneva, Switzerland, which, as of this writing, has pushed protons to an energy of 3.5 TeV and
is scheduled to push them to 7 TeV (a combined energy of 14 TeV when they collide).* (Even 14 TeV is still very much smaller than 1 joule.)
In the particle world, it is common to dispense with c2 and mea sure masses in energy units. Thus the mass of an electron is 0.511 MeV; of a proton, 938 MeV; and of a top quark, about 172 GeV (more than three hundred thousand times the mass of an electron). There is good reason for this practice. Mass- to- energy and energy- to- mass conversions are com-monplace for particles. Mass and energy are constantly being stirred to-gether. When an unstable particle decays, for example, some of its mass energy goes into the masses of its decay products and the rest into their kinetic energy as they fl y apart. How the initial energy is divided varies greatly from one particle to another. When a neutron decays, 99.9 percent of its mass reappears in the masses of the product particles, whereas when a muon decays, only 0.5 percent of its mass shows up in the masses of the products. Also, when particles crash together in accelerators, there are likely to be big conversions between mass and energy. Often a great deal of the energy poured into the particle being accelerated goes into creating new mass.
For one kind of particle transformation, 100 percent of the mass is converted to energy. When a positron (an antielectron) meets an elec-tron, both can vanish as their mass energy is totally transformed to the kinetic energy of massless photons. In the fi ctional Starship Enter-prise, annihilating antimatter provided the energy to drive the ship.
Unfortunately, there is no realistic prospect that this source of motive power will ever be realized.
20. What is electric charge? If you were asked to describe a par tic u lar
20. What is electric charge? If you were asked to describe a par tic u lar