by Colin Pask
Pauli goes on to say that his solution may sound improbable, “but only who dares wins,” and the alternative advice he got was, “the best thing to do is not to talk about it, like the new taxes.” He ends his letter by explaining why he cannot come to the conference: “Unfortunately I cannot come personally to Tübingen, because I am necessary here for a ball that will take place in Zürich the night from 6 to 7 December.” One of the greatest announcements in physics plays second fiddle to a ball!
I must immediately clarify one point: Pauli's “neutron” is not the same as the one discovered by Chadwick. Pauli said his neutron has small mass and is difficult to detect. It became known as the neutrino—little neutron—a name coined by the Italian physicist Enrico Fermi. Thus the suggestion is that the β-decay process takes the form:
neutron → proton + electron + neutrino.
Now the energy and momentum are spread over the three particles; there are extra terms to be included in equations (11.2), and solving them leads to a spread of possible electron energies. Thus the calculation of β-decay spectra led to the remarkable prediction of the existence of a totally new kind of fundamental particle. And so I add calculation 42, why we must have a neutrino to my list.
The neutrino has very small mass (possibly even zero in some schemes) and has only a minute chance of interacting with other particles. (The neutrino story continues in section 11.7.) Poor Pauli was distressed by this, as the astronomer Fred Hoyle amusingly recounts:
The astronomer Walter Baade told me that, when he was dining with Pauli one day, Pauli exclaimed, “Today I have done the worst thing for a theoretical physicist. I have invented something that can never be detected experimentally.” Baade immediately offered to bet a crate of champagne that the elusive neutrino would one day prove amenable to experimental discovery. Pauli accepted, unwisely failing to specify any time limit, which made it impossible for him to win the bet. Baade collected his crate of champagne (as I can testify, having helped Baade to consume a bottle of it) when, just over twenty years later, in 1953, Cowan and Reines did indeed succeed in detecting Pauli's particle.8
Clyde L. Cowan and Frederick Reines were awarded the Nobel Prize for the success of the experiments in what they called their “Project Poltergeist.” Calculation 42, why we must have a neutrino, not only led to the discovery of one of the weirdest of all particles, it also led to a poem by John Updike: The poem is called “Cosmic Gall” (see Updike's Collected Poems), and it celebrates the fact that the weird neutrino can pass right through the earth (and stallions and lovers, Updike notes) without any deviations at all.
11.2.5 Revelation Two: A Left-Handed Universe
The full quantum theory for β-decay was given by Enrico Fermi (1901–1954) and is now known as the theory of weak interactions (a thousand times weaker than electromagnetic interactions). This theory eventually led to a very big surprise: the supposed universal left-right symmetry of the universe breaks down in weak interaction processes like β-decay (technically called the nonconservation of parity). Look at a right hand in a mirror and you see a left hand. It was believed that everything in the physical world had its mirror equivalent. However, in 1956, a paper by T. D. Lee and C. N. Yang gave a theory for weak interactions in which that left-right symmetry is broken, and, in 1958, their predictions were verified experimentally by C.-S. Wu. Pauli had offered to bet a sum of money that parity would be conserved, but luckily for him, this time nobody took up the challenge! Discussion of left-right, or inversion, symmetry is now prevalent in many fields, and the aptly named book Right Hand, Left Hand by McManus is an entertaining and broad introduction to the subject.
11.3 COSMIC RAYS AND A TEST FOR EINSTEIN
I mentioned earlier that Einstein's special theory of relativity is an essential background for the topics in this chapter, especially the relationship between energy and mass. Einstein's theory also makes predictions about time and reference frames that appear quite strange on a first encounter. The observation of particles originating in cosmic rays allowed a test of the theory.
We are used to traveling in trains and airplanes and finding life goes on just the same as when we were at rest on the earth. Galileo used the example of life in the closed cabin of a moving ship. These ideas were built into Newton's mechanics and in his Principia, where he writes:
The motion of bodies included in a given space are the same amongst themselves, whether that space is at rest, or moves uniformly forward in a right line without any circular motion.9
Today, we speak about inertial frames of reference. Newton gave the example of a sailor walking along the deck of a uniformly moving ship to explain how speeds are changed depending on the objects to which we refer them (the ship or the earth in the sailor example). Newton's example tells us how to switch between frames of reference, and this established the basis for classical mechanics ever after. Things changed after about two hundred years when Einstein came on the scene.
11.3.1 Special Relativity
Einstein also stated that physics did not depend on the chosen inertial frame of reference, and he went further by including electromagnetic effects in that. Maxwell's theory of electromagnetism (see section 9.4) naturally introduces the speed of light, and thus (following Lambourne's presentation—see bibliography) we have:
The first postulate of special relativity:
The laws of physics can be written in the same form in all inertial frames.
The second postulate of special relativity:
The speed of light in a vacuum has the constant value, c = 3 × 108m sec–1, in all inertial frames.
The second postulate represents a major step in physics, and, under its influence, Newton's rules for moving between inertial frames must be modified. This makes the change from classical mechanics to relativistic mechanics.
Suppose we have a given inertial frame of reference S and a second inertial frame S' which is moving with speed V in the x direction as shown in figure 11.1. (For simplicity, I keep to the xy-plane—Lambourne gives the full details.) Relativity theory tells us that the change from the coordinates in frame S to those in frame S' is made using a set of formulas known as the Lorentz transformations, and they include the way in which times are specified:
Letting c become infinitely large, so that γ(V) = 1, gives transformation formulas that Newton would agree with. The inverse transformations tell us how to go from frame S to frame S', that is from x' and t' to x and t.
Figure 11.4. A frame of reference S' is in uniform motion with speed V in the x direction relative to the inertial frame S. Figure created by Annabelle Boag.
Using the Lorentz transformations makes many changes to Newtonian science which are now well verified. One particular change is perplexing and revolutionary for many people; it concerns time intervals.
11.3.2 Calculating a Time Dilation
Suppose two events occurring at the same place in frame S' have a time interval Δτ, then for someone in frame S the time interval ΔT is calculated to be
The transformation factor γ(V) defined in equation (11.4) is always bigger than 1 (becoming exactly 1 only for V = 0 or when the speed of light c is taken as infinite). The observed time interval ΔT is bigger than the interval Δτ.
If we turn the above statement around and say Δτ is smaller than ΔT, we can use the result to state the popular conclusion: moving clocks run slower than those at rest. This is known as time dilation. It is also popularly illustrated in the twins paradox: if one twin goes on a long journey into space on a fast moving rocket, she will be younger than her sister when the two meet up again on her return. Surely few calculations can have led to such a bewildering conclusion! Of course, it is that ratio V/c that is important, and for any significant effect to occur, the rocket's speed V must approach the speed of light c or the transformation factor γ(V) must differ only minutely from 1.
11.3.3 The Reality of Time Dilation
There is no doubt that the calculation leading to the time dilation fo
rmula is correct, but can we be sure about its meaning? Remember that the postulates of relativity mean that all physical processes must satisfy the theory. Conceptually, it is not easy to appreciate the workings of special relativity. (For the matter of time dilation, readers will find a careful, and largely nonmathematical, discussion in chapter 13 of David Mermin's book It's About Time. Mermin assures us that time dilation is real and that there is no intellectual trick in the theory.
Time dilation has now been measured many times (the Wikipedia article Time Dilation gives a string of references). Using modern atomic clocks and particle accelerators confirms the theory to high accuracy (see the recent papers by Reinhardt and Saarthoff, for example).
The earliest relevant experiments were those on the decay of muons (or mesotrons, as they were called in the early days of this research). Muons occur in cosmic rays and may be detected as they move through the earth's atmosphere at very high speeds. It is worth noting that for great speeds, say 90, 95, and 98 percent the speed of light, we calculate
γ(V = 0.9c) = 2.29, γ(V = 0.95c) = 3.20, γ(V = 0.98c) = 5.03.
Bruno Rossi observed muons at various heights on Mount Evans in the United States, and his general article “On the Decay of Mesotrons” gives an entertaining description of his exploits and an introduction to the importance of time dilation for his measurements. His detailed results were published in a 1940 paper. An experiment performed by Arthur Greenberg and his colleagues measured the lifetime of moving charged pions and found time dilation confirmed to an accuracy of 0.4 percent. Einstein set out a fundamental part of physics in his special theory of relativity, and testing it was of the utmost importance; calculation 43, decays and time dilation must be on my list of important calculations.
Although it might seem that relativistic effects are only to be found in the domain of specialized physical experiments, it should be noted that the global positioning system on which we rely more and more today must take account of those effects for its accurate operation (see the review by Ashby).
11.4 PARTICLE MADNESS
Early in the twentieth century, the atomic world seemed simple; there were electrons (e–) orbiting a nucleus of protons (p), and changes in the electron orbit followed photon emission or absorption. Then came the discovery of the neutron (n) as a component of the nucleus. Even today, that picture (a world of electrons, protons, neutrons, and photons) is sufficient to explain the properties of atoms, molecules, solids, and so on. (Michael Fayer beautifully shows how in his 2010 book titled Absolutely Small: How Quantum Theory Explains Our Everyday World.)
But refining the theory and probing deeper revealed a richer (potentially more confusing) world. Refining the quantum theory led Dirac to the positron (e+) and the idea that every particle also has an antiparticle. Then the decay of a neutron was explained only with the help of the neutrino (ν). Cosmic rays included a heavy version of the electron called the muon (μ). Quantum electrodynamics explained the electrical interaction of charged particles in terms of photons leading Hideki Yukawa to suggest in 1935 that protons and neutrons interact by exchanging particles called pi mesons. Pi mesons come in three charged kinds (π–, π0, and π+), and unlike photons, they have mass (about a seventh of the proton mass). Later, heavier K mesons were added to the list.
As particle accelerators and detection devices improved, more and more particles were discovered. There seemed to be a zoo of particles, and physicists looked a little like biologists trying to classify their new findings. In fact, Enrico Fermi quipped that “if I could remember the names of these particles, I would have been a botanist.”10 Some groupings were becoming clear; for example, the electron, its heavier versions (the μ and τ particles), their neutrinos (νe, νμ, and ντ), and all the antiparticles form a group of twelve spin one-half particles known as the leptons. Another large collection of particles are called hadrons, and they divide into the baryons (which include the proton and neutron) and the mesons (including the pions, which mediate the nuclear force). What was needed was a classification showing an order in the same way that Mendeleev's periodic table and the quantum theory of the atom brought order into the world of atoms.
11.4.1 Finding Patterns
Particles were obviously labeled by their mass, charge, and intrinsic spin (as discussed in section 11.1.1). Then it was discovered that new “quantum numbers” could also be attached to particles which now had “isospin” and “strangeness” as properties. (Dunlap's table 2.3 gives a summary of these quantum numbers and their conservation properties.) In the 1960s, Murray Gell-Mann and Y. Ne'eman discovered that arranging particles according to their properties could produce various particular patterns, an idea that became known as the eightfold way. If the spin one-half baryons (which includes the proton and neutron) are arranged in a plane according to their isospin and strangeness properties, a hexagonal pattern is obtained. Similarly, for spin three-halves baryons, a triangular pattern emerges (see figure 11.5). Similar hexagonal patterns emerge for the spin zero and spin one mesons. The triangular pattern including the particle called the Ω– became particularly famous because this particle had yet to be observed when the pattern was first given.
Figure 11.5. Particles plotted in a plane with axes (not shown) measuring isospin and strangeness. Each ■ represents a particle, and the name given is attached. On the left are the spin one-half baryons, and on the right are spin three-halves baryons. Figure created by the author.
The pattern organization led to formulas linking particles and their properties such as mass. Patterns are analyzed using a branch of mathematics known as group theory. (For example, all the possible two-dimensional patterns are classified using the seventeen “wallpaper groups,” and the possible types of crystals are catalogued using the space groups. In fact I almost included such calculations as a member of my list—going up from 50 calculations would certainly make room for them.) There is a group-theory structure that underpins the patterns of elementary particles and helps us to understand how the patterns form. But most importantly it also points to an underlying simpler structure from which the patterns may be built. That structure predicts a new type of particle called a quark.
11.4.2 Quarks
Elementary particles called quarks were introduced by Gell-Mann. He and George Zweig showed that by using a quark model for mesons and baryons, patterns involving particle masses, lifetimes, and spins could emerge. It now appears that quarks are those elusive unbreakable particles of matter that have been sought ever since the time of the ancient Greeks. (Of course, nobody can be certain—remember, it was once the atom that was taken as the most fundamental unit of matter.)
Current theories involve just six quarks:
up (u) with charge +⅔ down (d) with charge–⅓ strange (s) with charge–⅓
charm (c) with charge +⅔ bottom (b) with charge–⅓ top (t) with charge +⅔
There are also the six antiquarks: ua, da, sa, ca, ba, and ta. Note that these particles carry fractional charges (when measured in terms of the electron's charge of minus one.) I indicate three quarks x, y, and z, bound together as |xyz>. Pairs of quarks and antiquarks form mesons. Here are some examples:
|uud> gives the proton, p |udd> gives the neutron, d |sss> gives the Ω–
|uda> gives the π+ |uad> gives the π– |usa> gives the K+
All of the known baryons and mesons are formed using similar combinations of quarks.
The quarks interact by interchanging particles (bosons) known as gluons, and just as photons come from quantum electrodynamics, gluons come from a branch of quantum theory known as quantum chromodynamics. The theory shows that quarks are confined in their combinations so that no free individual quark can be observed (at least at currently envisaged energies).
This model of fundamental particles goes a long way in explaining the diversity and properties of the old “elementary particles.” This is an extremely technical, difficult, and involved theory, but it does appear to be successful. For many peopl
e, the crucial test is a calculation of the hadron (baryon and meson) masses in terms of the quark parameters as they are used to fit a small set of data. This was achieved by S. Dürr and his eleven collaborators and reported in a 2008 Science paper titled “Ab-Initio Determination of Light Hadron Masses.” Their major result is reproduced in figure 11.6. (Note that the masses are measured in terms of the energy unit MeV, or million electron volts, and N denotes a nucleon, which may be a proton or a neutron.) I take this wonderful result as calculation 44, quarks tell us particle masses. (This is not a simple area of science; the interested reader may like to consult The Lightness of Being: Mass, Ether and the Unification of Forces by Frank Wilczek, Nobel Prize winner in 2004 for his work on quark confinement.) It does appear that we now have an answer to that question of where do all those wretched particles come from: they are the quantum states of a small number of quarks.
It appears now that we have reached the “standard model” for the fundamental units making up the physical world. We have the electron and the other leptons and their neutrinos, and the quarks. (See the short review by Olsen for results about particles comprising more than two or three quarks.) There are also the particles like the photon and the gluons that carry the interaction between leptons and quarks. The search goes on to understand how all of the various particles in the standard model are related to one another and to find an understanding for the basic parameters (like the electron mass) that must be fed into the theory. It has been a long time since Lucretius introduced his “particles of an invincible hardness”11 and Newton suggested “that God in the beginning form'd Matter in solid, massy, hard, impenetrable, moveable Particles,”12 and it would be a brave person who declared that the search for the fundamental entities of matter is over, but the many calculations made using the standard model are most persuasive.