I think that Anderson’s argument was generally discounted because it was based on analogies with phenomena like superconductivity which are non-relativistic (i.e., these are phenomena that occur in domains in which Einstein’s special theory of relativity can be safely ignored). But the inevitability of massless Nambu-Goldstone particles had been shown, apparently rigorously, by Goldstone, Salam, and me, in a 1962 proof that relied on the manifest validity of relativity theory. Particle theorists were prepared to believe that Anderson was right in the non-relativistic context of superconductivity, but not in elementary particle theory, which necessarily incorporates relativity. The work of the 1964 papers made it clear that the proof by Goldstone, Salam, and myself did not apply to quantum theories with force-carrying particles, because although physical phenomena in such theories do satisfy the principle of relativity, the mathematical formulation of these theories in the context of quantum mechanics does not.
This problem with relativity was also the reason I was unable after 1967, despite strenuous efforts, to prove what Salam and I had conjectured, that nonsensical infinities that appeared in the electroweak theory all cancelled out, in the same way that similar infinities had already been shown to cancel in the quantum theory of electromagnetism alone. Relativity had been essential in demonstrating the cancellation of infinities in electromagnetism. The proof of cancellation by Gerard ‘t Hooft in 1971, described by Baggott in Chapter 5, used techniques that ‘t Hooft had worked out with Martinus Veltman, in which the principles of quantum mechanics are stretched to allow the theory to be formulated in a way that is consistent with relativity.
A second point: Baggott suggests in Chapter 4 that I did not include quarks in my 1967 paper proposing the electroweak theory because I was concerned about the problem that the theory might predict processes involving so-called ‘strange’ particles that were not in fact observed. I wish that my reason had been that specific. Rather, I did not include quarks in the theory simply because in 1967 I just did not believe in quarks. No-one had ever observed a quark, and it was hard to believe that this was because quarks are much heavier than observed particles like protons and neutrons, when these observed particles were supposed to be made of quarks.
Like many other theorists, I did not fully accept the existence of quarks until the 1973 work of David Gross and Frank Wilczek, and David Politzer. They showed that in the theory of quarks and strong nuclear forces known as quantum chromodynamics, the strong force gets weaker with decreasing distance. It then occurred to some of us that in that case the strong force between quarks would counter-intuitively get stronger as the quarks get farther apart, perhaps so much so as to prevent quarks from ever being separated from one another. There still is no proof of this, but it is generally believed. Quantum chromodynamics is by now a very well tested theory, and yet no-one has ever seen an isolated quark.
I was very glad to see this book begin in the early twentieth century with Emmy Noether, who realized before anyone else the importance of symmetry principles in nature. This helps to remind us that the work of scientists today is just the latest step in a grand tradition, of trying to guess how nature works, always subjecting our guesses to the test of experiment. Jim Baggott’s book should give the reader some of the flavour of this historic enterprise.
Steven Weinberg
6 July 2012
PROLOGUE
Form and Substance
What is the world made of?
Simple questions such as this have been teasing the human intellect for as long as humankind has been capable of rational thought. For sure, the way we ask this question today has become much more elaborate and sophisticated, and the answers have become much more complex and costly to provide. But, make no mistake, at heart the question remains a very simple one.
Two and a half thousand years ago, all the Ancient Greek philosophers had to go on was their sense of beauty and harmony in nature and their powers of logical reasoning and imagination applied to the things they could perceive with their unaided senses. With hindsight, it is quite extraordinary just how much they were able to figure out.
The Greeks were careful to distinguish between form and substance. The world is made of material substance which can take a variety of different forms. The fifth century BC Sicilian philosopher Empedocles suggested that this variety could be reduced to four basic forms, what we know today as ‘classical elements’. These were earth, air, fire, and water. The elements were judged to be eternal and indestructible, joined together in rather romantic combinations through the attractive force of Love and split apart through the repulsive force of Strife, to make up everything in the world.
Another school of thought originating with the fifth century BC philosopher Leucippus (and most closely associated with his pupil, Democritus) held that the world consists of tiny, indivisible, indestructible material particles (called atoms) and empty space (void). The atoms represented the building blocks of all material substance, responsible for all matter. Atoms were necessary as a matter of principle, so Leucippus argued, because substance surely could not be divided indefinitely. If this were possible, then we would be able to divide substance endlessly into nothing, in apparent contradiction with what seemed to be an unassailable law of the conservation of matter.
About a century later, Plato developed a theory which explained how atoms (the substance) are structured to make up the four elements (the forms). He represented each of the four elements by a geometrical (or ‘Platonic’) solid, and argued in the Timaeus that the faces of each solid could be further decomposed into systems of triangles, representing the elements’ constituent atoms. Rearrange the patterns of triangles – rearrange the atoms – and it was possible to convert one element into another and combine elements to produce new forms.*
It seems logical that there should be some ultimate constituents, some undeniable reality that underpins the world we see around us and which lends it form and shape. If matter is endlessly divisible, then we would reach a point where the constituents themselves become rather ephemeral – to the point of non-existence. Then there would be no building blocks, and all we would be left with are interactions between indefinable, insubstantial phantoms which give rise to the appearance of substance.
Unpalatable it may be but, to a large extent, this is precisely what modern physics has shown to be true. Mass, we now believe, is not an inherent property or ‘primary’ quality of the ultimate building blocks of nature. In fact, there is no such thing as mass. Mass is constructed entirely from the energy of interactions involving naturally massless elementary particles.
The physicists kept dividing, and in the end found nothing at all.
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It was not until the development of a formal experimental philosophy in the early seventeenth century that it became possible to transcend the kind of speculative thinking that had characterized the theories of the Ancient Greeks. The old philosophy had tried to intuit the nature of material substance from observations contaminated with prejudices about how the world ought to be. The new scientists now tinkered with nature itself, teasing out evidence about how the world really is.
The questions were still primarily concerned with the nature of form and substance. The concept of mass – a measure of the amount of matter as manifested in the dynamical movements of objects – became central to our understanding of substance. An object’s resistance to acceleration is interpreted as inertial mass. When kicked with the same force, a small object will accelerate much faster than a large one.
An object’s ability to generate a gravitational field is interpreted as gravitational mass. The force of gravity generated by the moon is weaker than the force generated by the earth, because the moon is smaller and so possesses a smaller gravitational mass. Inertial and gravitational mass are empirically identical, although there is no compelling theoretical reason why this should be so.
The scientists also exposed the secrets of nature’s great variety of form
. The fundamental Greek ‘element’ water was found to consist not of geometrical solids composed of triangles, as Plato had surmised, but of molecules composed of atoms of the chemical elements hydrogen and oxygen, in a combination we write today as H2O.
This more modern use of the word ‘atom’ initially evoked the interpretation lent to it by the Greeks, as an indivisible building block of matter. But, even as the reality of atoms was being hotly debated, in 1897 English physicist Joseph John Thompson was discovering the negatively charged electron. It seemed that atoms, in their turn, possessed constituent, subatomic, parts.
Thompson’s discovery was followed in the years 1909–11 by experiments in the Manchester laboratory of New Zealander Ernest Rutherford. These experiments showed that atoms consist, for the most part, of empty space. At the centre of the atom sits a tiny, positively charged nucleus, around which the negatively charged electrons orbit much like planets orbit the sun. Most of the mass of the atoms that make up the elements of material substance is concentrated in their atomic nuclei. It is therefore in the nucleus that form and substance come together.
This ‘planetary’ model of the atom remains a compelling visual metaphor even today. But it was immediately obvious to the physicists of the time that such a model actually makes no sense. Such planetary atoms were expected to be inherently unstable. Unlike planets moving around the sun, electrically charged particles moving in an electric field radiate energy in the form of electromagnetic waves. Such planetary electrons would exhaust their energy within a fraction of a second, and the internal architecture of the atom would then collapse.
The solution to this particular puzzle emerged in the guise of quantum mechanics in the early 1920s. The electron is not just a particle – which we might visualize as a tiny ball of negatively charged matter – it is simultaneously both wave and particle. It is not ‘here’ or ‘there’, as might be expected of a localized bit of stuff, but literally ‘everywhere’ within the confines of its ghostly, delocalized wavefunction. Electrons do not orbit the nucleus as such. Instead their wavefunctions form characteristic three-dimensional patterns – which we call ‘orbitals’ – in the space around the nucleus. The mathematical form of each orbital relates the probability of finding the now wholly mysterious electron at specific locations – ‘here’ or ‘there’ – inside the atom (see Figure 1).
The quantum revolution was a time of unprecedented fertility in both theoretical and experimental physics. When in 1927 English physicist Paul Dirac combined quantum mechanics with Albert Einstein’s special theory of relativity, out popped a new property called electron spin. This was a property already known to experimentalists, and tentatively interpreted in terms of an electron spinning on its axis like a spinning top, much as the earth rotates on its axis as it orbits the sun (see Figure 2).
FIGURE 1 (a) In Rutherford’s ‘planetary’ model of the hydrogen atom, a single negatively charged electron occupies a fixed orbit around a nucleus consisting of a single positively charged proton. (b) Quantum mechanics replaced the orbiting electron by an electron wavefunction, which for the lowest energy (1s) configuration is spherically symmetric. (c) The electron can now be ‘found’ anywhere within the confines of the wavefunction, but has the highest probability of being found at the distance predicted by the old planetary model.
But this was another visual metaphor that was quickly found to have no foundation in reality. Today, we interpret electron spin as a purely ‘relativistic’ quantum effect, in which electrons may take up one of two possible ‘orientations’, which we call spin-up and spin-down. These are not orientations along specific directions in conventional, three-dimensional space, but orientations in a ‘spin-space’ which has only two dimensions – up or down.
Each orbital in an atom was found to contain two – and only two – electrons. This is Austrian physicist Wolfgang Pauli’s famous exclusion principle, which he developed in 1925 and which states that electrons are forbidden from occupying the same quantum state. The principle derives from the mathematical form of the wavefunction for any composite state consisting of two or more electrons. If the composite state were assumed to be created with two electrons which have precisely the same physical characteristics, then the wavefunction has zero amplitude – such a state could not exist. For the wavefunction to exist with a non-zero amplitude, then the two electrons must somehow be different. In an atomic orbital, this means that one electron must have a spin-up orientation and one must have a spin-down orientation. In other words, their spins must be paired.
FIGURE 2 In 1927 Dirac combined quantum mechanics and Einstein’s special theory of relativity to create a fully ‘relativistic’ quantum theory. Out popped the property of electron spin, imagined as though the negatively charged electron were literally spinning on its axis thereby generating a small, local magnetic field. Today we think of electron spin simply in terms of its possible orientations – spin-up and spin-down.
It is wise to resist the temptation to imagine what these different orientations might actually look like. Their effects are real enough, however. Spin determines the amount of angular momentum carried by the electron – the momentum associated with the ‘rotational’ motion of its spin. Spin also governs how the electron interacts with a magnetic field, effects that can be studied in detail in the laboratory. But in quantum mechanics we appear to have crossed the threshold between what we can know of the origin of these effects, and what we cannot.
Dirac’s relativistic quantum theory of the electron also yielded up twice as many solutions as he thought he had needed. Two of the solutions correspond to the spin-up and spin-down orientations of the electron. So what did the other two solutions correspond to? He had some ideas of his own, but finally conceded in 1931 that they had to represent the spin-up and spin-down orientations of a previously unknown positive electron. Dirac had discovered anti-matter. The ‘positron’, the anti-particle of the electron, was subsequently found in experiments on cosmic rays, formed high in the earth’s atmosphere by collisions involving high-energy particles.
In 1932 it seemed that the final piece of the puzzle had been found. English physicist James Chadwick discovered the neutron, an electrically neutral particle which sits snugly alongside the positively charged proton inside the atomic nucleus. It seemed that physicists now had all the ingredients to formulate a definitive answer to our opening question.
The answer went something like this. All the material substance in the world is made of chemical elements. These elements come in a great variety of forms which make up the periodic table, from the lightest, hydrogen, to the heaviest-known, naturally occurring element, uranium.*
Each element consists of atoms. Each atom consists of a nucleus composed of varying numbers of positively charged protons and electrically neutral neutrons. Each element is characterized by the number of protons in the nuclei of its atoms. Hydrogen has one, helium two, lithium three, and so on, to uranium, which has 92.
Surrounding the nucleus are negatively charged electrons, in numbers which balance the numbers of protons, so that overall the atom is electrically neutral. Each electron can take either a spin-up or spin-down orientation and each orbital can accommodate two electrons provided their spins are paired.
It is a very comprehensive answer. With fundamental building blocks of protons, neutrons, and electrons and Pauli’s exclusion principle, we can explain why the periodic table has the structure that it has. We can explain why matter has shape and density. We can explain the existence of isotopes – atoms with the same numbers of protons but different numbers of neutrons in their nuclei. With a little effort, we can explain all of chemistry, biochemistry, and materials science.
In this description, mass is no real mystery. The mass of all material substance can be traced back to its constituent protons and neutrons, which account for about 99 per cent of the mass of every atom.
Imagine a small cube of ice, formed from triply distilled water. Its sides measure 2.7 centime
tres in length, or a little over an inch. Pick it up. It’s cold and slippery. It’s not heavy, but you are conscious of its weight in the palm of your hand. So, where does the mass of the ice cube reside?
The molecular weight of water is simply calculated from the total number of protons and neutrons in the nuclei of the two atoms of hydrogen and one atom of oxygen that make up H2O. The nucleus of each hydrogen atom consists of just one proton, and the nucleus of the oxygen atom contains eight protons and eight neutrons, making 18 ‘nucleons’ in total. The cube of pure ice you hold in your hand will weigh about 18 grams,* equal to the molecular weight in grams. The cube therefore represents a standard measure of solid water known as a ‘mole’.
We know that a mole of substance contains a fixed number of the atoms or molecules that make up that substance. This is Avogadro’s number, a little over six hundred billion trillion (6 × 1023). Here, then, is the answer. The weight of the ice cube that you feel in the palm of your hand is the combined result of the masses of six hundred billion trillion molecules of H2O, or about 10,800 billion trillion protons and neutrons (see Figure 3).*
It had to be accepted that atoms were no longer indestructible, as the Greeks had once thought. Atoms could be transmuted, turned from one form into another. In 1905, Einstein had used his special theory of relativity to show that mass and energy are equivalent, through what was to become the world’s most famous scientific equation, E = mc2: energy is equal to mass multiplied by the speed of light squared. However, far from undermining the concept of mass, the notion that mass represents a vast reservoir of energy somehow made it even more substantial.
Higgs:The invention and discovery of the 'God Particle' Page 2
