The Universe Within (20 page)

The Higgs mechanism lies at the heart of Glashow, Salam, and Weinberg's theory, in which the electroweak Higgs field
φ
is responsible for separating Maxwell's electromagnetic force out from the weak nuclear force, and fixing the basic masses and charges of the matter particles.

The last term, the Higgs potential energy,
V
(
φ
), ensures that the Higgs field
φ
takes a fixed constant value in the vacuum, everywhere in space. It is this value that communicates a mass to the quanta of the force fields and to the matter particles. The Higgs field can also travel in waves — similar to electromagnetic waves in Maxwell's theory — that carry energy quanta. These quanta are called “Higgs bosons.”
(click to see photo)
Unlike photons, they are fleetingly short-lived, decaying quickly into matter and antimatter particles. They have just been discovered at the Large Hadron Collider in the
CERN
laboratory in Geneva, confirming predictions made nearly half a century ago
(click to see photo)
.

Finally, the value of the Higgs potential energy,
V
, in the vacuum also plays a role in fixing the energy of empty space — the vacuum energy — measured recently by cosmologists.

Taken together, these terms describe what is known as the “Standard Model of Particle Physics.” The quanta of force fields, like the photon and the Higgs boson, are called “force-carrier particles.” Including all the different spin states, there are thirty different force-carrier particles in total, including photons (quanta of the electromagnetic field), W and Z bosons (quanta of the weak nuclear force field), gluons (quanta of the strong force), gravitons (quanta of the gravitational field), and Higgs bosons (quanta of the Higgs field). The matter particles are all described by Dirac fields. Including all their spin and antiparticle states, there are a total of ninety different matter particles. So, in a sense, Dirac's equation describes three-quarters of known physics.

When I was starting out as a graduate student I found the existence of this one-line formula, which summarizes everything we know about physics, hugely motivating. All you have to do is master the language and learn how to calculate, and in principle you understand at a basic level all of the laws governing every single physical process in the universe.

· · ·

YOU MAY WELL WONDER
how it was that physics converged on this remarkably simple unified formula. One of the most important ideas guiding its development was that of “symmetry.” A symmetry of a physical system is a transformation under which the system does not change. For example, a watch ticks at exactly the same rate wherever you place it, because the laws governing the mechanism of the watch do not depend on where the watch is. We say the laws have a symmetry under moving the watch around in
space
. Similarly, the watch's working is unchanged if we rotate the watch — we say the laws have a symmetry under
rotations
. And if the watch works just the same today, or tomorrow, or yesterday, or an hour from now, we say the laws that govern it have a symmetry under shifts in
time
.

The entry of these ideas of symmetry into physics traces back to a remarkable woman named Emmy Noether, who in 1915 discovered one of the most important results in mathematical physics. Noether showed mathematically that any system described by an action that is unchanged by shifts in time, as most familiar physical systems are, automatically has a conserved energy. Likewise, for many systems it makes no difference to the evolution of the system exactly where the system is located in space. What Noether showed in this case is that there are three conserved quantities — the three components of momentum. There is one of these for each independent direction in which you can move the system without changing it: east–west, north–south, up–down.

Ever since Newton, these quantities — energy and momentum — had been known, and found to be very useful in solving many practical problems. For example, energy can take a myriad forms: the heat energy stored in a boiling pan of water, the kinetic energy of a thrown ball, the potential energy of a ball sitting on a wall and waiting to fall, the radiation energy carried in sunlight, the chemical energy stored up in oil or gas, or the elastic energy stored in a stretched string. But as long as the system is isolated from the outside world, and as long as spacetime is not changing (which is an excellent approximation for any real experiment conducted on Earth), the total amount of energy will remain the same.

The total momentum of a system is another very useful conserved quantity, for example in describing the outcome of collisions. Similarly, Benjamin Franklin's law of electric charge conservation, that you can move charge around but never change its total amount, is another consequence of Noether's theorem.

Before Emmy Noether, no one had really understood why
any
of these quantities are conserved. What Noether realized was as simple as it was profound: the conservation laws are mathematical consequences of the symmetries of space and time and other basic ingredients in the laws of physics. Noether's idea was critical to the development of the theories of the strong, weak, and electroweak forces. For example, in electroweak theory, there is an abstract symmetry under which an electron can be turned into a neutrino, and vice versa. The Higgs field differentiates between the particles and breaks the symmetry.

Noether was an extraordinary person. Born in Germany, she faced discrimination as both a Jew and a woman. Her father was a largely self-taught mathematician. The University of Erlangen, where he lectured, did not normally admit women. But Emmy was allowed to audit classes and was eventually given permission to graduate. After struggling to complete her Ph.D. thesis (which she later, with typical modesty, dismissed as “crap”) she taught for seven years at the university's Mathematical Institute, without pay.

She attended seminars at Göttingen given by some of the most famous mathematicians of the time — David Hilbert, Felix Klein, Hermann Minkowski, and Hermann Weyl — and through these interactions her great potential became evident to them. As soon as Göttingen University's restrictions on women lecturers were removed, Hilbert and Klein recruited Noether to teach there. Against great protests from other professors, she was eventually appointed — again, without pay. In 1915, shortly after her appointment, she discovered her famous theorem.

Noether's theorem not only explains the basic conserved quantities in physics, like energy, momentum, and electric charge, it goes further. It explains how Einstein's equations for general relativity are consistent even when space is expanding and energy is no longer conserved. For example, as we discussed in the previous chapter, it explains how the vacuum energy can drive the exponential expansion of the universe, creating more and more energy without violating any physical laws.

When Noether gave her explanation for conserved quantities and more general situations involving gravity, she did so in the context of classical physics and its formulation in terms of Hamilton's action principle. Half a century later, it was realized — by the Irish physicist John Bell, along with U.S. physicists Steven Adler and Roman Jackiw — that quantum effects, included in Feynman's sum over histories, could spoil the conservation laws that Noether's argument predicted.

Nevertheless, it turns out that for the pattern of particles and forces seen in nature, there is a very delicate balance (known technically as “anomaly cancellation”) that allows Noether's conservation laws to survive. This is another indication of the tremendous unity of fundamental physics: the whole works only because of all of the parts. If you tried, for example, to remove the electron, muon, and tanon and their neutrinos from physics and kept only the quarks, then Noether's symmetries and conserved quantities would be ruined and the theory would be mathematically inconsistent. This idea, that Noether's laws must be preserved within any consistent unified theory, has been a key guiding principle in the development of unified theories, including string theory, in the late twentieth century.

Noether's dedicated mentorship of students was exemplary — she supervised a total of sixteen Ph.D. students through a very difficult time in Germany's history. When Hitler came to power in 1933, Jews became targets. Noether was dismissed from Göttingen, as was her colleague Max Born. The great mathematical physicist Hermann Weyl, also working there, wrote later: “Emmy Noether — her courage, her frankness, her unconcern about her own fate, her conciliatory spirit — was, in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace.”
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Eventually, Noether fled to the United States, where she became a professor at Bryn Mawr College, a women's college known particularly as a safe haven for Jewish women. Sadly, at the age of fifty-three she died of complications relating to an ovarian cyst.

In a letter to the
New York Times
, Albert Einstein wrote: “In the judgement of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods that have proved of enormous importance in the development of the present-day younger generation of mathematicians.”
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Emmy Noether was a pure soul whose mathematical discoveries opened many paths in physics and continue to exert great influence.

PAUL DIRAC WAS ANOTHER
mathematical prodigy from a humble background. His discoveries laid the basis for the formula for all known physics. A master of quantum theory, he was largely responsible for its current formulation. When asked what his greatest discovery had been, he said he thought it was his “bra-ket” notation. This is a mathematical device which he introduced into quantum theory to represent the different possible states of a system. The initial state is called a “ket” and the final state a “bra.” It's funny that someone who discovered the equation for three-quarters of all the known particles, who predicted antimatter, and who made countless other path-breaking discoveries would rate them all below a simple matter of notation. As with many other Dirac stories, one can't help thinking: he can't have been serious! But no one could tell.

A recent biography called Dirac “the strangest man.” He was born in Bristol, England, to a family of modest means. His Swiss father, Charles Dirac, was a French teacher and a strict disciplinarian. Paul led an unhappy, isolated childhood, although he was always his father's favourite. He was fortunate to attend one of the best non-fee-paying schools for science and maths in England — Merchant Venturers' Technical College in Bristol, where his father taught.

At school, it became clear that Paul had exceptional mathematical talent, and he went on to Cambridge to study engineering. In spite of graduating with a first-class degree, he could not find a job in the postwar economic climate. Engineering's loss was physics' gain: Dirac returned to Bristol University to take a second bachelor's degree, this time in mathematics. And then, in 1923, at the ripe old age of twenty-one, he returned to St. John's College in Cambridge to work towards a Ph.D. in general relativity and quantum theory.

Over the next few years, this shy, notoriously quiet young man — almost invisible, according to some — made a series of astonishing breakthroughs. His work combined deep sophistication with elegant simplicity and clarity. For his Ph.D., he developed a general theory of transformations that allowed him to present quantum theory in its most elegant form, still used today. At the age of twenty-six, he discovered the Dirac equation by combining relativity and quantum theory to describe the electron. The equation explained the electron's spin and predicted the existence of the electron's antiparticle, the positron. Positrons are now used every day in medical
PET
(positron emission tomography) scans, used to track the location of biological molecules introduced into the body.

When someone asked Dirac, “How did you find the Dirac equation?” he is said to have answered with: “I found it beautiful.” As often, he seemed to take pleasure in being deliberately literal and in using as few words as he could. His insistence on building physics on principled mathematical foundations was legendary. In spite of having initiated the theory of quantum electrodynamics, which was hugely successful, he was never satisfied with it. The theory has infinities, created by quantum fluctuations in the vacuum. Other physicists, including Richard Feynman, Julian Schwinger, and Sin-Itiro Tomonaga, found ways to control the infinities through a calculational technique known as “renormalization.” The technique produced many accurate predictions, but Dirac never trusted it because he felt that serious mathematical difficulties were being swept under the rug. He went so far as to say that all of the highly successful predictions of the theory were probably “flukes.”

Dirac also played a seminal role in anticipating the form of our formula for all known physics. For it was Dirac who saw the connection between Hamilton's powerful action formalism for classical physics and the new quantum theory. He realized how to go from a classical theory to its quantum version, and how quantum physics extended the classical view of the world. In his famous textbook on quantum theory, written in 1930 and based upon this deep understanding, he outlined the relationship between the Schrödinger wavefunction, Hamilton's action, and Planck's action quantum. Nobody followed up this insightful remark until 1946, when Dirac's comment inspired Feynman, who made the relation precise.