This working picture of left-handed particles is not exact, because if you think about it, you can also turn a left-handed particle into a right-handed particle by simply moving faster than the particle. In a frame in which a person at rest observes the particle zipping by, it may be moving to the left. But if you hop in a rocket and head off to the left and pass by that particle, then relative to you, it is moving to the right. As a result, only for particles that are massless—and are therefore moving at the speed of light—is the above description exact. For, if a particle is moving at the speed of light, nothing can move fast enough to pass the particle. Mathematically, the definition of left-handed has to take this effect into account, but this complication need not concern us any more here.
Electrons can spin in either direction, but what the V-A interaction implies mathematically is that only those moving electrons whose currents are left-handed can “feel” the weak force and participate in neutron decay. Right-handed currents don’t feel the force.
What is more amazing is that neutrinos only feel the weak force, and no other force. As far as we can tell, neutrinos are only left-handed. It is not just that only one sort of neutrino current engages in the weak interaction. In all the experimental observations so far, there are no right-handed neutrinos—perhaps the most explicit demonstration of the violation of parity in nature.
The seeming silliness of this nomenclature was underscored to me years ago when I was watching a Star Trek: Deep Space Nine episode, during which a science officer on the space station discovers something wrong with the laws of probability in a gaming casino. She sends a neutrino beam through the facility, and the neutrinos are observed to be coming out only left-handed. Clearly something was wrong.
Except that is the way it really is.
What is wrong with nature? How come, for at least one of the fundamental forces, left is different from right? And why should neutrinos be so special? The simple answer to these questions is that we don’t yet know, even though our very existence, which derives from the nature of the known forces, ultimately depends on it. That is one reason we are trying to find out. The elucidation of a new force led to a new puzzle, and like most puzzles in science, it ultimately provided the key that would lead physicists down a new path of discovery. Learning that nature lacked the left-right symmetry that everyone had assumed was fundamental led physicists to reexamine how symmetries are manifested in the world, and more important, how they are not.
Chapter 13
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ENDLESS FORMS MOST BEAUTIFUL: SYMMETRY STRIKES BACK
Now faith is the substance of things hoped for, the evidence of things not seen.
—HEBREWS 11:1
Borrowing from Pauli, we can say Mother Nature is a weak left-hander. With the shocking realization that nature distinguishes left from right, physics itself took a strange left turn down a road with no familiar guideposts. The beautiful order of the periodic table governing phenomena on atomic scales gave way to the mystery of the nucleus and the inscrutable nature of the forces that governed it.
Gone were the seemingly simple days of light, motion, electromagnetism, gravity, and quantum mechanics. The spectacularly successful theory of quantum electrodynamics, which had previously occupied the forefront of physics, seemed to be replaced by a confusing world of exotic phenomena associated with the other two newly discovered weak and strong nuclear forces that governed the heart of matter. Their effects and properties could not easily be isolated, despite that one force was thousands of times stronger than the other. The world of fundamental particles appeared to be ever more complicated, and the situation was getting more confusing with each passing year.
• • •
If the discovery of parity violation created shadows of confusion by demonstrating that nature had completely unexpected preferences, the first rays of light arose from the realization that other nuclear quantities, which on the surface seemed quite different, might, when viewed from a fundamental perspective, be not so different at all.
Perhaps the most important discovery in nuclear physics was that protons and neutrons could convert into each other, as Yukawa had speculated years earlier. This was the basis of the emerging understanding of the weak interaction. But most physicists felt that it was also the key to understanding the strong force that appeared to hold nuclei together.
Two years before his revolutionary work with T.-D. Lee, exposing the demise of the sacred left-right symmetry of nature, C.-N. Yang had concentrated his efforts on trying to understand how a different type of symmetry, borrowed from quantum electrodynamics, might reveal an otherwise hidden beauty inside the nucleus. Perhaps, as Galileo discovered regarding the basis of motion, the most obvious things we observe about nature are also the things that most effectively mask its fundamental properties.
What had slowly become clear, not only from the progress in understanding neutron decay and other weak effects in nuclei, but also from looking at strong nuclear collisions, was that the obvious distinction between protons and neutrons—the proton is charged and the neutron is neutral—might, as far as the underlying physics governing the nucleus is concerned, be irrelevant. Or at least as irrelevant as the apparent distinction between a falling feather and a falling rock is to our understanding of the underlying physics of gravity and falling objects.
First off, the weak force could convert protons into neutrons. More important, when one examined the rates of other, stronger nuclear reactions involving proton or neutron collisions, replacing neutrons by protons and vice versa didn’t significantly change the results.
In 1932, the year the neutron was discovered, Heisenberg had suggested that the neutron and proton might be just two states of the same particle, and he invented a parameter he called isotopic spin to distinguish them. After all, their masses are almost the same, and light-stable nuclei contain equal numbers of them. Following this, and after the recognition by the distinguished nuclear physicists Benedict Cassen, Edward Condon, Gregory Breit, and Eugene Feenberg that nuclear reactions seemed to be largely blind to distinguishing protons and neutrons, the brilliant mathematical physicist Eugene Wigner suggested that isotopic spin was “conserved” in nuclear reactions—implying an underlying symmetry governing the nuclear forces between protons and neutrons. (Wigner had earlier developed rules demonstrating how symmetries in atomic systems ultimately allowed a complete classification of atomic states and the transitions between them, for which he later won the Nobel Prize.)
Earlier, when discussing electromagnetism, I noted that the net electric charge doesn’t change during electromagnetic interactions—i.e., electric charge is conserved—because of an underlying symmetry between positive and negative charges. The underlying connection between conservation laws and symmetries is far broader and far deeper than this one example. The deep and unexpected relationship between conservation laws and symmetries of nature has been the single most important guiding principle in physics in the past century.
In spite of its importance, the precise mathematical relationship between conservation laws and symmetries was only made explicit in 1915 by the remarkable German mathematician Emmy Noether. Sadly, although she was one of the most important mathematicians in the early twentieth century, Noether worked without an official position or pay for much of her career.
Noether had two strikes against her. First, she was a woman, which made obtaining education and employment during her early career difficult, and second, she was Jewish, which ultimately ended her academic career in Germany and resulted in her exile to the United States shortly before she died. She managed to attend the University of Erlangen as one of 2 female students out of 986, but even then she was only allowed to audit classes after receiving special permission from individual professors. Nevertheless, she passed the graduation exam and later studied at the famed University of Göttingen for a short period before returning to Erlangen to complete her PhD thesis. After working for seven years at Erlangen as an instru
ctor without pay, she was invited in 1915 to return to Göttingen by the famed mathematician David Hilbert. Historians and philosophers among the faculty, however, blocked her appointment. As one member protested, “What will our soldiers think when they return to the university and find that they are required to learn at the feet of a woman?” In a retort that eternally reinforced my admiration for Hilbert, beyond that for his remarkable talent as a mathematician, he replied, “I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, we are a university, not a bathhouse.”
Hilbert was overruled, however, and while Noether spent the next seventeen years teaching at Göttingen, she was not paid until 1923, and in spite of her remarkable contributions to many areas of mathematics—so many and so deep that she is often considered one of the great mathematicians of the twentieth century—she was never promoted to the position of professor.
Nevertheless, in 1915, shortly after arriving at Göttingen, she proved a theorem that is now known as Noether’s theorem, which all graduate students in physics learn, or should learn, if they are to call themselves physicists.
• • •
Returning once again to electromagnetism, the relationship between the arbitrary distinction between positive and negative (had Benjamin Franklin had a better understanding of nature when he defined positive charge, electrons would today probably be labeled as having positive, not negative, charge) and the conservation of electric charge—namely, that the total charge in a system before and after any physical reaction doesn’t change—is not at all obvious. It is in fact a consequence of Noether’s theorem, which states that for every fundamental symmetry of nature—namely for every transformation under which the laws of nature appear unchanged—some associated physical quantity is conserved. In other words, some physical quantity doesn’t change over time as physical systems evolve. Thus:
• The conservation of electric charge reflects that the laws of nature don’t change if the sign of all electric charges is changed.
• The conservation of energy reflects that the laws of nature don’t change with time.
• The conservation of momentum reflects that the laws of nature don’t change from place to place.
• The conservation of angular momentum reflects that the laws of nature don’t depend on which direction a system is rotated.
Hence, the claimed conservation of isotopic spin in nuclear reactions is a reflection of the experimentally verified claim that nuclear interactions remain roughly the same if all protons are changed into neutrons and vice versa. It is reflected as well in the world of our experience, in that for light elements, at least, the number of protons and neutrons in the nucleus is roughly the same.
In 1954, Yang, and his collaborator at the time, Robert Mills, went one important step further, once again thinking about light. Electromagnetism and quantum electrodynamics do not just have the simple symmetry that tells us that there is no fundamental difference between negative charge and positive charge, and that the label is arbitrary. As I described at length earlier, a much more subtle symmetry is at work as well, one that ultimately determines the complete form of electrodynamics.
Gauge symmetry in electromagnetism tells us that we can change the definition of positive and negative charge locally without changing the physics, as long as there is a field, in this case the electromagnetic field, that can account for any such local alterations to ensure that the long-range forces between charges are independent of this relabeling. The consequence of this in quantum electrodynamics is the existence of a massless particle, the photon, which is the quantum of the electromagnetic field, and which conveys the force between distant particles.
In this sense, that gauge invariance is a symmetry of nature ensures that electromagnetism has precisely the form it has. The interactions between charged particles and light are prescribed by this symmetry.
Yang and Mills then asked what would happen if one extended the symmetry that implies that we could interchange neutrons and protons everywhere without changing the physics, into a symmetry that allows us to change what we label as “neutron” and “proton” differently from place to place. Clearly by analogy with quantum electrodynamics, some new field would be required to account for and neutralize the effect of these arbitrarily varying labels from place to place. If this field is a quantum field, then could the particles associated with this field somehow play a role in, or even completely determine, the nature of the nuclear forces between protons and neutrons?
These were fascinating questions, and to their credit Yang and Mills didn’t merely ask them, they tried to determine the answers by exploring specifically what the mathematical implications of such a new type of gauge symmetry associated with isotopic spin conservation would be.
It became clear immediately that things would get much more complicated. In quantum electrodynamics, merely switching the sign of charges between electrons and positrons does not change the magnitude of the net charge on each particle. However, relabeling the particles in the nucleus replaces a neutral neutron with a positively charged proton. Therefore whatever new field must be introduced in order to cancel out the effect of such a local transformation so that the underlying physics is unchanged must itself be charged. But if the field is itself charged, then, unlike photons—which, being neutral, don’t themselves interact directly with other photons—this new field would also have to interact with itself.
Introducing the need for a new charged generalization of the electromagnetic field makes the mathematics governing the theory much more complex. In the first place, to account for all such isotopic spin transformations one would need not just one such field but three fields, one positively charged, one negatively charged, and one neutral. This means that a single field at each point in space, like the electromagnetic field in QED, which points in a certain direction in space with a certain magnitude (and is called a vector field in physics for this reason), is not sufficient. The electric field must be replaced by a field described by a mathematical object called a matrix—not to be confused with anything having to do with Keanu Reeves.
Yang and Mills explored the mathematics behind this new and more complex type of gauge symmetry, which today we call either a non-abelian gauge symmetry—arising from a particular mathematical property of matrices that makes multiplying them different from multiplying numbers—or, in deference to Yang and Mills, a Yang-Mills symmetry.
Yang and Mills’s article appears at first glance to be an abstract—or purely speculative—mathematical exploration of the implications of a guess about the possible form of a new interaction, motivated by the observation of gauge symmetry in electromagnetism. Nevertheless, it was not an exercise in pure mathematics. The paper tried to explore possible observable consequences of the hypothesis to see if it might relate to the real world. Unfortunately the mathematics was sufficiently complicated such that the possible observable signatures were not so obvious.
One thing was clear, however. If the new “gauge fields” were to account for and thus cancel out the effects of separate isotopic spin transformations made in distant locations, the fields would have to be massless. This is equivalent to saying that only because photons are massless can the force they transmit between particles be arbitrarily long-range. To return to my chessboard analogy, you need a single rulebook to tell you how to properly move over the entire board if I have previously changed the colors of the board randomly from place to place. But having massive gauge fields, which cannot be exchanged over arbitrarily long distances, is equivalent to having a rulebook that tells you how to compensate for changing colors only on nearby squares around your starting point. But this would not allow you to move pieces across the board to distant locations.
In short, a gauge symmetry such as that in electromagnetism, or in the more esoteric Yang-Mills proposal, only works if the new fields required by the symmetry are massless. Amid all the mathematical complexity, this one fact i
s inviolate.
But we have observed in nature no long-range forces involving the exchange of massless particles other than electromagnetism and gravity. Nuclear interactions are short-range—they only apply over the size of the nucleus.
This obvious problem was not lost on Yang and Mills, who recognized it and, frankly, punted. They proposed that somehow their new particles could become massive when they interacted with the nucleus. When they tried to estimate masses from first principles, they found the theory was too mathematically complicated to allow them to make reasonable estimates. All they knew was that empirically the mass of the new gauge particles would have to be greater than that of pions in order to have avoided detection in then-existing experiments.
Such a willingness to throw their hands in the air might have seemed either lazy or unprofessional, but Yang and Mills knew, as Yukawa had known before them, that no one had been able to write down a sensible quantum field theory of a particle like the photon, but one that, unlike the photon, had a mass. So it didn’t seem worthwhile at the time to try to solve all the problems of quantum field theory at once. Instead, with less irreverence than Jonathan Swift, they merely presented their paper as a modest proposal, to spur the imagination of their colleagues.
Wolfgang Pauli, however, would have none of it. While he had thought of some related ideas a year earlier, he had discarded them. Moreover, he felt that all this talk about quantum uncertainties in estimating masses was a red herring. If there was to be a new gauge symmetry in nature associated with isotopic spin and governing nuclear forces, the new Yang-Mills particles, like the photon, would have to be massless.
The Greatest Story Ever Told—So Far Page 16
