that the particle/top is spinning about the direction of motion, the
particle is said to be right-handed. If you put your left hand out and
do the same thing, a left-handed particle would be spinning
clockwise to match the direction of your left-curled hand:
Just as viewing your left hand in a mirror will make it look like a
right hand, if you see a spinning arrow in the mirror, its direction of
motion will be flipped, so that if the arrow is moving away from you
in the real world, it will be moving toward you in the mirror, but the
spin will not be flipped. Thus, in the mirror a left-handed particle
will turn into a right-handed particle. (And so, if the poor souls in
Plato’s cave had had mirrors, they might have felt less strange about
the shadows of arrows flipping direction.)
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.
ͣͥ͝
C h a p t e r 1 3
E N D L E S S
F O R M S
M O S T
B E AU T I F U L :
S Y M M E T RY
S T R I K E S B A C K
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 instructor 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 electroma
gnetism 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
Lawrence Krauss - The Greatest Story Ever Told--So Far Page 19