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 is 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.
For these reasons, among others, the Yang-Mills paper made far
less of a stir at the time than the later Yang and Lee opus. To most
physicists it was an interesting curiosity at best, and the discovery of
parity violation seemed much more exciting.
But not to Julian Schwinger, who was no ordinary physicist. A
child prodigy who had graduated from university by the age of
eighteen, he received his PhD by the age of twenty-one. Perhaps no
two physicists could have been as different as he and Richard
Feynman, who shared the Nobel Prize in 1965 for their separate but
equivalent work developing the theory of quantum electrodynamics.
Schwinger was refined, formal, and brilliant. Feynman was brilliant,
casual, and certainly not refined. Feynman relied often on intuition
and guesswork, building on prodigious mathematical skill and
experience. Schwinger’s mathematical skill was every bit Feynman’s
equal, but Schwinger worked in an orderly fashion, manipulating
complicated mathematical expressions with an ease not possible for
ordinary mortals. He joked about Feynman diagrams, which
Feynman had developed to make what had previously been
perilously laborious calculations in quantum field theory
manageable, saying, “Like the silicon chips of more recent years, the
Feynman diagram was bringing computation to the masses.” Both of
them shared one characteristic, however. They marched to the beat
of a different drummer . . . in opposite directions.
Schwinger took the Yang-Mills idea seriously. The mathematical
beauty must have appealed to him. In 1957, the same year that parity
violation was discovered, Schwinger made a bold and seemingly
highly unlikely suggestion that the weak interaction responsible for
the decay of neutrons into protons, electrons, and neutrinos might
benefit from the possibility of Yang-Mills fields, but in a new and
ͤͥ͝
remarkable way. He proposed that the observed gauge symmetry of
electromagnetism might simply be one part of a larger gauge
symmetry in which new gauge particles might mediate the weak
interaction that caused neutrons to decay.
An obvious objection to this kind of unification is that the weak
interaction is far weaker than electromagnetism. Schwinger had an
answer for this. If somehow the new gauge particles were very heavy,
almost one hundred times heavier than protons and neutrons, then
the interaction they might mediate would be of much shorter range
than even the size of a nucleus, or even a single proton or neutron.
In this case, one could work out that the probability that this
interaction would cause a neutron to decay would be small. Thus, if
the range of the weak interaction was small, these new fields, the
<
br /> strength of whose intrinsic coupling to electrons and protons on
small scales could be comparable to the strength of
electromagnetism, could nevertheless, on the scale of nuclei and
larger, appear to be much, much weaker.
Put more bluntly, Schwinger proposed the outrageous idea that
electromagnetism and the weak interaction were part of a single
Yang-Mills theory, in spite of the remarkable and obvious
differences between them. He thought that perhaps the photon
could be the neutral member of a Yang-Mills-type set of three gauge
particles required by treating isotopic spin as a gauge symmetry, with
the charged versions conveying the weak interaction and being
responsible for mediating the decay of neutrons. Why the charged
particles would have a huge mass while the photon was massless, he
had no idea. But, as I have often said, lack of understanding is neither
evidence for God, nor evidence that one is necessarily wrong. It just
is evidence of lack of understanding.
Schwinger was not only a brilliant physicist but a brilliant teacher
and mentor. While Feynman had few successful students, probably
ͥ͜͝
because none of them could keep up with him, Schwinger seemed to
have a knack for guiding brilliant PhD students. In his life he
supervised more than seventy PhDs, and four of his students later
won the Nobel Prize.
Schwinger was sufficiently interested in relating the weak
interaction to electromagnetism that he encouraged one of his
dozen graduate students at Harvard at the time to explore the issue.
Sheldon Glashow graduated in 1958 with a thesis on the subject and
continued to explore the issue for the next few years as a National
Science Foundation postdoctoral researcher in Copenhagen. In his
Nobel lecture twenty years later, Glashow indicated that he and
Schwinger had planned to write a manuscript on the subject after
Glashow graduated, but one of them lost the first draft of the
manuscript, and they never got back to it.
Glashow was no clone of Schwinger’s. Refined and brilliant, yes,
but also brash, playful, and boisterous, Glashow did research that
was not characterized by mathematical acrobatics, but rather by a
keen focus on physical puzzles and exploring new possible
symmetries of nature that might resolve them.
When I was a young graduate student in physics at MIT, I was
initially drawn to deep mathematical questions in physics and had
written my admissions essay for my PhD application on just this
subject. Within a few years I found myself depressed by the nature of
the mathematical investigations I was pursuing. I met Glashow at a
summer school for PhD students in Scotland and became friends
with both him and his family—a friendship that continued to
blossom when we later became colleagues at Harvard. The year after
we met, he spent a sabbatical year at MIT. During this important
time for me, when I was considering alternatives, he said to me,
“There’s physics, and there’s formalism, and you have to know the
difference.” Implicit in this advice was the suggestion that I should
ͥ͝͝
pursue physics. When I saw the fun he was having, it became easier
to consider joining in.
I soon realized that for me to make progress in physics I needed
to work on questions driven primarily by physical issues, not ones
driven primarily by mathematical issues. The only way I could do
that would be to keep in touch with ongoing experiments—and new
experimental results. By watching Shelly and how he did physics, I
realized that he had an uncanny ability to know which experiments
were interesting, and which results might be significant or might
point toward something new. Part of this was undoubtedly innate,
but part was based on a lifetime of keeping in touch with what was
happening on the ground. Physics is an empirical science, and we
lose touch with that at our peril.
In Copenhagen, Glashow realized that if he wanted to properly
implement Schwinger’s proposal to connect the weak interaction
with the electromagnetic interaction, then simply making the
photon be the neutral member of a triplet of gauge particles, with
the charged members becoming massive by some as yet unknown
miracle, wouldn’t fly. This couldn’t explain the proper nature of the
weak interaction, in particular the strange fact that the weak
interaction seemed to apply only to left-handed electrons (and
neutrinos), whereas electromagnetic interactions don’t depend on
whether the electrons are left- or right-handed.
The only solution to this problem would be if another neutral
gauge particle existed—in addition to the photon—which itself
coupled to only left-handed particles. But clearly the new neutral
particle would also have to be heavy since the interactions it
mediated would have to be weak as well.
Glashow’s ideas were reported to the physics community by
Murray Gell-Mann at the 1960 Rochester meeting, as Gell-Mann
had by then recruited Glashow to Caltech to work in Gell-Mann’s
ͥ͝͞
group. Glashow’s paper on the subject, submitted in 1960, appeared
in 1961 in print. Yet, no sudden stampede occurred in response.
After all, two fundamental problems remained with Glashow’s
proposal. The first was the long-familiar problem of how one could
have the different masses of the particles needed to convey the
different forces, when gauge symmetries required all the gauge
particles to be massless. Glashow simply stated in the introduction of
his paper, following in a long line of such hubris, “It is a stumbling
block we must overlook.”
The second problem was more subtle, but from an experimental
perspective equally severe. Neutron decay, pion decay, and muon
decay, if they were indeed mediated by some new particles
conveying the weak force, all appeared to require only the exchange
of new charged particles. No weak interaction had been observed
that would require the exchange of a new neutral particle. If such a
new neutral particle did exist, calculations at the time suggested it
would allow the other known heavier mesons that decayed into two
or three pions (and were responsible for the original confusion that
led to the discovery of parity violation) to decay much more rapidly
than they were observed to decay.
For these reasons, Glashow’s proposal drifted into the
background as physicists became entranced with the new particle
zoo that was emerging out of accelerators, and the concomitant
opportunity for new discoveries. Yet several of the key theoretical
ingredients needed to complete a revolution in fundamental physics
were in place, but it was far from obvious at the time. That within
slightly more than a decade after Glashow’s paper was published all
of the known forces in nature save gravity would be unveiled and
understood would have seemed like pure fantasy at the time.
And symmetry would be the key.
͝
ͥ͟
C h a p t e r 1 4
C O L D,
S TA R K
R E A L I T Y:
B R E A K I N G
B A D
O R
B E AU T I F U L ?
From whose womb has come the ice? And the frost of heaven,
who has given it birth?
—JOB 38:29
It is easy to pity the poor protagonists in Plato’s cave, who may
understand everything there is to know about the shadows on the
wall, except that they are shadows. But appearances can be
deceiving. What if the world around us is just a similar shadow of
reality?
Imagine, for example, that you wake up one cold winter morning
and look out your window, and the view is completely obscured by
beautiful ice crystals, forming strange patterns on the glass. It might
look like this:
Photograph by Helen Filatova
The beauty of the image is striking at least in part because of the
remarkable order on small scales lurking within the obvious
randomness on large scales. Ice crystals have grown gorgeous
treelike patterns, starting in random directions and bumping into
ͥ͝͠
each other at odd angles. The dichotomy between small-scale order
and large-scale randomness suggests that the universe would look
very different to tiny physicists or mathematicians confined to live
on the spine of one of the ice crystals in the image.
One direction in space, corresponding to the direction along the
spine of the ice crystal, would be special. The natural world would
appear to be oriented around that axis. Moreover, given the crystal
lattice structure, electric forces along the spine would appear to be
quite different from the forces perpendicular to it: the forces would
behave as if they were different forces.
If the physicist or mathematician living on the crystal was clever,
or, like the mathematician in Plato’s cave, lucky enough to leave the
crystal, it would soon become clear that the special direction that
governed the physics of the world they were used to was an illusion.
They would find, or surmise, that other crystals could point in many
other directions. Ultimately if they could observe the window from
the outside on large enough scales, the underlying symmetry of
nature under rotations in all directions, reflected in the growth of the
crystals in all directions, would become manifest.
Lawrence Krauss - The Greatest Story Ever Told--So Far Page 20
