Daedalus winter 2002
All too often,
we ignore goals, genres, or values,
or we assume that they are so apparent
that we do not bother to highlight them.
Yet judgments about whether
an exercise–a paper, a project,
an essay response on an examination
–has been done intelligently or stupidly
are often difficult for students to fathom.
And since these evaluations
are not well understood,
few if any lessons
can be drawn from them.
Laying out the criteria
by which judgments of quality are made
may not suffice in itself to improve quality,
but in the absence of such clarification,
we have little reason to expect our students
to go about their work intelligently.
Twentieth-century physics
began around 600 b.c.
when Pythagoras of Samos
proclaimed an awesome vision.
By studying the notes
sounded by plucked strings,
Pythagoras discovered that
the human perception of harmony
is connected to numerical ratios.
He examined strings
made of the same material,
having the same thickness,
and under the same tension,
but of different lengths.
Under these conditions,
he found that the notes
sound harmonious precisely
when the ratio of the lengths of string
can be expressed in small whole numbers.
For example, the length ratio 2:1
sounds a musical octave,
3:2 a musical fifth
and 4:3 a musical fourth.
The vision inspired by this discovery
is summed up in the maxim
“All Things are Number.”
This became the credo
of the Pythagorean Brotherhood,
a mixedsex society that combined
elements of an archaic religious cult
and a modern scientific academy.
The Brotherhood was responsible
for many fine discoveries,
all of which it attributed to Pythagoras.
Perhaps the most celebrated
and profound is the Pythagorean Theorem.
This theorem remains a staple
of introductory geometry courses.
It is also the point of departure
for the Riemann-Einstein theories
of curved space and gravity.
Unfortunately, this very theorem
undermined the Brotherhood’s credo.
Using the Pythagorean Theorem,
it is not hard to prove
that the ratio of the hypotenuse
of an isosceles right triangle
to either of its two shorter sides
cannot be expressed in whole numbers.
A member of the Brotherhood
who revealed this dreadful secret
drowned shortly afterwards,
in suspicious circumstances.
Today, when we say √2 is irrational,
our language still reflects these ancient anxieties.
Still, the Pythagorean vision,
broadly understood–and stripped of cultic,
if not entirely of mystical, trappings–
remained for centuries a touchstone
for pioneers of mathematical science.
Those working within this tradition
did not insist on whole numbers,
but continued to postulate
that the deep structure of the physical world
could be captured in purely conceptual constructions.
Considerations of symmetry and abstract geometry
were allowed to supplement simple numerics.
In the work of the German astronomer
Johannes Kepler (1570–1630),
this program reached a remarkable apotheosis–
only to unravel completely.
Students today still learn about
Kepler’s three laws of planetary motion.
But before formulating
these celebrated laws,
this great speculative thinker
had announced another law
–we can call it Kepler’s zeroth law–
of which we hear much less,
for the very good reason
that it is entirely wrong.
Yet it was his discovery
of the zeroth law
that fired Kepler’s enthusiasm
for planetary astronomy,
in particular for the Copernican system,
and launched his extraordinary career.
Kepler’s zeroth law
concerns the relative size
of the orbits of different planets.
To formulate it, we must imagine
that the planets are carried about
on concentric spheres around the Sun.
His law states
that the successive planetary spheres
are of such proportions
that they can be inscribed
within and circumscribed
about the five Platonic solids.
These five remarkable solids
–tetrahedron, cube, octahedron,
dodecahedron, icosahedron–
have faces that are
congruent equilateral polygons.
The Pythagoreans studied them,
Plato employed them in the
speculative cosmology of the Timaeus,
and Euclid climaxed his Elements
with the first known proof that
only five such regular polyhedra exist.
Kepler was enraptured by his discovery.
He imagined that the spheres
emitted music as they rotated,
and he even speculated on the tunes.
(This is the source of the phrase
“music of the spheres.”)
It was a beautiful realization
of the Pythagorean ideal.
Purely conceptual,
yet sensually appealing,
the zeroth law seemed
a production worthy
of a mathematically
sophisticated Creator.
To his great credit as an honest man
and–though the concept is anachronistic–
as a scientist, Kepler did not wallow
in mystic rapture, but actively strove
to see whether his law accurately matched reality.
He discovered that it does not.
In wrestling with the precise observations
of Tycho Brahe, Kepler was forced to give up
circular in favor of elliptical orbits.
He couldn’t salvage the ideas
that first inspired him.
After this, the Pythagorean vision
went into a long, deep eclipse.
In Newton’s classical synthesis
of motion and gravitation,
there is no sense in which structure
is governed by numerical
or conceptual constructs.
All is dynamics.
Newton’s laws inform us,
given the positions, velocities,
and masses of a system
of gravitating bodies at one time,
how they will move in the future.
They do not fix a unique size
or structure for the solar system.
Indeed, recent discoveries
of planetary systems around distant stars
have revealed quite different patterns.
The great developments
of nineteenthcentury physics,
epitomized in Maxwell’s
equations of electrodynamics,
brought many new phenomena
within the scope of physics,
but they did not alter
this situation essentially.
There is nothing
in the equations of classical physics
that can fix a de finite scale of size,
whether for planetary systems,
atoms, or anything else.
The world-system of classical physics
is divided between initial conditions
that can be assigned arbitrarily,
and dynamical equations.
In those equations,
neither whole numbers nor
any other purely conceptual elements
play a distinguished role.
Quantum mechanics changed everything.
Emblematic of the new physics,
and decisive historically,
was Niels Bohr’s atomic model of 1913.
Though it applies
in a vastly different domain,
Bohr’s model of the hydrogen atom
bears an uncanny resemblance
to Kepler’s system of planetary spheres.
The binding force is electrical
rather than gravitational,
the players are electrons
orbiting around protons
rather than planets
orbiting the Sun, and the size
is a factor 10ˆ-22 smaller
[10ˆ-22 = 0,0000000000000000000001];
but the leitmotif of Bohr’s model
is unmistakably “Things are Number.”
Through Bohr’s model, Kepler’s idea
that the orbits that occur in nature
are precisely those that embody
a conceptual ideal emerged
from its embers, reborn like a phoenix,
after three hundred years’ quiescence.
If anything, Bohr’s model
conforms more closely
to the Pythagorean ideal than Kepler’s,
since its preferred orbits
are defined by whole numbers
rather than geometric constructions.
Einstein responded with
great empathy and enthusiasm,
referring to Bohr’s work
as “the highest form of
musicality in the sphere of thought.”
Later work by Heisenberg and Schrödinger,
which defined modern quantum mechanics,
superseded Bohr’s model.
This account of subatomic matter
is less tangible than Bohr’s,
but ultimately much richer.
In the Heisenberg-Schrödinger theory,
electrons are no longer particles moving in space,
elements of reality that at a given time
are “just there and not anywhere else.”
Rather, they define oscillatory,
space-filling wave patterns
always “here, there, and everywhere.”
Electron waves are attracted
to a positively charged nucleus
and can form localized
standing wave patterns around it.
The mathematics describing
the vibratory patterns
that define the states of atoms
in quantum mechanics is identical
to that which describes
the resonance of musical instruments.
The stable states of atoms
correspond to pure tones.
I think it’s fair to say
that the musicality Einstein praised
in Bohr’s model is, if anything,
heightened in its progeny
(though Einstein himself, notoriously,
withheld his approval from
the new quantum mechanics).
The big difference
between nature’s instruments
and those of human construction
is that her designs depend
not on craftsmanship
refined by experience,
but rather on the ruthlessly
precise application of simple rules.
Now if you browse through
a textbook on atomic quantum mechanics,
or look at atomic vibration patterns
using modern visualization tools,
“simple” might not be the word
that leaps to mind.
But it has a precise,
objective meaning in this context.
A theory is simpler
the fewer nonconceptual elements,
which must be taken from observation,
enter into its construction.
In this sense,
Kepler’s zeroth law
provided a simpler
(as it turns out, too simple)
theory of the solar system
than Newton’s, because
in Newton’s theory
the relative sizes of planetary orbits
must be taken from observation,
whereas in Kepler’s
they are determined conceptually.
From this perspective,
modern atomic theory
is extraordinarily simple.
The Schrödinger equation,
which governs electrons in atoms,
contains just two nonconceptual quantities.
These are the mass of the electron
and the so-called fine-structure constant,
denoted alpha, that specifies the overall strength
of the electromagnetic interaction.
By solving this one equation,
finding the vibrations it supports,
we make a concept-world that reproduces
a tremendous wealth of realworld data,
notably the accurately measured spectral lines
of atoms that encode their inner structure.
The marvelous theory of electrons
and their interactions with light
is called quantum electrodynamics, or QED.
In the initial modeling of atoms,
the focus was on their accessible,
outlying parts, the electron clouds.
The nuclei of atoms,
which contain most of their mass
and all of their positive charge,
were treated as so many tiny
(but very heavy!) black boxes,
buried in the core.
There was no theory
for the values of nuclear masses
or their other properties; these
were simply taken from experiment.
That pragmatic approach
was extremely fruitful
and to this day provides
the working basis
for practical applications
of physics in chemistry,
materials science, and biology.
But it failed to provide a theory
that was in our sense simple,
and so it left the ultimate ambitions
of a Pythagorean physics unfulfilled.
Starting in the early 1930s,
with electrons under control,
the frontier of fundamental physics
moved inward, to the nuclei.
This is not the occasion
to recount the complex history
of the heroic constructions
and ingenious deductions that at last,
after fifty years of strenuous international effort,
fully exposed the secrets of this inaccessible domain.
Fortunately, the answer is easier to describe,
and it advances and consummates our theme.
The theory that governs atomic nuclei
is quantum chromodynamics, or QCD.
As its name hints, QCD
is firmly based on quantum mechanics.
Its mathematical basis
is a direct generalization of QED,
incorporating a more intricate structure
supporting enhanced symmetry.
Metaphorically, QCD stands to QED
as an icosahedron stands to a triangle.
The basic players in QCD
are quarks and gluons.
For constructing an accurate model
of ordinary matter just two kinds of quarks,
called up and down or simply u and d,
need to be considered.
(Thereare four other kinds, at least,
but they are highly unstable
and not important for ordinary matter.)
Protons, neutrons, π mesons,
and a vast zoo of very shortlived particles
called resonances are constructed
from these building blocks.
The particles and resonances
observed in the real word
match the resonant wave patterns
of quarks and gluons
in the concept-world of QCD,
much as states of atoms match
the resonant wave patterns of electrons.
You can predict their masses and properties
directly by solving the equations.
A peculiar feature of QCD,
and a major reason
why it was hard to discover,
is that the quarks and gluons
are never found in isolation,
but always in complex associations.
QCD actually predicts
this “confinement” property,
but that’s not easy to prove.
Considering how much it accounts for,
QCD is an amazingly simple theory,
in our objective sense.
Its equations contain
just three nonconceptual ingredients:
the masses of the u and d quarks
and the strong coupling constant alpha s (for strong),
analogous to the fine structure constant of QED,
which specifies how powerfully
quarks couple to gluons.
The gluons are automatically massless.
Actually even three is an overestimate.
The quark-gluon coupling
varies with distance, so we can
trade it in for a unit of distance.
In other words,
mutant QCDs with different values
of as generate concept-worlds
that behave identically,
but use different-sized metersticks.
Also, the masses of the u and d quarks
turn out not to be very important, quantitatively.
Most of the mass of strongly
interacting particles
is due to the pure energy
of the moving quarks
and gluons they contain,
according to the converse
of Einstein’s equation, m = E/cˆ2.
The masses of the u and d quarks
are much smaller than the masses
of the protons and other particles
that contain them.
Putting all this together, we arrive
at a most remarkable conclusion.
To the extent that we are willing
to use the proton itself as a meterstick,
and ignore the small corrections
due to the u and d quark masses,
QCD becomes a theory with
no nonconceptual elements whatsoever.
Let me summarize.
Starting with precisely
four numerical ingredients,
which must be taken from experiment,
QED and QCD cook up a concept-world
of mathematical objects whose behavior
matches, with remarkable accuracy,
the behavior of real-world matter.
These objects are vibratory wave patterns.
Stable elements of reality–protons,
atomic nuclei, atoms–correspond,
not just metaphorically
but with mathematical precision,
to pure tones.
Kepler would be pleased.
This tale continues in several directions.
Given two more ingredients,
Newton’s constant and Fermi’s constant,
which parametrize the strength
of gravity and of the weak interaction, respectively,
we can expand our conceptworld
beyond ordinary matter
to describe virtually all of astrophysics.
There is a brilliant series of ideas
involving unified field theories
and supersymmetry
that might allow us to get
by with just five ingredients.
(Once you’re down to so few,
each further reduction marks an epoch.)
These ideas will be tested decisively
in coming years, especially as the
Large Hadron Collider (LHC) at CERN,
near Geneva, swings into operation around 2007.
On the other hand, if we attempt to do
justice to the properties of many exotic,
short-lived particles discovered
at highenergy accelerators, things get
much more complicated and unsatisfactory.
We have to add pinches
of many new ingredients to our recipe,
until it may seem that rather
than deriving a wealth of insight
from a small investment of facts,
we are doing just the opposite.
That’s the state of our knowledge
of fundamental physics today
–simultaneously triumphant,
exciting, and a mess.
The last word I leave to Einstein:
I would like to state a theorem
which at present can not be based
upon anything more than upon
a faith in the simplicity,
i.e., intelligibility, of nature:
there are no arbitrary constants . . .
that is to say, nature is so constituted
that it is possible logically to lay down
such strongly determined laws
that within these laws only rationally
completely determined constants occur
(not constants, therefore,
whose numerical value
could be changed
without destroying the theory).
_____
Frank Wilczek,
Herman Feshbach Professor of Physics at MIT,
is known, among other things,
for the discovery of asymptotic freedom,
the development of quantum chromodynamics,
the invention of axions, and the discovery
and exploitation of new forms of quantum statistics (anyons).
When only twenty-one years old
and a graduate student at Princeton University,
he and David Gross defined the properties of gluons,
which hold atomic nuclei together.
He has been a Fellow of the
American Academy since 1993.
Two years later from the writing
of this article Gross, Wilczek
and H. David Politzer won
the Nobel Prize in Physics
for the discovery of asymptotic freedom.
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