Paul Dirac published the first of his papers on
"The Quantum Theory of the Electron" seventy years ago this month. The
Dirac equation, derived in those papers, is one of the most important
equations in physics.
Each day, I walk past the road where Paul Adrien Maurice Dirac lived as
a child. It is pleasant to have even this tenuous association with one
of the greatest intellects of the 20th century. Paul Dirac was born at
15 Monk Road in Bishopston, Bristol, on 8 August 1902, and educated at
the nearby Bishop Road Primary School. The family later moved to Cotham
Road, near the University of Bristol, and in 1914 the young Dirac
joined Cotham Grammar School, formerly the Merchant Venturers.
Dirac was a student at Bristol University between 1918 and
1923, first in electrical engineering and then in applied mathematics.
Much later, he said: "I owe a lot to my engineering training because it
[taught] me to tolerate approximations. Previously to that I
thought...one should just concentrate on exact equations all the time.
Then I got the idea that in the actual world all our equations are only
approximate. We must just tend to greater and greater accuracy. In
spite of the equations being approximate, they can be beautiful."
Because Dirac was a quiet man - famously quiet, indeed - he is
not well known outside physics, although this is slowly changing. In
1995 a plaque to Dirac was unveiled at Westminster Abbey in London and
last year Institute of Physics Publishing, which is based in Bristol,
named its new building Dirac House.
It is hard to give the flavour of Dirac's achievements in a
non-technical article, because his work was so mathematical. He once
said: "A great deal of my work is just playing with equations and
seeing what they give."
Early days
When Dirac went to Cambridge in 1923, the physics of matter on the
smallest scales - in those days this was the physics of the atom - was
in ferment. It had been known for more than a decade that the old
mechanics of Newton - "classical" mechanics, as it came to be called -
does not apply in the microscopic world. In particular, evidence from
the light coming out of atoms seemed to indicate that some quantities
that in classical mechanics can take any values are actually restricted
to a set of particular values: they are "quantized". One of these
quantities is the energy of the electrons in an atom. This was strange
and shocking. Imagine being told that when your car accelerates from 0
to 70 miles per hour it does so in a series of jumps from one speed to
another (say in steps of one thousandth of a mph), with the
intermediate speeds simply not existing. It did not make sense, and yet
observations seemed to demand such an interpretation.
In the first attempts at a theoretical understanding,
physicists tried to find the general rules for imposing these
restrictions on classical mechanics - that is rules for quantization.
It seemed that in order to quantize, it was necessary first to identify
those quantities that do not change when their environment is slowly
altered. If a pendulum is slowly shortened, for example, it swings
farther and also faster, in such a way that its energy divided by its
frequency stays constant. These rules worked for simple atoms and
molecules but failed for complicated ones.
Dirac entered physics at the end of this baroque period. One of
his first papers was an attempt at a general theory of these unchanging
quantities. This is a delicate problem in classical mechanics, not
solved even now. It is amazing today to read that paper. In its
mathematics it is quite unlike any of Dirac's later works (for example,
he brings in fine differences between rational and irrational numbers),
and "pre-invents" techniques developed by other people only decades
later. (I say pre-invents because the paper was forgotten until
recently.)
At this time the situation in atomic physics resembled that at the
end of the 16th century, when the old Earth-centred astronomy had to be
made ever more elaborate in the face of more accurate observations. The
difficulties of the 16th and 20th centuries were resolved in the same
way: by a complete shift of thought. In atomic physics this happened
suddenly, in 1925, with the discovery by Heisenberg of quantum
mechanics. This seemed to throw out classical mechanics completely,
though it was built in as a limiting case to ensure that, on larger
scales, the new mechanics agreed with more familiar experience. The
quantum rules emerged automatically, but from a mathematical framework
that was peculiar. For example, it involved multiplication where the
result depends on the order in which the multiplication is done. It is
as though 2 multiplied by 3 is different from 3 multiplied by 2.
Heisenberg found this ugly and unsatisfactory. Dirac disagreed, and
just a few months after Heisenberg he published the first of a series
of papers in which quantum mechanics took the definitive form we still
use today.
The main idea is that the multiplied objects - objects that
represent variables we can measure in experiments - should be thought
of as operations. An experiment is an operation, of course, even though
its result is a number. With this interpretation, it is not surprising
that the order matters: we all know that putting on our socks and then
our shoes gives a result different from putting on our shoes and then
our socks. Dirac found the one simple rule by which a multiplied by b differed from b multiplied by a, and from which the whole of quantum mechanics follows.
The same unification was soon found to include Schrödinger's way of
doing quantum mechanics, where the state of a system is represented by
a wave whose strength gives the probabilities of the different possible
results of measurements on it. For a while this seemed completely
different from the framework that Heisenberg had used, but it quickly
emerged that in fact each represents Dirac's operators in a different
way. It seemed miraculous.
The Dirac equation
Although brilliant - in Einstein's words, "the most logically
perfect presentation of quantum mechanics" - this was a reformulation
of physics that had, admittedly only just, been discovered. Dirac's
main contribution came several years later, when (still in his
mid-twenties) he made his most spectacular discovery.
Before quantum mechanics, there had been another revolution in
physics, with Einstein's discovery in 1905 that Newton's mechanics
fails for matter moving at speeds approaching that of light. To get
things right, time had to be regarded as no longer absolute:
before-and-after had to be incorporated as a fourth co-ordinate like
the familiar three spatial co-ordinates that describe side-to-side,
forward-and-backward and up-and-down. Just as what is side-to-side and
what is forward-and-backward change when you turn, so time gets mixed
in with the other three co-ordinates when you move fast. Now, in the
1920s, came quantum mechanics, showing how Newton's mechanics failed in
a different way: on microscopic scales. The question arose: what is the
physics of particles that are at the same time small and moving fast?
This was a practical question: the electrons in atoms are small, and
they move fast enough for the new quantum mechanics to be slightly
inaccurate, since it had been constructed to have as its large-scale
limit Newton's mechanics rather than Einstein's. From the start people
tried to construct a quantum theory concordant with relativity, but
failed to overcome technical obstructions: in particular, their
attempts gave probabilities that were negative numbers - something that
is nonsense, at least in the usual meaning of probability. The question
boiled down to this: what are the right sort of quantum waves
describing electrons? And what is the wave equation that governs the
dynamics of these waves, while satisfying the requirements of
relativity and giving sensible physical predictions?
Dirac's construction of his wave equation for the electron - published in two papers in the Proceedings of the Royal Society (London)
in February and March 1928 - contained one of those outrageous leaps of
imagination shared by all great advances in thought. He showed that the
simplest wave satisfying the requirements was not a simple number but
had four components (see below).
This seemed like a complication, especially to minds still reeling from
the unfamiliarity of the "ordinary" quantum mechanics. Four components!
Why should anybody take Dirac's theory seriously?
First, and above all for Dirac, the logic that led to the
theory was, although deeply sophisticated, in a sense beautifully
simple. Much later, when someone asked him (as many must have done
before) "How did you find the Dirac equation?" he is said to have
replied: "I found it beautiful." Second, it agreed with precise
measurements of the energies of light emitted from atoms, in
particularly where these differed from ordinary (non-relativistic)
quantum mechanics.
There are two more reasons why the Dirac equation was
compelling as the correct description of electrons. To understand them,
you should realize that any great physical theory gives back more than
is put into it, in the sense that as well as solving the problem that
inspired its construction, it explains more and predicts new things.
Before the Dirac equation, it was known that the electron spins. The
spin is tiny on the scale of everyday but is always the same and plays
a central part in the explanation through quantum mechanics of the
rules of chemistry and the structure of matter. This spin was a
property of the electron, like its mass and its electric charge, whose
existence simply had to be assumed before quantum mechanics could be
applied. In Dirac's equation, spin did not have to be imported: it
emerged - along with the magnetism of the electron - as an inevitable
property of an electron that was both a quantum particle and a
relativistic one.
So, electron spin was the third reason for believing Dirac's
mathematically inspired equation. The fourth came from a consequence of
the equation that was puzzling for a few years at first. Related to its
four components was the fact that any solution of the equation where
the electron had a positive energy had a counterpart where the energy
was negative. It gradually became clear that these counterpart
solutions could be interpreted as representing a new particle, similar
to the electron but with positive rather than negative charge; Dirac
called it an "anti-electron", but it soon came to be known as the
positron. If an electron encounters a positron, Dirac predicted, the
two charges cancel and the pair annihilates, with the combined mass
transforming into radiation in the most dramatic expression of
Einstein's celebrated equation E = mc2.
Thus was antimatter predicted. When the positron was discovered by
Anderson in 1932, Dirac's immortality was assured. Dirac and
Schrödinger shared the Nobel Prize for Physics in 1933.
Nowadays, positrons are used every day in medicine, in PET
(positron emission tomography) scanners that pinpoint interesting
places in the brain (e.g. places where drugs are chemically active).
These work by detecting the radiation as the positrons emitted from
radioactive nuclei annihilate with ordinary electrons nearby.
The Dirac equation
Other achievements
Having explained spin, it was natural for Dirac to try to explain
electric charge, and in particular the mysterious fact that it is
quantized: all charges found in nature are multiples of the charge on
the electron. In classical electricity, there is no basis for this:
charges can have any value.
The 1927 Solvay Congress
In
1931 Dirac gave a solution of this problem in an application of quantum
mechanics so original that it still astounds us to read it today. He
combined electricity with magnetism, in a return to the 18th-century
notion of a magnet being a combination of north and south magnetic
poles (magnetic charges), in the same way that a charged body contains
positive and negative electric charges. That symmetry was lost in the
19th century with the discoveries of Oersted, Ampère and Faraday,
culminating in Maxwell's synthesis of all electromagnetic and - in
another example of getting out more than you put in - optical
phenomena. In its place came a greater simplicity: there are only
electric charges, whose movement generates magnetism (and now the
motive power for much of our civilisation). The absence of isolated
magnetic poles - magnetic monopoles - was built into classical
electromagnetism, and also the quantum mechanics that grew out of it.
Dirac wondered if there was any way that magnetic monopoles
could be brought into quantum physics without spoiling everything that
had grown out of assuming that they did not exist. He found that this
could be done, but only if the strength of the monopole (the "magnetic
charge") was linked to that of the electric charge, and if both were
quantized. This solved the original problem: for consistency with
quantum mechanics, the existence of even one monopole anywhere in the
universe would suffice to ensure that electric charge must be
quantized. The implication is compelling: to account for the
quantization of electricity, magnetic poles must exist. After this,
Pauli referred to Dirac as "Monopoleon".
Alas, no magnetic monopole has ever been found. Perhaps they do
not exist, or perhaps (and there are hints of this in the theory)
positive and negative monopoles are so tightly bound together that they
have not been separated. Much later, Dirac referred to this theory as
"just a disappointment". However, the mathematics he invented to study
the monopole - combining geometry with analysis - now forms the basis
of the modern theories of fundamental particles.
There were two other seminal contributions to physics in those
early years. I have space only to mention them. Dirac applied quantum
mechanics to the way light and matter interact. This made him realize
that it was necessary to quantize not only particles but the
electromagnetic field itself, and led him to the first consistent
theory of photons (which had been discovered several decades previously
in the beginnings of quantum mechanics). This led to the elaborate and
thriving quantum field theories of today.
Dirac also showed how quantum waves for many electrons had to
be constructed, incorporating the philosophically intriguing fact that
any two of these particles are absolutely identical and so cannot be
distinguished in any way. This produced the definitive understanding of
earlier rules about how quantum mechanics explains the periodic table
of the elements, and provided the basis for the theory of metals and
the interior of stars.
Like all scientists at the highest level, Dirac was not afraid
to descend from the pinnacle and discuss more down-to-earth matters.
Here are two examples. Much of our knowledge comes from light scattered
by matter; in particular, that is how we see. In a clever stroke of
lateral thinking, Dirac realized that the quantum symmetry between
waves of light and waves of matter implied that it is also possible for
material particles to be scattered by light, a ghostly possibility that
could be observed, as he showed in 1933 in a paper with Peter Kapitza.
This was observed for the first time about ten years ago and the
manipulation of atoms by laser beams is now a thriving area of applied
quantum mechanics - a fact recognized with a Nobel prize last year (Physics World November 1997 p51, print version).
The second example is his Second World War work. In the Manhattan
Project to develop the first nuclear bombs, it was necessary to
separate isotopes of uranium. One class of methods involved the
centrifugal effects of fluid streams that were made to bend. Dirac put
the theory of these techniques on a firm basis, and indeed his work in
this field has been described as seminal.
Dirac stories
It is not my intention to write about what sort of person Dirac was.
But I must mention the genre of "Dirac stories". He was so unusual in
the logic and precision of his interaction with the world, both in and
out of physics, that tales have become attached to him and have
acquired a life of their own. I suppose it matters to a historian
whether they are true or apocryphal (or as Norman Mailer says,
"factoids"), but to us they have a deeper resonance that transcends
fact. Resisting temptation, I retell just two less well known ones.
Like many scientists, Dirac was known to sleep during (other
people's) lectures, and then wake and suddenly make a penetrating
remark. Once, a speaker stopped, scratched his head and declared: "Here
is a minus where there should be a plus. I seem to have made an error
of sign." Dirac opened one eye and said: "Or an odd number of them."
Another time, Dirac was at a meeting in a castle, when another guest
remarked that a certain room was haunted: at midnight, a ghost
appeared. In his only reported utterance on matters paranormal, Dirac
asked: "Is that midnight Greenwich time, or daylight saving time?"
Dirac's writing was famous for its clarity and simplicity. Every physicist knows his Principles of Quantum Mechanics
- such a perfect and complete summary of his views that in later years
his lectures consisted of readings from it. There is the story that he
was once present when Niels Bohr was writing a scientific paper - with
many hesitations and redraftings, as was his custom. Bohr stopped: "I
do not know how to finish this sentence." Dirac replied: "I was taught
at school that you should never start a sentence without knowing the
end of it."
Many physicists have spoken of Dirac with awe. John Wheeler,
referring to the sharp light of his intelligence, said "Dirac casts no
penumbra." Niels Bohr said: "Of all physicists, Dirac has the purest
soul." He is also reported as saying (I cannot now find this
quotation): "Dirac did not have a trivial bone in his body."
The mathematician Mark Kac divided geniuses into two classes.
There are the ordinary geniuses, whose achievements one imagines other
people might emulate, with enormous hard work and a bit of luck. Then
there are the magicians, whose inventions are so astounding, so counter
to all the intuitions of their colleagues, that it is hard to see how
any human could have imagined them. Dirac was a magician.
Sir Michael Berry
is Royal Society Research Professor at the H H Wills Physics
Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK.
This is a version of a talk delivered in September 1997 at the official
opening of Dirac House, the headquarters of Institute of Physics
Publishing.
버
뮤다 삼각지대는 미국 남부에 위치한 플로리다 해협과 버뮤다섬, 푸에르토리코(혹은 아조레스 제도)를 잇는 삼각형 범위 안의 해역을
가리킨다. 지난 500년간 이 지역에서 일어난 선박과 항공기 실종 사고는 수백 건에 이르지만, 그 원인이 명확히 밝혀지지 않아
미스터리 실종 사건으로 기록되어 왔다.
