**Dirac's Contributions to Modern Physics**
Paul Adrien Maurice Dirac ranks among the greatest physicists of all time.
His successor at Cambridge,
Steven Hawking has said
"Dirac has done more than anyone this century, with the exception of Einstein,
to advance physics and change our picture of the universe".
However, his achievements, unlike Einstein's, are not well known to the general public.

Dirac's major contributions to physics began in 1925, with a paper on quantum mechanics.
1925 was the year that Heisenberg had developed quantum mechanics, in a form
now known as matrix mechanics. A matrix is the mathematical name for
an array of numbers. A fundamental feature of Heisenberg's quantum mechanics
was that observable quantities were the elements of matrices. And the particular
element depended upon how the experiment was prepared.

Dirac's paper highlighted another important property of this new physical theory,
namely that for matrices the commutative rule of multiplication
does not hold. That is for ordinary real numbers a*b = b*a but for matrices
this does not necessarily hold. Dirac's 1925 paper contained
an example of this where he showed that the matrix describing the position
of a quantum mechanical particle, X, and that describing its momentum
P, do not obey this commutative law (this was independently derived by
Born and Jordan). This failure to commute is the essence of another
famous piece of physics, the Heisenberg Uncertainty Principle.

Dirac's next major piece of work appeared alongside his PhD thesis in
1926. It was the first paper to apply the new quantum mechanics to
the field of statistical mechanics (the study of the behavior of large
numbers of particles). It was Dirac who clarified the different
statistical behaviors of particles with integer quantum spin, and
those with half-integer quantum spin. The former are said to obey
Bose-Einstein statistics, the latter Fermi-Dirac statistics.

This paper made use of the wave mechanical formulation of quantum
mechanics put forth by Schrodinger. In this formulation systems
of many particles are described by a single function, the
wavefunction, which we'll call F. This function depends
on the positions of all the particles, which we
notate F(x1,x2,x3,...). Dirac's paper showed that
the statistic properties of a system were related to the behavior of
this function under interchange of two particles. Say for example
we swapped particles one and three, that system would be
described by the function F(x3,x2,x1,.....). If this is the
same as the original function, then the particles obey the statistics
of Bose and Einstein (derived earlier using an early form of quantum theory).
If the swapped function is equal to minus the original one, the the particles obey
Fermi-Dirac statistics. This distinction is crucial to virtually all
of modern physics, it was Dirac who first realized it.

Immediately following his PhD Dirac did the work for which he would
earn the Nobel prize (shared jointly with Schrodinger). This
was his own general theory of Quantum Mechanics, known as the transformation
theory. Dirac's formulation was far more general than either Heisenberg's
or Schroedinger's, indeed one of the triumphs of this formulation
was showing how to relate the two less general forms to his own.
Although most introductions to quantum mechanics
start with the Schrodinger theory, it is the transformation
theory of Dirac that is used in virtually all modern work.

Dirac's work on transformation theory also introduced several
new concepts into quantum mechanics. The first, a technical device,
was a special mathematical tool known as the Dirac delta function.
The use of this function greatly simplifies many quantum mechanical
calculations. Along with this, the transformation theory showed
the way to take a classical physics theory, and turn it into
a quantum one. This procedure is called "quantization".

Quantization of the classical theory of electromagnetism was the
next subject that Dirac approached. In a pair of 1927 papers he showed
how to apply his quantization rules to Maxwell's theory of electromagnetism.
This work put the early work of Einstein on light quanta on much
firmer theoretical grounds. Along with his theory of electrons, this work
forms the foundation for modern quantum field theory, which underlies all
of our understanding of particle physics as well as much of our understanding
of condensed matter physics.

Dirac's most famous achievement was his theory of
electrons, summarized in the
equation which bears his name.
With the publication of his transformation theory, Dirac had completed the theory of the quantum mechanics of non-relativistic
particles. That is, the quantum mechanics of Heisenberg, Schrodinger and Dirac only applied to situations
where the particles were not moving quickly compared to the speed of light. A major open problem was to
unite quantum mechanics with Einstein's theory of Special relativity, in order to describe particles
traveling near the speed of light. A first attempt to do this had been made, independently,
by a few different people, however the equation that was derived (known now as the
Klein-Gordon equation) had a serious mathematical flaw. Among other things the
equation gave a way to predict the chance that a particle would be found in a certain
place. Unfortunately one could make it so the probability for a particle to
be found **anywhere**, which should be 100%, came out to be less
than 100%. This serious problem disturbed Dirac, and he set about to eliminate
it.

Dirac quickly identified the problem with the Klein-Gordon equation, and, by
insisting that his theory not have it, arrived at his quantum mechanical
equation for relativistic particles. Unfortunately in order to make his
equation work, Dirac was forced to generalize the wavefunctions of
Schrodinger. In the Schrodinger theory, the wavefunction was a
single number; input some coordinates; output
one single number: the wavefunction at those coordinates. The analogous
functions in Dirac's new theory (which were the solutions to his equation)
needed **four** numbers instead of one. Rather than
functions, these are now called Dirac Spinors (or just spinors) and the
four numbers are called the components of the spinor.

Dirac quickly identified the meaning of two of these four numbers. He looked
at his theory in the limiting case of very slow speeds, where he
knew it had to reproduce the Schrodinger theory. In this limit
two of the four components got very very small and could be ignored.
The equation that was left over was the original non-relativistic equation
of Schroedinger's, with a term due to Wolfgang Pauli. This equation
was known to describe slowly moving electrons, which had a
quantum mechanical spin of one half unit. Of the
two spinor components left in this limit, the first gave
the chance that the electron had "up" spin (think clockwise rotation)
and the second gave the chance that the electron had "down" spin.

When coupled to an electromagnetic field, the non-relativistic reduction of
Dirac's equation provided even more new things. It cleared up tiny
discrepancies between the theory of atomic spectral lines and the
measurements of them. It also successfully predicted the existence of an interaction
of the electron with the magnetic field that had previously been put into
the Schrodinger theory without good justification.

In this low speed limit, Dirac's theory was a huge triumph, however, the
fully relativistic theory still had problems. In this regime all
four spinor components were there. To make matters worse, the equation
appeared to predict that the two components that went away at low speeds
corresponded to a particles with an energy that was **negative**.

This would truly have been a disaster. If these negative energy states
existed then matter ought not to be stable. Typically systems of
particles go to a configuration which has the lowest possible energy.
In this case a collection of electrons could all go to negative energies by
emitting photons. Obviously this was a major problem since electrons are observed
to have positive energy.

Dirac's solution to this problem was inspired. Dirac supposed that all of the
negative energy states were already filled with electrons. That is, there's a massive
"sea" of filled states, everywhere. Because electrons obeyed the Pauli exclusion
principle, regular positive energy ones could not fall into already occupied
energy levels. So, Dirac reasoned, if all the energy levels were
filled, there would be no more problems with negative energy states.

This was not quite the end of the story though. If one considered interactions
with the electromagnetic field it was possible for a high energy photon to "kick"
one of the negative energy states hard enough that it's energy became
positive. For a short time, there would then be a hole in the sea of
filled states. This hole, Dirac found, would look like a positive energy
particle of the
same mass as the electron, but with opposite electrical charge.
At first, and for almost a year, Dirac tried to find an
alternative physical interpretation for this state. At one point
he thought it might be the proton, despite the huge mass difference. But
in May 1931 Dirac finally declared that this "hole" state would appear as
a new type of particle, the positron, or anti particle of the electron.
In this way was antimatter predicted. In late 1931 Carl Anderson discovered
evidence for the positron in cosmic ray photographs, and Dirac's prediction was
vindicated.

From 1931 until 1936 Dirac made many more fundamental advances.
His primary focus was to unify his electron theory with the quantum theory
of the electromagnetic field. The resulting Quantum Electrodynamics (QED)
is still with us today. Unfortunately QED had mathematical problems
which were not fully resolved until the late 1940's; however Dirac played a huge
role in formulating the theory and highlighting it's problems.

Following 1936, Dirac's scientific output began a slow, but steady, decline. He continued
to publish many papers though, and did a lot of work in many fields, but his days of making
revolutionary contributions were over.
In his later years he returned to the problems of QED time and time again. He was convinced that they
stemmed from problems in the classical, Maxwell, theory of electromagnetism.
His researches in that area, while not fixing QED in the way he would have liked,
illuminated the limits of the classical theory in new ways.

One further aspect of Dirac's career deserves special notice. His textbooks, particularly
The Principles of Quantum Mechanics
are generally considered among the best.

**A Few Dirac Stories**
Even among eccentric physicists, Paul Dirac was considered a bit odd. His most
obvious trait was his silence, he was a man of very few words. It appears that
this was primarily caused by his childhood, he once said:

My father made the rule that I should only talk to him in French. He thought that
it would be good for me to learn French in that way. Since I found that I couldn't
express myself in French, it was better for me to stay silent than to talk in English.
So I became very silent at that time - that started very early.

It appears that Dirac's childhood was not a happy one, at the time he
received the Nobel prize in 1933 he was not speaking with his father. When
his father died in 1936 he wrote to his wife "I feel much freer now".
Dirac occasionally revealed doubts about his work. About discovery in general he
wrote "Hopes are always accompanied by fears, and, in scientific research, the fears
are liable to become dominant". He was also dismissive of the quality of his
own work, shortly before his death (in 1984) in response to an invitation to
give a talk, he said "No! I have nothing to talk about. My life has
been a failure...". Clearly history does not share that judgment.
It should be noted that Dirac was a very private person, so negative comments
like these were quite rare. And he spoke very favorably of the
transformation theory of quantum mechanics, calling it "my darling".

A humorous Dirac story was told by Heisenberg, concerning their joint 1929 trip to
Japan. Heisenberg writes:

I liked to take part in the social life on the steamer and, so, for instance, I took
part in the dances in the evening. Paul, somehow, didn't like that too
much but he would sit in a chair and look at the dances. Once I came back from a dance
and took the chair beside him and he asked me, "Heisenberg, why do you dance?" I said,
"Well, when there are nice girls it is a pleasure to dance". He thought for a long
time about it, and after about five minutes he said "Heisenberg, how do you
know *beforehand* that the girls are nice?"

Other famous Dirac stories reflect this same love of precision in language. George
Gamow relates several, including one on classic literature

His friend, Peter Kapitza, the Russian physicist, gave him an English translation
of Dostoevsky's *Crime and Punishment*.
"Well, how do you like it?" asked
Kapitza when Dirac returned the book.

"It is nice," said Dirac, "but in one of the chapters the author made a mistake. He
describes the Sun as rising twice on the same day". This was his one and only comment
on the novel.

Paul Dirac was loved by all who knew him, he approached physics with single-minded
devotion, and achieved in the field what few others ever have. Neils Bohr once said "of all the physicists,
Dirac has the purest soul". His insights into quantum mechanics deeply influenced all subsequent thinking
on the subject, and his work on quantum field theory laid the ground for the entirety of
the standard model of particle physics. On a personal note, it is Dirac, above all others, that I would
regard as a role model. His writings on quantum mechanics made the subject clear to me for the first time,
the fields he opened up have provided many years of intellectual joy, and
his passion for physics and clarity of thought,
provide examples the purist dedication to physics.

Paul Adrian Maurice Dirac died on October 20, 1984, at the age of 82. On November 13th, 1995 a plaque
was laid in Westminster Abby, alongside the grave of Newton, commemorating his contributions to physics.

**References**

Abraham Pais, in *Paul Dirac The Man and his Work*, Peter Goddard (ed) Cambridge University Press (1998)

George Gamow, *Thirty Years that Shook Physics*, Dover reprint (1985)