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Matt's Particle Physics Column - Special Edition

By manobes in Science
Sat Aug 10, 2002 at 10:57:40 AM EST
Tags: Science (all tags)
Science

This is a special instalment of Matt's particle physics column, commemorating the life and work of Paul Adrian Maurice Dirac. The occasion is the centenary of Dirac's birth, which happened on August 8, the catalyst was another k5 article on the same subject. This short special edition will outline Dirac's contributions to modern physics, and relay a few famous stories about Dirac, to illuminate his unique personality. I will refrain from a full biography; readers who would like one can consult the the Nobel prize website.

Those unfamiliar with the terminology I use are urged to consult past editions of my column


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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)

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Poll
Greatest Physicist ever?
o Dirac 6%
o Newton 30%
o Einstein 20%
o Maxwell 12%
o Feynman 26%
o Landau 3%

Votes: 63
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Related Links
o another k5 article
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Matt's Particle Physics Column - Special Edition | 19 comments (9 topical, 10 editorial, 0 hidden)
Missing poll option (4.00 / 2) (#10)
by IHCOYC on Sat Aug 10, 2002 at 10:55:47 AM EST

Greatest physicist ever:

* Wile E. Coyote.
--
"Complecti antecessores tuos in spelŠis stygiis Tartari appara," eructavit miles primus.
"Vix dum basiavisti vicarium velocem Mortis," rediit Grignr.
--- Livy

particle man (2.00 / 3) (#11)
by Thinkit on Sat Aug 10, 2002 at 11:45:59 AM EST

good song.

Pais volume - and Kragh biography (none / 0) (#13)
by danny on Sun Aug 11, 2002 at 12:38:04 AM EST

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

I reviewed this... but it's not really a biography. Has anyone read the 1990 Kragh biography and would you recommend it?

Danny.
[900 book reviews and other stuff]

I love these articles (5.00 / 2) (#14)
by Shren on Mon Aug 12, 2002 at 03:19:31 PM EST

I rarely comment on the particle physics columns because my knowledge is lacking, but don't let the silence lead you to think you're unloved. I very much like the series.

I have no joke. (none / 0) (#15)
by epepke on Tue Aug 13, 2002 at 03:24:56 AM EST

But I'd just like to say that once I almost ran over P.A.M. Dirac with a '62 Chevy Impala near the crest of Chapel Drive in Tallahassee, FL.

But seriously, in addition to his incredible contributions to physics, he was also a nice man. Walked to work every day and always said "Hello," though little else.


The truth may be out there, but lies are inside your head.--Terry Pratchett


A few questions (none / 0) (#16)
by phliar on Wed Aug 14, 2002 at 09:46:44 PM EST

I hope that you can answer a couple of questions I had on reading your article.
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.
If I remember my physics, the Schrödinger equation is named Ψ (Psi) and gives you a complex number; |Ψ|2 (Psi-squared) gives you the probability distribution of a particle. When you say "the wavefunction was a single number," do you mean a single complex number or a single real? And is a Dirac spinor four reals? (Anything like a quaternion?)

...a special mathematical tool known as the Dirac delta function.

... take a classical physics theory, and turn it into a quantum one. This procedure is called "quantization".

I hope you'll say something more about these. (I know the definition of the Dirac delta, but I don't know how it's used in QM.) And what exactly is quantization? I thought you just start off by writing E = h (some-value) and go from there... like Einstein did in his explanation of the photo-electric effect, E = hν. Do Boltzmann and Planck get involved here at all? (I remember studying Boltzmann's explanation for black-body radiation in high-school: that harmonic oscillators' states were quantized, and how he didn't want to believe energy was quantized till the day he committed suicide.)

(Yes, I discovered recently that there are HTML entities for greek letters, both upper- and lower-case!)


Faster, faster, until the thrill of...

Answers... (none / 0) (#17)
by manobes on Wed Aug 14, 2002 at 10:14:20 PM EST

If I remember my physics, the Schr÷dinger equation is named Ψ (Psi) and gives you a complex number; |Ψ|2 (Psi-squared) gives you the probability distribution of a particle. When you say "the wavefunction was a single number," do you mean a single complex number or a single real?

Four complex numbers, with the Dirac spinor squared equal to 1 just like the Schrodinger wavefunction. That was the problem with the Klein-Gordon equation, the square of the wavefuntion could be less than one.

I hope you'll say something more about these. (I know the definition of the Dirac delta, but I don't know how it's used in QM.)

It would take me too long to list all the ways. The most obvious is for a position eigenstate. Say you have a particle, and you know for sure that at time T it's at point X. Then you use a delta function as the wavefunction, since you are 100% sure the particle is at point X.

And what exactly is quantization?

Erm, well, umm, it's a way of taking a classical theory and making it into a quantum one.

I'll assume you know what Hamiltonian mechanics is. If you write down a classical hamiltonian the variables are a set of coordinates X and a set of momenta P. Now in classical mechanics there's these things known as Poisson brackets. Take a look at the definition here. A and B are functions of the classical variables, X and P. Dirac's quantization idea was to tranlate any function of the classical vairables into a function of the quantum mechanical operators x and p (lower case can mean operators). Then, what he realized is that the commutators of the quantum theory, [a,b], would be -ihbar times the Poisson brackets of the classical theory. That is,

[a,b] = -ihbar {A,B}

Obviously this gives a prescription for going from the classical to the quantum theory. There's more to say, but first let me know that you followed that.

I discovered recently that there are HTML entities for greek letters, both upper- and lower-case

My turn for the question, how do you do that?


No one can defend creationism against the overwhelming scientific evidence of creationism. -- Big Sexxy Joe


[ Parent ]
Thanks! (none / 0) (#18)
by phliar on Thu Aug 15, 2002 at 01:33:02 PM EST

Thanks for all the pointers! I shall have to dig out the Bohm and crack it open again. (Or is there a book you'd recommend instead? I have a decent math background: graduate level analysis, abstract algebra etc.)

[HTML entities] My turn for the question, how do you do that?
If you use \TeX, same names but with ampersand and semicolon delimiters -- so upper-case theta, \Theta in \TeX, would be Θ in HTML.
ξ = A cos(θ), θ = ωt + α        !!!

Faster, faster, until the thrill of...
[ Parent ]

Well... (none / 0) (#19)
by manobes on Thu Aug 15, 2002 at 03:15:03 PM EST

Or is there a book you'd recommend instead?

If you haven't already done so read the first few chapters of Dirac's book, stunning stuff. For a modern perspective on quantum mechanics, try Sakurai's book (the one from the eights, not the mid sixties book, which is really a field theory textbook). Also there's a set of lecture notes by Schwinger, that was turned into a book (published posthumously by Springer). Schwinger was a master of the field, so those might be worth a look.

Unless you're real serious about relearning this stuff, I'd just read the first bit of Dirac. That's good stuff.

If you use \TeX, same names but with ampersand and semicolon delimiters -- so upper-case theta, \Theta in \TeX, would be Θ in HTML.

Σωεετ


No one can defend creationism against the overwhelming scientific evidence of creationism. -- Big Sexxy Joe


[ Parent ]
Matt's Particle Physics Column - Special Edition | 19 comments (9 topical, 10 editorial, 0 hidden)
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