Beautiful Science: The Other Einstein

This post is in remembrance of Paul A.M. Dirac who died this day, 8 August, 1984

A genius at work: Physics Nobel laureate, Paul A.M. Dirac in his thirties. Dirac died on this day, 8 August 1984, leaving a legacy of breathtaking achievements in theoretical physics.

Paul Dirac: The Man

Physicists and mathematicians are not the most talkative of people, but the British scientist, Paul Dirac is undoubtedly the “king of quiet” within this tribe. Here was a man who spoke so few words, and uttered each word so slowly and with utmost audible consideration, that to be able to hear him speak is a treat in itself. In contrast to his reticence, Dirac’s scientific career was voluminous, a productivity he maintained well into old age. His contributions to physics extend from quantum mechanics – for which he jointly won the 1933 Nobel Prize in Physics (being one of the youngest to do so) – to developing Einstein’s theories of relativity, and the discovery of antimatter. Among British physicists, Dirac’s contributions are thought to be some of the greatest since Sir Isaac Newton. The late Cambridge physicist, Stephen Hawking once said that: ‘Dirac has done more than anyone this century, with the exception of Einstein, to advance physics and change our picture of the universe.’ And yet, unlike Einstein who looms large in popular culture as a hallowed scientific genius, Dirac remains relatively unknown. This begs the question: why?

Dirac was born in his family home 15 Monk Road, Bishopston, Bristol, on the 8 August 1902. His father was a Swiss immigrant and the Head of Modern Languages at Dirac’s secondary school and his mother, a Bristolian, was a librarian at the Bristol Central Library just off College Green. Dirac’s taciturn nature is believed to be partly due to his strict upbringing. Dirac’s father was a disciplinarian who demanded that Dirac speak to him only in French and was quick to point out any mistakes he was making. His relationship with his father, at least in part, could explain why he became a reserved, introverted and emotionally austere character throughout his life.

] 15 Monk Road, Bishopston; the birthplace of Paul Dirac

Dirac went on to take a place at the University of Bristol to study electrical engineering. Upon graduating aged 21, he wanted to continue his studies at the University of Cambridge but couldn’t afford to study there. Instead, he took up a second degree in mathematics from Bristol, during which, he attended lectures on physics in addition to maths. It was there that he was introduced to Einstein’s recent theories of special and general relativity. Later, with enough money from scholarships, Dirac was able to move to study at Cambridge and complete a PhD in the fledging theories of relativity and the emerging field of quantum mechanics.

Attendees of fifth Solvay Conference on Physics held from the 24 to 29 October 1927. 17 of the 29 attendees went on to win a Nobel Prize. Paul Dirac, then 25, is pictured centre left, over the right shoulder of Albert Einstein | Institut International de Physique Solvay

The great theoretical physicist, Erwin Schrödinger had derived an equation that proved very successful in quantum mechanics, but Dirac saw a gap in the theory: it did not take account of Einstein’s relativity, and therefore is an imperfect description of nature.

One striking example of where Schrödinger’s equation fails is in the appearance of gold. Schrödinger’s equation predicts that based on the light that is absorbed and which is reflected off it, gold would appear silver in colour, rather than its distinctive yellow tone. Motivated by this puzzle, Dirac went on to formulate his own equation that not only correctly predicted the colour of gold but also united quantum mechanics with special relativity, and as a bonus, predicted the existence of a new particle: antimatter that is defined by their negative baryon number or lepton number (the antimatter for an electron for example is the “positron” which Dirac’s theory predicts). At the time, the suggestion that antimatter existed was a very bold idea but within two years, experiments confirmed its existence, an astounding achievement for which Dirac was awarded the Nobel Prize for Physics in 1933 at the age of 31!

The Dirac Equation

Here is the famous Dirac equation, regarded as one of the most beautiful equation in science.

Here is a very brief explanation of the Dirac equation (you can skip this paragraph if you wish and continue reading the rest of the post below)

Dirac’s equation is the relativistic description of an electron with rest mass, m. The non-relativistic description of an electron is described by the Pauli-Schrodinger equation. The non-relativistic electron has two spin states and we say that it is a two component equation. Dirac showed that there is no relativistic version of the Pauli-Schrodinger equation that has two components and that the minimal relativistic analogue is a four component equation, each with their own partial differential equation to the wave. The additional other components (or degrees of freedom) in the Dirac equation was ingeniously interpreted by Dirac as the antiparticles of the states.

Dirac’s equation, while less famous that Einstein’s {\displaystyle E = mc^2} , is certainly one of the greatest equations in the history of science (and to many mathematicians, one of the most beautifully concise equation in all of physics). Furthermore, it must have gratified Dirac to know years later that knowledge of antimatter is used in modern applied science. For instance, in medical imaging. PET (positron emission tomography) scans are used to detect diseases in humans and these contraptions use positrons to create light that can measured in the scan.

Dirac was also able to use his equation to pioneer the understanding of another area of physics, quantum electrodynamics (QED) which describes the interaction between light (via the theory electromagnetism developed by James Clark Maxwell) and matter. Indeed, Dirac is rightly credited as the first scientist to create quantum field theory by his pathbreaking work in quantizing the electromagnetic field, without which major advances in the theory from the 1940s onward till today would not have been possible.

The above achievements is arguable deserving of several Nobel Prizes. To top it off, these were achievements Dirac accomplished mostly during his twenties, being just 27 when he was elected to Britain’s Royal Society, 29 when he was appointed Lucasian Professor of Mathematics at the University of Cambridge – a post once held by Isaac Newton, and more recently, Stephen Hawking. He remained as the Lucasian Professor of Mathematics for over thirty years.

Dirac and Nobel Economist Gunnar Myrdal 1976 in Lindau. Source: Archive Pfeiffer.

Dirac spent his last decade as a faculy member of the Center for Theoretical Studies at Florida State University in Tallahassee to be near his daughter, Mary. There he died on October 20th, 1984, the quiet man who shook the world of physics with this elegant equation, now known the world over as the “Dirac Equation”.

Bonus: Excerpts of Dirac’s first Yeshiva lecture on Quantum Mechanics

In 1964, Dirac was invited by Yeshiva University in New York to give four lectures on quantum mechanics. Dirac used the occasion to expound the technical methods he developed to derive an approximate quantum field theory based on methods borrowed from classical mechanics. To reach out to a broader audience, I only reproduce portions of Dirac’s lecture which omits the more technical parts. I believe that even this truncated version of his lecture is sufficient to show the brilliance of the man both in his scientific approach and in his style of delivery. Along the way, I give brief explanatory notes of some of the technical terms.

Here is Dirac’s first lecture entitled “The Hamiltonian Method”

I’m very happy to be here at Yeshiva and to have this chance to talk to you about some mathematical methods that I have been working on for a number of years. I would like first to describe in a few words the general object of these methods.

In atomic theory we have to deal with various fields. There are some fields which are very familiar, like the electromagnetic and the gravitational fields, but in recent times we have a number of other fields also to concern ourselves with, because according to the general ideas of De Broglie and Schrodinger every particle is associated with waves and those waves may be considered as a field. So we have in atomic physics the general problem of setting up a theory of various fields in interaction with each other. We need a theory conforming to the principles of quantum mechanics, but it is quite a difficult matter to get such a theory.

One can get a much simpler theory if one goes over to the corresponding classical mechanics, which is the form which quantum mechanics takes one makes Planck’s constant tend to zero. It is much easier to visualize what one is doing in terms of classical mechanics. It will be mainly about classical mechanics that I shall be talking in these lectures.

Now you may think that that is really not good enough because classical mechanics is not good enough to describe Nature. Nature is described by quantum mechanics. Why should one, therefore, bother about classical mechanics? Well, the quantum field theories, are, as I said, quite difficult and so far, people have been able to build up quantum field theories only for fairly simple kinds of fields with simple interactions between them. It is quite possible that these simple fields … are not adequate for a description of Nature. (So) the success which we get with quantum field theories are rather limited …

In order to be able to start on this problem of dealing with more general fields, we must go over the classical theory. If we can put the classical theory into the Hamiltonian form, then we can always apply certain standard rules so as to get a first approximation to a quantum theory. My talks will be mainly concerned with this problem of putting a general classical theory into Hamiltonian form …

Note: In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. It is of fundamental importance to most formulations of quantum theory. For example, operating on the wavefunction with the Hamiltonian produces the Schrodinger equation.

So I take the point of view that the Hamiltonian is really very important for quantum theory. In fact, without using Hamiltonian methods one cannot solve some of the simplest problems in quantum theory, for example the problem of getting the Balmer formula for hydrogen, which was the very beginning of quantum mechanics …

I would like to begin in an elementary way and I take as my starting point an action principle. That is to say, I assume there is an action integral which I denote by

{\displaystyle {\mathcal {I}}=\int _{t_{1}}^{t_{2}}L\,dt,}{\mathcal  {I}}=\int _{{t_{1}}}^{{t_{2}}}L\,dt,

It is expressed as a time integral, the integrand L being the Lagrangian. So with an action principle we have a Lagrangian. We have to consider how to pass from that Lagrangian to a Hamiltonian. When we have got the Hamiltonian, we have made the first step toward getting a quantum theory.

Note: By convention, {\displaystyle L=T-V,}{\displaystyle L=T-V,} where {\displaystyle T} and {\displaystyle V} are the kinetic and potential energy of the system. So the action integral is the number associated with each path along a curve x(t) defined by the Lagrangian satisfying the given boundary conditions.

You might wonder whether one could not take the Hamiltonian as the starting point and short-circuit this work of beginning with an action integral, getting a Lagrangian from it and passing from the Lagrangian to the Hamiltonian. The objection to trying to make this short-circuit is that it is not at all easy to formulate the conditions for a theory to be relativistic in terms of the Hamiltonian. In terms of the action integral, it is very easy to formulate the conditions for the theory to be relativistic: one simply has to require that the action integral shall be invariant. One can easily construct innumerable examples of action integrals which are invariant. They will automatically lead to equations of action agreeing with relativity, and any developments from this action integral will therefore also be in agreement with relativity.

When we have the Hamiltonian, we can apply a standard method which gives us a first approximation to a quantum theory, and if we are lucky we might be able to go on and get an accurate quantum theory.

Could one not perhaps pass directly from the Lagrangian to the quantum theory, and short-circuit altogether the Hamiltonian? Well, for some simple examples one can do that. For some of the simple fields which are used in physics, the Lagrangian is quadratic in the velocities, and is like the Lagrangian which one has in the non-relativistic dynamics of particles. For those examples, …, people have devised some methods for passing directly from the Lagrangian to the quantum theory. Still, this limitation of the Lagrangian’s being quadratic in the velocities is quite a severe one. I want to avoid this limitation and to work with a Lagrangian which can be quite a general function of the velocities. To get a general formalism which will be applicable, for example, to the non-linear electrodynamics which I mentioned previously, I don’t think one can in any short short-circuit the route of starting with an action integral, getting a Lagrangian, passing from the Lagragian to the Hamiltonian, and then passing from the Hamiltonian to the quantum theory. This is the route which I want to discuss in this course of lectures.

Note: Dirac’s lecture continues from this point to flesh out the mathematical details of his method which I will omit. Readers who are interested in the full set of four lectures can get hold of the following book: Paul A. M. Dirac, Lectures on Quantum Mechanics, Snowball Publishing, 2012.

Further Study

I highly recommend the following book that superbly uncovers the mystique of Paul Dirac, the man and the scientist – The Strangest Man by Graham Farmelo, Basic Books, New York. First published: 2009.

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