Archive for Paul Dirac

My PhD Tree

Posted in Biographical, The Universe and Stuff with tags , , on June 23, 2015 by telescoper

Last week I discovered that somebody has kindly constructed my PhD Tree. I later discovered that similar things have been constructed for quite a few other scientists of my acquaintance. Perhaps even yours?

Anyway, here is my academic lineage. As you can see, I have some distinguished ancestors. In particular, my great-grandfather (academically speaking) was Paul Dirac

PhD Tree

Incidentally, you might like to see Dirac’s hand-written notes for his PhD Thesis, which you can find here. It dates from 1926. As far as I am aware this is the first PhD thesis ever written on the subject of Quantum Mechanics. It’s also worth mentioning the tremendous contribution to British science made by R.H. Fowler. Fifteen Fellows of the Royal Society and three Nobel Laureates (Chandrasekhar, Dirac, and Mott) were supervised by Fowler in Cambridge between 1922 and 1939.

The most beautiful equation?

Posted in The Universe and Stuff with tags , , , , on February 13, 2014 by telescoper

There’s an interesting article on the BBC website today that discusses the way mathematicians’ brains appear to perceive “beauty”. A (slightly) more technical version of the story can be found here. According to functional magnetic resonance imaging studies, it seems that beautiful equations excite the same sort of brain activity as beautiful music or art.

The question of why we think equations are beautiful is one that has come up a number of times on this blog. I suspect the answer is a slightly different one for theoretical physicists compared with pure mathematicians. Anyway, I thought it might be fun to invite people offer suggestions through the comments box as to the most beautiful equation along with a brief description of why.

I should set the ball rolling myself, and I will do so with this, the Dirac Equation:


This equation is certainly the most beautiful thing I’ve ever come across in theoretical physics, though I don’t find it easy to articulate precisely why. I think it’s partly because it is such a wonderfully compact fusion of two historic achievements in physics – special relativity and quantum mechanics – but also partly because of the great leaps of the imagination that were needed along the journey to derive it and my consequent admiration for the intellectual struggle involved. I feel it is therefore as much an emotional response to the achievement of another human being – such as one feels when hearing great music or looking at great art – as it is a rational response to the mathematical structure involved. But it’s not just that, of course. The Dirac Equation paved the way to many further developments in particle physics. It seems to encapsulate so much about the behaviour of elementary particles in so few symbols. Some of its beauty also derives from its compactness.

Anyway, feel free to suggest formulae or equations through the comments box, preferably with a brief explanation of why you think they’re so beautiful.

Dirac Lectures

Posted in The Universe and Stuff with tags , , on April 11, 2013 by telescoper

Earlier this year I posted a review of a book about the great theoretical physicist Paul Dirac. Presumably by a complete coincidence, on the very same day that I wrote that piece, somebody put the following video on Youtube. It’s very rare footage of the man himself giving some lectures in Christchurch, New Zealand in 1975 (when he was in his 70s). A great deal of conflicting stuff has been written about what Dirac was like as a lecturer – now you can see for yourself. The video isn’t very high quality – it breaks up entirely in a few places – but it’s nevertheless fascinating to hear Dirac talk physics!

My opinion? I’ve had worse!

p.s thanks to Ian Harrison (@itrharrison) for drawing this to my attention!

The Strangest Man

Posted in Literature, The Universe and Stuff with tags , , , , , on January 27, 2013 by telescoper

Since getting rid of my telly a few weeks ago I’ve reverted to a previous incarnation as a bookworm, and have been tackling the backlog of unread volumes sitting on my coffee table at home. Over the last couple of days I’ve spent the evenings reading The Strangest Man by Graham Farmelo, a biography of the great theoretical physicist Paul Dirac.

I’m actually quite ashamed that it has taken me so long to get around to reading this. I’ve had it for two years or more and really should have found time to do it before now. Dirac has long been one of my intellectual heroes, for his unique combination of imagination and mathematical rigour; the Dirac equation is one of the topics I most enjoy lecturing about to physics students. I am also immensely flattered to be one of his academic descendants: Paul Dirac was the PhD supervisor of Dennis Sciama, who supervised my supervisor John Barrow, making me (in a sense) his great-grandson. Not that I’ll ever achieve anything of the magnitude he did.

The book is pretty long, and I suppose one of the factors putting me off reading it was that I thought it might be heavy going. That turned out to be far from the case. It’s wonderfully well written, never getting bogged down in details, and cleverly interweaving Dirac’s life and scientific career together against a vivid historical backdrop dominated by the rise of Nazism in Germany and the tragedy of World War 2. It also beautifully conveys the breathless sense of excitement as the new theory of quantum mechanics gradually fell into place. Altogether it’s a gripping story that had me hooked from the start, and I devoured the 400+ pages in just a couple of evenings (which is quick by my standards). I’ve never read a scientific biography so pacey and engaging before, so it’s definitely hats off to Graham Farmelo!

Among the book’s highlights for me were the little thumbnail sketches of famous physicists I knew beforehand mostly only as names. Niels Bohr comes across as a splendidly warm and avuncular fellow, Werner Heisenberg as a very slippery customer of questionable political allegiance (likewise Erwin Schrödinger), Ernest Rutherford as blunt and irascible. I was already aware of the reputation of Wolfgang Pauli had for being an absolute git; this book does nothing to dispel that opinion. We tend to forget that the names we came to know through their association with physics also belonged to real people, with all that entails.

I was also interested to learn that Dirac and his wife Manci spent their honeymoon in 1937, as the clouds of war gathered on the horizon, in Brighton, which Farmelo describes as

..a peculiarly raffish town., famous for its two Victorian piers jutting imperiously out to sea, for the pale green domes of its faux-oriential pavilions, its future-robot and a host of other tacky attractions.

So in most respects it hasn’t changed much, although one of the two piers  has since gone for a Burton.

So what of Dirac himself? Most of what you’re likely to hear about him concerns his apparently cold and notoriously uncommunicative nature. I never met Dirac. He died in 1984. I was an undergraduate at Cambridge at the time, but he had moved to Florida many years before that. I have, however, over the years had occasion to talk to quite a few people who knew Dirac personally, including Dennis Sciama. All of them told me that he wasn’t really anything like the caricature that is usually drawn of him. While it’s true that he had no time for small talk and was deeply uncomfortable in many social settings, especially formal college occasions and the like, he very much enjoyed the company of people more extrovert than himself and was more than willing to talk if he felt he had anything to contribute. He got on rather well with Richard Feynman, for example, although they couldn’t have had more different personalities. This gives me the excuse to include this wonderful picture of Dirac and Feynman together, taken in 1962 – the body language tells you everything there is to know about these two remarkable characters:


Feynman is also an intellectual hero of mine, because he was outrageously gifted not only at doing science but also at communicating it. On the other hand, I suspect (although I’ll obviously never know) that I might not have liked him very much at a personal level. He strikes me as the sort of chap who’s a lot of fun in small doses, but by all accounts he could be prickly and wearingly egotistical.

On the other hand, the more I read The Strangest Man the more I came to think that I would have liked Dirac. He may have been taciturn, but at least that meant he was free from guile and artifice. It’s not true that he lacked empathy for other people, either. Perhaps he didn’t show it outwardly very much, but he held a great many people in very deep affection. It’s also clear from the quotations peppered throughout the book that people who worked closely with him didn’t just admire him for his scientific work; they also loved him as a person. A strange person, perhaps, but also a rather wonderful one.

In the last Chapter, Farmelo touches on the question of whether Dirac may have displayed the symptoms of autism. I don’t know enough about autism to comment usefully on this possibility. I don’t even know whether the term autistic is defined with sufficient precision to be useful. There is such a wide and multidimensional spectrum of human personality that it’s inevitable that there will be some individuals who appear to be extreme in some aspect or other. Must everyone who is a bit different from the norm be labelled as having some form of disorder?

The book opens with the following quote from John Stuart Mill’s On Liberty, which says it all.

Eccentricity has always abounded when and where strength of character has abounded; and the amount of eccentricity in a society has generally been proportional to the amount of genius, mental vigor, and courage which it contained. That so few now dare to be eccentric, marks the chief danger of the time.

Another thought occurred to me after I’d finished reading the book. Dirac’s heyday as a theoretical physicist was the period 1928-1932 or thereabouts. Comparatively speaking, his productivity declined significantly in later years; he produced fewer original results and became increasingly isolated from the mainstream. Eddington’s career followed a similar pattern: he did brilliant work when young, but subsequently retreated into the cul-de-sac of his Fundamental Theory. Fred Hoyle is another example – touched by greatness early in his career, but cantankerous and blinded by his own dogma later on. Even Albert Einstein, genius-of-geniuses, spent his later scientific life chasing shadows.

I think there’s a tragic inevitability about the mid-life decline of these geniuses of theoretical physics, because the very same determination and intellectual courage that allowed them to break new ground also rendered them unwilling to be deflected by subsequent innovations elsewhere. And break new ground Dirac certainly did. The word genius is perhaps over-used, but it certainly applies to Paul Dirac. We need more like him.

Is there only one electron in the Universe?

Posted in The Universe and Stuff with tags , , , , , , , , on February 1, 2012 by telescoper

I started teaching Nuclear and Particle Physics to the 3rd year Physics students today. I decided to warm up with a few basics about elementary particles and their properties – all pretty standard stuff and no hairy mathematics. Cue pretty picture:

This doesn’t show the whole picture, of course, because for every particle there is an antiparticle, so there are antiquarks and antileptons. The existence of these was first suggested by Paul Dirac in 1928 based on his investigations into relativistic quantum theory, basically because invariance of special relativity is compatible with the existence of both positive and negative energy states, i.e.

E^2 = p^2c^2 +m^2 c^4

has two sets of solutions, one with E>0 and the other with E<0. Instead of simply assuming the latter set were physically unrealistic, Dirac postulated that they might be real, but completely filled in “empty” space; these filled negative-energy states are usually called the “Dirac Sea”. Injection of an appropriate amount of energy can promote something from a negative state into a positive one, leaving behind a kind of hole (very similar to what  happens in the case of semiconductor). This process creates a pair consisting of a (positive energy) particle and a (negative energy) antiparticle (i.e. a hole in the Dirac Sea). In the case of electrons, the hole is called a positron.

The alternative, and even wackier, explanation of antimatter I usually mention in these lectures derives, I think, from Feynam who noted that in quantum (wave) mechanics the time evolution of particles involves things like

\exp(i\omega t)=\exp(i Et/\hbar),

which have the property that changing E into -E has the same effect as changing t into -t. This is, in essence, the reason why, in Feynman diagrams, antiparticles are usually represented as particles travelling backwards in time…

This is a useful convention from the point-of-view of using such diagrams in calculations, but it allows one also to raise the wacky bar to a higher level still, to a suggestion that, coincidentally, was  doing the rounds very recently – namely whether it is possible that there may really be only one electron in the entire Universe:

….I received a telephone call one day at the graduate college at Princeton from Professor Wheeler, in which he said, “Feynman, I know why all electrons have the same charge and the same mass” “Why?” “Because, they are all the same electron!” And, then he explained on the telephone, “suppose that the world lines which we were ordinarily considering before in time and space—instead of only going up in time were a tremendous knot, and then, when we cut through the knot, by the plane corresponding to a fixed time, we would see many, many world lines and that would represent many electrons, except for one thing. If in one section this is an ordinary electron world line, in the section in which it reversed itself and is coming back from the future we have the wrong sign to the proper time—to the proper four velocities—and that’s equivalent to changing the sign of the charge, and, therefore, that part of a path would act like a positron.”
—Feynman, Richard, Nobel Lecture December 11, 1965

In other words, a single electron can appear in many different places simultaneously if it is allowed to travel backwards and forwards in time…

I think this is a brilliant idea, especially if you like science fiction stories, but there’s a tiny problem with it in terms of science fact. In order for it to work there should be as many positrons in the Universe as there are electrons. Where are they?