Dirac Number = 2

Paul A. M. Dirac was one of the early quantum mechanics who, among his many significant contributions, predicted the existence of antimatter in 1928, as a consequence of finding the correct relativistically invariant wave-equation for the electron. He also formulated the original many-body description of how light interacts with matter, viz., quantum electrodynamics.
Figure 1: The man and his equation.
Although Dirac has not received the same degree of public awareness as say, Einstein or Feynman, biographical essays do exist [1] and several complete biographies have become available more recently [1,2,3,4]. My review of The Strangest Man [3] was published by the New York Journal of Books.

Contents

1  Dirac in Australia
    1.1  Who Took the Photos?
    1.2  Up a Gum Tree
    1.3  Dinner with Dirac
    1.4  Dirac Magnetic Monopoles
2  The Number
3  Eliezer's Dirac Paradox

1  Dirac in Australia

Dirac came to Australia in 1975. I don't recall why and he didn't come to Melbourne. My Master's thesis adviser, Prof. Christie Eliezer, was one of Dirac's few research students from 1941 to 1944 [4], so he and I flew to Adelaide to see Dirac while he was visiting the Physics Department at the University of Adelaide under the auspices of Profs. Angus Hurst and Herbert Green [6]. On the weekend we went for a hike (see Figures 2 and 3). It was rather cold, so people had their jackets and coats on. Only Cambridge types would be caught dead in a suit and tie in the Australian outback.
Figure 2: Dirac (left) and Eliezer (center). Unidentified person is not me (regrettably). Figure 3: This is me (2nd from right) and Eliezer (3rd from right).
I should also add that I was struck by how little was actually said between Dirac and Eliezer. Although some words were no doubt exchanged, there was never any real conversation. Given Eliezer's very affable nature and the various stories he had told me about working for Dirac, this looked really peculiar to me. In fact, a casual observer seeing this could only conclude that these two had never met before, let alone have a personal history together. But I was about to find out firsthand what was behind all that (see Section 1.3).

1.1  Who Took the Photos?

Update of Feb 2010:
Originally, I thought I had taken the photograph in Figure 2, thus making me the "genius" behind the camera who forever lost the opportunity to say "This is me with Dirac!" I now recall that, as a poor student, I didn't even own a camera at that time. Therefore, someone else must have taken that photograph.
In addition, I now remember that I was with a group of people immediately in front of Dirac and, based on the angle, it looks like the photographer was in our group. The good news is, I'm not quite as dumb as I had originally thought.
Update of Mar 2011:
Apparently, the unidentified person in Figure 2 is Prof. John Carver, who was then deputy chair of the Dept. of Physics at Adelaide University [7].
Update of Aug 2012:
I finally discovered a photo of me at the same event. See Figure 3. This clearly establishes that it was not me taking the photographs, but that I was present. Circumstances didn't put me in any shots with Dirac (or, at least, not one that I ended up possessing) but, by inference, being in a shot with Eliezer (and that inimitable white raincoat) does make the connection. Moreover, Prof. Carver can be seen walking away from the camera, at the far right.
An online review of a new biography of Christie Eliezer [4] contains some remarkable photographs of both Eliezer and Dirac, as well as other interesting historical documents.

1.2  Up a Gum Tree

Dirac supposedly had a thing about trees. I had read somewhere that someone went to visit Dirac at his home in Cambridge and his wife (Wigner's sister) answered the door. When the visitor enquired if Dirac was at home, she said that he was home, but he was up a tree.
During our walk (see Figure 2), I had some confirmation of this story. I was walking in front of Dirac with some other people when I suddenly heard something like an animal scurrying off to one side of the track. I assumed it was a rabbit or possibly a kangaroo. I turned around and discovered, to my great astonishment, that it was Dirac. He had taken off like a rocket up the slope (in his Sunday suite and shoes, mind you) toward a gum tree. Apparently, he was fascinated by a particular tree and wanted to get a closer look at it. I was very impressed at how he virtually sprinted up the hill, given that he was in his mid seventies!
Here is another explanation of Dirac's penchant for trees:
English physicist and Cambridge University professor Paul Dirac was an avid mountain climber and occasionally ascended such well-known peaks as Mount Elbruz in the Caucasus. In preparation for such excursions, Dirac would often climb trees in the hills just outside Cambridge - wearing the same black suit in which he was invariably seen around the university campus.

1.3  Dinner with Dirac

A dinner party was thrown for Dirac at the home of Prof. Hurst. It was buffet style. While Dirac's wife was holding court in the dining room (some of us thought she was quite the Brunhilde), Dirac snuck off into an adjacent, darkened room to eat his dinner. Since I was the only person who seemed to notice his stealthy departure, I soon decided to follow him and sat down beside him. I already knew from Eliezer, and reading other anecdotes, that Dirac did not do small talk. But I was now about to get schooled in the deeper significance of those warnings.
Figure 4: Dirac chewing on another tough problem.
To get around any silent treatment, I chose to avoid any small talk and cut to the chase by asking Dirac a physics question; about black holes and singularities, I believe. No answer. After what seemed like minutes, still no answer. Suddenly, it was impossible to know whether he hadn't heard the question, didn't have an answer (and was possibly embarrassed) or just didn't want to answer. It turned out to be none of these.
Small talk is a social lubricant, so it's extremely distressing not to get any response at all. You really don't know what is going on, because the expected natural flow has suddenly come to a screeching halt. Deciding what to do next is like trying to loosen a rusted nut—everything feels totally jammed up. And keep in mind, this is not just some old codger. He's one of the most eminent physicists of the century so, you also have that straightjacket to deal with. It's like falling off a bottomless cliff: you wish it would come to an end just to get rid of all the unbearable feelings. So, you don't know whether to keep trying harder or simply give up. Physicists don't usually give up easily.
While all this was swirling around in my head, I heard a small voice start up unexpectedly, after what seemed like eons and was quite possibly a minute or two. I eventually realized a series of very considered statements was now pouring forth. It sounded like this. Totally bewildered by all these embarrassing thoughts still spinning in my head, I suddenly realized it was Dirac responding to my question. By that time, however, I was so traumatized that I couldn't remember my own question, and I wasn't about to start the whole nerve-racking process over again. Sometimes, it's better to know when to give up.
More recently, I've come to learn about two additional things that may have been working against me:
  1. Dirac had digestive problems all his life. This wasn't remedied until 1980 [3]. Trying to strike up a conversation while he was eating was less than optimal timing on my part.
  2. There's some evidence that Dirac was not too impressed with Stephen Hawking and originially voted against him as his replacement for the Lucian chair [3]. My question about black holes may have been less than a stellar choice.
Perhaps I shouldn't feel too badly off. Dirac was a hero to Richard Feyman. When they finally met at a conference, Feynman was keen to get Dirac's response to his path integral formulation of quantum mechanics, which Feynman had developed out of Dirac's quantum Action concept. Dirac's only response was, "That's interesting." and then he walked off [2].

1.4  Dirac Magnetic Monopoles

Dirac gave three or four lectures [8, see videos] while he was in Adelaide; all of which were quite boring (to me at the time) because he talked about things he had done forty or fifty years earlier without providing any personal insight into how he developed the ideas.
Update of July 2013:
Apparently, the YouTube videos [8] of Dirac lecturing were recorded in 1975 while he was at the University of Canterbury in Christchurch, New Zealand. That's two additional places (UNSW and Canterbury) I didn't know he visited on this same trip. Is there anywhere else?
What I find most striking about these lectures now is that nary a scratch is made on a blackboard; although he does make some hand gestures. The only things that are written on the blackboard are all symbols, just like his monograph on quantum mechanics [9]. Yet, it's clear from his notion of regarding the vacuum as an infinite sea of electrons, or the time it takes light to cross a proton [10], and other visualizations, that Dirac must have had a vivid style of thinking that was not too dissimilar to that of Einstein. Dirac was trained as an engineer, after all. I'm therefore left to conclude that these visualizations were private (as an aid to himself). They were used to organize his ideas but never materialized as physical illustrations in their own right. Like Gauss, it seems Dirac preferred to cover his tracks that would otherwise have shown how he reached his conclusions.
In the second video on Quantum Electrodynamics [8], Dirac points out that the cutoff, imposed to avoid the well-known infinities that arise in certain integrals, is on the order of "1000 million volts" or 1 GeV as we would say today. This is clear evidience that he is not just a mathematician, but a real physicist, even though he goes on to discuss what constitutes "sensible mathematics" when it comes to renormalization in QED.
More exciting than the lectures on ancient quantum mechanics came the news that Blas Cabrera at Stanford was claiming to have measured a Dirac magnetic monopole. If such a thing really existed, it would account for the size of the electric charge. I remember Dirac was skeptical saying that magnetic monopoles should be lying around all over the place, if they existed. After phoning Luis Alvarez, the eminent experimental physicist at UC Berkeley, Dirac reported he didn't believe Cabrera's results. He was right, but a hint of irony remained: the inventor of the magnetic monopole didn't believe in his own invention.
Nonetheless, physics is an experimental science so, the search for magnetic monopoles has continued [11] but that may be about to come to an end [12].

2  The Number

Prof. Eliezer (my M.Sc. adviser) was one of Dirac's very few research students, having worked under Dirac in the early 1940s [4]. Since Eliezer worked with Dirac, he gets a Dirac number of 1; analogous to the popularized notion of six degrees of separation. I can claim whimsically the number 2, since I worked with someone who worked with Dirac.
The reason Dirac had so few students was, in part, due to his taciturn nature. Actually, taciturn does not capture the effect it could have on you. As I have already described in Section 1.3), you really had to experience it directly to feel its emotional impact. There is a famous story about Leopold Infeld who dreamt of working with Dirac. When he finally got to meet him as part of an interview for a research position, he came away in tears because nothing was said and Infeld knew immediately he could never cope with that total lack of personal interaction.

3  Eliezer's Dirac Paradox

Eliezer was fond of reminiscing about his time with Dirac. Of the many stories he told me, the one that intrigued me the most, I came to refer to as the "Eliezer Paradox." Although I later found out it was more generally known (by the name, Sommerfeld Puzzle or similar), I first heard it from my Professor, Christie Eliezer.
The story goes that Sommerfeld derived the eigenvalues corresponding to the fine structure of the hydrogen spectrum by incorporating relativistic corrections into the original Bohr planetary model of the atom. A beautiful account of this semi-classical approach can be found in Max Born's book [13], which was de rigueur reading and published right on the cusp of the arrival of the new matrix mechanics version of quantum theory [6]. Sommerfeld's work was published about a decade before Dirac derived his famous equation for the relativistic electron (see Figure 1) based on incorporating Pauli's concept of fermionic spin. The paradox is that the Sommerfeld solution contains no notion of electron spin. On the other hand, if you try to take spin out of the Dirac equation, it all falls apart. So, how can these two seemingly disparate approaches be consistent with one another?
Eliezer would have told me this story in particular, because he and I both knew that trying to resolve paradoxes of this type can often lead to quite deep results in physics (even decades later). However, although I was intrigued by this story, I never pursued it. If you're wondering why Dirac never pursued it either (or, certainly never published anything on it), he probably left it to others in the same way that he left Christie Eliezer to find the so-called runaway solutions to Dirac's classical extended-electron model [4, See essay by A. Pais]. Moreover, regarding Eliezer and me, I knew that neither of us was sufficiently up to speed on quantum field theory (I certainly wasn't at that particular time) so, I just assumed the Eliezer Paradox had probably already been resolved in the context of that more modern and elaborate mathematics.
I was wrong. Only in very recent years did I become aware that in 1968, Heisenberg (who was Sommerfeld's Ph.D. student) called the coincidence a miracle and suggested:
"It would be intriguing to explore whether this is about a miracle or it is the group-theoretical approach which leads to this formula."

The same suggestion Eliezer made to me.

In 1995, Weinberg declared it to be:
"Absolutely chance coincidence."
Even as recently as 2004, we find this doozy [14]:
"In the quasi-classical approximation, L2 = (l + 1/2)2 ℏ2 is a correct result, so that if Sommerfeld `did good science', then his formula would contain l + 1/2 instead of nφ."
The validity of some of these published remarks (and probably others) notwithstanding, they did rekindle my curiosity in this apparently still unresolved paradox. If it really had been resolved there wouldn't be this many points of view remaining.
For my part, I'm now less persuaded either by Eliezer's 1975 suggestion (which was a very good suggestion at the time I was working on my M.Sc.) or Heisenberg's 1968 comment, that the paradox might be resolved by an analysis of dynamical symmetries [15]. For one thing, the SO(4) symmetry of the Pauli hydrogen atom [5] is broken by relativistic modification of the Coulomb potential, thus lifting the (2l + 1)−2 spectral degeneracy. But that also leaves only the geometrical symmetry, i.e, the usual angular momentum states of SO(3). It's not clear how that smaller symmetry can support a group-theoretic derivation of the more complicated formula for the relativistic energy eigenvalues.
At long last, as a kind of postscript dedicated to Prof. Eliezer, my rather different resolution of the Sommerfeld-Dirac paradox [16] was finally presented at the American Physical Society global meeting in 2025. (Never give up!)

References

[1]
Abdus Salam and Eugene Wigner (eds.), Aspects of Quantum Theory, C.U.P., 1972
[2]
Helge Kragh, Dirac: A Scientific Biography, Cambridge University Press, 2005
[3]
Graham Farmelo, The Strangest Man: The Hidden Life of Paul Dirac, Basic Books, 2009.
[4]
Ranee Eliezer, Conquering Scientist: A Biography of Emeritus Professor Christie J. Eliezer AM 1918-2001, La Sha Prints Pty. Ltd, 2012.
[5]
W. Pauli, "The Hydrogen Atom Spectrum for the Standpoint of Quantum Mechanics," Z. Physik 36, 336, 1926. [This calculation was published prior to the corresponding Schrödinger paper.]
[6]
H. S. Green, Matrix Mechanics, Nooedhoff, 1965
[7]
Peter Lyster, private communication, March 5, 2011.
[8]
Video of Dirac's lectures has recently been uncovered and posted on YouTube (Jan 2013).
  1. Dirac Lecture: Quantum Mechanics.
  2. Dirac Lecture: Quantum Electrodynamics.
  3. Dirac Lecture: Magnetic Monopoles.
  4. Dirac Lecture: Large Numbers Hypothesis.
[9]
P. A. M. Dirac, Principles of Quantum Mechanics, Oxford University Press (Orig. 1930. Last revision 1967)
[10]
The current age of the universe, divided by the time it takes light to cross the radius of a proton, is about 1040. This number is also approximately equal to the ratio of the strengths of the electromagnetic and gravitational forces. Dirac felt that the approximate equality of these two large numbers was too unlikely to be accidental and that some physical process must be at work to maintain the equality. Since the first number clearly changes in time (because the age of the universe is increasing), Dirac proposed that the "fundamental constants of nature" entering the second number should also change in time, to maintain the equality. [Source: Dirac Large Numbers Hypothesis.]
[11]
· 2006 "Theoretical and Experimental Status of Magnetic Monopoles" http://arxiv.org/abs/hep-ex/0602040v1
· 2013 "Search for magnetic monopoles in polar volcanic rocks" http://arxiv.org/abs/1301.6530
[12]
Physicists discover hidden aspects of electrodynamics (April 11, 2017).
TL;DR:
  1. Maxwell's theory displays a symmetry called the EM duality.
  2. However, while electric charges exist, magnetic charges have never been observed in nature.
  3. If magnetic charges do not exist, the EM symmetry cannot exist.
  4. Claim: Gravity plus quantum effects disrupt the EM duality. Hence, magnetic monopoles don't exist.
[13]
Max Born, Mechanics of the Atom, Ungar, 1960 (Orig. German 1924. Orig. English 1927)
[14]
Ya. Granovskii, "Sommerfeld formula and Dirac's theory," Uspekhi Fiz Nauk, 47(5) 523-524, 2004
[15]
Neil J. Gunther, "Unitary declension of dynamical symmetries for the time-dependent harmonic oscillator," J. Math. Phys. 20, 1539 (1979)
[16]
Neil J. Gunther, "The Sommerfeld-Dirac Paradox Reexamined," APS Global Summit, Anaheim, California (2025)