“The δ function can be thought of as a rectangle of width a and height 1/a in the limit [as a goes to zero…]. The δ function is not like the usual function in mathematics because of its infinite discontinuity at the origin. It is one of a class of ‘generalized functions’ whose properties have been rigorously developed by mathematicians since they were first used by the physicist P. A. M. Dirac.”Dirac won his Nobel Prize for contributions to quantum mechanics. I bought a copy of his famous textbook The Principles of Quantum Mechanics when I was an undergraduate at the University of Kansas. Farmelo describes it as “never out of print, it remains the most insightful and stylish introduction to quantum mechanics and is still a powerful source of inspiration for the most able young theoretical physicists. Of all the textbooks they use, none presents the theory with such elegance and with such relentless logic.”

One of Dirac’s greatest contributions was the prediction of positive electrons, or positrons, a type of antimatter. His prediction arose from the relativistic wave equation for the electron, now called the Dirac equation. One interesting feature of the Dirac equation is that it implies negative energy states. The only time these negative states are observable is when an electron is missing from one of the states: a hole. Farmelo writes

“The bizarre upshot of the theory is that the entire universe is pervaded by an infinite number of negative-energy electrons – what might be thought of as a ‘sea’. Dirac argued that this sea has a constant density everywhere, so that experimenters can observe only departures from this perfect uniformity. […] Only a disturbance in Dirac’s sea—a bursting bubble, for example—would be observable. He envisaged just this when he foresaw that there would be some vacant states in the sea of negative-energy electrons, causing tiny departures from the otherwise perfect uniformity. Dirac called these unoccupied states ‘holes’. […] Each hole has positive energy and positive charge—the properties of the proton, the only other subatomic particle known at that time [1929]. So Dirac made the simplest possible assumption by suggesting that a hole is a proton.”We now know that these holes are not protons but are positrons, discovered experimentally in 1932 by Carl Anderson. Positrons are vital for understanding how x-rays interact with matter, as Russ and I describe in Section 15.6 of Intermediate Physics for Medicine and Biology

“A photon with energy above 1.02 MeV can produce a particle-antiparticle pair: a negative electron and a positive electron or positron. […] Since the rest energy (mcIn Sec. 17.14, Russ and I describe the crucial role positrons play in medical imaging.^{2}) of an electron or positron is 0.51 MeV, pair production is energetically impossible for photons below 2mc^{2}= 1.02 MeV.

One can show, using hν_{o}= pc for the photon, that momentum is not conserved by the positron and electron if Eq. 15.23 [conservation of energy] is satisfied. However, pair production always takes place in the Coulomb field of another particle (usually a nucleus) that recoils to conserve momentum.”

“If a positron emitter is used as the radionuclide, the positron comes to rest and annihilates an electron, emitting two annihilation photons back to back. In positron emission tomography (PET) these are detected in coincidence. This simplifies the attenuation correction, because the total attenuation for both photons is the same for all points of emission along each gamma ray through the body (see Problem 54). Positron emitters are short-lived, and it is necessary to have a cyclotron for producing them in or near the hospital. This is proving to be less of a problem than initially imagined. Commercial cyclotron facilities deliver isotopes to a number of nearby hospitals. Patterson and Mosley (2005) found that 97% of the people in the United States live within 75 miles of a clinical PET facility.”Another famous prediction of Dirac’s was magnetic monopoles. Russ and I only mention monopoles in passing in Section 8.8.1: “Since there are no known magnetic charges (monopoles), we must consider the effect of magnetic fields on current loops or magnetic dipoles.” Dirac predicted that magnetic monopoles could in fact exist. Farmelo tells the story:

“In Cambridge, during the spring of 1931, Dirac happened upon a rich new seam of ideas that would crystallize into one of his most famous contributions to science. […] As usual, Dirac appears to have said nothing of this to anyone, even to his close friends. In the early months of 1931, a quiet time for his fellow theoreticians, he was working on the most promising new theory he had conceived for years. The theory broke new ground in magnetism. For centuries, it had been a commonplace of science that magnetic poles come only in pairs, labeled north and south: if one pole is spotted, then the opposite one will be close by. Dirac had found that quantum theory is compatible with the existence of single magnetic poles. During a talk at the Kapitza Club, he dubbed them magnons, but the name never caught on in this context; the particles became known as magnetic monopoles.”Physicists have searched for magnetic monopoles, and once they even thought they found one. In 1982, physicist Blas Cabrera observed a signal consistent with the experimental signature of a monopole (Physical Review Letters, Volume 48, Pages 1378-1381), but it now appears to have been an artifact, as the result has never been reproduced. I have my own remote (indeed, very remote) connection with this experiment (and thus to Dirac). Cabrera’s PhD advisor, William Fairbank, was John Wikswo’s PhD advisor, and Wikswo was in turn my PhD advisor. Thus, academically speaking, I am one of Cabrera’s scientific nephews.

Dirac was known for saying little and behaving rather oddly (the title of the book is, after all, “The Strangest Man”), and Farmelo suggests a possible reason why: Dirac may have been autistic.

“[Dirac] always attributed his extreme taciturnity and stunted emotions to his father’s disciplinarian regime; but there is another, quite different explanation, namely that he was autistic. Two of Dirac’s younger colleagues confided in me that they had concluded this, each of them making their disclosure in sotto voce, as if they were imparting a shameful secret. Both refused to be quoted. […] There is not nearly enough detail in her [Dirac’s mother’s] comments or in reports of Dirac’s behaviour in school to justify a diagnosis that he was then autistic. His behavior as an a adult, however, had all the characteristics that almost every autistic person has to some degree—reticence, passivity, aloofness, literal-mindedness, rigid patterns of activity, physical ineptitude, self-centredness and, above all, a narrow range of interests and a marked inability to empathise with other human beings.”Whatever the cause of Dirac’s unusual behavior, he was a great physicist. Farmelo sums up Dirac’s enduring legacy at the end of his book.

“There is no doubt that Dirac was a great scientist, one of the few who deserves a place just below Einstein in the pantheon of modern physicists. Along with Heisenberg, Jordan, Pauli, Schrodinger and Born, Dirac was one of the group of theoreticians who discovered quantum mechanics. Yet his contribution was special. In his heyday, between 1925 and 1933, he brought a uniquely clear vision to the development of a new branch of science: the book of nature often seemed to be open in front of him.”

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