## Card bearing Feynman’s signature ‘Richard P. Feynman’, with one of his famous 'Feynman diagrams' below in his hand, and an affixed newspaper photograph of Feynman receiving the Nobel Prize in Physics 1965.

A rare example of Feynman’s signature, with a much rarer example of one of his eponymous diagrams in his hand. In fact, this diagram is almost identical to the first ever Feynman diagram that he drew in public, on the blackboard at the famous Pocono conference in the spring of 1948, where he first explained his diagrammatic approach to the problems of quantum electrodynamics. Widely regarded as the most brilliant, influential, and iconoclastic figure in theoretical physics in the post-World War II era, Feynman shared the Nobel Prize in Physics 1965 with Sin-Itiro Tomonaga and Julian Schwinger “for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles.” The rarity of any form of manuscript material by Feynman is well-known. When his autobiographical work *Surely You're Joking Mr. Feynman!*was about to be published, Feynman told his editor “I’m not going to go on TV and I’m not going to sign any books!” Requests for Feynman’s signature were referred routinely to his secretary, who returned instead a printed card stating firmly that ‘Professor Feynman has found it necessary to refuse all requests for autographs’. Feynman’s signature is here greatly enhanced by the presence of one of his iconic Feynman diagrams, which have since become ubiquitous in theoretical physics. Schwinger later wrote: “Like the silicon chip of more recent years, the Feynman diagram was bringing computation to the masses” (Brown & Hoddesdon, p. 329). In 1973, the great Dutch theoretical physicist and Nobel laureate Gerardus t’Hooft commented (CERN 79-9): “Few physicists object nowadays to the idea that diagrams contain more truth than the underlying formalism.” We know of only two other examples of Feynman’s signature accompanied by an autograph Feynman diagram, one on a copy of *Feynman’s Lectures on Physics* (commons.wikimedia.org/wiki/File:Feynman'sDiagram.JPG), and another on a copy of Feynman’s popular work *QED* owned by the physicist and bibliophile Jay Pasachoff (see: chapin.williams.edu/pasachoff/collecting.html).

“QED explains the force of electromagnetism − the physical force that causes like charges to repel each other and opposite charges to attract − at the quantum-mechanical level. In QED, electrons and other fundamental particles exchange virtual photons − ghostlike particles of light − which serve as carriers of this force. A virtual particle is one that has borrowed energy from the vacuum, briefly shimmering into existence literally from nothing. Virtual particles must pay back the borrowed energy quickly, popping out of existence again, on a time scale set by Werner Heisenberg’s uncertainty principle.

“Two terrific problems marred physicists’ efforts to make QED calculations. First, as they had known since the early 1930s, QED produced unphysical infinities, rather than finite answers, when pushed beyond its simplest approximations. When posing what seemed like straightforward questions − for instance, what is the probability that two electrons will scatter? − theorists could scrape together reasonable answers with rough-and-ready approximations. But as soon as they tried to push their calculations further, to refine their starting approximations, the equations broke down. The problem was that the force-carrying virtual photons could borrow any amount of energy whatsoever, even infinite energy, as long as they paid it back quickly enough. Infinities began cropping up throughout the theorists’ equations, and their calculations kept returning infinity as an answer, rather than the finite quantity needed to answer the question at hand.

“A second problem lurked within theorists’ attempts to calculate with QED: The formalism was notoriously cumbersome, an algebraic nightmare of distinct terms to track and evaluate. In principle, electrons could interact with each other by shooting any number of virtual photons back and forth. The more photons in the fray, the more complicated the corresponding equations, and yet the quantum-mechanical calculation depended on tracking each scenario and adding up all the contributions.

“All hope was not lost, at least at first. Heisenberg, Wolfgang Pauli, Paul Dirac and the other interwar architects of QED knew that they could approximate this infinitely complicated calculation because the charge of the electron (*e*) is so small: *e*^{2 }~ 1/137, in appropriate units. The charge of the electrons governed how strong their interactions would be with the force-carrying photons: Every time a pair of electrons traded another photon back and forth, the equations describing the exchange picked up another factor of this small number, *e*^{2}. So a scenario in which the electrons traded only one photon would ‘weigh in’ with the factor *e*^{2}, whereas electrons trading two photons would carry the much smaller factor *e*^{4}. This event, that is, would make a contribution to the full calculation that was less than one one-hundredth the contribution of the single-photon exchange. The term corresponding to an exchange of three photons (with a factor of *e*^{6}) would be ten thousand times smaller than the one-photon-exchange term, and so on. Although the full calculations extended in principle to include an infinite number of separate contributions, in practice any given calculation could be truncated after only a few terms. This was known as a *perturbative* calculation: theorists could approximate the full answer by keeping only those few terms that made the largest contribution, since all of the additional terms were expected to contribute numerically insignificant corrections.

“Deceptively simple in the abstract, this scheme was extraordinarily difficult in practice … by the start of World War II, QED seemed an unholy mess, as calculationally intractable as it was conceptually muddled.

“In his Pocono Manor Inn talk, Feynman told his fellow theorists that his diagrams offered new promise for helping them march through the thickets of QED calculations. As one of his first examples, he considered the problem of electron-electron scattering. He drew a simple diagram on the blackboard, similar to the one later reproduced in his first article on the new diagrammatic techniques” (Kaiser, pp. 156-8).

“The first published example of what is now called a Feynman diagram appeared in Feynman’s 1949 *Physical Review* article [‘The theory of positrons,’ Vol. 76, pp. 749-59]. It depicted the simplest contribution to an electron-electron interaction, with a single virtual photon (wavy line) emitted by one electron and then absorbed by the other. In Feynman’s imagination – and in the equations – this diagram also represented interactions in which the photon is emitted by one electron and travels back in time to be absorbed by the other, which is allowed within the Heisenberg time uncertainty” (https://physics.aps.org/story/v24/st3).

The first diagram Feynman drew at the Pocono conference is virtually identical to the one he drew on the offered card. The diagram represents events in two dimensions, with space on the horizontal axis and time on the vertical axis. The straight lines at bottom left and bottom right represent the paths of two electrons. In classical physics there is an electromagnetic force that causes the electrons to repel each other. In QED this interaction takes place via the exchange of a virtual photon, represented by the wavy line. After the virtual photon has been exchanged the subsequent motion of the electrons is represented by the straight lines at the top left and top right. At the left hand vertex, where the two straight lines and the wavy line meet, the energy and momentum of the left-hand electron changes. Since energy-momentum (technically, the relativistic 4-momentum) is always conserved, the change in the 4-momentum of the electron is balanced by the 4-momentum of the virtual photon emitted at this vertex. This virtual photon then interacts with the second electron at the right-hand vertex, where its 4-momentum is added to that of the electron, causing it to scatter.

But the Feynman diagram is much more than a pictorial representation of the interaction of the two electrons. It enables one to calculate a complex quantity called the ‘amplitude’ for the diagram. Its absolute square, apart from simple factors, is the ‘cross section’ describing the probability for the process to occur. (Strictly speaking, before taking the absolute square the amplitudes of all possible diagrams having the same initial and the same final states must be added, but as explained above only the first few diagrams need to be retained in practice.) To calculate the amplitude for the diagram one needs to know the ‘propagator’ for the virtual photon, which is the factor 1/*q*^{2} Feynman has written under the wavy line representing it. Here *q*^{2} is the squared length of the 4-momentum of the virtual photon (energy^{2} – momentum^{2}). In classical electrodynamics this would be zero, because it is equal to the square of the rest mass of the particle, which for a photon is zero. However, *q*^{2} need not be zero for a virtual photon (physicists say that virtual photons are ‘off shell’). To calculate the amplitude one also needs to know the ‘vertex factors,’ representing the likelihood that an electron would emit or absorb a photon. This is *eγ _{μ}*, where

*e*is the electron’s charge and

*γ*a vector of ‘Dirac matrices’ (arrays of numbers to keep track of the electron’s spin). Feynman has indicated these vertex factors by writing

_{μ}*γ*next to each of the two vertices of the diagram.

_{μ }“In this simplest process, the two electrons traded just one photon between them; the straight electron lines intersected with the wavy photon line in two places, called ‘vertices.’ The associated mathematical term therefore contained two factors of the electron’s charge, *e* − one for each vertex. When squared, this expression gave a fairly good estimate for the probability that two electrons would scatter. Yet both Feynman and his listeners knew that this was only the start of the calculation. In principle, as noted above, the two electrons could trade any number of photons back and forth.

“Feynman thus used his new diagrams to describe the various possibilities. For example, there were nine different ways that the electrons could exchange two photons, each of which would involve four vertices (and hence their associated mathematical expressions would contain *e*^{4} instead of *e*^{2}). As in the simplest case (involving only one photon), Feynman could walk through the mathematical contribution from each of these diagrams …

“By using the diagrams to organize the calculational problem, Feynman had thus solved a long-standing puzzle that had stymied the world’s best theoretical physicists for years. Looking back, we might expect the reception from his colleagues at the Pocono Manor Inn to have been appreciative, at the very least. Yet things did not go well at the meeting. For one thing, the odds were stacked against Feynman: His presentation followed a marathon day-long lecture by Harvard’s Wunderkind, Julian

Schwinger. Schwinger had arrived at a different method (independent of any diagrams) to remove the infinities from QED calculations, and the audience sat glued to their seats − pausing only briefly for lunch − as Schwinger unveiled his derivation.

Coming late in the day, Feynman’s blackboard presentation was rushed and unfocused. No one seemed able to follow what he was doing. He suffered frequent interruptions from the likes of Niels Bohr, Paul Dirac and Edward Teller, each of whom pressed Feynman on how his new doodles fit in with the established principles of quantum physics. Others asked more generally, in exasperation, what rules governed the diagrams’ use. By all accounts, Feynman left the meeting disappointed, even depressed” (Kaiser, pp. 159-160).

Feynman diagrams were eventually accepted largely due to the efforts of the British mathematician Freeman Dyson, who had regularly served as discussion partner to Feynman at Cornell and was probably the only person at that time who was really familiar with both Schwinger’s and Feynman’s theories. In his paper, ‘The radiation theories of Tomonaga, Schwinger and Feynman,’ (*Physical Review* 75 (1949), pp. 486-501), Dyson constructed a bridge between the two theories, showing that Feynman’s methods could be derived from the more traditional techniques used by Schwinger.

“Soon the [Feynman] diagrams gained adherents throughout the fields of nuclear and particle physics. Not long thereafter, other theorists adopted − and subtly adapted − Feynman diagrams for solving many-body problems in solid-state theory. By the end of the 1960s, some physicists even used versions of Feynman’s line drawings for calculations in gravitational physics. With the diagrams’ aid, entire new calculational vistas opened for physicists. Theorists learned to calculate things that many had barely dreamed possible before World War II. It might be said that physics can progress no faster than physicists’ ability to calculate. Thus, in the same way that computer-enabled computation might today be said to be enabling a genomic revolution, Feynman diagrams helped to transform the way physicists saw the world, and their place in it” (Kaiser, p. 156).

Richard Phillips Feynman was born on 11 May 1918 in the New York borough of Queens to Jewish parents originally from Russia and Poland. As a child, he was heavily influenced both by his father, Melville, who encouraged him to ask questions to challenge orthodox thinking, and his mother, Lucille, from whom he inherited the sense of humour that he maintained throughout his life. From an early age he delighted in repairing radios and demonstrated a talent for engineering. At Far Rockaway High School in Queens, he excelled in mathematics, and won the New York University Math Championship by a large margin in his final year there. He was refused entry to his first choice Columbia University because of the ‘Jewish quota’ and attended instead the Massachusetts Institute of Technology, where he received a bachelor’s degree in 1939, and was named a Putnam Fellow. He obtained an unprecedented perfect score on the graduate school entrance exams to Princeton University (although he did rather poorly on the history and English portions), where he went to study mathematics under his advisor John Archibald Wheeler (1911-2008). He obtained his PhD in 1942, with a thesis on the ‘path-integral’ formulation of quantum mechanics. During his time at Princeton, he married his first wife, Arline Greenbaum; she died of tuberculosis just a few years later in 1945.

While at Princeton, Feynman was persuaded by the physicist Robert Wilson to participate in the Manhattan Project. At Los Alamos Feynman immersed himself in the work on the atomic bomb, was soon made a group leader under Hans Bethe, and was present at the Trinity bomb test in 1945. During his time at Los Alamos, Niels Bohr sought him out for discussions about physics, and he became a close friend of laboratory head Robert Oppenheimer, who unsuccessfully tried to lure him to the University of California in Berkeley after the war. Looking back, Feynman thought his decision to work on the Manhattan Project was justified at the time, but he expressed grave reservations about the continuation of the project after the defeat of Nazi Germany, and suffered bouts of depression after the destruction of Hiroshima.

After the war, Feynman declined an offer from the Institute for Advanced Study in Princeton, New Jersey, despite the presence there of such distinguished faculty members as Albert Einstein, Kurt Gödel and John von Neumann. Instead he followed Hans Bethe to Cornell, where he taught theoretical physics from 1945 to 1950. Feynman then opted for the position of Professor of Theoretical Physics at the California Institute of Technology (partly for the climate, as he admits), despite offers of professorships from other renowned universities. He remained there for the rest of his career.

During his years at Caltech, he continued the work on quantum electrodynamics (the theory of the interaction between light and matter) he had begun at Cornell, and for which he was awarded the 1965 Nobel Prize in Physics. He developed an important tool known as Feynman diagrams to help conceptualize and calculate interactions between particles, notably the interactions between electrons and their anti-matter counterparts, positrons. Feynman diagrams, which are easily visualized graphic analogues of the complicated mathematical expressions needed to describe the behaviour of systems of interacting particles, have permeated many areas of theoretical physics in the second half of the twentieth century. He also worked on the physics of the superfluidity of supercooled liquid helium and its quantum mechanical behaviour; a model of weak decay (such as the decay of a neutron into an electron, a proton and an anti-neutrino) in collaboration with fellow Caltech professor Murray Gell-Mann; and his parton model for analyzing high-energy hadron collisions. At Caltech Feynman gained a reputation for being able to explain complex elements of theoretical physics in an easily understandable way – he opposed rote learning, although he could also be strict with unprepared students. His 1964 *Feynman Lectures On Physics* remains a classic.

In December 1959, Feynman gave a visionary and ground-breaking talk entitled ‘There's Plenty of Room at the Bottom’ at an American Physical Society meeting at Caltech. In it, he suggested the possibility of building structures one atom or molecule at a time, an idea which seemed fantastic at the time, but which has since become widely known as nanotechnology. He was also one of the first scientists to conceive of the possibility of quantum computers and played a crucial role in developing the first massively parallel computer, finding innovative uses for it in numerical computations, building neural networks and physical simulations using cellular automata.

Just two years before his death, Feynman played an important role in the Rogers Commission investigation of the 1986 Challenger Space Shuttle disaster. During a televised hearing, Feynman famously demonstrated how the O-rings became less resilient and subject to seal failures at ice-cold temperatures by immersing a sample of the material in a glass of ice water. He developed two rare forms of cancer, Liposarcoma and Waldenström's macroglobulinemia, and died on 15 February 1988 in Los Angeles.

Brown & Hoddesdon (eds.), *The Birth of Particle Physics*, 1983. Kaiser, ‘Physics and Feynman diagrams,’ *American Scientist*, Vol. 93 (2005), pp. 156-165 (http://web.mit.edu/dikaiser/www/FdsAmSci.pdf).

118 x 75 mm. In fine condition.

Item #4581

**
Price:
$15,000.00
**