Lectures on quantum field theory

" Dirac's life-long (1948-1984) and relentless pursuit for a quantum electrodynamics with a logical footing"

In the 1950’s in his search for a better QED, Paul Dirac developed the Hamiltonian theory of constraints (Cand J Math 1950 vol 2, 129; 1951 vol 3, 1) based on lectures that he delivered at the 1949 International Mathematical Congress in Canada. Although Dirac, like Einstein, would never jump on the band wagon, it is not simply true that he was not aware of particles called mesons ( see Farmelo G 2009 "The Strangest Man", London Faber & Faber). Dirac (1951 “The Hamiltonian Form of Field Dynamics” Cand Jour Math, vol 3 ,1) had also solved the problem of putting the Tomonaga-Schwinger equation into the Schrödinger representation ( See Phillips R J N 1987 “Tributes to Dirac” p31 London:Adam Hilger) and given explicit expressions for the scalar meson field (spin zero pion or pseudoscalar meson ), the vector meson field (spin one rho meson), and the electromagnetic field (spin one massless boson,photon) .

Dirac had met Feynman in 1961 and the two physicists talked about the non-existence of an equation “similar to the relativistic equation of the electron” describing a meson which apparently Feynman was supposed to be working on! Feynman started the conversation with the question “It must be good to have invented that equation ”.
J C Polkinghorne admitted how wrong he was about Dirac and contemporary particle physics in an article in Kursunoglu & Wigner (Ed) 1990 "Reminiscences about a great physicist" p228 Cambridge:CUP..quote... “All very clever”, we thought to ourselves, " but Dirac would probably not know a pion if he saw one”. The last laugh is where it ought to be, with the great and insightful not tossed about by every wind of physical fashion but profound in his understanding of the quantum field theory that he had invented. I realize now ,with hindsight, that I heard Dirac talk about monopoles and the quantum mechanics of constrained and of extended systems and the difficulties of quantizing gravity, all topics of the highest contemporary interest, to which he contributed the unique clarity and force of his understanding... unquote.

In 1956 C N Yang and T D Lee suggested that when particles interact weakly nature might choose to break the perfect symmetry between left and right, the so called parity symmetry (Framelo, 2009). Gravitational and electromagnetic interactions are ambidextrous.
Dirac had foreseen the possibility that parity symmetry might be broken in his paper “Forms of relativistic dynamics” (1949 Rev Mod Phys 21 392) in which he states that “I do not believe there is any need for physical laws to be invariant under these reflections (in space and time), although the exact physical laws of nature so far known (gravity and electromagnetism) do have this invariance.”
In a paper “Long range forces and broken symmetries” (1973 Proc Roy Soc 333 403 ) he discusses an important feature of Weyl’s geometry that leads to a breaking of the C (charge conjugation) and T (time reversal) symmetries with no breaking of P (parity change) or CT. The breaking of the C and T symmetries is a rare event and has been observed for the K-meson.
The Weyl interpretation of of the electromagnetic field as influencing the geometry of space and not something embedded in Riemannn space implies symmmetry breaking.
In the late 50’s he applied the Hamiltonian methods he had developed to cast Einstein’s general relativity in Hamiltonian form (Proc Roy Soc 1958,A vol 246, 333,Phys Rev 1959,vol 114, 924) and to bring to a technical completion the quantization problem of gravitation and bring it also closer to the rest of physics according to Salam and DeWitt. In 1959 also he gave an invited talk on "Energy of the Gravitational Field" at the New York Meeting of the American Physical Society later published in 1959 Phys Rev Lett vol 2, 368.

In 1964 he published his “Lectures on Quantum Mechanics” (London:Academic) which deals with constrained dynamics of nonlinear dynamical systems including quantization of curved spacetime. He also published a paper entitled “Quantization of the Gravitational Field” in 1967 ICTP/IAEA Trieste Symposium on Contemporary Physics.
In 1963 he apparently found from his old notes a rather novel method -using the Hamiltonian formalism that he had already used in 1958 to discuss the energy of gravitational waves and the gravitational polarization of spin 2 - of deriving the Schwinger term and the Lamb shift (1056.17MHz) without recourse to the usual “joining technique” of utilizing Bethe’s non-relativistic result adopted by a number of workers-including Weisskopf and French, Feynman and Schwinger-in quantum electrodynamics. The troublesome vacuum polarisation terms created problems for Feynman (1022.89 MHz) and Schwinger (1016.11 MHz) ; they calculated the Lamb shift without vacuum polarization and again with vacuum polarization which stretegies did not bring them any nearer to improvements in their calculations at the time when French and Weisskopf (1051.13 MHz) had the correct solution in hand already. It was revealed by French that the the source of their error was the way in which the joining to the Bethe non-relativistic result was done ! In other words French and Weisskopf were adopting the quantum field theory as developed by the European physicists whereas Feynman and Schwinger were reinventing new methods independent of the field theory! Weisskopf was rather unfortunate to miss the Nobel Prize with Lamb since he had put French his PhD student to work on the problem in October 1946,well before he heard of Lamb’s experiment.
Dirac’s work on QED is also based on his theory of constrained dynamics. He also gave a series of lectures on quantum electrodynamics in 1962/1963 at the Belfer Graduate School of Science, Yeshiva University and at Ban-Ilan University Israel, in 1965 (Jammer M 1966 "The Conceptual Development of Quantum Mechanics" London:McGraw-Hill). He again gave an invited talk on quantum electrodynamics at the New York Meeting of the American Physical Society in January 1965 which was later published in 1965 Phys Rev vol 139, 684. Do recall that his research activities took place before the “official” recognition of the work of Tomonaga, Feynman and Schwinger by the award of a Nobel Prize in October of 1965!. At that time Dirac must have known as did Hans Bethe that the correct result had been obtained by French and Weisskopf for the work they initiated before Lamb had done his experimental work!
In 1966 Dirac published his 1962/63 BGSS lectures on QED as “Lectures on Quantum Field Theory” (London:Academic). By that time Dirac must have explored the field of modern particle physics rather thoroughly. He had apparently been following the quark theory during this period and culminating in attending a full lecture course given by Gell-Mann at Cambridge round about 1966. In 1962 Dirac put forward the idea that the elementary particles might correspond to modes of vibrating membrane (Proc Roy Soc A 268, 57; Proc Roy Soc A 270, 354 ).Within the context of the string theory, the membrane idea could not be revived. In 1986, however, Hughes,Liu and Polchinski showed that a super-membrane could be introduced by combining membrane and supersymmetry. This work was based on constrained dynamics in which the first class constraints were used to construct the Hamiltonian . The fixation of the coordinates brought into the Hamiltonian theory some second-class constraints which reduced the number of degrees of freedom. The Dirac brane may be regarded as a source of a wormhole ( Einstein-Rosen bridge) configuration gluing two Reissner-Norstrom regimes. A point-like source (of co-dimension 3) is not compatible with Einstein’s field equations but the Dirac bubble being a co-dimension 1 extended object is a legitimate source for the gravitational field!(Davidson A & Rubin S 2009 Class Quantum Grav 26 235006)
In his introduction to the latest edition of Feynman’s “The Strange Story of Light and Matter” A Zee wrote:
“Ironically, it was Feynman himself who was responsible for this deplorable state of affairs. What happened was that students easily learned the "funny little diagrams" invented by Feynman. Julian Schwinger once said rather bitterly that "Feynman brought quantum field theory to the masses," by which he meant that any dullard could memorize a few "Feynman rules", call himself or herself a field theorist, and build a credible career. Generations learned Feynman diagrams without understanding field theory. Heavens to Betsy, there are still university professors like that walking around!”

According to Dyson, apparently, ( "George Green and Physics" ,Physics World p33 August 1993) Feynman and Schwinger did not deem it neccessary to use the annihilation and creation operators in their work for sometime. Bethe, Feynman and other American physicists had been successfully doing physics without recourse to formal quantum field theory. Although both Feynman and Schwinger did not get their “numbers” right at first their confidence in themselves and their genius won them the Nobel prize in 1965.
One can ,in a sense, understand why Dirac was moved to comment in his 1970 Physics Today April p29 article that “A theory with mathematical beauty is more likely to be correct than an ugly one that fits some experimental data”
Indeed as Farmelo remarked Dirac had developed it logically, lucidly, and simply- I would say beautifully- from its beginnings and spelt out the details of the computation that led to the Lamb shift and the anomalous magnetic moment of the electron in the hydrogen atom. I think I am right in saying that it is the first coherent and complete self -contained account comprising calculations of both the two important QED effects without recourse to looking up other peoples’ results eg Bethe’s result. No joining mistakes! It pays to work through the calculations to actually see how good Dirac must have been in teaching from his published work through out his life as a research physicist. I did and I think he’s truly a master. I found that one could actually get the good numerical results besides learning a good deal of quantum field theory, using this excellent book ! For collateral reading for actual computation I would recommend referring to section 34 page 337 of Heitler’s tome : 1954" The Quantum Theory of Radiation", Oxford: OUP and to the first of Weinberg’s trilogy: 1995 "The Quantum Theory of Fields", Cambridge: CUP section 14.3 page 579. Weinberg has very good sections also on constrained dynamics as applied to QED. Prof Sein Htoon Yangon University 29 January 2010.