Berlin: Springer, 1943; 1943; 1944.
First edition of Heisenberg’s S-matrix approach to the study of elementary particles, complete in three parts (a fourth part was written but not published). Although S-matrix theory was abandoned after the War due to the success of quantum electrodynamics and quantum chromodynamics, it again became very influential in the 1960s when it led to the development of string theory, which is the best-accepted approach to quantum gravity.
“Heisenberg’s prewar researches in quantum field theory, undertaken in part with Pauli, had led him to the study of cosmic rays, the highest energy particles then available for research. When an extremely high-energy cosmic ray strikes the earth’s atmosphere, it induces a shower of newly created particles and photons. This effect was to be explained on the basis of quantum field theory. Heisenberg’s researches had previously convinced him and others of the inadequacy of field theories for this task. Infinities and divergences plagued all three of the available theories - quantum electrodynamics, Fermi’s theory of beta decay (relating to what is now the weak force), and Yukawa’s meson theory (relating to what is now the strong, or nuclear, force).
“The small size of elementary particles and the close approach of the particles to each other in a cosmic ray collision – which triggered the particle shower – indicated to Heisenberg during the 1930s that the difficulties in quantum field theory could be resolved only if a universal minimum length, a new fundamental constant, were introduced into the theory… according to Heisenberg, quantum mechanics itself broke down when applied to events occurring within regions smaller than the size of an elementary particle…
“Pauli had already suggested that Heisenberg, as he did when formulating the 1925 breakthrough in quantum mechanics, should focus only on observable quantities and attempt to exclude all unobservable variables from the theory. Heisenberg now attempted to do so, at the height of the World War. His effort led to what became after the war his widely studied new theory of elementary particles, the so-called S-matrix theory.
“In his new approach, Heisenberg used this hypothetical fundamental length to define the allowed changes in the momentum and energy of two colliding high-speed elementary particles. This limitation would help identify the properties of the collision that were observable in present theories. Those at smaller distances were unobservable. For two colliding particles, this yielded four sets of observable quantities with which to work: two of these were the properties of the two particles as seen in the laboratory long before they collide with each other; and two were their properties long after the collision. During the collision they approach within a distance of less than the fundamental length and are thus unobservable. These four sets of observable properties could be arranged in a table, or in this type of work, a matrix, which Heisenberg called the scattering or S-matrix.
“Although Heisenberg could not actually specify the four elements of the S-matrix, he demonstrated that it must contain in principle all of the information about the collision. In his second paper, completed in October 1942, Heisenberg further showed that the S-matrix for several simple examples of scattering of particles yielded the observed probabilities for scattering. It also gave the possibility for his favorite phenomenon – the appearance of cosmic-ray explosion showers…
“… one evening in October 1943 Heisenberg presented his new theory to an informal colloquium in Kramers’s home near Leiden in the German-occupied Netherlands… During the discussion of Heisenberg’s talk, Kramers made the insightful observation that if the actual elements of the matrix could ever be determined without a complete theory, they would yield a so-called “analytic function” – that is, a function containing real and imaginary parts… Back in Berlin, Heisenberg wrote immediately that he had grown “more and more enthusiastic” about Kramers remark “ because I believe that with it one can really arrive at a complete model of a theory of elementary particles.” Heisenberg suggested to Kramers that they collaborate on a paper on the subject… But Kramers declined to write the suggested paper with Heisenberg… Heisenberg submitted his third S-matrix paper, with an acknowledgement to Kramers…
“Late in 1944 Heisenberg prepared a fourth installment of his theory – a paper dealing with not just two colliding particles but with many interacting particles, all giving rise to an even more complicated S-matrix. He presented the essentials of the paper in Zurich at the end of 1944, but the paper could not be published before the presses stopped at war’s end” (Beyond Uncertainty, pp. 347-9).
After the war interest in the S-matrix program declined when it was shown that the matrix elements could be calculated using the renormalized quantum electrodynamics of Feynman, Tomonaga and Schwinger. “In his paper ‘The S-matrix in quantum electrodynamics,’ Dyson carried out calculations of the matrix elements, including their renormalization. In the introduction, he remarked that “the Feynman method is essentially a set of rules for the calculation of the elements of the Heisenberg S-matrix and can be applied with directness to all kinds of scattering problems” … Thus, he established contact with the earliest application of the S-matrix scheme in particle physics” (Pions to Quarks, p. 567).
“S-matrix theory was largely abandoned by physicists in the 1970s, as quantum chromodynamics was recognized to solve the problems of strong interactions within the framework of field theory. But in the guise of string theory, S-matrix theory is still the best-accepted approach to the problem of quantum gravity. The S-matrix theory is related to the holographic principle and the AdS/CFT correspondence by a flat space limit… The most lasting legacy of the theory is string theory. Other notable achievements are the Froissart bound, and the prediction of the pomeron” (en.wikipedia.org/wiki/S-matrix_theory).
Brown, Dresden & Hoddesdon (eds.), Pions to Quarks: particle physics in the 1950s, 2009; Cassidy, Beyond Uncertainty, 2010; Cassidy, Werner Heisenberg: a bibliography of his writings, 1943a, 1943b, 1944a.
Pp. 513-538 in Zeitschrift für Physik 120. Band, 7.-10. Heft; pp. 673-702 in Zeitschrift für Physik 120. Band, 11. und 12. Heft; pp. 93-112 in Zeitschrift für Physik 123. Band, 1.-2. Heft. Three complete journal issues, 8vo (228 x 157 mm), pp. 413-672; 673-790, vii; [ii], 112. Original printed wrappers. A fine set.