## Die “beobachtbaren Grössen” in der Theorie der Elementarteilchen. Offprint from: Zeitschrift für Physik 120. Band, 7.-10. Heft. [With:] II. Die beobachtbaren Grössen in der Theorie der Elementarteilchen II. Offprint from: Zeitschrift für Physik 120. Band, 11. und 12. Heft.

Berlin: Springer, 1943.

First edition, extremely rare offprints, of Heisenberg’s S-matrix approach to the study of elementary particles. “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” (*Beyond Uncertainty*, pp. 347-9).

“In a series of papers during the period 1943-1946 Heisenberg proposed as an alternative to quantum field theory a program whose central entity was a matrix he denoted by S and termed the ‘characteristic matrix’ of the scattering problem … Heisenberg wanted to avoid any reference to a Hamiltonian or to an equation of motion and base his theory only on observable quantities. This emphasis on observables was a return to an idea which had proven useful in his earlier successful formulation of matrix mechanics. Heisenberg’s stated purpose in his seminal paper [I.], ‘The ‘Observable Quantities’ in the Theory of Elementary Particles’, was to abstract as many general, model-independent features of *S *as possible. In the abstract and introduction to that paper we read (p. 513):

‘The known divergence problems in the theory of elementary particles indicates that the future theory will contain in its foundation a universal constant of the dimension of a length, which in the existing form of the theory cannot be built in in any obvious manner without a contradiction. In consideration of such a later modification of the theory, the present work attempts to extract from the foundation of quantum field theory those concepts which are not likely to be discarded from that future, improved theory and which, therefore, will be contained in such a future theory.’

‘In recent years, the difficulty, which still stands in the way of a theory of elementary particles, has been pointed up in many ways. This difficulty manifests itself surprisingly in the appearancr of divergences (infinite self energy of the electron, infinite polarization of the vacuum, and the like), which hinders the development of a mathematically consistent theory and must probably be perceived as an expression of the fact that, in one manner of speaking, a new universal constant of the dimension of a length plays a decisive role, which has not been considered in the existing theory.’

“This paper is remarkable for the number of new ideas it introduces, many of which would be put on a firm mathematical basis only years later. Using a momentum-space representation, he defined the S-matrix as the coefficient of the outgoing waves in the scattering state … This paper contains (in an often symbolic and certainly non-rigorous fashion) the essential elements of formal time-dependent scattering theory, which would later be further developed, for example, by Lippmann and Schwinger (1950), by Gell-Mann and Goldberger (1953) and by Brenig and Haag (1959) … Heisenberg certainly brought the concept of the S**-**matrix to the attention of theoretical physicists. It has remained one of the central tools of modern physics …

“The Hermitian phase matrix *η *[related to the S-matrix by *S = *e* ^{iη}*] was the primary quantity to be determined in the theory, essentially by guessing at that time. Its determination would replace the equations of motion, such as the Schrödinger equation or the Hamiltonian formalism of quantum field theory. Heisenberg also indicated [I.] how the matrix

*η*could actually be calculated in those cases where a Hamiltonian

*H*was known, as in quantum mechanics and field theory. In his second paper [II.] he computed S for particular models of

*η*(i.e., essentially for certain enlightened guesses for the form of

*η*) …

“Some sense of the status of Heisenberg’s early S-matrix program can be gotten from Kramers’ remarks at a symposium held in Utrecht during the spring of 1944:

‘Heisenberg’s recent investigations concerning the possibility of a relativistic description of the interaction that is not based on the use of a Hamiltonian with interaction terms in a Schrödinger equation. Heisenberg considers only free particles and introduces a formalism (‘scattering matrix’) by means of which the result of a short interaction (scattering) between these particles can be described. Formerly the scattering matrix could be derived from the Hamiltonian, but now we are to consider the scattering matrix as fundamental. We do not care whether a Schrödinger equation for particles in interaction exists; we do care which correspondence requirements exist and how the scattering matrix can obey them. It is interesting that the scattering matrix is also able in principle to answer the question in which stationary states the particles considered can be bound together. These are related to the existence and the position of zeros and poles of the eigenvalues of the scattering matrix, considered as a complex function of its arguments. Heisenberg could already give a (very simple) model of a two-particle system, in which a perfectly sharply relativistically determined stationary state occurs, while there are no divergence difficulties whatsoever.

‘However promising, this is still only a beginning, and in particular with regard to a correct description of the electromagnetic fields of photons I expect difficulties, which the investigations in this direction will have to overcome. Fortunately, Heisenberg’s program is still open in several respects, and one may perhaps expect a great deal from a fortunate combination with further ideas’” (Cushing, pp. 112-116).

“The scattering matrix, or S-matrix, as it came to be called, had an interesting history during World War II. News of Heisenberg’s work was brought to Japan by German submarine in the form of a letter from Heisenberg to Nishina. A unitary S-matrix appeared in Japanese literature of the 1940s in analyses of microwave junctions by Tomonaga and his group. The US wartime microwave work also employed the S-matrix in descriptions of junctions. Around this time, Stückelberg independently introduced an analogue of the S-matrix. In the physics of antenna impedance matching, a transformation very similar to Stückelberg’s had been proposed even earlier, forming the basis of the frequently used *Smith chart*” (*Twentieth Century Physics*, p. 680).

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).

“In the 1960s, before the advent of the now standard quantum field theory of the strong nuclear forces, many theorists had given up on the idea of describing these forces in terms of any quantum field theory. Instead they sought to calculate the properties of nuclear particles and mesons through a positivistic program, known as ‘S-matrix theory’, that avoids referring to unobservable quantities like the field of the electron. In this program one proceeds by imposing physically reasonable conditions on observable quantities, specifically on the probabilities of all possible reactions among any numbers of particles … It turned out to be extraordinarily difficult to find any set of probabilities that satisfied all of these conditions. Finally, by inspired guesswork, a formula for reaction probabilities was found in 1968-9 that seems to satisfy all these conditions. Shortly after, it was realized that the theory that had been discovered was in fact a theory of strings” (*Twentieth Century Physics*, p. 2038).

“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).

A third part of Heisenberg’s work on the S-matrix appeared in 1944, studying its properties as an ‘analytic’ function. A fourth part was written later in the same year but the paper could not be published before the presses stopped at war’s end.

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. Cushing, ‘The importance of Heisenberg’s S-matrix program for the theoretical high-energy physics of the 1950s,’ Centaurus 29 (1986), pp. 110-149.

Two offprints, 8vo (230 x 157 mm), pp. [513], 514-538; [673], 674-702 (three small closed marginal tears to first leaf, first three leaves with crease in fore-edge margin, a bit soiled). Self-wrappers as issued.

Item #5971

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Price:
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