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Theory of the Scattering Matrix ( 1942-1946) Theory of the Scattering Matrix (1942- 1946)

An Annotation by Reinhard Oehme,

In the following pages, I will give a brief survey of Heisenberg's work on the scattering matrix (S-matrix). I am interested in the ideas which led him to con• sider the S-matrix, the essential results he obtained, and the reason why he abandoned it later. I will also briefly discuss the importance of the S-matrix for the later development of particle .

1. Introduction of the Scattering Matrix

Heisenberg introduced the scattering matrix in 1942 in the first of a series of papers on "the observable quantities in the theory of elementary particles" (paper No. 1, p. 611-636 below). In order to trace his reasons for the proposal, however, I have to go back a few years to Heisenberg's papers on the limitations of quantum theory and on the fundamental length (see papers No.6 and 7, Group 8, pp. 301-314, 315- 330 above). Shortly after the development of , Dirac, Heisenberg, Pauli, and also 1 ordan, Klein, and Wigner had shown that relativistic, noninter• acting wave fields could be consistently quantized by the canonical formalism and its generalization. However, with the introduction of nonlinear terms in the wave equations for the description of interactions, perturbation calculations in• volving the integration over four-momenta of virtual particles were found to give divergent results unless a high momentum cut-off was introduced. These difficul• ties appeared in quantum electrodynamics, in Fermi's theory of P-decay, and in theories of nuclear forces. Heisenberg proposed that they may all be an indica• tion of the existence of a universal length /0 • He viewed /0 as a fundamental con• stant with a significance similar to that of hand c: It should limit the applicability of intuitive concepts and the possibility of measurement. He considered it likely that the physics in dimensions which are small compard to /0 requires new ideas which are contained neither in quantum theory nor in the theory of special relativity. Within this framework, the masses of particles are not considered fundamental constants, but once /0 is built into the formalism, they should follow as energy levels of a universal system, somewhat like the levels of a complicated atomic system follow from the Hamiltonian. On the basis of the empirical knowledge available in the 1930s, Heisenberg tentatively proposed that the fundamental length is of the order of 10 -!3 em [1]. His view was strengthened by indications for the existence of multiparticle production is cosmic ray events [2], which he considered to be possibly associated with the existence of /0 •

605 Believing that the of elementary particles requires a major revision at high energies, possibly because of the existence of a universal length, Heisenberg asked which concepts of the existing theory of wave fields would survive in a more comprehensive theory. He concentrated on "observable quantities", listing the energy eigenvalues of closed systems and the probabilities for collisions and for absorption and emission. The latter quantities are asso• ciated with the asymptotic behavior of wave fields and can be characterized by a matrix labelled according to momenta, spin, and other quantum numbers of in• coming and outgoing noninteracting particle states. This is the S-matrix [3]. The notion of the S-matrix is, of course, a very general one, and in principle it has nothing to do with the divergence problems of field theories or with the ques• tion of a smallest length. Nevertheless, it was within this framework that Heisen• berg introduced the S-matrix. At the time when Heisenberg wrote the papers, he viewed this matrix as a primary quantity to be calculated directly. It was to replace the Hamiltonian, which had led to the divergence difficulties. Later, after it had been learned how to handle covariant , the S-matrix became the quantity to be calculated from quantum field theory. More generally, it became the physical quantity to be extracted from a field theory formalism, which itself may contain many unphysical elements as a price to pay for more mathematical simplicity.

2. Heisenberg's Papers

In the first two papers on observable quantities in the Zeitschrift fur Physik (Nos. 1, 2), Heisenberg explores in detail the implications for the S-matrix of Poincare in variance and of the conservation of probability. He extracts the energy-momentum-conserving ~-function and shows how to calculate cross-sec• tions for various processes in terms of matrix elements of R, where S = I+ R. The requirement of unitarity is derived, and a Hermitian 11 matrix is introduced by S = ei 17• The second paper (No. 2, pp. 637-666 below) contains special, simple Ansatze for 1'f, and the corresponding calculations of scattering cross• sections and, in particular, of production processes for many particles [4]. A new and important element is added in the third paper of the series, which was submitted in 1944 (No. 3, pp. 667-686). During a visit to Leiden in the fall of 1943, his friend H. A. Kramers had told Heisenberg about analytic properties of S-matrix elements as a function of momentum variables, with the physical amplitudes being given by the appropriate boundary values for real momenta. He also told him about the connection between single-particle states and simple poles in the appropriate matrix elements of S [5]. In binary scattering situations, these poles appear on the positive imaginary axis in the momentum variable, or below the physical threshold on the real energy axis. In the third paper (No. 3), Heisenberg incorporates analytic properties and single-particle poles in his S-matrix scheme. He studies these features in models with two- and three-particle channels. He also requires that the generalS-matrix should agree with the one obtained from conventional Hamiltonian formalism in the limit where the universal length can be considered very small.

606 There is more discussion of analytic properties in the letters exchanged between Pauli and Heisenberg during the years 1946-48. In September of 1946, Pauli reported to Heisenberg some results of "his Chinese collaborator Ma in Princeton" on the analytic structure of amplitudes for the scattering in an exponential potential. Ma finds that singularities on the positive imaginary k-axis appear, which are not related to stationary states and which are not present for cut-off potentials. Furthermore, Res Jost sent a letter to Heisenberg about "false" poles or zeros of the S-matrix. Because of these and other problems en• countered by Heitler and Hu, Pauli wrote in a letter dated July 1947: "I personal• ly consider the idea of an analytic continuation of the S-matrix to be a complete flop." [6] Yet, Heisenberg was not particularly worried about these apparent dif• ficulties. Writing to Mj~jller in June 1947, he expressed the hope that a future scheme for the construction of the S-matrix will make it possible to distinguish between "true" and "false" singularities. Today, we know that these "false" singularities, which are present (with vary• ing character) for all potentials with exponential fall-off, are just remnants of what appears as crossed-channel singularities in amplitudes obtained from relativistic field theories [7]. These are perfectly physical. They simply belong to another (crossed) channel described by the same analytic S-matrix. Heisenberg summarized his ideas concerning a finite S-matrix theory of elementary particles in his first postwar paper entitled "Der mathematische Rahmen der Quantentheorie der Wellenfelder" (The Mathematical Framework of the Quantum Theory of Wave Fields) in 1946 (No.5, pp. 699-713) and his 1947 Cambridge lecture [8]. With the essential restrictions obtainable from general principles taken into account, the S-matrix theory was still a very general scheme which had rather limited predictive power. In May 1946, Pauli wrote to Heisenberg: "I have read your papers on the S-matrix with much interest, but they remain only a program as long as no method is given for a theoretical deter• mination of the S-matrix" [9].

3. Later Developments

In the late 1940s, important progress was made in the treatment of Lagrang• ian field theories by Tomonaga, Schwinger, Feynman, and Dyson [10]. With extensive use of relativistic covariance, a renormalization scheme was formulat• ed, which made it possible to eliminate the ultraviolet divergences in the weak coupling perturbation theory. For renormalizable interactions, it then became possible to calculate the elements of the S-matrix as a (formal) expansion in powers of the . In the case of quantum electrodynamics, the comparison of these calculations with experimental results provided the well• known spectacular successes which added greatly to the confidence invested in relativistic quantum field theory. Late in 1949, when I came to Gottingen as a young student, I had just studied the papers of Dyson on the calculation of the S-matrix to arbitrary order in the coupling parameter [11]. Heisenberg was deeply interested in these results and asked me to give a number of lectures on the topic. He pointed out that, even

607 with the renormalization scheme, no satisfactory closed formulation of field theory was at hand. In particular, the power series expansion in the coupling parameter was inadequate for the strong interactions. On the other hand, Hei• senberg seemed, at that time, to have accepted the fact that his S-matrix scheme lacked the rules for explicit calculations, and that the restrictions from general principles and correspondences were insufficient. He soon embarked upon his attempts to formulate a unified field theory. Initially, his Ansatz again incorporated the idea of a universal length, and it was nonrenormalizable in the sense of weak coupling methods [12]. Already in his contribution prepared for the 1939 Solvay Conference [13], Heisenberg had pointed out the important difference between theories with dimensionless and with dimensionful coupling constants. After the development of renormalization methods, theories with dimensionless coupling parameters were shown to be generally renormalizable (or super-renormalizable in the case of positive mass dimensions). On the other hand, couplings with inverse mass dimensions were shown to lead to nonrenormalizability in the conventional sense [14]. I will close with a few words on later developments concerning the S-matrix, which Heisenberg followed with interest, but in which he played no part. In the early 1950s, important new results were obtained concerning the analytic proper• ties of S-matrix elements, in particular, scattering amplitudes. They were derived on the basis of general postulates underlying quantum field theory and did not involve perturbation expansions [15]. The most important postulate was micro• scopic causality: the commutativity (or anticommutativity) of Heisenberg field operators at points with spacelike separation. With S-matrix elements being Fourier transforms of retarded products of these operators, the support proper• ties resulting from microcausality and from spectral conditions allowed the derivation of analytic properties, including the important crossing relations which provide an analytic connection between amplitudes for particle-particle and particle- scattering. The resulting dispersion relations for forward and nearforward scattering became essential tools for the understanding of meson- and nucleon-nucleon interactions [16]. These relations gave a rather complete description of the low-energy properties of the amplitudes for these reactions. Furthermore, the exact sum rules obtained were the first success• ful theoretical results in strong interaction physics [17], and to this date, they are in excellent agreement with the results of experiments carried out at higher and higher energies. On the basis of the analytic properties discussed above, as well as more de• tailed ones abstracted from Feynman graphs, it became possible to introduce complex methods, which - under the name of Regge theory - gave a rather successful description of the high-energy behavior of hadronic amplitudes [18]. Duality relations between Regge trajectories and sequences of resonances in crossed channels led to meromorphic models of amplitudes with Regge-like asymptotic behavior away from the real axis [19]. Further sophistication of these formulations gave rise to a general theory of dual models [20], which was even• tually linked to theories of strings [21]. The latter were introduced, a priori, in order to account for quark confinement in . They were generalized to

608 include fermionic degrees of freedom, a development which led to an early appearance of in particle physics [22]. Consistent supersym• metric theories can be formulated in ten space-time dimensions [23], and they have massless spin 2 and spin 3/2 excitations in their spectrum. The reduc• tions to four dimensions can be achieved by Kaluza-Klein methods. Although string theories were not successful as theories of hadrons at a length scale of 10 -!3 em, it was recognized that they are very interesting candidates for a unified theory of all interactions, including gravity, with the characteristic length scale of the strings being of the order 10-33 em (Planck length) [24]. In contrast to local field theories, string theories are consistent quantum theories which necessarily contain gravity. The characteristic string features should be im• portant at the scale of the Planck length, whereas for much longer distances ordinary quantum field theory should be applicable as an effective theory. At the time of writing, candidates for superstring theories have been found which promise to be finite and physically interesting. Perhaps they will provide the key to Heisenberg's dream of a unified theory.

4. Acknowledgements

It is a pleasure to thank Peter G. 0. Freund in Chicago and Helmut Rechen• berg in Miinchen for helpful remarks. I am particularly indebted to H. Rechen• berg for providing me with copies of letters of Heisenberg, Pauli, Kramers, and others. Thanks are also due to S. A. Wouthuysen for an interesting exchange of letters, and to Laurie M. Brown for helpful conversations.

References

In his contribution to the issue of the Annalen der Physik published in honor of the 70th birthday of Max Planck (paper No.6, Group 8, pp. 301- 314 below), Heisenberg also mentions the Planck length /p = VG = 4 x 10-33 em, but considers it too small to be identified with /0 • 2 Some time later, there were indications, for a while, that the multiple production in nuclear colli• sions could be explained as a cascade process, but further experiments showed that there are definitely collisions where many particles are created in a single act. 3 As Heisenberg points out in paper No. 1, special cases of the S-matrix were previously used in nonrelativistic quantum mechanics, in particular in connection with the partial wave decompo• tion of amplitudes for scattering in a central potential. The S-matrix had first been used by J.A. Wheeler in more complicated many-channel situations in nuclear physics. [See Phys. Rev. 52, 1107 (1937)]. In a letter from Rome dated March 1943, Gian-Carlo Wick asks some questions about Heisenberg's first twoS-matrix papers, and he also informs him about the work of Gregory Breit on the S-matrix in SchrOdinger theory [Phys. Rev. 58, 1068 (1940)], which was aimed mainly toward a description of resonances in terms of complex poles in the energy variable. Heisenberg was apparently unaware of these papers. 4 The general connection between S-matrix elements and various cross-section had also been studied extensively by C. Ml')ller in Copenhagen, who corresponded with Heisenberg during and after the war [see K. Danske Vidensk. Selskab Mat.-Fys. Medd. 23, No. 1 (1945); 22, No. 19 (1946)]. 5 Kramers and Wouthuysen had discovered these properties for nonrelativistic scattering ampli• tudes in about 1940, as remarked in a letter by Kramers to Heisenberg written in April 1944.

609 6 The actual quotation from Pauli's letter is: "/ch personlich halte die Idee der analytischen Fort• setzung der S-Matrix fur einen vollstandigen Fehlschlag." 7 In the nonrelativistic limit of field theoretical scattering amplitudes, one generally finds that these crossed-channel singularities are related to branch cuts appearing in corresponding ampli• tudes obtained from Schrodinger theory with superpositions of Yukawa potentials: V(r) =fda e(a)(e- ur lr). In an S-wave amplitude, the Yukawa potential gives rise to logarithmic branch points on the negative energy axis. With the special choice e(a)ao'(a-a), one obtaines an exponential potential of range a- 1, and the logarithmic branch points become simple poles. Within the framework of field theory, we do not expect e(a) to be a o'-function. 8 W. Heisenberg: The present situation in the theory of elementary particles, in Two Lectures (Cambridge, University Press 1949), pp. 5- 25; reprinted in Collected Works B, pp. 444-449 9 The actual remark is: "Deine Arbeiten uber dieS-Matrix habe ich mit groj3em Interesse gelesen, aber sie bleiben so lange nur ein Programm, als keine Methode angegeben wird, um dieS-Matrix theoretisch zu bestimmen." 10 For references, see for example the collection of papers on Quantum Electrodynamics (1. Schwin• ger, ed., Dover Publications, New York 1958). 11 F. Dyson: Phys. Rev. 75, 476, 1736 (1949) 12 See for example W. Heisenberg: Der gegenwartige Stand der Theorie der Elementarteilchen, Die Naturwissenschaften 42, 637-641 (1955); reprinted in Collected Works B, pp. 547-551. 13 W. Heisenberg: Bericht iiber die allgemeinen Eigenschaften der Elementarteilchen (1939), in Col• lected Works B, pp. 346-358 14 S. Sakata, H. Umezawa, S. Kamefuchi: Progr. Theor. Phys. 7, 377 (1952) 15 For a review and references, see for example R. Oehme: Forward Dispersion Relations and Microcausality, in Quanta, ed. by P.G.O. Freund, C.J. Goebel, Y. Nambu, ( Press, Chicago and London 1970), p. 309; R. Oehme, Nuovo Cim. 4, 1316 (1956), (see appendix, in particular); N.N. Bogoliubov, D.V. Shirkov: Introduction to the Theory of Quantized Fields (Interscience, New York 1959) Chap. IX; H. Bremermann, R. Oehme, J.G. Taylor: Phys. Rev. 109, 2178 (1958); H. Lehmann: Suppl. Nuovo Cim. 14, 153 (1958). 16 For reviews and references, see for example M.L. Goldberger: In Relations de Dispersion et Particules Etementaires, ed. by C. de Witt and R. Omnes (Hermann, Paris 1960), pp. 15 -151; G.F. Chew: S-Matrix Theory of Strong Interactons (Benjamin, New York 1961); S. Mandelstam; Reports of Progress in Physics 25, 99 (1962). 17 M.L. Goldberger, H. Miyazawa, R. Oehme: Phys. Rev. 99, 986 (1955); M.L. Goldberger, Y. Nambu, R. Oehme: Ann. Phys. (New York) 2, 266 (1956) 18 For a review and references, see for example R. Oehme in: Strong Interactions and High Energy Physics, ed. by R.G. Moorhouse (Oliver and Boyd, Edinburgh and London 1964), pp. 164-256. 19 G. Veneziano: Nuovo Cim. 57A, 180 (1968); R. Oehme: Nucl. Phys. B16, 161 (1970) 20 For a review and references, see for example, Dual Theory, ed. by M. Jacob (North-Holland, Amsterdam 1974); J. Scherk; Rev. Mod. Phys. 47, 123 (1975). 21 Y. Nambu; in Proc. of the Intern. Conf. on Symmetries and Quark Models, ed. by R. Chand (Gordon and Breach, New York 1970) p. 269 22 P. Ramond: Phys. Rev. D3, 2415 (1971); A. Neveu, J.H. Schwarz: Nucl. Phys. B31, 86 (1971) 23 For a review and references, see M.B. Green, J.H. Schwarz, E. Witten: , 2 volumes (Cambridge University Press, Cambridge 1987); Unified String Theories, ed. by M. Green, D. Gross (World Scientific, Singapore 1986). 24 J. Scherk, J.H. Schwarz: Nucl. Phys. B81, 118 (1974); T. Yoneya: Nuovo Cimento Letters 8, 951 (1974)

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