(1) Typed letter in German signed to H. A. Kramers, one page, on letterhead of the Kaiser Wilhelm-Institut für Physik. Berlin, 10 January 1944. Stapled to (2) HEISENBERG. Die beobachtbaren Grössen in der Theorie der Elementarteilchen III. Typed manuscript with his autograph additions and corrections, 23 pages, undated.

[N.p. N.p., 1943-44].

Historic autograph letter from Werner Heisenberg (1901-76), the inventor of quantum mechanics and author of the famous uncertainty principle, to his fellow physicist and sometime collaborator Hendrik Anthony Kramers (1894-1952), discussing Heisenberg’s S-matrix approach to elementary particle physics and urging Kramers to collaborate with him on a joint paper on the subject. Heisenberg enclosed with this letter a typescript draft of this paper, with his autograph corrections. Kramers replied to Heisenberg’s letter after a three-month delay (on 12 April), ultimately declining Heisenberg’s request, but asking that his contributions be acknowledged in the published version of the paper, a request Heisenberg was happy to grant. In the early 1940s, dissatisfied with the limitations of quantum electrodynamics, Heisenberg had proposed replacing quantum electrodynamics with the ‘S-matrix’ approach as the fundamental theory of elementary particles. He introduced this proposal in ‘Die beobachtbaren Grössen in der Theorie der Elementarteilchen,’ the first two parts of which appeared in 1943 (Zeitschrift für Physik Bd. 120, pp. 513-38 & 673-702). “One evening in October 1943 Heisenberg presented his new theory to an informal colloquium in the Kramers home near Leiden in the German-occupied Netherlands … During the discussion of Heisenberg’s talk, Kramers made the insightful remark that if the actual elements of the S-matrix could ever be determined without a complete theory, they would yield a so-called analytic function . . . Back in Berlin, Heisenberg wrote immediately that he had grown ‘more and more enthusiastic’ about Kramers’s remark ‘because I believe that with it one can really arrive at a complete model of a theory of elementary particles’” (Cassidy, Uncertainty, pp. 478-9). “Heisenberg immediately set to work and on 10 January 1944 sent Kramers a draft of a new paper [offered here], which included a study of the analytic properties of the S-matrix for systems consisting of two and three particles (without taking into account annihilation or creation processes). This restriction was the result of a major difficulty encountered by Heisenberg: the S-matrices he had studied in his previous publication, in particular those that included explosions, could not be extended to the complex plane, due to ‘discontinuities in the scattering coefficients which do not seem compatible with the analytical properties of S’ [from the offered letter]. Heisenberg thus cooked up a new expression for S, which had simpler properties when extended to the complex plane. This S-matrix, however, only described the scattering of two particles and did not describe creation or annihilation processes … If the magnitude of the momentum [of the particles] is taken to be purely imaginary, one indeed gets a zero for the S-matrix. The total energy of the two-particle system for this imaginary momentum turned out to be less than the sum of the rest energies and was consequently interpreted by Heisenberg as the energy of a two-particle bound state. This was quite an accomplishment … Heisenberg had shown that the method of analytic extension worked in principle and that it was possible to build a theory on scattering amplitudes and still be able to describe the energies of bound states. This was indeed a major shift in perspective for quantum theory” (Blum, p. 21). In the letter accompanying his draft of part III of ‘Die beobachtbaren Grössen’, Heisenberg expressed his hope that Kramers would be able to help him with the ‘difficulties’ referred to above. The long delay in Kramers’ response to Heisenberg, and his ultimate refusal to collaborate, may reflect the complex political situation at the time: since the beginning of the war, Heisenberg had been heavily involved in the German uranium research program, and the Netherlands had been under German occupation since the spring of 1940. Heisenberg’s letter and draft are here accompanied by the published version of the three parts of Die beobachtbaren Grössen’ (journal issues in original printed wrappers).

The letter is translated from the German as follows:

Dear Kramers!

Unfortunately, your nice letter only reached me in a number of detours. In the meantime, following your discussion, I had calculated the problem of the η-matrix for a few more weeks and wrote a little work that is enclosed with this letter. This work is quite incomplete. There is also the question of whether it could be published in the foreseeable future because of the difficulties at the Leipzig printing works if it was sent to the magazine. So I would actually much prefer it if we could tackle the difficulties mentioned in the paper together and then maybe write a joint paper. The difficulties I mean are indicated on page 3 of the paper. The requirement that S should be an analytic function is in no way fulfilled by itself if one writes an analytic function for the η-matrix, as I did in Part II. For example, the prescription for η given in Part II, Eq. (36) leads to discontinuities in the scattering coefficients, which does not seem to be consistent with the analytical character of S. So one would have to answer the general question: What should a valid η-matrix look like? Probably not much else can be done than to work out such η-matrices for non-relativistic wave equations and then to study their properties. Perhaps you already have some kind of calculations from which you can learn about this problem. Anyway, I think it would be nice if we could both think about this problem and correspond. Despite the air raids on Berlin, my institute is still doing well for the time being. At Christmas I was with my family in Bavaria for 14 days. I wish you and your family all the best for the coming year.

With warmest regards, your Werner Heisenberg

“Study of the formation and propagation of S-matrix theory in the 1940s helps one to understand the rapid development, under radically different scientific and sociological conditions, of the theory of elementary particles in the fifties. The S-matrix is a two-dimensional array whose elements are the transition amplitudes between states of atomic or elementary particle systems …

“In a paper submitted in the summer of 1937, John A. Wheeler proposed to derive the properties of light nuclei, such as energy levels and transition probabilities, with the help of a unitary ‘scattering matrix.’ The elements of this matrix were connected with the transition from incoming to outgoing groups, consisting of a few protons and neutrons within the nuclei in question. Five years later, in a paper submitted to Zeitschrift fiir Physik in September 1942 [‘Die beobachtbaren Grössen I’], Werner Heisenberg introduced a similar, also unitary, S-matrix in the theory of elementary particles …

“Part of the background of Heisenberg’s introduction of the S-matrix was the unsuccessful effort during the 1930s to arrive at a satisfactory quantum field theoretical description of elementary particle properties and reactions. All the schemes considered then – whether quantum electrodynamics, Fermi’s β-decay theory, or Yukawa’s meson theory of nuclear forces – exhibited fundamental difficulties. Heisenberg, in thinking about radical means to remove these difficulties, proposed two ideas, which he assumed to be somehow connected: first, the existence of a universal length l0, such that the known quantum and relativity theory broke down at distances smaller than l0; second, the necessity of involving nonlinear features in the mathematical description of interacting elementary particles. In the later work on the scattering matrix, however, he did not want to propose a program incorporating those ideas; rather, as he expressed it in the introduction to his first publication, ‘the present paper attempts to isolate from the conceptual scheme of quantum theory of wave fields those concepts that probably will not be hit by the future alteration (of the theory of elementary particles) and which may therefore represent a constituent also of the future theory.’

“Because one did not yet know how to formulate a divergence-free theory describing elementary particles, he took the approach of asking ‘the question as to which concepts of the present theory might be kept in future, and this question is roughly identical with the other question as to what quantities in the present theory are observable.’ That is, Heisenberg tried to revive a philosophical program that had earlier led him from the old quantum theory to quantum mechanics, hoping now to replay the earlier success in elementary particle theory.

“Among the unobservable quantities to be kept out of the theory, Heisenberg mentioned distances smaller than 10-13 cm and time intervals smaller than 3 x 10-24 sec. On the other hand, he took as safely observable: (i) the discrete energy values of stationary states of closed systems; and (ii) the asymptotic behavior of wave functions in scattering, emission, and absorption processes. The properties (i) and (ii) could be related to each other, because the phase differences between the incoming and outgoing wave functions had to yield the discrete energy states as well …

“The mathematical formulation of this idea, when applied to relativistic systems of scattering elementary particles, led Heisenberg to introduce his unitary, ‘characteristic’ S-matrix. By writing this matrix as

S = e

he defined a Hermitian matrix η, which allowed a particularly simple description of the physical situation. The matrix η should then replace the defective Hamiltonian formulation of scattering problems in the quantum theory of wave fields; it contained only observable quantities. The obvious problem now was how to calculate the relativistically-invariant η-matrix.

“Several weeks after the submission of his first paper, Heisenberg sent a second with the same title to the Zeitschrift für Physik. In it, he discussed examples of relativistic η-matrices for

(i) a δ-function-like interaction leading to pure scattering of the particles;

(ii) a certain distance-dependent interaction resulting in finite cross sections for the scattering of particles with arbitrarily high energies; and

(iii) an interaction implying the creation of new particles.

“The last two examples could not be adequately represented in the known quantum field theories, nor did they possess an analogue in the classical theory. The S- or η-matrix theory thus provided a suitable tool to treat what Heisenberg called ‘explosive showers’ in cosmic radiation” (Rechenberg, pp. 551-3).

In the spring of 1943, Heisenberg was invited to visit the Netherlands. After some protracted negotiations, the visit took place that October. “The visit to Holland was a success in every respect. Not only did Heisenberg have the chance to speak with his various hosts, both experimentalists and theoreticians, but also he participated in seminars and colloquia (e.g., in the Utrecht colloquium of 24 October where Kramers spoke on ferromagnetism). He saw the new equipment in the Dutch laboratories, such as the high-voltage generator for nuclear reactions in Utrecht (not yet operating) and the cosmic-ray experiments in Amsterdam. His lecture on the S-matrix theory, presented on 21 October in Leiden, attracted guests from other universities (e.g., Adriaan Fokker from Haarlem). He made his deepest impression, however, on Kramers, who wrote him soon afterward: ‘First of all, I would like to tell you once more, how happy your visit has made me, again stimulating old ideals. Also the music has given me pleasure. The only regrettable thing is that we do not meet and see each other more often … I would very much like to discuss with you ideas on the problem and whole formalism of the S-matrix.’ He continued, mentioning an old calculation that he had made with his student S. A. Wouthuysen on the quantum mechanical scattering from a central potential, which could be used to determine the η-matrix in that case.

“The exchange of ideas with Kramers in October 1943 also excited new activity on Heisenberg’s part: in January 1944, Heisenberg wrote to Kramers [in the offered letter]: ‘Your nice letter has unfortunately reached me only after some detours … in following your discussion remark, I calculated for several weeks on the problem of the η-matrix and wrote up a smaller paper, which I enclose in this letter.’ He claimed that the paper was still ‘rather incomplete’ and might not come out in print because of war damage to the Leipzig printing companies. ‘Therefore I would really prefer, if we could approach the difficulties in the paper more closely and perhaps write a joint paper.’

“Kramers answered three months later. He had studied Heisenberg’s paper in detail only very recently, forced by the fact ‘that I have to talk the day after tomorrow on difficulties in the theory of elementary particles at a symposium of our Physical Society.’ He commented on various parts of Heisenberg’s manuscript and criticized, for example, the proof given there for the commutation relations of a generalized quantum field theory as being based, perhaps, on too specific assumptions. He suggested incorporating in the theory a kind of time-reversal symmetry. He also discussed Heisenberg’s three-body example, arguing in an appendix to his letter that a nonrelativistic calculation (involving an assumption that Heisenberg had made) seemed to yield wrong results. Finally, he raised the question of describing scattering problems in external electric and magnetic fields with the help of the η-matrix. On Heisenberg’s proposal of a joint publication, he said that ‘I can only rejoice that my remark on the analytic character of S has borne fruits for your considerations, and if you write this in a publication, my role in the thing is also described.’ Overall, he concluded that ‘the moment is not yet ripe for a joint publication.’ Having received this answer, Heisenberg submitted his manuscript, with a few alterations, to the Zeitschrift fur Physik (Bd. 123, pp. 93-112).

“The main progress in the third communication on observables in elementary particle theory consisted in answering the question whether or not the η-matrix of a given system, besides yielding all scattering cross sections, also described its bound states appropriately. ‘This gap existing in the present considerations is closed by a remark of Kramers, according to which one can treat the [S-]matrix as an analytic function of the state variables k’ and k” [the momenta of the incoming and outgoing particles] and derive the stationary states from its behavior in the complex plane,’ Heisenberg declared. He explained further: ‘The zeros of the matrix S for imaginary k’ provide the position of the stationary states,’ and ‘for the eigenvalues of the matrix η this result means that the poles of η lying on the imaginary k-axis determine the position of the stationary states.’

“The postulate of an analytic S-matrix restricted the possible forms of η-matrices describing given relativistic problems. Thus, Heisenberg found, in particular, that his examples treated in the second communication did not involve smooth enough functions. Hence, he had to construct more suitable models, and he presented in his new paper an η-matrix model theory for two particles having momenta k1 and k2, showing that the analytic continuation in the variable (k1 + k2) indeed provided the bound states. A similar model for the three-body case also seemed to give reasonable results; however the author commented in a footnote added in proof: ‘The treatment of the three-body problem presented below is, as I meanwhile found out, not consistent. Since, for external reasons, I cannot make larger corrections in the text, I shall discuss the situation more accurately in a later paper.’ In spite of such technical difficulties, Heisenberg was confident that more general examples, involving the production and annihilation of particles with spin and electric charge, could be treated as well. Finally, he emphasized that ‘the difficulty of the theory consists in the correct incorporation of the manifold of phenomena, but not in avoiding divergencies’” (Rechenberg, pp. 557-9). The ‘later paper’ Heisenberg referred to, part IV of ‘Die beobachtbaren Grössen’, was written but not published.

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,’ [Freeman] 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” (Rechenberg, 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 seemed 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).

The reasons for Kramers’ delayed response to Heisenberg’s letter, and his refusal to collaborate, cannot be known with certainty. The political situation may have played a role, as suggested above, but so may have the personal relations between Heisenberg and Kramers. The two men first met at Niels Bohr’s institute in Copenhagen in the spring of 1924. “When Heisenberg first arrived in Copenhagen he was overwhelmed by Kramers’ knowledge and sophistication … [but] Heisenberg had a rather difficult time getting along with him … The most important source of tension was Heisenberg’s ambition to replace Kramers as Bohr’s most favoured young scientist” (Dresden, p. 265). A further complication was related to the newly completed Bohr-Kramers-Slater (BKS) theory. “There was great optimism in the Bohr circles that many of the outstanding problems of quantum theory, including the photon-light wave controversy, could now be solved” (Dresden, p. 267). But the BKS theory was opposed by Einstein and Heisenberg’s mentor Pauli. “Heisenberg and Pauli, both as individuals and as a team, expressed increasing misgivings about Kramers’ approach to physics … it could become scathing and ferocious … Although Heisenberg and Pauli often amused themselves at Kramers’ expense, the reasons for their negative reactions varied widely. Pauli had always liked Kramers, he respected him as a person and as a physicist. But he disliked the BKS theory and became increasingly exasperated with Kramers’ dogmatic adherence to it … It is less clear why Heisenberg was so critical of Kramers and enjoyed making fun of him. It is unlikely that the reasons were purely scientific” (Dresden, pp. 267-8)

In the summer of 1924, while Heisenberg toured the Bavarian hills with his boys, Kramers realized that the formalism of the BKS theory could yield a new quantum theory of dispersion, the scattering of an incoming light beam by an atom into its different frequencies. When Heisenberg returned to Copenhagen in September, he joined in the effort. Heisenberg “was certainly anxious to cooperate with Kramers on dispersion-theoretic topics, if for no other reason than that he recognized the importance Bohr attributed to this approach. By the time Heisenberg arrived in Copenhagen, Kramers had obtained most of the results of that paper [Uber die Streuung von Strahlung durch Atome, Zeitschrift für Physik 31 (1925), 681-708]. Heisenberg did participate vigorously in the discussions … Kramers certainly originated the main questions considered in the Kramers-Heisenberg paper; he devised the methods and carried out most of the calculations. The interpretation was altogether due to him. So it is not surprising that Kramers always assumed that he would publish the paper by himself” (Dresden, p. 273). Its eventual appearance as a joint paper was probably due to Bohr: “there is little doubt that in the decision ultimately reached Bohr played some role … [Kramers said] that he put Heisenberg’s name of the paper as a pure courtesy … It was totally against his [Kramers’] style to fight for priorities or insist on his due” (Dresden, pp. 273-4)

“What is special about the Kramers-Heisenberg collaboration and the resulting paper is that this paper was the direct, immediate, and exclusive precursor to the Heisenberg paper on matrix mechanics [following which] Heisenberg’s reputation rose instantly to meteoric heights. Kramers’ reputation suffered a serious setback by the collapse of the BKS theory … These events influenced his [Kramers’] status in and personal approach to physics … in the inner circle everyone (including Kramers) felt and knew that an important phase in Kramer’ life had come to an end … Kramers was still a major, highly respected, and significant physicist, but he no longer was Bohr’s heir apparent” (Dresden, pp. 275-6).

Perhaps this earlier experience of collaboration with Heisenberg had made Kramers wary of repeating it.

Blum, ‘The state is not abolished, it withers away: how quantum field theory became a theory of scattering,’ Studies in the History and Philosophy of Modern Physics 60 (2017), pp. 46-80 (revised version: arxiv.org/pdf/2011.05908.pdf). Dresden, H. A. Kramers. Between Tradition and Revolution, 1987. Rechenberg, ‘The early S-matrix theory and its propagation (1942-1952),’ Chapter 39 in: Pions to Quarks: particle physics in the 1950s (Brown, Dresden & Hoddesdon, eds.), 2009. The three published articles are referenced in: Cassidy, Werner Heisenberg: a bibliography of his writings, 1943a, 1943b, 1944a.

Together 5 items. (1) TLS: 1 page (298 x 210 mm). Browned due to poor quality wartime paper, edges frayed, a few marginal tears. (2) Draft manuscript: 23 pages. Browned due to poor quality wartime paper, a few edges a bit frayed. Journal issues: 8vo (228 x 157 mm), pp. 413-672; 673-790, vii; [ii], 112. Original printed wrappers. Fine.

Item #5304

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