W. Heisenberg's 1943–44 papers on 'the observable quantities in the theory of elementary particles' proposed rebuilding quantum theory on the scattering matrix S=eiηS=e^{i\eta} alone, without a Hamiltonian, in order to incorporate bound states via analyticity and to escape the divergences of local quantum field theory through nonlocal nonlinear integral equations with invariant kernels A,B,C,…A,B,C,\dots. The third paper (Z. Physik 123, 93–112, 1944), the source paper, explicitly flagged what it left open: analyticity of SS was postulated, not derived; the additive three-body ansatz η=η(12)+η(23)+η(31)\eta=\eta(12)+\eta(23)+\eta(31) was arbitrary; the three-body treatment was approximate and acknowledged by the author as not fully correct; no kernels were shown to admit convergent, physically acceptable quantum solutions; no principle selected the correct kernels; and no explicit divergence-free interacting nonlocal theory was constructed. Later literature transformed parts of this program — the bound-state-as-pole correspondence and dispersion relations were established within local, Hamiltonian-based field theory, and genuine three-body interactions became standard few-body physics — but the Hamiltonian-free cores survive: the modern S-matrix bootstrap — the program that constrains scattering amplitudes from consistency conditions such as unitarity, analyticity and crossing alone, without a Lagrangian — still assumes the analyticity Heisenberg could not derive, and no unitary, causal, divergence-free interacting nonlocal quantum theory with a selection principle for its kernels has been constructed. These cores remain open.

Background

In the early 1940s, local quantum field theory — the framework in which particles are excitations of fields obeying differential wave equations with point-like (delta-function) interactions — was beset by divergences: higher-order corrections to scattering amplitudes came out infinite. Werner Heisenberg responded with a radical proposal published in three papers titled 'Die beobachtbaren Größen in der Theorie der Elementarteilchen' ('The observable quantities in the theory of elementary particles') in 1943–1944: abandon the (unobservable) wave function and Hamiltonian, and build the theory directly on the scattering matrix SS, the operator whose matrix elements connect the asymptotic free 'in' and 'out' states of a collision. Writing S=eiηS=e^{i\eta} with a Hermitian matrix η\eta (which makes unitarity of SS automatic), Parts I–II (Heisenberg, 1943a,b) introduced this η-matrix formalism. Part III (the source paper) addressed two gaps. First, whether the η-matrix determines the discrete stationary (bound) states of composite systems, not just scattering: following a suggestion of H. A. Kramers (as documented by Blum, 2017), Heisenberg postulated that SS is analytic in the particle momenta and showed that bound states correspond to poles of the analytically continued η-matrix. He demonstrated this in a toy model of a single neutral spin-zero boson species with conserved particle number: an explicit η-matrix (an arctangent of a relativistic invariant) whose eigenvalues give two-particle cross sections and whose poles give a two-body bound-state mass. Extending to three particles, he assumed the additive ansatz η=η(12)+η(23)+η(31)\eta=\eta(12)+\eta(23)+\eta(31) (each η(ij)\eta(ij) acting only on the pair), reduced the three-body problem to an integral equation, solved it approximately via a mean-value replacement of an integration variable, and extracted candidate three-body rest energies — a treatment the author himself flagged as flawed in a footnote of the published paper. Second, seeking a foundation for such η-matrices free of the divergences of local theory, he proposed replacing local wave equations by nonlocal nonlinear integral equations in momentum space, schematically

A(k)φ(k)+∫dk′ B(k,k′) φ(k′)φ(k−k′)+∫dk′ dk′′ C(k,k′,k′′) φ(k′)φ(k′′)φ(k−k′−k′′)+⋯=0,A(k)\varphi(k)+\int dk'\,B(k,k')\,\varphi(k')\varphi(k-k')+\int dk'\,dk''\,C(k,k',k'')\,\varphi(k')\varphi(k'')\varphi(k-k'-k'')+\cdots=0,

where A,B,C,…A,B,C,\dots are Lorentz-invariant integral kernels ('nonlocal' meaning the interaction couples fields at finitely separated spacetime points, smearing the point interaction over a fundamental length). He constructed a conserved energy–momentum tensor for such equations and quantized them without canonical commutators by requiring Jμφ(k)−φ(k)Jμ=−kμφ(k)J_\mu\varphi(k)-\varphi(k)J_\mu=-k_\mu\varphi(k) with JμJ_\mu the total four-momentum, leading to operator (matrix) equations for the field amplitudes (J′∣φ∣J′′)(J'|\varphi|J'') in which all particle masses — 'elementary' or composite — appear as discrete four-momentum eigenvalues. The paper itself acknowledged what remained open: the analyticity of SS was postulated, not derived, in the absence of a Hamiltonian; the additive three-body ansatz was arbitrary, and genuine many-body contributions to η were expected; the three-body treatment was approximate and (by the author's own footnote) not correct; no explicit kernels A,B,C,…A,B,C,\dots were shown to admit convergent, physically acceptable quantum solutions; no principle selected the correct kernels; and no concrete interacting, divergence-free nonlocal quantum theory was constructed. The later history shapes how these questions are now posed. Renormalized local quantum field theory — in which divergences are absorbed into a finite number of redefined physical parameters — became the standard framework, and the reduction formalism of Lehmann, Symanzik, and Zimmermann (1957) showed how scattering amplitudes are extracted from local field theory, within which analyticity properties could be established (dispersion relations, i.e. integral identities fixing the real part of an amplitude from its imaginary part, and crossing, i.e. the statement that amplitudes for rearranged processes are analytic continuations of one another). The bound-state-as-pole correspondence became foundational physics (resonances and the modern amplitudes program). But Heisenberg's specific demand — analyticity derived without assuming a Hamiltonian or local fields — remains unresolved: the modern S-matrix bootstrap — the program that constrains or determines scattering amplitudes from consistency conditions alone, taking unitarity, analyticity and crossing as inputs rather than deriving them — assumes analyticity and crossing as postulates (Correia, Sever, Zhiboedov, 2021). The nonlocal program foundered on the tension identified by Pais and Uhlenbeck (1950): it is 'difficult, if feasible' to reconcile convergence with positive-definite energy and causal behavior, a tension that persists in modern infinite-derivative ('ghost-free', i.e. free of negative-norm ghost states) field theories, which ameliorate ultraviolet divergences but exhibit acausal behavior (Green functions nonzero at spacelike separation) (Buoninfante, Lambiase, Mazumdar, 2019). The demand for a principle selecting the kernels — 'non-arbitrariness' — later drove the passage from S-matrix theory to the dual resonance models (the 1968–1974 Veneziano-lineage amplitudes that developed into string theory) and to string theory (van Leeuwen, 2024), and remains unresolved.

Problem Statement

Complete — or rigorously refute the possibility of — Heisenberg's Hamiltonian-independent theory of elementary particles based on scattering data alone, as left open by the 1944 paper. The single scientific objective is a complete, consistent, finite theory of interacting particles and their bound states determined by the SS-matrix (equivalently the η\eta-matrix in S=eiηS=e^{i\eta}) without assuming an underlying Hamiltonian, and underpinned by quantized nonlocal nonlinear wave equations. This comprises three linked requirements: (1) Analyticity: derive, from principles that assume no Hamiltonian and no local fields, the analyticity properties of SS and η\eta that license identifying the discrete stationary states with poles of the analytically continued scattering amplitudes — or prove that such a derivation is impossible without locality. (2) Many-body structure: define the η\eta-matrix elements for multiparticle states without the arbitrary additive ansatz η=η(12)+η(23)+η(31)\eta=\eta(12)+\eta(23)+\eta(31) (which, as Heisenberg himself conceded, leads only to two-body forces in a strict sense, its three-body bound state being saturated and non-interacting with the third particle), determine whether genuine non-additive many-body contributions — contributions to η\eta that are not expressible as a sum of pairwise terms each acting on an individual pair — are required, and determine the discrete (bound-state) spectrum of the resulting genuine many-body framework (the framework defined by these multiparticle η\eta-matrix elements), quantifying the impact of any non-additive contributions on that spectrum. (3) Nonlocal foundation: for the general nonlocal nonlinear momentum-space equation A(k)φ(k)+∫dk′ B(k,k′) φ(k′)φ(k−k′)+∫dk′ dk′′ C(k,k′,k′′) φ(k′)φ(k′′)φ(k−k′−k′′)+⋯=0A(k)\varphi(k)+\int dk'\,B(k,k')\,\varphi(k')\varphi(k-k')+\int dk'\,dk''\,C(k,k',k'')\,\varphi(k')\varphi(k'')\varphi(k-k'-k'')+\cdots=0 and its quantization via the total four-momentum commutator Jμφ(k)−φ(k)Jμ=−kμφ(k)J_\mu\varphi(k)-\varphi(k)J_\mu=-k_\mu\varphi(k), establish whether nontrivial invariant kernels A,B,C,…A,B,C,\dots exist whose quantized operator equations admit convergent, physically acceptable quantum solutions; construct at least one explicit interacting divergence-free example with physically meaningful particle content; and identify a physical principle that singles out the correct kernels uniquely (or up to finitely many free parameters), or prove that no such principle exists — making the theory predictive rather than a parametrization of possibilities. The terms used in requirement (3) are defined as follows. 'Nontrivial' means interacting: η\eta is not that of a free theory. 'Convergent' (equivalently 'divergence-free') means free of the ultraviolet divergence difficulties that motivated the extension to integral equations — finite SS-matrix elements — together with mathematical well-definedness of the quantized operator equations (existence/solvability of the coupled nonlinear integral equations). 'Physically acceptable' means unitary (S=eiηS=e^{i\eta} with η\eta Hermitian on the physical state space, positive norm, no ghost states), positive-energy, and causal under a causality standard whose minimum is the Pais–Uhlenbeck criterion of reconciling convergence, positive-definite energy, and causal behavior of the state vector — a standard that must exclude the acausal divergence-free infinite-derivative constructions cited above (a weaker bar such as SS-matrix macrocausality alone does not suffice); local commutativity (microcausality) is not imposed as the standard, since it is violated even by ghost-free infinite-derivative theories. 'Physically meaningful particle content' means discrete four-momentum eigenvalues interpretable as the masses of elementary and composite particles, per the source's quantization scheme. The three requirements jointly close the gaps the source paper itself flagged: without (1) the bound-state identification rests on a postulate; without (2) the spectrum of composite systems is undetermined; without (3) the η\eta-matrix has no finite, selectable foundation. All three requirements are to be established within one Hamiltonian-free framework, or the possibility of such a framework is to be refuted by a rigorous no-go result. By 'within one Hamiltonian-free framework' is meant that the three requirements are met by one and the same theory — the multiparticle η\eta-structure of requirement (2) and the analyticity of requirement (1) are those of the nonlocal theory of requirement (3) — not by three mutually independent results. The negative branch of requirement (1) — a proof that a Hamiltonian-free derivation of analyticity is impossible without locality — and the negative branch of requirement (3) — a proof that no admissible kernels exist — are each rigorous no-go results of the required kind; likewise, a requirement-(2) result proving that no consistent multiparticle η\eta-structure can be defined is a rigorous no-go, whereas determining whether genuine non-additive contributions are (or are not) required is part of requirement (2) rather than a negative branch. Every sub-part of the three requirements is mandatory; within requirement (3) an explicit interacting divergence-free construction also settles the question of existence of such kernels.

References

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