In his 1943 program paper 'Die beobachtbaren Größen in der Theorie der Elementarteilchen' (Z. Phys. 120, 513), Werner Heisenberg argued that the divergence difficulties of quantum field theory point to a universal length constant that threatens the Hamiltonian framework, and proposed to base the theory of elementary particles on the observable quantities alone: the discrete energies of confined stationary systems and the scattering matrix SS, with its Hermitian phase matrix η\eta defined by S=eiηS=e^{i\eta}. The paper's final section is devoted to relations between observables that do not follow from the general properties of SS, and the companion papers of 1943-44 explicitly study a theory in which, in place of a Hamiltonian, the η\eta-matrix is the given fundamental object, leaving open what conditions fix it. The problem: which fundamental, non-Hamiltonian conditions (principles, axioms, symmetry or regularity requirements) determine SS or η\eta within the framework of unitarity and Lorentz invariance? The core question survives and is actively open: the determination, or sharp structural characterization, of relativistic scattering amplitudes from consistency principles alone, especially in four dimensions, for massless particles, and for gravity.

Background

In the early 1940s, quantum field theory (QFT) — the quantum theory of systems with infinitely many degrees of freedom, normally formulated through a Hamiltonian operator generating time evolution of local fields — suffered from unremovable divergences: computed quantities such as the electron's self-energy came out infinite at every order. Heisenberg drew a radical conclusion in 'Die beobachtbaren Größen in der Theorie der Elementarteilchen' (Z. Phys. 120, 513 (1943)) and its sequels (Z. Phys. 120, 673 (1943); Z. Phys. 123, 93 (1944)): the divergences suggest that the future theory will contain a universal constant of the dimension of a length — a 'minimal length' of order 10−13 cm10^{-13}\,\mathrm{cm}, the then-known nuclear scale — which cannot be built into the existing theory without contradiction; exact localization of events in space and time may therefore lose its meaning, and only those quantities that survive without a local space-time description should be kept. The surviving 'observable quantities' are the discrete energy eigenvalues of spatially confined (bound) systems and the complete asymptotic behavior of scattering wavefunctions, encoded in the scattering matrix SS (first introduced for nuclear scattering by Wheeler in 1937). SS is the linear operator that maps the amplitudes of incoming asymptotic particle channels of a collision to the outgoing ones; conservation of probability is expressed by unitarity, S~∗S=1\tilde S^{*}S=1 (with the tilde denoting interchange of rows and columns and ∗* complex conjugation). For each fixed total energy and total momentum, the unitary SS can be written as S=eiηS=e^{i\eta} with a Hermitian phase matrix η\eta, whose eigenvalues are the observable scattering phases; Heisenberg set up the corresponding eigenvalue problem, intending stationary states and the rest energies of composite systems to emerge from SS alone, and verified that in first-order perturbation theory η\eta reduces to the matrix elements of the interaction energy whenever a Hamiltonian exists (Eq. (68) of Part I). Part II then studies a theory in which, in place of a Hamiltonian function, the η\eta-matrix is given, showing in concrete models that such a theory is free of divergence difficulties and can yield finite high-energy cross sections, with an eigenvalue spectrum of η\eta that comes out discrete for one interaction whose coefficients are normal-ordered (all particle-creation operators placed to the left of annihilation operators, so that vacuum self-contractions vanish) and continuous — with 'explosive' multiple-particle production — for a variant interaction; and it closes by noting that the form of η\eta itself must be constructed from entirely different conditions, such as further physical principles, and that the space-time (localization) content of such a theory is unclear. Part III studies equations with nonlocal kernels (interaction kernels coupling distinct space-time points rather than a single point), postulates (without proof) analyticity of SS, and lists the open problems of kernel existence and selection. The problem is thus the founding question of S-matrix theory: what, beyond unitarity and Lorentz invariance, fixes SS? The later literature supplies the context for its status. Mandelstam (Phys. Rev. 112, 1344 (1958)) introduced the double dispersion relations — the statement that scattering amplitudes obey integral ('dispersion') relations following from analyticity, the property that amplitudes are boundary values of analytic functions of the invariant kinematic variables, together with crossing, the statement that the same analytic function describes particle and antiparticle scattering in the crossed kinematic channels. Bros, Epstein and Glaser (Commun. Math. Phys. 1, 240 (1965)) rigorously proved crossing for two-to-two amplitudes of gapped local QFTs (theories whose lightest particle has a strictly positive mass); the general analyticity domain remains unproven. Chew and Frautschi (Phys. Rev. Lett. 7, 394 (1961)) turned the question into the bootstrap program ('nuclear democracy'): determine the strong-interaction SS-matrix by pure self-consistency of unitarity, analyticity and crossing; this did not converge to a unique theory and was superseded by quantum chromodynamics (QCD, the modern field theory of the strong interaction) in the 1970s. Constructive field theory (Glimm and Jaffe, Quantum Physics, 2nd ed., 1987) rigorously constructed interacting QFTs with genuine self-adjoint Hamiltonians in two and three space-time dimensions, partially answering the Hamiltonian-existence sub-question; four dimensions remains open. In 1+1 dimensions, Zamolodchikov and Zamolodchikov (Ann. Phys. 120, 253 (1979)) determined the exact factorized SS-matrices of integrable models (models with infinitely many conserved quantities, in which every many-particle collision decomposes into a sequence of independent two-particle scatterings) from unitarity, crossing and analyticity plus bootstrap (fusing) conditions — a positive partial answer in low dimension. Since 2016 the question has been revived as the modern S-matrix bootstrap (Paulos, Penedones, Toledo, van Rees, Vieira, arXiv:1607.06109, arXiv:1607.06110): one numerically or analytically maps the space of amplitudes allowed by unitarity, crossing and analyticity (with specified spectrum), obtaining rigorous bounds — refined analytically by Correia, Sever and Zhiboedov (arXiv:2006.08221) and extended to massless pions by Guerrieri, Penedones and Vieira (arXiv:2011.02802). The 2022 Snowmass white paper (Kruczenski, Penedones, van Rees, arXiv:2203.02421) records the state of the art and the open gaps, including analyticity in general kinematics, QCD as a distinguished point of the allowed space, and the massless and gravitational sectors.

Problem Statement

The problem, as posed by Heisenberg's 1943–44 S-matrix papers, is: what fundamental conditions — principles, axioms, symmetry or regularity requirements — other than a Hamiltonian determine the scattering matrix SS — the operator relating the incoming and outgoing asymptotic particle states of a collision, equivalently the Hermitian phase matrix η\eta with S=eiηS=e^{i\eta}, whose eigenvalues are the observable scattering phases — of a relativistic quantum theory of elementary particles, given that the framework itself supplies only unitarity S~∗S=1\tilde S^{*}S=1 (conservation of probability, with S~\tilde S the transposed matrix and ∗* complex conjugation) and Lorentz invariance? Concretely, the following open aspects sharpen the question (a derived concretization, not an enumeration found in the source papers; these aspects are scope notes, not separate deliverables): (i) whether, and under what assumptions, a time-local Hamilton function exists in the final theory at all, or whether the concept of the Hamilton operator must be given up; (ii) which additional physical conditions — freedom from divergences, intrinsic many-body force terms from the outset, new symmetry principles, or limiting requirements tied to a minimal length — could take the place of a Hamiltonian principle and fix the matrix η\eta uniquely; (iii) to what extent a constructive determination of η\eta from such alternative principles would yield the same physical spectrum, including the rest energies of composite systems, as a possibly underlying Hamilton operator. The same question admits a modern reformulation — a derived restatement of the same question, not the original wording — in which the given framework remains unitarity and Lorentz invariance and the additional conditions are drawn from candidates such as crossing (the statement that the same analytic function describes scattering in the crossed kinematic channels), analyticity (the property that amplitudes are boundary values of analytic functions of the invariant kinematic variables), further symmetry or regularity requirements, applied with or without specified spectral and symmetry data. The question remains open beyond the results already cited in Background — the exact 1+1-dimensional factorized $S-matrices and the published bootstrap bounds are known cases rather than a general determination. The four-dimensional, massless-particle and gravitational regimes of this question are currently unresolved.

References

  1. W. Heisenberg, Die „beobachtbaren Größen“ in der Theorie der Elementarteilchen, Zeitschrift für Physik 120, 513–538 (1943). DOI: 10.1007/BF01329800, https://doi.org/10.1007/BF01329800
  2. W. Heisenberg, Die „beobachtbaren Größen“ in der Theorie der Elementarteilchen. II, Zeitschrift für Physik 120, 673–702 (1943). DOI: 10.1007/BF01336936, https://doi.org/10.1007/BF01336936
  3. W. Heisenberg, Die „beobachtbaren Größen“ in der Theorie der Elementarteilchen. III, Zeitschrift für Physik 123, 93–112 (1944). DOI: 10.1007/BF01375146, https://doi.org/10.1007/BF01375146
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  7. J. Bros, H. Epstein and V. Glaser, A proof of the crossing property for two-particle amplitudes in general quantum field theory, Communications in Mathematical Physics 1, 240–264 (1965). DOI: 10.1007/BF01646307, https://doi.org/10.1007/BF01646307
  8. J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 2nd edition, Springer, New York (1987). DOI: 10.1007/978-1-4612-4728-9, https://doi.org/10.1007/978-1-4612-4728-9
  9. A. B. Zamolodchikov and Al. B. Zamolodchikov, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Annals of Physics 120, 253–291 (1979). DOI: 10.1016/0003-4916(79)90391-9, https://doi.org/10.1016/0003-4916(79)90391-9
  10. M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, The S-matrix Bootstrap I: QFT in AdS (2016). arXiv:1607.06109, https://arxiv.org/abs/1607.06109
  11. M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, The S-matrix Bootstrap II: Two Dimensional Amplitudes (2016). arXiv:1607.06110, https://arxiv.org/abs/1607.06110
  12. M. Correia, A. Sever and A. Zhiboedov, An Analytical Toolkit for the S-matrix Bootstrap (2020). arXiv:2006.08221, https://arxiv.org/abs/2006.08221
  13. A. Guerrieri, J. Penedones and P. Vieira, S-matrix Bootstrap for Effective Field Theories: Massless Pions (2020). arXiv:2011.02802, https://arxiv.org/abs/2011.02802
  14. M. Kruczenski, J. Penedones and B. C. van Rees, Snowmass White Paper: S-matrix Bootstrap (2022). arXiv:2203.02421, https://arxiv.org/abs/2203.02421