In his 1958 paper establishing the integral representation of causal commutators now called the Jost-Lehmann-Dyson (JLD) representation, Dyson proved the representation for functions vanishing in a region RR of momentum space bounded by two spacelike surfaces, and for unions of such regions whose components are pairwise timelike separated. In a closing footnote he asked whether any restriction on the shape of RR is necessary at all, conjecturing that for any open set RR admitting at least one hyperboloid lying completely outside it, every function of his class C (Fourier transform vanishing at spacelike positions) that vanishes in RR still admits the hyperboloid representation with spectral weight Ψ(u,s)\Psi(u,s) supported only on hyperboloids lying completely outside RR; he noted the striking geometric consequence that the closure of the nonzero set of every class-C function would then have to be contained in a union of hyperboloids lying completely outside RR. Every available statement of the JLD representation retains geometric restrictions on RR, and no proof is known for a general open set RR admitting at least one admissible pair, nor is any counterexample recorded. The problem is to prove the conjecture for a general open set RR admitting at least one admissible pair, or to exhibit an explicit class-C function and open set RR for which it fails.

Background

In axiomatic quantum field theory - the Lehmann-Symanzik-Zimmermann (LSZ) framework in which correlation functions of local fields are constrained by general principles such as Lorentz covariance and causality rather than by a specific Lagrangian - local commutativity (microcausality) states that field operators commute at spacelike separation, and this forces strong analyticity properties on the Fourier transforms of causal commutators (matrix elements of commutators of field operators between physical states). Dyson (1958), extending the equal-mass result of Jost and Lehmann (1957) to unequal masses, studied functions on four-dimensional momentum space q∈R4q\in\mathbb{R}^{4} of class C: functions f(q)f(q) whose Fourier transform f~(x)\tilde f(x) vanishes for spacelike xx (with metric signature (+,−,−,−)(+,-,-,-), spacelike means x2=x02−x⃗ 2<0x^{2}=x_{0}^{2}-\vec{x}^{\,2}<0), i.e. f~\tilde f is supported in the closed light cone. His key technical device was to identify class-C functions with rotationally symmetric solutions of the wave equation in six dimensions, which turns the analysis of ff into a Radon-type problem, that is, a problem of reconstructing a function from integrals over families of lower-dimensional surfaces, as in the classical Radon transform. The resulting Jost-Lehmann-Dyson (JLD) representation writes any class-C function as a superposition of invariant hyperboloid measures,

f(q)=∫d4u∫ds  ϵ(q0−u0)  δ((q−u)2−s)  Ψ(u,s),f(q)=\int d^{4}u\int ds\;\epsilon(q_{0}-u_{0})\;\delta\bigl((q-u)^{2}-s\bigr)\;\Psi(u,s),

where ϵ\epsilon is the sign function, ss is a real mass-squared-like parameter, and each hyperboloid H(u,s)={q:(q−u)2=s}H(u,s)=\{q:(q-u)^{2}=s\} is a level set of the invariant quadratic form. If ff vanishes in some open region RR of momentum space, one wants Ψ\Psi supported only on admissible pairs (u,s)(u,s), meaning the hyperboloid H(u,s)H(u,s) lies completely outside RR. Dyson proved this only for RR bounded by two spacelike surfaces (hypersurfaces all of whose tangent vectors are spacelike) and for unions of such regions whose components are pairwise timelike separated; his Theorem C gave a necessary and sufficient support condition, in terms of bounding functions m(u,s)m(u,s) and M(u,s)M(u,s), for these geometries. Bros, Messiah and Stora (1961) recast the question as a problem of analytic completion in several complex variables: the JLD statement is equivalent to computing the envelope of holomorphy - the largest domain to which every function holomorphic on a given domain extends - of the union of the forward and backward tubes T±={z=x+iy∈C4:±y0>∣y⃗ ∣}T^{\pm}=\{z=x+iy\in\mathbb{C}^{4}:\pm y_{0}>|\vec{y}\,|\} (the sets of complexified points whose imaginary parts lie inside the forward or backward light cone) with a complexified neighborhood G~\tilde G of the real domain GG, namely H(T+∪T−∪G~)=C4∖⋃(x′,λ)∈N(G){z:(z−x′)2=λ2}H(T^{+}\cup T^{-}\cup\tilde G)=\mathbb{C}^{4}\setminus\bigcup_{(x',\lambda)\in N(G)}\{z:(z-x')^{2}=\lambda^{2}\}, where N(G)N(G) parametrizes the complementary hyperboloids. This reformulation underlies all later rigorous treatments (Vladimirov 1965; Vladimirov and Zharinov 1970; Lazur and Khimich 1977; Lambert 1980; the Bogolubov-Logunov-Oksak-Todorov monograph of 1990). Physically, the JLD representation is the engine behind dispersion relations for unequal-mass scattering - identities expressing a scattering amplitude as an integral over its imaginary part, obtained from analyticity - and behind the Lehmann ellipse, the region of the momentum-transfer variable in which the scattering amplitude is analytic; it remains in active use, for instance in higher-dimensional and compactified field theories (Maharana 2020) and in the momentum-space conformal bootstrap (Gillioz 2025). Dyson himself described the question of arbitrary RR as mathematically interesting but irrelevant to the physical applications of the representation.

Problem Statement

Conventions: work in R4\mathbb{R}^{4} with the Minkowski quadratic form q2=q02−q⃗ 2q^{2}=q_{0}^{2}-\vec{q}^{\,2}; a vector xx is spacelike if x2<0x^{2}<0 and timelike if x2>0x^{2}>0; a spacelike surface is a hypersurface all of whose tangent vectors are spacelike. For u∈R4u\in\mathbb{R}^{4} and s∈Rs\in\mathbb{R} let H(u,s)={q∈R4:(q−u)2=s}H(u,s)=\{q\in\mathbb{R}^{4}:(q-u)^{2}=s\} denote the hyperboloid with vertex uu and parameter ss. Say that ff is of class C if ff is a (tempered) function or distribution on R4\mathbb{R}^{4} whose Fourier transform f~(x)\tilde f(x) vanishes for all spacelike xx. Say that a subset of R4\mathbb{R}^{4} is a union of hyperboloids if it is expressible as a union, finite or infinite, of complete hyperboloids H(u,s)H(u,s). Prove or disprove the conjecture of Dyson (1958): for every open set R⊆R4R\subseteq\mathbb{R}^{4} for which at least one admissible pair exists, that is, at least one pair (u,s)(u,s) with H(u,s)∩R=∅H(u,s)\cap R=\emptyset, and every ff of class C with f(q)=0f(q)=0 for all q∈Rq\in R, there exists Ψ(u,s)\Psi(u,s) (in a precisely stated function or distribution space, e.g. tempered in uu and suitably integrable in ss) such that

f(q)=∫d4u∫ds  ϵ(q0−u0)  δ((q−u)2−s)  Ψ(u,s),f(q)=\int d^{4}u\int ds\;\epsilon(q_{0}-u_{0})\;\delta\bigl((q-u)^{2}-s\bigr)\;\Psi(u,s),

with Ψ(u,s)≠0\Psi(u,s)\neq 0 only for admissible pairs, i.e. only when H(u,s)∩R=∅H(u,s)\cap R=\emptyset (the hyperboloid lies completely outside RR). A positive answer would have the striking geometric consequence that Dyson emphasized, namely that for every ff of class C the closure of the set {q:f(q)≠0}\{q:f(q)\neq 0\} must be contained in the union of the complete hyperboloids H(u,s)H(u,s) with H(u,s)∩R=∅H(u,s)\cap R=\emptyset; indeed, any representation of the displayed form confines the nonzero set {q:f(q)≠0}\{q:f(q)\neq 0\} to the union of the complete hyperboloids H(u,s)H(u,s) with Ψ(u,s)≠0\Psi(u,s)\neq 0, each of which is admissible. A negative answer consists in exhibiting an explicit open set RR and an explicit class-C function ff vanishing on RR, together with a rigorous proof that no admissible Ψ\Psi exists. The baseline against which progress is measured: the conjecture is known for RR bounded by two spacelike surfaces (Dyson 1958), for unions of such regions with every pair of points in distinct regions separated by a timelike interval (Dyson 1958), and, in the equivalent envelope-of-holomorphy formulation, for unions of spacelike-separated such components (Vladimirov 1965); the question is open for a general open set RR admitting at least one admissible pair, without causal or regularity restrictions on its boundary.

References

  1. F. J. Dyson, "Integral Representations of Causal Commutators", Physical Review (2) 110, 1460-1464 (1958). DOI: 10.1103/PhysRev.110.1460, https://doi.org/10.1103/PhysRev.110.1460
  2. R. Jost and H. Lehmann, "Integral-Darstellung kausaler Kommutatoren", Il Nuovo Cimento 5, 1598-1610 (1957). DOI: 10.1007/BF02856049, https://doi.org/10.1007/BF02856049
  3. H. Lehmann, "Analytic properties of scattering amplitudes as functions of momentum transfer", Il Nuovo Cimento 10, 579-589 (1958). DOI: 10.1007/BF02859794, https://doi.org/10.1007/BF02859794
  4. V. Ya. Fainberg, "On Analytic Properties of Causal Commutators", Soviet Physics JETP 9, 1066-1069 (1959). No DOI assigned; INSPIRE record: https://inspirehep.net/literature/1841770 (Russian original: Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 36, 1066 (1959))
  5. J. Bros, A. Messiah, R. Stora, "A Problem of Analytic Completion Related to the Jost-Lehmann-Dyson Formula", Journal of Mathematical Physics 2, 639-651 (1961). DOI: 10.1063/1.1703750, https://doi.org/10.1063/1.1703750
  6. O. W. Greenberg, "Heisenberg Fields which Vanish on Domains of Momentum Space", Journal of Mathematical Physics 3, 859-866 (1962). DOI: 10.1063/1.1724300, https://doi.org/10.1063/1.1724300
  7. V. S. Vladimirov, "On the construction of envelopes of holomorphy for regions of a special type and their applications", American Mathematical Society Translations, Series 2, 48, 107-150 (1965); translation from Trudy Matematicheskogo Instituta imeni V. A. Steklova 60, 101-144 (1961). DOI: 10.1090/trans2/048/07, https://doi.org/10.1090/trans2/048/07
  8. V. S. Vladimirov and V. V. Zharinov, "On a representation of the Jost-Lehmann-Dyson type", Theoretical and Mathematical Physics 3, 525-536 (1970). DOI: 10.1007/BF01028522, https://doi.org/10.1007/BF01028522
  9. V. Yu. Lazur and I. V. Khimich, "A proof of the Jost-Lehmann-Dyson integral representation for a causal commutator, within the limits of localizable theories", Ukrainian Mathematical Journal 29, 458-468 (1977). DOI: 10.1007/BF01089896, https://doi.org/10.1007/BF01089896
  10. André Lambert, "The Jost-Lehmann-Dyson representation for a wide class of local fields", Letters in Mathematical Physics 4, 387-397 (1980). DOI: 10.1007/BF00417406, https://doi.org/10.1007/BF00417406
  11. N. N. Bogolubov, A. A. Logunov, A. I. Oksak and I. T. Todorov, "The Jost-Lehmann-Dyson Representation", chapter in "General Principles of Quantum Field Theory" (English translation by G. G. Gould), pp. 170-196, Kluwer (1990). DOI: 10.1007/978-94-009-0491-0_4, https://doi.org/10.1007/978-94-009-0491-0_4
  12. J. Maharana, "Proof of dispersion relations for the amplitude in theories with a compactified space dimension" (2020). arXiv: 2003.14330, https://arxiv.org/abs/2003.14330
  13. Ulrich Armbrüster, "Construction of envelopes of holomorphy and QFT", Master's thesis, Karlsruhe Institute of Technology (2023). arXiv: 2309.06346, https://arxiv.org/abs/2309.06346
  14. M. Gillioz, "The momentum-space conformal bootstrap in 2d" (2025). arXiv: 2502.21227, https://arxiv.org/abs/2502.21227