Non-linear quantum processes and automorphism groups of C -algebras · 1970-mathphys-5

Quantum Theory · ICM 1970: Non-linear quantum processes and automorphism groups of C -algebras

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Problem statement

For the singular time-dependent perturbations arising in higher-dimensional nonlinear quantum field theory, let $H$ be the free self-adjoint Hamiltonian and let $V$ be the continuous bilinear form on $D_\infty(H)$ , the vectors infinitely differentiable with respect to $H$ . For each nonnegative $g\in C_c^\infty(\mathbb R)$ , define a distribution propagator relative to $H$ to be a mapping $S(g)$ into the unitary operators on the Hilbert space that belongs to the smallest class containing the propagators generated by bounded self-adjoint perturbations and closed under sequential convergence. Say that the adiabatic perturbation of $H$ by $V$ exists if there is such a distribution propagator for which, for every $g$ , the products $S(g/n)^n$ converge as $n\to\infty$ to the unitary evolution generated by a self-adjoint operator $H(g)$ whose associated bilinear form extends $\int V(t)g(t)\,dt$ .

Determine whether, for the nonlinear quantum-field-theoretic bilinear forms with a compactly supported spatial cutoff $f$ (and subsequently in the limit $f\to1$ ), the adiabatic perturbation exists uniquely at least as an automorphism of the corresponding Weyl $C^*$ -algebra, with the expected locality and covariance properties, and whether the hyperbolic limit produces an automorphism $\Sigma(g)$ whose limit as $g\to1$ is unitarily implementable by an essentially unique unitary $S$ -operator.

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Remaining questions in this record

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Question 1

Let $\mathcal F$ be the symmetric Fock space of a massive scalar field in $1+1$ -dimensional Minkowski space, let $H_0$ be its free Hamiltonian, and for every nonnegative $g\in C_c^\infty(\mathbb R^2)$ let

V_g(t)=\int_{\mathbb R} g(x,t):\!\phi^4(x,0)\!:\,dx, \qquad \widetilde H_g(t)=H_0+V_g(t)+M I,

where $M$ is chosen so that $\widetilde H_g(t)$ is positive. Let $U_g(t,s)$ be the unitary propagator solving

i\,\partial_tU_g(t,s)=\widetilde H_g(t)U_g(t,s),\qquad U_g(s,s)=I,

and define the Bogoliubov operator, when the strong limit exists, by

S(g)=\operatorname*{s-lim}_{\substack{t\to+\infty\\s\to-\infty}}e^{i(H_0+M I)t}U_g(t,s)e^{-i(H_0+M I)s}.

Prove or disprove that, whenever $g_1,g_2$ have spacelike separated supports, $S(g_1+g_2)=S(g_1)S(g_2)$ .

Source reference: Walter F. Wreszinski, Luiz A. Manzoni, and Oscar Bolina, “Existence of the Bogoliubov S(g) Operator for the $(:\!\phi^4\!:)_2$ Quantum Field Theory,” arXiv:math-ph/0306042, 2004, Section V (“Conclusion: Open Problems”), https://arxiv.org/abs/math-ph/0306042.

Question 2

In the $1+1$ -dimensional massive scalar $(:\!\phi^4\!:)_2$ theory, define $S(g)$ for $g\in C_c^\infty(\mathbb R^2)$ from the nonperturbative unitary propagator exactly as follows: $H_g(t)=H_0+\int_{\mathbb R}g(x,t):\!\phi^4(x,0)\!:\,dx$ , choose $M$ so that $H_g(t)+MI$ is positive, let $U_g(t,s)$ solve $i\partial_tU_g(t,s)=(H_g(t)+MI)U_g(t,s)$ with $U_g(s,s)=I$ , and set

S(g)=\operatorname*{s-lim}_{\substack{t\to+\infty\\s\to-\infty}}e^{i(H_0+MI)t}U_g(t,s)e^{-i(H_0+MI)s}

whenever this strong limit exists. Determine whether there is a unitary representation $U(a,\Lambda)$ of the $1+1$ -dimensional Poincaré group on Fock space such that

U(a,\Lambda)S(g)U(a,\Lambda)^{-1}=S(g_{a,\Lambda}), \qquad g_{a,\Lambda}(x)=g\bigl(\Lambda^{-1}(x-a)\bigr),

for every $g$ and every Poincaré transformation $(a,\Lambda)$ .

Question 3

In the $1+1$ -dimensional massive scalar $(:\!\phi^4\!:)_2$ theory, let $S(g)$ be the nonperturbatively constructed Bogoliubov operator for a nonnegative compactly supported smooth coupling function $g\in C_c^\infty(\mathbb R^2)$ , and let $g_\varepsilon(x)=g(\varepsilon x)$ for $\varepsilon>0$ , with $g(0)>0$ . Determine whether there exists a Poincaré-invariant dense subspace $\mathcal D$ of Fock space such that

\lim_{\varepsilon\downarrow0}S(g_\varepsilon)\Psi

exists for every $\Psi\in\mathcal D$ .

Source reference: Walter F. Wreszinski, “Conceptual Problems in Quantum Field Theory,” Brazilian Journal of Physics 34(4A) (2004), 1343–1348, DOI: 10.1590/S0103-97332004000700010, Section 2.1, https://www.scielo.br/j/bjp/a/j3xVfrzH4v5xPgZhyBtX6Zk/?lang=en.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. No citable result was found that proves the complete nonperturbative statement for the singular nonlinear interactions of higher-dimensional quantum field theory: existence and uniqueness of the adiabatic perturbation as a Weyl- $C^*$ -algebra automorphism with locality and covariance, removal of the spatial cutoff, and existence of a global adiabatic limit unitarily implemented by an essentially unique $S$ -operator.

Partial progress is substantial but does not cover the exact problem:

Irving E. Segal, “Construction of Nonlinear Local Quantum Processes. I,” Annals of Mathematics 92 (1970), 462–481, DOI: https://doi.org/10.2307/1970628. This constructs a class of nonlinear local quantum processes by singular-perturbation methods, but it does not establish the complete higher-dimensional adiabatic-perturbation and cutoff-removal program stated here.
Irving E. Segal, “Construction of Non-Linear Local Quantum Processes,” Inventiones Mathematicae 14 (1971), 211–241, direct URL: http://eudml.org/doc/142113. This extends Segal’s constructive program for particular nonlinear processes; it does not prove the asserted general existence, uniqueness, locality/covariance, and global $S$ -operator conclusions in higher dimensions.
Walter F. Wreszinski, Luiz A. Manzoni, and Oscar Bolina, “Existence of the Bogoliubov $S(g)$ Operator for the $(:\!\phi^4\!:)_2$ Quantum Field Theory,” Journal of Mathematical Physics 45 (2004), 2579–2593, arXiv: math-ph/0306042, DOI: https://doi.org/10.1063/1.1753666. This proves nonperturbative existence, unitarity, and time-separated causality for the cutoff Bogoliubov operator in the specific two-dimensional $(:\!\phi^4\!:)_2$ model, while explicitly leaving spacelike causality and Poincaré covariance open; it therefore does not settle the higher-dimensional statement.
Henri Epstein and Vladimir Glaser, “The Role of Locality in Perturbation Theory,” Annales de l’Institut Henri Poincaré, Section A 19 (1973), 211–295, direct URL: http://www.numdam.org/item/AIHPA_1973__19_3_211_0/. This gives a rigorous causal perturbative construction of time-ordered products and local scattering operators. Its conclusions are perturbative formal power series and do not provide the nonperturbative distribution propagator or the complete global adiabatic limit required here.
Michael Dütsch and Klaus Fredenhagen, “Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion,” Communications in Mathematical Physics 219 (2001), 5–30, arXiv: hep-th/0001129, DOI: https://doi.org/10.1007/s002200100400. This constructs interacting local algebras perturbatively and implements the algebraic adiabatic limit locally, avoiding the need for a global switching-function limit; it does not produce the exact nonperturbative Weyl-algebra automorphism and global unitary $S$ -operator requested here.
Paweł Duch, “Weak Adiabatic Limit in Quantum Field Theories with Massless Particles,” Annales Henri Poincaré 19 (2018), 875–935, arXiv: 1709.09907, DOI: https://doi.org/10.1007/s00023-017-0649-7. This proves existence of a weak adiabatic limit for a broad class of perturbative models under normalization conditions. A weak perturbative limit is not the strong, nonperturbative, unitarily implemented limit demanded by the statement.
Kasia Rejzner, Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians, Springer, 2016, DOI: https://doi.org/10.1007/978-3-319-25901-7. This gives a rigorous framework for perturbative interacting local observables and algebraic cutoff removal, but explicitly remains perturbative and therefore does not resolve the full exact problem.

Thus the literature supplies rigorous constructions for special models and rigorous perturbative or weak versions of several requested steps, but no verified result covers all hypotheses and conclusions of the exact statement.

References

[1]ICM Conjectures · 1970-mathphys-5 link
[2]ICM 1970: Non-linear quantum processes and automorphism groups of C -algebras link

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Source & evidence

1970-mathphys-5

ICM snapshot retrieved 8 September 2026. Source review state: unreviewed. The status text and 3 remaining questions are reproduced with attribution; no independent proof audit has been performed here.

Source date: 2026-08-18
Imported: 2026-09-08

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Difficulty: HardResisted sustained effort by specialists.
Posed: 1970
Source: Curated problem listICM ConjecturesICM 1970: Non-linear quantum processes and automorphism groups of C -algebrasOriginal statement ↗
Citation: ICM 1970: Non-linear quantum processes and automorphism groups of C -algebras
Added: 5 Nov 2026
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Mathematical physics