Geometric significance of modular objects for non-conformally covariant quantum fields

Determine whether the Tomita–Takesaki modular conjugation J and modular automorphism group Δ associated with the local von Neumann algebra N(O) and the vacuum vector Ω in algebraic quantum field theory possess a geometric action for non-conformally covariant quantum fields, and, if so, characterize the corresponding spacetime transformations (analogous to Lorentz boosts for wedges or conformal transformations for double cones).

Background

In Minkowski spacetime, the Bisognano–Wichmann theorem identifies the Tomita–Takesaki modular group for wedge algebras with Lorentz boosts, providing a clear geometric meaning to the modular objects in that setting. For conformally covariant theories, Hislop–Longo showed that modular objects associated with double cones implement conformal transformations; this has also been extended to de Sitter spacetime.

However, beyond the conformal case, it is unresolved whether the modular conjugation and modular flow for local algebras N(O) with vacuum Ω continue to admit a geometric interpretation for general (non-conformally covariant) quantum field theories. Establishing or refuting such geometric modular action would clarify the scope of modular covariance beyond the currently known cases.

References

The question if the Tomita-Takesaki modular objects of $\mathsf{N}(O)$, $\Omega$ have geometric significance for non-conformally covariant quantum fields is a longstanding question.

— Lecture Notes on Operator Algebras and Quantum Field Theory  (2507.00900 - Verch, 1 Jul 2025) in Remarks, Section 2.3 (Borchers’ Theorem)

It is therefore conceivable that one could identify a Hadamard state whose modular flow approximately acts as the boost around $l$ and that satisfies an approximate KMS condition --- though whether such a state exists and whether it would be sufficient to recover a Type II algebra globally, remain open questions that need further investigation.

— On Algebraic Background-Independence  (2610.22042 - Vinck et al., 15 Oct 2026) in Section 2.4, “Comments on Generalisation to Arbitrary Spacetimes”