Interacting QFT on curved spacetime
For a specified interacting field model and class of globally hyperbolic curved spacetimes, construct local observables and admissible states beyond formal perturbation theory; state any extra boundary, symmetry or preparation data used to select a state.
Problem statement
For a specified interacting field model and class of globally hyperbolic curved spacetimes, construct local observables and admissible states beyond formal perturbation theory; state any extra boundary, symmetry or preparation data used to select a state.
Lean 4
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Background
On a generic curved spacetime there is no Poincaré symmetry and hence no preferred vacuum or particle notion. The framework of locally covariant / algebraic QFT (Brunetti–Fredenhagen–Verch) provides a mathematically clean formulation, using Hadamard states to fix the short-distance structure and define renormalized observables like the stress tensor.
What is known: The perturbative theory is in excellent shape. Interacting fields on curved backgrounds have been constructed to all orders in perturbation theory via causal (Epstein–Glaser) renormalization and the operator-product/microlocal-spectrum machinery (Hollands–Wald, Brunetti–Fredenhagen), with a consistent, background-independent renormalization scheme. The class of Hadamard states and their properties are well understood.
Fewster–Verch's natural-state no-go theorem obstructs a universally covariant preferred-state assignment under its dynamical-locality and other assumptions. It is not a proof that Hadamard states or physically useful states do not exist. Restricting backgrounds or supplying symmetry, boundary or preparation data avoids demanding a universally natural vacuum.
The target is a model- and dimension-specific nonperturbative construction with controlled renormalized observables and state dependence. In four dimensions this inherits flat-space construction difficulties (QF1/QF3); curvature introduces additional issues, but neither every lower-dimensional case nor perturbative curved-spacetime QFT is being declared unsolved.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Brunetti, Fredenhagen, Verch, 'The generally covariant locality principle...', Comm. Math. Phys. 237 (2003) link
- [2]Hollands & Wald, 'Quantum fields in curved spacetime', Phys. Rep. 574 (2015) link
- [3]Fewster & Verch, 'Dynamical locality and covariance: What makes a physical theory the same in all spacetimes?' link
Recorded progress
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