Interacting Curved-Spacetime and Gauge Existence Problems

Interacting QFT on a curved background has a rigorous local perturbative construction on globally hyperbolic spacetimes. Gauge theories can be treated with BRST–BV methods in the same formal setting. Those results construct coefficientwise local observables and covariance maps; they do not generally construct a convergent interacting state, a preferred vacuum, a positive nonperturbative measure, or a global four-dimensional non-Abelian theory. “Curved,” “interacting,” “gauge,” and “nonperturbative” each add a separate obligation.

Required background. Interacting pAQFT on curved spacetimes supplies the formal local construction. Nonperturbative gauge measures supplies the positivity and limit target, while existence, uniqueness, and equivalence fixes the logical distinctions. Helpful background. Adiabatic limits and infrared obstructions, constructing Hadamard states, and low-energy constraints on UV completion locate the physical boundaries.

Local perturbative existence on curved spacetime

Let $M$ be oriented, time-oriented, and globally hyperbolic, and let $V_g$ be a compactly supported interaction. Renormalized time-ordered products define a formal local S-matrix

$$S(V_g)=\sum_{n\ge0}\frac1{n!} \left(\frac{i}{\hbar}\right)^nT_n(V_g^{\otimes n}).$$

Causal factorization and locally covariant renormalization produce relative S-matrices and interacting observables as formal series. If two cutoff functions agree on a causally closed neighborhood of a region $O$, the resulting local interacting algebras on $O$ are canonically related. Hollands and Wald prove existence of local covariant time-ordered products and classify their finite renormalization freedom 2002, Theorems 5.1–5.2, pp. 328–343. Brunetti and Fredenhagen supply the microlocal extension method on physical backgrounds 2000, §§3–6, pp. 640–660.

The coefficient ring is part of the result. A typical algebra is over $\mathbb C[[\lambda,\hbar]]$; its elements are formal series. Formal positivity means an ordered leading coefficient condition, not a countably additive measure at fixed nonzero $\lambda$. A formal state obtained by composing a free Hadamard state with a perturbative Møller map therefore does not prove convergence or a genuine GNS representation at physical coupling.

For Yang–Mills, one adds the gauge complex, ghosts, antifields, the renormalized quantum master equation, and anomaly cancellation. Hollands constructs renormalized quantum Yang–Mills fields on curved spacetime in perturbation theory under explicit covariance and cohomological hypotheses 2008, Theorems 1–3 and §§4–5, pp. 1058–1125. The output is a perturbative construction; it is not a nonperturbative gauge-field measure.

Local nets without distinguished states

A newer algebraic route uses relations inspired by local S-matrices to define a net of abstract $C^*$-algebras without treating the coupling as a formal coefficient inside each generator. This is a meaningful nonperturbative algebraic construction of local relations. Its present boundary is equally important: physically distinguished states such as a vacuum are not generally known, and on a generic curved spacetime there may be no global timelike symmetry from which to define one. Brunetti, Fredenhagen, and Rejzner state this distinction explicitly 2025, §§2–3 and 17, pp. 2–3, 16–17.

Thus “a $C^*$-net exists” and “an interacting physical model exists” are not identical. One still needs nonzero representations, locally normal states with the desired microlocal properties, covariance across the background category, and any claimed spectral or thermal structure.

First application: scalar versus non-Abelian gauge theory

Return to adiabatic limits and infrared obstructions. Compare compactly supported scalar $\phi^4$ pAQFT with non-Abelian Yang–Mills on the same globally hyperbolic $M$.

For the scalar theory, causal factorization, microlocal renormalization, and local covariance define the formal interacting net. Removing the compact interaction cutoff globally requires separate infrared control and a compatible state. For Yang–Mills, the same steps are preceded by a BRST–BV resolution and followed by anomaly and gauge-independence checks. Neither theory automatically has a preferred global vacuum, and the gauge theory additionally requires recovery of physical cohomology and positivity.

A useful comparison has four columns:

Output	Scalar pAQFT	Yang–Mills pAQFT	Missing nonperturbative step
Local observables	Formal microcausal algebra	Formal BRST/BV cohomology	Convergence or another genuine algebra/state construction
Covariance	Natural under admissible embeddings	Natural after gauge and anomaly conditions	Global representation compatible with the background class
States	Formal deformations or special model-dependent states	Harder cohomological positivity problem	Positive state at finite coupling
Global limit	Infrared- and geometry-dependent	Infrared plus gauge-sector dependence	Adiabatic/volume limit and physical Hilbert space

Boundaries and nonglobally hyperbolic geometries

Global hyperbolicity supplies advanced and retarded propagators and causal factorization. A timelike boundary or a nonglobally hyperbolic spacetime requires boundary conditions or another choice of dynamics. One must prove Green-hyperbolicity, control symplectic flux, and preserve positivity and covariance for the chosen background category. A theorem on globally hyperbolic manifolds without boundary cannot simply be reused after deleting that hypothesis.

Failure test: local time ordering as a measure

Take the existence of all renormalized $T_n$ and infer a convergent interacting probability measure or vacuum. The inference fails because the construction is coefficientwise, compactly supported locally, and supplies no summability estimate. The strongest surviving conclusion is the locally covariant formal interacting algebra with its classified finite renormalization freedom.

An independent check asks whether the proposed state is an ordinary positive functional, a formal state, or merely a collection of perturbative expectation values. If this type is not stated, the existence claim is incomplete.

Exercises

Why does the algebraic adiabatic limit not establish the global limit $g\to1$?

Solution

It compares cutoffs that agree near a bounded region and uses causal factorization to identify the corresponding local algebras. It does not control the interaction over all of spacetime, sum infrared contributions, construct one global state, or prove convergence of the formal series.

References

• Brunetti, Romeo, and Klaus Fredenhagen. “Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds.” Communications in Mathematical Physics 208 (2000): 623–661. DOI; Open PDF.
• Brunetti, Romeo, Klaus Fredenhagen, and Kasia Rejzner. “Perturbative Algebraic Quantum Field Theory and Beyond.” arXiv:2512.14227 (2025). arXiv.
• Hollands, Stefan. “Renormalized Quantum Yang–Mills Fields in Curved Spacetime.” Reviews in Mathematical Physics 20 (2008): 1033–1172. DOI; Open PDF.
• Hollands, Stefan, and Robert M. Wald. “Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 231 (2002): 309–345. DOI; Open PDF.