Algebraic Quantum Field Theory on Curved Spacetimes

Algebraic quantum field theory (AQFT) on curved spacetimes is a rigorous framework that generalises the axiomatic approach to quantum fields beyond flat Minkowski space, accommodating general relativity’s dynamical geometry. In this setting, the basic objects are nets of operator algebras assigned to causally convex regions of a globally hyperbolic manifold, respecting locality, covariance and causality without relying on a preferred vacuum state. Central to the programme is the locally covariant construction, which realises quantum field theory as a functor from the category of spacetimes and embeddings to the category of ∗-algebras, ensuring compatibility under isometric embeddings and causal disjointness. The microlocal spectrum condition characterises physically admissible states via their singularity structure, leading to the Hadamard condition as a substitute for the Poincaré-invariant vacuum. This formalism underpins rigorous treatments of semiclassical backreaction, black hole thermodynamics and perturbative construction of interacting fields in curved geometries, providing a bridge between mathematical physics and applications in cosmology and astrophysics.

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Algebraic Quantum Field Theory on Curved Spacetimes publication trend

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Technical terms

Algebraic quantum field theory: A framework in which observables are organised into nets of operator algebras indexed by spacetime regions, encoding locality and causality axiomatically.

Globally hyperbolic spacetime: A Lorentzian manifold admitting a Cauchy surface, ensuring well-posedness of hyperbolic field equations.

Hadamard state: A quantum state whose two-point function has a prescribed singularity structure characterised by the microlocal spectrum condition, generalising the vacuum in curved backgrounds.

C*-algebra: A Banach ∗-algebra with a norm satisfying the C*-identity, serving as an algebraic home for bounded quantum observables.

Locally covariant fields: A functorial assignment of algebras to spacetimes and of ∗-homomorphisms to embeddings, embodying general covariance in AQFT.

Microlocal spectrum condition: A constraint on the wavefront set of distributions defining n-point functions, ensuring the absence of unphysical singularities.

References

1. Quantum Gravity from the Point of View of Locally Covariant Quantum Field Theory. Communications in Mathematical Physics (2016).
2. Homotopy theory of algebraic quantum field theories. Letters in Mathematical Physics (2019).
3. A C*-algebraic Approach to Interacting Quantum Field Theories. Communications in Mathematical Physics (2020).
4. Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes. Mathematical Physics, Analysis and Geometry (2021).