nLab AQFT on curved spacetimes




algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT




Where the Haag-Kastler axioms formulate quantum field theory on Minkowski spacetime, known as algebraic quantum field theory (AQFT) there is a generalization of these axioms to curved spacetimes (Brunetti-Fredenhagen 01), also known as locally covariant algebraic quantum field theory. For the case of perturbative quantum field theory this is locally covariant perturbative quantum field theory, see there for more.

(This falls short of being a theory of quantum gravity, instead it describes quantum field theory on classical background field configurations of gravity.)

This is the mathematically rigorous framework for studying subjects such as the cosmological constant (see there), Hawking raditation or the cosmic microwave background (Fredenhagen-Hack 13).


Vacuum energy and Cosmological constant

The renormalization freedom in perturbative quantization of gravity (perturbative quantum gravity) induces freedom in the choice of vacuum expectation value of the stress-energy tensor and hence in the cosmological constant.

Review includes (Hack 15, section 3.2.1).

For more see at cosmological constant here.


To some extent the problem of AQFT on curved spacetime was formulated in

  • Freeman Dyson, Missed opportunities, Bulletin of the AMS, Volume 78, Number 5, September 1972 (pdf)


    [[ the Haag-Kastler axioms ]] taken together with the axioms defining a C*-algebra are a distillation into abstract mathematical language of all the general truths that we have learned about the physics of microscopic systems during the last 50 years. They describe a mathematical structure of great elegance whose properties correspond in many respects to the facts of experimental physics. In some sense, the axioms represent the most serious attempt that has yet been made to define precisely what physicists mean by the words “observability, causality, locality, relativistic invariance,” which they are constantly using or abusing in their everyday speech. [[]] I therefore propose as an outstanding opportunity still open to the pure mathematicians, to create a mathematical structure preserving the main features of the Haag-Kastler axioms but possessing E-invariance instead of P-invariance.

PP here denotes the Poincaré group, while EE denotes what Dyson calls the ‘Einstein group’, which is now called the diffeomorphism group.

General accounts of (perturbative, algebraic) quantum field theory on curved spacetimes include

See also:

Foundations for perturbative quantum field theory on curved spacetimes in terms of causal perturbation theory were laid in

The AQFT-style axiomatization via local nets on a category of Lorentzian manifolds (locally covariant perturbative quantum field theory) is due to:

Enhanced formulation in terms of stacks of categories (2-sheaves) of field theories :

Reviews with emphasis on the AQFT-local-nets point of view:

On the locally covariant pAQFT approach to effective quantum gravity and applications to experiment:

There is also a complementary approach via OPEs:

On the application of microlocal analysis:

  • Alexander Strohmaier, Rainer Verch, Manfred Wollenberg: Microlocal analysis of quantum fields on curved spacetimes: Analytic wavefront sets and Reeh-Schlieder theorems (arXiv).

Discussion of renormalization in AQFT on curved spacetimes includes

Discussion of the cosmology in the context of AQFT on curved spacetimes includes

Relation to 2d CFT:

Last revised on April 24, 2024 at 03:34:37. See the history of this page for a list of all contributions to it.