nLab AQFT on curved spacetimes

Idea

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).

Applications

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.

Related concepts

• general covariance

• causal structure

References

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.

$P$ here denotes the Poincaré group, while $E$ 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

• N. Birrell, P. Davies, Quantum Fields in Curved Space, Cambridge: Cambridge University Press, 1982

• Robert Wald, Quantum field theory in curved spacetime and black hole thermodynamics. Univ. of Chicago Press 1994 (ZMATH entry).

• Stefan Hollands, Robert Wald, Quantum fields in curved spacetime, Physics Reports Volume 574, 16 April 2015, Pages 1-35 (arXiv:1401.2026, doi:10.1016/j.physrep.2015.02.001)

• Christopher Fewster, Rainer Verch, Algebraic quantum field theory in curved spacetimes (arXiv:1504.00586)

See also:

• Bernard S. Kay, Quantum Field Theory in Curved Spacetime, Encyclopaedia of Mathematical Physics (2023) [arXiv:2308.14517]

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

• Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

• Stefan Hollands, Robert Wald, Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime, Commun. Math. Phys. 223:289-326,2001 (arXiv:gr-qc/0103074)

• Stefan Hollands, Robert Wald, On the Renormalization Group in Curved Spacetime, Commun. Math. Phys. 237 (2003) 123-160 (arXiv:gr-qc/0209029)

• Stefan Hollands, Robert Wald, Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes, Rev.Math.Phys. 17 (2005) 227-312 (arXiv:gr-qc/0404074)

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

• Romeo Brunetti, Klaus Fredenhagen, Rainer Verch, The generally covariant locality principle – A new paradigm for local quantum physics, Commun. Math. Phys. 237:31-68 (2003) (arXiv:math-ph/0112041)

• Romeo Brunetti, Klaus Fredenhagen, Quantum Field Theory on Curved Backgrounds , Proceedings of the Kompaktkurs “Quantenfeldtheorie auf gekruemmten Raumzeiten” held at Universitaet Potsdam, Germany, in 8.-12.10.2007, organized by C. Baer and K. Fredenhagen (arXiv:0901.2063)

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

• Marco Benini, Alastair Grant-Stuart, Alexander Schenkel, Haag-Kastler stacks [arXiv:2404.14510]

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

• Robert Wald, The Formulation of Quantum Field Theory in Curved Spacetime (arXiv:0907.0416)

• Robert Wald, The History and Present Status of Quantum Field Theory in Curved Spacetime (arXiv:gr-qc/0608018)

• Klaus Fredenhagen, Katarzyna Rejzner, QFT on curved spacetimes: axiomatic framework and examples (arXiv:1412.5125)

• Níckolas de Aguiar Alves, Nonperturbative Aspects of Quantum Field Theory in Curved Spacetime [arXiv:2305.17453]

• Bernard S. Kay, Quantum Field Theory in Curved Spacetime, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2308.14517]

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

• Romeo Brunetti, Klaus Fredenhagen, Kasia Rejzner, Locally covariant approach to effective quantum gravity, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2212.07800]

There is also a complementary approach via OPEs:

• Stefan Hollands, Robert Wald, Axiomatic quantum field theory in curved spacetime, Commun. Math. Phys. 293:85-125, 2010 (arXiv:0803.2003)

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

• Igor Khavkine, Valter Moretti, Analytic Dependence is an Unnecessary Requirement in Renormalization of Locally Covariant QFT, Communications in Mathematical Physics, March 2016 (arXiv:1411.1302, publisher)

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

• Klaus Fredenhagen, Thomas-Paul Hack, Quantum field theory on curved spacetime and the standard cosmological model (arXiv:1308.6773)

• Romeo Brunetti, Klaus Fredenhagen, Thomas-Paul Hack, Nicola Pinamonti, Katarzyna Rejzner, Cosmological perturbation theory and quantum gravity (arXiv:1605.02573)

• Thomas-Paul Hack, Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes, Springer 2016 (arXiv:1506.01869, doi:10.1007/978-3-319-21894-6)

Relation to 2d CFT:

• Marco Benini, Luca Giorgetti, Alexander Schenkel, A skeletal model for 2d conformal AQFTs (arXiv:2111.01837)