entry to be merged or else disambiguated with locally covariant perturbative quantum field theory
algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
From the physical point of view there are two different reasons to consider the Haag-Kastler approach on more general spacetimes than the Minkowski spacetime:
It is expected that such a theory, while not solving the problem to construct a theory of quantum gravity, would still have a wide range of validity.
From a conceptual viewpoint abandoning the special situation of Minkowski spacetime could lead to the development of new ideas and tools that turn out to be helpful to understand the concept of a quantum field theory better.
The first point deserves some elaboration: The curved manifolds under consideration are supposed to be solutions to the field equations of General Relativity, i.e. physically realistic spacetimes, so that gravity is modelled classically by the curvature of spacetime. A quantum field theory on such a spacetime should be able to model the situation of elementary particles that feel the effects of gravity while neglecting the effect that the particles themselves have on spacetime (the notion of “particle” is highly nontrivial and problematic in this setting and is to be understood in a metaphorical sense in the given context).
Example: If you let an electron drop from your hand to the ground, that would be a situation that the theory is supposed to describe. While a full theory of quantum gravity still eludes us, a theory of quantum fields on curved spacetimes could be useful as a kind of interpolation. In a certain sense this is already the case, since the laws of black hole thermodynamics were first discovered with the help of this setting.
Gereneral accounts of (perturbative, algebraic) quantum field theory on curved spacetimes includes
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)
Christopher Fewster, Rainer Verch, Algebraic quantum field theory in curved spacetimes (arXiv:1504.00586)
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)
Reviews with emphasis on this AQFT-point of view include
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)
Papers about the application of microlocal analysis include
Discussion of renormalization in AQFT on curved spacetimes includes
Discussion of the standard model of cosmology in the context of AQFT on curved spacetimes is in