Davies, Quantum Fields in Curved Space, Cambridge: Cambridge University Press, 1982 General accounts of (perturbative, algebraic) quantum field theory on curved spacetimes include P P here denotes the Poincaré group, while E E denotes what Dyson calls the ‘Einstein group’, which is now called the diffeomorphism group. ] 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. 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. They describe a mathematical structure of great elegance whose properties correspond in many respects to the facts of experimental physics. ] 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. To some extent the problem of AQFT on curved spacetime was formulated inįreeman Dyson,_Missed opportunities_, Bulletin of the AMS, Volume 78, Number 5, September 1972 ( pdf) Review includes ( Hack 15, section 3.2.1).įor more see at cosmological constant here. 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. Applications Vacuum energy and Cosmological constant
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). (This falls short of being a theory of quantum gravity, instead it describes quantum field theory on classical background field configurations of gravity.) For the case of perturbative quantum field theory this is locally covariant perturbative quantum field theory, see there for more. 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. Vacuum energy and Cosmological constant.Gravity as a BF-theory, Plebanski formulation of gravity, teleparallel gravity super Poincaré Lie algebra, supergravity Lie 3-algebra, supergravity Lie 6-algebraĮinstein-Hilbert action, Einstein's equations, general relativityįirst-order formulation of gravity, D'Auria-Fre formulation of supergravity.Main theorem of perturbative renormalization Osterwalder-Schrader theorem ( Wick rotation) Order-theoretic structure in quantum mechanics Renormalization group flow/ running coupling constants Stückelberg-Petermann renormalization group
DIVERGENCE IN CURVED SPACE SERIES
Star algebra, Moyal deformation quantizationĬanonical commutation relations, Weyl relationsĬausal perturbation theory, perturbative AQFTįeynman diagram, Feynman perturbation series State on a star-algebra, expectation valueĬollapse of the wave function/ conditional expectation value Operator algebra, C*-algebra, von Neumann algebra Quantum mechanical system, quantum probability Geometric quantization of symplectic groupoidsĪlgebraic deformation quantization, star algebra Algebraic quantum field theory ( perturbative, on curved spacetimes, homotopical)Ĭlassical, pre-quantum, quantum, perturbative quantumĮuler-Lagrange form, presymplectic current
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