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 Finally, we use these relationships to motivate a family of gap/overlap-free and curvature-free tow-steered patterns.Algebraic quantum field theory ( perturbative, on curved spacetimes, homotopical)Ĭlassical, pre-quantum, quantum, perturbative quantumĮuler-Lagrange form, presymplectic current This analysis leads to the conclusion that layups featuring such patterns require strict constraints on bounds to ensure satisfaction of manufacturability requirements. We also explore the relationship between curl and divergence of rotated tow patterns. We demonstrate the developed constraint formulations on two analytical examples, as well as on a structural optimization. These relationships also lead to a constraint on the minimum cut/add length of a tow for a given tow-steered pattern. This mathematical formulation provides a relationship between tow path curvature, gap/overlap propagation rate, vector curl, and divergence. To develop these constraints, we consider a general tow-steered layer pattern as a 2D unit vector field, where the field streamlines represent the tow paths laid down by the automated fiber placing machine. In this work, we consider two manufacturing constraints: tow path curvature and gaps/overlaps. Numerical optimization can address this complexity, though it is critical that constraints are enforced to ensure that the resulting optimal tow-steered designs are manufacturable using current automatic fiber placement machines. The additional freedom provided by tow-steered composites has the potential to increase structural performance however, this approach comes with added design complexity. Automatic fiber placement machines have made it viable to manufacture composites where fiber angles vary continuously-tow-steered composites.