6533b870fe1ef96bd12d06cc
RESEARCH PRODUCT
Gravity, Non-Commutative Geometry and the Wodzicki Residue
W. KalauM. Walzesubject
High Energy Physics - TheoryPhysicsResidue (complex analysis)General Physics and AstronomyFOS: Physical sciencesGeometryCosmological constantGeneral Relativity and Quantum Cosmology (gr-qc)Riemannian manifoldDirac operatorGeneral Relativity and Quantum Cosmologysymbols.namesakeGeneral Relativity and Quantum CosmologyTensor productHigh Energy Physics - Theory (hep-th)Einstein–Hilbert actionsymbolsGeometry and TopologyCommutative propertyMathematical PhysicsHeat kerneldescription
We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator $D$ on an $n$ dimensional compact Riemannian manifold with $n\geq 4$, $n$ even, the Wodzicki residue Res$(D^{-n+2})$ is the integral of the second coefficient of the heat kernel expansion of $D^{2}$. We use this result to derive a gravity action for commutative geometry which is the usual Einstein Hilbert action and we also apply our results to a non-commutative extension which, is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 1993-12-21 |