D Quantum Gravity with Torsion, Dilaton Theory and Black Hole Formation

The main motivation to study general dilaton theories comes from spherically reduced gravity (SRG) ¹ or string theory ^2,3, although the dilaton black hole has a null-complete singularity ⁴. Previous studies of general dilaton theories ⁵ in the conformal gauge only could treat quantum theory in a semiclassical way ⁶ through the addition of the Polyakov action ⁷. For the example of a 2d model with torsion ^8,9 the crucial advantages of a physical gauge ¹⁰ like the light cone gauge for the Cartan variables ^11,12 were realised which, for that model, even allowed an exact path integral ¹³ and exact Dirac quantization ¹⁴. Because the light cone gauge for Cartan variables implies Eddington-Finkelstein gauge for the metric the global features of all 2d theories could be analysed in great detail ¹⁵. Motivated by the Hamiltonian analysis ¹⁶ and an analogous formulation in string theory ¹⁷ the importance of a first order formulation ¹⁸ was realized because of its local and global equivalence to the usual formulation as a dilaton theory ^14,19. Even in the presence of matter ²⁰ an absolute conservation law, generalizing the notion of the ADM-mass ²¹ to an energy flux relation ²² and a related novel type of Noether symmetry ²³ were found. The concept of the "Poisson-Sigma-Model" ²⁴ emerged in this context which now also finds an application in string theory ²⁵ and can be used to create all possible 2d supergravity theories ^26,27. The main power of the first order formulation was noticed in connection with the path integral formulation of 2d gravity where matter can be included by systematic loop-wise expansion ¹⁹. Based upon the formalism of extended Hamilton theory ²⁸ with contributions to the constraints also including fermions ²⁹ it was shown recently ³⁰ that as one of the results of 2d quantum gravity a "virtual black hole" appears in the scattering of scalars. Planned applications include reduced generalized Einstein theories ³¹.