Observables in quantum gravity
Abstract
We study a family of physical observable quantities in quantum gravity. We denote them W functions, or nnet functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the npoint functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the threegeometry to threegeometry transition amplitude. The W functions are scalar under fourdimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural npoint functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sumoverfourgeometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a "free" model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries.
 Publication:

arXiv eprints
 Pub Date:
 April 2001
 arXiv:
 arXiv:grqc/0104034
 Bibcode:
 2001gr.qc.....4034P
 Keywords:

 General Relativity and Quantum Cosmology
 EPrint:
 25 pages, typos corected