Symmetry in Nonlinear Mathematical Physics - 2009

Eyo Ita (United States Naval Academy, Annapolis, USA & University of Cambridge, UK)

Finite states in four dimensional quantized gravity

The semiclassical-quantum correspondence (SQC) is a new principle which has enabled the explicit solution of the quantum constraints of GR in the full theory in the Ashtekar variables for gravity coupled to matter. The solutions, which constitute the physical space of states implementing the quantum dynamics of GR in the Dirac quantization procedure, include a special class of states known as the generalized Kodama states (GKod). The GKodS can be seen as an analogue of the pure Kodama state (Kod) when quantum gravity (QGRA) is coupled to matter fields quantized on the same footing.

The criterion for finiteness stems from a precise cancellation of the ultraviolet singularities stemming from the quantum Hamiltonian constraint, allowing for an exact solution. This signifies the following developments for 4D QGRA:
(i) Equivalence among the Dirac, reduced phase, geometric and path integration approaches to quantization for GKods;
(ii) A generalization of topological field theory to include matter fields via the instanton representation of GKod;
(iii) A possible mechanism to establish 4D QGRA, via tree networks, as a renormalizable theory;
(iv) A direct link from QGRA to Minkoswki spacetime physics, which would enable tests of 4D QGRA without the necessity to access the Planck scale;
(v) A third-quantized analogy to second quantized spin network states implementing the quantum dynamics of GR.

The aforementioned algorithm is designed to construct explicit solutions to the constraints of the full theory by inspection, while implementing any desired ‘boundary’ conditions on the states necessary to reduce to the appropriate semiclassical limit. Conversely, the finite states of 4D QGRA can place severe restrictions on phenomena occurring in the weak gravitational limit below the Planck scale.

While we demonstrate this for the GKodS in this talk, the procedure can be applied to obtain a family of states labeled by two arbitrary functions of position, which possess the requisite Hilbert space structure in the limit where the matter fields are turned off. Remaining areas of research in progress include the illumination of the Hilbert space structure of the GKodS, analysis of various models for which the SQC can produce tractable solutions, in the full theory and in minisuperspace, and the addressal of any issues of interest regarding the mathematical rigor of the states.