No abstract

This is the first in a series of papers outlining an algorithm to explicitly construct finite quantum states of the full theory of gravity in Ashtekar variables. The algorithm is based upon extending some properties of a special state, the Kodama state for pure gravity with cosmological term, to matter-coupled models. We then illustrate a prescription for nonperturbatively constructing the generalized Kodama states, in preparation for subsequent works in this series. We also introduce the concept of the semiclassical-quantum correspondence (SQC). We express the quantum constraints of the full theory as a system of equations to be solved for the constituents of the 'phase' of the wavefunction. Additionally, we provide a variety of representations of the generalized Kodama states including a generalization of the topological instanton term to include matter fields, for which we present arguments for the field-theoretical analogue of cohomology on infinite-dimensional spaces. We demonstrate that the Dirac, reduced phase space and geometric quantization procedures are all equivalent for these generalized Kodama states as a natural consequence of the SQC. We relegate the method of the solution to the constraints and other associated ramifications of the generalized Kodama states to separate works.

Within the context of the Ashtekar variables, the Hamiltonian constraint of four-dimensional pure General Relativity with cosmological constant, Λ, is reexpressed as an affine algebra with the commutator of the imaginary part of the Chern-Simons functional, Q, and the positive-definite volume element. This demonstrates that the affine algebra quantization program of Klauder can indeed be applicable to the full Lorentzian signature theory of quantum gravity with non-vanishing cosmological constant; and it facilitates the construction of solutions to all of the constraints. Unitary, irreducible representations of the affine group exhibit a natural Hilbert space structure, and coherent states and other physical states can be generated from a fiducial state. It is also intriguing that formulation of the Hamiltonian constraint or Wheeler-DeWitt equation as an affine algebra requires a non-vanishing cosmological constant; and a fundamental uncertainty relation of the form ∆V V ∆Q ≥ 2πΛL 2 P lanck (wherein V is the total volume) may apply to all physical states of quantum gravity.

In an ever-expanding spatially closed universe, the fractional change of the volume is the preeminent intrinsic time interval to describe evolution in General Relativity. The expansion of the universe serves as a subsidiary condition which transforms Einstein's theory from a first class to a second class constrained system when the physical degrees of freedom (d.o.f.) are identified with transverse traceless excitations. The super-Hamiltonian constraint is solved by eliminating the trace of the momentum in terms of the other variables, and spatial diffeomorphism symmetry is tackled explicitly by imposing transversality. The theorems of Maskawa-Nishijima appositely relate the reduced phase space to the physical variables in canonical functional integral and Dirac's criterion for second class constraints to non-vanishing Faddeev-Popov determinants in the phase space measures. A reduced physical Hamiltonian for intrinsic time evolution of the two physical d.o.f. emerges. Freed from the first class Dirac algebra, deformation of the Hamiltonian constraint is permitted, and natural extension of the Hamiltonian while maintaining spatial diffeomorphism invariance leads to a theory with Cotton-York term as the ultra-violet completion of Einstein's theory.

In this paper, we construct the generalized Kodama state for the case of a Klein-Gordon scalar field coupled to Ashtekar variables in isotropic minisuperspace by a new method. The criterion for finiteness of the state stems from a minisuperspace reduction of the quantized full theory, rather than the conventional techniques of reduction prior to quantization. We then provide a possible route to the reproduction of a semiclassical limit via these states. This is the result of a new principle of the semiclassical-quantum correspondence (SQC), introduced in the first paper in this series. Lastly, we examine the solution to the minisuperspace case at the semiclassical level for an isotropic CDJ matrix neglecting any quantum corrections and examine some of the implications in relation to results from previous authors on semiclassical orbits of spacetime, including inflation. It is suggested that the application of nonperturbative quantum gravity, by way of the SQC, might potentially lead to some predictions testable below the Planck scale.