Finite states in four-dimensional quantum gravity: the isotropic minisuperspace Asktekar–Klein–Gordon model

Abstract

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.

Finite states in four-dimensional quantized gravity

“…where in (31) we have suppressed the label T of the spatial hypersurface T forming the boundary ∂M. So, the quantum state (31) and the semiclassically determined state (14) coincide to all orders with no quantum corrections. We will define this property, the 'semiclassical-quantum correspondence' (SQC).…”

“…We will show in this section an important representation of the generalized Kodama state GKod in analogy to the instanton F ∧ F term for the pure Kodama state Kod generalized to the case when matter fields are present. We must first show, in analogy to the steps leading from (26) to (31), that the matter-coupled case does in fact correspond to a wavefunction defined only on the three-dimensional boundary T at time T and is in fact independent of the gravitational and matter fields in the bulk M of spacetime. Complementarily to the matter-free case previously discussed, we shall proceed in a more direct manner.…”

“…One possible criteria for the elimination of this freedom in f is to demand that it corresponds to δ eff [φ]/δφ α for the effective action eff [ φ] of the theory of matter quantized in the absence of gravity (lim G→0 ), such as to produce the correct semiclassical limit. We motivate this concept briefly in[11,31].…”

Reconsiderations on the formulation of general relativity based on Riemannian structures

“…On the other hand, other important fact to remark is that the results found in the Sects. 9-10 could yield us to find a close relation among alternative models that describes classical-canonical gravity in three or four dimensions, and the generalized Chern-Simons state for both quantum gravity [51,52] or topological invariants [53]. However, all these lines are in progress and will be the subject of forthcoming works.…”