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Register a new userExact solutions of the Wheeler-DeWitt equation and the Yamabe construction
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Exact solutions of the Wheeler-DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrodinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.
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The Quantum Wheeler-DeWitt operator can be derived from an affine commutation relation via the affine group representation formalism for gravity, wherein a family of gauge-diffeomorphism invariant affine coherent states are constructed from a fiducial state. In this article, the role of the fiducial state is played by a regularized Gaussian peaked on densitized triad configurations corresponding to 3-metrics of constant spatial scalar curvature. The affine group manifold consists of points in the upper half plane, wherein each point is labeled by two local gravitational degrees of freedom from the Yamabe construction. From this viewpoint, here we show that the translational subgroup of affine coherent states constitute a set of exact solutions of the Wheeler-DeWitt equation. The affine translational parameter $b$ admits a physical interpretation analogous to a continuous plane wave energy spectrum, where the curvature constant $k$ plays the role of the energy. This result shows that the affine translational subgroup generates transformations in the curvature constant $k$ from the Yamabe problem, while $k$ is inert under the kinematic symmetries of gravity.
We investigate the quantum evolution of the universe in the presence of two types of dark energies. First, we consider the phantom class ($omega<-1$) which would be responsible for a super-accelerated cosmic expansion, and then we apply the procedure to an ordinary $Lambda>0$ vacuum ($omega=-1$). This is done by analytically solving the Wheeler-DeWitt equation with ordering term (WdW) in the cosmology of Friedmann-Robertson-Walker. In this paper, we find exact solutions in the scale factor $a$ and the ordering parameter $q$. For $q=1$ it is shown that the universe has a high probability of evolving from a big bang singularity. On the other hand, for $q = 0$ the solution indicates that an initial singularity is unlikely. Instead, the universe has maximal probability of starting with a finite well-defined size which we compute explicitly at primordial times. We also study the time evolution of the scale factor by means of the Hamilton-Jacobi equation and show that an ultimate big rip singularity emerges explicitly from our solutions. The phantom scenario thus predicts a dramatic end in which the universe would reach an infinite scale factor in a finite cosmological time as pointed by Caldwell et al. in a classical setup. Finally, we solve the WdW equation with ordinary constant dark energy and show that in this case the universe does not rip apart in a finite era.
In a theory of quantum gravity, states can be represented as wavefunctionals that assign an amplitude to a given configuration of matter fields and the metric on a spatial slice. These wavefunctionals must obey a set of constraints as a consequence of the diffeomorphism invariance of the theory, the most important of which is known as the Wheeler-DeWitt equation. We study these constraints perturbatively by expanding them to leading nontrivial order in Newtons constant about a background AdS spacetime. We show that, even within perturbation theory, any wavefunctional that solves these constraints must have specific correlations between a component of the metric at infinity and energetic excitations of matter fields or transverse-traceless gravitons. These correlations disallow strictly localized excitations. We prove perturbatively that two states or two density matrices that coincide at the boundary for an infinitesimal interval of time must coincide everywhere in the bulk. This analysis establishes a perturbative version of holography for theories of gravity coupled to matter in AdS.
We reexamine the relationship between the path integral and canonical formulation of quantum general relativity. In particular, we present a formal derivation of the Wheeler-DeWitt equation from the path integral for quantum general relativity by way of boundary variations. One feature of this approach is that it does not require an explicit 3+1 splitting of spacetime in the bulk. For spacetimes with a spatial boundary, we show that the dependence of the transition amplitudes on spatial boundary conditions is determined by a Wheeler-DeWitt equation for the spatial boundary surface. We find that variations in the induced metric at the spatial boundary can be used to describe time evolution---time evolution in quantum general relativity is therefore governed by boundary conditions on the gravitational field at the spatial boundary. We then briefly describe a formalism for computing the dependence of transition amplitudes on spatial boundary conditions. Finally, we argue that for nonsmooth boundaries, meaningful transition amplitudes must depend on boundary conditions at the joint surfaces.
In this paper, we study the changes of quantum effects of a growing universe by using Wheeler-DeWitt equation (WDWE) together with de Broglie-Bohm quantum trajectory approach. From WDWE, we obtain the quantum modified Friedmann equations which have additional terms called quantum potential compared to standard Friedmann equations. The quantum potential governs the behavior of the early universe, providing energy for inflation, while it decreases rapidly as the universe grows. The quantum potential of the grown-up universe is much smaller than that required for accelerating expansion. This indicates that quantum effects of our universe cannot be treated as a candidate for dark energy.
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