Quantum geometrodynamics of the Bianchi IX cosmological model

Abstract

The canonical quantum theory of gravity -Quantum Geometrodynamics (QG) is applied to the homogeneous Bianchi type IX cosmological model. As a result, the framework for the quantum theory of homogeneous cosmologies is developed. We show that the theory is internally consistent, and prove that it possesses the correct classical limit (the theory of general relativity). To emphasize the special role that the constraints play in this new theory we, compare it to the traditional ADM square-root and Wheeler-DeWitt quantization schemes. We show that, unlike the traditional approaches, QG leads to a well-defined Schrödinger equation for the wave-function of the universe that is inherently coupled to the expectation value of the constraint equations. This coupling to the constraints is responsible for the appearance of a coherent spacetime picture. Thus, the physical meaning of the constraints of the theory is quite different from Dirac's interpretation. In light of this distinctive feature of the theory, we readdress the question of the dark energy effects in the Bianchi IX cosmological model for highly non-classical quantum states. We show that, at least for this model, for any choice of the initial wave function, the quantum corrections will not produce the accelerated expansion of the universe.

Quantum Kasner transition in a locally rotationally symmetric Bianchi II universe

“…(30) Obtaining the operators R1 and R2 might be a very complicate task, so we will proceed order by order. Up to third-order in moments, one can construct these conserved operators just by directly promoting the conserved classical quantities (14) to operators:…”

“…However, it is still unclear how this scenario can be modified when quantum effects are taken into account. In the literature, there are several approaches to quantum models of Bianchi IX, focused on different features and questions (as the avoidance of the singularity or the survival of the chaotic behavior) [12][13][14][15][16][17][18][19][20][21][22][23], as well as simpler approaches to Bianchi II quantum models [24][25][26][27].…”

Some examples for different descriptions of energy–momentum density in the context of Bianchi IX cosmological model

“…In 2005, the authors in Ref. [20] applied the canonical quantum theory of gravity (Quantum Geometrodynamics) to the homogeneous Bianchi type IX cosmological model, and accordingly they developed the framework for the quantum theory of homogeneous cosmology. In 2009, Bakas et al [21] considered spatially homogeneous (but generally non-isotropic) cosmologies in the Hořava-Lifshitz gravity and compared them to those of GR using Hamiltonian methods.…”

On the Dynamics of Bianchi IX Cosmological Models

“…The problem of time is the most known difficulty of the Wheeler-DeWitt (WDW) quantum geometrodynamics which is a theoretical basis for modern quantum cosmology [18], [19]. Time in quantum mechanics and general relativity are drastically different from each other; in quantum mechanics time is a global and absolute parameter, in general relativity a local and dynamical variable [20].…”