[Paper Review] A New Approach in Quantum Gravity and its Cosmological Implications

Simone Mercuri, Giovanni Montani|arXiv (Cornell University)|Jan 26, 2004

Noncommutative and Quantum Gravity Theories18 references3 citations

TL;DR

This paper proposes a new canonical quantization approach to quantum gravity by fixing the (3+1) slicing of spacetime through a kinematical action, which eliminates the 'frozen formalism' problem by introducing a dynamical reference frame. The method yields Schrödinger-like evolution equations by transforming Hamiltonian constraints into parabolic form, and in the semiclassical limit, the reference frame corresponds to a non-geodesic dust fluid, leading to cosmological implications such as a dynamical dark matter component in the WKB limit.

ABSTRACT

This work concerns a new reformulation of quantum geometrodynamics, which allows to overcome a fundamental ambiguity contained in the canonical approach to quantum gravity: the possibility of performing a (3+1)-slicing of space-time, when the metric tensor is in a quantum regime. Our formulation provides also a procedure to solve the problems connected to the so called "frozen formalism". In particular we fix the reference frame (i.e. the lapse function and the shift vector) by introducing the so called "kinematical action"; as a consequence, the new hamiltonian constraints become parabolic, so arriving to evolutive (Schroedinger-like) equations for the quantum dynamics. The kinematical action can be interpreted as the action of a pressureless, but, in general, non geodesic perfect fluid, so in the semi classical limit our theory leads to the dynamics of the gravitational field coupled to a dust which represents the material reference frame we have introduced fixing the slicing. We also investigate the cosmological implications of the presence of the dust, which, in the WKB limit of a cosmological problem, makes account for a "dark matter" component and could play, at present time, a dynamical role.

Motivation & Objective

• To resolve the fundamental ambiguity in canonical quantum gravity regarding (3+1) slicing when the metric is in a quantum regime.
• To overcome the 'frozen formalism' problem in the Wheeler-DeWitt equation by introducing a dynamical reference frame.
• To preserve the geometrical nature of gravity while enabling evolutive quantum dynamics through a fixed lapse and shift vector.
• To explore cosmological implications of the introduced reference fluid, particularly its role as a dynamical dark matter component in the WKB limit.

Proposed method

• Introduce a kinematical action that fixes the lapse function and shift vector, thereby breaking time-diffeomorphism invariance in a controlled way.
• Transform the Hamiltonian constraints into parabolic form, enabling Schrödinger-like evolution equations for the quantum state.
• Interpret the kinematical action as describing a pressureless, non-geodesic perfect fluid, which becomes the material reference frame in the semiclassical limit.
• Apply the formalism to a Bianchi type IX minisuperspace model with a self-interacting scalar field to test the dynamics.
• Derive a quantum evolution equation where the label time 't' acts as a dynamical parameter, with the wave function evolving via a Hamiltonian operator involving spatial derivatives and a potential.
• Enforce consistency by requiring the initial wave function phase to satisfy a Hamilton-Jacobi equation, ensuring semiclassical consistency.

Experimental results

Research questions

• RQ1 How can the (3+1) slicing ambiguity in quantum gravity be resolved when the metric is in a superposition?
• RQ2 Can a dynamical reference frame be introduced in a way that restores evolutive quantum dynamics without breaking general covariance?
• RQ3 What is the cosmological role of the reference fluid introduced via the kinematical action in the WKB approximation?
• RQ4 How does the proposed formalism compare to the multi-time approach in terms of geometrical consistency and physical interpretation?
• RQ5 Can the kinematical action lead to a physically meaningful dark matter-like component in cosmological models?

Key findings

• The proposed formalism transforms the Hamiltonian constraints into parabolic equations, enabling Schrödinger-like evolution and resolving the 'frozen formalism' problem.
• The kinematical action corresponds to a non-geodesic, pressureless perfect fluid in the semiclassical limit, which serves as a dynamical reference frame.
• In the WKB limit, the reference fluid contributes a dynamical dark matter component, potentially explaining cosmological observations.
• The model is applied to a Bianchi IX universe with a scalar field, yielding a consistent quantum evolution equation in terms of label time 't'.
• The initial wave function phase must satisfy a Hamilton-Jacobi equation, ensuring consistency with classical dynamics in the semiclassical regime.
• The formalism preserves the geometrical origin of the 3-metric while allowing evolutive dynamics, unlike the multi-time approach which treats volume as a time variable.