[Paper Review] Vacuum CGHS in loop quantum gravity and singularity resolution

Alejandro Corichi, Javier Olmedo+1 more|arXiv (Cornell University)|Aug 22, 2016

Noncommutative and Quantum Gravity Theories3 citations

TL;DR

This paper presents a complete loop quantum gravity quantization of the vacuum CGHS model using new variables analogous to those in spherically symmetric LQG. By employing a polymeric representation and a redefined Hamiltonian constraint with structure constants, the work constructs a physical Hilbert space where singular geometries are ruled out due to non-vanishing eigenvalues of the dilaton and metric operators, while preserving the black hole mass observable and introducing a new bulk observable for the dilaton squared.

ABSTRACT

We study here a complete quantization of a Callan-Giddings-Harvey-Strominger (CGHS) vacuum model following loop quantum gravity techniques. Concretely, we adopt a formulation of the model in terms of a set of new variables that resemble the ones commonly employed in spherically symmetric loop quantum gravity. The classical theory consists of two pairs of canonical variables plus a scalar and diffeomorphism (first class) constraints. We consider a suitable redefinition of the Hamiltonian constraint such that the new constraint algebra (with structure constants) is well adapted to the Dirac quantization approach. For it, we adopt a polymeric representation for both the geometry and the dilaton field. On the one hand, we find a suitable invariant domain of the scalar constraint operator, and we construct explicitly its solution space. There, the eigenvalues of the dilaton and the metric operators cannot vanish locally, allowing us to conclude that singular geometries are ruled out in the quantum theory. On the other hand, the physical Hilbert space is constructed out of them, after group averaging the previous states with the diffeomorphism constraint. In turn, we identify the standard observable corresponding to the mass of the black hole at the boundary, in agreement with the classical theory. We also construct an additional observable on the bulk associated with the square of the dilaton field, with no direct classical analog.

Motivation & Objective

• To extend loop quantum gravity techniques to the CGHS vacuum model, a 2D dilaton gravity theory with black hole solutions.
• To address the problem of spacetime singularities in quantum gravity by constructing a unitary, background-independent quantization.
• To identify physical observables, including the black hole mass and a new bulk observable for the dilaton field squared.
• To ensure consistency with classical physics in the semiclassical limit, particularly for the mass observable.

Proposed method

• Adopt a reformulation of the CGHS model in terms of new canonical variables resembling those in spherically symmetric LQG.
• Introduce a redefined Hamiltonian constraint with structure constants to facilitate Dirac quantization in a polymeric representation.
• Implement a polymeric representation for both geometric degrees of freedom and the dilaton field, ensuring discrete spectra.
• Construct an invariant domain for the scalar constraint operator and explicitly solve for its kernel states.
• Apply group averaging to the solutions of the scalar constraint to project onto states satisfying the diffeomorphism constraint.
• Identify the standard black hole mass observable at spatial infinity and construct a new bulk observable corresponding to the square of the dilaton field.

Experimental results

Research questions

• RQ1 Can the vacuum CGHS model be consistently quantized using loop quantum gravity techniques with a polymeric representation of geometry and matter fields?
• RQ2 Does the resulting quantum theory resolve spacetime singularities by preventing the metric and dilaton from vanishing locally?
• RQ3 How are physical observables, such as the black hole mass and a new bulk observable for the dilaton squared, recovered in the quantum theory?
• RQ4 Is the classical limit of the quantum theory consistent with the known classical behavior, particularly for the mass observable?
• RQ5 What is the role of the redefined Hamiltonian constraint with structure constants in enabling a consistent quantization procedure?

Key findings

• The eigenvalues of the dilaton and metric operators are strictly non-zero in the quantum theory, ruling out singular geometries locally.
• The physical Hilbert space is constructed via group averaging of states satisfying the scalar constraint, ensuring diffeomorphism invariance.
• The standard black hole mass observable is recovered at spatial infinity, confirming consistency with classical general relativity.
• A new bulk observable associated with the square of the dilaton field is constructed, which has no direct classical counterpart.
• The redefined Hamiltonian constraint with structure constants enables a consistent Dirac quantization approach in the polymeric representation.
• The solution space of the scalar constraint is explicitly constructed, providing a foundation for further analysis of quantum dynamics.