Physics

One of the qualitatively distinct and robust implication of Loop Quantum Gravity (LQG) is the underlying discrete structure. In the cosmological context elucidated by Loop Quantum Cosmology (LQC), this is manifested by the Hamiltonian constraint equation being a (partial) difference equation. One obtains an effective Hamiltonian framework by making the continuum approximation followed by a WKB approximation. In the large volume regime, these lead to the usual classical Einstein equation which is independent of both the Barbero-Immirzi parameter γ as well as . In this work we present an alternative derivation of the effective Hamiltonian by-passing the continuum approximation step. As a result, the effective Hamiltonian is obtained as a close form expression in γ. These corrections to the Einstein equation can be thought of as corrections due to the underlying discrete (spatial) geometry with γ controlling the size of these corrections. These corrections imply a bound on the rate of change of the volume of the isotropic universe. In most cases these are perturbative in nature but for cosmological constant dominated isotropic universe, there are significant deviations.

In this second paper on loop quantization of Gowdy model, we introduce the kinematical Hilbert space on which appropriate holonomies and fluxes are well represented. The quantization of the volume operator and the Gauss constraint is straightforward. Imposition of the Gauss constraint can be done on the kinematical Hilbert space to select subspace of gauge invariant states. We carry out the quantization of the Hamiltonian constraint making specific choices. Alternative choices are briefly discussed. It appears that to get spatial correlations reflected in the Hamiltonian constraint, one may have to adopt the so called 'effective operator viewpoint'.

In this second paper on loop quantization of Gowdy model, we introduce the kinematical Hilbert space on which appropriate holonomies and fluxes are well represented. The quantization of the volume operator and the Gauss constraint is straightforward. Imposition of the Gauss constraint can be done on the kinematical Hilbert space to select subspace of gauge invariant states. We carry out the quantization of the Hamiltonian constraint making specific choices. Alternative choices are briefly discussed. It appears that to get spatial correlations reflected in the Hamiltonian constraint, one may have to adopt the so called 'effective operator viewpoint'.

We study the coupling of the gravitational action, which is a linear combination of the Hilbert-Palatini term and the quadratic torsion term, to the action of Dirac fermions. The system possesses local Poincare invariance and hence belongs to Poincare gauge theory with matter. The complete Hamiltonian analysis of the theory is carried out without gauge fixing but under certain ansatz on the coupling parameters, which leads to a consistent connection dynamics with second-class constraints and torsion. After performing a partial gauge fixing, all second-class constraints can be solved, and a SU (2)-connection dynamical formalism of the theory can be obtained. Hence, the techniques of loop quantum gravity can be employed to quantize this Poincare gauge theory with non-zero torsion. Moreover, the Barbero-Immirzi parameter in loop quantum gravity acquires its physical meaning as the coupling parameter between the Hilbert-Palatini term and the quadratic torsion term in this gauge theory of gravity.