Mathematical structure of loop quantum cosmology (original) (raw)

1 2 A pr 2 01 2 Group theoretical Quantization of Isotropic Loop Cosmology

2021

We achieve a group theoretical quantization of the flat Friedmann-Robertson-Walker model coupled to a massless scalar field adopting the improved dynamics of loop quantum cosmology. Deparemeterizing the system using the scalar field as internal time, we first identify a complete set of phase space observables whose Poisson algebra is isomorphic to the su(1, 1) Lie algebra. It is generated by the volume observable and the Hamiltonian. These observables describe faithfully the regularized phase space underlying the loop quantization: they account for the polymerization of the variable conjugate to the volume and for the existence of a kinematical non-vanishing minimum volume. Since the Hamiltonian is an element in the su(1, 1) Lie algebra, the dynamics is now implemented as SU(1, 1) transformations. At the quantum level, the system is quantized as a time-like irreducible representation of the group SU(1, 1). These representations are labeled by a half-integer spin, which gives the min...

An Introduction to Loop Quantum Gravity with Application to Cosmology

2014

The development of a quantum theory of gravity has been ongoing in the theoretical physics community for about 80 years, yet it remains unsolved. In this dissertation, we review the loop quantum gravity approach and its application to cosmology, better known as loop quantum cosmology. In particular, we present the background formalism of the full theory together with its main result, namely the discreteness of space on the Planck scale. For its application to cosmology, we focus on the homogeneous isotropic universe with free massless scalar field. We present the kinematical structure and the features it shares with the full theory. Also, we review the way in which classical Big Bang singularity is avoided in this model. Specifically, the spectrum of the operator corresponding to the classical inverse scale factor is bounded from above, the quantum evolution is governed by a difference rather than a differential equation and the Big Bang is replaced by a Big Bounce.

Is loop quantization in cosmology unique

Physical Review D, 2008

We re-examine the process of loop quantization for flat isotropic models in cosmology. In particular, we contrast different inequivalent `loop quantizations' of these simple models through their respective successes and limitations and assess whether they can lead to any viable physical description. We propose three simple requirements which any such admissible quantum model should satisfy: i) independence from any auxiliary structure, such as a fiducial interval/cell introduced to define the phase space when integrating over non-compact manifolds; ii) existence of a well defined classical limit and iii) provide a sensible "Planck scale" where quantum gravitational effects become manifest. We show that even when it may seem that one can have several possible loop quantizations, these physical requirements considerably narrow down the consistent choices. Apart for the so called improved dynamics of LQC, none of the other available inequivalent loop quantizations pass above tests, showing the limitations of lattice refinement models to approximate the homogeneous sector and loop modified quantum geometrodynamics. We conclude that amongst a large class of loop quantizations in isotropic cosmology, there is a unique consistent choice.