Diffeomorphisms

We propose an approach to the quantization of the interior of a Schwarzschild black hole, represented by a Kantowski-Sachs (KS) framework, by requiring its covariance under a notion of residual diffeomorphisms. We solve for the family of Hamiltonian constraint operators satisfying the associated covariance condition, in addition to parity covariance, preservation of the Bohr Hilbert space of Loop Quantum KS and a correct (naïve) classical limit. We further explore imposing minimality of the number of terms, and compare the solution with other Hamiltonian constraints proposed for Loop Quantum KS in the literature, with special attention to a most recent case. In addition, we discuss a lapse commonly chosen to decouple the evolution of the two degrees of freedom of the model, yielding exact solubility of the model, and we show that such choice can indeed be quantized as an operator densely defined on the Bohr Hilbert space, but must include an infinite number of shift operators. Also, we show the reasons why we call the classical limit “naïve”, and point this out as a reason for one limitation of some present prescriptions.

The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries.
This dissertation develops methods for the accurate computation of high-order polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05].

1In this dissertation we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. To impose the symmetry at the quantum level, on both the connection and its conjugate momentum, the method used necessarily has similiarities to the Gupta-Bleuler method of quantizing the electromagnetic field. Lastly, a strategy for embedding states of loop quantum cosmology into this new homogeneous isotropic sector, and using this embedding to compare the physics, is presented.