Lectures on Loop Quantum Gravity

Abstract

Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures.Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radical perception that gravity is geometry.Minimality here means that one explores the logical consequences of bringing together the two fundamental principles of modern physics, namely general covariance and quantum theory, without adding any experimentally unverified additional structures such as extra dimensions, extra symmetries or extra particle content beyond the standard model. While this is a very conservative approach and thus maybe not very attractive to many researchers, it has the advantage that pushing the theory to its logical frontiers will undoubtedly either result in a successful theory or derive exactly which extra structures are required, if necessary. Or put even more radically, it may show which basic principles of physics have to be given up and must be replaced by more fundamental ones.QGR therefore is, by definition, not a unified theory of all interactions in the standard sense since such a theory would require a new symmetry principle. However, it unifies all presently known interactions in a new sense by quantum mechanically implementing their common symmetry group, the four-dimensional diffeomorphism group, which is almost completely broken in perturbative approaches.

Lorentz and Galilei invariance on lattices

“…In addition to providing a convenient tool for obtaining quark confinement and regularizing divergencies in quantum theories [2,3,4,5], an elementary length, related to the Planck scale l P = √ k = 10 −33 cm, may actually exist. Indeed, this is consistent with several approaches to quantum gravity [6,7,8,9]. Qualitative arguments support the idea that gravity may provide a natural cut-off in the integration of virtual loops in the UV regime of field theories.…”

Loop cosmological dynamics and dualities with Randall-Sundrum braneworlds

“…Modifications to the Friedmann dynamics appear also in loop quantum cosmology (LQC) [4] which is the symmetry reduced quantization of homogeneous and isotropic spacetimes based on the loop quantum gravity (LQG) [5]. LQG which is one of the background independent and non-perturbative candidate theories for quantizing Einsteinian gravity in four dimensions predicts that at quantum level, classical spacetime contin- * e-mail address: singh@gravity.psu.edu uum is replaced by a discrete quantum geometry and operators corresponding to length of a curve, area of a surface and volume of an enclosed region have discrete eigenvalues.…”

Semiclassical states, effective dynamics, and classical emergence in loop quantum cosmology

“…A fundamental input of loop quantum gravity (LQG) [1] to our understanding of quantum spacetime is that it is inherently discrete and spatial geometry is quantized. Quantum features of spacetime become evident in the regime of very high curvature whereas continuous spacetime emerges as a large eigenvalue limit of quantum geometry.…”