Covariant Loop Quantum Gravity

Abstract

Quantum gravity is among the most fascinating problems in physics. It modifies our understanding of time, space and matter. The recent development of the loop approach has allowed us to explore domains ranging from black hole thermodynamics to the early Universe. This book provides readers with a simple introduction to loop quantum gravity, centred on its covariant approach. It focuses on the physical and conceptual aspects of the problem and includes the background material needed to enter this lively domain of research, making it ideal for researchers and graduate students. Topics covered include quanta of space; classical and quantum physics without time; tetrad formalism; Holst action; lattice QCD; Regge calculus; ADM and Ashtekar variables; Ponzano-Regge and Turaev-Viro amplitudes; kinematics and dynamics of 4D Lorentzian quantum gravity; spectrum of area and volume; coherent states; classical limit; matter couplings; graviton propagator; spinfoam cosmology and black hole thermodynamics.

Observational issues in loop quantum cosmology

“…It should also be pointed out that recent results from the full spinfoam theory give strong indications in favor of singularity resolutions [125]. In addition, first attempts to derive Friedmann equations, at first order, from group field theory, are also very promising [126].…”

A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons

“…(ii) Holonomy discretization, mostly inspired by the ideas of quantum gravity [16], where connections along the edges are discretized through their holonomy. This approach requires mathematical precisions by fixing a preliminary gauge on the 1-vertices of the discretized manifold, but in a final analysis, only theories depending of secondary characteristic classes (e.g., Chern-Simons theory) can give rise to pathologies in gauge covariance, whereas gauge theories arising from primary characteristic classes (e.g., Yang-Mills theories) are fully gauge covariant when discretized [3].…”

“…This enables the (physics) definition of holonomy present, for example, in [16] that we apply to the edges [ , ] and identify with the coefficients , :…”

“…This paper is a companion work to [1][2][3] where, after a tentative in [4], the extension of the notion of classical PC matrices to matrices with coefficients in a group is considered, partially motivated (in my case) by the striking similarities with mathematical constructions in discretized gauge theories. A not complete list of reference about PC matrices is [5][6][7], oriented in our perspective, and a very partial list of references about gauge theories and their discretized forms is [8][9][10][11][12][13][14][15][16][17][18][19].…”

“…The link with gauge theories is performed via holonomy, which appears in discretizations described in [3,16]. In order to make a link with questions raised by computer scientists in private communications, we have to precise that any matrix group is a finite dimensional Lie group and that most finite dimensional groups can be expressed as matrix groups, for inversion and multiplication of matrices.…”