LQG for the Bewildered: The Self-Dual Approach Revisited

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Springer, Oct 7, 2016 - Science - 138 pages

This primer offers a concise introduction to Loop Quantum Gravity (LQG)
- a theoretical framework for uniting Quantum Mechanics (QM) with General Relativity (GR). The emphasis is on the physical aspects of the framework and its historical development in terms of self-dual variables, still most suited for a first, pedagogical encounter with LQG.

The text starts by reviewing GR and the very basics of Quantum Field Theory (QFT), and then explains in a concise and clear manner the steps leading from the Einstein-Hilbert action for gravity to the construction of the quantum states of geometry, known as spin-networks, and which provide the basis for the kinematical Hilbert space of quantum general relativity. Along the way the various associated concepts of tetrads, spin-connection and holonomies are introduced.

Having thus provided a minimal introduction to the LQG framework, some applications to the problems of black hole entropy and of quantum cosmology are briefly surveyed. Last but not least, a list of the most common criticisms of LQG is presented, which are then tackled one by one in order to convince the reader of the physical viability of the theory. A set of appendices provides accessible introductions to several key notions such as the Peter-Weyl theorem, duality of differential forms and Regge calculus, among others.

The presentation is aimed at graduate students and researchers who have some familiarity with the tools of QM and GR, but are intimidated by the technicalities required to browse through the existing LQG literature. This primer aims at making the formalism appear a little less bewildering to the uninitiated and helps lower the barrier for entry into the field.

 

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Contents

1 Introduction
1
2 Classical GR
5
3 Quantum Field Theory
15
4 Expanding on Classical GR
29
5 First Steps to a Theory of Quantum Gravity
53
6 Kinematical Hilbert Space
63
7 Applications
76
8 Discussion
99
Appendix D 3 + 1 Decomposition of the Palatini Action
111
Appendix E Duality
114
Appendix F Path Ordered Exponential
122
Appendix G PeterWeyl Theorem
123
Appendix H SquareFree Numbers
127
Appendix I BrahmaguptaPell Equation
129
Appendix J Kodama State
133
Appendix K 3j Symbols and 6j Symbols
135

Appendix A Conventions
103
Appendix B Lie Derivative
105
Appendix C ADM Variables
107
Appendix L Regge Calculus
137
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