First Steps to a Theory of Quantum Gravity
As discussed in the previous section, we wish to attempt to canonically quantise GR, which means turning the Hamiltonian, diffeomorphism and Gauss constraints into operators and replacing Poisson brackets with commutation relations.
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LQG for the Bewildered The Self-Dual Approach Revisited
LQG for the Bewildered
Deepak Vaid Sundance Bilson-Thompson •
LQG for the Bewildered The Self-Dual Approach Revisited
123
Sundance Bilson-Thompson School of Chemistry and Physics University of Adelaide Adelaide, SA Australia
Deepak Vaid Department of Physics National Institute of Technology Surathkal, Karnataka India
ISBN 978-3-319-43182-6 DOI 10.1007/978-3-319-43184-0
ISBN 978-3-319-43184-0
(eBook)
Library of Congress Control Number: 2016947774 © Springer International Publishing Switzerland 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 3 4
2 Classical GR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Parallel Transport and Curvature . . . . . . . . . . . . . . . . . . . . . . 2.2 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Changes of Coordinates and Diffeomorphism Invariance . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Covariant Derivative and Curvature . . . . . . . . . . . . . . . 3.2 Dual Tensors, Bivectors and k-Forms . . . . . . . . . . . . . . 3.3 Wilson Loops and Holonomies . . . . . . . . . . . . . . . . . . . 3.4 Dynamics of Quantum Fields . . . . . . . . . . . . . . . . . . . . 3.4.1 Lagrangian (or Path Integral) Approach . . . . . . 3.4
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