Quantum Field Theory Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter

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Quantum Field Theory

Quantum Field Theory Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter

Lukong Cornelius Fai

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2020 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-0-367-18574-9 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify this in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Fai, Lukong Cornelius, author. Title: Quantum field theory : Feynman path integrals and diagrammatic techniques in condensed matter / Lukong Cornelius Fai. Description: Boca Raton, FL : CRC Press, Taylor & Francis Group, [2019] | Includes bibliographical references and index. Identifiers: LCCN 2019000921| ISBN 9780367185749 (hbk ; alk. paper) | ISBN 0367185741 (hbk ; alk. paper) | ISBN 9780429196942 (ebook) | ISBN 0429196946 (ebook) Subjects: LCSH: Quantum field theory. | Feynman integrals. | Feynman diagrams. | Condensed matter. Classification: LCC QC174.45 .F35 2019 | DDC 530.14/3–dc 3 LC record available at https://lccn.loc.gov/2019000921 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents Preface....................................................................................................................... xi About the Author................................................................................................... xiii

1

Symmetry Requirements in QFT........................................................................1

2

Coherent States. . ................................................................................................17

1.1

2.1 2.2

2.3

2.4

Second Quantization..........................................................................................................................1 1.1.1 Fock Space���������������������������������������������������������������������������������������������������������������������������1 1.1.2 Creation and Annihilation Operators............................................................................ 6 1.1.3 (Anti)Commutation Relations......................................................................................... 9 1.1.4 Change of Basis in Second Quantization......................................................................10 1.1.5 Quantum Field Operators................................................................................................ 11 1.1.6 Operators in Second-Quantized Form..........................................................................12 1.1.6.1 One-Body Operator........................................................................................12 1.1.6.2 Two-Body Operator........................................................................................14

Coherent States for Bosons..............................................................................................................17 Coherent States and Overcompleteness........................................................................................18 2.2.1 Overcompleteness of Coherent States........................................................................... 20 2.2.2 Overlap of Two Coherent States......................................................................................21 2.2.3 Overcompleteness Condition......................................................................................... 22 2.2.4 Closure Relation via Schur’s Lemma............................................................................. 23 2.2.5 Normal-Ordered Operators............................................................................................ 25 2.2.6 The Trace of an Operator................................................................................................ 25 Grassmann Algebra and Fermions............................................................................................... 26 2.3.1 Grassmann Algebra......................................................................................................... 26 2.3.1.1 Differentiation over Grassmann Variables................................................. 27 2.3.1.2 Exponential Function of Grassmann Numbers.........................................31 2.3.1.3 Involution of Grassmann Numbers............................................................. 32 2.3.1.4 Bilinear Form of Operators........................................................................... 32 2.3.1.5 Berezin Integration........................................................................................ 33 2.3.1.6 Grassmann Delta Function........................................................................... 34 2.3.1.7 Scalar Product of Grassmann Algebra........................................................ 34 2.3.2 Fermions��������������������������������������������������������������������������������������������������������������������������� 34 Fermions and Coherent States....................................................................................................... 36 2.4.1 Coherent State Overcompleteness Relation Proof....................................................... 39 2.4.2 Trace of a Physical Quantity............................................................................................41 2.4.3 Functional Integral Time-Ordered Property............................................................... 42 v

vi

Contents

2.5

2.6

3

Gaussian Integrals........................................................................................................................... 45 2.5.1 Multidimensional Gaussian Integral............................................................................ 45 2.5.2 Multidimensional Complex Gaussian Integral...........................................................46 2.5.3 Multidimensional Grassmann Gaussian Integral....................................................... 47 Wick Theorem for Multidimensional Grassmann Integrals............................................................................................................................................ 48 2.6.1 Wick Theorem ............................................................................................................... 49

Fermionic and Bosonic Path Integrals............................................................. 51 3.1 3.2 3.3 3.4 3.5 3.6

Coherent State Path Integrals.........................................................................................................51 Noninteracting Particles................................................................................................................ 56 3.2.1 Bare Partition Function................................................................................................... 56 3.2.2 Inverse Matrix of S(α)........................................................................................................ 60 Bare Green’s Function via Generating Functional..................................................................... 62 3.3.1 Generating Functional.................................................................................................... 63 Single-Particle Green’s Function................................................................................................... 65 3.4.1 Matsubara Green’s Function.......................................................................................... 65 Noninteracting Green’s Function................................................................................................. 67 Average Value of a Functional....................................................................................................... 68

4

Perturbation Theory and Feynman Diagrams.................................................71

5

(Anti)Symmetrized Vertices............................................................................ 93

6

Generating Functionals.. ................................................................................. 101

4.1 4.2 4.3 4.4

5.1

6.1 6.2 6.3 6.4

Representation as Diagrams...........................................................................................................71...

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