Skyrmions: A Theory Of Nuclei

322 Pages·2022·17.135 MB·English

by  Nicholas S. Manton

SSKKYYRRMMIIOONNSS
A Theory of Nuclei
QQ00336688__99778811880000661122447711__TTPP..iinndddd 11 1111//11//2222 44::5577 PPMM
B1948 Governing Asia
TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk
BB11994488__11--AAookkii..iinndddd 66 99//2222//22001144 44::2244::5577 PPMM
SSKKYYRRMMIIOONNSS
A Theory of Nuclei
Nicholas S Manton
University of Cambridge, UK
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
QQ00336688__99778811880000661122447711__TTPP..iinndddd 22 1111//11//2222 44::5577 PPMM
Published by
World Scientific Publishing Europe Ltd.
57 Shelton Street, Covent Garden, London WC2H 9HE
Head office: 5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
Library of Congress Cataloging-in-Publication Data
Names: Manton, Nicholas, 1952– author.
Title: Skyrmions : a theory of nuclei / Nicholas S. Manton.
Description: Hackensack, New Jersey : World Scientific, [2022] |
Includes bibliographical references and index.
Identifiers: LCCN 2021062774 | ISBN 9781800612471 (hardcover) |
ISBN 9781800612488 (ebook for institutions) | ISBN 9781800612495 (ebook for individuals)
Subjects: LCSH: Skyrme model. | Atomic structure. | Topology. | Nuclear physics--History. |
Quantum theory--History.
Classification: LCC QC793.3.S8 M36 2022 | DDC 539.7/4--dc23/eng20220218
LC record available at https://lccn.loc.gov/2021062774
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2022 by World Scientific Publishing Europe Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.
For any available supplementary material, please visit
https://www.worldscientific.com/worldscibooks/10.1142/Q0368#t=suppl
Printed in Singapore
SShhiiYYiinngg -- QQ00336688 -- SSkkyyrrmmiioonnss..iinndddd 11 2222//22//22002222 99::4444::4400 aamm
January27,2022 9:17 ws-book9x6 SKYRMIONS–ATheoryofNuclei WSSky/0 pagev
Foreword
by Prof. K. K. Phua
Chairman and Editor-in-Chief, World Scientific
World Scientific is honoured to publish Skyrmions – A Theory of Nuclei
by Professor Nick Manton of Cambridge University to celebrate the cente-
nary in December 2022 of the late Professor Tony Skyrme’s birth.
I got my Ph.D. from the Department of Mathematical Physics at the
University of Birmingham in early 1970, and had very close contact and
some very good discussions with Professor Skyrme about different areas of
research in physics.
IrememberveryclearlythatProfessorSkyrmecameanddiscussedwith
me whether SU(3) and his work on Skyrmions are in any way related or
connected. We had some good discussion about this problem (which is
considered in Chapter 14 of this book) and frankly speaking, I learnt and
benefitted a lot from him.
Professor Skyrme was quite familiar with science and research in
Malaysia and Singapore. He was Head of the Mathematics Department
at the University of Malaya for about three years in the early 1960s and
visitedSingaporeafewtimes.Hence,hegottoknowthelecturersandstu-
dents in Malaysia and Singapore well, which is one of the reasons why the
DepartmentsofPhysicsandMathematicsattheUniversityofBirmingham
attractedquitealargenumberofAsianstudentsfromSingapore,Malaysia
and Hong Kong.
In summary, Professor Skyrme was one of the greatest physicists of the
20th century. He was a deep thinker and had many original ideas.
v
B1948 Governing Asia
TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk
BB11994488__11--AAookkii..iinndddd 66 99//2222//22001144 44::2244::5577 PPMM
January27,2022 9:17 ws-book9x6 SKYRMIONS–ATheoryofNuclei WSSky/0 pagevii
Preface
I would like to thank J. Martin Speight for encouraging me to write this
book.Forawhile,Ihadtoomanydutiesanddistractions,butanopportu-
nitytostartitwastheclosureofmyuniversitydepartmentinMarch2020,
part of wider lockdowns and the need to work from home because of the
Coronavirus pandemic.
Much of this book is based on my published papers from 1986 on-
wards, often written jointly with my Ph.D. students and other collabora-
tors.IwouldparticularlyliketothankTrevorSamols,RobertLeese,Bernd
Schroers,PaulSutcliffe,RichardBattye,KimBaskerville,ConorHoughton,
PatrickIrwin,SteffenKrusch,BernardPiette,OlgaManko,StephenWood,
TheodoraIoannidou,DankradFeist,P.H.ChrisLau,ChrisHalcrow,Chris
King, Dave Foster, Mareike Haberichter, Jonathan Rawlinson and Derek
Harland for their contributions, and for discussions about Skyrmions over
very many years. They originally carried out numerous analytical and nu-
merical calculations whose results are presented here, and they produced
most of the figures. I also wish to record my gratitude for the inspiring col-
laboration and discussions with the late Sir Michael Atiyah, whose interest
in Skyrmions and related ideas in theoretical physics ran from about 1987
until the end of his life in 2019.
PartofthisbookisareworkingofsomechaptersinTopologicalSolitons,
the CUP monograph I wrote jointly with Paul Sutcliffe. I thank Paul and
Cambridge University Press for permission to reuse this material. I am
also most grateful to Paul, and also to Chris Lau and Chris Halcrow, for
reading substantial parts of a draft of this book, and for suggesting several
improvementsandcorrections,manyofwhichhavebeenimplemented.Last
but not least, I warmly thank Shi Ying Koe and Rozita Osman at World
vii
January27,2022 9:17 ws-book9x6 SKYRMIONS–ATheoryofNuclei WSSky/0 pageviii
viii SKYRMIONS – A Theory of Nuclei
Scientific for the efficient and helpful way they have turned my manuscript
into a book ready for publication.
This book is dedicated to the memory of Tony Skyrme, the centenary
of whose birth falls in 2022. I was fortunate to have met him once.
January27,2022 9:17 ws-book9x6 SKYRMIONS–ATheoryofNuclei WSSky/0 pageix
Contents
Foreword v
Preface vii
1. Introduction 1
2. Fields and Particles 11
2.1 The Classical Notions of Particles and Fields . . . . . . . 11
2.1.1 Quantum theory . . . . . . . . . . . . . . . . . . . 13
2.2 Quantum Field Theory and the Standard Model . . . . . 14
2.3 Pions and Nucleons . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Nuclei and Isospin . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Effective Field Theory . . . . . . . . . . . . . . . . . . . . 24
2.6 Skyrme’s Effective Field Theory . . . . . . . . . . . . . . . 27
3. Lagrangians and Symmetries 29
3.1 Finite-Dimensional Systems . . . . . . . . . . . . . . . . . 29
3.2 Symmetries and Conservation Laws . . . . . . . . . . . . . 32
3.3 Lagrangian Field Theory . . . . . . . . . . . . . . . . . . . 33
3.4 Noether’s Theorem in Field Theory . . . . . . . . . . . . . 37
4. Skyrme Theory 41
4.1 SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 The Skyrme Lagrangian and Field Equation . . . . . . . . 44
4.3 Skyrme Field Topology . . . . . . . . . . . . . . . . . . . . 47
4.3.1 Topological degree of a map . . . . . . . . . . . . 48
ix