Table Of Content

10367_9789813207110_tp.indd 1 23/11/17 1:53 PM 10367_9789813207110_tp.indd 2 23/11/17 1:53 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Arai, Asao, 1954– author. Title: Analysis on Fock spaces and mathematical theory of quantum fields : an introduction to mathematical analysis of quantum fields / by Asao Arai (Hokkaido University, Japan). Description: New Jersey : World Scientific, 2017. | Includes bibliographical references and index. Identifiers: LCCN 2017039521 | ISBN 9789813207110 (hardcover : alk. paper) Subjects: LCSH: Hilbert space. | Quantum theory. Classification: LCC QA322.4 .A73 2017 | DDC 530.1201/515733--dc23 LC record available at https://lccn.loc.gov/2017039521 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2018 by World Scientific Publishing Co. Pte. 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 http://www.worldscientific.com/worldscibooks/10.1142/10367#t=suppl Printed in Singapore LaiFun - 10367 - Analysis on Fock Spaces.indd 1 06-11-17 10:19:50 AM October30,2017 10:56 ws-book9x6 BC:10367-AnalysisonFockSpaces 2ndReading arai pagev Preface The notionofquantumfield wasintroducedinphysics inthe late of1920’s tounifytheparticleandthewavepictureofquantumparticlessuchaselec- tronsandphotonswhichhavetheso-calledwave-particleduality. Afterthe pioneering work on quantization of wave functions (the so-called “second quantization”)by Dirac (1927),Jordan& Klein(1927),Jordan(1927)and Jordan & Wigner (1928), Heisenberg and Pauli (1929, 1930)1 presented a generaltheoryofquantumfieldswhichunifiestheforegoingtheories,where a quantum field is “defined” by the “canonical quantization” of a classical wave field. A basic class of quantum fields is called free quantum fields which describe free (non-interacting) elementary particles. As far as the free quantum fields are concerned, the quantum field theory (QFT) by Heisenberg and Pauli was successful, giving a unification of the wave and the parti- cle picture of elementaryparticles with a generaltheoreticalframeworkfor quantumsystemsinwhichthenumberofelementaryparticlesmaychange. But, if oneapplies the formalismofQFT toaninteractingsystemofquan- tum fields and calculates physical quantities of interest in an approximate way (formal perturbation theory), then one encounters with divergent in- tegrals and hence meaningless results. This is the so-called “difficulty of divergence” in QFT.2 Later in 1940’s, a prescription to avoid the difficulty of divergence and to obtain meaningful results which canbe comparedwith experiments was 1Heisenberg,W.andPauli,W.,ZurQuantendynamikderWellenfelderI,Z.Phys. 56, pp.1–61; ibid. II,Z. Phys. 59,pp.168–190. 2There are two kinds of divergences: an ultraviolet and an infrared divergence. The former(resp. latter)isadivergencewithrespecttolargemomenta(resp. smallmomenta inthecaseofmasslessquantum particles). v October30,2017 10:56 ws-book9x6 BC:10367-AnalysisonFockSpaces 2ndReading arai pagevi vi Analysis on Fock Spaces and Mathematical Theory of Quantum Fields inventedbyFeynman,SchwingerandTomonaga. Theprescriptioniscalled “renormalizationtheory”,withwhichQFTas a physical theoryrevived. In particular, quantum electrodynamics (QED), a QFT describing the inter- action of electrons (the quantumelectron field) and photons (the quantum radiation field), explains experimental results with surprisingly high preci- sions.3 From the mathematically rigorous point of view, however, the difficulty of divergence in QFT still remained unsolved; it was unclear what kind of mathematics can justify QFT with renormalization theory.4 Inthesesituations,mathematicallyrigorousinvestigationsofQFTwere startedin1950’stogiveasoundmathematicalbasistoit. Sincethenmath- ematicalstudiesonquantumfields haveformedamajorstreamofresearch in modern mathematical physics. Many interesting mathematical theories have been born from studies on mathematical problems in QFT and are stilldevelopingtovariousdirections,havingcloserelationsmutually. These developments show richness of mathematics that the concept of quantum field has in itself and suggest that QFT may have something to do with almost all areas in mathematics explicitly or implicitly. Quantum fields are so fascinating, but not so easy to understand in their totality. The purpose of this book is to describe and explain for beginners basic parts of mathematical theory of quantum fields. In mathematical studies on QFT, there are two approaches basically: axiomatic one and constructive one. The former assumes a set of axioms that a quantum field should obey and investigates what results can be derived from the axioms. This approach is useful in analysis of general aspects ofQFT whichareindependent ofindividualquantumfieldmodels. QFT based on this approach is called axiomatic QFT. On the other hand, the latter approach is concerned with showing the mathematical existence of concrete models in QFT and deriving their properties in a mathematically rigorous way. This kind of QFT is called constructive QFT.5 But this classification of QFT is just for convenience; these two approaches are complementary in fact. Rather, by employing these two approaches, 3See,e.g.,Kinoshita,T.(ed.),QuantumElectrodynamics,1990,WorldScientific,Singa- pore. ThethreephysicistsmentionedabovewereawardedtheNobelprizeforphysicsin 1965fortheircontributionstoconstructionofQEDwithrenormalizationtheory(Nobel Lectures, Physics Vol.4,1972,ElsevierPublishingCompany,AmsterdamNewYork). 4At present (as of January 16, 2017), this problem is not solved yet for QFT in 4- dimensionalspace-time. 5Originallythename“constructive QFT”wasusedformathematical approaches con- structingconcretelyrelativisticinteractingquantumfieldmodels. But,here,weusethe wordinawidersense. October30,2017 10:56 ws-book9x6 BC:10367-AnalysisonFockSpaces 2ndReading arai pagevii Preface vii one may arrive at deeper, higher and wider cognition on the nature of the microscopic world described by quantum fields. This book consists of two parts. Part I, which is from Chapter 1 to Chapter 7, is devoted to giving a detailed description of theory of Fock spaces which are basic spaces in mathematical theory of quantum fields. From a purely mathematical point of view, Fock space is a general and natural type of Hilbert space in the sense that it is defined as the infinite direct sum of tensor products of a Hilbert space (see Chapters 4–6 for details), i.e., in Fock space, the concept of sum is incorporated with the concept of product in a general and natural form in terms of Hilbert space with the concept of infinity (recall that “sum” and “product” are fundamental concepts which appear in almost all places in the mathematical world, taking various forms). Historically, as its name suggests, first concrete forms of Fock space were presented by Fock6 in 1932 in relation to the aforementioned “second quantization”. In the context of QFT, a Fock space is used to describe state vectors of a quantum system in which infinitely many elementary particles may exist. It is well known that there are two kinds of families of elementary particles, i.e., bosons and fermions. Corresponding to this fact, there are three types of Fock spaces. A Fock space describing state vectors of a quantum system consisting of the same kindofbosons(resp. fermions)iscalledaboson(resp. fermion)Fockspace and a Fock space describing state vectors of a quantum system consisting ofthe samekindofbosonsandthesamekindoffermionsiscalledaboson- fermion Fock space.7 In fact, a boson Fock space (resp. a fermion Fock space) is identified with a closed subspace of a larger Hilbert space, called a full Fock space. From this point of view, in this book, description of the theory of Fock spaces starts with a full Fock space in Chapter 4. Theory of boson Fock space (resp. fermion Fock space, boson-fermion Fock space) is discussed in Chapter 5 (resp. Chapter 6, Chapter 7). In particular, in Chapter 7, a theory of infinite-dimensional Dirac type operators on a boson-fermion Fock space, which is a new theory developed by the present author in connection with supersymmetric QFT, is described in detail. As far as the author knows, this is the first time that the theory is presented as a part of a book. In Part II, which is from Chapter 8 to Chapter 14, QFT is described 6Fock,V.,KonfigurationsraumundzweiteQuantelung, Z. Phys. 75,pp.622–647. 7ThereisanothertypeofFockspace,calledaq-Fockspace,whichinterpolatesaboson FockspaceandafermionFockspace. But,inthisbook,wedonotdiscussit. October30,2017 10:56 ws-book9x6 BC:10367-AnalysisonFockSpaces 2ndReading arai pageviii viii Analysis on Fock Spaces and Mathematical Theory of Quantum Fields as applications of mathematical theories in Part I. Except Chapter 8 in which general theories of QFT including axiomatic QFT are described, concrete quantum field models and their basic properties are discussed. In Chapters 9–12, we construct four kinds of free quantum fields: (i) de Broglie field—non-relativistic matter field (Chapter 9); (ii) Klein-Gordon field—relativistic scalar field (Chapter 10); (iii) radiation field (Chapter 11); (iv) Dirac field—relativistic spinor field (Chapter 12). Free quantum fields do not describe interacting elementary particles. In Chapter 13, we treat a simple interacting quantum field model, called the van Hove model or the van Hove-Miyatake (vHM) model. This model is a prototype for interacting quantumfield models andhas been extensively studied. In this book, an abstract version of the model is defined and analyzed in detail. This approach makes clear general structures behind the concrete vHM model. It is shown how the problem of ultraviolet divergence and infrared divergence is solved in the abstract vHM model. In the last chapter, we presenta list ofinteracting quantumfield models,which aremore complex and/or realistic than the vHM model, and give a short description to each model. To make the book as self-contained as possible, appendices are added. This book is based on the author’s books “Fock Spaces and Quantum Fields I,II” (2000,Nippon-Hyoron-Sha,Tokyo)writteninJapanese,butit is not just an English translation of them: in this English edition, enlarge- mentsaswellasalterationshavebeenmade(e.g.,Chapter1isnewlyadded and Chapter 0 (heuristic arguments on QFT in physics) in the Japanese editionisnotincluded). Thisbookisprimarilyintendedtobeanintroduc- torytextbook,notamonographassoaretheJapaneseeditions. Hencethe bibliographyinthebookispartialandfarfromacompleteone. Itincludes also textbooks and monographs written in Japanese, some of which are cited in the Japanese editions. They may be helpful for readers who can readJapanese. Theauthorwouldlike tothank the readersofthe Japanese editionsfortheircomments,remarksandquestions,whichhavebeenuseful in writing this revised and enlarged edition. Asao Arai Emeritus Professor Hokkaido University Sapporo January 2017 October30,2017 10:56 ws-book9x6 BC:10367-AnalysisonFockSpaces 2ndReading arai pageix Preface ix Reader’s Guide Chapter 1 is a preliminary chapter which is devoted to description of el- ements of the theory of linear operators (in particular, unbounded self- adjoint operators) on Hilbert spaces. Those who are famliar with these topics can skip Chapter 1 and start reading in Chapter 2. In the present book,Chapter7isnotnecessarilyneededtoreadChapters8–14. Therefore, for the first reading, one can skip Chapter 7. October30,2017 10:56 ws-book9x6 BC:10367-AnalysisonFockSpaces 2ndReading arai pagexi List of Symbols Mathematical Symbols A:=B A is defined by B def A = B A is defined by B for all ∀ , anti-commutator: A,B :=AB+BA { } { } [ , ] commutator: [A,B]:=AB BA − exist ∃ , inner product of an inner product space H H h i , inner product h i H norm of an inner product space H k·k norm k·k , n tensor product ⊗ˆ, ⊗ˆn algebraic tensor product ⊗ ⊗ n anti-symmetric tensor product ⊗âns algebraic anti-symmetric tensor product ⊗as n symmetric tensor product ⊗ˆsn algebraic symmetric tensor product ⊗s N , direct sum ⊕j=1 ⊕∞n=0 s-lim strong limit Tr trace u-lim uniform limit w-lim weak limit , p exterior product (wedge product) ∧ ∧ k x Euclidean inner product: k x:= d kjxj for · · j=1 k=(k1,...,kd) and x=(x1,...,xd) P (kj,xj R,j =1,...,d) ∈ xi