Table Of Content10367_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.
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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
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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 de-
scribe free (non-interacting) elementary particles. As far as the free quan-
tum 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
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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 dif-
ficulty 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 mathemat-
ically 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.
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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 infi-
nite 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 mathemat-
ical 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.
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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
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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.
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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
⊗ˆans 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
