Table Of ContentSPRINGER BRIEFS IN MATHEMATICAL PHYSICS 35
Fumio Hiroshima
Ground States of
Quantum Field
Models
Perturbation
of Embedded
Eigenvalues
123
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Fumio Hiroshima
Ground States of Quantum
Field Models
Perturbation of Embedded Eigenvalues
123
Fumio Hiroshima
Faculty of Mathematics
Kyushu University
Fukuoka,Japan
ISSN 2197-1757 ISSN 2197-1765 (electronic)
SpringerBriefs inMathematical Physics
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To my family and friends
Preface
Thebookaimstogiveasurveyoftheexistenceofthegroundstateinquantumfield
theory from an operator theoretical point of view. In particular, I focus on inves-
tigating quantum interaction systems between non-relativistic quantum mechanical
matters and quantum fields. Mathematically the energy Hamiltonian of a quantum
interaction system can be realized as a selfadjoint operator H acting in a Hilbert
spaceH.Theselfadjointoperatorsassociatedwithquantuminteractionsystemswe
consider are Schrödinger-type operators coupled to quantum fields. The problem
reviewed in this book can be reduced to studying perturbations of eigenvalues
embedded in the continuous spectrum of the selfadjoint operators. In particular,
stabilityandinstabilityofthelowesteigenvalueoftheselfadjointoperatorsarethe
main subjects.
If the bottom of the spectrum of a selfadjoint operator is an eigenvalue, we say
thatthereexistsagroundstateoftheselfadjointoperatorortheselfadjointoperator
has a ground state.
In quantum field theory, there appears infraredproblem owingto the vanishing
mass of bosons. On account of infrared problem, it is not trivial to show the
existence of the ground state of H. Nevertheless, several versions of proofs of the
existence of the ground state of H have been developed since the end of
the twentieth century, and huge amount of related papers have been published to
date. It is, however, impossible to make a complete list of them in this book. The
reader may find descriptions and references in, e.g., Arai [4], Gustafson and Sigal
[22], and Spohn [55].
Anew pointofview ontheexistenceoftheground state emerged around 1995
withworksbyBach,Fröhlich,andSigal[6–9]inwhichtheexistenceoftheground
stateoftheso-calledPauli-Fierzmodelinnon-relativisticquantumelectrodynamics
was shown, and renormalization group method based on Feshbach maps was
introduced to study resonances. Simultaneously and independently, Arai and
Hirokawa [5] also proved the existence of the ground state of generalized
spin-boson models. In 1999, Spohn [54] succeeded in proving the existence of the
ground state of the so-called Nelson model by a functional integral method and
simultaneously Gérard [17] by compact operator arguments. In particular, [17]
vii
viii Preface
considers an abstract model including the Nelson model. In [17, 54], the existence
of the ground state is proven, no matter how large a coupling constant is.
In this period, I was spending brilliant days as a postdoctoral student (JSPS
fellow) in Universität Bonn (February 1998–January 1999) and Technische
UniversitätMünchen(February1999–February2000).Inthepostdoctoralperiodin
Germany,Iwasinspiredby[5,9,17,54],andstartedstudyingspectralanalysisof
Hamiltonians related to quantum field theory and collaborated with Herbert Spohn
in München. I attended “Les Houches Summer School of Physics 1998” in
Chamonix where my research was encouraged by an encounter with a lot of dis-
tinguished researchers coming from all over the world and tried to climb up the
Mont Blanc in holidays during the summer school. Several participants of the
summer school asked me how the top of the Mont Blanc was, but nobody unfor-
tunatelyaskedmemyownscientificresearch.Intheautumnof1998,Igaveatalk
inamathematicalphysicsseminarinTechnischeUniversitätBerlin,whereaftermy
talk Volker Bach taught me to be able to avoid infrared divergence in
non-relativisticquantumelectrodynamics.Thissurprisedmeverymuch.Iwasalso
invited to an international conference held in Lille in France in 1999, where
ChristianGérardgaveatalkontheexistenceofthegroundstatejustbeforemytalk.
It was too impressive to talk for me. In 2001, Griesemer, Lieb and Loss [21] and
LiebandLoss[43]provedtheexistenceofthegroundstateofthePauli-Fierzmodel
undertheso-calledbindingcondition,inwhichacompactembeddingofaSobolev
space was applied.
As was stated above, spectral analysis of the Pauli-Fierz mode, the Nelson
model,and related models was progressed veryquickly attheend ofthe twentieth
century. Without a break in the twenty-first century, a lot of important papers
concerned with non-relativistic quantum electrodynamics, etc. were also produced
at an increasing tempo. I may say that results accumulate and eventually reach a
critical state nowadays.
In July 2015, I attended ICMP2015 held in Santiago in Chile and just after my
short talk concerned with the semi-relativistic Pauli-Fierz Hamiltonian, a staff of
Springer invited me to publish a book about ground state “not about functional
integration” which is my scientific hometown. I first intended to make my own
results as a book, but I thought my own results were insignificant extensions of
results established in [5, 9, 17, 21, 54] mentioned above. What then one really
needs is a single source that combines fundamental results on the existence of
ground states obtained in the end of the twentieth century with a guide to how to
usethemtoothermodelsreadermayconsider,becauseIcouldnotfindsatisfactory
discussions in literature. Then I decided to write up my own account of the exis-
tence of ground states.
However, any book is a compromise between deadlines and my striving for
perfections. Without deadlines no book would be published, this means a final
versionwhichisperfectexistsonlyinmymind.Duetotimeandspaceconstraints,
Ididnotincludefundamentalmathematicaltoolsbutthebookiswrittenaspossible
asself-consistently.Thepresentbookcomprisesmaterialwhichcanpartlybefound
Preface ix
inotherbooksandpartlyinresearcharticles,andwhichforthefirsttimeisexposed
from a unified point of view.
The book consists of six chapters. Chapter 1 is overview of models studied in
this book without rigor. Here, we introduce several strategies on how to show the
existenceofgroundstates.Chapter2isdevotedtoaminimalintroductionoftoolsI
used in this book. Compact operators, compact sets in Lp space, compact embed-
dingsofSobolevspaces,andbosonFockspacearereviewed.Chapter3isdevoted
to studying the Pauli-Fierz model and Chap. 4 the Nelson model with/without
cutoffs.Inbothchapters,weprovetheexistenceofgroundstatesbyapplicationsof
compactness. In Chap. 5, I study spin-boson model by path measures. Finally in
Chap. 6, I overview enhanced bindings for the Pauli-Fierz model and the Nelson
model.
Fukuoka, Japan Fumio Hiroshima
Acknowledgements Iwouldliketoveryheartilythankallthosewhohelpedmefortheproductof
book. I am grateful for helpful comments and discussions from Thomas Norman Dam, Takeru
Hidaka,MasaoHirokawa,JózsefLőrinczi,OliverMatte,JacobSchachMøller,ItaruSasaki,Akito
Suzuki,andHerbertSpohn.IamfinanciallysupportedbyJSPSKAKENHI 16H03942,CREST
JPMJCR14D6,andJSPSopenpartnershipjointresearchwithDenmark1007867.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Quantum Field Theory and Ground States . . . . . . . . . . . . . . . . . . 1
1.2 The Pauli–Fierz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Nelson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Spin-Boson Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Infrared Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Enhanced Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6.1 Enhanced Binding for the Pauli–Fierz Hamiltonian . . . . . . 11
1.6.2 Enhanced Binding for the Nelson Hamiltonian . . . . . . . . . 12
1.7 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Compact Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Compact Operators Basics . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.2 Spectrum of Compact Operators. . . . . . . . . . . . . . . . . . . . 17
2.1.3 Compact Operators on L2ðXÞ . . . . . . . . . . . . . . . . . . . . . . 18
2.1.4 Carleman Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Compact Embeddings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Strong Compactness in Lp Spaces. . . . . . . . . . . . . . . . . . . 21
2.2.2 Sobolev Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.3 Compact Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Boson Fock Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Abstract Boson Fock Spaces . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Field Operators and Second Quantizations. . . . . . . . . . . . . 31
2.3.3 Localizations in FðWÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Boson Fock Space over L2ðRdÞ. . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 Field Operators and Second Quantizations. . . . . . . . . . . . . 36
2.4.2 Compact Operators on FðL2ðRdÞÞ . . . . . . . . . . . . . . . . . . 38
2.4.3 Localizations in FðL2ðRdÞÞ. . . . . . . . . . . . . . . . . . . . . . . 39
xi
