Rigorous Quantum Field Theory: A Festschrift for Jacques Bros

322 Pages·2007·7.05 MB·English

by  Anne Boutet de Monvel, Detlef Buchholz, Daniel Iagolnitzer, Ugo Moschella (eds.)

Progress in Mathematics
Volume 251
Series Editors
H. Bass
J. Oesterlé
A. Weinstein
Rigorous
Quantum Field Theory
A Festschrift for Jacques Bros
Anne Boutet de Monvel
Detlev Buchholz
Daniel Iagolnitzer
Ugo Moschella
Editors
Birkhäuser Verlag
Basel (cid:120)(cid:3)Boston (cid:120)(cid:3)Berlin(cid:3)
Editors:
Anne Boutet de Monvel Daniel Iagolnitzer
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Detlef Buchholz Ugo Moschella
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Contents
1 Introduction
................................................................ 1
2 Local Counterterms on the Noncommutative Minkowski Space
Dorothea Bahns ................................................. 11
3 Massless Scalar Field in a Two-dimensional de Sitter Universe
Marco Bertola, Francesco Corbetta, Ugo Moschella ................... 27
4 Locally Covariant Quantum Field Theories
Romeo Brunetti.................................................. 39
5 Asymptotic Abelianness and Braided Tensor C∗-Categories
Detlev Buchholz, Sergio Doplicher, Giovanni Morchio, John E. Roberts
and Franco Strocchi .............................................. 49
6 Yang–Mills and Some Related Algebras
Alain Connes and Michel Dubois-Violette ........................... 65
7 Remarks on the Anti-de Sitter Space-Time
Henri Epstein ................................................... 79
8 Quantum Energy Inequalities and Stability Conditions in Quantum
Field Theory
Christopher J. Fewster............................................ 95
9 Action Ward Identity and the Stückelberg–Petermann
Renormalization Group
Michael Dütsch, Klaus Fredenhagen ................................113
10 On the Relativistic KMS Condition for the P(φ)2 Model
Christian Gérard, Christian D. Jäkel ...............................125
VI Contents
11 The Analyticity Program in Axiomatic Quantum Field Theory
Daniel Iagolnitzer ................................................141
12 Renormalization Theory Based on Flow Equations
Christoph Kopper ................................................161
13 Towards the Construction of Quantum Field Theories from a
Factorizing S-Matrix
Gandalf Lechner .................................................175
14 String-Localized Covariant Quantum Fields
Jens Mund .....................................................199
15 Quantum Anosov Systems
Heide Narnhofer .................................................213
16 DFR Perturbative Quantum Field Theory on Quantum Space
Time, and Wick Reduction
Gherardo Piacitelli ...............................................225
17 On Local Boundary CFT and Non-Local CFT on the Boundary
Karl-Henning Rehren.............................................239
18 Algebraic Holography in Asymptotically Simple, Asymptotically
AdS Space-times
Pedro Lauridsen Ribeiro ..........................................253
19 Non-Commutative Renormalization
Vincent Rivasseau and Fabien Vignes-Tourneret .....................271
20 New Constructions in Local Quantum Physics
Bert Schroer ....................................................283
21 Physical Fields in QED
Othmar Steinmann...............................................301
22 Complex Angular Momentum Analysis and Diagonalization of the
Bethe–Salpeter Structure in Axiomatic Quantum Field Theory
Giovanni Alberto Viano...........................................311
1
Introduction
AfterthegreatscientificrevolutionsofSpecialRelativityandofQuantum
Mechanics in the first half of the twentieth century, Relativistic Quantum
Field Theory (QFT) was introduced to provide a synthesis of these two new
paradigms(in Kuhn’s words).As such, QFT is one of the majoradvances for
theoreticalphysicsin the secondhalf ofthe lastcentury.The mainreasonfor
emphasizingthe importanceofQFTliesintheimpressiveeffortthatitrepre-
sentstowardsaunifiedunderstandingofthestructureofmatteratsubatomic
scales,asitemergesinthecollisionphenomenaofparticlephysicsintherange
of energies presently explored in particle accelerators. But another reason is
the extraordinary mathematical wealth of this theoretical framework, which
makes it a fascinating domain of research for mathematical physics and may
also stimulate the interest of mathematicians.
Foreseen by Dirac in 1933 and finaly established discovered by Feynman
around 1949 for the treatment of quantum electrodynamics, the so-called
“path-integral formalism” of quantum field theory is also now considered as
a powerful tool for providing perturbative methods of computation in many
problems of theoretical physics, namely in statistical mechanics and in string
theory. However, the use of relativistic quantum fields as a basic concept of
mathematical physics underlying all the phenomena of particle physics in a
very large range of energies represents a much more ambitious program.
This program was indeed stimulated by the success of the quantum elec-
trodynamics (QED) formalism for computing the electron-photon, electron-
electron and electron-positron scattering amplitudes. Even today, it is by no
means understood why the perturbative expansion of QED in powers of the
coupling parameter, that is the electric charge of the electron, provides such
a spectacular agreement with experimental data, although in practice it is
reducedtothe computationofthe veryfirsttermsofthe series(the onlyones
which are computable). This is even more surprising since we now know that
the series cannot be convergent, so that, paraphrasing Wigner’s words, one
may wonder about the “unreasonable effectiveness” of perturbation theory in
four-dimensional QED.
2
In the 1950s, however, the success of these computations was credited to
thesmallnessofthecouplingparameterofQED,andthissituationstimulated
research for other methods of investigation of QFT, which could apply to
quantum field models with large coupling: it was in fact the very concept of
a quantum field which appeared as the most powerful and promising one for
explaining the phenomena of strong nuclear interaction of particle physics.
During all that period the study of phenomena of weak nuclear interaction
also benefitted, like QED, from the success of a perturbative QFT formalism
reduced to very few terms.
The demand for a more general nonperturbative treatment of QFT mo-
tivated an important community of mathematical physicists to work out
a model-independent axiomatic approach to relativistic quantum field the-
ory[2,8,9,11,14].Theirfirsttaskwastoprovideamathematicallymeaningful
concept of a relativistic quantum field in terms of “operator-valued distribu-
tion in the Hilbert space of quantum physical states”, in such a way that the
notionofingoingandoutgoingparticlestatescouldbe introducedintermsof
appropriate asymptotic forms of the field operators.
The precise mathematical formulation of fundamental physical principles
of relativistic quantum theory in terms of field operators considered as ba-
sic entities of the system allowed the possibility to offer a general model-
independent framework, in which one could propose a theoretical treatment
of high-energy collision phenomena.
These phenomena involve not only the usual processes of “elastic scatter-
ing” preserving the number and nature of the particles, but also all possible
processes of creation and annihilation of particles permitted by the kinemat-
ical laws of special relativity and the conservation laws of basic quantum
numbers called “charges”. In this mathematical framework, all the collision
processes are encoded in a certain general unitary scattering operator S in
the Hilbert space of states, according to the concept originally introduced by
Heisenberg, and all the corresponding scattering kernels S of (m → n)-
m,n
particle collision processes are related to general N-point structure functions
of the quantum fields (with N =m+n) via the so-calledLSZ “reduction for-
mulae”.Thisformalismwasfully justifiedatleastforprocessesinvolvingonly
massive particles; the inclusion of massless particles, such as the photons of
QED, soon revealed the existence of hard mathematical problems of various
types.
This general approach of QFT was followed by a conceptually important
variant, in which the quantum fields are supposed to generate “algebras of
localobservables” attachedtoarbitraryregionsofMinkowskispacetime.This
is the algebraic approach to quantum field theory (Haag,Kastler1964),whose
most importantdevelopments between1960and 1990havebeen presentedin
a book by Haag (1992) under the name of “local quantum physics” [6].
From both mathematical and physical viewpoints, the visionary works of
VonNeumann, whichwerecontemporaryto the birthofquantummechanics,
are at the origin of all these developments; however, the algebraic structures
1 Introduction 3
involved in relativistic QFT refer to the most advanced developments of the
theory of Von Neumann algebras. Just as an example, it is worthwhile to re-
callthat, in contrastwith the caseof nonrelativistic quantummechanics,one
isnecessarilyconfrontedbyfactorsoftypeIII inthestandardclassificationof
VonNeumannalgebras.Thisoccursalreadyinthesimplestmodelsofrelativis-
ticQFT,namelythefree-fields.Moreover,importantconceptualprogresswas
madebythealgebraicQFTviewpointwithrespecttotheprimitivequantum-
mechanical notion of the Hilbert-space of states of a given physical system.
It consists in introducing algebras of local observables of the system as ab-
stract C∗−algebras (and evenmore general ∗-algebras)and then representing
all possible physical states of the system as linear functionals on such al-
gebras. In this more general viewpoint, various types of physically relevant
Hilbertian representationsof the algebrasmay be introducedmathematically
via GNS-type (Gelfand-Naimark-Segal) [6] constructions. As a typical illus-
tration of this viewpoint, one can mention the Borchers-Uhlmann algebraic
presentation of the original Wightman axiomatic QFT [3].
In these various general approachesto relativistic QFT aiming to provide
a theoretical understanding of particle physics, a central role is played by a
certain field-theoretical formulation of Einstein causality together with the
relativisticprincipleofpositivityofenergy(alsocalledspectral condition)and
an appropriateexpressionof the covarianceof the system under the Poincaré
group.
The interplay of these postulates has generated rigorous proofs of impor-
tantphysicalproperties,thanks tothe discoveryofveryrichanalyticity prop-
ertiesoftheN-pointstructurefunctionsofthefields(elaboratedinsuccessive
works by Ruelle, Steinmann, Araki, Bros, Epstein, Glaser, Stora). As a mat-
ter of fact, all these functions enjoy two associated analytic structures, which
coexistinthecomplexifiedspacetimevariablesz andintheFourier-conjugate
complexified energy-momentum variables k; this double analytic structure is
equipped with apeculiar type ofalgebraicrelationsbetween boundaryvalues
from tube domains and relevant (multiple) discontinuities, including various
interconnections by Fourier-Laplace transformations in several variables. In
bothz-spaceandk-space,nontrivialproblemsofholomorphyenvelopesofthe
so-called “edge-of-the-wedge” type which emerge from this double analytic
structure have only been very partially solved. As typical illustrations of the
latter,onecansaythatontheonehandtheexploitationofthez-spaceanalytic
structure hasprovidedproofs ingeneralQFT ofsuchimportantpropertiesof
particlephysicsasthe PCT-symmetryandthe spin-statisticsconnection.On
theotherhand,theexploitationofthe k-spaceanalyticstructurehasbrought
a general justification in QFT of important analytic continuation properties
of the scattering kernels on the so-called complex mass shell manifold of var-
ious collision processes. As a typical example, let us mention the crossing
property between the “particle-particle” and “particle-antiparticle” scattering
processes,whichcanbegeometricallyexpressedastheexistenceofadomainof
analyticityonthecomplexmassshellmanifoldrelatingtwodifferent“physical
4
regions” on the real mass shell submanifold (Bros, Epstein and Glaser). Such
successful results, which were mostly obtained in the 1960s, correspond to
the “golden age” of the so-called dispersion relations in particle physics. It
is rather remarkable that such model-independent Cauchy-type integral rela-
tionswerefoundtobeinsatisfactoryagreementwiththeexperimentaldataof
two-particlecollisionprocessesandthatinthisanalyticframework,important
“high-energy bounds” could also be derived from general principles in basic
works by Froissart and Martin [10].
Ofcourse,as a naturalcomplementto this conceptualprogress,there was
(and there is) the need to produce nontrivial models of relativistic quantum
fields:forthatmatter,abasiccriterionofnontrivialityistherequirementthat
the associated scattering operator S be different from the identity operator.
This accounts for the occurrence of another school of mathematical physi-
cists, called “Constructive quantum field theory” and founded essentially by
GlimmandJaffeinthe late 1960s[5],whosepurposewasto producerigorous
existence proofs together with properties of the corresponding solutions for
modelsofrelativisticQFTassociatedwithLagrangiansofsuitabletype.Inge-
niousmethodsof“clusterexpansions” wereinventedwhichallowedonetotreat
successfully various classes of nontrivialmodels in two and three-dimensional
Minkowski spacetime, thus displaying that (at least for these dimensions)
the axiomatic framework was not a pure abstraction, but it contained a lot
of mathematically well-defined Lagrangian theories. It was even shown that,
for such models, the traditional perturbation expansion, defined as a formal
power series in terms of a small coupling parameter, had Borel-summability
properties which allowed one to reach the exact solution of the model [5,7].
However, it became patent that all the models of interacting relativistic
quantum fields in the physical four-dimensional spacetime are afflicted by
very hard mathematical problems which have to do with unavoidable short-
distance as well as long-distance divergent behaviours of the structure func-
tions of the fields. These difficulties are closely relatedto the renormalization
problemsinthestandardperturbativeapproachofQFT.Inthe1980s,inspite
of the fact that no completely rigorous proof of it could be produced, there
appeareda wide consensus among theoreticalphysicists that the QFT model
with quartic interaction, that is the “simplest” Lagrangian model of scalar
fields to be considered,probably did not exist in four-dimensionalspacetime,
but only reduced to a pure formal series in the sense of the perturbative
approach.
The same consensualopinion also applies to QED, which therefore makes
it still more thrilling to have in mind the physical success of its perturbation
series.TheadditionalintriguingaspectofQEDisthefactthatitincorporates
the fundamental physical notion of electric charge. As a matter of fact, the
horizonsofQFTwereconsiderablyrenewedinthe1970sbythegeneralization
ofthenotionofcharge inparticlephysicsandtheassociatedconceptofgauge
symmetry groups for the quantum field description of charged states. From