Singularities of integrals: homology, hyperfunctions and microlocal analysis

218 Pages·2011·3.313 MB·English

by  Pham, Frâedâeric

Singularities of integrals
Universitext
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SheldonAxler
SanFranciscoStateUniversity
VincenzoCapasso
UniversitàdegliStudidiMilano
CarlesCasacuberta
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AngusJ.MacIntyre
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KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
CNRS,ÉcolePolytechnique
EndreSüli
UniversityofOxford
WojborA.Woyczynski
CaseWesternReserveUniversity
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Frédéric Pham
Singularities
of integrals
Homology, hyperfunctions
and microlocal analysis
FrédéricPham
LaboratoireJ.-A.Dieudonné
UniversitédeNiceSophiaAntipolis
ParcValrose
06108Nicecedex02
France
EDPSciencesISBN:978-2-7598-0363-7
TranslationfromtheFrenchlanguageedition:
‘Intégralessingulières’byFrédéricPham
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DOI10.1007/978-0-85729-603-0
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Contents
Foreword ...................................................... IX
Part I Introduction to a topological study of Landau singularities
Introduction................................................... 3
I Differentiable manifolds ................................. 7
1 Definition of a topological manifold ...................... 7
2 Structures on a manifold ............................... 7
3 Submanifolds ......................................... 10
4 The tangent space of a differentiable manifold............. 12
5 Differential forms on a manifold ......................... 17
6 Partitions of unity on a C∞ manifold .................... 20
7 Orientation of manifolds. Integration on manifolds......... 22
8 Appendix on complex analytic sets ...................... 26
II Homology and cohomology of manifolds ................. 29
1 Chains on a manifold (following de Rham). Stokes’ formula . 29
2 Homology ............................................ 31
3 Cohomology .......................................... 36
4 De Rham duality ...................................... 39
5 Families of supports. Poincar´e’s isomorphism and duality ... 41
6 Currents ............................................. 45
7 Intersection indices .................................... 49
III Leray’s theory of residues................................ 55
1 Division and derivatives of differential forms .............. 55
2 The residue theorem in the case of a simple pole........... 57
3 The residue theorem in the case of a multiple pole ......... 61
4 Composed residues .................................... 63
5 Generalization to relative homology...................... 64
VI Contents
IV Thom’s isotopy theorem ................................. 67
1 Ambient isotopy....................................... 67
2 Fiber bundles ......................................... 70
3 Stratified sets ......................................... 73
4 Thom’s isotopy theorem................................ 77
5 Landau varieties....................................... 80
V Ramification around Landau varieties.................... 85
1 Overview of the problem ............................... 85
2 Simple pinching. Picard–Lefschetz formulae............... 89
3 Study of certain singular points of Landau varieties ........ 98
VI Analyticity of an integral depending on a parameter.....109
1 Holomorphy of an integral depending on a parameter ......109
2 The singular part of an integral which depends on a
parameter ............................................114
VII Ramification of an integral whose integrand is itself
ramified .................................................127
1 Generalities on covering spaces ..........................127
2 Generalized Picard–Lefschetz formulae ...................130
3 Appendix on relative homology and families of supports ....133
Technical notes ................................................137
Sources........................................................141
References.....................................................143
Part II Introduction to the study of singular integrals and
hyperfunctions
Introduction...................................................147
VIII Functions of a complex variable in the Nilsson class .....149
1 Functions in the Nilsson class ...........................149
2 Differential equations with regular singular points .........154
IX Functions in the Nilsson class on a complex analytic
manifold.................................................157
1 Definition of functions in the Nilsson class ................157
2 A local study of functions in the Nilsson class .............159
Contents VII
X Analyticity of integrals depending on parameters ........163
1 Single-valued integrals .................................163
2 Multivalued integrals ..................................164
3 An example...........................................167
XI Sketch of a proof of Nilsson’s theorem ...................171
XII Examples: how to analyze integrals with singular
integrands ...............................................175
1 First example .........................................175
2 Second example .......................................183
XIII Hyperfunctions in one variable, hyperfunctions in the
Nilsson class.............................................185
1 Definition of hyperfunctions in one variable ...............185
2 Differentiation of a hyperfunction........................186
3 The local nature of the notion of a hyperfunction..........187
4 The integral of a hyperfunction..........................188
5 Hyperfunctions whose support is reduced to a point........189
6 Hyperfunctions in the Nilsson class ......................189
XIV Introduction to Sato’s microlocal analysis................191
1 Functions analytic at a point x and in a direction..........191
2 Functions analytic in a field of directions on Rn ...........191
3 Boundary values of a function which is analytic in a field
of directions ..........................................193
4 The microsingular support of a hyperfunction .............196
5 The microsingular support of an integral .................197
A Construction of the homology sheaf of X over T .........201
B Homology groups with local coefficients..................205
Supplementary references, by Claude Sabbah..................207
Index..........................................................215
Foreword
Many times throughout the course of their history, theoretical physics and
mathematics have been brought together by grand structural ideas which
have proved to be a fertile source of inspiration for both subjects.
The importance of the structure of holomorphic functions in several vari-
ables became apparent around 1960, with the mathematical formulation of
thequantumtheoryoffieldsandparticles.Indeed,theircomplexsingularities,
known as “Landau” singularities, form a whole universe, and their physical
interpretation is part of a particle physicist’s basic conceptual toolkit. Thus,
the presence of a pole in an energy or mass variable indicates the existence
of a particle, and singularities of higher complexity are a manifestation of a
ubiquitous geometry which underlies classical relativistic multiple collisions
and also includes the creation of particles, which lies at the heart of quantum
interaction processes.
On the pure mathematics side, one can safely say that this branch of
theoretical physics genuinely contributed to the birth of the theory of hyper-
functions and microlocal analysis. (For example, one can mention the initial
motivation in the work of M.Sato at the time of the “dispersion relations”,
which came from physics, and also the work of A.Martineau, B.Malgrange,
and M.Zerner on the “edge-of-the-wedge” theorem, which came out of the
first meetings in Strasbourg between physicists and mathematicians).
For a mathematical physicist, these holomorphic structures in quantum
field theory have a deep meaning, and are inherent in the grand principles of
relativistic quantum field theory: Einstein causality, the invariance under the
Poincar´e group, the positivity of energy, the conservation of probability or
“unitarity”, and so on. But it is in the “perturbative” approach to quantum
field theory (whose relationship with “complete” or “non-perturbative” the-
ory can be compared to the relationship between the study of formal power
series and convergent series), that the holomorphic structures which gener-
ate Landau singularities appear in their most elementary form: namely, as
holomorphic functions defined by integrals of rational functions associated to
“Feynman diagrams”.
X Foreword
It is to Fr´ed´eric Pham’s great credit that he undertook a systematic anal-
ysis of these mathematical structures, whilst a young physicist at the Service
de Physique Th´eorique at Saclay, using the calculus of residues in several
variables as developed by J.Leray, together with R.Thom’s isotopy theo-
rem. This fundamental study of the singularities of integrals lies at the inter-
face between analysis and algebraic geometry, and culminated in the Picard–
Lefschetzformulae.Itwasfirstpublishedin1967intheM´emorialdesSciences
Math´ematiques (edited by Gauthier-Villars), and was subsequently followed
by a second piece of work in 1974 (the content of a course given at Hanoi),
where the same structures, enriched by the work of Nilsson, were approached
using methods from the theory of differential equations and were generalized
from the point of view of hyperfunction theory and microlocal analysis.
It seemed to us that, because of the importance of their content and the
wide range of different approaches that are adopted, a new edition of these
textscouldplayanextremelyimportantrolenotonlyformathematicians,but
also for theoretical physicists, given that the major fundamental problems in
the quantum theory of fields and particles still remain unsolved to this day.
First of all, we note that the methods developed by Fr´ed´eric Pham have
had wide-ranging impact in the non-perturbative approach to this subject,
by allowing one to study holomorphic solutions to integral equations in com-
plex varieties with varying cycles. Such equations are inherent in the general
formalism of quantum field theory (they are known as equations of “Bethe–
Salpeter” type, and are intimately connected to general unitarity relations).
Inthisway,itispossible,forexample,inthecaseofcollisionsbetweenmassive
particles which are described in a general manner by the structure functions
of a quantum field theory, to disentangle the singularities of a “three-particle
threshold”, which appears as an accumulation point of “holonomic singulari-
ties”(herewerefertoaclassificationofthesingularitiesofcollisionamplitudes
considered by M.Sato).
Someextremelytoughproblemsremainopenconcerningthistypeofstruc-
ture in the case when massless particles are involved in collisions, which is of
considerable physical importance. The type of analysis that we have just de-
scribedabove(thecaseofcollisionsinvolvingonlymassiveparticles)concerns
properties of field theory which are independent of “problems of renormaliza-
tion”, which are a different category of crucially important problems. These
appear on the perturbative level as the need to redefine primitively-divergent
Feynman integrals, and have recently been cast in a new light by the work of
Connes and Kreimer, who brought a Hopf algebra structure into the picture.
Nonetheless, on a general, non-perturbative level, the problems of renormal-
ization still appear to be at the very source of the problem of the existence of
non-trivial field theories in four-dimensional space-time.
This “existential problem” was brought to light by Landau in 1960 in the
resummationofrenormalizedperturbationseriesinquantumelectrodynamics
and is exhibited, at a more general level, by the simplest scalar field theory
with quartic interaction term. This phenomenon involves the “generic” cre-