Hyperfunctions and Theoretical Physics: Rencontre de Nice, 21–30 Mai 1973
222
Lecture Notes in Mathematics Edited by ~ Dold and B. Eckmann Series: Universite de Nice Adviser: J. Dieudonne 449 Hyperfunctions and Theoretical Physics Rencontre de Nice, 21-30 Mai 1973 Edited by F. Pham Springer-Verlag Berlin-Heidelberg • New York 19 7 5
Transcript of Hyperfunctions and Theoretical Physics: Rencontre de Nice, 21–30 Mai 1973
Lecture Notes in Mathematics Edited by ~ Dold and B. Eckmann
Series: Universite de Nice Adviser: J. Dieudonne
449
Hyperfunctions and Theoretical Physics Rencontre de Nice, 21-30 Mai 1973
Edited by F. Pham
Springer-Verlag Berlin-Heidelberg • New York 19 7 5
Prof. Fr6deric Pham Institut de Math@matiques et Sciences Physiques Universite de Nice Parc Valrose F-06034 Nice-Cedex
L i b r a r y of Cong re s s Ca ta log ing in Publ ica t ion Da*a Main entry under title:
~yperfunctions and theoretical physics.
(Lecture notes in mathematics ; 449) English or French. Bibliography: p. Includes index. l . Mathematical physics--Congresses. 2. Hyperf~mc-
tions--Congresses. 3- Quantum field theory--Congresses. I. Pham~ Fr@d6ric. II. Series: Lecture notes in mathematics (Berlin) ; 449. QA3.L28 no. 449 [QC19.2] 510'.8s [530.1'5] 75-9931
AMS Subject Classifications (t970): 32D10, 46F15, 81A17, 81A48
ISBN 3-540-07151-2 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-07151-2 Springer-Verlag New York • Heidelberg • Berlin
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A la mEmoire de A. MARTINEAU
Le a6jour & NICE, pendant l'annEe 1972-1973, de M. SATO et Oe sea El#yes
T. KAWAI et M. KASHIWARA donna l'occasion & des mathEmaticiens nigoie d'organiser~
en liaison avec des physiciens th@oriciens, un colloque sur la th@orie des hyper-
functions st 8ur quelques probl~mes connexes rencontres en Physique msth@matique,
Au cours de ce colloque~ qui a r#uni une quaranteine de participants du 21 au
30 Mai 1973, on a pu entendre d'une part des exposes plus ou moin5 init~tiques
(lea mathEmaticiens e'adreseant aux physiciens) sur la thEorie Oes hyperfonctions,
d'autre part des exposes sur divers probl~mes de Th~orie quantique Relativiste en
rapport avec la th@orie des hyperfonctions : probl~mes de thEorie des champs d'une
part, thEorie de la matrice S d'autre part (rappelons, pour lea situer rapidement,
qua ces deux theories s'intErassent aux memes phEnom@nes, & savair lea interactions
des partieules lourdes & haute Energie ~ mais que la premiere essaye de b~tir une
axiomatique autour Cu concept de " champ ", inspir6 de l'ElectroOynamique quantique,
tandis que la seconde reste plus " en surface des ph~nom~nes " avec l'entitE Oirec-
tement mesurable qu'est " la matrice S " ] .
Les textes groupEs Cans ce volume sont ~es r~daetions d'expos@s faits au Col-
loque. Malheureusement, noue n'avone pas rEussi & obtenir de r#daction des exposes
~e M. SATO, ni de la suite d'exposEs de M. KASHIWARA intitul6e
" ReBularity of h~perfunctions ~ applications : dust points~ edge of the weq~e;
relation between Support and singular support " ,
ni de la suite d'expos@s de d. BROS intitulEe
" Repr~sen~atign de Fourier locale des hyperfonctions et miorofonctions ;
thEor&mes du t~pe edge of the wedge "
(dont une pattie toutefois eat rEsumEe par l'Appendice ~e l'artic&e Ce O. IAGOLNIT-
ZEB) .
TABLE DES MATIERES
PART I : HYPERFUNCTIONS
Introduction aux hyperfonctions 1
A. CEREZO, J. CHAZARAIN, A. PIRIOU
Pseudo-differential operators acting on the sheaf of microfunctions 54
T. KAWAI
Micro-h~perbolic pseudo-differential qperators 70
M. KASHIWARA, T. KAWAI
PART II : S-MATRIX
M icroanalyticit@ de la m atrice S 83
F. PHAM
Macrocausality~ Physic al-re$ion analyticity and independence
property in S-matrix theory D. IAGOLNITZER
Appendi x : Microlocal essential support of a distribution and
decomposition theorems - An introduction
D. IAGOLNITZER
Unitarity and discontinuity formulae
D. OLIVE
102
121
133
PART III : THEORY OF FIELDS
Geometry of the n-point p-space function of quantum field theory
H. EPSTEIN, V. GLASER, R. STORA
Some applications o~ ~he Jost-Lehmann-Dyson theorem to the study
of the global analytic structure of the n-point function of
quantum field theory
R. STORA
Quelques aspects globaux des probl~mes d'ed~e of the wedge
J. BROS, H. EPSTEIN, V. GLASER, R. STORA
143
163
185
I N T R O D U C T I O N AUX H Y P E R F 0 N C T I 0 N I.S I
A , CEREZO, d , CHAZARAIN, A. PIRIOU .....
D@partement de Math@matiques, NICE
CHAPITRE 0 INTRODUCTION
Depuis longtemps, on a fait lea deux remarques suivantes :
I] I1 est utile d'introduire et d'@tudier des " ~onctions g6n@ralis~es "
2] Celles-ci ont souvent int@r@t & @tre consid@r6es camme " valeurs au
bord " de fonctions tr~s " r@guli6res " [holomorphes, harmonlques .... ].
Par exemple, sur IR une distribution T 1 1
~ > au bord de la fonction ~[z] = 2±~( < Tx, x-z
~ - s u p p T , a u s e n s d e s d i s t r ~ b u t $ o n s :
[~R [~[x+ie] - *[x-ie)] ~[x] dx +>
support compact est valeur
, holomorphe dens
0
C'est SATO qui le premier a @tudi~ des fonctions g@n@ralis@es d6finies
a prior~, comme " valeurs au herd " de fon ctions holomorphes, obtenant ainsi la
clasae la plus large de fonctions g~n6ralis@es loealisables, les hyperfonotions.
A une variable, les choaes sent trbs simples, car la notion de valeur au
bord est imm@diate : une hyperfonctlon sur ~ est d@~inie par une fonetion holo-
morphe dana ~ -IR , et deux telles fonctions d6finisaent la m~me hyperfonction
si at seulement si leur diff@rence est holomorphe dens tout ~ [intuitivement,
elles ont m@me saut sur IR ] .
A pluaieurs variables, la notion de valeur au herd est plus d61ieate, car
il faut ma~trlser des ph6nom~nes nouveaux de prolongement analytique [par exemple
toute fonctlon holomorphe dens [2 _ ~2 l'est en fait dens ~2 ].
SATO a ~t@ amen6 naturellement & rechercher la bonne d6finition de veleur-
au-bord-de-fonetion-holomorphe (par exemple sur tR n ), dens la cohomologle rela-
tive de IR n dens n & coefficients dana (~ , o'est-~-dlre dens la colonne de
gauche de la suite exacte de cohomologie relative :
(~ H 1 (~:n (..q,) IR n
Nn
0 > C~ (~n) ------> (~ (cn_ ~n)
- - > HI [~ n, (.lr) ------,> HI(~: n- IRn,~)
. , , >
. . . . . . . ------> Hn - l ( $ n_ IR n, ~ )
C ;H n (~n, ~ ------> H n(E n, ~0 ------> H n(~n_ IR n, £~] ~ > LR n
En felt, il a montr@ que tousles termes de la premiere colonne sent nuls
saul le dernier, d'oQ la d~finition des hyperfonctions sur ~n ,
~ ( ~ n ) H n (En ~ ) IR n
e t plus g~n~ralement, s i ~ est un ouver t de ~n et U un euver t de ~n dens
l eque l £ est contenu e t ferm6 :
Bien entendu, ces fonctions g~n~ralis~es n'ont pas en g~n~ral de " valeur "
en un point au sens usuel : comme pour les distributions, on ne sait d6flnir que
leur restriction & un ouvert. Mais les hyperfonctions sent localisables, en ce sens
qu'elles sent d~termin~es par leurs restrictions aux ouverts d'un recouvrement.
Aussi le langage des ~aisceeux s'impose.
Le ~alsceau ~ des hyperfonctlons contient celul des distributions et
a l'avantage d'etre flasque : toute hyperfonction sur un ouvert peut se prolonger
& un ouvert plus grand.
En calculant ces groupes de cohomologie relative & l'aide de la cohamolo-
gie de Cech, on fait appara~tre naturellement toute hyperfonction comme somme de
" valeurs au bord " de fonctions holomorphes dens des tubes du type IR n + ir ,
oQ r est un cBne convexe ouvert propre de IR n . Cette representation permet de
d~composer la singularit~ d'une hyperfonetion suivant les directions cotangentes
~n , et SATe constrult ainsi le faisceau ~ des microfonctlons, sur le fibr~
en spheres cotangentes. Le support de la microfonction associ~e ~ une hyperfonction
n'est autre que son spectre singulier (wave front).
Toutes ces constructions sont naturelles, et beaucoup d'op6retions d6-
finies sur les fonct±ons holomorphes se prolongent donc naturellement aux hyper-
fonctions et aux microfonctions.
SOMMAIRE .
CHAPITRE I HYPERFONCTIONS A I VARIABLE (A. CEREZ0)
Il concerne le cas de la dimension 1 , sur lequel on ne s'~tend que
pour introduire des notions utiles aux autres chapitres. II contient aussi ce
qu'il faut de th~orie des faisceaux pour en utiliser le langage.
CHAPITRE II . HYPERFONCTIONS A UN NOMBRE @UELCONOUE DE VARIABLES
[A. PIRIOU)
On donne la d@finition des hyperfonctions au moyen des groupes de cohomo-
logie relative, et on montre que route hyperH%nction est une somme finie de va-
leurs eu bord de fonctions holomorphes dens des tubes locaux.
CHAPITRE III FAISCEAU ~ [J. CHAZARAIN)
On expose la notion de spectre slngulier d'une hyperfonction et on d~fi-
nit le faisceau ~ des singularit~s.
CHAPITRE IV APPLICATIONS [J. CHAZARAIN)
On donne un @nono@ hyperfonction du th@or~me " edge of the wedge " puis
on d~crit les operations que l'on peut d~finir dens certeins cas sur les hyper-
fonctions,
R~f6rences . Le propos de ces expos6s est purement " p6dagogique ", aussi on n'y
trouvera rien de nouveau au s ujet des hyperfonctions. On a utilis~ de nombreuses
sources, ~ commencer par les publications et les exposes de SATO et de ses deux
~l~ves KASHIWARA et KAWAI. On s'est 6galement inspir6 des presentations de la
th~orie per KOMATSU ainsi que par MORIMOTO. Enfin, pour le lien entre distribu-
tions et hyperfonctions, on renvoie aux travaux de MARTINEAU et au livre de SHA-
PIRA,
On t r o u v e r a ces r ~ f ~ r e n c e s darts :
M. SATO
M. SATO
A. MARTINEAU ,
P, SHRPI RA
Theory of hyperfunctions I and II - 5~ ~ac. of Sc£encsS Uniw Tokyo
(1959) & (1980)
R e g u l a r i t y o$ h y p e r f u n c t i o n s o l u t i o n s o f p a r t i a l d i f f e r e n t i a l equa-
t i o n s - P r o c . NICE Congress , 2 , G a u t h i e r s - V i l l a r s , P a r i s [1970)
Les articies de SATO-KAWAI-KASHIWARA, H. KOMATSU, M. MORIMOTO
dans Hyperfunctlons and pseudo-differential equations , Proc. Conf,
Katata, Lecture Note n ° 287, Springer (1973)
Distributions et valeurs au bord des $onctions holomorphes -
Proc. Inter. Summer Inst. Lisbon [1964)
Th~orie des hyperfonctions - Lecture Note N ° 126, Springer (1970)
CHAPITRE I - LE CAS D'UNE VARIABLE
A, CEREZO
I - DEFINITION
La suite exacte de cohomologie relative se r@duit dens le cas d'une seule va-
riab le ~ :
0 .... > ~[~) ---> ~[[4R) .... > ~C~,[]
st on d6finit done les hyper~onctions sur R par
[~-~] 1
b> 0
D st plus g~n6ralement, si I est un ouvert de IR , st
D un voisinage complexe de I [c'est-~-dire un ouvert I . . . .
de [ dens lequel I est contenu et ~erm~],
O'[o-l) [ = H~[O, ~[I) 0 (0)
O]] i
Une hyper~onction sur I est donc la donn~e d'une fonction holomorphe dens
D-I , o~ D est un certain voisinage complexe de I , modulo les ~onctions qui
se prolongent en qonctions holomorphes dens tout D , Intuitivement, une hyper-
fonction sur I , c'est le saut sur I d'une fonction hoiomorphe aupr~s de I
Le th~or~me d'excislon a~irme que ~(I) est ind@pendant du voislnage complexe
D choisi : si 01 et D 2 sont deux vo±sinages complexes de I , on a un iso-
morphisme eanonique
(~(D1-1) ~ (~(02-I] >
OlD I] (~[D 2]
isomorphisme qui, lorsque D I ~ D 2 • est induit par la restriction naturelle
(~ [DI-I] ~ > (~[D2-1)
Remarque - On d@finirait de fagon analogue les hyperfonctions sur une vari@t@
anelytique r@elle de dimension 1
Il est facile de restreindre une hyperfonction ~ un ouvert plus petit
I' C I , par restriction d'une fonction repr@sentative : si I'C I et
0'- I' C D - I , la restriction ~(O-I) --> O CO'- I'] induit une appli-
cation ~[I] ~ ~ [I'] , appei@e restriction let qu'on note comme teile].
On v@rifie imm@diatement que ces restrictions sont transitives : si
I" C I' C I et f ~ Fj~ (I]
fli,Ii,, = fli,,
On a ainsi d6fini un pr@faisceau ~ sur IR , qui se trouve ~tre un faisceau.
II - QUEL@UES NOTIONS UTILES SUR LES FAISCEAUX
La donn@e, pour tout ouvert U d'un espace topologique X , d'un espace vecto-
riel (sur ~ ) F~ [U) appei~ espace des sections au-dessus de U , et, pour
tout couple d'ouverts [U,U') tels que U' C U , d'une application lin@aire ap-
pel6e restriction de ~[U] dens ~:[U'] , de telie sorte que
la restriction de U dens U soit l'identit@
si U"C U' C U le diagramme ~ (U] - - > ~'~[U"]
est com~utatif (id est ,' Vf e ~CUl flU,, = flU, IU,, )
constitue ce qu'on appelle un pr@falsceau (d'espaces vectoriels sur [ )
~: sur la base X
Un tel pr6faisceau est dit ~tre un faisceau si " on peut recoiler les sections
locales de mani@re unique ", c'est-~-dlre si :
Pour tout recouvrement d'un ouvert U par une famille d'ouverts (U ]
et pour toute donn@e de sections # ~ ~[U ) " compatibles ~ [c'est-~-dire
= telies que f IU ~ UB fBI U~ ~ UB pour tous ~,B dens A ) , il existe une
et une seule section f E ~'[U] telle que flU = # pour tout
Exemples (sur Rn ) _ Le faisceau des fonctions continues, celui des fonc-
tions ind@finiment derivables, celul des distributions
le faisceau des fonctions queleonques [& valeurs complexes)
le faisceau (3~des fonctions enalytiques, le faisceau (~ (sur ~n )
des fonctions holomorphes
le pr@faisceau des fon ctions bornees n'est pas un faisceau.
THEOREME - 1) Le pr~faisceau ~ des hyperfonctions sur ~ est un feisceau
2~ Ce {aiscceau est flasque (id est : toutes les restrictions sent
surjectives).
On admettra Icl le premier point qui equivaut essentiellement au lemm~ de Cousin.
Le second est facile : toute hyperfonctlon f sur I' peut se repr@senter par une
fonction ~ holomorphe dens [O - (I-I')] - I' , puisque le crochet
] est un voisinage complexe de I' ; ~ est holo-
iI mcrphe dens D-I et d#finit donc une hyperfonction
sur I dont ie restriction & I' est @videmment f .
femilles ( f )
lent :
Remarque - De tousles examples cites ci-dessus, seul le faisceau des fonctions
quelconques est flasque. On sait en partl culier qu'une distribution sur un ouvert
ne se prolonge pasen g@neral & un ouvert plus grand.
Ouand on e un prefaisceau qui n'est pas un faisceau, on sait construire un fais-
ceau associ6 qui lui est attache de mani~re naturelle : au-dessus de cheque ouvert,
on prend comma sections " toutes les sections locales qui se recollent " : si
(U) est un recouvrement de U , on consid#re l'espace vectorial ~ ((U J) des
de s e c t i o n s eu-dessus des cove r t s du recouvrement , qui se r e c o l -
$ ~ "}"(U) et f a l U N u B fBlu~ N u~
On c o n s t r u i t ~ ( [ U J] pour t o u s l e s recouvrements ( U ) de U , e t dens la
reunion des ~J~((U )) , on identi~le deux familles qui coincident localement
(c'est-&-dlre par restriction aux ouverts d'un recouvrement encore plus fin], Le
resultet c'est
cO la limite inductive est prise sur tousles recouvrements de U
est alors l'espace des sections au-dessus de U d'un prefaisceau
Si r~ ~ t a i t un p r# fa i s ceau separ~ [ c ' e s t - ~ - d i r e s i t ou te s e c t i o n loca lement n u l l e
de ~ est nulleJ, il se trouve que ~ est un falsceau, et c'est lui
qu'on appelle le feisceau associ6 b ~ , (Sinon, il faut recommencer, et con-
sid@rer ~ , mais peu importe ici, c~r tousles pr@falsceaux qu'on consi-
dErera dens la suite seront s6par6s).
On peut r6sumer carte construction en disant qu'intuitivement, les sections
du faisceau assoc±6 sent les gens qui sent locelement des sections du pr@faisceau
{au moins sice dernier est s6par6j.
Example - Le faisceau associ6 au pr@faisceau des fonctions born@es {sur ~n)
est le faisceau des fonctions localement born6es {e'est-b-dire born~eS sur tout
compact).
--> ~ , oO ~ et ~ sent deux Un morphisme ~s pr~faisceaux pr@faisceauX sur la m@me bose X , e'est la donn@e pour tout ouvert U de X
d'une application lln@aire de r~ (U) dens q{U) de telle sorte que ces ap-
plications commutent eux restrictions : si U'~ U le diagramme
r~ Cu) > ~ (u)
L~ notion de faisceau associ@
I)
2)
est commutatif,
un pr6faisceau est naturelle en ce sans qua :
I1 existe un morphlsme canonique ~ --> ~ , qul n'est autre qua
l'identit@ si ~ est un faiseeau.
Un morphisme de pr6faisceaux ~ --> ~ se prolonge en un morphisme des
falsceaux asseci@s ~ --~ ~ et le diagramme
I est commutatif
En particulier, si ~
en un morphisme
est un fa isceau, tou t morphisme --> ~ se prolonge
Remarque - On peut parler du f~isceau asSoci6 ~ un pr@faisceau dent on ne s'est
donn@ les sections qu'au-dessus de certains ouverts, ferment une base d'ouverts
de X : on pourra en effet trouver un reeouvrement plus ~in
qu'un recouvrement donn@ de X et qui soit form~ d'ouverts de la base,
et il sufflt donc de consid6rer los %emilles de sections au-dessus de ces ouverts
lb qui se recollent.
Le faisceou en question est, si 1'on veut, le fois ceau associ~ au pr@faisceeu
obtenu en choisissant {0} comma espece de sections eu-dessus des outres ouverts.
Cat ebus de langage est bien pratique, et sere commie syst6metiquement dons
la suite.
III- PREMIERS CALCULS SUR LES HYPERFONCTIONS
Puisque ~J2J[l] ~[O~l) ~(D) , toute hyperfonction f[x] sur I se repr6sente
par une fonction ~ (z) holomorphe dens O-I , ce qu'on ~crira :
On sait multiplier une hyperfonction, pgr une fonction an alytlque : si
e[x) £ (%(I) , ella se prolonge en fonction holomorphe dens un voisinoge complexe
D' de I , O n O' est encore un volsinege complexa de I , et on pose
a(x) f [x ) [aCz) ~ (Z ) ] z=x
On salt oussi d@river une hyperfonction
Plus g@n@ralement, si
cients enelytiquas sur
P[x, d_) f(x) dx
d'-~ f×) ~ [z) z=x
d P(x, ~ ) est un op~rateur diff~rentiel lin~aire ~ coeffi-
I , on possra
[ P(z' d----] ~ [z ) ] z = x d z
oO la fonction entre crochets est holomorphe dens un certain voisinage complexe
de I (l'intersectlon de D et de voisineges oO les coefficients de P se pro-
longent an fonctions bolomorphes).
Injections des fonct,ions analytiques :
1 si Im z > 0
On pose e (z) = 0 si Im z < 0
r = ~ 0 si Im z > 0
L - I si Im z < 0
10
et
fonctions sur I [pour tout I , ~[I]
S[x] C ~t[I] }, > a[x].1[x]
c'est-&-dire qu'on identifie la fonction
£onction d@finie par
I ou
ICx] = [s(Z]]z=x [=[~(Z]]z=x puisque ~ - ~ est analytique partout].
On a alors une injection des {onctions analytiques sur I dens ies hyper-
÷ (~[I)) :
= [e[z) ~[Z)]z=x
a[x) analytique sur
~_ ~ [ I ]
I a v e o l ' h y p e r -
Ii s'agit d'une injection de feisceaux (~ -->
IV MORPHISFES DE FAISCEAUX
On sait restreindre & un ouvert plus petit une section d'un pr@faisoeau [par
exemple une distribution ou une hyper~onction), mais on ne salt pasa priori d6-
~inlr sa vsleur en un point. On peut tout au plus lui donner un sens de ls mani@re
suivante : si x E U et f ~ F~(U] , on regarde les restrictions de f Q des
ouverts de plus en plus petlts contenant x . On identifie ~ et g si e11es
coincident dens un petit voisinage de x , Autrement dit, on oonsid~re l'image
de f dens
×
et on eppelle cette image ~erme de f en
diagramme commut atif
x
x . D~s que x ¢~ V C U , on a u n
L'espace ~-- s'appelle la fibre de ~ en x , et ses ~l~ments sont les x
germe,,s,,,,de sections de ~ en x . @uand on aun morphisme de pr6faisceaux
÷ ~4~ [c'est-~-dire pour tout U , ~[U) --~ ~ (U] ,,.] on en d~duit
naturellement des morphismes entre les fibres : pour tout x , ~ x x
evec des diagremmes commuteti~s
t'~ C U) ,~ > ~.'~i U ]
~'~ x x
d@s que x ~ U
On dit qu'un morphisme de faisceaux ~-e ~ est une injection, ou une sur-
jection, si et seulement si c'est vrai sur chaque ~ibre (pour tout x
~ est injectif, surjectif). On pourra parler eussi d'une suite exacte <J~x x de faisceaux > ~jm __u> ~ v> h --> : ge veut dire qu'en cheque
point x la suite
- -> ~ x _.~,,u > t,~x ...,~_v > hx
v~ri~ie v (g) = 0 si est exeote (o'est-~-dire Im u = Ker v ] : g ~- x
et seulement s'il existe f ~ ~ telle que u(f) = g x
u V w
× x x
d'une section de
En termes de sections, g se repr@sente par
section de ~ dens un voisinage V de une
x , et l'ex~ctitude en x signifie que l'i-
mage par v de cette section est nulle ioee-
lement [pas forc~ment dens V mais eu moins
dens un voisinege W de x peut-@tre plus
petit) si et seulement si elle est dens l'imege
au-dessus d'un voisinage U de x iocalement [c'est-~-dire
qu'elles peuvent ne co~ncider ni dens V ni dens U , mais peut @tre seulement
dens un voisinage encore plus petit de x ).
On notero 0 --> --> une in ection et --> --> 0 une
surjection, perce que les suites @crites sont exectes si 0 signifie le feisceeu
nul (dont tous l~s espaces de sections, et donc toutes les fibres sent {0} )
Par exemple, le suite 0--> ~)~ --> ~ est une injection de faisceeux
sur ~ , qu'on compl~tere bientet en une suite excate
12
V - AUTRES CALCULS
Donnons maintenant un sans pr@cis ~ la notion de Valeurs au bord :
s i ~ e oco-z~ , o n p o e e ~C×+ io~ =[eC~ ~Cz~]
et on appelle l'hyperfonction ~ {x + io] [~[x - io]) la valeur au bord
par dessus [par dessous] de
Avec ceS notations
[ ~ [ Z ) ] z = x ~ [× + i o ) - ~ (x - i o ) " s a u t de ~_~ "
Ces notations concr~tisent l'id@e intuitive qua I'hyperfonotion repr@sent@e per
n'est pes autre chose que le saut de ~o , o'est-&-dire la dif~@rence de
ses valeurs au bord par dessus et par dessous.
Support ; int@~ration des. hyperfonctions ~ support compact •
comme ~ est un faisceau, on a la notion de suppor ~ d'une hyperfonction : c'est
le compl@mentaire du plus grand ouvert [ou de le r@union des ouverts) cO sa res-
triction est nulle.
0
En bref :
Si f = [~] ~ ~ [I) a son support dens un ferm@ F
et si x E I-F , il existe I' C I voisinage de x
tel que f|i, = 0 .
Si D' C 0 est un voisinage complexe de I' tel qua
O' - I' ~ O - I , flI' est d@finie par ~ [O'- I'
Oonc ~ est en fait holomorphe dens O' tout en-
tier.
supp f C F ~ ~ E (-~ CO-F)
Soit meintenant K un compact de I et f = rL~] une hyperfonction sur I
support dans K ce qui se note f £ ~ [I] . On d@finit l'int@grale de f sur K
I par la formule
13
[ z) dz
cO y est un contour trac@ dens D - K et entourant
K une 9ois dens le sans inhabituel, Ii 9aut remarquer que grace au th@or~me de
Cauchy la d@#inition na d@pend nl du choix du rapr@sentant ~ , ni du choix
du contour y . Cette d@ginitlon deviant intuitive si l'on resserre y autour
de K et qu'on se souvlent qua 9 n'est autre qua le saut de k~
Examples d'hypergonctions :
Si e[z) C (~[D] , la 9ormule de Cauchy agfirma qua
2i~ j~ Z-Z ° D
lorsque y est un contour entourant le compact {z ] o
donc conduit & d69inir l'hyperfonction de Oirac par
I o 1 8 [ x] 2i--~
comma c i -dessus, On est
de sorte qua l'on a : f
C ~ {o}@R] , }~ 8 [x ] dx = 1 , a t plus @@n@ralement
I ~ [ x ] , 6 [x ] dx ~[0] d@s qua ~ E d [ I ) , I contenant l ' o r i g i n e . I
On remarquera qua dens la notation par valeurs au bard, on e :
1 1 1 6(x] [ )
21~ x+io x-io
On obtient en d~rivant 8[n][x] [ - 1 ) n + I n l [ 1 ]
2i~ 7
at an int@grant l'hypergonct±on de Heaviside :
= 2 i -~ l o g I - z ]
14
oQ log est la d~termination principale, holomorphe sauf sur la demi-droite
r~elle n6gative.
On se contentera d'un seul autre exemple : si f est une fonction m@romor-
phe sur I , on d~finit une hyperfonetion " ~artie finie de f " par
I [ f l x ÷ i o ) + f(x-io)] Pf f i x ] = 7
[¢[x+io) - f(x-io) serait une hyperfonction port~e par les pBles, c'est-&-dire
une somme de combinaisons lin6aires finies de d~riv~es de 6 en ees points).
En perticulier pf 1 _ 1 ( I~__ 1 x 2 x+iO + x~-io )
d'o~ la formule bien eonnue
1 P f ~ $ i w ~ [ x ] x±io x
~hang,ement de variables .
En fait, on peut sous certaines conditions, d6finir beaueoup plus g@n@ralement
l'image r~eiproque ou l'image directe d'une hyperfonction par une application ana-
lytique, et ce sera fair dens le Chapitre IV . On se contente ici du cas parti-
culler d'un diff~omorphisme : si ~ : 11 + 12 est un diff@omorphisme analytique
de 11 sur 12 [c'est-&-dire si (' ne s'annule pas), ( se prolonge en une
bijection holomorphe d'un voisinage D I de 11 sur un voisinage D 2 de 12 .
Si a l o r s g = [ ~ [ Z ) ] z = x £ ~ ) [ I 2) , o~ ~ ~ 0 (O 2- 12) , on d @ f i n i t
l ' i m a g e r@cip roque f de g p a r ~ p a r la f o r m u l e :
f ( x ) (~ g (x ) = ( s i g n e de ~' ) [ $ ( ( C Z ) ) ] z = x
qui g@n6ralise l'image r6ciproque d'une fonction analytique.
Plongement des distributions :
Si K e s t un compact de I , e t f ~ ~ K [ I )
tion representative k_~ de f holomorphe dans 0
#
1 | f [ x ) o (z) 2'i~ ] ~ x - z dx
K(~R] , on o b t i e n t une f o n c -
- K en posan t :
Ion montre que [~o] = f ] "
C'est aussi la manibre dent on plonge les distributions darts les hyperfonctiens :
si T ~ ~'(R) , on pose •
15
1 1 (z] = 2i--~ < Tx, --×-z >
et l'application T ~ > [%0] ainsi obtenue est une injection de ~'[R]
dens ~(~) qui conserve le support. On peut toujours d~composer une distri-
bution sur ~R en somme iocalement finie de distributions A support compact, iden-
tifier celles-ci A des hyperfonctions par la m6thode pr~c6dente, puis som~r dens
la famille ainsi obtenue, qui est ~ supports Iocalement en nombre ~ini,
puisque ~ est un faisceau. On obtient einsi une injection de faisceeux
0 - -> %' >
Convolution -
On se contente Icl de signaler qu'on peut d~finir le produit de convolution de
deux hyperfonctions sur R sous les m~mes conditions de support que pour des dis-
trlbutions, et que ce produit jouit de propri@t~s (cammutativit~, support .... ) en
tous points semblables.
VI - SPECTRE SINGULIER D'UNE HYPERFONCTION
D +
D-
On utillse la notation
f ~ [~] = ~÷[×÷io) - ~f_(x-io)
oO f ~ ~(I) t~ enest un repr@sentent
holomorphe dens D-I , D-I est la r~union de
deux moreeaux D + et D- contenus l'un dens le
demi-plan sup6rieur, l'autre dens le demi-plan
inf@rieur (ce qui suppose qu'on les a distin-
gu6s), e t ~ - ° ~ l o *
DEFINITIONS - On dit que f est analytique en x [ E I] si t~+ et 0
se prolongent en fonctions holomorphes au voisinage de x o
............ + i~ (x - i~] On dit que f est microan.a.lytique en x ° o
+ (~ ] se prolonge en fonction holomorphe au voisinage de x °
16
Remarques . 11 Ces propri@t@s ne d@pendent pas du repr@sentant ~ de f
2] La premiere d6finition est coh6rente avec l'injeotion
~ > ~ d@J~ d@finie : si f est analytique en x ° , il exists un voisinage
I ° de Xo etun voisinage oomplexs Oo de Io tels que ~ + et ~ _ sont
hclomorphes dams 0 •donc f{xl ~ ~+(x+iOJ - ~ (x-ioJ = ~+[x+io) - ~ [x+io} O
= E[~+(zJ - ~ [zJJ~{Z)] dens I , - 0
est dams l'image de ~{I ] --> ~ (I) et c'est dire que fli ° o o
31 Dire q,,u,e, +P est microanal~tique en x+i~ [x-i~] ~ c'est dire
.qu'eu voisinage de x ° f est one vale ur au bord par-dessous (par-dessus] :
en effet, une restriction oonvenable de { s'@cr±t
~+(x-io) ~_(x- io ] (~+{x+io] - ~~(x+io])
Pour rendre compte de la mioroanalytioit@, on est amen@ ~ consid@rer deux
exem@laires de I • not@s I+i~ et I-i~ +et on notera S~I leur r@union dis-
joints [I+i~ ~ {I-i~) {Bans le cas des hyper~onctions de plusieurs varia-
bles sur un ouvert ~ , c'est en effet le flbr@ en sphbresootengentes S~8 que
l'on introduit pour @tudier le microsnalytieit@ ~ iciles espaoes tangent at cotan-
gent en un point de I sont des droites et leurs spheres sont done r@duites cha-
curie ~ deox points}.
Le s~ectre sin~ulier de f , not@ SS% est, par d@finition, le compl@men-
taire dens S~I de l'ensemble des points cO f est microanelytique : un point
_
~cmJaJ~_J
_ ~ I-i~
"~(BSf}
x±i~ de I±i~ est dans le spectre singulier de
f si et seulement si ~ ne se prolongs pas
au voisinage de x
SSf est @videmment un ferm@ de S I , et si
: S~I --> I est la projection oanonique
{c'est-~-dlre l'application x±i~ ~'~ > x) , on
a visiblement :
= supp sing f
cO supp sing f est le support singuller (analytique) de f , c'est-~-dire le
compl@mentaire dams I de l'ensemble des points cO f est analytique.
17
Romarque . Le spectre singulier d'uno hyperfonction porte aussi quelquefois
dens la litt@rature les noms de support essentiel (not@ SE ) , ou de Wave front
(not@ WF) , ou m~me de s u p p o r t s i n g u l i o r , b i e n qua co d e r n i e r f iermo c r6e une con-
fusion regrettable avec le supp sing d@fini ci-dessus, qui enest la projection
s u r l a base ,
1 { 0 } supp I Exemples: SS ! = {O + i~} supp sing x+io x+io X+iO
SS 6 = { O + i ~ , O - i ~ } supp s i n g 6 {0 } supp 6 = {O}
Applicatlon~ .
On en verra plusieurs dens les chapitres suivants. On so contente ici d'une seule,
titre d'exemple.
Los hyperfonctlons soot des @tres trop singuiiers pour qu'on sache los multiplier
entre olios on g@n@ral. Mais on le peut sous oertaines conditions qui portent sur
leur spectre singulier.
Soit a l'epplicetion antipodale de S ~ I , qui ~change les deux exemplaires
de I : a ( x ± i ~ ) = x ; i ~
THEOREME . Si f et g sont deux hyp.orfonctign.s sur I , et sous l'hypoth~se
SS f ~ a(SS gJ = ~ ,
l o p r o d u i t f , g E . ~ { I J e s t d ~ f i n f n a t u r e l l e m e n t . {En p a r t i c u l i e r , ce p r o d u i t
prolonge celui des fonctions analytiqueS, ou d'une fonction analytique par une
hyporqonctlon),
Puisquo ~ est un Yaiscaau, il suffit en offer do d@finir co produit locale-
ment puis do recoller. Or localement, l'hypoth@so du th~or@me implique qua l'on e
l'uoe des quetre situations suivantes (los h&chures figurent la non-microeneiytici-
t@ d'un c6t#) :
/ j / / j / - I / J l / i / l l z l l / 2 1 ; l
et g f et g f et g , f et g " f / / / / J z / / / z f s / t / / z - , . l l l , . , o l
Dens les deux d e r n i e r s ces , l ' u n e des deux h y p e r f o n c t i o n s e s t a n a l y t i q u e , e t Ie
p r o d u i t a d@j~ ~ t ~ d ~ f i n i . Dens les deux p r e m i e r s cas , f e t g s o n t des v a l e u r s
au b o r d du m~me cSt~ ~ p e r example dens le p r e m i e r caS, on a f { x ) = ~ ( x + i o ) o t
g ( x ) = ~ ( x + i O ] . I 1 s u f f i t a l o r s de p o s e r
f . g ( x ) = ~ { x + l o )
18
Par example une expression comme ~2 est bien d~finis : c'est [8 un coef%icient +
pros) l'hyperfonction
(x+~o ]2 1 1 7 s[z) = 7 ~<z)
VII - IMAGES DE FAISCEAUX
On n'a d~flnl de morphismes de faisceaux qu'entre faisceaux sur une m~me base. Ii
est n6eessaire de saveir transporter un faisceau d'une base sur une autre per une
application continue. Cela se fair dens les deux sens.
Faisqeau image d i r e e t e :
S o i e n t X e t Y deux espaces t o p o l o g i q u e s , X ' ~,,,.> Y une a p p l i c a t i o n c o n t i n u e ,
e t ~ un f a i s c e a u su r X , On d ~ f i n i t un p r@fa isceau u ~ su r Y en posan t
u ( ~ [ V ] = ~ - [ u - I [ v ) ] pou r t o u t o u v e r t V de Y , On v @ r i f i e lmm@diatement qua
u ~ e s t un f a i s c e a u , qu 'on e p p e l l e le f a i s c e a u image d i r e c t e de ~ pa r u .
Falsqeau 19age inverse :
Soient X, Y, u comma pr~e~demment et
du faisceau inverse se complique du felt qua l'image par u
n'est pas un ouvert. On obtient quand mgme un pr6faisceau
pour tout ouvert U de X
~(U] li m ~r (v] V D u[U]
(Au l i e u de p rend re les s e c t i o n s de ~ su r u[U]
un faisoeau sur Y . La construction
d'un ouvert de X
sur X en posant,
qu i n ' e s t pas o u v e r t , on a
prls toutes les sections de ~Y sur tousles voisinages v de u[U] , en iden-
tifiant callas qui coincident au voisinage de uCU) . Si u[U) est ouvert, on
retrouve bien ~[U) = ~[u[U)] ) .Mais le pr~faisceau ~ nPest pasen
g6n~ral un faisceau, et c'est le feisceau associ~ & 4 qu'on appelle le faisceeu
image r~clproque de ~ par u , et qu'on note u -1
Ses sections sent dcnc compliqu6es & d~crire en g6n~ral, mais on se consolera
en remarquant qua ses ~ibres, elles, s'expriment simplement ; on aen effet :
[u-1 "~Y ]x : " ~ u[x)
~9
VIII - LE FAISCEAU ~ DES MICROFONCTIONS
On introduit ici de nouvelles notations dent l'utilit6 se fera surtout sentir
dens lecas de plusieurs variables [chapitres-suivants). I1 s'agit de construlre
plusieurS faisceaux sur la base S~ ~R = [~+i~)LJ (~-i~]
- D'abord, comme l'application ~ : S ~IR ÷IR est ici partieuli~rement
S" simple, ii est ais~ de construire le faisceau -1~ sur ~ : les sec-
tions de w -I ~ au-dessus d'un ouvert de la forme I+i~ [c'est-~-dire contenu
dens IR+i~ ou dens IR-i~ ) sent tout simplement les hyperfonetions sur I
8ien entendu, sur un ouvert de la forme [l+ioo) t] (J-ion] , l'espace des sections
de -1 0D ~ est le somme directe ~ (I) (~ ~ (J)
On construit ensuite un sous-faisceau £-~* du pr~c@de~t en consld~rant
le pr6falsceau sur S IR dent les sections sur un ouvert de la forme I+i ~
(ou I-l~) sent les hyperfonctions sur I qui sent microan'elytiques en tousles
points de I+i~ [ou I-i ~] . Le faisceau associ~ ~* ales m@mes sections
au dessus des ouverts de carte forme, et pour sections au dessus d'un ouvert quel-
conque i~ somme directe des sections au dessus de chmque composente. On a visible-
S ~ ment une injection de faisceaux sur ~ :
0 > ~ -1
E n f i n le f a i s c e a u
e s t le f a i s e e a u - q u o t i e n t de 7 -1
faiseeau quotient, dent 1as sections
-I
~.~c u)
Soit ici, si U [l+i ~) kJ [J-i °°)
-1 ~(U] "~(I] ( ~ ~(J)
~(u)
[dent les sections sent appel~es micro%onctions]
par ~ . On construit d'abord le pr@-
au-dessus de 1'ouvert U sent !~ quotient
• j c I~ ® 0 5 u ) .
En g ~ n ~ r a l , quand on construit ainsl l e pr~faisceau quotient d'un faisceau
p a r un s o u s - f a i s c e a u , on n ' o b t i e n t pas un f a i s c e a u , e t ce s e r e le cas , p a r exam-
p i e , quand on f e r e l a c o n s t r u c t i o n a n a l o g u e p o u r tes h y p e r f o n c t i o n s de p l u s i e u r s
variables, C'est elors le faisceau associ~ qu'on appelle faisceau quotient.
Ici, per centre, on obtient tout de suite un ~alsceau, le faisceau ~ des
microfonctions.
20
Par exemple, l'espace ~ {I+i~) , des microfonctions sur I+i~ , est
l'espaca des hyperfonctions sur I consid6r~es modulo callas qui sont microana-
lytiques sur I+i~ [ou encore modulo callas qui sont valeurs au bard par dessous).
II r@sulte de la construction m@me du feisceau ~ qu'on a une suite exacte
de faisceaux sur S ~ I :
0 ' > ~ ....... > 7 - 1 ~ b > ~ 6 - - - > 0
qui d'ailleurs peut se transporter ici en une suite exacte de faisceaux sur
0 -----> ~-~ --> ~
-1 Les q u a t r e f a i s c e a u x ~ ~ ~ ,
~D-" Cni EL!
L'application " spectre " , not@e sp
pondre ~ une hyperfonction sa " singularit@ " qui est une microfonotion
Enfin, il est clair par construction qu'on a l'~galit~ suivente, pour une
hyper%onction f :
SS f = supp [sp $)
le second membre est le support de la microfonotion sp f , en tent qua section
du Saisceau ~ ,
On aoh~ve ce chepitre en ~nonqant une proposition [locale] et son corollaire
imm~diat [~lobal] ; la premi6re n'est qu'une autr8 mani~re de proDonoer l'@galit~
ai-desSUs.
I :
sp > ~ ~ ~> 0
, ~T ~ sont flasques, mais pas
de ~ dens ~ ~ ,fait corres-
: en 84-
SS $ ~ x - i- (x+i~)
COROLLAIRE - U_me hyper~onction $ su__r I est de la forme k~ [xtio~ si et
seulement si
SS f C I _+ i~
PROPOSITION - Une hyper$onct.ign f est valeur au..bord,, p, ar dessus (par dessous)
au voisinage d'un point x , si at seulement si
CHAPITRE I I - HYPERFONCTIONS A UN NOWBRE
QUELCONQUE DE VARIABLES
A. PIRIOU
I - INTRODUCTION HEURISTIQUE
On a vu dens l ' e x p o s @ pr@c@dent q u ' u n e h y p e r f o n o t i o n s u r ~ e s t un @l@ment
de ~( ~ )= ~tCC-j ~ " ) - soieot u l ,u 2 ~ * ( ~ 1, r~pr~seot~e~ respeotivement per
~ I ' ~ 2 E O(C - ~ ) ; i l e s t n a t u r e l de c h e r c h e r & d 6 f i n i r l ' h y p e r f o n c t i o n (de deux
v a r i a b l e s ) U ( X l , X 2 ) = u 1 ( x l } u 2 ( x 2 } au moyen de Ze f u n c t i o n ~ ( z l , z 2 ) = ~ I ( Z I ]
~ 2 ( z 2 ] ; r e m e r q u o n s qua ~ e s t h o l o m o r p h e dens (C - ~ ) x (C - ~ ) , e t q u ' e l Z e e s t
d ~ f i n i e modu lo O(Cx (C - ~ ) ) + O ( ( C - ~ )xC) p u i s q u e @ I e t ~2 ne s e n t d # f i n i e s
que modulo O(C) - A i n s i , on e s t c o n d u i t & d ~ f i n i r u comme @l@ment de
ll+ I,J- comma on l e v e r r a p l u s l o i n , qua H 2 (C2mod C 2 - ~ 2 , 0 ) second g r o u p e de c o h o m o l o -
g i e r e l a t i v e de C 2 modu lo C 2 _ ~ 2 & v a l e u r s dams l e f e i s c e e u 0 , e t q u ' o n n o t e
aussi H2~ 2 (C2,0).
D'eutre pert, si on reprend 1'interpretation d'une hyperfonction sur ~ en
termes de veleurs eu bord abstraites on a :
ul (× I ) = ~t (x~+io) - ~1 ( x l - i ° )
u2 (×2) = ~2 (×2 + i ° ) - ~2 (×2 - i ° )
et ~ est netu~eZ d ' ~ o ~ e u(×~,x~) = u~(x~) u~(x~) s o ~ re,me d'une eomme de
q u a t r e " v a l e u r s au b o r d " :
~(×1 ,×2) = ~(x 1 + io, ×2 ÷ io ) - d x 1 - i e , ×~ + ±o)
+ ~ ( x I - i o , x 2 - i o ) - q0(x 1 + i o , x 2 - i o )
- Yl
WI°,,_
22
Dens ce qui suit~ nous allons d@finir lee hyperfonctions de n variables au moyen
de la cohomologie relative, et prEciser la notion de valeur au bord, qui nous
fournira un moyen commode de repr@senter lee hyperfonctions, Pour cela, il nous
faut tout d'abord rappeler bri~vement la cohomologie de C~ch. On d@signere par
le faisceau (sur C n) des fonctions bolomorphes.
II - RESUME DE LA COHOMOLOGIE DE C~CH
Soient U, U' deux ouverts de C n, avec U' c U. Appelons recouvrement relatif
(~,~') de (U,U') la donn@e d'un recouvrement ouvert~= (U)~ E I de U, et d'un
sous - recouv remen t ~ ' = [O )~ E I ' ( a v e c I ' c I) de U ' . Pour p 6 ~ , l e groupe
a d d i t i f C p ( ~ , ~ ' , 0 ) des p - cocha ines altern@es ( r e l a t i v e s ) de ( ~ , ~ ' ) ~ c o e f f i -
c i e n t s dams ~ est constitu@ par les familles ~ = ( ~o .... ~p) (~o .... ~p) 6 I p + I
telles qua
m~o, 60(U ..~ ), SO on apos@ U ..~ = O N..N O . • " ' ~p o p ~o p o p l a)
b) "'' ~i .... ~j .... ~p "'' ~j .... ~i .... ~p
I o) = O si a s .... ~p E I' q~o ' " " ' O~p
On d@finit ensuite les m grphismes de cobord 6
6 : c P ( ~ , ~ ' , O ) > C p+I ( ~ , ~ ' , 0 )
~ ,, > 6%o par : p+1 -Z C-~) j
[ 6~P)~O . . . . ~p+l j=O ?~o . . . . ~ j . . . . ~ p+1
On v 6 r i f i e f a c i l e m e n t qua 6o6 = O, d'oO un complexe de groupes :
0 - - > c ° ( ~ , ~ ' , o ) ...... ,6> c I ( . ~ , ~ , , ~ ) ....... ~ > . . . 8 > c p - I ( ~ , ~ , , o ) 6> cP(~,~ ',o) 8 > cp+1 (~,~,,~) > .,,
dont on consid~re les grgupes de cohomologie
H p (~,~',O)= Ker (C p (~,~X',O) 6 > O p+I (~,~',O),) .
,~m (c p-I (~,~ ' ,o) - ~ > c p ( ~ , ~ ' , o ] )
23
Soit maintenant ( ~ , ~ ' ) un autre recouvrement relatif de (U ,U ' ] , aveo
= , ~ ' = ( V ~ ) j , ( J ' c J) (V~) B e J B E " Nous dirons qua ls recouvrement (~ , ~') est plus fin que le recouvrement (~,~')
si :
a) Pour tout B 6 J, il exists G E I tel que V B o U
b) Pour tout 8 6 J', il existe G E I'tel que V B c U
Soit alors une fonstion de choix G = ?(B ) ; dEfinissons lea morphismes
C p (~,~ ' ,0) ?~ ~ C p (~,~ ' ,0)
~ > T~ par :
(T~-~)~o ' = ~ . . ,Bp 9T(~o) . . . . m(Bp]. " Lss morphismes s , qui commutent @videmment
avec les morphismes de cobor~ 8, induisent des morphismes
H p (~,~',O) ....... > H p (~,~',©) qui sont canoniques,
en ce sans qu'ils ne dependent plus de la fonction de choix m utilis@e.
ieme DEFINITION I - On appalls p groupe de cohomologie relative de U modulo U'
& valeurs dans le faisceau O le groupe H p (U mod O', O) = !~m--> HP(~,~',O) , ( ~ ,~ ' )
o~ la limits inductive est prise salon les recouvrements plus fins.
Signalons qua ce groupe est aussi note H~_U, (U,O] ; dens lecas o0 U'= ~ ,
on le note simplement H p (U,O].
PrEcisons que, dans la definition I, le passage & la limits inductive signifie
ceci : soient (~,~') et (~,~') deux recouvrements relatifs de (U,U'), alors
E H p (~,~',O) et ~6 H p (~,~',O) sont identifies s'il exists un recouvrement
relatif (b,b') de (U,U'), plus fin qua les deux pr6c@deots , tel que m st T
aient la m@me image par lea morphismes canoniques
H p (gJ,~' ,0) H p ( ~ , ~ ' ,O)
H p ( l f , l r ' ,O) /
n ~ a r q u o n e qu~ s± bX,~X') ~ s t un ~ s c o u w e ~ e n t r ~ i a t i f qu~ i conqu~ d~ ( U , U ' ) , on a
des morphismes canoniques H p ( ~ , ~ ' , 0 ) . . . . . . . . . > H p (U,U' ,O) •
24
Mais si on suppose H q (U o .... % 0) = 0 pour tout q ~ I, pour tout p ~ o, et
pour tous ~o .... ~p E I, on salt alors (th@or~me de Lera/) que ces morphismes
sont en felt des isomorphismes. Ces conditions sont en pertioulier v@rifi6es
si, pour tout ~ 6 I, U est un ouvert d'holomorphi e de C n, On a donc le
THEOREME I - Soit (~,~') un recouvrement relatif de (U,U'), avec
~2 = (U)~ ~ Iet U ouvert d'holomorphie pour tout ~ E I
#&ors H p (U rood U',O) ~ H p (~, ~', 0),
III- DEFINI~ON DES HYPERFONCTIONS
n Soit Q un ouvert de ~ , et consid@rons un voisinage complexe U de O,
c'est-&-dire un ouvert U de C n tel queQsoit contenu et ferm@ dens U. Alors
U' = U - ~ est ouvert dens U, et on peut consid@rer les groupes de cohomologie
reletive~
UC
P (u , e ) .......... ~ n H p (U mad U - O, O) = H 0
I °
S a l t me in tenan t m o u v e r t dens 0 , e t V un r a i s i n . s compZexe de ~ t e l qus
V c U, V - ~ c U - ~ ; s i ( ~ , ~ ' ) e s t un recouv remen t r e l a t i f de (U, U-O), on
o b t i e n t un recouv remen t r e l a t i f ( ~ , ~ ' ] de (V,V - w) en posan t V = U N V ; p a r cpff( G
restriction, on obtient des morphismes C p ( ~,~',0] .... > ~,~',0),
qui induisent des morphismes H p (~,~',0) ..... > H p ( ~,~',0],
d'ob finalement des morphismes P PO,w : HP(o mod U - O , O ) - - - - > HP(v mod V-m,O].
Dmns Ze oas oD w = ~, on montre (th@or~me d ' e x c i s i o n ) que pP es t un
i somsrph isme, e t done que H p (U mod U-~,O) ne d@pend qus de p e t de O, e t non
pas du v o i s i n a g e compLexe U u t i l i s @ pour ~ ; pour p f i x 6 , on peu t ma in tenan t
cons id@rer l e p r 6 f a i s c e a u qu i , & t o u t o u v e r t ~ de ~ n e s s o c i e l ' e s p a c e
v e c t o r i e l H p (U mod U-O,O), l e s morphismes de r e s t r i c t i o n @tenfi l e s morphismes
P d~finis plus haut, On a alors le r@sultat fondamental suivant : PO,
25
THEOREME et DEFINITION 2
I) H p (U mud U - ~,0) = 0 si p ~ n , ~ ouvert de ~n
2) Le pr@faisceeu d@fini par le donn@e, pour tout ouvert ~ de ~ n, de l'espece
H n (U mud U 0,0), et per les morphismes de restriction n - P~,w est un
faisceeu flesque qu'on appelle feisceeu 8 (sur ~q n) des hxperfonctions.
Donc si 0 est un ouvert de ~ n les hyperfonotions sur 0 sont les 61@ments de
1'espece.
t ' 1 (u, e) a(~)=H ~(e~odu-~,o)=H~
oO U est un voisinage complexe arbitraire de ~.
Dens tout ce qui pr@c~de, on peut remplacer ~ n par une yari@t~ anel~tique
r@elle M de dimension n , et C n par une complexifi@e X de M,
IV - REPRESENTATIONS D'UNE HYPERFONCTION PAR DES FONCTIONS HOLOMORPHES
Suit ~ un ouvert de ]R n ; consid@rons un voisinege comple×e U de 0 tel que
U d'holomorphie
(reppelons & ce propos que, d'epr~s le th6or~me de Grauert, tout ouvert de &qn
admet on syst@me fundamental de voisineges d'holomorphie dens C n ).
On va expliciter 8(O) = H n (U mud U - ~, 0) en utilisant le th@or@me I.
Premier,,, exemple de recouvrement :
On prend le recouvrement relatif (~,~') de (U,U-Q) d@fini'par :
= (Uo, 01 .... Un}
~ '= ( U I , . . , Un]
U = U O
U.=j U n (z = (z I . . . . Zn) E C n I ~mzj / 0 ] ( j = I , . . , n)
26
En appliquant les ddfinitions du ~, 2, on voit que :
C ~+~ ( ~ , ~ ' , 0 ) = {o ]
C ~ [ ~ , ~ ' , 0 ] ~O(U n [O - ~ )n)
C n-1 [ ~ , ~ ' , 0 ) = ~ r.p = (~Pj)j=I . . . . n I ~j E (} ( U I . . 3 . . n) }
et que 5 : C n-1 (~,~' ,0) > C n (~,~' ,0) est alors d@fini par
j = l
Puisque U , U 1
donnent
U sont des ouverts d'holomorphie, les th@or~mes I e t 2 n
(.) o [un[o - m )~)
~ o ( u 1 . , 3 . . ~ ) j = l
( avec U l , , 3 , , n = {z E U ! ~rnz k ~ 0 pour k ~ j ] )
e t l ' o n re t rouve a i n s i l ' e space -quo t i en£ ± n t r o d u i t au § fl dens l e ces D = &q 2 U C 2
D@signons les 2 n composantes connexes de (C - ~q)n par ~n + i£ o~ (I 1
a = _ (01 .... ~n ] _ , avec ~- 3 = -- ÷ I , et o~ ~q est le cBne convexe ouvert
£o = [y E lq n I ~j yj > 0 pour j = I .... n} , La formule(~)s'6cr±t
[*,) • @(un (lqn + ££~])
n
~ o [uI.4... ) j=1
de sorte qu'une hyperfonction f E ~(~) peut ~tre ,repr~semt@e par 2 n fonctions
% ~O[Un (Inn + i r )]
27
Deuxi~m,,%exemple de recoq,,vrement
Soient n + I vecteurs ~I,,,, ~n+1 de ~n - [o] tels que les n + 1 demi-espaces
ouverts L = [ y E ~ n I y . ~j > 0 ] ~j
recouvrent ~ n _ {o] , c'est-&-dire tels que 0 soit int@rieur & l'enveloppe
convexe de {~I ~ I] . x
33
~1 ~ o " • ~2 On consid~re alors le recouvrement relatif (~,~') de (U,U - ~] d4fini par :
= ( ~ , v I . . . . v~+1)
~'= (v I . . . . Vn+1)
V = V o
v.= u n ( ~ ~ + i ~ ) ( j=1 . . . . ~+ I ) J
On a :
C n+l ( ~ , ~ ' , ~ ) = [o ] puisque ~I n . . . n ~n+1 = ¢ n+l
C n ( ~ , ~ ' , 0 ) ~ j~1 ~ ( V 1 . , 3 . . n + 1 ]
c ~ - I ( ~ , ~ ' , o ) ~ ¢ o C v ~ . . ~ . . 3 . . ~ + l ) 1~i<j ~n+ I
et 8 : c n-1 (m,~z' ,o) > C n (~,~',0) est d@fini par
c6 lj = _ 7 _ + c- l . 1~i<j j<i~n+1
Puisque V, V1,.., Vn+ I sont des ouverts d'holomorphie, on obtien% en posant
V1..O,,n+1 = Un(3Rn + irj], et en d@signant per r. le cBne ouvert convexe J
r j = ~I n..n~a n, ,~n+1 ,
28
l'6galit@
(**)
n+ l ® O(Un (]B n 4- i r j ] ]
j =1
> ~ o(v~. . ,~. .3. .~+~ ) I ~i<j ~n+ I
de sorte qu'une hyperfonction f E ~ (O) peut @tre repr4sent6e par n+1
fonctions ~j E O(Un ( ~ n + i r j ] ) ( j = I . . . . n+1 ) ,
R.pmarque..1 : Pour v@rifier s±, avec les notations ci-dessus, m = (~]~ et
= (~j]j=1 .... n+1 repr@sentent le m@me hyperfonction f E 8(0], il faut
cons±d@rer un recouvrement (~,~') plus fin que (~,~'] et (~,~'], pu±s vo±r
si ~ et Y ont, par les morphismes canon&ques d@erits au § 2, la m&me image
dans H n (~,~',~].
Exemple emprun t~ & A. CEREZQ : on p r e n d 0 = ~ 2 , U = C 2, e t on d~s±gne p a r
f = - 4 ~ 6 ( x l , x 2 ] l ' h y p e r f o n e t i o n r e p r ~ s e n t 6 e , au moyen de ( ~ ] , p a r l a
~ o ( ( c - ~ )~). fonotion ~(zl,z2] = zl z2
On ~finit maintenan£ le recouvrement (~,~'] par
v = c 2 v I [z E c 2 t ~ z I > e ~ v 2 = [z ~C 2 I ~ z 2 > o ], O ~ = ~ ~ :
V3 = [ z E C 2 I ~mz I + ~mz 2 < o ] , e t on c h e r c h e & r e p r e s e n t e r
pa~ (~I ' ~z' %), ~ o ~ . ~o(i~ ~ + i r j ) ( j=1,2,3) J
/z//2."x~//.~. Yl Comme recauvrement ( b , b ' ) plus f i n que
( ~ , ~ ' ) et ( ~ , ~ ' ] , prenons par exempie
. I//~ / b = (W o , W I , W 2 , W 3 , W 4 ]
b ' = (~1 ' W2 ' W3 ' W4 ) ' avec
W ° = C , W 1 = V I , W 2 = V 2 , W 3
W 4 = [ z ~ C 2 I ~ z 2 < e , ~ z t + ~ z 2 < 0
= {:~ e c ! ~ z 1 < o , ~ 1 + ~ z 2 < ° ~. ,
29
Les inclusions W ~ U , V o o o
W I ~ U 1 , V I
W 2 ~ U 2 , V 2
W 3 c U 1 , V 3
W 4 c U 2 , V 3
permettent d'expliciter les morphismes canoniques
H 2 ( ~ , ~ ' , 0 ) ~
H 2 ( t , t , , o ) ~ ~ H 2 ( ~ , ~ ' , o )
et on obtient que (~I,~2,73) repr~sente f si et seulement siil existe des
f o n c t i o n s £'a E 0 (Wj) ( j = 1 , 2 , 3 , 4 ) t e l l e s que :
1 zlz 2 ~3 = e 2 - el
- ~ 2 = 8 3 - 81
1 ZlZ 2 T2 B 4 - 61
I Z lZ 2 ~I e 3 - e2
- ~ I = e 4 - e2
1
La derniere ~qu~tion, du type "lemme de Cousin" , admet par exemple la solutiol
I
I
Si on choisit per exemple @I = o, @2 = o , on obtient finalement que
f = - 4 ~ 8(Xl,X2) est repr@sent@e par
- ~ ~3 = - 1 ~ 2 t ~ ~ 1
~t - Zl ~ Z2(ZI+Z2 ) ' z l z 2 84 = Z l ( Z l + Z ~ ' 3 = Z l - -~7-
30
V - VALEURS AU BORD
On suppose d~sormais que Iq nest erientd . Soient ~) un ouvert de 3q net Fun
cBne ouvert convexe (de sommet o] de lq n. Soit U un voisinage eomplexe
d'holomorphie de {) , avec O0 &q n = ~et soit r@ une fonction holomorphe dans
le tube local on ( IR n+ir) ; on va associer & m une valeur au bord selon[ ,
÷ Pour eels, on reprend le deuxi~me
exemple de recouvrement expos~ au
~ U c ~
paragraphe pr6cEdent, en choisis-
sant lee vecteurs ~q'~2 . . . . ~n+1 tels que
r l = ~2 N , , , N _ ~n+ l a @
Consid@rons l'hyperfonction f sur
\~ ~ 0 repr@sent~e, au moyen de (~) ,
L par les fonctisns
#j e o(un (~n+irj)) (1~j~ n+1)
d~finies ainsi :
T I = ¢ ~IFI! , avec e = ~ I selon que l'orientation de la
base ( ~2 ~n+1) de ~n est positive OU
n@gative
~, = 0 si j = 2 , , , , n + l O
A l o r s , en p roe@dant comme dans l ' e x e m p l e t r a i t @ & l a f i n du §4, on o b t i e n t :
THEOREME ET DEFINITION 3
L'hyperfonction f E ~ (Q) construite ci-dessus ne d@pend pas des vecteurs
~I,~2,,,,~n+ I util/s@s. On l'appelle valeur au bord de ~ selon ~, e t on la
note ~(x+i r o),ou b r (~).
31
Remarque 2 :
a) Si £' est un cBneouvertconvexe dans F, et si ~ EO[UN [ ~n+ir]) , il r@sulte imm@diatement de la d~finition pr@c@denfle que
(x+~r,o) :d×+~ro) b) De re@me, si f E ~ (~) est repr@sent@e, au moyen de [**) par des
fonctions Y E 0 [UR [Iq n Fj • +i )) on~ O n+1
:>i (-II j-1 ~j (×+ir. ol j=1 J
c) Sif E ~[~) est repr@sent@e, au moyen de (¢,~') par des fonotions
~d EO (UR [~qn+ EFt)), on peut v@rifier
f = 3---. Sd ~0 Cx + i r ~ ) , oh SC~ ffl " ' dYn ' cy
ce qu i g@n@rmlise l a foPmule donn@e dans l ' i n t r o d u o t i o n .
Exemple : en u t i l i s a n t l a remarque 2, on peu t j u s t i f i e r t r ~ s s imp lemen t l e
changement de repr@sentation donn~ ~ la fin du § 2 pour l'hyperfonction
f = - 4 ~ 6[x I , x2). En effet, on a, d'apr~s c)
f b~1,1
F I
F - I , - I
1 I I 1 ZlZ2 - b~ + b~
Y2 T I , 1 = r 3
f ( - t = b r l ~ z2 ®3 ] - bc2
e t b) mon t re a l o r s que
_1
Si ®3 ' ®4 sont telles que
Yl
sj ~o(wj] ( j = 3 , 4 ] { 1 { _ - % do°
OF a
b[, 1 _ = - I , - 1 zl 72 b r - 1 , - 1
e t donc, d ' a p r ~ s a ] :
r I I z 1 ~ ®4 ) + bF 3 z 1 z 2
fi est repr@sent~e par
84 - b F - 1 , - 1
I 1 . . . . . % ' ~2 ~I~ ®~' % =~-~ -
,
32
Les parties b) et o) de la remarque 2 montrent que toute hyperfonction est one somme finie de valeurs au bord - Plus g@n~relement, on peut @tablir le
THEOREME 4
r ~ y~ Soient ~I,°. Fm de c~ne convexes ouverts de ~n tels que I,.., m recouvrent
Soient 0 un ouvert de ~q n , f E 8 (0] , et U un voisinege eomplexe d'holomorphie
de 0 tel que U 0 ~ n = O, Alors il existe des functions mj E 0 (U n [ ~ n + ±rj))
(j = I .... m) telles que m
j = l
Enfin on d@montre le th@or~me suivant, du type "edge of the wedge" :
THEOREBE 5
On suppose O connexe. Alors le morphisme "valeur au bord"
b r : ~ ( Un (~n + it)) ------> 8 (Q) est injectif
Vl -INdECTION8 de ~I dens ~ .
D@signons par Q le faisceau (sur ]Rn) des functions analytiques.
Soient 0 un ouvert de iqn et @ E C (0).
Oonsid@rons un voisinage oomplexe d'holomorphie U de 0 (avec UN iR n = O) tel
que (p se prolonge en function holomorphe ~ dens U. 8i ~ est un oboe ouvert
convexe (de sommet o] arbitraire de ]Rn oe qui pr@c~de montre que l'hyperfonc-
tion ~ (x + iFo) sur 0 ne d@pend ni de O , ni de F, et que les morphismes
m~- ...... > ~ (x + ire)
sent ±njeotifs ; on obtient ainsi une ~ de faisceaux G ...... > ~,
Suit maintenant m E C' (~n) . La function ~ E 0 (( C - ~ ]n) d@finie par I I
(z I .... Zo) = U2~O~- < T (tl,..tn) , (t1_zl)...(tn_Zn ~ repr@sente, ao
moyen de (¢~) une hyperfonotion ft E ~ (~n), et on montre que le support de la
distribution Test @~al au support de l'hyperfonction fT" Soient maintenant
0 un ouvert de ~n, et T E ~ (0) ," d~composons Ten une somme loealement
finie ~Z Tj de distributions T i 6 C' (~n) ; la propri@t@ pr@c@dente des J
33
supports montre que -~-- fT. est aussi localement finie et d@finit un 61~men~
J
fT de 8 (~) qui ne d@pend pas de la d~composition utilis~e pour T, et enfin
que le5 morphismes ~D~(~] ....... > ~ (~) sent injeotifs
T ~ ......... > fT
on obtient ainsi une injection de fsiseeaux ~' .... > ~ , qui prolonge
l'injection G .... > ~ . On a done les sous-f~isceaux O c~' c ~ .
CHAPITRE III - FAISCEAU "~
J, CHAZARAIN
Dons ce chapitre, on expose pour los hyperfonctions ~ un nombre
quelconque de variables, los notions de spectre singulier et de faisceau des
slngulorit~s qui ont 6t@ introduites ~ la fin du chapitre I dons le cos d'une
variable.
Saul indication contraire, on suppose toujours qua M est un ouvert
delR n, un voisinage complexe de M est slurs un ouvert X de C n tel que M suit
farm@ dons X. 0'autre part, F d6signera toujours un cBne ouvert convexe et F ~
son cBne dual = {ql<~,~ > 2 0 taut ~ F}
I - SPECTRE SINGULIER D'UNE HYPERFONCTiON.
On a d6fini au chapitre pr@c6dent l'injection i des functions ana-
lytiques r@elles dons les hyperfonctions, de sorts que l'on a une suite exaote
de faisceaux sur M
i O }~. ~ B .
Roppelons bri~vement lo d@finitlon de l'hyperfonction i[~J ~ B[M) assooi@e A
une function anolytique ~6~{M). Suit X un voisinage complexe de M cO ~ se pro-
lunge en une function holomorphe et suit Fun cBne convexe ouvert de IR n, on
pose
f = i ( ~ J = ~ [ x + i F O) avec ~ G ~ [XQtR n + i F )
e t on odmis au c h a p l t r e I I qua c ' 6 t a i t i n d 6 p e n d o n t au c h o i x de F •
C o n s i d 6 r o n s uns h y p e r f o n c t i o n de l a fo rms
f = g ~J~" (x + i r . O} avec ~@j e ~ ( X R I R n + i F . ) , finis J J
slurs si los ~j sont holomorphes au voisinage de x o~ M, ~ est analytique dons
un voisinage de x , Ceci est r@sum6 dons Io : o
35
PROPOSITION 1.fl. -
nege de x ° & M, si et seulement si Bile peut s'6crire
f = ~ . ~ j (x + i £ j 0) avec ~3 ~ " £ ~ [X~ IRn + i r . ] finie J
avec des ~j holomorphes dens un voisinage de ×O"
O'autre part, on salt que toute hyperfonction f & B[M] peut s'@crire sous 18
forme
f = ~ ~ j [x + i r j O] avec ~ j E ~ [ X ~ I R n + i £ . ] finie J
pour des cBnes £, convenebles [el. chapitre II]. J
Soit %@B[M], cette hyperfonction est analytique au voisi-
On est alors conduit & poser la :
DEFINITION 1.1. - Soit un covecteur non nul qo e [4Rn] ~ et solt x o£ M.
On dit qu'une hyperfonction f 6 B(M] est migro-analytique au voisinage du
point Ix o, in o] si elle peut s'6crire :
f = Z ~j [x + iF. O] avec ~j £ ~[XOIR n + i r.] finie J J
oO les ~j sont holomorphes au voisinage de x ° sl le c6ne rj v@rifie
r , c { ~ l < ~ ,n o > > o} = {no } ~ . 3
Remamque 1 . 1 , - On v ~ r i f i e imm~dia tement que l ' e x p r e s s i o n " f e s t m i c r o - a n a l y -
t i q u e eu v o i s i n a g e de [x o , i n o ] " peu t se r e f o r m u l e r en dLsan t que su r un v o i s i
nage o u v e r t de x dans M, on a O
oO tousles
f = z . ~j C× + i r. o) finie J
r. v6rifient J
rjc{~ I <~,qo> < O} .
36
Remarque ~.2. - Ii est clair que cette d69inition coincide avec celle d6j&
donn@e darts le eas de une variable.
Notons que ce qui importe, ce n'est pas rant le covecteur
noE[~n] ~ - {0}, que la demi-droite qu'il d@ginlt que l'on note no~ et que
l'on consid~re eomme Element de la cosph@re ORn] × - {O}/[homothEties > O] = S ×.
O'autre part, le piongement NClRnc ~n = LR n + i IR n permet d'interprE-
ter les covecteurs de la ~orne iq
@n. De 9a~on plus dans ~n not6 T M
exaete d'espaces vectorlels
0 ( T x IR n ~_ p T x ~n ( X X
ql ~ ....... [ql÷iq2 ]
comme des @IEments du fibre conormal & M
]rEcise, Etant donne x @ M on a une suite
I" ....... -7
', T~ @n] ', ( 0 L__[' .... xj
[d69inition]
× cn Ce qui permet d'identigier T M ~ N x i ~Rn) x et en passant aux 9ibr6s
en cosph~res associEs, on obtient :
X ~n ~ M x ± S x S x S M _ = i M
qui est appel@ le 9ibr@ en cosph~res conormeles.
37
On appelie ~ la projection
canonique [x, i q]~i S × M
x M
On peut alors poser la :
0 ',,]
X
iSXM
DEFINITION 1.2. - Soit f~ B[M], l'ensemble des [x, in ~)~ i S x MoO f n'est pas
micro-analytique est un farm@ de i S × M ; on l'appelle spectre singulier de f
et on le note S.S.f.
Note :IOn dit aussi : support singulier [S.Sc], support spectral, support
I essentiel, analytic wave front set.
II - FAISCEAU ~.
On commence par d6finir le faisceau des microfonctiens analytiques
les ouverts de la forme U x i£ o~ U est un ouvert de Net r d6signe la trace
sur S ~ d'un cBne ouvert convexe (encore not~ F ] de @Rn] ×, ferment une base
d'ouverts de i S ~ M. A l'ouvert U x ir on associe
fLe sous espace des hyperfonctions f 6 8[U]
[U x ir ] = Ltelies qua S.S.f ~ (U x iF ) = J
et soit 6L ~ le faisceau associ6 sur i S × M, on l'appelle le falsceau des mi-
crofonct, ions analytiques (ses sections au dessus de U x i£ sent bien callas
ci-dessus].
O'apr~s la d6finition de l'image inverse d'un faisceau [of. chapitre I],
il est clair que [7 -1 B] [U x ir ] = B[U], ce qul permet d'interpr~ter O~ ×
-I comma un sous-faisceau de ~ B :
0 ~ ~ ~ - 1 B,
On d~init alors le faisceau ~en eompl~tant carte suite e×acte sur
i S ~ M :
38
DEFINITION 2.1o - Le ~eisceau ~ est d~fini sur i S X M comme le quotient de
-1 B par 0_ X, ce qui s'@crit en une suite exeote :
0 .......... ~ O~ > -1 S > ~ > 0 (sur i S X m)
C'est & dire qua le faisoeeu~ est assocl@ au pr@feisoeau d@~ini par les
quotients
R~' [0 x ir ) sEo) / 0 ~ (0 x ir ]
oQ l ' o n " t u e " $es h y p e r f o n c t i o n s m i c r o - e n a l y t i q u e s dens l es c o d i r e o t i o n s
de F , r e s t e donc l es s i n g u l e r i t 6 s .
On a cZars l e r ~ s u Z t e t f o n d e m e n t a l s u i v a n t qua l ' o n e d m e t t r e
THEOREME 2 .1 - Le f a i s c e a u ~ e s t f l e s q u e .
On p r o j e t t e e n s u i t e l e s u i t e e x a c t e p r ~ c ~ d e n t e s u r M p a r l ' e p p l i c a t i o n ~ e t
l ' o n d~mont re qua l ' o n a encQre une s u i t e e x e c t e
sp o ........ ~ CL > B > ~x~ > o,
ce qui signifie qua B/~% ~ ~. x
Le morphisme de faisceaux, not@ sp, permet de d6finlr l'application
f ~ BCM) ~ sp f E (7 ~ ] [ M ) = ~ ( i S X M) X
et en p r e n a n t i e s u p p o r t de c e t t e s e c t i o n du f e i s c e e u ~ on o b t i e n t l a
PROPOSITION 2 , 1 , - S a l t f ~ B ( M ] , a l o r s on a
supp, (sp { ] = S , S , f
DEMONSTRATION, - S o i t z ~ i S X M, on a l e s ~ q u i v a l e n c e s s u i v a n t e s :
z ~ s u p p , ( s p { ) < ~ ) i l e x i s t e un v o i s i n a g e ~ F= U x iF de z teZ qua
[ s p f ] ~ = 0
39
±i existe un voisinage "LF' = U' x i F' de z tel que
f / u ' ~ O ~ (0 ' x i r , )
{---=> z ~ S.S.f.
COROLLAIRE 2.1. - Soit f E B(MJ, alors on a
supp. sing a f = ~{S.S.f)
DEMONSTRATION. - On sait que B/CL ~ ~ ~ , or le support de f dens le faisceau
8/~ est pr6cis~ment le support singulier analytique de fet eomme d'autre part,
le support dans ~×~ est la projection par ~ de supp. spfj on en d~duit le
corollelre.
Remarque 2.1. - Bien entendu, SAT0 donne aussi une d~finition plus intrlns~que
du faiseeau ~, mais elie n6c#sslte une grosse mechlnerie d'alg~bre homologique,
C a Remarque 2,2. - Dens ie cadre des fonctions et des distributions, HORMANDER
(Fourier Integral Operater I, Acre Math. 1971J ~ introduit par anaiogie un
faisceau des singularit~s ~ partir du pr#faisceau suivent sur i S ~ M, au dessus
de i'ouvert U x ir on prend
~ ' ( u ) / { f E e ' ( U ) et WF(f)Q F = ~} .
III- FAISCEAU ~ ET VALEURS AU BORD.
Pour pr6ciser 3e lien entre le faisceau ~ et l'op~ration de veleur
au bord, on va introduire un nouveau faisceau sur le fibr~ en spheres normales
i S M (= N x ± SJ.
On d ~ f i n i t un p r~ fe i sceau sur i S Men posant pour t o u t ouve r t de la
forme U x iF C M x i S
~I(U x iF ) = iim ind ~(X ~ 8R n + iF']J
XDM
oO la limite inductive est prise pour les voisinages complexes X de U dens ~n
40
Soit~le £aisceau associ6 sur i S M, on l'appelle le faisceau des
"vaieurs au bord id~alas de fonctions holomorphes".
On d@montre que les sections de& au dessus d'un ouvert de la forme
U x iF sont donn@es par l'expression
~(O x ir ] = lim proj lim ind ~(X N 8R n + i£' ] ]
r ' c r . r x3 M
e6 la limite projective est prise pour les oSnes r' ~ base relativement com-
pacte dens F . Autrement dit, un 61~ment fC ~[U x iF ]est d6~ini pour tout
r'ccr par la denn@e d'une ~onction holomorphe f~ ~[X(~[U x it']] o0 x est
un certain volsln~ge complexe de U qui d6pend de fet de F', ce qua l'on indi-
qua par le sh@ma ci-dessous :
/ / ~ ~ ~I dessin dens l'espace des
' 0 " ~ . . . / ~ parties imaglnaires].
L'applicatlon "valeur au bord" passe la limite
~[U x ir ) __ b___) B(U) (T -I B) [U x it')
O[x (~[u x it,])
et d@{in i t un morphisme de {a isceeux sur i S N qui est encore i n j e c t i f :
0 ) & b ) T-I B.
On a alors le r@sultat fondamental suivant
THEOREME 3,1. Soit run cene ouvert convexe et r& - son dual.
Soit f~B[U), on a l'6quivalence entre a) et b)
Ii axlete ~e ~_(u x it) telle que f = b
S.S.£C U x i [ ~ ) .
41
II est utile de disposer d'une forme plus g6n@rale de ce th@or@me relative &
des ouverts Z quelconques de i S M,
Auperavant, on d~finit pour ZC i S M son ortho~oq,e,l Z ± dens i S X Men
posant
Z ~ = { [ x , i q ~ ] [ < ( , q> ~ 0 t o u t ( t e l que ( x , iC O) e Z }
-1 Une p a r t i e Z C i S M e s t d i t e convexe s i cheque f i b r e T (X] N Z e s t l a t r a c e
s u r S d ' un c~ne convexe de ~ n
On pout slots ~noncer l'important
THEORENE 3,2, - Soit Z un ouvert convexe de i S Met soit f ~ B[~Z]. On
a l'@quivmlenoe entre ~ et b) :
%
e) II existe ~6~[Z] tells que b~ = f
b) S.S. ~C Z ~.
IV - EXEMPLES.
Pour illustrer ces derniers th@or@mes, terminons par quelques exemples,
EXEMPLE 1. -
1 <x,a> + iO
Pour d6{inir d'hyperfonction f 1 <x,a>+ iO ~ B[~n] n
[o0 <x,e> = ~ xj aj] on proc~de de la fa~on suivante :
1 I la ~onction (z)
<z,a> <x ,a> + i <y ,~>
e s t ho lomorphe dens 3 ' o u v e r t ~ C sn qu i e s t d ~ i n i pa r
= { z ~ IR n + i l R n j <x ,a> # 0
<x,a> 0 et <y,a> >
et soit Z la trace sur i S M de cet ouvert conique, alors
= b~ @ BORn).
0
~[Z] et on pose
42
De plus le thief@me 3.2., donne S.S.fc Z ~, c'est & dire
1 ~c {{x,ia~B I <x,~> = o} S.S. [ <x ,a> + i0
Oonnons une g~n6r~lisation utile de cet exemple.
EXEMPLE 2. -
1 ~ [ x ) 3 iO
S o i t ~ : IR n ~ ~ , une f o n c t i o n a n a l y t i q u e r 6 e i l e t e l l e qua dO[x) ~ 0 3 ~ 9~
quand O{x ) = 0 (oO d e = ( - ~ . . . . . ~--~-)), 1
Alors on a f 1 ~ B~R n} et ~ [ x ) ~ ±o
S , S . f + C { [ x , ~ i d C [ x ) ~ ) I O [ x ) = O} .
En e f f e t , on s a i t que l a f o n c t i o n ~ admet un p ro l ongemen t ho lomorphe dens un
v o i s i n a g e complexe X deIR n, On pose
~ [ z ) = Re ~ [ z ) + i ~m OIz )
et on applique la ~ormule de Taylor entre x + iy et x :
Re O(x + i y ] = ~ [ x ] + i < y , dG[x) > + IY l ~ [ x ]
~m O[X + i y ) = IYl g [ x )
d,oO
~ [ z ] = O[x) + i < y , d G [ x ) > + ]yl ~ ( x ) .
1 Per cons6quent l a f o n c t i o n ~ [ z ] = O['z")'" e s t ho lomorphe dens l ' o u v e r t
~ {z x + i y £ ~n I O[x] ~ o ~I = = ou 0 X'
~x) = 0 et < y, d (x) > ~0
06 X' est un "petit" voisinege complexe de IR n,
Alors 1'assertion sur le support singulier de f = b+~ d~coule du th~or~me 3.2.
EXERCICE : Appliquer ceci ~ l'hyperfonction 2 2 p -m + iO
43
EXEMPLE 3.
Avec les m~mes conditions sur : ~ n IR on d~finit
- I I I _ _ ] 6(~ ) = 2 " ~ ( ~ + iO ~ - iO ~ BORn)
ce qui est naturel d'apr~s le representation de
~(t) = -1 (~ 2 I__.2__._] a B{IR). 2~i + iO t - iO
Et d'apr@s l'exemple 2, on a
S.S. 6 (~) C { ( x , ~ i d ~ ( x ] ~ ] t G ( x ) = 0 } .
RESUME DU CHAPITRE
On ales suites exactes :
i S M i SXM
M
o ) G _ - - - ~ s > ~ - - > o x
sp
b - I + - - - I o--~ o~ > T a - - + ,-->o
Soit Z un ouvert convexe de i S Met f~ B(TZ), alors
I+=b~ " Z ( - ) S . S . f C z £
L~ -~ ~
(sur M)
[sur i SxM)
[sur i S M).
CHAPITRE IV - APPLICATIONS
J. CHAZARAIN
Oans ce chapltrs, on expose la formulation hyperfonction du th@or~me
du "Edge of the wedge", puis on d@erit los op6rations usuelles sur los hyper-
#onctions (multiplication, restriction .... ] que l'on peut d@finir grace ~ des
hypoth@ses sur le spectre singulier.
I - EDGE OF THE WEDGE.
Commengons par une formulation avec deux cBnes :
THEOREME 1.1. : So±ent £I,F2 des eBnes ouverts convexes deiR n - {O} , Soient
~j ~EgR n + iFj] j = 1,2 telles que b~1 = ~2' alors il existe~@Rn÷i r I U£ 2)
tells que ~ prolongs ~I et ~2" En particulier si £1 = -£2' elors ~ est holo-
merphe dens on voislnege deIR n.
(~d~signe l'enveloppe convexe de la pattie A).
DEMONSTRATION : Posons f = b~1 = b~2~ B@Rn), le th#or@me 3.1 [chap. III)
montre que :
S.S. f C 8R n + i F 1 ) ~ ~ n + iF2) = iRn ÷ i [r 1 UF2]dono on a f ~ b ?
avec ~ e ~ RR + i r 1 UF2).
Et d ' a p r # s l ' i n j e c t i v i t @ de b, on en d@duit que / @ p r o l o n g e -- ~1 e t 2 '
45
Pour ~noncer la g~n~ralisation avec un nombre fini de cSnes,
on introduit la notion de partie propre de i S N. On dit que Z ~ i S M
est propre si Z~Z a = ~, oO Z a d~signe l'image de Z par l'application anti-
podale sur i S M :
a • [ x , i ~ ] ) [ x , - i ~ ] .
On a le
THEOREME 1.2. (Edge of the wedge). Soient des cDnes ouvarts convexes propres
Z N de i S r~- z I .... z Net soient des ~j ~ (L[ j) telles que ~ b ~j = O. j=l
Alors il existe des ~j,K E ~(Zj U Z K) telles que
N = ~ ~ pour tout j = 1 .,N
J k=l j ,k ' "
et
.~ j , k = -~k- , j tout j , K
Avant d'indiquer la d~monstration de ce th6or~me, indiquons aussi la
N PROPOSITION 1.1, - Soit f E B(M) et supposons que SISI ~ U Z. O~ les Z.
j=1 J J %
sent des ouverts Nc°nvexes propres de i S M. Alors il existe des ~j ~ (LIZ.)j
telles que f = ~ b -- ~j I
Ceci pos6, le th6or~me 1.2 et la proposition 1.1. sent (ainsi que me l'a
indiqu~ M. Kashiwara) des corollaires imm6diats du r6sultat suivent de th~orie
des faisceaux.
THEOREME 1.3. -Soit ~" un faisceau flasque sur un espace X, et soient des
ferm~s F. j = I,..,N de X. Alors on ala suite exacte de greupes J
!
o < ru F (x ,~) ( ~ T T FF.(XT-~" ~ ( B jTT K FFj~FKCX,~ ~ j J j
46
oO N
J% N
6 : ( ~ j , h ) j , T - - } (K~ I ~ j , h ].J
et H' d6signe l 'espace des [~ j ,h ) _ avec --J~,h + ~ , j = 0 j , h
Demonstration de ce th@or&me :
j._-,,O@monstration de la surjectivit@ de ~ ,
On precede par r6currence sur N, Commengons par N = 2, soit done
~ FF I U F2[X~ et construisons ~j £F FjcX,~]
talle~ qua ~o ~I ~ %"
On pose
, { ~ sur X \ F 2
0 sur X X F 1
I1 est c l a i r que c 'es t compatible sur X \ [F l~ E 2) at comme~est
~la~que, o~ pa-t prolonger ~ e~ u~e sectio~ I ~ e P F l c X , T ~ . 0~ pose eos- i te
~2 = ~ - f f l et an v~ri{ie imm@diatement que ~2 6 r ? 2 [ X , ~ ,
Dens le eas N qualconque, on se ram@ne imm@diatamant au cas N = 2
en 6crivant FIU,,.UF N = [FIU..UFN_ 1] U F Net on termine ~r~ea ~ l'hypgth~se
de r6curranee.
2 - O6monstration de l'axactitude de la suite en ~ rF.(X,~]. J J
D@signons par (B N) la propri~t@ de ~ d'etre exacte quand on a N
~erm6s, On d@montre que l'on a (BNI pour tout N en proc@dant par r6currence
sur N,
Pour N = 2, c'est imm@diat car l'hypoth@sa ~I +~2 = 0 implique
~1 =-~2 @ FZI~ z2[X'~] ce qui permet de poser~l 2 =~I eta21 =~2"
47
Montrons que (BN_I) --> (6N).
Soit donc (~j) telles que j=1 .... N
on ~ o r i t , f f l + i f2 + ' ' *~N~ : o
N d'oO supp,T1 C U (F 1 ~ F K)
k=2
et la surjectivlt@ de ~ permet d'6crire
N
~1 = k~ 2 ~ l , k avec ~1,k ~ FZ1~ Zk(X'~)
et on poae ~K,I = -T1,k"
N Introduisons ~k = -- ~K + -- ~l,k k = 2 .... N, de sorte que l'on a
2 alors l'hypoth@se de r6currence permet d'@crire
N
= 2~ erZk a CX,~) et ~ h,j + ~j h = 0 ~K ~ k,j avec ~K,j Zj j
~k = O,
j,h ,> 2.
On v@ri~ie alors finalement que les [~j,h)j,h I N
bien antisym@trlques en [j,h] et que
N
ainsl construits sont
La d6monstration de la proposition 1.1. d@coule alors de l'e×actitude
de ~ appliqu6e au ~aisceau ~(~ et la demonstration du th@or@me 1.2. d~coule,
compte tenu du th6or@me 3.2. Chap. II~ de l'exactitude de ~ appllqu@e avec
II - OPERATIONS SUR LES HYPERFONCTIONS.
Il est possible, moyennant des hypoth@ses sur le spectre singulier,
d'@tendre aux hyperfonctions certaines op6rations usuelles sur les ~onctions
(multipllcation, restriction, substitution, int6~ration).
48
a) M u l t i p l i c e t , , i o n :
THEOREME 2.2. - Soient des hyperfonctiens f, g ~ B[M) telles que
S.S.@ ~ IS.S.g) a = ~, alors on peut d@finir de fagon naturelle le produit
f . g ~ B[M) et on a
S . S . [ f , g ) C S,S.$ U S.S.g U ( S . S , f + S . S . g ] .
1 L) DEMONSTRATION : On recouvre S,S.f ~resp S.S,g) par des $erm~s Z T (resp. Z~
oO les Zj, Z~ sont des ouverts propres convexes de i S Met tels que
(Z~%) a = ~, ce qui est possible gr&ee & l'hypoth@se sur les spectres z j
singuliers. La proposition 1.1. permet d'6crire les d6eompositions
f = g b~j ~j ~ ~L[Zj)
Z'
et on pose par d6finition
$.g : ~ b ~ej.~D._ - , ,
j , k
C e l a abi~n u n ~ o ~ c a r . ~ j ' . ~ ' ~ CL(z.jn z,)~ et z . j n z , ~ ~ p u i s q u e
J
De plus, cette d6~inition montre que
s C d j , k ..L ..L 3.. ~
d'oO l'inelusion annonc6e pour S.S.(f.g] en ra{finant les recouvrements,
Bien entendu, il Saut s'assurer que cette d@finition est ind@pen-
dante do choix des ~j, ~. Soit ,j un autre ohoix de ~j meis relati@ au
m~me recouvrement z , , J
49
On a done Z. b(~j_ - ¢~j) = f - f = 0 J
et le th6or~me 1.2 permet alors d'6crire
f f j ~ j = g ~ j k avec ~ j , K ~ (L.[Zj U Z K) k '
d 'oO 3,'ZK b(~j 9~,) - j,k~: b(*j~A]
j ,nE, k b [ ~ j ,h" ~ I~)
= O, oa r ~ j , h + ~ h , j = O.
L ' i nd@pendance p a r r a p p o r t aux r e c o u v r e m e n t s se d@montre
& des r e e o u v r e m e n t s p l u s { i n s .
en p e s s e n t
EXEMPLE. P I
On peut d@finir [ I ] = b[ , ] avec les notations de ~ [ x ) + i O O . [ z ]p
l'exemple 2 [chap. III], et on a
I P S.S. ( ~ ] C { ( x , i d ~ ( × ] ~ } I • [ x ) = O} .
En revanche, {6 [~])P n'est pas d~fini pour p ~ 2 car 1' hypoth@se
du th6or~me 2,1, n'est pas satisfaite.
b] - Restriction d'une hyperfonction & une sous-vari@t6,
Soit NC IR n, une sous-vari6t@ analytique r6elle de oodimension p,
on peut l'@erire iocalament sous la forme
= = C x ) = o } N = { x l O t c × ~ , . , Op
avec des fonctions anelytiques r~elles ~. telles que les dif{@rentielles J
d~1[x],..,d~oCx] soient ind@pendantes quand x@ N. Le sous espace de TmiR nx m
engendr@ par ces formes, s'appelle l'espace conormal ~ N an x at est not@
(T~ |Rn) x. On rappelle la suite exacte :
P 0 ( ....... T ~ N ~ T ~ iR n ~ [T~ An? ( 0 [x~ N)
x × x
P Col = qI ~ ~ n I T N
×
50
et soient S m N, SXIR n, SNIRn les fibr6s en cosph~res associ@s. On peut alors
@noncer le
THEOREME 2.2. - Soit N une sous-vari@t6 analytlque r~elle deiR n d@#inie par
des 6quations ~(x) = O, j = I .... P. Soit f~B&R n] telle que J
jR n ~, alors on peut d@finir de faqon naturelle la restriction S.S. f A i S N =
N de f soit
fl ~B(N) et on a N
S ' S ' ( f l N ) C I { [ x ' i p ( n ) ~ ] J [x , iq~) ~ S .S . f et xC-N}.
Remarque 2.1. - Bien entendu, on peut donner un @nonc6 intrins@que de ce
th6orbme mels cela n6cessite quelques digressions sur les fibr6s en cosph~res
normales.
O~monstration du th@or@me. -
C'est un r@sultat local, il suffit donc de le d6montrer quand N
est un sous-espace vectoriel de lR n, dens o8 ces on a
X TNIRn = { [ x , q ) / x~N et q~ Ni}.
Soit #&BOR n ) telle que {(x,in~)~S.S.f et x~N}~>q~N ~'. On recouvre S.S . f
i . sont des o~nes ouverts convexes pro- par des ferm6s Zj = ( U j x i r .)oO ies r j pres de IR n tels que rj A N i =3~.
o~ n Soit f = ~ b ~I ~i ~d_QR + Jr j) , une d6composition associ@e de f. Commen-
gons par d~montrer le
LEMME- Soit ~e&am n + ir~ a~ee r~n N% ~.
Alors on peut d6finir de fagon naturelle
~J 6 ~[N + i[rAN)]. iSN
DEMONSTRATION. - Pour tout T'CC r , il existe, d'apr@s ia d@finition de
~Rn + ±2'], un voisinege complexe X delR net ~(z] appartenant
~X A(~n + ir')) qui represente ~. O'autre part, on v6rifie que si 2' est
assez volsln de £ , l'intersection r'n Nest un cane non r@dult & {0} car
F~N Ni= ~.
51
Par cons@quent, la restriction --'In + i [ N N r ' ) ~ q X
de ~-(N + iirN N]) qua l'on
note ~ IiSN"
d@finie un @16ment
P'n
Revenons ~ { , on pose par d~finition
f l l = Z b[~j ] ~ B{N]. N J iSN
On e done
s.s. IfiN~ c U m . + icrjn NI ~ J
o~ i' signi~ie l'orthogonal dens le sous-espace N, cat orthogonal est ~gal
{ P [ n ] l q 6 r ~ } , et il v i e n t J
S.S. ( { { ] C { [ x , i p [ q ] ~ ] ( x , i q ~ ] 6 U @ R n + i ~ , ] } N j 3
et x6N.
d'oO le th6or@me. La d@monstration de i'ind@pendance de carte d6~inition relative-
ment aux Zj et ~j se ~alt comma dens le th~or@me de multiplication.
Donnons une application de ce th6or@me @ une situation {r@quente.
Suit q E B@R n] telle qua S.S.~ ne contient aucun point de la ~orme [x',O;O,q n]
oQ x = [ x', x n] et n = [q',qn ), alors on peut d6~inir £I et on a (x = O)
n S.S. [ { I )C{ ( x ' ' i O ' ~ ) l ] ( x ' , O ; i ( q ' , q n ] ~ ] 6 S . S . f }.
× =0 n
Per exemple, on salt que l'hypoth@se sur S.S.{ est v@ri{i@e quand { v6ri{ie
une ~quation aux d@riv6es partielles
P ( x , O ] { = 0
t e l l e qua l ' h y p e r p l a n x = 0 ne s u i t pas c a r a c t # r i s t i q u e p o u r l ' o p 6 r a t e u r P, on n
p e u t donc d o n n e r un sans aux t r a c e s s u r x = 0 de { e t de sea d@r i v#es . n
52
c} Chan~ement de variabies (ou image r6ciPiiroque] pour les hyperfonctions.
Enongons sans d@monstration le
THEOREME 2.3. - Suit Nun ouvert de IR net ~ une application analytique
r@elle N > IR m. Solt f6B{IRm), alors on peut d~finir de fa~on naturelle
la compos@s
nl sous l 'hypoth@se ; {~ (x } , iS~} 6 S .S . f '.'.::~ g nj dG (x} ~ O.
j~1 J
Et on a alors
S I S I [~X$ ] ~ { { X , 1 (2 n d G . [ x } ] °°] I C O [ x ) , i q ~ ) ~ S . S . f } . J J
EXEMPLE. - Avec ~" : IRn--~--~ IR comma ~ l ' e x e m p l e 2 ( c h a p , I I I } , on
montre qua 0 × 6 = 6 ( ~ } , o
iRemarque 2.2. -Bien entendu, on peut donner une formulation intrins~que
de oe th@or~me relatif au cas oO ~est une application entre deux vari@t~s
analytiques r6elles.
d) Int6~ratlon {ou ima~e directe) pour les hlperfonctions.
Enon~ons sans d6monstration le
THEOREME 2.4. - Soient N, M des vari@t@s analytiques r@elles et ~une
submersion analytique N ) M (o'est-~-dire qua l'application tangente
~'(x) est surjective pour tout x}. On supp~ose 6galement que la restriction
de ~ au support de fest propre (supp. f) .) M. Suit f6 B(N] ~ WN
une hyperfonotion ~ valeurs dans les formes de degr6 maximum sur N, alors on
peut d@finir de fagon matureile son int@grale sur les fibres de ~ : f
J % " E B[M} ~ ~m" ((~× f) (x] = _q(x )
Et on a :
s .s . {@. f i e { {O[x ) , in~] I [x , i~O ' [x]q~]E s . s . f } ; . (
53
m oO t S ' [ x ) q s i g n i f i e dans une ca r t e de M : ~ nj d ~ ( x ) .
Remarque 2,3 . -
il faut alors rajouter ~ S.S.C~ f~ l'ensemble ~-I (X)
{(~(x),iq~) I t~'[x)n = 0 et x6supp. # } "\ ~ ~ L J
" r f - J
o - - l [ x )
3~
Oonnons un cas perticulier important de ee th@or@me, e'est lecas o,', ~9" est
une projection
On peut supprimer l'hypoth@se que~ est une submersion, mais
N = T x M ~ ( t , x ]
H ~ x
alors #6 B(N) @ ~ s'@crit #ormellement f[t,x) dt dx et on a (
[ ~ f ) (x) dx = [ ] f ( t , x ] d t ] dx T
avec
S.S, ( ~ f ) C
Terminons par un
{ [ x , i n ~ ] j ] ( t , x ; i [ O , q ) ~ ] 6 S . S . f } .
EXEMPLE. - On ala repr@sentation suivents de
ondes planes".
(n-l] ! ~ . w[~]
6° ~2~i)n / J ~ J = 1 (<x,~> + io) n
n A avec w(~) = j ~ l [ -1] j ~j d~l A . . . A d~j A . . . A d~ n
et o~ 1'int@grale signi#ie l'int@grale de l'hyper~onction
w(~) 6 BQR n x S ~) sur la sph@re J~I = 1.
(<x,~> + i0) n
Cette repr@sentation de 6 joue un r@le important dans la d@finition des op@ra-
teurs pseudo-di##@rentiels.
6 B@R n] dire "repr@sentation en o
PSEUDO-DIFFERENTIAL OPERATORS ACTING
ON THE SHEAF OF ~ICROFUNCTIONS
Takahiro KAWAI
Research Institute for Mathematical Sciences,
Kyoto University
and
D~partement de Math&matiques, Nice
This report is intended to give an intuitive explanation of
the theory of pseudo-differential operators in hyperfunction
theory. The reader is refer~ed to Sato-Kawai-Kashiwara [i] for
further details of the theory and its applications. Hereafter
Sato-Kawai-Kashiwara [I] will be quoted as S-K-K for short. In
this exposition the present speaker wants to lay great emphasis
on the following two points.
I. The pseudo-differential operator which we want to
manipulate is of infinite order.
II. The pseudo-differential operator acts on the sheaf of
microfunctions as a sheaf homomorphism.
55
The present speaker hopes that the importance of these
properties will be recognized clearly by this exposition.
To begin with, we recall the definition of linear differential
operators in hyperfunction theory.
Intuitively speaking, a linear differential operator (with
real analytic coefficients) on a real analytic manifold M is a
special kind of integral operators whose kernel functions have
their support in the diagonal set AM={(x,y )EM×M; x=y}. (See S-K-
K Chapter I §2.1 for the precise definition.) Here we use hyper-
functions as kernel functions. This fact corresponds to the fact
that there apper linear differential operators of infinite order.
One of the typical examples of such operators is given by
~o i dn cosh d = X ~ (~~)
n=O
One of the essential points in our argument is the following: in
order that an infinite sum of differential operators of finite
order should make sense as a differential operator of infinite
order thus defined, very servere condition should be posed.
For example,
ex ~ d ir d~n
n=0 •
never gives rise to a differential operator of infinite order.
In fact the kernel function corresponding to exp(~) is ~(x-y-l)dy,
whose support is not contained in A£={x=y}. The precise condition
will be clarified later (see condition (3)) and here we only
mention one example which shows the advantage of employing pseudo-
56
differential operators of infinite order.
We have the following relation:
Y
l --~) -1 ~x 1
x
~x 2 ~x I
~x 1
0
/ cosh(xl~) ~j o" 2
0) ~x I
?) 3~2 sinh (Xl )
c°sh(Xl
Here 1
Clearly
2n+l = n~=0 (2n+l)' ( )n and so on.
I cosh(_Xl ~2 )
sinh(-XlJ~2x2-)
1 sinh (_Xl
-x 3 cosh( l~xx 2 ) I
o .f
~
" ~
i
t~
0
×
0 "?,
? 0 C~
I -~
IZV
"7
,-7
X
0
0
i ~ .
J
x
0 111
X 0
J
58
are transformed each other by an inner automorphism. Especially
t h i s f a c t t e l l s us t h a t t he d i f f e r e n t i a l e q u a t i o n ( z- )u = 0
22 ~Xl is equivalent to the equation 7-u = 0. Such
~x
a transparent result can be obtained only after the introduction
of differential operators of infinite order, The enlarged version of
this fact plays its essential role in discussing the theory of
general system of pseudo-differential equations. (See S~K-K Chapter
II §5.3 and Chapter III ~2.) Here we would like to call the reader's
attention to the fact that the operators cosh(x 1
- - sinh( 1 ) etc. give rise to a sheaf homomorphism
;x 2
b e t w e e n s h e a v e s of h y p e r f u n c t i o n s by t h e i r d e f i n i t i o n . T h i s i s a
c r u c i a l p o i n t i n ou r a r g u m e n t b e c a u s e we want t o a n a l y z e t he
s t r u c t u r e of ( s y s t e m s o f ) p s e u d o - d i f f e r e n t i a l e q u a t i o n s ZoeaZZy.
(See S-K-K C h a p t e r I I §5 and C h a p t e r I I I §2 . )
Now a l i n e a r d i f f e r e n t i a l o p e r a t o r a c t s on t h e s h e a f ~ M o f
h y p e r f u n c t i o n s and t h e s h e a f ~ g of r e a l a n a l y t i c f u n c t i o n s as
a s h e a f homomorphism by t h e d e f i n i t i o n , he nc e i t a c t s on t he
q u o t i e n t s h e a f d~ M/(~,g as a s h e a f homomorphism. On t h e o t h e r
hand we know t h a t ~ M/~M --~ "~"~M h o l d s . Here C M d e n o t e s t h e
s h e a f o f m i c r o f u n c t i o n s and ~ d e n o t e s t h e c a n o n i c a l p r o j e c t i o n
from /=T S*M t o M. (See t h e e x p o s i t i o n o f C h a z a r a i n o f t h i s
i s s u e and S-K-K C h a p t e r I . ) Hence i t i s n a t u r a l t o ask w h e t h e r
a l i n e a r d i f f e r e n t i a l o p e r a t o r a c t s t h e s h e a f o f m i c r o f u n c t i o n s
as a s h e a f homomorphism or n o t . I f one r e c a l l s t h e d e f i n i t i o n
of t h e s h e a f of m i c r o f u n c t i o n s , one w i l l e a s i l y f i n d t h a t t h e
answer is affirmative.
59
Now, we know the following theorem due to Sato [i].
Theorem. Let P(X,Dx) be a linear differential operator of
finite order m defined on a real analytic manifold M. Then
P(x, Dx) gives rise to a sheaf isomorphism between sheaves of
microfunctions outside its real characteristic variety V={(x,i~ ~)E
~=I- S~M; Pm(X,in)=0}. Here Pm denotes the principal symbol of
P(x, Dx). Recall that the principal symbol of a linear differential m
operator (of finite order m) P(X,Dx)=j~ 0 pj(x,Dx) (pj is
homogeneous of order j with respect to Dx) is by definition the
function obtained from Pm(X,Dx) by substituting the cotangent
vector n for D x. It is well-known that the principal symbol
has an intrinsic meaning independent of the choice of local
coordinate system. Note that it is also well-defined on ~ S*M
since it is homogeneous with respect to n •
In view of this result, it is natural to seek for a class of
operators which includes the inverse of a differential operator
defined outside its characteristic variety and whose element acts
on the sheaf of microfunctions as a sheaf homomorphism. Such an
object should be defined (locally) on q~ S*M by the above
requirements. In order to find such a suitable class we use the
well-known decomposition of n-dimensional 6-function into plane
waves (after fixing a local coordinate system on M ), i.e.
~(x-y) : (-2~i) n (<x-y,n>+iO) n Inl:l
Now let P(X,Dx) be a linear differential operator of finite
60
m
order m, i.e. P(X,Dx) = ~ pj(x,Dx), where pj(x,Dx) j=0
homogeneous of order j. Then we clearly have
is
m pj (x, ~) (i) P(X,Dx) 6(x-y) = ~ (n+j-l): I ~(n)
j=0 (-I) n+j(2~i) n (<x-y,q>+i0) n+3
Taking account of these formula we will try to find the
inverse of P(X,Dx) assuming its principal symbol Pm(X,n)~0
near (x0,~0), that is, we want to find a class of "differential
d operator of negative order". Then the relation ~-~ log (x+i0) =
1 x-g-i-0- suggests us that we should introduce some logarithmic factor
as a kernel function. For this purpose we introduce a family of
-I ~ Clearly auxiliary functions ~X(z) defined by ~ (_T)~+ I .
~x(<x-y,~>+i0) = (-l)~X! 2~i(<x_y,~>+i0)X+l if l is a positive integer
and this is just the function used in (i). Moreover if l is a
negative integer ¢i(~) stands for
-~-i i i -~-i " g " ~ i ( X - 1 ) ' T ( l o g ( - ~ ) - ( ~: G - Y ) } "
~---1
(Here y denotes the Euler constant.) Using ~X(~), we consider
the following hyperfunction K in (x,y,n).
K = . ~ pj(x,~) ~j(<x-y,~> + i0),
where pj(x,~) satisfies the following:
(2) pj(x,n) is holomorphic in a complex neighborhood U of
(x0,~ 0) and homogeneous of degree j with respect to n.
61
(3) jlim'+~ j~fs~p [pj(x,n)[ = 0
(4) lim - ~ -J~ sup IPj (x,q)[ < ~. j÷-~ ] V
Since ~ pj(x,n)¢j(<x-y,q>) converges in W = {(x,y,n);
Ix-x01,[y-y0[,[q-~0[<<l, Im<x-y,q>>0} by the above conditions,
the hyperfunction K is well-defined as a boundary value of this
holomorphic function. The above conditions (3) and (4) are
natural ones of the sort so that the series ~pj(x,q)¢j(<x--y,q>)
converges in W.
It is easily verified that the microfunction defined by
(5) fK(x,y,n)m(q)dy In-n°[<<l
Inl=m has i t s suppo r t i n A a = { ( x , y ; i ( n , q ' ) ~ ) ; x=y , n=-~ ' } i n a n e i g h b o r -
hood of (x 0, y0., i(n0,-q0)~) and gives rise to a sheaf homomor-
phism of ~ M near (x 0, iq0~). (These facts follow from the
theorems concerning such operations on microfunctions as integration
and multiplication. See S-K-K Chapter I ~2.)
The integral operator whose kernel (micro-)function is given
as above will be called a pseudo-differential operator. Clearly
it is a natural generalization of the notion of linear differential
operators so that it should be suitable for the (local) analysis
of microfunctions. Moreover Theorem 1 given below will tell us
that the inverse of a linear differential operator P(X,Dx) of
finite order is a pseudo-differential operator as long as its
62
principal symbol does not vanish. Thus far we have discussed pseudo-
differntial operators by fixing the local coordinate system.
However, it is not difficult to verify that they have an intrinsic
meaning. (See S-K-K Chapter II 51.) In fact they constitute a
sheaf of rings on VrT~ S*M. Their composition rule can be expressed
concretely in the following fashion.
When the kernel function of P(X,Dx) has the form (5), we
sometimes denote it by J~- pj (X,Dx). Using this convention, the
composite R(X,Dx)= ~ r~(x,Dx) of two pseudo-differential ~=-~
Z p (x,D x) and Q(x,D x) = ~ qj(x,D x) operators P(X,Dx) j=_~ J j _~
can be calcul~ed by the following rule:
r~(x,n) = X z=j+ki~1 l a : D ~ pj(x,n) D~ n x qk (x'n)
(D~ = al a n " x al 3hi "''~n SXl "" ~x~ n
).
As in the case of differential operators, the principal
symbol Pm(X,n) of a pseudo-differential operator of finite order
m P(X,Dx) = .~ pj(x,Dx) is well-defined on the cotangent bandle.
Since the verification of the growth order conditions (3)
and (4) on {pj(x,~)}j=_~ is not so easy in general, we introduce
a formal norm Nm(P;t ) of a pseudo-differential operator P of
finite order m after Boutet de Monvel and Kr~e [i] as follows.
63
N ~ ( P ; t ) = k>O
a ,~0
2 ( 2 n ) _ k k : a B , t 2k+la+t31 (..Ic~l+k):(igl+k.)~) sup ID~D~P~_I.(x n) I
This is a formal power series in the auxiliary variable t
with non-negative coefficients. It is easily verified by Cauch's
formula that N£(P;t)<~ for 0<t<~ is equivalent to say that
estimate (4) holds. The most important property that Nz(P;t)
enjoys is the following
Proposition (Boutet de Monvel and Kree [i]) Let P9(x,D x)
be a pseudo-differential operator of finite order £ . (~=1,2).
Then we have N~I(PI; t) N~2(P2;t)>>N£1+~2(PIp2;t), i~e.
N£I(PI;t)Nz2(P2;t) is a majorant series of NZl +22(PIP2; t).
Using the formal norm of pseudo-differential operator of
finite order we can prove the following important theorem.
Theorem i. Let P(X,Dx) be a pseudo-differential operator
of finite order m. Assume that its principal symbol Pm(X, i~)
never vanishes on tiC/r1- S*M. Then there exists a unique
pseudo-differential operator E(X,Dx) of finite order defined on
which satisfies
PE -- EP = I.
Proof. Once we have proved the existence of left and right
inverse of P locally in ~, the assertion of the theorem can be
easily verified by using the associative law for composition of
pseudo-differential operators. Now we will show the local existence
64
of left inverse. Firstly we define a pseudo-differential operator
Q_m(X,Dx) , homogeneous of order -m, by defining Q_m(X,~) =
i/Pm(X,n ). Clearly R(X,Dx) = I-Q_mP is a pseudo-differential
operator of order at most (-I). Then one can easily see by using
the formal norm of pseudo-differential operators that S= R z
realy makes sense locally as a pseudo-differential operator of
finite order. The pseudo-differential operator S(X,Dx)Q_m(X,D x)
defines a left inverse of P(X,Dx) locally, since I = S(I-R) =
SQ_mP holds by the definition,
Theorem 1 implies that we have to investigate the behavior
of the microfunction solutions of the pseudo-differential equation
of finite order P(X,Dx)U=0 only on the characteristic variety
V={(x,i~)E~S*M; Pm(X,i~)=0}, since outside V P(x,D x) gives
rise to a sheaf isomorphism.
In passing the characteristic variety V enjoys many
geometrically interesting properties as a subvariety of ~ S'M,
which is a contact manifold. For example, V will be generated
by the bicharacteristic strips attached to Pm(X,i~) if V is
real and regular. Hence here comes the following question
naturally:
To what extent can the "geometry" ~ the characteristic
variety --control the "analysis" the structure of micro-
function solutions -- ? In other words, to what extent can the
commutative objects control the non-commutative objects?
The situation is surprisingly plain: Generically, i.e. at
the regular point of V, V completely determines the local structure
65
of the pseudo-differential equation, ~ fortiori, the local structure
of microfunction solutions. This is also the case for the general
overdetermined systems under some moderate algebraic condition on
the system (i.e., for the purely dimensional systems). (See S-K-K
Chapter II Definition 5.3.6 as for the definition of purely
dimensional systems. Note that any system with one unknown is
purely dimensional.)
This is the main result obtained in S-K-K Chapter II and
Chapter III. See Theorem 2.4.1 in Chapter III and Theorem 5.3.7
in Chapter II especially.
The proof of the above statement is very long and needs a
great deal of preparation. However, its basic idea is straight-
forward and we present it here rather schematically. See S-K-K
for details.
The proof will be divided into two steps.
The first one deals with the geometrical investigation of
the characteristic variety V, which is a subvariety of pure
imaginary contact manifold 4r-J7 S~M. Here the important point is
the following observation originally due to Maslov [i] and Egorov
[i] (see also Hormander [i] and S-K-K Chapter II ~3.3 and §4.3):
We can associate a transformation of pseudo-differential operators
to any contact transformation so that the principal symbol of the
operators are transformed according to the contact transformation.
(For the precise statement, see S-K-K Chapter II Theorem 3.3.3 and
/or Theorem 4.3.1 for example.)
Therefore obtaining the standard form of V under the (real)
contact transformation reduces the problem to the case where the
66
principal symbol of the pseudo-differential operator takes such a
simple form as either HI, nl+i~2 or ~l~iXl~2. (In the case of
over-determined system, their combination appears.) This topic
is discussed in S-K-K Chapter III §2. See also Sato, Kawai and
Kashiwara [2].
Here it is essential that we consider all problems locally
on /zT S'M, not locally on M. Here appears one of the greatest
advantages of employing the notion of microfunctions.
The next step is to deal with the lower order terms once we
have obtained the standard form of the principal symbol, llere the
use of pseudo-differential operators of infinite order is very
crucial as is easily seen from the example discussed at the begin-
ning of this talk. In fact we cannot transform the equation
~2 ~ ~2 ( I~ ~x 2) u = 0 to the equation ~ Ul = 0 if we ~x ~x
use only pseudo-differential operator of finite order. The fact
that we need not pay attention to the lower order terms at the
generic point of V can be established only by the aid of pseudo-
differential operators of infinite order. This is the result
established in S-K-K Chapter II §5. Note that we have used
pseudo-differential operators of infinite order only as an auxiliary
tool in transforming the complicated equation to a simpler one.
Plainly speaking, the theory of pseudo-differential equations of
infinite order is quite beyound us.
This is the main idea in establishing the desired result:
"Commutative object" completely determines the "non-commutative
object" at the regular point of the characteristic variety V.
67
Thus far we have mainly discussed the pseudo-differential
operators from the view point that they act on the sheaf of
microfunctions as sheaf homomorphisms. However, the notion of
pseudo-differential operators itself is not inherent to the real
manifold. In fact it can be defined on the complex manifold,
though the sheaf of microfunctions cannot be defined there. Hence
we can talk about the structure of pseudo-differential equations
even in the complex domain. This is the topic discussed in S-K-K
Chapter II.
In order to make this point clearer we call the reader's
attention to the following fact: Pseudo-differential operators
cannot exhaust all integral operators that give rise to sheaf
homomorphisms on the sheaf of microfunctions. In fact any operator
a {(x,x' ; whose kernel (micro-) function has its support in AM=
i(n,n')~)C ~-r S*(M×M); x=x ', ~=-n'} does enjoy such a property.
Of course the class of all integral operators that give rise to
sheaf homomorphisms is too wide and is not suitable for concrete
calculation, which is possible in the case of pseudo-differential
operators. However, the employment of such operators -- called
micro-local operators in S-K-K -- is still very important in
some points. In fact, the celebrated counter-example of Lewy [I]
is best iHustrated by employing micro-local operators and it is
also crucial in investigating the structure of microfunction
solutions of general (system of) pseudo-differential equations.
See S-K-K Chapter III ~2.3 and §2.4 for details. Here we explain
only the Lewy phenomena briefly.
68
Let P be the Lewy operator, i.e., 8zP iz ~t~ - ~(FI ~ +i ~)
~ +iz ~ Here (z t) denotes a - i(x+iy) and let Q be P = ~z 3-T "
point in £ × ~ ~3. Let us define the following kernel function
K by
1 2
IT (t-t' + i(IzI2-1z'12-Zzz ') + i0) 2
It is easily verified that the support of K as a microfunction
in (z,z',t,t')~3 × ~3 is contained in &a R3 •
If we denote by ~ the micro-local operator defined by the
kernel function K, then we have the following exact sequence
on ~ = {(z,t;i(~,~)~) ~ ~ S~ 3" ~ >0} ,
P 0---~ Q C ~ ~ '~ 70
This exact sequence tells us that the Lewy operator P is
surjective on ~, while its complex conjugate Q is injective
on ~. Moreover the range of Q and the null-space of P are
characterized by the micro-local operator K. Thus we have
found the concrete obstruction against the solvability of the
equation Qu = f. It is also easily seen in the same way that
we can find the analogous exact sequence in ~-= { (z,t;i~ ,z~);
< 0 } by replacing the roles of P and Q, that is, P is
injective but not surjective there and that Q is surjective but
not injective there. These observations have given the lucid
explanation for the Lewy phenoma and at the same time they show
69
clearly the advantage of employing the theory of microfunctions in
the theory of linear differential equations.
References
Boutet de Monuel, L. and P. Kr6e: [i] Pseudo-differential operators
and Gevrey classes, Ann. Inst. Fourier, i__~7(1967) 29S-323.
Egorov, Yu. V.: [i] On canonical transformations of pseudo-
differential operators, Uspehi Mat. Nauk, 2_~5(1969) 235-236.
H6rmander, L.: [i] Fourier integral operators, I, Acta Math.
127(1971) 79-183.
Lewy, H.: [I] An example of a smooth linear partial differential
equation without solution, Ann. of Math. 66(1957) 155-158.
Maslov, V.: [I] Theory of Perturbation and Asymptotic Method,
Moscow State Univ. 1965(Russian, also translated into French
by Lascoux and Seneor (Dunod-Ga thier-Villars, 1972).
Sato, M: [I] Hyperfunctions and partial differential equations,
Proc. Intern. Conf. on Functional Analysis and Related Topics,
Univ. of Tokyo Press, 1969, pp.91-94.
Sato, M., T. Kawai and M. Kashiwara: [I] Microfunctions and
pseudo-differential equations, Proc. Katata Conf., Lecture
Notes in Math. No.287, Springer, 1973, pp.263-529.
: [2] On the structure of single linear pseudo-
differential equations, Proc. Japan Acad. 48(1972) 643-646.
MICR0-HYPERBOLIC PSEUDO-DIFFEReNTIAL OPERATORS
Masaki KASHIWARA and Takahiro KAWAI
Research Institute for Mathematical Sciences,
Kyoto University
and
D@par~ement de'~ath@matiques, Nice
Study of (fundamental solutions of) hyperbolic differen-
tial equations has a long history. See for example Riemann [I],
Hadamard [I], Courant-Hilbert [i] and references cited in
Courant-Hilbert [i], [2]. In such a long history the works of
Petrowsky [i] and G~r~ng [I] are clearly outstanding milestones
from the view point of the general theory of differential
equations because of the generality of their results. Leray [I]
has also influenced much the later development of the theory of
hyperbolic differential equations by establishing the existence
and uniqueness theorems of the solutions in an elegant and
far-reaching way. See also Friedrichs-Lewy [i] and Friedrichs
[i]. On the other hand HBrmander [i], [2] gave good existence
and uniqueness theorems for real operators of principal type
along these lines. One of the reasons for the success of
HSrmander seems to us to be the fact that such operators are
71
micro-hyperbolic. The notion of micro-hyperbolicity was intro-
duced in Kashiwara-Kawai [i] in full generality. Hereafter
Kashiwara-Kawai [i] will be often referred to as K-K for short.
In the case of linear differential operators with constant
coefficients the same notion was introduced by Andersson [I]
influenced by the work of Atiyah-Bott-G~rding [I]. (See Kawai
[2] and G~rding [2].) In most of the above quoted papers the
fundamental solution plays its essential role. As for the con-
struction and investigation of the fundamental solutions for
hyperbolic operators or real operators of principal type, we
refer to Courant-Lax [i], Lax [i], Leray [I], [2], Ludwig [i],
Mizohata [I], HSrmander [3], Kawai [I] and Duistermaat-HSrmander
[I]. Note that these works assume the regularity of character-
istic variety of the operator. In order to treat the operators
whose characteristic variety is not simple, the employment of
hyperfunctions is very crucial as is shown by Bony-Schapira [I],
[2]. (See Mizohata [2], where a necessary condition for hyper-
bolicity is discussed. See also Leray-Ohya [l],Mizohata-Ohya
[I], Chazarain [i] and the references cited there about the
operators with constant multiple characteristics.)
The purpose of this report is to explain in a sketchy way
the idea of Kashiwara-Kawai [I], whose results cover all the
above quoted (local) existence and uniqueness theorems as long
as the operators under consideration have analytic coefficients.
Note that the theory developed in Kashiwara-Kawai [I] is also
related to Egorov [i], [2], Nirenberg-Treves [i] and rreves [2].
72
(Theorem 3 in the below.)
The topics of this report are completely restricted to the
existence and uniqueness of the (fundamental) solutions of linear
(pseudo-) differential equations and other topics of hyperbolic
equations are not discussed here, though some of them are im-
portant and also expected to be closely related to the topics
discussed here, e.g. hyperbolic mixed problems. (As for such
problems we refer to the exposition of Chazarain [2] for example.)
Now we will sketch the idea of the proof of the existence
of fundamental solutions for partially micro-hyperbolic pseudo-
differential operator P(X,Dx).
A pseudo-differential operator P(X,Dx) is said to be
partially micro-hyperbolic at (x 0, i~0)~ /:TS~M with respect
to the direction <~,dx>+<p,d~> if Pm(X+i~p, i~+~p)~0 for
every (x,~) sufficiently close to (x 0, t 0) and for 0<E<<I.
(See K-K §I for the precise definition. See G~rding [2], [3]
also.) Then using the "quantized" contact transformation (Sato-
Kawai-Kashiwara [i] Chapter II ~3.3) we can easily reduce the
problem to the case where P has a matrix form D -A(x,D') so Xl
that A is a square matrix of pseudo-differential operators of
order ~ 1 which commute with x I and that all eigenvalues of
its principal symbol Al(X, i~') have non-negative real part.
(The above quoted report of Sato, Kawai and Kashiwara will be
referred to S-K-K [i] hereafter.)
The first step in our arguments is to construct formally a
solution a (x)D 'e as an infinite sum of pseudo- R(x,D')=E
73
differential operators so that PR=0 and that RIxI=0=I. This
part of the proof is not difficult to perform. In fact we need
not use the assumption of partial micro-hyperbolicity of P at
this stage. Cf. Treves [I]. What we need here is that {Xl=0}
is non-characteristic with respect to P. (See Propos~ion2.2 ~K-K
§2.) The essential difficulty comes in at the next step. Gener-
ally the existence domain of R is so small that R cannot be
endowed with the meaning as a kernel microfunction of a funda-
mental solution of P. In order to overcome this difficulty
we rewrite the equation PR=0 by using the "defining function"
of P and R. (Lemma 4.1 in K-K §4.) Then we try to extend
the domain of definition of the defining function G of R by
the partial micro-hyperbolicity of P. In extending the domain
of definition the following Lemma 1 is crucial. Note that the
partial micro-hyperbolicity of P=DxI-A(x,D' ) at (Xl,X'; i(~l,
0' 0' ~')~)=(0,x ; i(~l,~ )~)=(0~ i(~l,0 ..... 0,i)~) for every real
gl with respect to the direction x I implies that
g(xl,z',~l,~') = det (~i - Al(Xl'Z''~'))
never vanishes on
{ ( X l , Z , , ¢ l , ¢ , ) r : ~ " × £n-1 x £ x £n ; 0 <Xl <__ 6,
Iz,I < ~, I(~ 2 .... ,¢n_l) l=l{"l < ~lCnl, -Im(¢i/¢ n)
n-I > M(]y[ + y~ [Im(cw/<n) l. }
~=2
74
Here the assumption of the partial micro-hyperbolicity of P
for every real ~I is not restrictive in application because
we can easily localize the problem with respect to ~i by the
preparation theorem of Weierstrass for pseudo-differential
operators (S-K-K [i] Chapter II §2.2.) See the arguments in
the proof of Theorem 5.2 in K-K §5 for details.
After the above observation concerning the implication of
partial micro- hyperbolicity, we state the following lemma.
Lemmal. Let ~(Xl,Z',Z' ) be a positive valued real ana-
lytic function defined on U={(Xl,Z')=(Xl,X'+iy' ) ; 0<Xl<61,
[x'I<62, ]y'l<63 with 622+632<612}. Assume that ~ satis-
fies the following:
(1)
(2) n-I 6
I~-~I< ~ on u.
Suppose that G can be extended to V={z; 0<Xl<61,
Iz'l<62, yn>~(xl,z',~')}. Then G can be extended to a holo-
morphic function defined on an open set V' which contains
{z; 0 < x I < 61, Iz'] < 62, Yn >= ~(Xl'Z''7')}"
Once we have proved this lemma, we can easily prove the
following Theorem 2 by a suitable choice of ~, while the proof
75
of the above lemma is reduced to the invertibility of elliptic
pseudo-differential operator. (See S-K-K [I] Chapter II §2.1
Theroem 2.1.1. See also the exposition of Kawai of this issue.)
Theorem 2. There exist 60>0 and M 1 such that G(Xl,Z')
is holomorphic
{[Xl,Z')¢ R×cn; 0 < x I < 60, Iz'L < 60 ,
n - 1 > [ IIm )} Im z n MlX 1 ( z [
~=2
As for the details of the proof of Lemma 1 and Theorem 2 we refer
to K-K ~4.
Now Theorem 2 a l l o w s us to d e f i n e t he b o u n d a r y v a l u e w(x) +
of a h y p e r f u n c t i o n G ( x ! , z ' ) = Y ( x ~ G ( ~ , z ' ) w i t h h o l o m o r p h i c p a r a -
m e t e r s z ' d e f i n e d on
n-I , ' > M ] x l ] ( ~ Jim z ] ) ] . {(x I z ) ; Ix 1 [ , [ z ' [ < 6, ]m z n
x~=2
(See S-K-K Chapter I §3.2 about the notion of taking the boundary
value of hyperfunctions with holomorphic parameters.)
It is readily verified that the singular spectrum u(x) of
w(x) satisfies Pu=6(x) and that Supp uC{(x; i~); Xl~0,
l~l!MXlI~nl (~=2 ..... n-y, IXn[~VXl}. Using this fact one can
easily show the existence of fundamental solution of the partia-
lly micro-hyperbolic pseudo-differential operator P. (See
Theorem 5.2 in K-K 55.)
78
Once one gets a fundamental solution, it is easy to show
the existence or (propagation of) regularity of solutions. The
results are listed up in ~6 of K-K and we omit the details here.
However, we would like to touch the following theorem without
proof. This theorem will show why we have treated the partially
micro-hyperbolic operators, not the micro-hyperbolic operators.
In fact, Dxl+iXl2kDx2 (Mizohata [3]), the easiest and most
typical example that can be covered by Theorem 3, is not micro-
hyperbolic, though it is partially micro-hyperbolic. (See Sato-
Kawai-Kashiwara [2] also.)
Theorem 3. Assume that the real characteristic variety V
of P(X,Dx) is defined by a(x,q)+ -~b(x,q)=0 where (/~-l)-ma(x,
vrl--l~) and (/~--~)-mb(x,~TT~) are real for (x,~n) in /~-TS*M
near x0*=(x0, -~0) and that grad(x,q)a(x,~ ) and ~ are
linear~independent there. (Here m denote the degree of a
and b with respect to q.) Assume further that ( -~)-mb(x,
/Z-ln) is positive (or negative) on each real bicharacteristic
strip of (~)-ma(x,~---Tn) and not identically zero there.
Then P(X,Dx) has an inverse in the ring of micro-local opera-
tors.
We refer to S-K-K [i] Chapter I ~2.5 and the exposition of
Kawai of this issue about the notion of micro-local operators.
Note that the above theorem implies not only the micro-local
solvability of the equation Pu=f but also the "micro-local"
analytic-hypoellipticity of P. We also note that a more general
result is given in K-K. (Theorem 6.6 in §6.)
77
At the end of this exposition the speakers would like to
lay stress on the following point as a summary:
The employment of hyperfunctions and microfunctions has
made the theory of linear hyperbolic differential equations
very lucid and thrown the light to the nature of a class of
hypoelliptic operators from the view-point of "hyperbolicity."
The essential idea in showing these is "taking the boundary
value of pseudo-differential operators defined in the complex
domain." In fact P(X,Dx) is invertible when Pm(X,n)~0
and the partial micro-hyperbolicity of P(X,Dx) means the
invertibility of P on a conical set which is tangent to the
real axis. Therefore what we have done may be summarized as
a justification of the procedure of "taking the boundary value
of pseudo-differential operators."
78
References
Andersson, K. G.: [i] Propagation of analyticity of solutions
of partial differential equations with constant coefficients,
Ark. Mat., 8 (1971) 277-302.
Atiyah, M. F., R. Bott and L. G~rding: [i] Lacunas for hyper-
bolic differential operators with constant coefficients
I., Acta Math., 124 (1970) 109-189.
Bony, J. M. et P. Schapira: [i] Probl~me de Cauchy, existence
et prolongement pour les hyperfonctions solutions d'@quations
hyperboliques non strictes, C. R. Acad. Sci. Paris, 274
(1972) 188-191.
: [2] Solutions hyperfonctions du probl~me de Cauchy,
Hyperfunctions and Pseudo-differential Equations, Lecture
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New York, 1973, pp. 82-98.
Chazarain, J.: [I] Op6rateurs hyperboliques ~ caracteristiques
de multiplicit~ constante, to appear.
: [2] An article which will appear in S~m. Bourbaki (1972/
1973).
Courant, R. u. D. Hilbert: [I] Methoden der Mathematischen
Physik, II, Springer, Berlin, 1937.
: [2] Methods of Mathematical Physics, II, Interscience,
New York, 1962.
Courant, R. and P. D. Lax: [I] The propagation of discontinuities
79
in wave motion, Proc. Nat. Acad. Sci. U.S.A., 42 (1956)
872-876.
Duistermaat, J. and L. Hormander: [I] Fourier integral operators
II, Acta Math., 128 (1971) 183-269.
Egorov, Yu. V.: [i] Conditions for the solvability of pseudo-
differential operators, Dokl. Akad. Nauk USSR, 187 (1969)
1232-1234. (In Russian.)
: [2] On subelliptic pseudo-differential operators, Dokl.
Akad. Nauk USSR, 188 (1969) 20-22. (In Russian.)
Friedrichs, K. O.: [I] Symmetric hyperbolic system of linear
differential equations, Comm. Pure Appl. Math., 7 (19S4)
345-392.
Friedrichs, K. O. u. H. Lewy: [i] Uber die Eindeutigkeit und
das Abhangigkeitsgebiet der L~sungen beim Anfangsproblem
linearer hyperbolischer Differentialgleichungen, Math.
Ann., 98 (1928) 177o195. o
Garding, L.: [i] Linear hyperbolic partial differential equations
with constant coefficients, Acta Math., 85 (1950) 1-62.
: [2] Local hyperbolicity, Israel J. Math., 13 (1972) 65-81.
: [3] A note which will appear in Israel J. Math. as a
supplement to GSrding [2].
Hadamard, J.: [I] Lectures on Cauchy Problem in Linear Partial
Differential Equations, Yale Univ. Press, 1923. Reprinted
by Dover, New York.
HSrmander, L.: [I] On the theory of general partial differential
80
operators, Acta Math., 94 (1955) 161-184.
: [2] Linear Partial Differential Operators, Springer,
Berlin-Heidelberg-New York, 1963.
: [3] On the singularities of solutions of partial
differential equations, Proc. Int. Conf. Functional
Analysis and Related Topics, Univ. of Tokyo Press, Tokyo,
1970, pp.31-40.
Kashiwara, M. and T. Kawai: [I] Micro-hyperbolic pseudo-
differential operators I, to appear.
Kawai, T.: [I] Construction of local elementary solutions for
linear partial differential operators with real analytic
coefficients (1)-- The case with real principal symbols
--, Publ. RIMS, Kyoto Univ., 7 (1971) 363-397.
: [2] On the global existence of real analytic solutions
of linear differential equations (I), J. Math. Soc. Japan,
24 (1972) 481-517.
Lax, P. D.: [I] Asymptotic solutions of oscillatory initial
value problems, Duke Math. J., 24 (1957) 627-646.
Leray, J.: [I] Hyperbolic Differential Equations, The Institute
for Advanced Study, Princeton, 1952.
: [2] Un prolongement de la transformation de Laplace qui
transforme la solution unitaire d'un eperateur hyperbolique
en sa solution 61ementaire, Bull. Soc. Math. Fr., 90
(1962) 39-156.
Leray, J. and Y. Ohya: [i] Equations et syst~mes non-lin~aires,
81
hyperboliques non-stricts, Math. Ann., 170 (1967) 167-205.
Ludwig, D.: [i] Exact and asymptotic solutions of the Cauchy
problem, Comm. Pure Appl. Math., 13 (1960) 473-508.
Mizohata, S.: [I] Analyticity of solutions of hyperbolic
systems with analytic coefficients, Comm. Pure Appl. Math.,
14 (1961) 547-559.
: [2] Some remarks on the Cauchy problem, J. Math. Kyoto
Univ., 1 (1961) 109-127.
: [3] Solutions nulles et solutions non-analytiques, ibid.
271-302.
Mizohata, S. and Y. Ohya: [I] Sur la condition d'hyperbolicit~
pour les ~quations a characteristiques multiples, II,
Japanese J. Math., 40 (1971) 63-104.
Nirenberg, L. and F. Treves: [I] On local solvability of
linear partial differential equations -- Part II.
Sufficient conditions, Comm. Pure Appl. Math., 23 (1970)
459-510.
Petrowsky, I. G.: [I] Uber das Cauchysche Problem f~r Systeme
yon partiellen Differentialgleichungen, Mat. Sbornik,
44 (1937), 815-868.
Riemann, B.: [i] Uber die Fortpflanzung ebener Luftwellen
von endlicher Schwingusweite, Mathematische Werke,
Dover, New York, 1953, pp. 156-175.
Sato, M., T. Kawai and M. Kashiwara: [I] (Refered to as S-K-K[1])
Microfunctions and pseudo-differential equations,
82
Hyperfunctions and Pseudo-differential Equations, Lecture
Notes in Mathematics. No. 287, Springer, Berlin-Heidelberg-
New York, 1973, pp. 265-529.
: [2] On the structure of single linear pseudo-differential
equations, Proc. Japan Acad., 48 (1972), 643-646.
rreves, F.: [i] Ovcyannikov Theorem and Hyperdifferential
Operators, I. M. P. A., Rio-de-Janeiro ~Brasil), 1969.
: [2] Analytic-hypoelliptic partial differential equations
of principal type, Comm. Pure Appl. Math., 24 (1971)
537-570.
MICROANALYTICITE DE LA MATRICE S
Fr@d@ric PHAM
D@partement de Wath@matiques, NICE
La physique des particules 61@mentaires s'int@resse & des "proeessus de collis±on",
qu'on a l'habitude de noter de ~agon analogue ~ des r@actions chimiques, p.ex.
~T + P == > Ko + Ko + _ ~o
type de ~ormule que nous r@sumerons par la notation g6n@rique
ou plus simplement [IJ]
I '> J
II sere aussi commode de noter un tel processus par un graphe, p. ex.
~ K °
Si l'on ne tient compte que des "interactions fortes" [n@gligeant les interactions
@leetromagn@tiques et autres interactions "faibles"], et si l'on ne consid~re qua
des processus de collisions entre particules stables ("stables" s'entend relative-
msnt aux interactions fortes], on peut admettre qua longtemps avant [resp.longtemps
apt@s] la collision, les particules entrantes [resp.sortantes) sent des particules
~} II sere commode d'interpr@ter les lettres I, J de fagon "ensembliste" : dens l'exemple ci-dessus, I = {m@son w-, proton} , J = {ler m@son K °, 2e m@son K °, hyperon 5o} ; la num@rotation des particules de m~me type (les deux m@sons K ° dens dens cet exemple) n'est qu'un artifice math6matique, absolument vide de sans physique en vertu du prinoipe d'indiscern~bilit@ des particules.
84
libres (car les interactions fortes sont &courte portee). Rappelons que l'@tat
cin@matique d'une particule libre psut ~tre caract@ris@ [an oubliant Is nombre
quantiqus discret qu'est le "spin") par sen quedri~veqteur d'impulsioo-6nsrgie,
P = (P(o)" ~) E IR x ~ 3 = ~4
dont la composente P(o) > 0 rapt@santa l'@nergie, et dont le cerr@ scelaire dens
la m@trique de MinkowsKi est @gel au cart@ de la masse de la particule
2 2 +2 2 P ~ P~o ~t J - p = m
lies unit@s sont choisies de fagon que is vitesse de la lumi@re vale I]. Nous note-
runs M , st appellerons couche de masse de la particule, la napps d'hyperbololde
ainsi de#inie :
M : {pErR 4 I p2 2 = m ' P{o) > O}
En vertu du "~rincipe de superposition" (principe fundamental ds touts la physique
quantlque], touts l'informatien qu'il est possible d'extraira d'exp@riences de
collision I ~ J est contsnue dens ce qu'on appelle l'@l@ment de metrics S
du processus I ~ J : c'est une distribution SIj • ~ valeurs complexes, sur
la vari@t@ "T 1 M i (produit des couches de masse de toutes las partioules i ~ IUJ
en jeu). A partir du principe d'invariance par t, Faqslat,,i~,~ ' des lois de la physique,
il est facile de montrer qua cette distribution a son support dens la sous-vari@t@ m)
M[Ij) = {(Pi )E I I M i I ~ Pi = ~ Pj} i ~ I/-L] i~l jEJ
[c'est la ioi de "conservation de l'impulsion-@nergie") ; plus pr~cis@ment• on
montre qua SIj est de la forme
SIj = 6[ ~ Pi - E Pj ) S[Ij] i~ i j~J
~) Comma les autres vari@t@s alg@briques qua nous introdulrons par la suits• cette "vari@t@" est lisse pour des valeurs g@n@riques des masses• mais ells acquiert des slngularit@s pour des valsurs des masses qui• bien qua "particuli@res", se ren- eontrent effectivement en physique - p.ex. cheque lois que les masses des parti- cules sortentes sent 6gales ~ celles des particules entrantss• ce qui est lecas des "processus 61estiques" {processus I --~I). Css cas "particuliers" conduisent
des difficult@s techniques qua - faute de les avoir r@soluss - j'exorciserai par l'incantation suivante : "on se ram@ne au cas gen@rique par perturbation des masses",
85
o~ ~ est la distribution de Dirac [ou plut~t le produit de 4 distributions de Dirac,
une per composante de quadri-vecteur), et S[Ij) est une distribution sur la sous-
vari6t6 M[Ij) , distribution appe16e amplitude de diffusion du processus I -->J.
Dens ce qui suit, on ve ~noncer des '_~h,~poth~ses d,e,,,~microanalyt,i, cit~" des amplitudes
de diffusion. On ne cherchera pas ~ justifier physlquement ces hypotheses [ceei sere
feit dens l'expos~ de Iegolnitzer), mais on t~chera de les pr6senter sous une forme
meth6metique telle que l'lnterpr6tetion physique apparaisse "en filigrene".
I1 ressort des progr~s r~cents de la "th6orie.exiomatique de la metrice S" que ces
hypoth6ses de microanelyticit~ (ou les hypotheses 6quivalentes de "macrocausalit~"
dont perlera Iagolnitzer) peuvent @tre prises eomme axiomes fondamentaux de cette
th~orie, En th~orie axiomatique des ch~mps, o~ la metrice S n'est pas l'ingr6dient
de d~part mais un sous-produit, ces hypotheses "devraiemt" pouvoir @tre d6montr~es
comme consequence des axiomes [mais tout le traveil reste ~ feire !).
86
I, MICROANALYTICITE DES AMPLITUDES DE DIFFUSION.
1.1, E yBace cotangent ~ la vari@t@ M[Ij) .
L'espace de Minkowski des quadrivecteurs d'impulsion 6nergie peut ~tre mis en
dualit@ avec l'espace de MinKowsKi des translations d'espace-temps : sl p est
un quadriveoteur d'impulsion-6nergie et u une translation d'espace-temps, le
produit scalaire p.u (dens la m~trique de MinKowsKi) est un invariant de Lorentz
qui ala dimension d'une action [produit d'une @nergie par un temps) ~ or il
existe dens la nature une constente fondamentale qui ala dimension d'une action,
la "constente de PlancK" ; p.u divis@ par la constante de Planck est done un
scalaire au sens math@matique du terme.
Par cette dualit6, il est 6vident que l'espace cotangent & la cauche de masse
en un point p ~ M s'identifie eu quotient de l'espace des translations
d'espace-tamps par le sous-espace (& une dimension) des translations parall@les
p ,Autrement dit, si l'on appelle "tra~ec~oire libra" d'impulsion-~nergie p
taute draite de l'espace-temps parell@le ~ p , l'espace cotangent ~ M en p
peut s'interpr6ter comme l'espace vectorial des translations des trajectoires
l l, b,res d'im~ulsion-6nergie p .
~ la varlet6 De fa~on analogue, on v6rifiera qua l'espace cotangent T[p]M[lj)
M[ij) en un point [p) = (Pi)iEiJ.~ de cette vari@t6 peut s'interpr6ter comma
l'espace vectorial des "translations relatives" d'une {amille de trajectoires
fibres d'impulsions;@nergies Pi (i C I J.~J) , o,A,d, les translations modulo i)
les translations d'ensemble
1,2. C on{igurqtion,@l@mentaire associ6e &_un point [p) ~ M(Ij]
°
~ LK ° . . . . . . . . > [axe des temps)
gP ~ L g
Nous appe l le rons c o n { i g u r a t i e n ~ l~menta i re associ~e ~ un processus ( IJ ] t ou te
{amille de demi-droites orient6es [~i]i~ l llJ de l'espace-temps, incidentes
• ) Pour arriver ~ eette interpretation, il ~aut d~finir la dualit6 dens l'espace [~4]l~IJ par la formula : (p].{u) = ~ Pi,Ui - ~ pj,uj , Evidemment, faute
i@l j~J d'entrer dens des justifications physiques d~taill~es, notre seul argument pour justifier ce choix est la simplicit@ du r6sultat.
87
un mBme point 0 , et telles que
I ° ) chaque ~. est de genre temps, et orient@e dans le sans du temps j l
2°J ~i ale point 0 comma extr@mit6 ou comme origine selon que ± 6 I ou
3 °j si l'on d@signe par Pi l'unique quadrivecteur parall~le @ ~. tel que 2 2
Pi = m. {m. = masse de la particule i] , on a 1 1
Pi ~ Pj
iE T j£ J
J~
Ii est clair par d6finition qua les configurations 61~mentaires associ6es
un processus {IJ) sont, modulo le choix du point 0 , en correspondance bi-
jective avec les points (p) E H{ij)
1.3. Configuration causale.
Nous appelons configuration causale toute famille finie de trajectoires (dans
l'espace-temps) de particules suivant les lois de la cin~matique relativiste
classique et interagissant ponctuellement : l'exemple le plus simple de confi-
guration causale est la "configuration @l@mentaire" que nous venons de d@finir ;
plus g@n6ralement, une configuration causale est une famille de trajectoires
reatilignes (droites, demi-droites, ou segments de droites, du genre temps et
orient@es dans le sans du temps) telle qua si l'on appelle "sommet" de la con-
figuration tout point oO l'une des trajectoires commence ou se termine, la
configuration coincide au vaisinage de chaoun de ses sommets avec une configura-
tion 61~mantaire.
Les trajectoires born6es (segments de droites) de la configuration sont appel6es
"lianas internes" , les autres sont les "lianas externes".
1.4. Covecteur causal.
Soit {u) un vecteur cotangent ~ N(IJ) en un point [p) E M[Ij) . Un tel 4
covecteur peut @tre represent6 par une famille de translations (u~)i~IJ_LJ "
En faisant agir les translations u, sur les lignes Z. de la configuration 1 i
61~mentaire associ~e ~ [p) , on obtient une famille de demi-droites ui(~ i) •
I1 peut se faire que cette famille de demi-droites coincide avec la famille des
88
lignes externes d'une con#izuration causele C (#igure ci-dessous].
u2 (%2 Y"- " 5[g5] 41 \i
Configuration causale C
Si c'est lecas (pour un c h o i x e o n v e n a b l e du r e p r ~ s e n t a n t ( u . ~ N 4] du c o v e c t e u r 1
(u] ] , nous dirons que le covecteur (u] et la ,configuration Causale C sont
associ@s.
Nous appellerons covecteur causal tout covecteur associ@ ~ une conqlguration
caueale,
On observera que, puisque la notion de con#i~uration causale est stable par
dilatation, le caract@re causal ou non d'un covecteur se conserve quend on
multiplie ce covecteur par un scaleire positif, autrement dit, ne d@pend que
de la direction de ce coveeteur.
On notere S~M[Ij] le fibr6 (sur M(Ij)] de toutes les directions de covec-
teurs [c.O.d. le fibr~ en spheres assoei@ au fibr@ cotangent T*M(Ij])
Tout ce qui pr6c~de nous permet d'@noncer i'
HYPOTHESE DE MICROANALYTICITE :
L'amplituide,,,de i,d,iffusion S(Ij] est microan@lytique dens la direic,ti,on de tout
covecteur (u]~ T~M[Ij] , ~ l'exception des covecteurs causaux.
Cette hypoth@se permet en principe de d@limiter le support spectral de S [IJ]
[dens S M(Ij]] & partir de la seule donn@e du "spectre de masse" des patti-
89
1.5.
cules existant dans la nature, c.~.d, du sous-ensemble ~'C ~+ form@ de
toutes les masses de particules existantes. On suppose g@n@ralement que ce
spectre de masse est discret et non adherent ~ z~ro [hypoth@se d'une "masse
minimale"]. Oans ces conditions, on ale r6sultat suivant [H.P. Stapp]
Convenons que deux configurations causales sont de m~me t~pe s'il existe un
hom@omorphlsme de l'espace-temps ~4 qui transforme l'une en l'autre de telle
fa~on que chaque 11gne soit transform@e en une ligne porteus.e de la m~me masse
(si plusieurs particules suivent la m~me trajectoire, la masse port@e est la
somme des masses), Alors, llensemb~e des tyBes de configurations caugalgs est
localement fini sur M(Ij] [ou sur S~M[Ij]),
En particulier un covecteur causal ne peut correspondre qu'Q un nombre fini de
types de con£iguratlons causeles. S'il ne correspond qu'~ un seul type, je
dirai que o'est un covecteur causal simple. La question suivante me semble
naturelle : "presque tout" covecteur causal est-il simple ? (je ne connais pas
la r@ponse),
2. SINGULARITES DE LANDAU.
2.0. Oublions la structure m@trique [dans l'espace-temps) d'une configuration causa-
le pour n'en retenir que la structure combinatoire : on obtient le "graph9 de
.diffusion mu!tipiSi associ@ ~ la configuration causale ; c'est un graphe
[abstrait) orient@, sans circuits, dont chaque ligne interne ou externe porte
le nom d'une particule et dont chaque sommet a pour @toile le graphe d'un
processus de collision. Nous dirons qu'un graphe de diffusion multiple est
associ~ ~ un covecteur [u)ET~MIij] s'il est associ@ ~ une configuration cau-
sale associ@e & (u)
REMARQUE : Le lecteur peut se demander quelle difference il y a entre la donn6e
d'un "graphe de dif~usion multiple" et la donn~e d'un "type de configuration
causale". Bien sOr~ le fair que les lignes du graphe ne portent pas seulement
des masses mais des noms de particules est une distinction un peu byz~ntinw
Une di~@renee plus importante est illustr@e par la ~igure ci-dessous
[N = 1,2 .... ]
qui repr6sente une Infinlt6 de graphes tous associ#s au m6me type de configu-
90
ration causale, Ce phenomena, qui se retrouve evidemment cheque ~ols qu'un
graphe a des lignes multiples, n'e aucune importance pour les conslderetions
du present paragraphe ~], mais eonduira ~ des difficult~s techniques [qua nous
eluderons) dens les 6nonc6s du paragraphe suivent.
2.1. R6~ion physique et veri~t6 de Landau d'~n ~£aphe de diffusion multiple.
Salt G un graphe de diffusion multiple, dent l'ensemble de lignes [internes
et externes) sere not6 IGI . Formons le produit i~G| Mi de toutes les
couches de masse des particules en jeu, et coupons le per lee plans [de codi-
mansion 4) d6flnis par la conservation de l'impulsion-energie ~ cheque sommet
de G : on obtient ainsi une veriet~ algebrique ["g~nerelement"lisse] qua nous
noterons M G , et qua nous appellercns co uche de masse du gFaphe G . Si I
resp. J d6signe l'ensemble des lignes externes entrantes rasp. sortantes du
grephe G , on a une projection 6vidente ~G : MG ~ M[Ij] d~finie en
"oubliant" les impulsions-6nergies des lignes internes. I1 est facile de voir
que ~G est un morphisme propre (ear en bornant les ~nergies des partlcules
externes du grephe on borne 1as 6nergies des particules internes]. L'image de
ce morphisme est donc un ferme semi-elg6brique de M[ij] , qui sere appale
r6gion physique du ~raphe G . Nous noterons FG 1'ensemble critique de ~G
[ensemble des points o~ l'application tangente n'est pas surjective] et L[G]
le "contour apparent" de ~G ' c.&.d, l'image de FG par ~G " Ce "contour
apparent" L[G) est un ~erm6 semi-alg~brique [nulle part dense] de M[Ij] •
qua nous appellerons "vari~te de Landau" du graphe G , et dent lee points
seront appel~s "points de Landau".
J) LfG]
Pour calculer l'ensemble critique de ~G ' ii sere plus commode de raisonner en
termes d'applicetlon cotan~ente : naus ellons voir que le noyau de l'application
cotangente en un point critique (noyeu dent la dimension est le "coran~"du point
critique) admet une interpretation simple en termes de "configurations" dens
l'espace-temps,
I] Les objets d6finis ci-apr~s ne d~pendent en fait qua du type de configuration causale associ~e ~ G •
91
2.2, Con f igu ra t i ons criti.q,u,e,,s, et covecteurs criti.que,s, d 'un sraphe de diff,u,>,,tq, n
m u l t i p l e .
On appellera configuration libra touts famille de droites de i'espace-temps,
de genre temps et orient@as dens le sens du temps. En particulier & tout point
(p) E M(Ij) est essoci@e une "configuration fibre ~l~mentaire", d@duite de
la configuration 1.2 en prolongeant les demi-droites par des droites. D'apr~s
la description 1.1 de l'espace cotangent, on peut identifier tout covecteur
{u) ( T?p)M{Ij)_ ~ une translation relative des lignes de cette configuration
libra, qui la transforme en une eutre configuration libra {dire "associ@e &[uJ').
On appellera configuration critique d'un graphe de diffusion multiple g touts
configuration libra, dont les lignes sont index@es par IGI , et qui ~ l'@toile
de cheque sommet de G fait correspondre une configuration libre @16mentaire
(c.~.d. des droites eoncourantes satisfaisant ~ la loi de conservation de
l'impulsion-~nergie).
Un covecteur (u] E T(p)M(Ijj sere appel6 cevecteur critique de G si la
configuration libra qui lui est assoei@e est la famille des lignes externes
d'une configuration critique de G
PROPOSITION : L'ensemble des covecteurs critiques de G n'est autre que le
noyau de l'applicetion cotangents @ ~G
La d6monstration de cette proposition est immediate une lois qu'en a compris
la description suivante de l'espace cotangent ~ M o en un point {p) ~ M G :
Pour cheque "@toils" e {proeessus El6menteirel du graphe G on censid~re le
point (P)e ~ Me , image de {p] par la projection ~vidente, et l'on associe
ce point une configuration libre 61@mentaire L Ida la fa~on d~j& indiqu@e); e
soit L = I I L la configuration libre d6finis par l'union disjoints Ipour e
e ~ 6t G routes les ~toiles de G) de ees configurations @l@mentaires ; il faut bien
noter qua cheque ligne interne du graphs G apparent deux lois dens L , une
lois pour l'@toile d'oO ella sort etune lois pour l'~teile cO ella entre :
ca qui fair qu'& cheque ligne interne de O est associ@ dens L un couple de
droites parall@les. L'espace cotangent & M G en {p) peut s'identifier
l'aSpace des translations des lignes d'une telle configuration L , modulo
translation d'ensemble de cheque L et modulo translation d'ensemble de chacun e
des couples de droites parall~les dont nous venons de parler.
92
2.3 . Configurations causales "associ@es" et "subprdoon~es" ~ u n ~raphe.
Parmi les configurations critiques qua nous venons d'introduire, certaines sont
"acausales" : nous entendons par l& qua leurs sommets sont dispos@s dans l'es-
pace-temps suivant un ordre temporal non conforms @ l'orientatlon du graphe G .
Les autres, tout en respectant l'ordre temporal, peuvent @ventuellement ~tre
"d@g@n@r@es" en ce sens que plusieurs sommets de G peuvent s'envoyer sur le
mBme point de ~4 ; neus les appellerons configurations causales subordonn@es
au iraEhe G ; parmi elles, celles quine d@g@n@rent pas seront appel@es
configurations causales associ@es ~ G : ce sont les configurations libres d6-
dultes [par prolongement des lignes) de configurations oausales du type 1.3.
Oe fagon g@n@rale, ~ toute configuration causale suborOonn@e & G correspond
une "contraction" G' de G (si deux summers de G ont m~me image dans ~4 ,
on Identifle ces deux sommets et l'on contracte les llgnes qui les joignaient)
tells que la configuration causale puisse ~tre consld@r6e comma associ@e & G'
[en effagant les droites qu± correspondaient aux llgnes contract@as). On voit
ainsi que tout coveeteur causal subordonn~ & G est un covecteur causal associ@
une contraction G' de G .
2 , 4 . Structure de la singulerit6 de Landau au voisinage d'un ~oint c ausa~ de cqran~ 1
Suit Pc ~ MG un point critique causal de corang 1 associ~ au graphs G :
~G est un on entend par l& un point critique de coran@ I de ~G (i.e. Ker Tpc
espace vectoriel @ une dimension) tel que la configuration critique oorrespon-
dante [unique ~ dilatation pr@s) suit une configuration causale associ6e ~ G .
Suit PL (M(IJ) le point de Landau, image de Pc "Oans ces conditions on peut
d~montrer la
PROPOSITION :
I ° ) La fibre de ~G au-dessus de PL se r6duit au seul point Pc ' Au voi-
sinage de PL ' la vari6t6 de Landau L(G) est une hypersurfaoe lisse, image
isomorRhe par ~G de l'ensembls critique r G .Cette hypersurface lisse horde
la r@~ion physique de G , c.&.d, que l'Image de ~G est situ6e d'un saul cBt@
de cette hypersurface ; plus pr@cis6ment, si l'on choisit le signe d'uns 6qua-
tion locale s de L[G) de telle fagon que ]6 covecteur ds soit causal ,
l:,image de ~G se trouve du c~t6 s ~ 0
2 ° ] Au voisinage de Pc " l'application ~G est du type "pli" de Thom ,
c.Q.d, qu'on peut choisir sur M G des coerdonn@es analytlques locales
x I ..... x n telles que l'application ~G s'6crive dens ces coordonn@es
93
Yl = Xl .,,
YK = Xh
2 2 s = XK+1+ ... + x n ( o u = 0 d e n s l e c a s n = K)
[Yl' .... YK sont des coordonn@es locales sur L[G) ]
2.5. La structure des points critiques de corang > I est beaucoup moins bien
oonnue. Cependant la pr@@minence du corang Iest attest~e par le r@sultat
suivant :
PROPOSITION : Si G est un graphe connexe ,l'ensemble des eovecteurs causaux
subordonn@s ~ G en un point PL E L(G) est un c6ne convexe & base poly6drale,
dont lee points extr@maux (sommets du poly~dre] sont des covecteurs causaux
de coran~ 1 assoei@s Q l'un,,e des contractions de G ,
NOTATION •
L+(G] = partie oausale de L[G) = ensemble des points de Landau d'oO part au +
moins un covecteur causal subordonn6 ~ G [L [G) est un sous-ferm~
semi-alg@brique de LCG]) .
+
LI{G] = pattie c~usele de coran~ I de L(G] = ensemble des points de Landau
du type 2.4 .
+
REMAR@UE . LI(G) peut @tre vide : si le grephe G a beaucoup de lignes
internes et peu de cycles, on peut avoir dim M G < dim M[Ij) I , et alors
aucun point n'est de corang I .
Cependant la proposition pr6c@dente montre qua ,Rout tout graphe G connexe
+
L+(G)C U L 1
G'
G' de G
( G ' ) , o0 l'union porte sur toutes les contractions
2.6. Singularit~s $6n~riques des amplitudes de diffusion.
Si nous appelons point causal de MIIj) tout point d'oO part un covecteur
causal, 1'hypoth~se de microanalyticit6 du §I nous dit qu'en un point non
causal l'amplitude de diffusion S[Ij] est microanalytique dens toutes les
directions, done analytique :
+ +
Ca) SCIj) est anal~tique dans M[Ij) - LCIj) , o0 LCIj) d@signe la partie .........
causale de M(Ij] .
94
+ = U L + Or les consid6rations qui pr6c6dent nous montrent qua L(Ij) G (G) ,
60 l'union peut @tre consid@r~e comma loc~lement <inie grace ~ 1.5 , ÷
IIen r@sulte qua L(Ij) est un ~erm@, Ioealement semi-alg~brique, nulle part
dense dans M[Ij) .
Demandons nous maintenant quel type de singularlt~ peut avoir S(Ij) en un
point g~n6rique de L~Ij) . Ii r@sulte de 2.4 et 2,5 quesi l'on oublie la
partie de L~Ij) qui provient des graphes non connexes ["oubli" qui trouvera +
sa justiTication dans l'Appendice) , L(Ij) est au voisinage de presque tout
point une hypersur~ace lisse, d'@quation locale s , telle qu'en tout point de
cette hypersurface ds edit l'unique covecteur causal ~) . L'hypoth@se de
microanalyticit@ se traduit alors de la @aqon suivante :
(hi
+ l'amplitude de diffusion au voisina~e d'un tel point "$@n@rique" de LIIj) ,
SCIj) est valeur au bord d'une Tonction 4+ analytique "du cot@" Imds > 0 .
Joint ~ [a] , ce r@sultat [b] montre en particulier qua les deux fonctions
anaiytiques qua d6finit S[Ij] dans les deux demi-espaces r@els U O{s < O}
et UN{s > O} [U = voisinage d'un point causal g~n~rique dans M[Ij]) se
d6duisent l'une de l'autre par prolongement analytique dans un "tube local" + ~+
U x ~ dont la base imaginaire est un ouvert du demi-espace Im s ~ O •
"tangent" & l'hyperplan Im s = 0
Im s
"//////// Im Yl ..... Im YK
3. SPECTRE DES AMPLITUDES DE DIFFUSION.
L'hypoth~se de microanalyticit6 ne nous donnait de renseignements qua sur la
position des singularit6s de l'amplitude de diffusion (dans le fibr6 cotangent).
"L'hypoth6se spectrale" qui va suivre nous renseignera sur la nature de ces sin-
gularit~s.
~) Si ~eux graphes G~ et G 2 ont ClocalemBnt) m~me vari@t~ de Landau L+=L+[GI )= = L {Go), il est ~cile de voir qua leurs r@gions physiques sont du m~me c8t6 d~ L + , de sorte qua si ds est covecteur causal pour G 1 c'est enco- re ds Cat non -ds) qui est causal pour G 2
95
OEFINITIONS-NOTATIONS
3.1 On appelle 61~ment de matrice S d'un graphe de diffusion multiple G , et
l'on note S<G > , la distribution sur la vari6t6 ~ M i [produit des couches
de masse de toutes ies particules internes et externes] d@finie, dens le cas
oO G n'a pas de l i@nes mq~itiples, comme le produit des @16ments de matrice S
de teus les pracessus ~i6mentaires dont est constitu6 le graphe G [~toiles
des sommets de G] . Si G a des lignes multiples il y a lieu de modifier
16g@rement cette d6finition, d'une fa~on que nous ne pr6ciserons pas ici,
3.2 Soit I(resp. J) l'ensembie des lignes externes entrantes (resp. sortantes) du
graphe G . On note AIj[G] la distribution sur ~ M i d6finie en in- i 6 IIL]
t@grant S<G > sup le produit des couches de masses des particules internes de
G [cheque couche de masse M. est munie de la mesure invariante 1
~[p2 _ m~] dP[o]dP[1)dP[2]dP(3 )]
3.3 Ii est clair que les distributions d6finies ci-dessus peuvent ~tre divis~es
par une fonction ~ de Oirac exprimant la conservation de l'impulsion-6nergie.
On ~crira
S<G > o 6( ~ Pi ~ Pj~ SG i6I jmJ
AIj(G] = ( ~ Pi - ~ Pj] A[Ij](G] i(I j eJ
oQ S G [resp. A[Ij][G]] est une distribution d@finie sur la couche de masse
M G (resp. M[i J] .
Pour abr@ger on notera
A(G] = A[Ij](G]
qu'on appellera partie absorptive de l'amplitude de diffusion S[Ij] , releti-
vement au graphe G .
Remarquons que A(G] s'obtient en int@$rant S G le lon~ des fibres de la
projection ~G : MG .. > M[13] (avec une mesure inveriante facile ~ ~crire].
En partieulier A[G] a. son support dens 1 e r6gion physique du ~raphe G .
3.4 REMARQUE : Les d@finitions ci-dessus comportent des op@rations a priori "dan-
gereuses" s'agissent de distributions : produits, int6greles le long des ~i-
bres..,
96
Heureusement, gr@ce & ce que nous savons du "support spectral" de ces distri-
butions (hypoth~se de microanslyticit6] des arguments standard de th6orie des
hyperfonctions [cf. expos@s de Chazarain] permettent de montrer facilement que
toutes les op6rations cl-dessus ont un sans, et de pr@ciser le "support spectral"
des distributions ainsi obtenues. On obtient notsmment le r~sultat suivant
PROPOSITION : La pattie absorptive A[G] est microenalytique partout, saul
dens les codirections associ@es @ des configurations obtenues en "@clatent
de fa~on causale" les sommets de G : on entend par l& les configurations cri-
tiques ~ associ@es & des graphes ~ tels qua
1 °] G soit donn6 comma contraction de ~
2 °] cheque sous-graphe de G qui se contracts suivant une @toile [procassus
~l@mentaire] de G est assoei6 dens C ~ une sous-configuration causale.
En particulier, si [u] ~ T~M{Ij] est un covecteur causal simple [eu sans de
1.5] associ6 eu graphs de diffusion multiple G , il r~sulte de la proposition
ci-dessus qua le support spectral de A[G] coincide a__u voisinage de [u] avac
l'ensemble des covecteurs critiques de G [tous causaux puisque voisins de
Cu]]. Autrement dit, les supports spectraux de A[G] et de S[Ij] co~nciCent
au voisinage de Cu] . "L'hypoth@se spectrale" ci-dessous dit qua non seulement
les supports spectraux mais aussi les "spectres" {ausens de SATO, c.~.d, les
microfonctions correspondantes] sont 3es m~mes.
HYPOTHESE SPECTRALE :
Soit [u] ~ T~M[Ij] un covecteur causal simple [ausens 1.5] associ6 un g raphe
d__e diffusion multiple G etun saul. Alors la distribution S[Ij] - A[G] est
microanalytique dens la direction du covecteur [u] .
3.5 REMARQUE : Em r6alit@, l'hypoth@se spectrale n'est pas ind@pendante de l'hypoth@-
se de microanalytlcit@, mais peut en @tre d6duite & l'aide de la propri@t6 fon-
damantale d'uniterit@ de la metrics S .[c-F. expos~ d'OLIVE]. Si nous averts pr@-
f@r@ presenter ces deux "hypoth@ses" sur un pied d'~galit~, c'est parce qua rou-
tes deux sent @galement fondamentales du point de vue de l'interpr6tation physi-
que [voir l'expos6 de Iagolnltzer).
3.8 FORMULES DE DISCONTINUITE.
L'hypoth@se spectrala vanous permettre de pr6ciser les in%ormations donn@es au
n ° 2.6 sur les singularit@s de l'amplitude de diffusion S[Ij] au voisinage +
d'un point de Landau PLE Lq[G] , moyennant l'hypoth6se suppl6mentalre qua
97
PL n'est point de Landau d'aucun autre graphs que G
Sous cette hypoth@se, non seulement l'amplitude de diffusion S[Ij] n'est
singuli@re que dens la direction du covecteur causal [u] , mais la pattie
absorptive A[G] n'est singuli@re (Proposition 3.4] que dans Iss deux co-
directions [u] et [-u] [codirections associ6es aux seules configurations
critiques de G : la "causale" [u] et "l'anticausale" [-u]]. Par cons@quent
S[Ij] - A[G] ne peut a priori ~tre singuli@re que dans ces deux codirections,
doric en r@alit@ seulement dans la codirection [-u] puisque l'autre est
exclue par l'hypoth@se spectrale.
O'oO le r6sultat [avec les notations 2.6) :
[ c ) au voisinage de PL • la distribution S[Ij] - A[G]
fonction f ,,qnalytique "du c6t6" Im d,,~, ,< 0 .
est valeur eu bord d'une
Or on salt d'apr@s 3.3 que A[G] s'annule "en dessous du seuil" de G ,c.&.d.
du c6t@ s< 0 ,oompl~mentaire de la r@gion physique de G . Les deux {onctions
analytiques f+ (d@finie en 2,6 [b]) et f [d6finie par [c] ci-dessus} se
prolongent donc en une m~me fenction analytique f dont on peut se repr6senter
le domaine d'analyticit@ comme sur la figure suivante :
A plan des Imp ~ / / ~
pour Re pen dessous
du seuil [c.a,d.
Re s < 0].
Plan des Imp ,
pour Re p en
dessus du seuil
[c.~.d. Rs s > 0].
Im s
f /•/, Im Yl .... , Im YK
Im s
f= f
Im Yl ..... Im YK
La pattie absorptive A[G] peut donc dans ce cas s'interpr@ter comme une
"discontinuit@"
A [ G ] = b + [ ~ + ] - b [~ ]
[o0 b+ resp. b_ signifie "valeur au berd" du cSt~ Im ds > O resp. < O] .
98
APPENDICE : DECOMPOSITION EN PARTI~S
CONNEXES
Les configurations causales d6finies en 1,3 ne sent pas n6cessairement connexes.
Parmi les configurations causales non connexes, les plus slmples sent celles
qui n'ont pas de li~nes internes : ce sent tout simplement des unions disjointes
de configurations ~16mentaires.
4t Soit [u] ~ T[p]M[Ij] un covecteur associ~ & une telle configuration
~J 1
translation relative
I J .................. ~ 12 [u]
J2 Configuration 616mentaire La m~me apr~s action du
Qssoci@e & ( p ] covecteur [U ]
[:~]
Un tel covecteur est associ6 & un graphe E que nous Qppellerons "d@composition
61@mentaire" de [IJ] • et qui est tout simplement une union disjointe de gra-
phes de processus ~16mentaires :
I I E = J ] l l [ I e e ' e
i = I I I~ , J : l J ~ - 8
e 8
L'hypoth6se spectrale dens ce cas particulier tr~s simple dit qu'au voisinage
d'un tel covecteur [u] , on a la relation "mierofonctionnelle" :
SIj ~Tsi j e e8
est microanalytique .
Cette relation donne tout son sel au lemme [purement combinatoire] suivant :
LEMME :
Supposons donn6e une famille d'objets [SIj][Ij]~ index~e par l'ensemble ~
de tousles processus @16mentaires [IJ] . A une telle famille est associ6e de
99
C C facon unique une famille {SIj)[Ij) @~ , cO les SIj sont des @l@ments de
l'alg@bre libra engendr~e par les SIj , telle qua pour tout {IJ]E ~ on air
o = ~ ~ S I J
SIj • e ~ 6t(E] e e
E ~ d6c{IJ)
cO d6cCIJ) d6slgne l'ensembie des d@compositions 616mentaires du processus
[IJ) , et 6t(E) d6slgne l'ensemble des ~toiles Ic,~.d. des parties connexes)
d'une telle d6composition E
PREUVE DU LEMME :
En isolant dens le membre de droite de (~) le terme correspondent ~ la
"d6composition triviale" E = {IJ) , on trouve une formule du type
C + SIj = SIj •,.
c cO I at cO les .,, repr@sentent des sommes de produits de termes SI J • e e e
J sont des sous-ensembles striets de I et J ; le lemme s'~n d6duit Imm@dia- e
tement par r@ourrence sur le nombre d'@16ments des ensembles I et J
c , et la formula OEFINITION: Les SIj s'appellent "parties connexes" des SIj
C~m) s'appelle "for mule de d6composition en partiss connexes".
Si l'on veut @tendre ce lemme (purement combinatoire dens l'6nonc6 ci-dessus)
notre situation cO les SIj sont des distributions, le probl@me se pose de
savoir si la multiplication de ces distributions a un sens. La r6ponse est encore
une lois donn@e, comma au n ° 3.4 , par l.lh~poth~se de microanalyt3e±t~, qui
nous permet non seulement de donner un sans au produit des distributions mals
m@me de calculer le support spectral des nouvelles distributions ainsi obtenues~
on volt einsi sans difficult6 qua Supp Spec S~j~ Supp Spec SIj
Hais il y a mieux : l'hwpoth6se spectrale et notamment la relation (m) permet o
de voir que l'inclusion ei-dessus est stricte, le support spectral de SIj
n'6tant eonstitu@ qua de covecteurs associ@s ~ des configurations causales
oonnexes. De ~agon pr@clse, on peut d6montrer la
PROPOSITION :
Ii y a 6quivalence entre
i °) La donn~e d'une ~amille de distributions S = 6[~-~ p i ~ - ~pl)S[ij]
.IJ .i~l ..~ J~I:4) alnsi [IJ)E ~ , satis~aisant & l'hypoth~se de microaoazy~ici~e
I O0
qu'~ l'hypoth@se spectrale [3,4) , et
2 ° ) la donn@e d'une famille de distributions
c SI Jc : 6(~iei Pi - jeJT~ pj) s(ij] , (iJ~e~ ,
satisfaisant & deux hypotheses analogues mais oQ les "configurations cau-
sales" sont partout remplac@es par les "configurations causales connexes '~.
Ces deux f,~milles ,,d,e donn6e sont reli@es entre e!!es par ,1,a formule de
"d@composltloo,, en parties connexes" [~m] j qui est bien d6finie "au sens
des distributions",
101
NOTE BIBLIOGRAPWIQUE
Los axiomes de la matrice S exposes ici sont le r~sultat d'une lente mature-
tion dont on pout voir l'aboutissement dens l'article de
• O. IAGOLNITZER et H.P.STAPP - Macroscopic causality and physical region analyti- city in S-matrix theory, Comm. Math, Phys.14,15
[1968) ,
On en trouvera un excellent expos~ dens le livre de
• 0. IAGOLNITZER - Introduct,ion to S. Matrix Theory.
(Association pour la diffusion de textes scientifiques et
iitt6raires, Paris 1973).
Je n'ai fair ici qua reprendre los m~mes id6es en remarquant que le langege
"mlcro~onctionnel" de SATO permet de leur donner une forme particuli~rement
concise : pour apprendre ~ parlor ce langage, lira
• Introduction aux hyperfonctions par A.CEREZO, A.PIRIOU, J.CHAZARAIN (dens ce
volume)
Voici quelques r~f6rences compl~mentalres (avec en regard le j de won expos6
auxquelles elles se rapportent],
(Introduction : "principe de superposition"] - R,P. FEYNMAN, R.B. LEIGHTON, M. SANDS
The Feynman lectures in Physics, vol I, chap, 37 (Addison-Wesley 1969).
(§1 .5) H.P. STAPP - Finiteness of the number of positive ~ Landau surfaces in
bounded portions of the physical region - J. Math, Phys. 8,6 (1967) .
(~2.4] F. PHAM - Slngularit6s des processus de diffuslon multiple
Ann. Inst. Henri Poincar6 6, 2 (1967) •
[ §3 ,5 ] O. OLIVE [dens ce vo lume) ,
MACROCAUSALI~, PHYSICAL-REGION ANALYTICITY
AND INDEPENDENCE PROPERTY IN S-MATRIX T~EORY
D. IAGOLNITZER
DPh-T CEN Saelay BP n°2 91190 Gif-sur-Yvette FRANCE
ABSTRACT
The equivalence between macrocausality and physical-region analyticity
properties of the S-matrix, which has first been proved in reference I, is described
here in an improved version derived from Chapter II of reference 2.
This version follows from various mathematical developments (references
3,4) which have allowed to give a somewhat better statement of the macrocausality
property and to complete the results of reference 1 in various ways.
It turns out, in particular, that the independence property, which was
originally presented (in some situations) as a supplementary assumption, can always
be derived from macrocausality.
INTRODUCTION
As mentioned in the previous lecture by Pham [6], the basic quantities
in the quantum relativistic physics of systems of massive particles (m > O) with
short-range interactions~ such as the "strong" interactions, are the scattering
amplitudes SIj between sets I of initial particles and sets J of finalpartieles.
We first briefly recall, for the non specialized reader, the physical [2]
meaning Of SIj o (For details, see for instance ).Since the spin variables are
unessential in the topics discussed, we shall only consider spinless particles.
A "pure" (i.e. "completely determined") state of a free particle with
mass m is represented in quantum relativistic physics by a "wave function" ~ , up
to multiplication by a complex constant ; ~ is a vector of L2(M) , Joe. is a
function of the (real) energy-momentum variable P ~- Po ' ~ ' p ~ M (Po) 0 ,
103
2 2 ~ 2 m2), P = PO - p = and is square ~ntegrable with respect to the measure
d~(p) = ~(p2-m2) 0(Po) d4p ~ dp 2 pv~TjJm :
2 r
llq Oll = I lq~(p) 12 d~(P) < ~ 0 (i)
Now, before and after the scattering processes take place, the physical
systems under consideration can be asymptotically identified with free particle
states. The basic principles of quantum theory then entail the existence, for any
given sets I,J, of the corresponding "scattering amplitude" Sij :2SIj is a func-
tional which acts on the sets [%1 = ([*i}i(i , [~jlj(j) , ~ ( L (M k) (vk) , is
linear with respect to each variable ~k ' and is such that the transition probabi-
lity (~) from a set I of initial particles in the states ~i ' i (I to a set J
of final particles in the states ~j , j(J , can be written in the form :
2
k
where m denotes complex conjugation.
The inequality W < I , i.e~ :
k (3)
ensures in particular that SIj is a tempered distribution in momentum space. g A
energy-momentum conservation <~j and of Eqo(3), SIj can be written in In view of
the form :
i¢I j(J
where StIj)~ is defined on the manifold M(IJ)
X pj ; Pk ~ Mk ~ Vk ~ and Pi z / . M(Ij )
i~I j~J of the scattering process I ~ J.
of the points p = Ipkl such that
is called the physical-region
( ~ ) . 10eo t h e p r o b a b i l i t y o f d e t e c t i n g a s e t J o f f i n a l p a r t i c l e s i n t h e
states ~j , starting with initial particles in the states ~i o
o o relation V = ~-~ cannot be satisfied (~)i e W(I~kl) = O if the Pi Pj
with each Pk in the support of ~k ' ~L i~I j(j
104
c Finally it turns out to be useful to introduce the "connected parts" Sij
(*) of the functionals SIj ; they can also be written in the form :
= Z Pi- Pj i~l j~J
where TIj is defined, as S(Ij) , on M(Ij) .
[7 Maerocausality L7~ is an appropriate mathematical expression of a certain
classical limit of quantum theory, namely of the principle that any energy-momentum
transfer over large distances which cannot he attributed to (stable) physical par-
ticles according to classical ideas, gives effects that are damped exponentiall~z
with distance. (This includes in particular the idea of the short-range character
of the interactions).
To get a better understanding of the classical ideas involved, it is
first necessary to abandon quantum theory and to study, as a guide, a classical
model of point particles. This model has already been introduced in PHAM's lecture.
We review it in section I, where we indicate the main definitions and
results about the +~ - Landau surfaces and the causal displacements [8]
In section II, we return to quantum theory, and indicate how the clas-
sical ideas can be adapted, asymptotically, in the quantum case. The macrocausality
property is then stated in the form of exponential fall-off properties of the tran-
sition probabilities, under appropriate conditions, when subgroups of initial and
final particles are displaced from each other
We show that this property amounts to the following basic result : the
"essential support" o£ S(Ij) , respo of TIj , at a point p n Ipkl of M(Ij) is
(contained in) the set o£ causal directions at p as determined in the classical
model of section I, resp. is (contained in) the set of causal directions at p
associated with connected configurations.
(~)
where the sum
(It,J t) and
The connected parts are defined, by induction, by the formulae :
Sc ~' S c SIj n IJ + ~_ '9
t--i ItJt
runs over all non trivial partitions of (I,J) into subsets
is essentially the tensorial product.
105
The definition of the essential support of a distribution is recalled
in Appendix I, where the main results needed below are described° (For more details,
see [3]).
iv .
We also see there that the essential support coincides with the slnz~/lar
I, ,t support , also called "singular spectrum , introduced independently, by very diffe- [ ]
rent methods, by professor SATO and coworkers L5j, and that the micro-analyticity
property indicated by PHAM therefore coincides with the above essential support
property of S(Ij) , resp. TIj .
In section III, we finally describe the analyticity properties implied
by, and as a matter of fact equivalent to, the above essential support property :
a) For any given l,J, there exists a unique analytic function FIj (defined
in a domain of the complexified manifold M(Ij) of M(Ij)) to which TIj is equal
at all non +~ Landau points, and from which it is a "plus ie" boundary value at
almost all +~ -Landau points.
Besides a few exceptional points, the remaining +~ -Landau points are
(some of) those which lie in the intersection of several +s-Landau surfaces with no
"common parent"° For them, the following property was still derived in [I] :
b) In the neighborhood of a +a-Landau point p of the latter type, TIj is a
su_.~m of boundary values (in the sense of distributions) of analytic functions G~ .
Each of these boundary values is obtained from "plus is" directions associated with
one of the "parent" surfaces involved at p .
The "independence property in its original form was the assertion "that
each G~ has moreover, in the neighborhood of p , the same analytic properties as
if no other parent surface was involved, "(i.e. G~ is for instance analytic at
all real points p' which do not lie on the parent surface considered, or possibly
on its own "daughters"). This property was proved in [I] only in special situations,
this being due to the lack of general information on the links between the various
decompositions of a distribution, obtain4d at different points p , into sums of
boundary values of analytic functions.
More recent mathematical results [4] provide this information and the
independence property easily follows. As a matter of fact t these results also pro-
vide global decomposition properties of TIj in M(Ij).
106
I - CAUSALITY,,,,,IN A CLASSICAL MODEL L8" [] : + ~ - LANDAU,,,,,,,,,,SURFACES AND CAUSAL DISPLACEMhhNTS
The classical model considered obeys the following laws :
i) A pure state of a free particle is characterized by one energy-momentum
2 m 2 ) 4-vector p, p ~ M (Po > O, p z , and by the position x of the particle at o
one time t . The particle has a well defined space-time trajectory which is the o
line parallel to p and passing through the point (to,Xo) , according to the pro-
pagation law
where v is the velocity.
AT -~ -+
Po
if) Particles may "interact" when their space-time trajectories meet at some
point. Then, new outgoing particles, which replace the incoming ones, may emerge
from the interaction point with the same properties as before, until their space-
time trajectory meets again that of another particle, and so forth.
At each interaction point, energy-momentum must be conserved (i.e. the
sum of the incoming energy-momentum 4-vectors must equal that of the outgoing ones),
and the number of outgoing (as well as incoming) particles must be larger than or
equal to 2o
Being given sets I and J of initial and final particles and a correspon-
ding set p ~ Ipk I of 4-momenta, we then consider sets ~ ~ {Ukl of space-time
displacements : starting from trajectories which all pass through the origin, the
displaced trajectory of particle k now passes through u k .
U is said to be causal at p , or the set (p,U) is said tb be causal,
if energy-momentum can be transferred from the initial to the final particles,
possibly by intermediate particles, according to the laws i) and if) above. A causal
U is moreover said to be connected if it corresponds to a connected configuration
of all (initial, final and intermediate) particles involved.
The remainder o£ this section is devoted to the study of the connected
causal sets.
+~- Landau surfaces
If (p,U) is causal, p ~st clearly belong to the submanifold
~4( [I I+IJ ) defined by the conditions M(Ij)
of
107
Pk ~ Mk : Pk 2 -- mk ,(Pk)0 > 0 , Vk
Pi A.~ Pj i¢I jeJ
(4)
M(Ij) is a smooth manifold except at the exceptional points (when they exist) where
all initial and final 4-momenta are colinear. These points will be excluded below.
To study the connected causal sets (p,~), one first considers all connec-
ted topological graphs G with oriented external and internal lines such that :
i) Their incoming, resp. outgoing~ external lines are in a 1-I corres-
pondence with the elements i6I , resp. jeJ, and, for each point p m Ipkl considered,
with the corresponding 4-momenta Pk "
ii) To each internal line ~ , is attributed the mass m 6 of a physical
particle.
Finally, there are at least 2 incoming and 2 outgoing lines at each ver-
tex.
A point p ~ ipkl is said to belong to the +~ Landau surface L+(G) ,
L:(G)u , if it is possible to flnd a set of 4-momenta k6 for each internal resp.
line 6 , k 6 ~ N 6 (k~ z m~ ,(k£) ° > O) and a set of ~6 ~ 0 , respo ~6 > 0 , such
that :
a) energy-momentum is conserved at each vertex of G
b) K z(6) ~6 k6 z 0 for each closed loop z of G : z(4) z 0 if z
does not contain line ~ ~ z(~) z +I, resp.-it if it contains it with the correct,
resp. opposite, orientation° Eqs (a)(b) are called the +g -Landau equations of G.
It can be checked that the surfaces L+(G) (if not empty) are analytic o
submanifolds of eodimension l (m) of M(Ij) , and that their union is no_~t dense in
M(Ij) , but divide it into sectors.
Each surface L+(G)o has also a well defined "physical side" in M(Ij)
from which it is the boundary, as described in PHAM's lecture (~) •
(~). 1.e0 their dimension is that of M(Ij) minus one.
(m~)The "physical side" lies in the "physical region of G" introduced by PHAM when the latter does not reduce to L+(G) itself. This last situation occurs
for graphs G without closed loops, in wh°ich case the "physical side" of L~(G) is easily defined directly .
108
. physical side of
L ,G) o
Figure i
L~(G)
Connected causal displacements
Being given a point p ~ Ipk I , the displacements of the form
T z (kkp k + al are called trivial at p:clearly if ~ = [Ukl is causal at p, ~+T
is also causal since the displacement kkp k does not change the trajectory of par-
ticle k , and a is a global translation of all trajectories° It will therefore be
useful to introduce, at each point p = Ipk I , the vector space of displacements
defined only up to addition of trivial displacements at p o
As a matter of fact, the trivial sets T at p are the sets ~ ~ [Ukl
which are eonormal at p to the manifold M(Ij) , if the scalar product <~,~> of
a vector U = lukl with a vector ~ = I~k~ in ener~-momenfum space is defined
by :
(U,~> = ~-~ Uio~ i - L uj0~ 'J (5)
iel j~J
where Uk.~ k = (Uk) ° (~k)o - ~k~i (see figure 2).
As mentioned by PHAM, the above space of displacements U
correspondingly identified with the cotangent vector space T * p M(Ij)
M(Ij) •
at p can be
at p to
The basic facts about the connected causal sets ~ z lUkl at a point
p ~ ipkl of M(Ij) can be stated as follows :
a) if p lies on no +a -Landau surface, then there is no (non zero)
connected causal ~ at p (ioeo all connected causal ~ are trivial).
(It is in fact immediately checked that the +~- Landau equations
follow from the kinematical equations of the model).
b) If p lies on only one surface L+_(G) (of u q2
only one connected causal direction I# at p (i.e. one
M(Ij)) , then there is
up to multiplication
109
by a positive scalar k). This direction is normal at p to L+(G) o
( 5 ) ) ( ~ ) , and i s o r i e n t e d t o w a r d s t h e p h y s i c a l s i d e o f L+(G) o o
(in the sense of
This situation is represented (locally) in Figure 2, where M(Ij) is
(locally) represented by the plane (x,y) and )Mf~j)(p)[i by the z-axis .(K-space
has been identified with the space ~4( II I+IJI where M(Ij) is embedded, the
axis (pk)u being identified (according to (5)) with ~k(Uk)~ , ~ z 0,1,2,3,
where ~k ~ +I if k is initial, ~k ~ -i if k is final).
z
x
Figure. 2
Y
c) If p lies on one surface L+(G) and on one or more other surfaces _(*~)
L+(G ') such that all G' are "contractions" of ~ , then the set of connected o
causal ~ at p is the set of linear combinations, with positive coefficients, of
the various J associated with each L+(G ') as in b). o
It can be proved that the convex cone of directions obtained is always
strictly contained in an open half-space o
(~) 10eo any representative U of ~ is normal at p to L+(G) , in ~4(Ii i+ [~l) o
( ~ ) A contraction of G is a graph obtained from G by removing some
internal lines and then identifying the end-point vertices of each line removed.
This situation occurs when one subset, or various different subsets, of coefficients ~6 associated with the lines of G may vanish at po
110
d) Finally, if plies in the intersection of several +~ Landau surfaces
L+(G'), where the graphs G' are not contractions of a common "parent graph" G , o
then the set of connected causal U is the union of the sets associated with each
graph, or parent graph, involved at p.
In some situations~ the directions~ or convex cones of directions, asso-
ciated with each graph, or parent graph, involved at p , are disjoint ; p is then
"t e " " called a yp I-polnt . It is called a "typeH-point" otherwise.
In either case, the set of connected causal directions is no longer always
contained in a convex (salient) cone. Examples of these situations are easily obtai-
ned (~), and some of them have been exhibited inE9Jor ~
II - MACROCAUSALITY PROPERTY
Macroscopic space-time localization of free particles [I][2]
In quantum relativistic physics, a pure state of a free particle is no
longer represented by a given 4-vector p 6 M and a given space-time trajectory
parallel to p , but turns out to be instead represented, up to multiplication by
a complex constant, by a vector of an irreducible representation space of the cove- (~)
ring group of the Poincar@ group . These representations are in general labelled
by two numbers, the mass m , which is strictly positive in the physical eases that
we consider, and the spin s which is integer or half-integer.
For spinless particles (szO) , the representation space is the space L2(M) u
introduced at the beginning o f this text, and the representation ~ ~ ~ of the
(~) Take for instance a point p z IP. I of M(Ij) such that energy-momentum
conservation is satisfied by two subsets of i~itial a~a ~inal 4-momenta
i ~ Pi - /~' PJ -- ~ Pi- ~ PJ z 0 ' Ii U I2 z I ' II ~ I2 z ~
i~Ii J~J l icI2 J~J2 J1 ~J J2 z J , J1 ~ J 2 -- ~
/ The p o i n t p i s t h e n i n g e n e r a l a " t y p e I I - p o i n t " ~ whose s e t o f c o n n e c t e d c a u s a l directions is not contained in a convex salient cone.
(~) The Poincar@ group is a semi-direct product of the group of space-time
translations and of the group of Lorentz transformations°
111
space-time translation by a 4-vector u is given by :
U(p) z ~(p) e ip'u (6)
(where p.u ~ PoUo - ~ ~). As a matter of fact, the .space-time translation..s, of free
particle states have been assumed at the outset to be physically well defined for
any 4-vector u , and u is the wave function representing the state obtained
from ~ after translation by u .
Now, let us first consider the non relativistic quantum case. Then ~ is
-+ f a function of the 3-momentum p and the following facts are well known : i ~ has
a unit norm ~(p)[ dp _- ~ , the,, I~(p)l i s the probability density for de- \ J / -9
tecting the particle with momentum p (independently of its position), and
[~(x, t ) [ 2 where
-~ r -~ -i( t '-~ -~ 2/2m) -~ q)(x~t) z j ~ ( p ) e eIP x dp , (7)
is the probability density for detecting, at time t , the particle at
dently of its momentum).
x ( indepen-
In the relativistic quantum case, l~(p) l 2 is again the probability den-
sity with respect to momentum° On the other hand,the concept of a well-defined space-
time ~osition of a particle on the microscopic level is now to be abandoned (for
physical and related mathematical reasons). However one may still define a space-
time wave function, analogous to ~/ through the formula :
j ,x) (8) f(x) = ~(p) e -±p'x d~(p) (x = x 0
(d~(p) z ~(p2-m2) O(p. O) d4p) , or other related quantities and, although these quan-
tities have no longer any interpretation, in general, in terms of probability den-
sities, some features of the non relativistic case do remain valid, on the macroscopic.
level. We below describe the properties which will be needed later and which are a
refined expression of the idea that the probability of finding the particle in ma-
croscopic space-time regions is "negligibly small" if f(x) is itself "negligibly
small" in these regions.
Consider a set of wave functions %9~(p) of the form :
q)~(p) : X(P) e - ~ (P-P) (9)
112
where X is C a (i0e. infinitely differentiable), has a compact support around
the point P z ( ~ , ~) , and is moreover locally analytic at P , and where Y
is a positive constant (y > O) .
above.
The corresponding space-time wave functions will be denoted by f as
Let V(p)be the line issuing from the origin in space-time and parallel
2 = m 2) and let V(X) be the (closed) set to a given 4-vector p 6 M (Po > O , p
of all V(p) with p in the support of X ; V(X) is called the ~e!ocit[ cone of X :
time
/ ~ / V(X)
Space
Figure 3
Using the methods of lenmla la) in Appendix I, one checks "I~-2"[ ][ ] that, being
given any open region R in space-time whose closure does not intersect V(X) , the
following bounds are satisfied for all positive integers N and all sufficiently
s~ll Y (o ~ y < YR"fR ~ o) :
D N -~ Max If (x) l < ' e , ~ > 0 (10) x c R I+T N '
where R~ is the region obtained from R by the transformation x ~ T x o
around P
The constant ~ > O depends only on the real analyticity domain of X
The constants D N (and 7 R) depend on X and R, but not on 7 or ~.
We note that a similar bound is also obtained at N : 0 whenever the
closure of R does not intersect the line V(P) itself :
-13 ¥,~ ~ax ]%(x) l < D e ; ( o ~ , < , ~ )
x ~.R T
( 1 0 ' )
113
where ~ > 0 may now depend on R , (if the closure of R intersec~V(x)).
If we next consider the displaced wave functions ~u obtained from ~
by the translation ~u , where u is a given 4-vector, (see Eq.(6)), it is readily
checked (in view of the identity f~u (%'u') n f (~(u'-u))) that f~u again satis-
fies the bounds (I0)(I0') whenever the closure of R does not intersect the dis-
~..laced velocity cone vUfx) , whose apex is now the point u instead of the origin,
respo the displaced line vU(p) .
According to the idea on macroscopic space-time localization mentioned
earlier, it is correspondingly assumed that the probability ~(R ) of finding the ~u
particle whose wave function is ~ in the region R (which is macroscopic for
large ~) also possesses analogous exponential fall-off properties in the ~ ~ ~ limit.
[7] Macroscopic causality
~u
Consider a set " - [~k kl
the above mentioned form :
of displaced initial and final wave functions of
~kT Tuk (Pk) z Xk(Pk ) e-YT(Pk-Pk ) el~(Pk°Uk) (ll)
Then, tile initial and final particles have, in the ~ ~ ~ limit, pro-
parties which are analogous to those of the classical model of section I, with the
4-n]omentum Pk being possibly replaced by the set of all 4-momenta Pk in the
support of Xk :
v i) The probability of finding particle k with a 4-momentum Pk vanishes
v if Pk does not lie in the support of ~k ' resp0 decreases exponentially as
-2~(Pk-P k) ' lies in that support but is still diffe- e j in the z ~ ~ limit~if Pk
rent from P k
,. t~
ii) The space-time trajectory of particle k is the displaced velocity cone
u k u k V (X k) , rasp° V (Pk) , in the sense of the above mentioned exponential fall-off
properties of Pk(R~) when the closure of R does not intersect it.
On the other hand, the interaction laws also become analogous, in the
~ ~ limit, to those of the classical model, in view of the physical assumption
that any energy-momentum transfer which cannot be attributed to (stable) physical
particle~ according to the classical laws gives effects that are damped exponentially
114
with distance, (i.e. with the space-time dilation parameter ~)o
Semi-classical arguments [IO] then lead to the following first statement
of the maerocausality property :
Macrocausality (I)
Let K be any given compact set in U-space of displacements U n lUkl
which are all non causal at p n {pkl , (in the sense of section I), whenever each
Pk lies in the support of. X k , resp° are non causal at P = Ipk} o Then the tran-
sition probability W for the scattering process has the following bounds for all
positive integers N and all sufficiently small 7 (0 ~ y ~ YK ' YK > O) :
N , > 0 k ~ i + •
(12)
resp., has the bounds :
w(I~k~ }) < c' e / k
~t , > 0 , (12')
' C',a'ty~may depend on K and [Xk} ~ but are independent of Y and ~ • where C N ,
Remarks :
~Uk(~) 12 i) The left-hand side of (12)(12') is ISij(I~k~ }) (see Eqo(2)).
2) In view of the bound (3) (W < i), it turns out [2] that the bound (12') ca_~n
be derived from the bound (12) (for a different set of functions Xk with smaller
supports around Pk ) .
Conversely, it can also be proved that (12') implies (12) [11] .
Therefore either one of the two bounds may be removed from the above
statement.
if ~ is causal, but is not connected at p z Ipkl , (resp . at Now,
P n IPk}) , i.e. corresponds to a causal configuration which is composed of several
disconnected parts linking the external (initial and final) particles of various
subsets (I,J) of (I~J) , then the physical ideas discussed earlier lead to
the further requirement that the remainde.r W -H ~ W~ should again satisfy bounds
of the form (12)(12')0
It is proved in the second paper of [I] that a slight generalization
of this property (which involves the same physical ideas) yields an analogous
115
property, without phases, for
~Uk(*) ~Uk(~) SIj([~)k, [ ] ) - ~ S _ ([q)k'~
I J ( I ,J ) ]k c ( I ' ,~ ) ) '
and that the latter, yields (if a few exceptional situations are excluded) the
following bounds for the connected amplitudes (defined in the Introduction) :
Macrocausality (II)
When there is no connected causal U in K at p z Ipkl , with Pk
in the support of Xk , reSpo at P z [pkl ~ then the following bounds are satisfied
for all sufficiently small y (O < y ~ 7K , 7K > O) :
ISIj TUk (~) C ( [q:'k'~ I t < N -~ I+~N e (13)
resp. Is II < c , ( 1 3 ' )
where C N ~ C and ~ are independent of y (and T).
The bound (131 ) can again be derived from (13), and conversely.
Macrocausality-II is slightly stronger than macrocausality-I, and in
fact implies it, as easily seen°
Essential s ppport properties
The bounds (12) and (13) directly provide the following essential support
properties :
The essential support of Stij~<j , resp. TIj , at a point P z Ipkl of
M(Ij) , is (contained in) the set of causal directions ~ at P (in the sense of
section I), resp. is (contained in) the set of causal directions ~ at P which
correspond to connected configurations.
zUk(~) e (I~k ~ }) (for instance) can be written To see this, we note that SIj
116
in the form (~) :
TIj(p) ~(p) e-y ~@(p) ei<~ U,p> x
icI j(J k
(14)
where p z [pkl , X(p) z £ Xi (pi)I~ Xj(Pj), ~(p) z /~ (~k-~k)2
i(I jcJ k
Since functions X with arbitrarily small supports around P n~y be chosen (by
using functions Xk with sufficiently small supports), the above essential property
for TIj directly follows from the definitions of Appendix I-B (see also lemma 4 at
the end of section E).
III- PHYSICAL-REGION ANALYTICITY PROPERTIES
In view of Theorems 1 and 2 of Appendix I-C, the essential support pro-
perty of TIj and the results on causal displacements given in paragraphs a)b)c)d)
at the end of part I, directly progide the following analytieity properties :
a) If p = Ipkl is not a +a - Landau point, then TIj is analytic
at p .
b) In the neighborhood of a +a -Landau point p which lies on one sur-
face L+(G)o , TIj is the boundary value of an analytic function from well defined
"plus i s'' directions.
More precisely, let q z lq~l , ~ = 1 .... 3( [I I+ IJ I-4) . he any system of
real analytic local coordinates of M(I J) at p , and let ~ be a real analytic
function of q such that i) L+(G) is locally represented by the equation 6(q)~O, o
and ii) 6(q) > 0 on the physical side of L+(G) o Then, in a neighborhood N of p o
T z lim F(q+i~) (15) IJ
t i e c
in the sense of distributions, where the directions ~ are those of a cone C
(~) The 0-functions of the energies can be removed from the integrand of
(14) in the neighborhood of a given P in M(Ij) o
117
strictly contained in the half-space V~p, . ~ > 0 . This cone can be chosen arbi-
trarily close to this half-space if N is chosen sufficiently small.
c) In the situation of paragraph c) at the end of section I, the same
result holds with the half-space V$o ~ > 0 being now replaced by the (non empty)
intersection of the half-spaces associated with the various contractions G' of G.
Since the remaining points belong to lower-dimensional submanifolds of
M(Ij) , the above results immediately provide (theorem 2 of Appendix I) :
Property
For each given physical process I ~ J , there exists a unique analytic
function FIj(~) to which TIj is equal at all non + ~ -Landau points, and from
which it is a "plus i~" boundary value at all the above mentioned +a - Landau points.
d) Finally, in the neighborhood of a point p which lies at the inter-
section of several +~ -Landau surfaces with no common parent, TIj can be decom-
posed as a sum of boundary values f~ (in the sense of distributions) of analytic
F~ , each of which from the directions of a cone C~ associated with one of
the graphs, or parent graphs, involved at p o
Independence propert[
When p is a "type I" point (i.e. when the various cones C~ of causal
directions at p are disjoint), it was moreover (partly) proved in [I] that
each f~ "has the same analytic properties, in the neighborhood of p, as if no
other parent surface was involved", namely that i) f$ is analytic at all real
points which do not belong to the corresponding parent surface L+(G~) or to the
surfaces L:(G~) of the "contractions" G~ of G~ i~volved at p, and if) that
it obeys the "plus i8" rule of paragraphs b) and c) at these points.
When the cones ~ C~ are disjoint, this is in fact a direct consequence
of properties a) to d) above (see Appendix I-D).
Consider for instance a point p' (of the neighborhood of p)
which lies on no +~ -Landau surface, and at which the essential support of TIj is
therefore empty (TIj is analytic)° From property d), the essential support of f~
(~)
Defined in a do1~in of the complexified manifold M(Ij) of M(Ij) .
118
:v at p' is contained in the dual cone C~ of C~ .On the other hand, since :
f~ = TIj ~ f~' ~'~
it is also contained in the union, for ~' ~ ~ , of the dual cones ~, of
C~, o Since the sets C~ and U C~, are disjoint (for ff sufficiently
small) the essential support at p' ~-he ~ e , l~ ~ relor empty, and f~ is analytic
there°
In view of theorem 3 of Appendix I , this "independence property" also
follows at type II points. Moreover, as a result of this theorem, global decomposi-
tion properties of TIj into sums of boundary values (in the sense of distributions)
of analytic functions can be derived in M(Ij) . The analytic functions involved are
independent of p in M(Ij) , although the directions ~ from which their boundary
value is obtained may depend on it. They are moreover analytic at all real points
outside given +~ -Landau surfaces.
A detailed analysis of these decomposition properties, which are the
generalization of the above property i), when the points p of paragraph d) are
included, will be given elsewhere°
119
REFERENCES
[1]
[2]
[3]
[4 ]
[5]
[6]
[7]
D. IAGOLNITZER and H~P. STA-PP ; Comm. Math. Phys. 14, 1 5 (1969)
Further related results, also obtained in collaboration with Dr. H.P.STAPP,
have been given in
Do IAGOLNITZER ; in Lectures in Theoretical Physics, ed. by K. Mahanthappa
and W° Brittin, Gordon and Breach, New-York (1969), p.221.
- D. IAGOLNITZER ; Introduction to S-matrix theory, AoD.T. 21 rue Olivier-
Noyer, Paris 75014, France (1973)o
Jo BROS and D0 IAGOLNITZER ; Local Analytic Structure of Distributions ;
I - Generalized Fourier transformation and essential supports (In this
volume).
The results of this work, which are also contained in a large extent
in Ch. II.C of reference 2, follow from an elaboration of the previous re-
sults of reference 1 and of :
J. BROS and D. IAGOLNITZER ; in Proceedings of the 1971Marseille meeting
on Renormalization Theory, and Ann. Inst. Poincar~, Vol. 18, nO2 (1973),p147,
J. BROS and D. IAGOLNITZER ; Local Analytic Structure of distributions;
II - General decomposition theorems°
The results of this work which are extensions of those of references 1
and 3 are also very closely linked to, and have been inspired ~o a large
extent by, the results of reference 5 below :
Mo SATO~ T. KAWAI, Mo KASHIWARA ; in "Hyperfunctions and Pseudo-Differential
equations" - Lecture Notes in Mathematics,Springer-Verlag, Heidelberg (1973).
The basic results of Professor Mo Sato and his coworkers have been
presented in the course by
J. GHAZARA~, A, CEREZO, A. PIRIOU (In this volume)
Fo PHAM ; Micro-analyticit~ de la matrice S (in this volume).
- The statement of the macrocausality property given here is essentially that
of reference 2, which is a slight improvement of that given in reference I.
t 2 0
I s ]
[9] -
[ l o ] -
[ii] -
[12] -
The latter was itself inspired from previous macrocausality properties pre-
sented in various situations in :
C. CHA~NDLER and H.Po STAPP ; J. Math° Phys. 90, 826 (1969)
R. OMNES ; Phys. Ray0 146~ 1123 (1966)
F. PHAM ; Ann. Insto Henri Poincar6, Vol.6, n°2~ 89 (1967)
Earlier pioneer works on maerocausality are due to :
G° WANDERS ; Nuov. Cim° 14~ 168 (1959) and Helv~ Phys. Acta 38, 142 (1965)o
On the present physical status of macrocausality, the interested reader may
also be referred to :
HoP. STAPP ; S-matrix theory, in preparation and to Lawrence Berkeley Labo-
ratory reports by the same author, in particular° "Macrocausality and its
role in Physical Theories", and "Foundations of S-matrix theory".
The interest of this classical model and the possibility of its relevance
to quantum relativistic physics have been for instance emphasized by
C. COLEMAN and R. NORTON ; Nuovo Cim. 38, 438 (1965)
who have noticed that the + a - Landau equations are equations of classical
kinematics.
The basic results on the +a- Landau surfaces and the causal displace-
ments at + ~ -Landau points are due to
C. CHANDLER and H.P. STAPP, op°eit0
F. PHAM, ON. cit.
A detailed presentation of the model, with simple examples,will also be found
in ChoII-B of reference 2
C. CHANDLER ; Helv. Phys. Acta 42, 759 (1969)
A more detailed physical discussion is given in references I and 2 . See
also further aspects of this discussion in the works by Dr. H.P. STAPP
mentioned at the end of reference 7.
Microlocal essential support of a distribution and decomposition theorems,
an Introduction, Saelay preprint, and in this volume.
The variables p and iU correspond to the variables x and ~ respectively
of this mathematical Appendix.
This follows from results proved in Jo BROS and Do IAGOLNITZER (in prepara-
tion).
APPENDIX
MICROLOCAL ESSENTIAL SUPPORT OF A DISTRIBUTION
AND DECOMPOSITION THEOREMS - AN INTRODUCTION
Do IAGOLNITZER
DPh-T CEN Saclay BP n°2 91190 Gif-sur-Yvette
FRANCE
This text is a short mathematical introduction to the notion of essential
support of a distribution in the "microlocal" sense and to the corresponding decom-
position theorems of distributions into sums of boundary values of analytic func-
tions. Results concerning the multiplication of distributions and their restrictions
to submanifolds are also mentioned at the end° The details are given in references i,
2 (and in the references quoted therein), in which a more general "non microlocal"
notion of essential support and corresponding results are also given°
It can be checked (see section C) that the microlocal essential support
coincides with the "singular support" or "singular spectrum" introduced independently
by very different methods (and for general hyper~unctions) by Professors SATO,
KAWAI, KASHIWARA [a 114 ](m)
A - GENERalIZED FOURIER TR&NSFOR[~ATION
Let f be a tempered distribution defined in the n-dimensional real vec-
n of a variable x z (xl,,. . Xn ) . For our purposes below, we assume tor space R(x ) _
for simplicity that f has a compact support°
The generalized Fourier transform Wo (f) at a point Xo (of R(x)n ) is n+l
defined in the real space R(~,~o) of the variables ~ z ~i' °'' ~n and of a sup-
plementary variable ~o by the formula :
f -i~.x -~o ~ (x-x) lWo(f) 1 (~,~o) = f(x) e o dx , (I)
It may also coincide with the "analytic wave front set" introduced in [5]~ although the corresponding analyticity properties have not so far been exhibited
~n general t to our knowledge~ in this latter case.
122
n 2
w h e r e ~ ( x ) : x 2 : ~ x . ° / z
i : l
The function ~ may also be chosen to be a~ other function with analogous
local properties in the neighborhood of x z O : ~ is analytic, ~(z ~) n (~(z)) ~
where z z x+iy = (zl~°.oz n) and ~ is complex conjugation, ~(0) z O and
has a critical point (V#(O) z O) "with positive signature" at x = O . For simpli-
2 city, the reader may keep in mind the function ~ z x and the dependence of W °
on ~ will be left implicit.
We note that lWo(f)l (~,O) is the usual Fourier transform of f . The
first following lemma holds :
Lemma i : Let g be a C ~ function with compact support around x ° Then g o
is locally analytic at x if and only if its generalized Fourier transform Wo(g) o
at Xo satisfies the following bounds with uniform constants ~ > 0 , _Yo > 0 , and
C < ~ : N
c - Yl l l o(g) l < " (=) +let" e
for all positive Y less than Yo ' and all positive in%egers N o
The bounds (2) at Y : O express the C ~ character of g o It is well
known that the local anal yticity at x ° cannot be expressed in terms of an exponen-
tial decrease of the usual Fourier transform of g . Lemma 1 tells us that it can be
characterized in terms of the above exponential decrease of the generalized Fourier
transform°
Pro o~ : a) The proof that local analyticity implies (2) is obtained by dis~or-
ting the part 0 < ~ (x-x) < a of the integration domain in (I) (with a suffi- - o - (~)
cie ntly small) in the an alyticity domain of g
For N > 0 , appropriate integrations by parts are also used.
b) For the converse proof, it is useful to consider, together with
W (g) , a set of n functions W (g)(k z 1 °on) of ~'~o ' and z , defined by O k ~"
with b
(m)Use a distorted contour lying of the surface
y + b (Re ~(x - x o + iy) -a) = O
appropriately chosen (for each direction of ~) and Ibl sufficiently small.
123
f [email protected]' -@ ~(x'-x o) [Wk(g)}(~,~o,Z ) r i g(x') e o Pk(X'-Xo,Z-Xo) dx' (3)
where ~k(Z,Z') = z k + z~ if ~ = z 2 ; more generally ~k is an analogous[locally
analytic)function such that
n
~ ( z ) - ~ ( z ' ) -- ~ . p k ( z , z ' ) (z k - Z k ) o (4) kzl
In view of this property, one checks that the following differential form of degree
n in (~,~o) -space :
n
WX (g) = e i ~ ' x +~o~(X-Xo) y ( _ l ) k Wk(g)(~,~o,X)d~oA . . . d ~ k A . . . d ~ n (5)
k'~-O
where the n o t a t i o n d~k means t ha t t h i s f a c t o r i s omi t t ed , i s c losed (dW x = O) f o r
each value of x . On the other hand :
/ Wx(g) i - /Wo(g)(~,O) ei~'x dx 1
~ =o o
-- (2%) n/2 g(x) . (6)
We show below that all functions Wk(g) satisfy bounds of the form (2)
if Wo(g) does~ with constants CN,ap7 ° which can be chosen independent of k z O,...n
and of z in any bounded complex neighborhood g of x z 0(~)o
In view of these bounds and of Stokes theorem, the integration surface
~o 0 in (6) can be distorted on the surface ~o z 7oI~ I when ~(X--Xo)< a , and the
local analyticity of g at x is in turn derived from the exponential bounds of o
the functions W k on this surface.
Finally, the above result on Wk(g) is proved by considering it as the
Fourier transform (see (3)) of g e -~°(I-'1)~ -~°~ x Pk ~ e where 0 < ~ < 1 and
is a C ~ function with compact support, which is equal to one in the support of g o
Correspondingly :
IWk(g)}(~,~o,Z) ~ / d~' Wo(g)(~-~',(1-~) do )
x Wo(Pk~) (~',m ~o ) (7)
function
(.) Sgme restrictions on the width of N may have to be added for a general (with no inconvenience for the purposes of this text).
124
The announced result is obtained from the assumed bounds on W ° (g) and
from the fact that Wo(~k ~) satisfies analogous bounds (which are shown to be inde-
pendent of z in N) in view of the local analytieity of pk ~ at x ° with respect
to x' (see part a) of the proof>° To see this, divide for instance the integration
domain in (7) into two parts : I~' I < ~ I~I, I~' I > ~ I~I o
B - ESSENTIAL SUPPORT OF A DISTRIBUTION (possibly defined on a manifold).
n C ~ Being given a distribution f defined on ~(x) ' and functions X
with compact support around x ° , locally analytic and different from zero at x ° ,
n we denote by Zx(f) the set of directions ~ in R(~) along which Wo(Xf ) does
not satisfy bounds of the type (2) : namely, a direction $ is in the complement o
of Zx(f) if there exists a neighborhood U of ~o , constants ~ > O, 7o > 0 ,
and C < ~ such that the bounds (2) be satisfied by I W (Xf) l for all points N ~ o
The following lemma holds :
Lemma 2 : Z X (f) C ZxI(f) whenever X2 has its support in the region where
X1 ~ O around 2 x O
(This is proved by writing X2f n Xlf x X2XI l-
gument analogous to that used at the end of section A).
and using a convolution ar-
Definition : The essential support Z (f) of f at x , is the limit of x o
~x(f) when the width Isupp XI of the support of X around x O tends to zero.
In view of lemma ~, this definition is clearly independent of the sequence
of functions X considered. Theorem I of section C also shows that it is independent
of the function ~ considered , since the analyticity properties
involved there do not depend on ~ °
If f is defined on a manifold M rather than on ~n ~x , (f) is defined
o . as above for any system of local coordinates at x O 0 Theorem I of sectlon C again
ensures that it is a well defined subset of the cotangent vector space T ~ M at x o
x to M , independent of the choice of local coordinates. o
The essential support (in the "microlocal" sense) Z(f) of f is the
closed subset of the (sphere) cotangent bundle T~M defined as
U x × Zx(f) . (8)
X6 M
125
q Remark : Z (f) has been defined above as a subset of directions ¢ o We
x o n
also identify it(sometimes)with the corresponding cone in R(~) with apex at the
o r i g i n , f r o m w h i c h t h e o r i g i n i s r e m o v e d .
C - A LOCAL DECOMPOSITION THEOREM
We first fix some notations : C~ will denote an open convex salient cone
in ~-spaee with apex at the origin , and C~ will denote its open dual cone in
y-space. Then the following theorem, which is an extension of lemma 1 holds :
Theorem I : There is equivalence between the two following properties
i) ~x (f) is contained in the union of a finite family of o
c o n e s C~ °
ii) There exists a neighborhooa ~ of x , and boundary values o
f~ in U (in the sense of distributions) of analytic functions F~ , from the
respective directions y of the cones C~ , such that :
f = f8 in
F
Proof : a) The proof that ii) implies i) iS obtained by introducing the
function : 1
~(x-x )-~ h(x) = e o when O < #(x-x ) <
o -
n O outside this region (9)
with ~ such that the region 0 ~ ~(X-Xo)<~ lies in ~ . The fact that ?h(f~) is
contained in C~ , is obtained as in lemma 1 by distorting this region in the ~na-
lyticity domain of F~ (ana of n) (~~) o
b) For the converse proof, we introduce a C ~° function X , locally
analytic and different from zero at Xo , with support such that Wo(Xf ) satisfies
the bounds (2) outside [) ~ with uniform constants CN '~'%/o ~ As in lemma I
(~) ~n is ~ne closure o~ C~ ; i.e. the directions y are those of open cones slightly larger than C8 .
(~e~) h is analytic in a complex (open) neighborhood of O < ~(X-Xo)<~ , and
one checks that hF ~ as well as its derivatives, is bounded on the distorted con-
tours, in view of the properties of h .
126
it can be shown that all functions Wk(Xf) also satisfy analogous uniform bounds
outside U C~ o We now consider~ as in Eq.(6), the equality :
kf(x) = / w (Xf) (io) J x
t ° =o
The contribution to this integral of the complement of U C~ in the
hyperplane ~o m O is again treated by distorting this domain in ~he region to > Oo
It provides a C ~ function which is a sum of boundary values (in the sense of func-
tions) of analytic functions FI~ from the directions y of the cones C~ ~ when
x is in the neighborhood of Xo(~) o The contribution of ~ C~ in the hyperplane
~o m O is treated by the usual Laplace transform theorem and easily provides a
sum of boundary values (in the sense of distributions) of analytic functions F2~ ,
again from the directions y of the cones C~ 0 The result is therefore provedo(~)
Remarks : l)[l]Being given a function ~ , a given point x ° , a given
> 0 and the corresponding set ~ (O < ~(x-x¢)<~) , a notion of fl -essential sup-
port in (~,~o)-Space, which generalizes the set ~Xo(f) in ~-space, can be intro-
duced by methods which are similar to those used above° Theorem 1 then appears as
a limit case of a more general decomposition theorem, which provides corresponding
decompositions of f in ~ into sums of boundary values of analytic functions
whose analyticity domains can be specified. The cones C~ in y-space from which
the boundary values are obtained are here independent of x in ~
A corollary of ~his result is the usual generalized edge-of-the wedge
theorem, over any bounded set fl (with smooth boundary), with possibly precise
specifications of the analyticity domains involved (see refol)o
2) According to a definition given in [4] , f is said to be "micro-
(Xo,~ o analytic" at a point (Xo~ o)~ or ) is said to be in the complement of the
"singular spectrum" of f, if f can be loce, lly decomposed, in the neighborhood ^
of x ° , as a sum of boumdary values f~ of analytic functions F~ from
directions y which all lie in the region ~o0Y < O o
(~)
In contrast to lemma i, this contribution is not analytic, because of
the parts of the surface joining the boundary of U C a ~ at ~o = 0 ~ to the surface
t o =
To obtain cones "slightly larger" than the cones C~ , replace the cones
C~ by "slightly smaller" cones whose union still contains the closed set ~×o(f) .
127
The part ii) ~ i) of theorem I ensures, as easily checked, that o
is then also in the complement of the "essential support" Z (f) of f at x x o o
Theorem 1 then provides the following lemma :
Lemma 3 : Being given any compact set K of directions ~ such that f is
m i e r o - a n a l y t i c a t ( x , ~ ) f o r a l l ~ i n K ( i n t h e s e n s e o f t h e a b o v e d e f i n i t i o n ) , o
f c a n b e d e c o m p o s e d , i n t h e n e i g h b o r h o o d o f x , a s a sum o f b o u n d a r y v a l u e s f e o P
o f a n a l y t i c f u n c t i o n s F~ , w h i c h a r e i n d e p e n d e n t o f ~ i n K ( ~ ) .
D - GENERAL (MICROLOCAL) DECOMPOSITION THEOREM
Being given a distribution f with essential support ~(f) , theorem 1
provides decompositions of f into sums of boundary values of analytic functions at
each point x , but gives so far no information on the links between decompositions o
obtained at different points° Theorems2 and 3 below provide this information in
various situations,
Theorem 2 below is a direct corollary of theorem 1 (in view of arguments
of analytic continuation).
Theorem 2 : The following properties are equivalent :
i) the fiber ~ (f) of Z(f) at all points x of an open set x
a closed convex salient cone Cx (which may be depend on x is contained in and
may be empty at some points)°
ii)" f is the boundary value, in fl , of a unique analytic func-
tion F from directions y which are those of the dual cone C of ~ , at each x x
point x (F is analytic at x ~ if ~ is empty), x
We next state :
Theorem 3 : Let ~(f) be contained in the union of a finite family of closed
subsets Z~ of T~ , whose fibers (Z~) x are closed convex salient cones for all
values of ~ and x (in ~.
(~) For any $o in K^, each boundary value f~ can still be obtained
from directions y such that ~o0y < O o
128
support
each f~
Then f is equal, in fl , to a sum of distributions f~ whose essential
Z(f~) is contained in Z~ , for each value of ~ o According to theorem 2,
is the boundary value of a corresponding analytic function F~ o
We give an idea of the proof below in the case when the fibers ( Z ~ ) x are
disjoint for every point x in fl ((g~) x ~ (Z~,)x z ~ V ~,~',~ ~ ~' , g x • ~).
Consider a given point x ° in O and a family of disjoint cones C~
c o n t a i n i n g t h e r e s p e c t i v e s e t s ( 2 ~ ) x o T h e o r e m 1 e n s u r e s t h e e x i s t e n c e o f a n e i g h -
b o r h o o d ~ o £ x w h e r e o o
f = E f~ '
w i t h E x ( f ~ ) C ~ , V~ , V x • ~ ,
~t We now show that £x(f~) C (E~) x : being given any family of (disjoint) C~
containing the sets (Z~) x , theorem I again ensures the existence of a neighborhood
U C U of x such that :
with
Therefore, one has :
add sinee the cones C~ are disjoint, it clearly follows that
empty, for every ~ o Hence :
Zx(f ~) c Zx<g ~) c ~ , v~
z x (f~-g~) is
Since the cones
theorem 3 is proved in Uo
-4 C~ may be chosen arbitrarily close to the sets (Z~) x ,
The proof of theorem 3 in ~ then results from an application of Cousin
theorem (see part V reference i).
129
When the cones (Z~) x are no longer disjoint, the proof is somewhat more
complicated and we refer the interested reader to [2] .
Remark : Theorem 3 is very closely linked with basic results of reference 3 on [ ]
the "sheaf of mlcrofunctlons . It seems possible ~6 to derive somewhat more 9efined
results of the latter type, which, in particular, also hold in the framework of
distributions,by making use of the generalized Fourier transformation, together with
an application of Cousin theorem which is adapted from [3] o
E - MULTIPLICATION OF DISTRIBUTIONS AND RESTRICTIONS TO SUBMANIFOLDS [7]
a) Product of two distributions
Theorem 4 : A sufficient condition for the product
b~tions to be well defined is
~ (fl) ~] ( - Z ( f2) t --
flf2 of two distri-
(12)
The fiber Ex(flf 2) of E(flf 2) at x is contained in the set
~x fl ) ÷ Zx(f2) of vectors ~ of the form ~i + E2 ' E1 ~ Zx(fl) ' ~2 ~ Ex(f2) "
Proof : This problem is clearly a local problem in the neighborhood of each
point x . It is easily solved by using theorem i and the natural definition of the
product of two distributions which are boundary values of analytic functions from
comn~n directions (see [I] )o
Alternatively, the product Xlf I x x2f 2 , where XI,X 2 are C ~ functions,
locally analytic and different from zero at x , and with sufficiently small support
around x is defined as the inverse Fourier formula of :
(Xl f l ) (~ ' ) (X2f2) (E-~') dE' (13)
where xif i is the Fourier transform of Xif i .
In view of the slow increase of (Xlfl) and (X2f 2) and of their rapid
decrease outside Zxl(fl ) and ~x2(f2) respectively (bounds (2) at 7 z 0), this
integral is in fact convergent for all E , and defines a slowly increasing function
of E , whose Fourier transform is therefore a well defined distribution. To cheek
130
that Wo(Xlflxx2f2 ) does satisfy bounds of the type (2) in all directions ~ which
do not belong to Z (fl) + Zx(f 2) , write (in the s%me may as in (6)) : Xl
lWo(Zlf1×x2f2)l (f,~o) = /dC lwo(x~f~)I (4',(~-~)~o)×lWo(X2f2)l(f-f',~ o) (~4)
where 0 < ~ < i , and use the bounds (2) for Wo(Xlfl ) and Wo(X2f 2) outside
ZxI(f I) and Zx2(f 2) ; the result is ensured by the fact that the intersection 6f
neighborhoods (in ~'-space) containing the cones ~Xl(fl) (with apex at the origin)
and ~-ZX2 (f2) (with apex at ~) is empty°
The announced property of E (f f ) follows from the fact that XI,X 2 x i 2 , .
can be chosen with arbltrar~ly small supports o
b) Restriction of a distribution to a submanifold and related results
Theorem 5 : Let M be a submanifold of on open set Q of R n and let
N be the conormal bundle to Mo A sufficient condition for a distribution f on Q
to have a well defined restriction rim to M (as a distribution) is :
S ( f ) ~ N = ~ (15)
The fiber ~x(flM) of this restriction at a point x of M is contained
in the quotient Z (f)/g of Zx(f) by the vector space ~ conormal at x to Mo x x x
P~o~ : it is also sufficient here to solve the problem locally, in the
neighborhood of each point x o As in subsection a) the proof may be derived from o
theorem 1 ( s e e [ 1 ] ) o A l t e r n a t i v e l y , l e t M be d e f i n e d i n t h e n e i g h b o r h o o d o f x o
by a set of g 6quations L.(x) = O (i = 1,0oog) where each L. is a real analytic l 6 l
function of x , and let 6(M) = ~ 6(Li(x)) . It can be checked that this is (local- . . i Z
l y ) a w e l l d e f i n e d d i s t r i b u t i o n w~ os e e s s e n t i a l s u p p o r t a t x o i s g x o . A c c o r d i n g
t o t h e o r e m ( 1 5 ) , ~(M)× f i s t h e r e f o r e ( l o c a l l y ) a w e l l d e f i n e d d i s t r i - 4 a n d t o
b u t i o n , a n d E x ( ~ ( M ) x f ) Q 2 x ( f ) + g x o o o
The definition of /~IM follows. To see that EXo(flM) C Z x (f)/Nx
o o one may for instance choose a set [ql of n-~ local coordinate of M at x
o among the set of n variables Xl " .. x n , and consider the subspace r of ~ )
( * ) It is easy to cheek that the definition of flf2 "at x" does not
d e p e n d on t h e c h o i c e o f X 1 a nd X2 o
of
131
vectors whose f~ components, corresponding to those excluded above~ are fixed at
zero. It is easily seen that r (h N z 0 and that there is a well defined i-I x
n o c o r r e s p o n d e n c e be tween 1" and N(~) /N x . The announced r e s u l t t h e n e a s i l y f o l l o w s o
from the definition of the essential support at x of distributions defined in n_y ~ o
~(q) , the needed bounds (2) being ensured by the corresponding bounds on Wo(f~(M))(~),
The last part of the proof also provides the following lemma :
Lemma 4 : Let f be a distribution defined on M, f 5(M) be its product with
n ;T}tea 5(M) ( d e f i r ~ d in N ( x ) ) and 2Xo ( fS(M)) i t s e s s e n t i a l s u p p o r t a t x °
2 ( f S ( N ) ) be i n v a r i a n t by a d d i t i o n o f v e c t o r s i n g af~d Z ( f ) i s t he quo- X X O J X o o
tient Z (f6(M))/N~ ° @
x o
n (~) Note that the restriction of a function ~ defined in ~(x) to M
n-6 which belongs there to the class described below provides a function ~ in ~(q)
(~q.l).
132
REFERENCES
[1]
[2]
[3 ]
[4]
[~]
[6 ]
[7]
Jo BROS and D. IAGOLNITZER ; Local Analytic Structure of Distributions
I - Generalized Fourier transfornmtion and essential supports (in ~his
volume). The method of the generalized Fourier transformation was first
introduced in :
D. IAGOLNITZER and HoP. STAPP ; Comm. Math. Phys. 14, 15 (1969)
where lemmas i and 4, and theorems 1 and 2 o~ the present text were essen-
tially proved, in a less elaborate form(and in the framework of S-matrix
theory)°
Further developments have been given in :
J. BROS and D. IAGOLNITZER ; in Proceedings of the 1971Marseille meeting
on Renormalization Theory, and Ann° Inst. Poincar&, Vol. 18, n°2 (1973)p.147
Jo BROS ; in Comptes-Rendus de la RoCoP. 25, Novo1971, Strasbourg, CNRS
D. IAGOLNITZER, in Ch0II-C of Introduction to S-matrix theory, A.DoT., 21
rue Olivier-Noyer, Paris 75014~ France (1973).
Jo BROS and D. IAGOLNITZER ~ Local Analytic Structure of distributions ;
II - General decomposition theorems.
The results of this work which are extensions of those of references 1
and 3 are also very closely linked to, and have been imspired to a large
extent by, the results of reference 3 below :
M. SATO, To KAWAI, Mo KASHIWARA ; in "Hyperfunctions and Pseudo-Differential
equations" - Lecture Notes in Mathematics, Bpringer-Verlag, Heidelberg (1973).
The basic results of Professor M. Sato and his eoworkers have been
presented in
- J. CHAZARAIN~ A. CEREZO, A. PIRIOU (In this volume)
- Lo HORMANDER ; Comm. Pure Appl. Math. 24, 671 (1971)
Jo BROS and D. IAGOLNITZER ; in preparation
Analogous results have been proved in references 3 and 4o
These results can also be extended to the case of "non mierolocal" essential
supports by using somewhat more refined arguments, J. BROS and D. IAGOLNITZER
in preparation.
UNITARITY AND DISCONTINUITY FORMULAE
David OLIVE
CERN, Geneva, Switzerland
I. Introduction
I would like to thank the organizers for enabling me to be here this week in
Nice to hear about the new developments concerning hyperfunction theory which seem
so well suited to rendering more precise and rigorous certain ideas and results in
S-matrix theory which I shall describe. However, first I must apologize for the
fact that I have not myself worked on these aspects of S-matrix theory since six
years ago, and secondly for the fact that I am not a mathematician and consequently
have not fully understood the new mathematical language that I have heard this week.
I want first to make a historical digression to explain my own point of view,
which is somewhat more old fashioned than that of Pham [12] and Iagolnitzer [9] in
the preceding talks.
Study of the S-matrix p~r 8e began its activity in the early 1960's [3,17,14,
8,15]. The S-matrix was expected to satisfy many interesting properties, unitarity
and analyticity among them, but its detailed structure was not well understood. It
was believed by some physicists that the general requirements were so restrictive
that there might be a unique solution -- that which describes and contains all use-
ful information about the physical world. Study of these requirements might then
lead to this solution. I do not think I believed this then, and probably very few
do today, yet in a sense it has become much more plausible with the advent of the
dual theory of the S-matrix -- but that is another story, on which I currently work,
with its own intriguing brand of mathematics [7,18]. What I did believe then was
that the study of these general properties could lead to a more detailed understand-
ing of the structure of many-particle S-matrix elements (which are nowadays observed
experimentally) and, in particular, of the singularity structure (in the sense of
complex variable theory). You have already heard [12,9] a lot about how this has
become true.
The first obvious question was whether the known properties of the S-matrix
(known in the sense of having been abstracted from quantum field theory) were inde-
pendent or interdependent; were there a few basic properties which could be justi-
fied by direct physical argument and used to derive the others? It soon became clear
[8,15] that the two crucial properties were tmitarity and the analytic behaviour of S-
matrix elements in terms of the particle momenta for values of these momenta close to
the physically permitted ones, i.e. "in the neighbourhood of the physical region".
This latter is what we shall now discuss.
134
2. Postulates of analyticit X in the nei~hbourhood of the physical region
Pham has explained [12] one such postulate which is certainly the most recent
and seems to be the most precise, refined, and subtle one, and which in his notation
reads
I Microanalyticity .
The scattering amplitude SIj is microanalytic in the direction of each covector
u e T MIj, except for covectors which are causal.
lagolnitzer has explained another postulate [9], the macrocausality postulate,
which is more physical in content and can apparently be shown to be equivalent to
the microanalyticity postulate [13,10], i.e. microanalyticity ~ macrocausality.
Ten years ago ignorance, both physical and mathematical, prevented one from
making such detailed postulates -- one just did not know or understand the analytic
structure of the physical region well enough. Because of this I was forced in my
work [15] to make a more vague and less precise postulate:
II ie postulate .
The scattering amplitude is analytic in the neighbourhood of all physical points ex-
cept for those lying on certain curves: the amplitudes defined on either side of
these exceptional points are analytically related by paths of continuation which be-
come infinitesimally imaginary. Any closed path of continuation can be contracted
to zero since the physical scattering amplitude is single valued.
Actually it is now known that this is not quite correct; one must allow the
possibility of a linear decomposition into parts satisfying the above. This illu-
strates the fact that this it postulate can probably be formulated much more cleanly
in terms of hyperfunction language.
Later I shall explain more about another fundamental postulate, one which is
universally accepted,
III the unitarity of the S-matrix .
A fourth statement is the spectral assumption of Pham [12] which, he argued, is
equivalent to what physicists would call the Cutkosky discontinuity formulae [5]
as applied to physical region singularities. So I shall write
IV Spectral assumption ~ Cutkosky formulae .
The current point of view seems to be that I and III constitute the most satis-
factory choice of axioms since they can now be stated precisely and have a direct
physical significance. From my point of view this is not satisfactory because it
seems to me that the two statements I and II are not independent of each other. It
follows that one must show that
135
i) I and III are consistent .
However, I think it would be even better to find a weaker version of I which
could be valid in conjunction with III to prove I. This is why I have insisted on
presenting my relatively vague assumption II because I had originally hoped to show
something like [15]
ii) II + III~I .
It could be that in order to formulate II better, one should abstract the in-
formation contained in the linear program of axiomatic quantum field theory, but
this has not yet been done completely.
Finally one would like to show
iii) I + III~IV .
Of course if (ii) and (iii) had been demonstrated, we would have
iv) II + III~I + IV .
This was what I originally wanted to do, and it would seem to be the ideal situa-
tion in that the maximum amount of information is deduced from the minimal reasonable
assumptions. In fact a lot of work has been done to this end, to prove (iv), and
later on I shall illustrate the arguments used, trying to show how the mathematical
concepts and techniques developed relate to the recently developed hyperfunction
theory. I shall also make the point that it is probably possible now to construct
rigorous proofs of statement (i) and (ii) by combining the existing (physicists) argu-
ments with the more refined and precise methods of hyperfunction theory.
3. Connectedmess structure of the S-matrix and the unitaritx e~uations
Phamhas mentioned the S-matrix. A particular matrix element referring to given
initial and final configurations of particles is the quantum mechanical probability
amplitude [6,16] for the transition between these states. The conservation of pro-
bability, i.e. the fact that unity must be the total probability for some outcome to
a scattering experiment, leads to the fact that S is unitary:
SS ~ = SiS = i .
In terms of S-matrix elements these equations are fairly complicated (and this
will be important -- in fact where hyperfunction theory can help). The index sum-
mation implied in thematrixproduct involves a sum over the possible sets of parti-
cles in the intermediate states and also an integration over the momentum of each
intermediate particle, consistent with it always being on its mass shell:
I ~d4p @(pO)6(p2 _ m 2) (2~) 3 J
t
136
Owing to energy and momentum conser~ation, this su~aation and integration is of
finite range since it is restricted by the energy and momenta of the outside particles.
There are really even more terms in the unitarity equations than I have mentioned,
since the S-matrix elements themselves break up into separate parts describing the
different possibilities of subsets of particles colliding or missing each other al-
together. This can best be represented by introducing a "bubble" notation, which is
a sort of pictorial representation of the scattering process [15]. The S-matrix ele-
ment is written
1 ~ i + 1 after before
collision ~ : collision
i i + j
and the "cluster decomposition" just mentioned can be illustrated by
: Zr
I °
A straight-through line--represents the possibility that the particle concerned is
not deflected and contributes a three-d~ensional Dirac ~-f~ction representing this.
S c is a new matrix (the "co~ected part" of the S-matrix) which cabot be deco~osed
a~ ~rt~r in this way and so is free of such 6-functio~ i~icating that a particle
is ~deflected.
As Phammentioned~ in all these collisions, total energy and moment~must be
co~e~ed, and this is represented ~ t~ fact that S c contains as a factor just one
four-d~ensional 6-function ~aranteeing over-all energy ~d m~ent~ conse~ation:
I j ~J
The matrix elements of this new operator A are the fundamental quantities of the
theory. They are the hyperfunctions defined on the mass shell manifold M(Ij) and at
many physical points are in fact locally analytic.
137
Inserting these definitions of A into the unitarity equations, one finds the
following relations :
-
for example,
m ~ if (2 m) 2 < (Pl + P2 )2 < (3 m) 2
-- ~ ÷ ~ if (3 m) 2 < (Pl + P2 )2 < (4 m) 2
4 ~ ~ if (3 m) 2 < (Pl + P2 + P3) 2 < (4 m) 2 .
These are just two examples of the infinite system of non-linear integral equa-
tions satisfied by the hyperfunctions A(p) -- the examples we shall use later. In
terms of the hyperfunctions A, the way to read off the terms of the above equations
is given by the following rules [15]:
i) I ~ J ÷ A(Pl,p J)
ii) I ~ J + A(pj,pl)*
iii) each internal line + -2~i @(p°)~(p 2 - m 2)
iv) each independent loop f id~k
1 v) ~DD where S D is the symmetry number of the diagram (n'. for n
identical particles)
These equations, together with the ones we have not written down, have such a
rich structure that it is perhaps not very surprising that we can deduce so much about
the singular spectrum of the hyperfunctions A.
4. The pole singularity
This is the simplest example with which to illustrate the basic ideas.
Consider the causal configuration ~ I~~T---~ 654
We notice that in the mlitarity equation there is a term
138
which by our rules is
A(p2,P3; P2 + P3 - P6,P6)(-2~i)@(P2 + P~ - P6)6[(P2 + P~ - P6) 2 - m2]
A(p~,Ps; P2 + P3 - P6,Pl)* •
This is microanalytic except in the codirections u and -u associated with the above
diagram, because of the 6-function.
If we tried to assume that ~ were analytic at (P2 + P~ - P6) 2 = m2, it
would follow that ~ was also analytic there, and also in fact all the other terms
of the equation except for the one above. This would then constitute a contradiction
and illustrates our previous statement that microanalyticity and unitarity are not in-
dependent.
Now let us assume that ~ satisfies the microanalytic hypothesis I. Then
at (P2 + P3 - P6) 2 = m2 it is microanalytic in all codirections except u. This is
the sanm as saying in the old-fashioned language that it has a +is prescription in
the variable (P2 + P~ - P6) ~. We can now examine all the other terms in the unitarity
equation and find the following:
microanalytic in all directions except u -u u -u
or in the old +is -is +is -i~ language
u and -u 0
no ic analytic prescription
- -
So ---k?~L
This means that the difference between the two sides of the equation is microanalytic
in the u-direction (since the terms omitted are microanalytic in fact in all direc-
tions except the -u direction).
Now multiply on the right by
(this does not affect the microanalytic properties of the above terms) and use
_ ZnSI7 _
to f i n d
139
As Pham explained, this implies
disc ~ - - -
the o ld- fash ioned way of wr i t ing the r e s u l t of the above argument, which dates from
1963 when s t a t ed in the old language as due to myself [15]. There i s another l a t e r
way of rearranging the argument due to Coster and Stapp [4]: First do the multi-
plication on the right and then examine the analytic properties.
5. The triangle singularity
This is slightly more complicated and illustrates some new features [ii].
Consider the causal configuration
5 2 3 6
The corresponding values of PI' PJ just satisfy an equation in the three variables
(P2 + P3) 2, (Ps + ps)2, and (Pl - P4) 2" Let us fix (Pl - p~)2 at some physical
(negative) value and look at the real values of the other two variables.
(Ps + P6) 2 , t
B
I
I
J
"" (P2 + P3) 2
The causal (positive ~) arc A is an arc of a hyperbola subtended between the lines
(P2 + P3) z = (2m) 2 and (Ps + Ps) 2 = ( 2m)2 and should be singular. The other, non-
causal arcs, B and C, have mixed ~'s, and should be non-singular (according to the
microanal~ticity postulate I).
When we look at the unitarity equation we find three terms each of which might
be singular on any of the above arcs, A, B, and C; namely,
Then we analyse t h e i r ana ly t i c s t ruc tu re and make arguments analogous to those made
fo r the pole s i n g u l a r i t y , and f ind
140
disc on A
on B and C
even if we make the weaker i~ assumption II. Thus all of statements (i)-(iv) can be
checked in this case. More details are given in reference ii.
Thus we have seen how the various statements (i), (ii), and (iii) can be de-
duced in special cases, i.e. for specific S-matrix elements in specific parts of
their physical region. These arguments have been considerably generalized [1,2],
and there are always, as above, two main parts to the argument:
a) the analysis of the analytic properties of the unitarity integrals;
b) algebraic rearrangements using unitarity relations.
To achieve (a) one must know how to multiply hyperfunctions, integrate the result
over a manifold (phase space), and study the microanalytic properties of the result.
The physicists' solution to this problem is given in [I] together with earlier re-
ferences. The fact that one works in the neighbourhood of the reals is continually
exploited. I understand that analogous and no doubt improved results now emerge
from hyperfunction theory.
Step (b) is in general messy since there lacks a sufficiently good notation for
dealing with it. There is a choice of doing step (a) or step (b) first; step (a)
was habitually done first in Britain [2] and step (b) in America [4]. The second
step is then simplified (relatively).
To sum up, the unitarity of S-matrix theory constitutes a very interesting set
of non-linear equations for the S-matrix. It seems that the resultant singular spec-
trum can be evaluated by techniques akin to hyperfunction theory, and that the more
precise methods of this theory could be used to tidy up the arguments.
141
References
[I] M.J.W. Bloxham, D.I. 0live and J.C. Polkinghorne, S-matrix singularity structure in the physical region: I. Properties of ~itiple integrals, J. Math. Phys. i0 (1969) 494-502.
[2] M.J.W. Bloxham, D.I. Olive and J.C. Polkinghorne, S-matrix singularity structure in the physical region: If. Unitarity integrals, J. Math. Phys. I0 (1969) 545-552. Ill. General discussion of simple Landau~ingularities J. Math. Phys. IO (1969) 553-561.
[3] G.F. Chew, S-matrix theory of strong interactions (W.A. Benjamin, New York, 1961).
[4] J. Coster and H.P. Stapp, Physical region discontinuity equations for many- particle scattering amplitudes: I. J. Math. Phys. iO (1969) 371-396, II. J. Math. Phys. II (1970) 1441-1463.
[5] R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429-433.
[6] R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix (Cambridge University Press, 1966).
[7] P. Goddard, Recent progress in the theory of dual resonance models, Proc. of the Aix-en-Provence Int. Conf. on Elementary Particles, 1973.
[8] J. Gunson, Unitarity and on-mass-shell analyticity as a basis for S-matrix theories, I. J. Math. Phys. ~ (1965) 827-844. II. J. Math. Phys. ~ (1965) 845-851.
D. lagolnitzer, Talk given in these proceedings. See also, D. lagolnitzer, Introduction to S-matrix theory, ADT, 1973.
D. Iagolnitzer and H.P. Stapp, Macroscopic causality and physical region ana- lyticity in S-matrix theory, Comm. Math. Phys. 14 (1969) 15-55.
P.V. Landshoff and D.I. Olive, Extraction of singularities from the S-matrix, J. Math. Phys. ~ (1966) 1464-1477.
F. Pham, Introduction ~ la microanalyticit6 de la matrice S, Talk given in these proceedings.
F. Pham, Singularit6s des processus de diffusion multiple, Ann. Inst. Henri Poincar# 6A (1967) 89-204.
J.C. Polkinghorne, Analyticity and unitarity, Nuovo Cimento 23 (1962) 360-367; If. Nhovo Cimento 25 (1962) 901-911.
D. Olive, Exploration of S-matrix theory, Phys. Rev. 135 B (1964) 745-760.
D. Olive, Singularities in relativistic quantum mechanics, Proceedings of Liverpool Singularities Symposi~ II, Lecture Notes in Mathematics 209 (Springer Verlag, 1971), 244-269.
[9]
[lO]
[11]
[12]
[13]
[14]
[zs]
[16]
142
[17] H.P. Stapp, Derivation of the CPT theorem and the connection between spin and statistics from postulates of the S-matrix theory, Phys. Rev. 125 (1962) 2139-2162.
[18] J. Schwarz, Dual resonance theory, Physics Reports 8 C (1973) 269-335.
GEOMETRY OF THE N POINT P SPACE FUNCTION
OF QUANTUM FIELD THEORY
H. Epstein V. Glaser R] Stora
I.H.E.S. - Bures sur Yvette CERN - Geneva CE~N - Geneva and
C.N.R.S. ~arseille
The geometry of the n point p space function of quantum field 2)
theory was first simultaneously developed by Araki I) and Ruelle , following
previous but less extensive work by Polkinghorne 3) and Steinman 4) The
conStruction of these authors which can be found in original papers an@ reviews
is intuitively based on a preliminary study of the one-dimensiona~ space-time
situation - conventional non-relativistic quantum mechanics - which leads to
the relativistic situation studied by these authors.
A more recent approach 5) is suggested by perturbation theory
studies 6) as well as experience gained in the study of local analyticity 7)
properties of the Scattering ~nplitudes in qu~qtum field theory One
natural way to supplement the Wightman axioms 8) _ or, with minor modifications
which do not affect analyticity statements, to exploit the Haag-Araki 9)
axioms - is to assume the existence of operator valued distributions depending
on n space-time points (Xl...Xn) = X, conventionally called time-ordered
products of n field operators, whose products leave stable a dense domain
in Hilbert space, which contains the vacuum vector/'/ . Time-ordered products
are ass~ned to fulfil the causal factorization property 6) :
T (x ) = T ( z ) T ( z
if I > I' where I' = X\l and I 2 I' = (x I xi-x j ~ F- ~i~ I, j ~: I').
It is the hatter condition that undergoes a minor modification in the Haag-
Araki formulation of Q.F.T. One can thus naturally introduce the partially
retarded operator 7)
with support
( x ) - - r ( x ) _ v- f z ' ) -/- C I J
( x l x i - x j e F + some i e I , j E I ' ) . "Let now
Applying spectrum properties to Fourier transforms, one obtains
144
SI= where ~0~ Vl, V I being the closed convex hall of the lower sheet
of a hyperboloid of mass ~I depending on seleetioz, rules and spectrum.
These properties can be extended to more general expressions which
w~ shall now construct.
1. - THE ALGEBRA GENERATED BY TIME-ORDERED PRODUCTS
One considers the vector space ~ spanned by ordered monomials
T~T(Ik,),TC~)...-T(E~,~ .z;,o.,,uZ',,=X, Z, nS'- ¢, k,~ 4 i , = 4 . . . v j
whose g e n e r i c e l e m e n t w i l l be dep le t ed ~ . I f I ~ X ~ one d e f i n e s
by l i m e a r e x t e n s i o n o f
k--~ k.--
2 TT r(r~;= TT T(z,~;) "T(r'~ z,) ~.-r /< -'#
with the convention £ = ~ = I , T(~) = I . One has the following properties :
~) I -__I.
( i i i )
.r, < - z,z~ <=> Z,=__T, o,, Z,~_Z.,
-~,., TT % ; : K~ z, ~ _~,.,
~ , , , 4 . r -,4
(iv) same as (iii) with U ~
145
.~ ~-,= f , (vi) T(X) is cyclic :
TC~<.,]... T ( < , ) : ~ . . . I~ r ( x )
This allows %o define a partial order on (.~ :
T(~,~... T(K,,) > T(L,).. .-7-c%) if the partition (LI...L ~ ) refines the partition (KI...K ~ ).
(vii) If
h " ' ~ A
(ix) Eigenstat es of ~ :
~e=e ~ e~occ.n. vectors of the form
®_- Fr
with I k C I or I k CI'
rr=~) ~elon~ to a~I).
Conversely, all elements of ~(I) are of this form :
146
In other words, either a monomial is non-decomposable 5n ~(I~,
and is an eigenstate of ~ with eigenvalue I, or it is decomposable
~in ~(I~ and there corresponds to it an eigenstate of ~ with
eigenvalue 0 - such eigenstates of ~ are thus put in a one-to-one
correspondence with all partitions of X and thus span the whole
of ~ .
2. - ~CE~LS 5),m)
We now look for genecalizations of the r l'S, with support
properties in x space, and with coincidence properties in p space with
7(P).
..per 5)
A preoell ~ is a set of proper parts of
i )
2)
~ ! ~ - f , t h e n ~ r ' ~ .
then -T~UI~ E ~.
3) if
the
~a consequence of I) and'2) applied to ~
x (~¢, x) such that
De~
147
Pr~£~l~_Lemm~ 5) (elementary)
Then ~ / / ~ ~ ~ 0 "t'~ ~ U ~ E
D_~ (paracell) ~0)
A paracell is a set
precell lemma holds.
of proper parts of X
Th
If ~ is a paracell, the product
is independent of the order of the factors.
f o r w h i c h t h e
~}}}£~_~_~£~
Let I1 . . . I N be an arbitrary ordering of ~ , that for any permutation ~"
we have to show
Suppose ~ is the transposition N ~ N-]. The property is true by (ii),
(iii), (iv). Suppose the property has been proved for all ~ 's such that
The passage from p to p+1 follows easily from the induction hypothesis
and application if (±i), (iii), (iv). In fact the proof shows the following.
L emma
L e t A ] o . . A N be a s e q u e n c e o f n o t n e c e s s a r i l y d i s t i n c t p r o p e r
s u b s e t s o f X, s u c h t h a t f o r e v e r y p a i r i , j , t h e r e e x i s t s k s u c h t h a t
o f 1 . . . N A k = A iU or A. ~ A. l'~A~ then for every permutation Aj ~ i .3 - -
148
- Support properties
L emma
(-~_.~) ®,,,_- o hence support % C q f l - "J = x x j _ , 9 ~ V, , .so.he ~ , j~
(use causal factorization property).
- Corollary
has support contained in
O~q~ I%@' 3,~z'
This support can be analyzed as follows.
such that h(1) ~ I. Let
With this notation
S O c..- U c" IE~ i~z ~J l h Z'~ -.t" j e l '
A choice 5),7) is a map I~X
149
Choices h which yield the support can be restricted to choices compatible 5)
with 4 as follows. Let
@-_ x \ --3*= ~ '* One can show 5) that for each choice h, there exists ~ "compatible choice"
h' such that
h ' ( z ) E --3 + ~ I~
and
C f r o m w h i c h t h e r e f o l l o w s
It e~,,,pa~,'t, le 5) One c~ furthermore show that each C~ is of the form
where ~h ~ J ~ is a tree graph with vertices i ~ X and links
j ~ ~-. Such trees were first discovered by Bros 11)
(ij), ia~ +,
- ~nalyticity properties
It is easy to see that the Fourier transform of a tr~slation
invariant distribution with support in C h can be extended into a holomorphie
function defined on the hyperp±ane
and analytic in the tube with imaginary basis
the par~etric form I) ,9)
Ch dual to Ch, given by
150
' "7,j>: - % , - - " + ; D '
T h i s p a r a m e t r i c fo rm i s due to A r a ~ i 1 ) , 9 ) and has the f o l l o w i n g n i c e p r o p e r t y :
i f a t r a n s l a t i o n i n v a r i a n t d i s t r i b u t i o n has suppor t C ~ ~ C ~ , i t s F o u r i e r
transform can be extended into the tube with imaginary basis
whore _+~ G V + and S are the co-ordinates of exposed one-dimensional
facets of the convex hull
~: ~ m ~ ~ s+~ ~,j~ + (~_+J~,~Dj' ~,j)'
Alternatively, the tube corresponding to C h can be described by the set
of n-1 conditions of the type Im PI(ij) ~ V+' I(ij) ~ ~, obtained by
writing
.,%+., (.. , +.,,j + + ++ +=, P<.. -A d e s c r i p t i o n o f t he l a t t e r k i n d does no t e x i s t f o r convex h u l l s , in gene ra l °
151
3- - SPECTRUM PROPERTIES OF T~
We look at Pourier transforms of vacuum expectation values,
obtained for ® = ~-(x) .
Expanding the product
and looking at the last factor of each term, we see that
~ .z- s -S
~4 (P)
4. - DISCONTINUITY FORMULAE
De.__Lf
If is a preeell, I is called a boundary of ~ (I ~ ~)
is a precell.
_Bo_~n__d_a_~_~_e~_a 5)
I is a boundary of ~ if
(i) L , . , z ~ : z , L n ~ ¢
--, z ; , -3 (ii)
L ~ _r, _r., ~ rf
p t / / # / I ; o I~ = gb , I; ~I , ._r., E
if
t52
(i) and (ii) are equivalent to (i) and :
( i i i ) I , n / ~ . , I , I . u I . - X ,
If I is a boundary of ~ we shall define
Using the oommutativity o f ~ (I-~) ~qd leaving I as the last factory
one easily gets
Hence the commutator formula
%,_%.,: z J~ffz Write now
J : ~ . . ~ : ~-¢,:r. o-,-,~"
as a direct product which allows to write~ if K ~ I~ K' C I'
~r "T'(K') -re,<' )= :r~..~ -T'CK) TO,<')= ~ -rcK).~. T(K~
where
T(K)= TCKh,rnl) T('l<~:'~I,)
~x' T(K')- T(K'~J~ Z') T(K'~,J'r~ I')
153
Let now
These are preceils from the preeell property of ~ and the boundary propert
of I.
Now if J ~ ~I' either Jff~ ]1 Ls in ~l or J /~ [' is :in
~I' (or both) because if it were not so, both Jf~ I and J~ [' would
be in ~' and since they are disjoint, their unton~ J~ would be in ~'
which is not true. Hence J is of one of the foLLowing types :
(~) K U ~<' KE Z ~ K% Z z I j i ~'
( i i ) ~< ,d ~ ,' l<~ Z . ,
(iii) ¢ g K' K' , ~ ~ ,
(v)
<vi)
( v i i )
"Y u K' K ' ~ Z z, j •
, L ' c z ' C r.', I ' , ¢ ) K U L ' I< '~ ~rz , , •
m u K ' r, c I (~ , :~ - r ,~ ) , K ' ~ z , , . J I
Using again the precell property of ~ , the commutativity of
and the idempotency identities
@..^'" j ETa_3 (i-3)
w e c,'ra'~ w : r J t e
T7 (~ ,~)__ TT (~_ ~, ~ • ... ~'6 Zz"
154
L t ; 1"
= ~ (~-~')(~-~) w h e r e we h a v e u s e d K'I£:EI"
I < c z r
(~.<.~)(~-~)
~ - ~'. ~) (~- ~'~ o (~- ~'i
on account of the idempotency
K _-.- K ,, K =
The co~lutativity of the factors (I-K), (1-K') allows to split these factors :
~'r: zz ,
K~Z~, J K~zz
This is the so-called Ruelle discontinuity formula 2),5)
155
5. - STEINM~NN's IDENTITIES 4),I),2)
Let I , J such that I ~ J , I C J i , J ¢ I, J ¢ I , I ' ¢ J l, J ~ I ' , J ' ¢ I ' Consider four cells admitting I, J,
-r:r
lJ
Then
: • r uluJ= J
hence I' J,
as boundaries :
"~z U J'= %,U.T
- 4 , o j - : % <, ~:,
, J'_- z '
where
Similarly
Z z I< K c Z K ~
s~nce Z
-FG+ ;, _ - - F ,
where
Z;,o ~ince ' ¢. I '
I~ c I , KE ~ t
l<c I ', K~G~}
156
Hence
and
W-
-C, - , - - -r v T
The so-called Steinmann-Ruelle relations ~ )' ~ )
=O
6. - CELLS i),2)
The interpretation of ~(P) as a piecewise boundary v a l u e of
analytic functions goes via the consideration of cells.
De_if 5)
A precell ~ is a cell if, in
SX = ~ Si = O, the set of conditions i
is non empty. This set will be called ~ .
S I < 0 if I ~ ~'.
R n, on the hyperplane
Clearly, if S E 7"~
One can easily see that Steinmann-Ruelle relations can only connect
together cells whose ~'s are within a hypercell I~l : S i > 0 i ~ I,
S i < 0 i ~ I', i.e., which all contain i, i ~ I and
one dimensional boundaries of I~l
S~j = ~ - ~, we already met.
the admissible C~h are such that
(g)~
contain ~ .
~i~ ' i ~ I ' The 7
are the vectors Sij ~ i ~I~ j ~I',
One can prove that for a cell ~ ,
rh :
157
such that
where the ~h
dual of Ch),
A more detailed result (Bros 11)) shows that for all ~'s
P ~ c F I there is a decomposition of the fore
& are h31omorphic iL the tube with imaginary basis U~h (the
where
The foregoing results are best summarized within the framework
of the following geometrical construction.
hyperplane
Consider an n dimensional space { S = (S1,...,Sn) } and the
~S[ = 0
One draws all the hyperplanes
which bound geomet r i ca l c e l l s ~ w i th in which a l l S l ' s have a p r e s -
c r ibed sign. One can easily read off this diagra~n the various relevant
S's in the preceding formulae. 13
Examples are shown on Figs. I, 2, 3.
158
7. - TRUNCATION
One has still to make a slight modification in order to make the
foregoing construction useful. The regions in p space where the ~ (P)
coincide with ~(P) do not cover ~he whole of P space because of the
occurrence of the vacuum contribution to the spectr~-~ Si. One can
however, define connected distributions ~c(P), ~ ~,c(P) ~hich turn out
to be identical to ~ (P~ such that all previously established properties
hold. The regions where ~c(P) coincide with the ~ (P)~s cover all of
+ being then reduced to P space when ~ goes over all possible cells, S I
V I.-+ Although this is quite important, we shall not give any more detail here.
The bridge can be made with the construction of Haraki and Ruelle
by establishing the following identity 5)'12)
.T '= .~kj . . • ~--~1
from which one easily recovers the spectrttm properties~ the discontinuity
formula and the Steinmazn identities, but from which support properties
are hard to get.
159
REFERENCES
I) H. Araki - J.~ath.Phys. 2, 163 (1961).
2) D. Ruelle - Nuevo Cimento 19, 356 (1961)-
3) J.C. Pslkinghorns - Nuovo C imento ~, 216 (1956).
4) O. Steinmann - Helv.Phys.Acta 33, 257 (1960) ; ibid 33, 347 (1960).
5) J. Bros, H. Epstein, V. Glaser and R. Stora - to be published.
6) H. Epstein and V. Glaser - CERN Prsprint TH. 1400 (1971).
7) J. Bros, H. Epstein and V. Glaser - Helv.Phys.Acta 45, 149 (1972).
8) R. Jost - "The General Theory of Quantized Fields", A.M.S. Providence
(1965).
9) H. Araki - ETH Lectures, ZGrich, unpublished.
10) Reference 5) introduces the more general notion of paracells which is
a special ease of the notion of cycle introduced by D. Ruelle,
Ref. 2).
11) J. Bros - Thesis, Paris (1970) ; Lectures at RCP 25, StrasOourg, Mathematics Department, Vol. VIII (1969).
12) The first step in deriving this formula from the definition of Refs. I),
2), was taken by C. Itzykson (1963), unpublished.
s 3~~,\
s 21 $1
o
FIG
1 n
=3
S 1 -
~S 3
=-($
2+S
~
f
S
I I \
53
..,.,-
- .,,
,,,,- \
\ \
j'
J ...
. I
S
/ /
/ /
"I
-./L
.
I
S I +
S 4
=- (
$2+S
3)
S~
+ S
2=
-(S
3+
SL
)
Ste
lnm
ann
FIG
. 2
n=
4
Iden
tity
wl
G~
m_
=r
0 --4
I/I
~ N
"
nmTmm~m-
.,
n
CU
~o
]>
N C]
SOME AP~ICATIQNS OFTHE JOST-LEH~L4/~N-DYSON THEOREM
TO THE STUDY OF THE GLOBAL ANALYTIC STRUCTURE
O.F THE N P01NT FUNCTION OF QUANTUM FIELD THEORY
R. Stora
CERN - Geneva and
C.N.R.S. - Marseille
I~TRODUCTION
Whereas ~he theory of hyperf~muctions and microfunctions I) is
able to produce local information on the problem posed by the strocture of \
the p space n point function of quantum field theory 2) it is quite
obvious that the problem physicists are confronted with is )f a globo_l nature.
Most global resurts that are known at present can be derived from
a few ingredients :
- the tube theorem,
- the semi-tube theorem,
- the edge-of-the wedge theorem,
- the vanishing of the first cohomology group of a Stein manifold,
several combinations of which are able to yield partial answers of a global
nature, in some cases, more sophisticated applications of the continuity
theorem were able to produce non-trivial analytic completions.
Standing aside is a beautiful piece of work due to Jost, Lehmann
and Dyson (JLD) 3), which is by itself able to produce many of the results
known by the previously mentioned methods and has, besides, a touch of origi-
nality and exotism which has made it almo~t impossible to gene~alize~ excepting
an unpublished work by Glaser. This can be taken as a proof that, if there
is a general idea behind it, no one has been sc far able to grasp it.
We shall furthermore see here some applications 4) of JLD's trick
which have so far not been recovered by the more conventional methods.
In what follows, we shall first give a sununary of JLD's work,
then make the bridge between some of its consequences and results which can
be obtained by analytic methods, and finally give so~e applications which,
at present, cannot be treated otherwise.
164
I. - THE JOST-LE[LV~ANN-DYSON (JLD) REPRESENTATION
A) - Let C be a temperate distribution in Minkowski configuration (x)
space, with support the union of two closed opposite light cones
The Fourier transform ~ of C is the restriction to the hyperplane ~= 0
of a tempered distribution r in five-dimensional momentum Minkowski space
(p, ~), where ~ is an additional space-like co-ordinate, even under
reflections through the hyperplane ~= O, and solution of the wave equation
[] 7 :0 ~,~
where
Conversely , every s o lu t i on r
a r e s t r i c t i o n to the hyperpl~ue
V+U V-- C's and ~ 'S
respondenee
of the wave equation that is even in ~4. has
= 0 whose Fourier transform has support
with the stated properties are in a one-to-one eor-
A*) This situation can be generalized to the case where V± is replaced
by V M = x I _~ M 2, <~ 0 , the wave equation being replaced by
the Klein-Gordon equation
<G o .
The variable ~ can be replaced by a set ~ of space-like
variables and the evenness requirement by that of invariance under
the orthogonal group in ~ space.
165
(i) ~o2__£__!£_~_
Let (x, ~) be the variables conjugate to (p, ~). The product
C(x) cos~x 2 exists as a distribution in (x,~6), even in ~ ,
which is infinitely differentiable in ~6 , polynomi.ally bounded in ~6
and coincides With C for /~ = O, because cosJ6~x 2 is polynomially
bo~Lnded together with all its derivatives with respect to x, on the
support of C, and the support of C is regular, r is defined as
the Fourier transform with respect to x of C(x) cos ~x 2 and
obviously is an even solution of the wave equation.
Since ~ fulfils the wave equation, its Fourier transform ~ has
its sLipport within the union of two opposite light cones in (p, K.) space.
Hence, from the regularity property of this support, one can decompose
according to
fpositive ~ can ~ having its support on the closed light cone. [negative
thus be written} as the difference of boundary values of two polynomially
bounded analytic functions holomorphic in tubes with these two self-
dual cones as imaginary bases. The restrictions of these analytic
functions to the analytic hyperplaae ~1<= 0 exist, are polynomially
bounded and analytic, in the remaining variables, in tubes with light
cone basis, hence the support properties of the Fourier transforms of
their boundary values.
B) - We now wish to take into account support properties of C, in p
space, and derive from it support properties of r.
I f ~ v a n i s h e s i n a r e a l n e i g h b o u r h o o d o f a t i m e - l i k e segment :
166
• vanishes in the double cone subtended by it :
U s i n g t h e wave e q u a t i o n a n d t h e e v e n n e s s o f r , o n e f i n d s t h a t ~'~ i s z e r o
t o g e t h e r w i t h a l l i t s d e r i v a t i v e s w h e r e ~ v a n i s h e s . One s a n t h e n u s e t h e
d e c o m p o s i t i o n o f F a s a d i f f e r e n c e o f b o u n d a r y v a l u e s o f f u n c t i o n s h o l o -
m o r p h i c i n o p p o s i t e t u b e s a n d t h e K o l m - N a g e l v e r s i o n o f t h e e d g e - o f - t h e - w e d g e
t h e o r e m 1) t o g e t h e r w i t h a c o m p l e t i o n by d i s k s b e l o n g i n g t o h y p e r b o l a e g o i n g
through the end points of the time-like segment and asymptotic to light-like
directions as indicated on Fig. I.
One thus gains points by application of the continuity theorem,
up to completely filling in the expected double cone. This is a geometric
version of a proof first constructed by Borehers 5)
B.2 . - H~y~h~_p~P~
Knowing some support properties for F ~ one solves the 3auchy
problem on a hypersurfaee best adapted to the support of E : let
be a hypers~rface which is spacelike, so that it is compactly intersected by
an arbitrary light cone (one may even allow lightlike pieces provided they
are bounded). Let ~+,~_ be the closed half-spaces defined by ~- . One
can decompose ~ according to
~ having support in ~. Then
o r . - Hence
mr.
167
where (~ has its support on ~" . If a different decomposition is chosen,
(~ ch~ges into ~+ ~ , where ~ has its support on ~W- . We call
mod ~ the Cauchy data of ~ on ~- • They are distributions with
support ~" defined modulo the Dalembertian of distributions with support
~, and one can check that~ locally where ~- is a manifold, this notion
coincides with the usual one. If ~ vanishes in some open set ~, it
is clear that one can choose (~ to vanish in ~ . Let now Dret,
Dad v be the elementary retarded and advanced solutions of the wave operator.
One can solve the inhomogeneous wave equations :
O
a&v
where ~! are solutions of the homogeneous wave equation which, under the
geometrical assumptions we made, have supports ~-~, bounded from below or
above, hence identically zero. We thus obtain the representation
where D = Dre t- Dad v vanishes outside the ~qion of two opposite light cones.
As expected, ~ only depends on its Cauchy data (~mod D~ , since
~D = 0. The representation of r is able to yield further support pro-
perties : r vanishes outside the future and paSt shadows of the support
of its Cauehy data (Huyghens principle), an information which has to be
combined with the double cone property.
A more refined property 6) ~s the following. D is real analytic
except on the skin of the light cone. Hence, if ~'~ vanishes on an open
subset ~ of a connected region ~ where no characteristic from a point
of the support of ~ goes, it vanishes identically in ~ (el. Fig. 2).
Whereas the double cone and Huyghens principle are enough to obtain
complete information when the input is the vanishing of ~ between two space-
like surfaces (the problem solved by JLD 3)), the analyticity principle was
applied by Greenberg 6) to cases where the region of vanishing of C is non
connected.
As is well known, the JLD problem is connected with the following
edge-of-the-wedge problem : let us decompose C into
C = Support R = V+, support A = V .
168
Then R, A are boundary values of functions analytic in opposite tubes, and
the statement
C = O i~ .~
is converted into
b.v. t~ = b.v. "~ ~.~ _C'Z,
a statement to be exploited by use of the edge-of-the-wedge theorem and
analytic completion. It turns out that all analyticity properties derivable
by the JLD procedure could be obtained by analytical geometric methods
including those which require the use of the analyticity principle. Some
known results can only be obtained by geometrical methods 7). Whatever
analyticity follows from the JLD theorem can be obtained as follows : D
can be decomposed in a non-unique way as the difference of boundary values
of two functions analytic in tubes :
~D= D% D c-~
It turns out) however) that these functions have as sole singular-
ities the points belonging to the hyperboloids
A careful choice of the decomposition of
analytic functions ~+, r- as
allows to express the corresponding
which yields the analyticity described in Section 2, after investigation of the
support of (~. This way, the JLD procedure solves an analytic problem "with
growth", whereas the geometrical methods solve it without growth 7)
169
2. - PROBLEMS OF ANALYTIC COMPLETION 7)
The analytic result that stems from JLD's analysis can be summarized
as follows. Let ~ be the tubes with imaginary basis V i, ~ a coincidence
region, open, connected, bounded by two spacelike surfaces. The holomorphy en-
velope of the associated edge-of-the-wedge problem (f~ holomorphic in ~±,
bvf + = bvf- in ~) is the complement of the set of complex points of real
"admissible" hyperboloids ~u~ :
which do not intersect ~.
We now look at a few special cases.
A) - ~ iS unbounded from above (cf. ~ig. 3). The only admissible
hyperboloids are spacelike planes which do not intersect ~.
Hence ~ has to be convex ("re-entrant nose" phenomenon). The domain of
analyticity is made up with :
- the real points of the convex hull of ~ ;
- all complex points of the real straight lines which intersect
(no such point lies in a real plane not intersecting ~.)
B) - The previous result can be applied to the following situation :
~as above, but V ~ replaced by arbitrary (convex) cones.
Using two-dimensional hyperplanes and putting together all analytieity points
deduced from A), one has the same result aS in A). In particular, V ~ are ~future
enlarged to the [past asymptotic cones of ~ .
C) - Same as B) but ~= ~ ~ (cf. Fig. 4). The union of the
domains pertaining to ~+, ~- is natural and is the complement
of the real and complex points of real hyperplanes intersecting neither ~+
nor ~- (Hahn-Banaeh).
Remark
All these domains are of the following type : given an open set
~in real space, the set of complex points of straight lines intersecting
, together with the real points of ~ is natural if ~ is convex.
If the real points of ~ are deleted, one obtains a generalization of two
opposite tubes.
170
From this point of view, the re-entrant nose phenomenon is not
essentially different from the double cone phenomenon : in two dimensions +
let = u (ef, Fig. 5)
Within the domain pertaining to r~, take the complex points of
straight lines intersecting ~a : make a real projective transformation
which takes a~ into the straight line at ~ and apply the double cone
theorem to a half-straight line in ~+b" One could, of course, also apply
the continuity theorem to a family of straight lines parallel to ab, starting
high enough.
D) - A more general application 7) allows to describe in geometrical
terms the holomorphy envelope associated with two arbitrary double
cones in two dimensions, combining projective tranSformations and inversions
ESee E~ which transform finite double cones into infinite double ccnes. For
instance, in the example shown on Fig. 6, the JLD result cs~ be applied to
situation If. There, the unioo of the two holomorphy envelopes is natural.
In situation I, the adapted admissible hyperbolae go through the apex of the
infinite cone. The solution to this problem was found during an attempt to
understand the analytic equivalent to the analyticity principle in the JLD
approach 7)
E) - r + , ( - ) i s a t o p o l o g i c a l p r o d u c t o f V + ( - ) ' s , ~ = r ~ . The
inversion
transforms r+ into the double cone of diagonal ab if b is in the past a F k
of a. It preserves the tubes of imaginary basis r +,Q-J. it transforms
straight lines
i n t o c u r v e s w h i c h w e r e c a l l e d Q c u r v e s . I n t h e JLD e a s e , t h e Q c u r v e s
a r e h y p e r b o l a e w h i c h a l l o w t o d e s c r i b e t h e d o m a i n f r o m i n s i d e . The ~LD d o m a i n
c o n s i s t s o f a l l c o m p l e x p o i n t s o f d o u b l y i n a d m i s s i b l e h y p e r b o l a e , i . e . , r e a l
h y p e r b o l a e a s y m p t o t i c t o t h e l i g h t c o n e w i t h a n u p p e r s h e e t a n d a l o w e r s h e e t
which both intersect ~, together with its suitable completions (by double
cone and re-entrant nose enlargements).
171
In the more general case of a double cone, one describes the
domain in terms of complex points of doubly inadmissible Q curves. The
description of the domain from outside, in more general cases, is due to
Epstein and Glaser.
3- - SOME PROBLEMS OF GLOBAL DECOMPOSITIONS
All the problems which are dealt with here are solved with the
help of the JLD theorem : they s~e problems "with bounds" but no solution is
so far known without bound. They are all connected with the structure of the
p space n point function of quantum field theory 2) summarized in these
proceedings.
The starting point is the decomposition of the generalized retarded
functions associated with a given channel ("hypercell") in terms of Bros trees 2)
The next step is to use the discontinuity formula for neighbouring geometric
cells : the notations being the ss~me as in the lecture by Epstein, Glaser and
Stora (this volume), the disccntinuity formula
contains both x Space and p space support information, thus suggesting the
use of the JLD theorem. Similar problems were first treated by Streater 8)
in the context of the structure of Wightman functions. From Bros' tree decom-
position, one sees that the difference t ~+-t ~ involves only trees for
which one of the defining equations is S I ~ 0 or S I < 0. Thus, one has the
following situation :
/b~ =0 ~'e'3'+ -- /~..'7" = ' I .Z
where the f~ i are holomorphic in simplieial tubes with imaginary basis con-
tained in ImPi ~ V i and M I is a mass associated with the spectrum properties
of the theory. Analyticity is furthermore deduced from x space support
properties.
172
Let us choose a set of variables, one of which we denote ~ ,
= x.-x. where i e I, j E l'. By JLD, ~I is the restriction to z j
~( = 0 of a distribution ~ in all variables together with 7~ , solution
of the Klein-Gordon equation
Using the double cone principle, w~ see that at least in the applications
given here~
because all other variables being fixed, the support in ~ is a u/%ion of
Vi's with different origins~ whose complement contains arbitrary large
timelike segments between the upper arid lower boundary and all apices of V +
cones are in the past of all apices of V- cones. We ean therefore splat
2=7__ z _Z r- i6J + ' {6Y" '
where
Hence
KG r = Z_ < _ S KG I " . - = 0 L
There exist therefore distributLons ~ij = - 4 j i ' i ' j ~ 3~ ' / ~-
, y~ .r'oa- V
where
,i = supp. n sapk a
such that
173
hence
where the c o n v o l u t i o n e x i s t s ~ has i t s s u p p o r t c o n t a i n e d i n the suppo r t o f
r.~ hence~ also that of r .. Thms support being bounded front above or 1 o 1
from below~ fi vanishes identically, l°utting pieces together, we get
+'= ++ z+ <:. with restriction to ~ = 0 equal to . A suitable decomposition
D = D ret - D adv yields a common analytic decomposition in p spaee~ for
= / ~ . ¢ . je:r"
It is of the form (in p space)
where f . is holomorphic in the tube convex hull of the tubes defining z,] 2 2
the trees fi~ fj indented by the cut Pl = MI + ~ ' p real positive.
The ambiguity of this decomposition is of the form
where fijk is the co~:~on analytic continuation of
,4" ~'j~ , k ,
t he s u p p o r t o f A i j k be i ng t he i n t e r s e c t i o n o f t hose c o r r e s p o n d i n g to
trees fi ~ fj~ fk" Hence fijk is analytic in the convex hull of the 2 +~.
tubes defining fi~ fj, fk~ indented by the cut p2 = MI
We now list a few applications of this decomposition property to
low values of n. We Shall not write down supports and their duals in detail
but rather show the s space corresponding cell. We shall write equations
involving cells, each cell symbol representing a function analytic in the
corresponding Araki tube - wiggly lines represent cuts. In the following,
E.W. means edge-of-the-wedge, JLD means Jost-Lehmann-Dyson.
174
A) - Three-point function, s spece is shown in ~ig. 7. Each couple of
neighbouring cells gives rise to a holomorphy envelope 8) ;
E.w. s~
B) - Four-point function, s space is shown in Fig. 8.
I)
v s~ 1 S~ _ s., ~ + ~ 5~
~ ~ + 2) the quartet 8) : V
Jw J~
~LD
5. s~
175
Hence the common decomposition for all four members of the quartet
hypercell :
S~
3) putting together both decompositions we get :
I hence :
S~
$ e ~s
= holomorphy envelope
176
defining s, , ~ t s J
we get :
Y &
hence the global decom
o #
losition I/~11 + -~3
Unfortunately the next step involves the relationship
"~i + ~&':~ = ' ~ " + ' ..... "~' __//
177
C) - Five-point function. Two types of Bros trees are shown in Fig. 9 : ~
ABCD~ BCDE ; there are six of each kind within the hypercell represented
by ABCDEF (ef. Fig. 7). Their convex hull ABCDE indented by the cut
BCD will be denoted ABCDE. On our drawing, there is another similar
object : FAB_~. Let (EF) denote the holomorphy envelope of these
two domains (which is not explicitly known). There are six such
domains, each one corresponding to a diagonal of a vertical facet.
In terms of these six functions, one has the global decomposition,
valid for the analytic continuation of any member of the hypersell :
CONCLUSION AND OUTLOOK
The partial answers exhibited here pose the more general following
problems :
- For general n, can one reduce the problem which arises from the Steinmann
identities within s hypercell to the construction of well-defined holo-
morphy envelopes ?
- When looking at neighbouring hypercels, one is faced with problems of
the following type : let fi be holomorphic in ~i' i ~ I, let
what are the implications of such identities ? Do these problems belong
to some part of eohomology theory ?
- In other words, can the problem posed by the structure of the n point
p space function of quantum field theory be reduced to the construction
of well-defined holomorphy envelopes ?
178
REFERENCES
This list is not exhaustive, but mostly gives a sample of
refergnces where more bibliography can be found, besideS few originals
which are not very well known.
I) "Hyperfunctions and Pseudo-Differential Equations", Lecture Notes in
Mathematics, Vol. 287, Springer Verlag (1973) ;
"Th4orie des Hyperfonctions", by P. Shapira, Lecture Notes in Mathematics,
Vsl. 126, Springer Verlag (1970) ;
A. Martineau - Lectures at the Meetings of'RCP 25, Strasbourg Mathematics
Institute, Vol. 3 (1967).
2) J. Bros and R. Stora - Lectures at the Meetings of NCP 25, StraSbourg
Mathematics Institute, Vol. 3 (1967) ;
H. Epstein - Brandeis Summer Institute (1965) ; Gordon and Breach, New
York (1966) ;
H. Epstein, V. Glaser and B. Stora - These Proceedings.
3) A.S. Wightman - Lectures at Les Houehes Summer School of Theoretical
Physics (1969) ; Hermann, Paris (1961)
V.S. Vladimirov - "Methods of the Theory of Functions of Several
Complex Variables", Cambridge M.I.T. Press (1966).
4) R. Stora - unpublished 11964), summarized in Lectures at the Meetings of
RCP 25, Strasbourg Mathematics Institute, Vol. 2 (1966).
5) H. Borchers - Nuovo Cimento 19, 787 (1961).
6) 0.W. Greenberg - J.Math.Phys. ~, 859 (1962).
7) J. Bros, A. Messiah and R. Stora - J.Math.Phys. ~, 639 (1961) ;
H. Botchers and R. Stora - unpublished (1962).
Per another type of solutions with growth, see :
R. Seneor - Comm~un.Math.Phys. 1_~1, 233 (1968).
8) R.F. Streater - Proc.Roy. Soe. A2_~, 39 (1960).
~ po
ssib
le
1 _
~,
2 ~
r
sect
ion
of
the
dom
ain
by
hype
rbol
ic
disk
s
"4
FIG
.1
Ana
Lytic
pro
of
of t
he
doub
le
cone
th
eore
m.
FIG. 2
If I has null Cauchy data on R it is real analyt ic in , ~ .
180
FIG 3
-1"I
6")
0"1
I::1
_
-G3
-rl
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182
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1
QUELQUES ASPECTS GLOBAUX DES PROBLEMES D'EDGE - OF - THE - WEDG E
J. BROS~ H. EPSTEIN~ V. GLASER et R. STORA
Contribution au Colloque sur les Hyperfonctions et leurs Applications,
Nice, Mai 1973 - (Pr4sent4e par H. EPSTEIN)
Cet expos~ d~crit bri~vement quelques r4sultats obtenus entre 1961
et 1963. Certains d'entre eux ont ~t~ publi4s dans [i], [2], [3~ .
D'autres n'ont pas encore 4t~ publi4s. L'accent est mis sur l'aspect
global des probl~mes du type d'edge-of-the-wedge. L'exploitation "loca~',
ou, plus exactement, infinit4simale, du th4or~me e.o.w, en th4orie des
champs est en effet presque achev4e [5] cependant l'utilisation de la
"positivit~" pourrait ouvrir de nouvelles possibilit4s;(voir l'expos4
de Glaser) ; tout au contraire les r~sultats globaux connus sent tr~s
rares et h4t4roclites. Cela est d'autant plus regrettable que le plus
c~l~bre d'entre eux (dent il ne sera pas question ici ; voir l'expos~ de
Stora), dO ~ Jest, Lehmann et Dyson (JLD) [6,7] s'est montr4 extr~mement
utile dans un grand nombre de probl~mes de la th4orie des champs. On
commencera par traiter un exemple tr~s simple dent on peut d4duire une
@tonnante vari~t4 d'applications. Saul exceptions, toutes les valeurs
aux bords seront prises au sens des fonctions C ~ , l'extension aux
distributions et hyperfonctions ne posant, en g4n4ral, pas de probl~me
maj eur.
186
I.- EXEMPLE SIMPLE.
Variables : z I = x I + iy I E ~ , z 2 = x 2 + iY 2 E
I °) Soient deux fonctions :
fl(Zl,X2), C ~ dans {xl,Yl,X2:O~y I ~ ~}, analytique en
z I pour 0 < Yl < ~7 ;
f2(xl,z2), C=dans {Xl,X2,Y2:O~Y2 ~ ~}, analytique en
z 2 pour O < Y2 < ~
2 ° ) On suppose de plus que, dans leur domaine de d~finition, ces
fonctions satisfont & :
Ifj(zl,z2) I <- C (l+IZll) -I (l+Iz21) "I
3 ° ) Enfin on suppose que :
fl(Xl,X2)=f2(xl,x2) pour tout (Xl,X 2) E ~2
Ixl+x2 I < e (o~ e est > O)
tel que
Dans ces conditions
187
Lemme i : il existe F holomorphe dans H , et C ~ dans H , o~
H = [(Zl,Z 2) E ~2 : O < YI' O < y2,y I + Y2 < ~ ] '
telle que F(Zl,X 2) = fl(Zl,X2) pour O ~ Yl ~ ~ ' x2 ~ ~ '
F(Xl,Z 2) = f2(xl,z2) pour O < Y2 N ~ ' x I 6 ~ ; en particulier
si (Xl, x2 ) ~ ~2 , on a F(Xl,X 2) = fl(Xl,X2) = f2(xl,x2).
D~monstration
Posons, pour z = ( Zl, z 2) C H ,
F(Zl'Z2 ) = 12r/ f ~ fl(zl+z2 - ~' 9 2 ) 8 2 - z2 de ~I ~ f2(''Zl+Z2-')@l - Zl d@l
-=o -~
Des raisonnements classiques montrent que F E @ (H) et que F est
C ~ , ~2 iXl+X2 sur H . Choisissons (x I x2) E tel que I < ¢ •
Alors
F(Xl,X2) = lim [~_.r~ / fl(xl+x2 - e2 + 2i~q ' ~2) d@2 .... r~O @2-x2-ii] ~>0
f fl(Xl - = lim I
mo ~>o
i / f2(el, Xl + x2 - @i + 2i~) 1 + ~ ~_xl.i~] d~
e + 2i~] , x 2 + @) d8
I f f2(xl -@' x2 + @ + 2i~]) ] 2ui J @ + i~] d@ J
et il est facile de voir que cette limite est ~gale &
188
lim ~o ~]>0
1 / [fl(Xl 8 ' x2 + 8) f2(xl-D ' x2 + 8)] @- i~] - @ + i~] d
Puisque J(Xl-e) + (x2 + e) I = IXl + x2 I < e , ceci est 4gal &
fl(Xl,X2) = f2(xl,x2) . Par prolongement analytique il en d4coule
fl(Zl,X 2) = F(Zl,X 2) pour 0 ~ Yl ~ ~ ' x2 E ~ , et
f2(xl,z2) = F(Xl,Z 2) pour x I 6 ~ , 0 ~ Y2 ~ ~ " Le lemme est d4montr4.
Introduisons maintenant une transformation eonforme tr&s utile :
l'application
~-~ z = log 7 + ~ ( O~ ~ est > O) (dont l'inverse est ~= Tth ~ )
applique biunivoquement le plan coup4
{~: Im ~# 0 ou I~l < T } =
= [~: - r~ <Arg ~ < ~ ]
sur la bande
z : - 17< Im z <7}
En particulier l'image du domaine (lun~le circulaire)
D(~, CO = {~E ~ : 0 <Arg T+~ T-~
(o~ 0 < 7 , 0 < C~ -< ~ ) est la bande
[ z E { : 0 < Im z < c~ ]
189
i Ttg
- T 0 T
Fig.l : le domaine D(T,Q) .
Cette remarque nous permet de "localiser" le lemme i :
Notons CI = % + i~l ' % = ~2 + i~2 deux variables complexes.
Soient ~i et ~2 deux fonctions de CI et C2 telles que :
i °) Cpl (CI , C2) est d4finie et C m quand C2 = ~2 ~ IR et
~i ED(TI'C~I) , et analytique en ~I dens D(TI, O I) pour cheque
~2 = ~2 r6el ; ( on suppose 0 < T I , O < QI -< ~ ) ;
2 ° ) Q0 2 ales m&mes propri4t4s en 4changeant ~i et 4 2 et en
rempla~ant TI ' ~i par T2 ' ~2 ;
3 ° ) Ii existe dans B2 un voisinage V de
[(gi,¢2) 6 m2 ~1 52
tel que q°l (~i ' ~2 ) = CP2 (~i ' ~2 ) dans V.
190
Dans ces conditions :
Lemme 2
~I et ~72 coincident dans tout le rectangle
= , 2 [~ (~i ~2 ) 6 : IEI~ < TI ' I~21 < T2}
et sont les valeurs aux bords d'une m%me fonction G( ~i,~2 )
dans
analytique
H I = [~= (¢I,~2) : 0 < Im ~i ' O < Im % ,
i TI + ~i i T2 + ~2 -- Im + - I m - < I}
I 71 - ~,I ~2 T2 - ¢ 2
et C ~ dans HI
D~monstration
En rempla~ant ~ par
on est ramen4 au cas o~ V
o~
• j (l'e) Tj (J = 1,2 ; 0 < e < i)
contient l'ouvert r4el :
gl g2 W = {gl'g2 : Igjl < ~j (j=l,2), I-~i+-~ I <K }
0 < K . Censid4rons les transformations conformes:
191
T' + Cj j
z. = -- log (j = 1,2) j c~.j T'j - C+j
f+ = T' th J J ~j j 2~
Quand z I = x I , z 2 = x 2 , r~els, on a
OlXl ~2 x2 C~IXl+~2x2 i < l°~iXl + ~x2! Ith ~ + th --~--- I ~ 2 Ith 2TT TT
Donc la bande
{(Xl,X2) E ]R 2 : l~x I' + C~2x2i < ~ K}
est dans l'image de W . De plus les fonctions
fl(Zl,X 2) = (z I + i) -I (x 2
f2(Xl,Z2) = (x I + i) -I (z 2
+ i) -I ~l(Ti =h , +~ th--y-- )
v6rifient les hypoth&ses i °) et 2 ° ) du lemme i. Ii en est afortiori
de m&me pour
~2 = f2(_~ Xl' "~ z2 ) Fl(Zl,X 2) = fl(-~ z I, -~ x 2) , F 2 62
Le lemme i montre que ces derni~res coTncident sur tous lesr~els ;
donc fl et f2 satisfont & ~outes les conditions du lemme I . En
revenant aux variables ~ et ~ et en faisant tendre T! vers ' j
Tj , on en d~duit le lemme 2. Remarquons qu'en faisant tendre T I
192
et T 2 vers l'infini en maintenant T tg ~j borne, on g~n~ralise J
aussi le lemme I au cas o~ fl,2 ne sont pas n~cessairement born~es
l'infini, et oh, initialement, elles ne coTncident que sur un voisi-
nage quelconque de la droite {Xl,X 2 : x I + x 2 = O} . Les g~n~ralisations
au cas oO on a n fonctions fk(Xl ..... Xk_l, Zk, Xk+l,..., x n) ,
chacune analytique lorsque Im z.j = 0 (j # k), 0 < Im z k < ~ , et co~nci-
n dant dans un voisinage de l'hyperplan {x = (Xl,...,xi): Z x. = O}
j=l ] peut se faire sans difficult~ :dans le cas borne, on part de la formule
n f F(z I ..... Zn) = Z i fk(~ ..... %_l,Sk+j=l zJ '~+I' k=l (2i~) n-I ....
n
6( E 8.) • (8 - Zr )-i ~ dS~ j=l ] r~k r /=l
e)x n
On continue en faisant les m~mes transformations conformes que pour
n=2.
Conclusions :
i) Le th~or~me du tube est applicable ~ des situations "aplaties" :
c'est le r~sultat de MALGRANGE-ZERNER (1961).
2) Ii y a des ph~nomgnes int~ressants d'agrandissement de la r~gion de
coincidence dans les r~els. Nous y reviendrons plus loin.
193
II.- LES PROBLEMES D'E.0.W. & 2 TUBES.
lls se posent ainsi : deux fonctions fl et f2 sont analytiques
respectivement dans 31 = ~n + i C I et 32 = ~n + i C 2 ; C I et
C 2 sont deux cDnes convexes dans ~n qui peuvent &tre soit ouverts,
soit "aplatis" ; les valeurs aux bords de fl et f2 dans les r4els
coYncident dans un ouvert r4el ~ de ~n ("r4gion de coincidence").
Ii s'agit de trouver (ou tout au moins d'4tudier) le "domaine"
H(~ I U ~2 U ~) o~ elles ont un prolongement commun. En pratique
on rencontre essentiellement deux cas
a) C 1 = - C 2 (probl~mes d'e.o.w, oppose)
b) C 1 N (-C 2) = @ (e.o.w. oblique)
Ii y a de plus des cas, importants en pratique, d'une nature "semi-locale":
on a g trouver H((g 1 U ~2 U ~) N ~) , o~ ~ est un voisinage ouvert
complexe donn~ de ~ . Comme il a d~j& ~t~ dit, il y a peu de r~sultats
g~n~raux. Mais il existe des exemples solubles assez int~ressants.
194
II.i.- E.O.W. OPPOSE.
A. (CDnes poly~draux simpliciaux).
C+ [y = (YI' ,yn ) E ~n = _ = "'" : Yl >O'''''Yn >0} C
~+ = ]R n +_ i C+
= I x E IR n : ~xj I < 1 pour tout j = I ..... n}
Dans ces cas on peut facilement calculer H(~+ U g_ U ~ : la trans-
formation
1 + z. i ~j = log J , z. = th
I - z . 3 2 ]
remplace le probl~me par celui de trouver H(g$ U 1~' U ]R n) oO
~, = _ ~, = [~ E~ n + _ : o < Im ~ < ~]
Vues les remarques faites & propos du lemme Iet de ses g4n4ralisations,
on peut appliquer le th4or~me du tube convexe, et on trouve
H ( ~ U ~' U ]Rn) = U [~ -' "O < Im ~j < T[ - ~)'} - 0 ~ O <rr
Donc
l+z, H(g, U ~ U ~,) = U [z 6 ~n : - O <Arg 3 < 17- O}
T 0 ~ 8~17 l-z.
3
Le domaine peut donc &tre d4crit comme une union de polydisques :
195
Fig.2 : H(g+ U g U ~ ) = union de polydisques.
Ii est clair qu'on obtient de m~me, par un changement de variables
r~el-lin~aire, le domaine H(g U g U C(a,b)), o3 C(a,b) est le +
double -c6ne (ici parall~l~pip~de)
C(a,b) [x 6 ~R n = : x - a E C+ , b - x E C+}
B. ~ = C I .
En particulier il est int~ressant de consid~rer le cas o~
et b. = += pour tout j , c'est-~-dire le cas o3 ~ = C 3 +
peut ~tre ramen~ au th~or~me du tube en posant
a=O
• Ce cas
j = log zj , z. = e ~j 3
et on trouve :
H(g+ U g U C+) = U {z E ~n : _ e < Arg z. ~ rr-~)}
U e -ie U sg + = 0<-@~ g+ = s:Im s<O
Ii est utile de donner une autre representation g~om~trique de ce
domaine .
196
Les points non r~els de H(g+ U ~" t.I C+) sont exactement les points
non r~els de toutes les droites "r~elles" (comme vari~t~s alg~briques)
dont la trace r~elle rencontre C+ . En effet tout point du domaine
s'~crit
z = (@- i) (x 0 + iy O) , YO E
ou
= (i + ~2) Y0
et l'~quation
z = (i + 2,) YO
C+, ~ E ~
+ (~- i) (x 0 - ~yo )
+ ~ (x O - ~ yo ) , • E
est bien celle d'une droite "$ coefficients r~els" dont la trace r~elle
rencontre C+ (en particulier au point (I + ~2) YO )" La r~ciproque
est aussi vraie.
C. ~= ~£+ C+
C+
Consid~rons maintenant le cas o~
, i.e. oN
f~+ C+ =
est une union de translates de
Danc ce cas
a E ~ O~e~
197
qui contient ~ son tour
U (a- i~,) = i ~n +
a C
L'enveloppe d'holomorphie de ce tube est son enveloppe convexe ;donc
n'est naturelle que si elle est convexe. En particulier, si on se
donne initialement une r~gion de coincidence ~ non convexe (mais telle
que ~+ C+ = ~) , elle s'agrandit automatiquement ~ son enveloppe con-
vexe : c'est ig un deuxi~me ph~nom~ne d'agrandissement de la r~gion
r~elle de coincidence. (Ce ph~nom~ne est li~ ~ celui des "reentrent
noses" d~couvert par Dyson [7] ). Lorsque ~ est convexe et v~rifie
+ C+ = ~ , on voit facilement que U H(~+ U g_ U (a+C+)) est
aER un domaine d'holomorphie : c'est en effet le compl~mentaire de l'union
des (n-l)- plans analytiques g coefficients r~els dont la trace r~elle
ne rencontre pas ~ (utiliser Hahn Banach et la description du domaine par
les droites r~elles).
D. G~n~ralisation ~ des CDnes convexes quelconques.
Soit Fun cbne convexe ouvert dans ~n , posons g~ = ~n! iF .
Soit ~ un ouvert de ~n tel que ~ ~ r =~ On peut toujours,
par un changement de coordonn6es, se ramener au cas o~ r contient
C+ . Ii en r~sulte que i °) H(g+ F U ~ U ~ ) = H(~+ U ~_ U conv.~ ) ;
2 ° ) H(g+ F U $r_ U conv.~ ) = H(~+ U ~_ U conv.~ ) = union des points non
r~els des droites r~elles qui rencontrent conv.~ et de conv.~ = compl~-
mentaire des (n-l)-plans analytiques r~e~qui ne coupent pas
t98
Finalement remarquons que si ~ est un ouvert r~el (non n~cessai-
rement tel que ~ + F = ~ ) contenant un cDne ouvert canve×e F I tel
que F C F I on a
H((~n+ iF)U(I~ n - iDU~) ~H((~n+ iF)U(I% n- ir)~F I) =
= H((IRn+ iq) U (~n + i]71 ) U I~i )
et par suite H((~n+ir)U(~R n- ir)U~) = H((~n+il~l)U(]~n-irl)U~) .
Tous ces ph~nom~nes ont des versions locales (visibles par transformations
conformes par ex.) mais leur description n'est pas aussi simple.
E. Retour su~ le eas o~ ~ est un double-cDne.
Revenons au domaine H(g+ U ~_ UC(a,b)), (n °- A),
~+ -~ = [x + iy E C n = : yj > 0 pour tout j}
C(a,b) = {x E ~ n : a i <x i <bj pour tout j}
avec
Le changement de variables :
b - a. zj - bj z~ = i - J J ....... (I ~ j ~n) ,
3 g . - a . z , - a J J 3 J
applique ~+ dans lui m~me et la r~gion r~elle a + C+ sur
[x E ~n ' : x. < 1 (i ~ j ~ n)} . II applique en particulier J
sur C . On en d~duit la description suivante du domaine
C(a,b)
199
H(g+ U g_ U C(a,b)) =
Z~ - b.
-- : = t Im >0 Imt>O l~j ~n } z. - a. ' ' ' J J
g. " b.
J J = t~ (l~j ~n) o~ les ~j sont fixes et Les n ~quations z. - a.
J 3 o~ t d~crit C sont les ~quations param~triques d'une courbe alg~brique.
Ces courbes seront appel~es "courbes Q" et jouent un rDle important dans
les probl~mes du type H(g+ U ~_ U C(a,b)). Elles passent par a et b .
F. R~ions de eolnqidenc e convexes.
On peut montrer que, dans le cas o~
dence est connexe et naturelle, on a
n = 2 , si la r~gion de coTnci-
u u m = U w(a,b) a et b E~ b - a EC+
(C'est un r@sultat de la representation JLD, mais on peut ~galement r~ob-
tenir JLD de cette fagon). Ce r~sultat n'est plus vrai pour n > 2
Mais il reste valable si ~ est convexe. Le principe de la d~monstration
est le suivant. On suppose d'abord ~ born~e, convexe, et donn~e par
= {x : f(x) < 0 , g(x) > O}
o~ f (resp g) est une fonction ~ telle que gradf (resp gradg) E C+ ;
on suppose que
200
R+ = {x : g(x) > O} et R_ = [x : f(x) < O }
sont convexes et telles que ~ + + c + = ~+ , ~ _ + c = ~
- - f ( ~ ) = o
Figure 3
(Le passage au cas g~n~ral, par passage ~ la limite, est facile).
Notons
W(R) = a,b~ R W(a,b)
L'~tude, tr~s longue et fastidieuse, de la fronti~re de
duit aux r~sultats suivants :
w(~) , con-
i) la fronti~re est compos~e de points de ~+ U ~g_ et d'un ensem-
ble not~ 6W . Celui-ci comprend une partie "g~n~rique" 61W et des
parties d~g~n~r~es dont, en fait, les points sont adherents ~ 61W
201
2) 61W est une union de "courbes Q" & coefficients r~els. Tout point
z = (Zl,...,z n) non r~el de 61W v~rifie :
J ~J zj - TIj = ~,jt
O~ : ~ et ~ sont des points r~els tels que
f(~) = g(~ = 0 , ~j - ~]j > 0 Vj ,
N = grad f(~) , n = grad g(~ ,
Xj = cj n~/---~ , (ej = + I) ,
J
n ~ j ' E Wj(gj "T]j) = 000 ~j = £j nj
j=l
t E ~ et Imt > 0
pour tout j ;
Le calcul d'enveloppe qui conduit & ces conditions montre qu'en
un tel point z , l'hyperplan tangent au domaine est donn~ par
o£I
n
{~ : ~ Im a (~j - z.) = 0 } j=1 J J
T ak =-- Pk ' Pk = Nk(l " & t)2 = Nk (~k ~k )2 Imt (z k - ~k) 2
202
En particulier cet hyperplan contient l'hyperplan analytique :
n [~ : ~ Pj(~j - z.) = O}
j=l J
On peut chercher l'enveloppe de ce dernier lorsque t varie. On trouve
la vari~t~ d'~quation :
~ (N in j - ~ i ~ j ) ( ~ - ~ ) ( ~ j - 1 1 j ) = 0
ou (avec un produit scalaire ~vident) :
(N,~-0(n,¢- ~ - (~,~-~(~,~-~ = 0 ,
ou encore, puisque (W, ~ = 0
( N , ¢ - @ ( ~ , ¢ - @ - ( ® , ~ _ © 2 = o
Pour tout x r4el dans ~ on a (par convexit4)
(N,x-~) < O , (n,x-~ > 0 ,
tandis que si x appartient & la quadrique pr4c~dente, on a
(N,x-~)(n,x-@ = (~,x-~) 2 e 0
La trace r~elle de la quadrique ne coupe donc pas ~ . On v~rifie
facilement que la quadrique (complexe) ne coupe pas g+ U ~
203
Elle ne peut donc couper H(g+ U ~_ U ~) . Ceci, joint ~ l'~tude
des autres points de la fronti~re, montre que
i =) H(g+ U ~ U @~) = W(~ H
oO K est un ensemble de families A = {A } I It~L~2 de coefficients
r~els tels que la forme homog~n~isfie du polyn~me q u a d r a t i q u e ~ A z ~
ai~ au plus 3 carr~s, et que ce polyn~me ne prenne aucune valeur posi-
rive dans le domaine.
G. R68ions convexes et cDnes quelcqnques.
Le r~sultat de (F) peut s'~tendre au cas de tubes ~+ = ~n! iX
o~ F est un cDne ouvert convexe quelconque par la "m6thode des varia-
bles surabondantes" : ehoisissons des vecteurs el,...,e N dans ~ .
Si f est analytique dans H(~+ U gF U. ~ ) , la fonction
N
F(~ I ..... {N ) = f( E {j ej) j=l
est en particulier, analytique dans l'union des deux tubes
[~ E ~N : Im ~j > 0} et {~ E {N : Im ~j < O} ,
et au voisinage de
N
j=l J
204
Si ~ est convexe, ~ l'est aussi ; on peut lui appliquer la th~orie
de (F) et, au prix de quelques efforts suppl~mentaires, on obtient
finalement le th~or~me suivant :
THEOREME 3.
Soit F un c5ne ouvert convexe de ~n tel que F O (-F) = @ .
Soit ~ un ouvert convexe de ~n tel que si a E ~ , b E ~ , x-a E r ,
b-x 6 F , alors x E ~ . Notons W(a,b) = H(~. U gF U F(a,b)) ,
gF mn+ iF, F(a,b) [x E ~n = = : x - a E F , b - x E F} . Alors +
U W(a,b) i°) H(~ U ~']f" u ~) = a,b E ~
2 ° ) H(g~ UT" gF U_ R) est le compl~mentaire de l'union d'une
famille de quadriques ~ coefficients r4els ayant au plus
3 carr4s (en coordonn4es homog~nes).
H. Canes relativistes.
On consid~re ~4n (resp. C 4n) comme produit topologique de n
lois l'espace de Minkowski r~el (resp.complexe). On notera si z 6 {4n
ni par
~ ~. ~ ° ~ ~o z0 ~ ~ z~ ~ ~ z~ z~ Zk~ g~v Zk
Dans l'espace de Minkowski ]114 , le c5ne V + = -V" est d~fi-
205
x O v÷={x={x °,x~: >iTi}
On pose g+ = ~4 + iV + . Parmi les automorphism~analytiques de
g+ figure l'inversion relativiste :
= [z~] ~ - (z~z) '
(noter que (z,z) ne s'annule pas dans g+) . On en profite pour
calculer simplement le domaine e.o.w. H((~) n U (~-)n U F (a,b))
o~
F(a,b) {x E ~4n = : x - a 6 (V+) n , (x - b) E (V+) n}
En effet la transformation
Z. - a
z. ~z'. =- J "] J 3 (zj-aj ,zj-aj)
(j = I ..... n)
conserve les tubes (g*)n et applique F(a,b) sur
b. - a. [x = (Xl, ,x n) E ~4n I J ... : Yj, xj + (bj-aj,bj-aj) E v-]
D'apr~s la th~orie g~n~rale (voir n 9_ D) l'enveloppe
H((~+) n U (g+)n U c - (V+) n) est donn~e par
{z' : z'. - c. = t~. , Imt >0 , ImP. EV + ( i ~ j ~ n)} 3 J J J
donc
o~
H((~+) n U (~-)n U F(a,b)) =
tk. - c. _ = .I .I = [z : Vj, Zj a.3 - (t~j_cj,tkj_cj) , Imt >O,Im~j E V +}
(bj - aj) = .
• -a ,bj-aj) cj (bj j
206
De plus le compl~mentaire du domaine est l'union des inverses
des hyperplans analytiques ~ coefficients r~els qui ne rencontrent pas
c +(v')ndans les r~els. En particulier, si n = I on obtient des hy-
perbololdes : l'inverse du plan {z' : (h,z') + k = 0 } s'~crit en
effet
(h~ z q a) (z-a,z-a) + k = 0 , ou encore :
k(z-a, z-a) - (h, z-a) = 0
C'est un hyperbololde (si k ~ O) ayant pour cane asymptotique le
cDne de lumi~re {z : (z,z) = O~ et passant par a . En outre le
domaine a pour points non r~els tousles points non r~els appartenant
des inverses de droites r~elles (c'est-g-dire des hyperboles r~elles
asymptotiques ~ des directions du genre lumi~re) dont la trace r~elle
rencontre I~a,b). Ces remarques permettent, en fait, de retrouver le
r~sultat de Jost-Lehmann-Dyson.
207
II.2.- E.O.W. OBLIQUE.
Dans ce sous-parggraphe, on notera C I et C 2 deux cBnes non
vides convexes dans ~n tels que C I N (-C2) = ~ , C+ = CI-C 2 = -C ,
~I = ~n + iCl, ~2 = IRn + iC2 ' g+ = ]Rn ! iC+
L'exemple trait~ au §I. est un probl&me d'e.o.w, oblique. La
solution peut ~tre interpr4t~e eomme une intersection :
H([(Zl,Z2) E C 2 : Im z 2 = 0 , O <Ira z I < ~}[[Zl,Z2EC2:ImZl=O,O<Imz2 < ~}U~2) =
= H({(Zl,Z2) 6 2 : 0 <ImZl< TT, 0 <-Imz2< 17}U{Zl,Z 2 6 ~2 : 0 <-Imz I < ~,
O < Imz 2 < ~} U ~2) A {Zl,Z2E ~2 : O < Imz I < ~, O < Imz 2 < ~} .
En faisant les changements de variables indiqu4s au ~I. on en d4duit que
H([(Zl,Z 2) Eg 2 : Imz 2 = O, -T <z 2 < ? , Imz I >O} U
U [(Zl,Z 2) E ~2: ImZl=O, -~ <z I < T , Imz 2 >0}
U {(Xl,X 2) E jR2: ~T <x I < ~ , -T <x 2 < T} =
H([(zl,z 2) E ~2 ~2:imzl< O = : Imz I >O , Imz 2 <O} U [(Zl,Z2)6 ,Imz2>O }
U [(xl,x 2) E ~2 Ixll < T, Ix21 < ~}.
En d'autres termes si C I est le cone
C 2 = [(yl,Y2 ) E ]R 2 : Y2 > O, Yl = O} et
{x 6 ~2 : iXll < • , iXmi < T } on a
~2 [(Yl,Y2 ) E : Yl >O ' Y2 = O},
est le double-cOne
H(~ 1 u~2 ua) = H ( ~ + U ~ U~) n (~ l + ~2 )
208
Ici le "double-cDne" doit ~tre interpr~t~ comme fabriqu~ avec C+
Ii est facile de constater que l'~quation ci-dessus reste vraie, ~ deux
dimensions, si C I et C 2 sont des c~nes ouverts convexes quelconques
tels que C I n (-C2)= ~ et si ~ est naturelle pour 3+ i.e. si
Ce th~or~me g~n~ral peut d'ailleurs se d~montrer directement en appli-
cant la solubilit~ du probl~me de Cousin dans les domaines d'holomorphie
(voir [I]) . Nous avons vu au § I. un ph~nom~ne d'agrandissement de la
r~gion de coincidence. II est clair qu'A deux dimensions on obtient im-
m~diatement le lemme suivant :
Lemme 4.
Soient C I et C 2 des cbnes convexes dans 2 , non r~duits
{0} , tels que C I D (-C 2) = ~ ; si ~ e~t un ouvert r~el de ~2
contenant un segment de droite d'extr~mit~s aet b telles que
b - a E C+ (=C I - C 2) alors ~= H(~ I U 5 2 U ~) ~ IR 2 contient le
double ebne (a + C+) n (b - C+)
Montrons maintenant que le deuxi~me ph~nom~ne d'agrandissement
de la r~gion de coincidence (voir II.I.c) se produit aussi. Consld~rons
nouveau le cas C I = [YI' Y2 : 0 < YI' Y2 = O} , C 2 = {YI' Y2:0 <y2,Yl=O}
et prenons ~ = C+ = {YI' Y2 : Yl > O, Y2 <0} . Le th~or~me d'inter-
section s'applique ~ ce cas (cas limite d'un double-cbne) done
= ~ ( ~ + u ~; u ~ ) n {~1' =2 : ~m~l > ° , Im= 2 > o }
209
En particulier les points non r4els du domaine sont les points non r4els
de route droite r4elle dont la trace r4elle rencontre ~= C+) et dont
la par tie imaginaire est contenue dan s C I + C 2 . Supposons maintenant
que ~ est une union de translates de C+ . Soit ~ =Conv. ~ . Si
x E ~ , toute droite r4elle passant par x rencontre ~ . Ii en r4sulte
que H(Z i U ~2 U ~) contient l'intersection d'un voisinage complexe
de x avec ;31 + 32 et que la r4gion de coincidence s'4tend jusqu'~
T . Ii est 4vident qu'~ deux dimensions, ce r~sultat s'~tend ~ des
C I , C 2 convexes non vides quelconques tels que C I n (-c 2) = @ .
Pour n > 2 le th4or~me d'intersection n'est plus vrai en
g~n4ral. Toutefois on ale th4or~me suivant.
THEOREME 5.
Soient C et C 2 deux cBnes ouverts convexes non vides de i
]R n tels que C I N (-C 2) = ~ ; on pose C+ = C I - C 2 , 31 = An+ iC I ,
~2 = ~n+ iC2, ~+ = ~n iC+ . Soit ~ un ouvert eonvexe tel que
+ C+ = ~ . Alors
D~monstration : Elle se fait par la m~thode des variables surabondantes
et ne sera pas donn~e en d~tail.
D'autres exemples de cas o0 le th~or~me d'intersection est vrai
se rencontrent en th~orie des champs : voir par ex. [3] et [4] .
Nous terminerons ces remarques sur l'e.o.w, oblique par le
th~or~me suivant (cf.[2] . Pour le cas de l'e.o.w, oppose, le th~or~me
est d~ ~ Dyson [7 ] , Borchers [8 ] , Vladimirov [9 ]).
210
THEOREME 6.
Soient C I et C 2 deux cSnes ouverts convexes non vides de
1R n , C+ = C I- C 2, 51 = IR n + iC I , ~2 = ~n+ iC 2 . Soient aet b
deux points de ~n tels que b - a E C+ , V un voisinage ouvert
r~el du segment (ouvert) (a, b) , et W un voisinage ouvert complexe
du "double-cDne" :
K {x E •n = : x - a E C+ , b - x E C+}
Si fl et f2 sont des fonctions holomorphes dans gl n W et
~2 n W ,respectivement, et ont des valeurs aux bords dans K qui
coincident dans V , elles coincident dans tout le double-cOne K
Remarque : La condition que C I et C 2 soient ouverts est essentielle
dans le cas n e 3 : on connaSt des contre-exemples lorsqu'elle n'est
pas satisfaite, m~me si C+ est ouvert.
D~monstration :
a) Consid~rons d'abord un cas particulier. Soient p et
entiers > I ; un point courant de ~P × ~q sera not~ (z,w)
z = x + iy , w = u + iv , x = (x l,...,xp) et y = (YI' .... yp) E
u = (Ul,...,Uq) et v = (Vl,...,Vq) E I~ q . On suppose que fl
holomorphe dans l'ouvert D I de ~P X ~q donn~ par
D I = {(z,w) : zj ED(T,~) pour tout j = i ..... p ; lUkl < • et
Vk < ~Yl pour tout k = l,...,q}
et fl est holomorphe dans
D 2 = {(z,w) : w k E D(~,~) pour tout k = i ..... q ; Ixj I < ~ et
yj < ev I pour tout j = i .... ,p}
q deux
avec
~P
e s t
211
f. est C = dans D. (j = 1,2) et les valeurs aux bords sur les reels J J
de ces deux fonctions coincident dans
V = {(x,u) E ]R p X ]R q : Ixj - Xll < 6 , Ixj I < T pour tout j ,
lUk + Xll < 6, lUkl < T pour tout k}
La conclusion est que fl et f2 coincident dans tout le double-cane
{(x,u) 6 IRe X IR q : Ixj [ < T , fUel < T , Vj , Vk } .
Pour le voir posons
et
T+E. = Tth J = + i•j = ~ log ) zj , T-Zj
~ i T+Wk w k = Tth --~-- , (~ = ~ + i~ k = ~ log ~_--~k )
f.(Tth ~ ~i . Tth ~) gj (~,w) = ] "7-- .... , j = 1,2
P q
r~=l(Cr+i)= k=l~- (~+i)
On v4rifie facilement que les domaines d'analyticit4 de gl et g2
t! contiennent respectivement les domaines D~' et D 2 donn4s par
D[= { C , X : O < % < i , 0 < % < i ,
D 2''=[¢,x:o<~j<I,0<%< l ,
o <e-~i%1% < ~, % e'~l%l
212
(o~ e' e 2 = ~ cos ~ ) . De plus gl et g2 coT ncident aux points
reels (~, Q) de V'
V'= [(~,p): l~j-~ll < 6 I% + ~i I < 6 Ct~ ' -~ (j=l .... p, k=l .... q)] .
~OSOIIS
¢o
• t_ h gl ( q+h "t ..... $+~l-t, t, ~2- h +t .... kq" h+t)
I f d___!_s g2 (s, ~2_ ~i+ s ..... Cp_ ~l+S, ~+~I_ s .... kq+~l_S) " s-g I
Ii est facile de v~rifier que h est holomorphe dans
-4M(~, p) D 3" = {~ X : O < ~j < ~ , , O < Ok < ½ , O < Ok - Ol < e'(~l+Ol)e
o~
O < ~qj-~l < e'(r~l+Ol )e-4M(~'p)
M(~,p) = max (l~j [ ' IPkl )
, Yj = i .... p , Vk = i .... q}
[Toutes ces estimations reposent sur les in~galit~s suivantes : si
y = Im th-~ (~+i~ , 0 < Ct < rT , 0 < I] < 1 ,
on a "
~ e - Ict~isin Ct < ½ sin C~l]e- IOL~I < y < 4e "0~I~I tg~2 < 4~]e "C~i~[tg "~ ] .
Les domaines D~ et D i (resp. D~ et D~) ont une intersection non
vide qui contien% pour tout (~,p) , l'intersection d'un voisinage com-
plexe de (~,p) avec un tube & base conique. De plus, par un calcul tout
fait analogue & celui du SI., on constate que
213
h(~,~) = gl(~,~) = g2(~,@) dans le domaine V' ° Done
h(~,p) , gl(~,p) et g2(~,~) coTncident partout et notre assertion est
v4rifi4e.
b) Cas $4n4ral.
On se place dans les hypotheses du th4or~me 5 en supposant que
a = 0 , et b est fix4 dans C+ . Choisissons des vecteurs
el,..., ep 6 C I , ep+l,..., e N E - C 2 de telle sorte que
N P b = ~ e. . On note F I = ~ = ~ ~. ej ~. > O}
j=l 3 j-i 3 ' 3 '
N
F 2 = ~ = ~ ~j ej , gO < O} , et F+ = F I - F 2 . Le double cDne j=p+l
K est donn4 par
E ~n : x ~ C+ , b-x E C+}
D~finissons deux nouvelles fonctions analytiques dans des domaines de
~N par :
N
F.(~) = f ( Z % e k) J J k=l
En supposant qu'on a l~g~rement r~duit les domaines d'analyticit~ de
f l e t f2 on se ramgne au e a s og F 1 ( r e s p . F 2) e s t a n a l y t i q u e dans
l'intersection de l'ouvert :
[6 6 ~N IArg SJ I < (~ , i ~ j ~ N}
avec un tube ~ base cbnique eonvexe ouverte contenant tousles points non
r~els de :
{~ E ~N : Im ~j
(resp. [ ~6 ~N : Im ~j
e0 pour i ~ j ~p , Im ~j
= 0 pour i ~ j <_ p , Im ~j
= 0 pour p+l ~ j ~ N}
0 pour p+l ~ j _< N]) .
2'i4
De plus les valeurs aux bords de F 1 et F 2 aux points r~els coincident
dans un voisinage ouvert r~el de tout segment r~el de la forme
[~ E : ~j = 0(I+Tj) , O ~ e g i} ,
o~ les ~j sont des hombres r~els tels que i + Tj > 0 pour tout j et
N Tj e. = O . Soit U l'ensemble des T E ~R N poss~dent ces propri~t~s.
j=l 3
En appliquant le cas (a) on volt que F I et F 2 coincident dans l'union
lorsque T parcourt U , des ouverts r~els
[~ E ~N : ~j = e ( I +T j ) o < e < 1} .1 ' j '
et que fl et f2 coincident donc dans l'union, lorsque T parcourt
U , des ouverts r~els
N Ix E IR n : x = Z e (i+~.) e. , O < e< I} .
j=l j 3 J 3
Cette union n'est autre que le double cDne
Ix E ]Rn : x E r +, (b-x) E ~} , oh ~+= Fl-r2,
N ~+=[x E ]Rn : x = I ~. e. , ~. > O Vi} .
i= I z 1 l
N En effet si x = E 0.(I+T.) e. avec I+T. >O , 0 < O < i , et
j=l 3 J 3 3 J
E T ej = O, on a 6videmment x = ~ ~.e. avec ~. > O donc x E ~6 et j J j J J J
N N d'autre part E e.-x = E [(I-~.)-~ T }ej = E (I-6.)T e. donc
j 3 j=l S J J j=l 2 J J
• R~ciproquement si x E ~ et b-x E r + on a
N
= ~ (l-uj) e o~ ~.>O , u. >O ~jej j = i J 2 J
b - x E ~ +
N x = E
j = l
N Donc j=iZ (~.3 + u.j - i) e.j = O . Posons ~j = ~'3 + uj-I , et
E e (l+T.)e Oj = ~j+uj ; on a x = . . j 3 J J
En f a i s a n t t e n d r e F 1 e t F 2 v e r s C 1 e t C 2 on o b t i e n t l e
th~or~me.
215
III. - EXTENSIONSPAR DES GROUPES ANALYTIqUES.
Ce dernier paragraphe n~a que des rapports lointains avec les
questions de valeurs aux bords euc..° ; il figure ici parcequ'il pr4-
sente un m4¢anisme, tr~s important en th4orie des champs , d'agrandisse-
ment des domaines. Commengons par un lemme tr~s simple :
Lemme 7.
Soit D un domaine de ~n × ~m de la forme suivante :
D = [(Z,W) 6 ~n X m : z E ~ , w E ~(z)} ,
o~ fl est un domaine de C n et A(z) est, pour cheque z 6 fl , un
~omaine non vide de ~m . S'il existe un ouvert % dens ~ tel
que z 6 % = ~(z) = ~m (c'est ~ dire si D contient % X ~m)
l'enveloppe d'holomorphie de D contient ~ X ~m
D~monstration
a) un c.as particulier
Supposons que ~ = ~z : Izj + iRj I < Rj , 1 ~ j ~ n} ,
% = irj , . < R. , et : [z : Izj + I < rj I ~ j ~ n} , avec rj 3
D = (% X ~m) u ([~ × [w : lWkl < ~k } )
..... 1 -iw' k L'inversion z~ ~ z'. et la transformation w~ = e
3 ce domaine en un tube :
transforment
D'={z',w':Imz'j>~-rl } U {z',w':Imz'j >__~_I Imw' k <log Tk } • 2 R . ' '
3 3
216
dont l'enveloppe d'holomorphie est : {z',w' : Imz'j >~.} = D"
J De plus toute fonction holomorphe dans D' et p6riodique en Rew' k
se prolonge dans D" en une fonction p6riodique. Iien r6sulte que
l'enveloppe d'holomorphie de D est bien ~ X Gm
b) Le cas g~n~ral est obtenu en construisant dans ~ des
chaines finies de polydisques (s'intersectant 2 ~ 2) commen~ant dans
et en utilisant le cas (a). Les d~tails sont laiss~s au lecteur.
THEOP~ME 8.
Soit G un groupe de Lie complexe, ~ son alg6bre de Lie
et notons X ~ exp X l'application exponentielle habituelle. Sup-
posons donn6e une application
(g,z) ~'~ g.z
G X ~n ..~ ~n
holomorphe et telle que l.z = z , h.(g.z) = (hg).z . Soit ~ un
domaine de ~n ; supposons qu'il existe un ouvert ~ ~ ~ tel que
G.~ ~ ~ . Alors l'enveloppe d'holomorphie de ~ est un domaine de
Riemann dont la projection sur ~n contient G.~ , Plus pr~cis~ment
soit G le groupe de recouvrement de G , ~ l'homomorphisme naturel
A de G sur G . Si f est holomorphe dans ~ , la fonction
(h,z) ~ ~ f(T(h).z) ,
a un prolongement unique
A G X
F holomorphe sur le produit topologique
217
D4monstration.
Consid4rons une suite finie XI,.o.,X m d'414ments non nuls
de ~ et posons
HX I ..... X m (~i ..... ~m;Z) = f(exP~iXl.exP~2X 2 .... eXP~mXm. Z>
D'apr&s le lemme 6 cette fonction a un prolongement analytique unique
dans ~m X ~ . Le th4or&me s'obtient par des raisonnements de mono-
dromie faciles qui ne seront pas reproduits ici (voir [i0] )
En pratique on utilise plutDt une autre version du th~or~me 7 :
THEOREME 9.
Soit G un groupe de Lie complexe, Q son alg~bre de Lie, et
(g,z)~'~ g.z une application holomorphe de G X ~n dans ~n telle
que l.z = z et h.(g.z) = (hg).z. Soit ~ une base de Q . Soit
un domaine de ~n tel que~ pour tout X E ~ , il existe un ouvert
VX C~ tel que (exp ~X).V X ~ ~ pour tout ~ E ~ . Alors on a la
m~me conclusion que dans le th~or~me 8.
HXI (Dans ce cas, pour d~montrer l'analyticit~ de ,...,X m
on montre d'abord que HXI(~ ; z) est analytique dans ~ X ~ . On
consid~re alors HXI(~ ; exp ~X2.z) = HXI,X2(~I, ~ ;z) qui est ana-
lytique dans ~ X ~ × Q , etc .... Le reste de la d~monstration est
inchang~).
Ces th~or~mes sont appliques, en th~orie des champs, au cas o~
G est le groupe de Lorentz complexe connexe. La version donn~e ici suit
et g~n~ralise l~g~rement celle de [i0] . Voir [ii] et [12]
218
REFERENCES
[i] J. Bros. Les probl~mes de construction d'enveloppe d'holomorphie
en th4orie quantique des champs, S4minaire Lelong 4~me ann4e
n o 8 , 1962 .
[2 ] H. Epstein dans: Axiomatic Field Theory, Chretien & Deser 4diteurs,
Gordon & Breach, New York 1966.
[3] J. Bros, H. Epstein et V. Glaser, Nuovo Cimento 3__1, 1965 (1964).
[4 ] H. Epstein, V. Glaser et A. Martin, Commun. Math. Phys. 13, 257
(1969).
[5] J. Bros, H. Epstein et V. Glaser, Helv. Phys. Acta .
[6] R. Jost et H. Lehmann, Nuovo Cimento ~, 1598 (1957).
[7] F.J. Dyson, Phys. Rev. ii0, 1460 (1958).
[8] H.J. Borchers, Nuovo Cimento 19, 781 (1961).
[9] V.S. Vladimirov, Trudy Mat. Inst. A.N. SSSR, 6__O0, i01 (1961).
[lO ] J. Bros, H. Epstein et V. Glaser, Commun. Math. Phys. ~, 77
(1967).
[ii] R.F. Streater, J. Math. Phys. ~, 256 (1962) .
[12 ] R. Jost, General Theory of Quantized Fields, American Math. Soc.
Providence, R.I. 1965.
