Differential geometric methods in mathematical physics: proceedings of an international conference held at the Technical University of Clausthal, FRG, August 30-September 2, 1983

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1139 Differential Geometric Methods in Mathematical Physics Proceedings of an International Conference Held at the Technical University of Clausthal, FRG, August 30-September 2, 1983

Edited by H. D. Doebner and J. D. Hennig

Spfinger-Verlag Berlin Heidelberg New York Tokyo

Editors

Heinz-Dietrich Deebner JSrg-Dieter Hennig Institut fSr Theoretische Physik A, Technische Universit&t Clausthal 3392 ClausthaI-Zellerfeld, Federal Republic of Germany

Mathematics Subject Classification (1980): 49H, 53B, 53C, 58C, 58F, ?0E, 81 C, 81G, 83D, 73C, 55N, 20F. ISBN 3-540-15666-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15666-6 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned, specificallythose of translation,reprinting,re-useof illustrations,broadcasting, reproductionby photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than privateuse, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

Preface

The XII. Methods the

International

Clausthal, by H.D.

for T h e o r e t i c a l

Germany

Doebner,

The D G M - s e r i e s

F.R.,

was

initiated

in Bonn

(1976),

Clausthal

and J e r u s a l e m

series

ferential-geometrical

They

of such

cal

physics

methods

1983.

and H.D.

Salamanca

at

of

It was o r g a n i z e d

and G. Denardo

(Trieste).

Doebner

1971

in

(1974,1979),

(1979),

the a p p l i c a t i o n

and their

and to e x p l o i t Over

Trieste

(1981)

ones

the years

and the

hidden

and

gathered

quite

of m a t h e m a t i c s /

grew c o n s i d e r a b l y .

in d e v e l o p i n g

the g e o m e t r i c a l

dif-

of com-

geometry

the c o n f e r e n c e s in this b r a n c h

interest

with

especially

for the m o d e l l i n g

field of the D G M series

in c o n n e c t i o n

of geometrical,

interplay,

the often

researchers

also an i n c r e a s i n g

new mathema-

structure

of p h y s i -

systems. of the XII.

lowing key words,

-

2,

Geometric

took place

University

Aix-en-Provence

and - t o p o l o g i c a l

systems.

techniques

The topics

-

by K. B l e u l e r

(1978,1980),

of p r o m i n e n t

stimulated

tical

(Clausthal)

is to promote

systems,

a large n u m b e r mathematical

(DGM-series)

Technical

30 - S e p t e m b e r

(1973,1975),

and a l g e b r a i c a l

plex p h y s i c a l symme t r y

A,

"Differential

(1982).

idea of the

analytical

August

series

Physics"

Physics

S.I. A n d e r s s o n

continued

Warsaw

The

of the

in Mathema-tical/Theoretical

Institute

Bonn,

Conference

Momentum Aspects

Mappings

The a r t i c l e s

pedagogical time,

cover

(48 lectures).

to these

with a very the

of Special

in this volume

requirement

also

Integrability

Modelling

at the c o n f e r e n c e

finit e l y

by the fol-

Theories

Systems,

- Geometrical

applies

described

of the c h a p t e r s

and I n v a r i a n t s

of Gauge

- Non-Linear

papers

are r o u g h l y

the titles

of Q u a n t i z a t i o n s

- Structure

editorial

DGM conference

w h i c h are also

and F o l i a t i o n s Systems.

only part The editors

of h o m o g e n e i t y proceedings.

strong bias

form of a pure exposition.

agree

in a lecture

Hence

towards

research

of the m a t e r i a l

with the general notes

volume,

it was not p o s s i b l e

physics

or papers

announcement,

Some of the m a n u s c r i p t s

some of the m a t e r i a l

presented

to include

having

de-

a pure r e v i e w

were

is or will be p u b l i s h e d

which

not r e c e i v e d

elsewhere.

or a in

IV

Concerning

the d i s c u s s i o n

of

which

systems

refer

M. R a s e t t i , tions

were

specifically

that

lopment

M.

not

to the

Epstein

such

of the

geometrical

in the

focus

lectures

et al.

and

investigations

of the g e o m e t r i c a l

and

of p r e v i o u s

of G.

Casati,

R. K e r n e r .

will

topological

DGM conferences

G.

There

contribute

background

a n d G.A. are

to the

we

Lassner,

strong future

indicadeve-

approach.

Acknowledgements

We w i s h

to e x p r e s s

persons

for g e n e r o u s

r i n g the

conference

our

gratitude

financial and these

- Der Nieders~chsische

support

Minister

von Humboldt-Stiftung

- Deutscher

Akademischer Stiftung

- Technische Prof.Dr.

fur

and Foreign

to t h a n k

also

matters

of p u b l i c a t i o n .

Last but

not

Physics,

Clausthal,

students

least

of the

and

chairman

We w a n t

and

Clausthal,

the

Mrs.

its O f f i c e

Prof.Dr.

M.

whose

and

assistance

rende-

und Kunst

DAAD

especially

H.

the

DSE Rektor

for Continuing

Education

Quade.

for t h e i r

Ilgauds,

for t h e p r e p a r a t i o n Institute

for o t h e r

Entwicklung,

Springer-Verlag

we t h a n k

organizations

possible

fur W i s s e n s c h a f t

Internationale

Schottlaender, Studies,

and

Austauschdienst,

Universit~t

St.

following

proceedings

- Alexander

- Deutsche

to the

Institute

of t h i s

help made

kind

volume

the

assistance

for T h e o r e t i c a l and the members

organization

smooth

and efficient.

Clausthal, The

in

~anuary

Editors

1985

TABLE

OF C O N T E N T S

V

Preface Table

of C o n t e n t s

I.

The

Work

S.M.

II.

Vll

of

STEVEN

PANEITZ,

Mcmentum

Mappings

M.

PANEITZ

Indecomposable Finite Dimensional R e p r e s e n t a t i o n s of t h e P o i n c a r @ Group and Associated Fields ........... and

Invariants

R. H.

CUSHMAN, KNORRER

The Energy Momentum Mapping of the L a g r a n g e T o p . . . . . . . . . . . . . . . . . .

12

Y.

KOSMANNSCHWARZBACH

On the M o m e n t u m M a p p i n g in Field Theory .........................

25

f

J.M.

III.

MASQUE

Aspects

G.

An Axiomatic Characterization of the P o i n c a r @ - C a r t a n F o r m for S e c o n d O r d e r Variational Problems .................

74

of Q u a n t i z a t i o n s

CASATI

Energy Level Distributions and Chaos in Q u a n t u m M e c h a n i c s . . . . . . . . . . . . . . . . .

86

G. L A S S N E R , G.A. L A S S N E R

Quasi-*-Algebras and General Weyl Quantization .........................

108

A.

G e o m e t r y of D y n a m i c a l S y s t e m s w i t h Time-Dependent Constraints and Time-Dependent Hamiltonians: An Approach towards Quantization .....

122

R e g u l a r i t y A s p e c t s of the Q u a n t i z e d P e r t u r b a t i v e S - M a t r i x in 4 - D i m e n s i o n a l Space-Time ...........................

136

LICHNEROWICZ

I.E.

SEGAL

VI

IV.

Structure

of G a u g e

Curvature Forms with Singularities and Non-Integral Characteristic Classes ..............................

152

Y a n g - M i l l s A s p e c t s of P o i n c a r 6 Gauge Theories .......................

169

Supermanifolds and Berezin's New Integral .........................

189

S. R A N D J B A R DAEMI

Spontaneous Compactification and Fermion Chirality ....................

199

A.

O f f - S h e l l E x t e n d e d S u p e r g r a v i t y in Extended Superspace ..................

214

A. A S A D A

J.D.

Y.

V.

HENNIG

NE'EMAN

ROGERS

Non-Linear

P.F.

A.M.

N.

Systems,

DHOOGHE

DIN

A.M. A.H.

VI.

Theories

NAVEIRA, ROCAMORA

s~NC~EZ

Geometrical

Integrability

and Foliations

C o m p l e t e l y I n t e g r a b l e S y s t e m s of K d V T y p e r e l a t e d to I s o s p e c t r a l P e r i o d i c Regular Difference Operators .........

236

N o n - L i n e a r T e c h n i q u e s in T w o D i m e n s i o n a l G r a s s m a n n i a n S i g m a M o d e l s .....

253

A G e o m e t r i c a l O b s t r u c t i o n to the E x i s t e n c e of t w o T o t a l l y U m b i l i c a l C o m p l e m e n t a r y F o l i a t i o n s in Compact Manifolds ....................

263

Einstein Equations without Killing Vectors, Non-Linear Sigma Models and Self-Dual Yang-Mills Theory ..........

280

Modelling

of

Special

Systems

M. E P S T E I N , M. ELZANOWSKI, J. S N I A T Y C K I

L o c a l i t y a n d U n i f o r m i t y in Global Elasticity ....................

300

R.

D i f f e r e n t i a l G e o m e t r i c a l A p p r o a c h to t h e T h e o r y of A m o r p h o u s S o l i d s .......

311

T h e I s i n g M o d e l on F i n i t e l y G e n e r a t e d G r o u p s a n d the B r a i d G r o u p . . . . . . . . . . .

328

M. G.

KERNER

RASETTI, D'ARIANO

I.

Steven

M.

Paneitz

The W o r k

presented

his

Asymptotics

of

Connection"

in the a f t e r n o o n

Immediately

after

together

seconds

night.

The

he

next

departed

While

to h i s

Steven

Paneitz

and mathematical

for h i s

Clausthal ces

of the

We w i l l

and

Equations

well

for a b a t h he

found

friend.

was an outstanding

family,

scientists.

contribution

September

and very

swimming

Divers

on

PANEITZ

his body of the

It w a s

1983.

in a s m a l l got

in the

lecture lake

he w e n t

near

the

into difficulties, lake

conference

agreed

"Sharp

a n d the C o n f o r m a l

I,

received

suddenly

on

during

paid

the

respect

to d e d i c a t e

to

this

memory.

tician

of

session

the p a r t i c i p a n t s

colleague

M.

to the Y a n g - M i l l s

excellent

sank.

day

Steven

conference

participants

building.

within

volume

his

with other

conference

our

Solutions

of

his

friends

He v i s i t e d

several series

remember

times

and most

physicist.

His

Institute

and participated

on D i f f e r e n t i a l with

but

also

loss

actively

not

for the

for T h e o r e t i c a l

Geometric honour

mathema-

is a h e a v y

and collaborators

the

him always

talented

death

community

Physics

in the

last

only

in

conferen-

Methods.

as a s t r o n g y o u n g

mathematician

a n d as a f r i e n d .

H.D.

Doebner,

J.D.

Hennig

PANEITZ

was a man of e x c e p t i o n a l

A l t h o u g h he chose genera]

to work m a i n l y

field of F u n c t i o n a l A n a l y s i s

have a c o n s i d e r a b l e central

mathematical

theoretical physical

consequences description relations

of causality,

range.

and stability.

that had

the

The very b r i e f

a r o u n d these themes,

field and p a r t i c l e

spaces

in the

his p u b l i c a t i o n s

the m a t h e m a t i c a l

and their

theory.

He r e s o l v e d and d e v e l o p e d q u e s t i o n s

groups and h o m o g e n e o u s

unusual breadth. direction

They are all c o n n e c t e d w i t h

p r o b l e m of d e v e l o p i n g

given here will be o r g a n i z e d

Causality:

coherent

and A p p l i c a t i o n s ,

symmetry,

to m a t h e m a t i c a l

originally

f l e x i b i l i t y and

in a r a t h e r

about c a u s a l i t y

in

i n t e r e s t e d a group at M.I.T.,

in c o n n e c t i o n w i t h t h e o r e t i c a l p h y s i c a l

issues.

Typically,

P a n e i t z b o t h p l u m b e d the depths of the o r i g i n a l

issues c o n n e c t e d with

4-dimensional

in n o n - u n i q u e n e s s

space-times,

causal

structures

of the

stability

spin~

in the

treated

He a d a p t e d

the case of n o n l i n e a r stract level

group, cone

the n a t u r a l

of

SU(2,2) , and the d e p e n d e n c e

for wave e q u a t i o n s generalization

This work

given curved,

on their

to a r b i t r a r y

s t a b i l i t y t h e o r y of the K r e i n

i n v a r i a n t wave e q u a t i o n s ,

served also to r e s o l v e theory,

non-static,

vacuum,

in a d d i t i o n

the

and w i t h p e n e t r a t i n g a p p l i c a t i o n

linear q u a n t i z a t i o n

canonical

energy)

surprises

Lie groups.

Stability:

cases.

local causal

(or p o s i t i v e

and e x h a u s t i v e l y

semisimple

w i t h major

as e.g.

or e q u i v a l e n t l y ,

a g a i n b o t h at an ab-

to i n t e r e s t i n g p a r t i c u l a r

a major outstanding problem

the case of wave e q u a t i o n s

Lorentzian

to the q u a n t i z e d

school to

manifold,-

the d e t e r m i n a t i o n of a

c r e a t i o n and a n n i h i l a t i o n

field o p e r a t o r s

in

on a

operators,

that had e a r l i e r b e e n

established. Symmetry: homogeneous particle poral

Intensive

s y s t e m a t i c w o r k on the h a r m o n i c a n a l y s i s of

vector bundles

theory.

Not m e r e l y a m a t t e r of group

l a b e l l i n g of v e c t o r s

the f o r m a t i o n of local PANEITZ'

w o r k here

brilliant

over s p a c e - t i m e s a p p l i e s b o t h to field and

display,

and his w o r k may

forward,

but

i m p l i c a t i o n s not o t h e r w i s e

lutions

are

the f i n i t e n e s s (Lorentzian!)

and the s e l f - a d j o i n t n e s s sion of the

S-matrix

spatio-tem-

is c r u c i a l

for

of the

and may well c o n t i n u e attainable.

for g e n e r a l

on M i n k o w s k i

invariant quantized

re-

for

fields

so-

space;

l e a d i n g t e r m in the p e r t u r b a t i v e

for c o n f o r m a l l y

interaction representation.

equations

straight-

to have,

A m o n g these,

integrated action

Yang-Mills

of the

he e s c h e w e d any k i n d of

in part a p p e a r d e c e p t i v e l y

it u l t i m a t e l y has had,

of the

the

spaces

i n t e r a c t i o n s and other p h y s i c a l p u r p o s e s .

v e r g e d on the m o n u m e n t a l ;

markable example,

theory,

in the r e p r e s e n t a t i o n

expanin the

Overall, mathematician more than event,

P A N E I T Z was the m o s t

impressive

I have known.

joint w o r k he u s u a l l y c o n t r i b u t e d

I did,

and e s p e c i a l l y

his a c c i d e n t a l

significant

In our

to p r e c i s i o n

and p r o d u c t i v e y o u n g

and c o m p l e t e n e s s .

d e a t h at the age of 28 was a t e r r i b l e

loss to the m a t h e m a t i c a l

and r e a l l y

community.

I.E.

In any

Segal

PUBLICATIONS

OF STEPHEN

M. PANEITZ

I.

U n i t a r i z a t i o n of s y m p l e c t i c s e q u a t i o n s in H i l b e r t space.

and s t a b i l i t y for causal d i f f e r e n t i a l J. Funct. Anal. 41 (1981), 315-326.

2.

Invariant c o n v e x cones and c a u s a l i t y in s e m i s i m p l e and groups. J. Func. Anal. 43 (1981), 313-359.

3.

Q u a n t i z a t i o n of wave e q u a t i o n s and h e r m i t i a n structures in p a r t i a l d i f f e r e n t i a l varieties. Proc. Natl. Acad. Sci. USA 77 (1980), 6943-6947. (With I.E. Segal.)

4.

E s s e n t i a l u n i t a r i z a t i o n of s y m p l e c t i c s and a p p l i c a t i o n s q u a n t i z a t i o n . J. Func. Anal. 48 (1982), 310-359.

5.

C o v a r i a n t c h r o n o g e o m e t r y and extreme distances: E l e m e n t a r y particles. Proc. Natl. Acad. Sci. 78 (1981). (With I.E. Segal, H.P. Jakobsen, B. Crsted, and B. Speh.)

6.

A n a l y s i s in space-time bundles. I. General c o n s i d e r a t i o n s and the scalar bundle. J. Func. Anal. 47 (1982), 78-142. (With I.E. Segal.)

7.

Analysis J. Func.

8.

S e l f - a d j o i n t n e s s of the Fourier e x p a n s i o n field L a g r a n g i a n s . Proc. Natl. Acad. Sci. 4595-4598. (With I.E. Segal.)

9.

The Y a n g - M i l l s e q u a t i o n s on the u n i v e r s a l cosmos. J. Func. Anal. 5_33 (1983), 112-150. (With Y. C h o q u e t - B r u h a t and I.E. Segal.)

10.

D e t e r m i n a t i o n of a p o l a r i z a t i o n by n o n l i n e a r scattering, and e x a m p l e s of the r e s u l t i n g q u a n t i z a t i o n . Lec. N o t e s in Math. No. 1037, Ed. S.I. A n d e r s s o n and H.D. Doebner (Proceedings, Clausthal, 1981), S p r i n g e r - V e r l a g , Berlin, 1983.

11.

Determination A r k i v f. mat.

12.

All linear r e p r e s e n t a t i o n s of the P o i n c a r 6 group up to dimension 8. Ann. Inst. H. P o i n c a r 6 (Phys. Theor.) 40 (1984), 35-57.

13.

P a r a m e t r i z a t i o n of causal and global h y p e r b o l i c i t y .

14.

Analysis J. Func.

15.

Global solutions of the h y p e r b o l i c Y a n g - M i l l s e q u a t i o n s and their sharp asymptotics. P r o c e e d i n g s of the Amer. Math. Soc. Summer Institute on N o n l i n e a r F u n c t i o n a l A n a l y s i s and A p p l i c a t i o n s (Berkeley, 1983), in press.

16.

I n d e c o m p o s a b l e finite d i m e n s i o n a l r e p r e s e n t a t i o n s of the P o i n c a r & group and a s s o c i a t e d fields. These p r o c e e d i n g s (Clausthal, 1983).

in space-time bundles, II. Anal. 49 (1982), 335-414.

of invariant convex 21 (1983), 217-228.

The

cones

spinor

Lie algebras

and form bundles.

of q u a n t i z e d i n t e r a c t i o n USA 80 (1983),

in simple

Lie algebras.

actions of u n i v e r s a l c o v e r i n g J. Func. Anal., in press.

in space-time bundles. III. Anal. 54 (1983), 18-112.

to field

Higher

groups

spin bundles.

17.

I n d e c o m p o s a b l e r e p r e s e n t a t i o n s of the P o i n c a r 6 group and a s s o c i a t e d fields. Proc. XII. I n t e r n a t i o n a l Coll. Group T h e o r e t i c a l M e t h o d s in Physics, Trieste, 1983 (Posth. p r e s e n t a tion), Lecture Notes in Physics, Vol. 201 (1984), 84-87.

INDECOMPOSABLE FINITE DIMENSIONAL REPRESENTATIONS OF THE POINCARE GROUP AND ASSOCIATED FIELDS

Stephen M. Paneitz Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 USA

Introduction The idea that 'true' space-time may deviate in the large from Minkowski space without sacrificing group-eovariance has led to an empirically accurate, parameterfree,

and theoretically

the redshift fundamental far

less

facet and

of

[1,2].

program

natural

the standard

R,

G

that

model

G

phenomena such as

group-covariance and

is relevant

the

that are thus

bein Z explored

[4,5].

of space-time

~ = RIxsu(2).

determined

at that point.

meaning

fermions,

which differs

from

deforms into them as an invariant

'radius

of the universe',

tends to causal

The group representation is assumed to be

that the transformation rules for the fields

by the transformation

space)

One

here concerns an apparently more rigid

yet mathematically as

is essential)

are currently

fields are presumed to transform under the 15-dimensional

in a finite-dimensional C

of cosmological

[6] for the fundamental

interpretable

an induced representation, are completely

(where

clear-cut

spinor fields

unit

Fundamental group

physics

quantitatively this

description

However, the idea also has implications noted long ago [3] for

particle

equally

distance

satisfying

at a point

of field values

p ~ M

Now the isotropy group of

(assumed to lie

under the isotropy group

0 ×-I

G P coincides with the Poin-

care group extended

by scale transformations

acting on Minkowski

space M0, such that M 0 is regarded as embedded in M by e.g.

(an ll-dimensional group denoted P)

conformal compactification and covering transformations (cf.[4, Part I]). Thus

~-covariant

fields

=(R IxSL(2,C)) ~H(2) Conventionally translations

R

is

are

questioned.

representations by spins

The

representations

R

of

H(2)-subgroup,

trivial

on

the

point is the

position that also notes

next

section

of P restricting

(s+,s_),

inducing

assumed

the action of accessible translations be

by

H(2) = 2x2 hermitian matrices).

when the inducing

a more conservative

determined

(~ = semi-direct product,

s+,s_

representing

'point at infinity'

Ox-I.

the

Yet from

the large distance scale rendering

relatively unobservable, this assumption may

shows that the mathematical

possibilities

to given representations of SL(2,C)

half-integral)

is highly restricted

there is a unique representation of SL(2,C)~H(2)

[7].

for

(determined For example,

(up to contragredience) restric-

ting to the direct sum of half-spin representations (½,0) and (0,½); fields induced

from this incompletely

reducible

on C 4 with a suitable

representation

conformal

dimension or weight have been dubbed spannor fields (a 'wrenched' spinor, as via a spanner

[6]).

A means

of

determining

the

'special'

conformal

weights

(defined

below) is sketched in the third section.

Determination of Indecomposable Representations of According

to the first result (cf.[7,8]),

tion of the Poincare group Theorem n,

i.

Let

where

n_

g

tation

be any

real

is the maximal

finite-dimensional

solvable

to n_.

"" "+ Vn

ideal

representa-

form.

Lie algebra such that

of f~.

Let

h

[f~,n]

be any semisimple

Then, given any finite-dimensional

@ of f~ in a complex vector space V = V1 +

any finite-dimensional

~O may be put in block-upper-triangular

subalgebra of #i complementary

tion

~O

represen-

V, there exists a direct sum decomposi-

such that the Vj are invariant and irreducible under @(h_~

and such that p(n) V. J Thus a finite-dimensional mined H(2)

C --

Z k<j

Vk

for

representation

p of

j = 1 .....

n.

P--O-"S L ( 2 , C ) ~ H ( 2 )

is deter-

by a finite sequence of spins into

{(s+,s_)]n .and maps (for l -- 0 Figure

relations

Pa )

~2 775

is the s e m i a l g e b r a i c

a,b

(which d e f i n e

the p r o -

18

reduced

Hamiltonian

Ha, b E

I ([FI ' ~ 2 ' ~ 3 ) - - > ~ I

: P a , b - - > ER : [ =

is a r e g u l a r

Ha,b(T)

is

= E

value

of

Ha, b

is not t a n g e n t

if and o n l y

to

P a , b ~ that is,

~I

to the fold c u r v e value

E,

fore for a r e g u l a r

value

(E,a,b)

is d i f f e o m o r p h i c ~

-I (E,a,b)

3.

RELATIVE

In this of

Ha, b.

to a t h r e e

posite

s e c t i o n we find the c r i t i c a l

b ; which

points

of H

Ha, b

points

is s h a p e d

and critical

correspond

of the L a g r a n g e

top.

F r o m the c r i t i c a l

values

of the a n g u l a r

like a b o w l

values

to the r e l a t i v e of

~2T[ is

momentum

and a t h r e a d w h i c h

values joins op-

sides of the bowl.

lying

points";

two types

of c r i t i c a l

in the s m o o t h p a r t of

and the s i n g u l a r p o i n t s

gular critical

points"

of

Pa,b

critical

{ ~3

= 0 ] ~ Pa,b"

point

of

Ha, b

of the v a r i e t y

The p o i n t if a n d o n l y

case.

is t a n g e n t

at

is t a n g e n t

of

Ha, b

on

"nonsingular Pa,b

Pa,b" critical

we a l s o call

: 2I

~ 0 = T/0(a,b)

All c r i t i c a l 0 0,0 ) = (~I'~2

(U0,0)

if an a f f i n e

I

W

points

we call

"sin-

Ha, b.

F i r s t we t r e a t the n o n s i n g u l a r lie in

that

There-

~22[-1(E,a,b)

[5] . In p a r t i c u l a r ,

we s h o w that the set of c r i t i c a l

We d i s t i n g u i s h Those

compare

S I.

to

the f i b e r

I

EQUILIBRIA

H

and

~29[ ,

T 3,

a,b the u n i o n of the g r a p h of a f u n c t i o n a

from figure

is d i f f e o m o r p h i c

of

torus

easily

is c o n n e c t e d .

The c r i t i c a l

of

b2

It f o l l o w s

H -I a,b(E)

equilibria of the H a m i l t o n i a n values

line

= E~= E - 2i 3 4.

that for a r e g u l a r

plane

the a f f i n e

I

+

I b2 [[I + ~ I

+ ~

if the a f f i n e

I

2-~I ~ 2 is not t a n g e n t

~2

points

of

Ha, b

is a n o n s i n g u l a r

line f r o m the f a m i l y

= Et I

~2 + Z

~I

to the s m o o t h p a r t of the f o l d c u r v e

to the g r a p h of the f u n c t i o n 2 (a-b [E I ) ~2 = ha,b(~) = I- Ttlz

where

I~1}
0

is a critical point of

: 0 ,

(compare Ha, b

Iff ll ~ l [2], p. 155, eqn. 55).

if and only if

20

U 01 6 [-I ,I] a zero

of

Clearly that

is a d o u b l e the

~1:1 ~i=I

if a n d

of f

is a d o u b l e is n o t

only

zero

discriminant

if

g

0 =

~g

zero

a double has

f.

In o t h e r

of

f.

of

zero

a double

words

First

f

if a n d

of

f,

zero,

: a 4 + 2(a-3B)a 2 +

only

then

that

~ 01 6 [-1,1]

suppose

f

a=b.

is

Then

if

~=B.

has

a double

Suppose zero

is

(B+cO,

real we

6

roots

obtain

has

if

two

~-B

implies

is an i s o l a t e d

=

of

a

4

+2(e+36)

~+3~>0, IB=const]

-- ~+

~

f

if a n d o n l y

if a n d o n l y

that

point

the mapping

z e r o of

zero

~3

a2

a2

(I+Tfl)g(Tfl)

if

a=-B.

If

~#-B

if

+

2

(u-k)

6>0

if

6(a)

~#~.

{3 { £i f : 0 ]

to fig.

=

3(a)

.

Hence

(a,~)

(see Fig. gives

fig.

=

3(a)).

(0,6) Applying

3(b).

a = -b

Fig. 3(a): a=-b, ~:const.

Fig. 3(b) : a=-b cross section of

slice of discriminant

image of energy momentum mapping.

locus

~o where So = (0,E)

/kf=0 0 (0,~) is

an isolated point .

an isolated point °

So we

see:

The

critical

Lagrange-top

This lysis

sides.

The

values

which

is o n l y shows

a very

of

set of the

energy-momentum

looks

a bowl

image

like

of

~ ~

geometric

that

singular

the

a thread

set of a l l

description,

for Fig.

mapping

~

of the

joining

energy

opposite

momentum

this bowl.

section

points

with

is the

lie on or a b o v e

crude

for e x a m p l e

a = ~ b consists

b 2 is -Z + ~13

a more

detailed

of the b o w l w i t h

ana-

the D l a n e s

the bowl. 4: C r i t i c a l

set of e n e r g y

mentum

mapping

is a b o w l w i t h

thread

joining

opposite

sides.

moa

22

4.

MONODROMY

In this

section

of the L a g r a n g e

top h%s

Monodromy

in the e n e r g y

set

of r e g u l a r

bundle

with

points thread

T 3.

We

which

lie a b o v e

the

which

joins

to an

S I, /~ class

T3

itself

into

vial

bundle,

both

of

~

that

is,

~

has H

monodromy

We n o w

determine

that

- ~

to a

T3

Following

the

~

to

Hence

class If

SIxT 3,

then

is n o n t r i v i a l ,

spherical

energy

momentum

topology

of the

~

level

the

iso-

argument

of

is a triwe

say t h a t

t h e n we used

pendulum

mapping

a

of a m a p

of ~ .

an a n a l o g o u s

in the

excluding

S I.

homotopy

~

set of

is c o n t r a c t i b l e

over

monodromy

the

is a f i b e r

function ~

fiber

consider

(~)

is the o p e n

like

is d i f f e o m o r p h i c

the

= ~-I

bundle

if

of n o n t r i v i a l

Since

by the

geometric

critical

symmetry left

shows larger

value

is the point

on

E

say

for p r o -

(see

[6] or

of the L a g r a n g e

sets

top

of the H a m i l t o n i a n

bundle

definite.

9 C

V,

Since

of

S0(3)

argument In the

of c r i t i c a l

,

the

fiber

to

the fiber of 9

is

to S I x S 2 x T 2.

Since

T 3. Suppose that

~

is

is i n v a r i a n t

under

of the M o r s e

lemma

E

greater

than

group,

slightly the m a x i -

to the u n i t

H=K+V TISO(3)

a certain

~29Zsuch

~

H

Physically

its b o d y

for all

since

and

K

is

is a t r i v i a l

Let

let

•

that ~

bundle

coming

be a c y l i n d e r the

center

= ~22Z-I (C).

from

with

line

of 4~

Over

every

T 2, while over every point in C - ~ ~ ,

is a trivial fibration, that is, ~

(Note

H.

$2xS0(3) .

[~Z

values.

with

version

is,

of

corresponding

is d i f f e o m o r p h i c

that

of

of

of

diffeomorphic

action.

SO(3) ,

shows

~

SI

slxs 4

is a Lie

image

in the b o u n d a r y

left

that

H-~ (E)

TISO(3)

sets

H

is at r e s t

to

large,

level

of

downward.

is d i f f e o m o r p h i c

following

~C

top

of the

value

an e q u i v a r i a n t

very

of the p o t e n t i a l

thread

of the

the

vertically

For

is n o n t r i v i a l .

boundary

orbit when

applying

type

minimum

is d i f f e o m o r p h i c

.

and hence

The E~

-~

sphere

positive bundle

action, H-I (E)

than

tangent

occurs

pointing

SI

that

topological

critical

orbit

axis

the

is an a b s o l u t e

to a n o n d e g e n e r a t e this

mum

mapping

changes.

Observe

the

of a b o w l

of the bowl.

of m o n o d r o m y

because

momentum

precisely,

~

in 3. that

otherwise,

has m o n o d r o m y .

s h o w that

energy

More

Then

graph

edges

~

that

vina the e x i s t e n c e

mapping. ~ .

showed

the

has no m o n o d r o m y ;

the

the a p p e a r a n c e

is d e t e r m i n e d

called

~

[7]) we w i l l

of

is i s o m o r p h i c

morphism

that

with

momentum

values

fiber

show

monodromy.

is c o n c e r n e d

bundles ~

we w i l l

that

is

23

over every point

p

of a simple curve

in each component

of

~

SIxT 2,

if

p

pt. 1 xT 2,

if

p

with fiber

(see. fig. and

~- =

[8],

S2xT2.]

of

y

Now split ~

~

which has an endpoint

is diffeomorphic

or is diffeomorphic ¥.

~ 2 Z - I ( y ) = S 2 x T 2.

Since

(~-)

on ~ ,

~Z-I(¥)

is an endpoint of

5(a)). ~-I

¥

the fiber

is not an endpoint

bundle c o n s t r u c t i o n S]

,

to

to

Therefore

by Smales reduced

Hence

is a bundle over

~

into two closed half cylinders

is trivial,

are diffeomorphic

the fibrations

~

+=

~ ~

~T(-I(~)

fibrations. +

-- ~

E:E +

8~

t

g ~ ~

E=E Fig.

Fi~.

5(a):

cylinder

5(b) : Heavy curve

the p r o j e c t i o n

Splitting of ~ around thread

E=E +-. Shaded region

% .

jection of

Avoiding disc

the thread we can move

~2

where

in the E=const.

E

is close - X

= EZ

and

~ ~

is pro~-

~ -+ •

by an isotopy onto the closed two

slice of the image of ~)~Z(see fig.

E£

is

of 8t +- on

is very large.

Since the

E±

5(a)),

slice

is the disjoint union of two half open discs D~, D~ and the close4two -2 -I ± + I+ z + + discs Dm ,H ( E ) is the disjoint union of ~?, ~ and ~ where ~$= = ~D~-I(D~).

Now ~

is isotopic

ettaching maps of ~ - by hypothesis, this is false, homology

since H-I(E+),

Let contained

geometric

~ + is homeomorphic

equivalent

is topologically

f~ be a noncontr~ctible

circle on ~ - ~ C

.

bundle

where

S2xT 2

T3=SIxT 2

fiber of ~

of ~

with

over ~ , SI

monodromy mapping of

we may extend it monodromy

different

S xS 4. Therefore

is a trivial in the

to

to H-I (E-). But

fibration.

Since the geometric SIxT 2,

Since

to 8x-. Also the

which istopologically S2xSO(3),has

groups than H-I(E -) which

~-I(c) S 2.

are homotopic.

H-] (E +) is h o m o t o p i c a l l y

is not a trivial

on

to

(and hence homeomorphic)

Suppose that SIxT 2

is

being the equator

of

~-1(p)

to the identity map on

is the identity

S2xT 2.

is the identity which contradicts

Thus the the nontri-

24

viality of ~

.

Therefore the bundle

~2T~-I (P)

is not trivial,

and

we have p r o v e d the

Theorem:

The energy m o m e n t u m m a p p i n g of the Lagrange top has monodromy.

A more careful a r g u m e n t ( b a s e d on the o b s e r v a t i o n that the reduced system on P0

coincides with the H a m i l t o n i a n system d e s c r i b i n g the spherical

pendulum) w h i c h we do not give, mapping

of E)2~ -I (p)

is

shows that the geometric m o n o d r o m y

II ~ 0I 0 o ) 001

REFERENCES

[i]

Arnold, V., M a t h e m a t i c a l methods of classical mechanics, Springer-Verlag, New York, 1978

[2]

Goldstein, H., Classical mechanics, Reading, Mass. 1959

[3]

Ratiu, T. and van Moerbeke, P., The L a g r a n g e rigid body motion, Ann. Inst. Fourier, Grenoble 32 (1982), 211-234

[4]

Cushman, R., Normal form for H a m i l t o n i a n vector fields with p e r i o d i c flow, p r e p r i n t II 255, R i j k s u n i v e r s i t e i t Utrecht, 1982

[5]

Jacob, A., Invariant m a n i f o l d s in the motion of a rigid body about a point, Rev. Roum. Math. Pure et. AppI. 16 (1971), 1497-1521

[6]

Duistermaat, J.J., On global action angle coordinates, Appl.Math. 33 (1980), 687-706

[7]

Cushman, R., Geometry of the energy m o m e n t u m mapping of the spherical pendulum, C e n t r u m voor W i s k u n d e en Informatica N e w s l e t t e r I (1983), 4-18

[8]

Smale, S., T o p o l o g y and M e c h a n i c s 305-331

[9]

Holmes, Ph., Marsden, J., H o r s e s h o e s and A r n o l d d i f f u s i o n for H a m i l t o n i a n systems on Lie groups, Indiana University, Math. Journ. 32 (1983), 273-309.

Ist ed., A d d i s o n Wesley,

I, Inv. Math.

Comm. Pure

10 (1970),

ON THE MOMENTUM MAPPING IN FIELD THEORY

Yvette Kosmann-Schwarzbach

U.E.R. de Math~matiques Universit~ de Lille I 59655 Villeneuve d'Ascq, France

I.

INTRODUCTION

Noether's theorem and its several generalizations state the relationship - let us call it the Noether mapping - between the infinitesimal symmetries of a Lagrangian system of partial differential equations and some of its conservation laws. For Hamiltonian systems, roughly speaking, it is the momentum mapping that plays the role of the Noether map. Momentum mappings for Hamiltonian systems with symmetries, on finite-dimensional symplectic manifolds, were introduced by Souriau [38] and their properties have been expounded in many articles and treatises such as [1] and [28], [29]. Momentum mappings for Hamiltonian systems on infinite-dimensional symplectic manifolds have been extensively studied by Marsden and others, see [61 and [30], while some versions of the infinite-dimensional Hamiltonian Noether theorem appear in the mathematical physics literature

[21], [12] and [42].

The classical notions of Lagrangian and Hamiltonian systems have several generalizations to the infinite-dimensional

setting. We explain the situation on an

example, that of the nonlinear Klein-Gordon equation on Minkowski space (§2), and we return to this example in 4.10 and 6.11 in order to illustrate the main theorems. In this paper the various "fields" are considered as sections of various fibered manifolds with finite-dimensional base and fiber. In all applications these fibered manifolds will be vector bundles, usually trivial ones such as ~4×~d (this is the typical case for the theory of §4) or ~3x~d×(~d)~

(as in the typical applications

of §6). After some background material introduced in §3, we define and study the infinite-dimensional Lagrangian systems over a fibered manifold in §4. This is what

26

can be called a relativistic

situation since all the independent

well as space, play analogous

roles. The symmetries

sense, see def. 4.4) give rise to conservation Euler-Lagrange

equations.

laws (conserved

currents)

form, one obtains associated

systems which are not of evolution (see (4.|5)).

Hamiltonian

systems and one obtains conservation

One can reformulate

in the relativistic

Lagrangian

Noether's

can be considered

type of these "evolution" Gardner,

obviously

The theory of evolution Hamiltonian [23] (also see

Hamiltonian

is the Korteweg-de

the geometric

systems on fibered manifolds

[27] ,144] , [3~ and

constructions

identical

to

formalism. equations

systems.

The proto-

Vries equation which

and Zaharov and Faddeev wrote as an evolution Hamiltonian

Kupershmidt

Dorfman

systems

is one-

theorem for the relativistic

laws that are

as infinite-dimensional

Hamiltonian

a

relativistic

We then explain in §5 how certain systems of partial differential of evolution

for the

form unless the base manifold

dimensional

those obtained

time as

(in a generalized

This is the Lagrangian Noether theorem 4.7. Introducing

generalized Hamilton-Poincar~-Cartan Hamiltonian

variables,

of the Lagrangian

system in 1971.

is the work of

~19]), while the algebraic version of

is to be found in a series of papers by Gel'fand and

[9], [101 for the case of evolution equations with one space variable.

Kupershmidt

introduced

which constitutes

the notion of fibered manifolds with Hamiltonian

a generalization

rather than of the symplectic,

to the infinite-dimensional

manifolds.

structure,

case of the Poisson,

It seems that this "contravariant"

approach

is indeed better suited to the applications. We shall show how, on a fibered manifold with Hamiltonian define Hamiltonian Hamiltonian

Lie algebra actions and infinitesimal

systems.

generalizations

to the infinite-dimensional

case of the

case.

The infinite-dimensional a Hamiltonian

Hamiltonian

Noether theorem 5.6 states that whenever

Lie algebra action leaves a Hamiltonian

is a conserved

functional

One particular

case is of special

invariant,

for the associated Hamiltonian interest because

The cotangent bundle

V E

canonical Hamiltonian

structure which we describe

of any fibered manifold

(i.e.,

If the fibered manifold

a Lagrangian Hamiltonian

is given on

3-dimensional E

E

is a fibered manifold with a

in §6. We derive explicit

Euclidean

VE

transformation

of a relativistic

Lagrangian)

transformation"

expressions

They involve inte-

space in the case of field

is a Riemannian bundle,

E, then the vertical bundle

gent bundle by an "evolution Legendre

system of evolution.

it is the setting of field theory.

of

structure obtained from the canonical Hamiltonian

from the Legendre

its momentum mapping

dynamical

for the momentum mapping for lifted actions on cotangent bundles. grals over the base space theory).

one can

of evolution

This leads to a momentum density mapping and a momentum mapping

which appear as natural classical

structure,

symmetries

or, more generally, E

possesses

if

a

structure of the cotan-

(which is, in general,

associated with the Hamilton-Poincar~-Cartan

distinct form

; the momentum mapping on the tangent bundle yields

27

conserved quantities obviously identical to those obtained in the evolution Hamiltonian formalism. The two main versions of Noether's theorem,

the one for relativistic

Lagrangian systems and the other for evolution Hamiltonian systems, are essentially different. Yet they are not unrelated. When the time and space coordinates are distinguished,

a time-independent relativistic Lagrangian gives rise to an evolution

Hamiltonian on a cotangent bundle, and the time components of the conservation laws obtained from the relativistic Lagrangian Noether theorem become the momentum densities of the corresponding Hamiltonian equation of evolution.

Seen in this perspec-

tive, the case of classical mechanics appears as a degenerate case in which several notions that are distinct in the general case -- relativistic and evolution Lagrangian systems, relativistic and evolution Hamiltonian systems, and three kinds of Legendre transformations

that relate them -- all become identical. We sketch this discussion in

§7, where we also study the Legendre

AN EXAMPLE

2.

transformation of the Korteweg-de Vries equation.

: THE NONLINEAR KLEIN-GORDON EQUATION

The Klein-Gordon equation can be seen in four different ways

: as a "relativis-

tic Lagrangian system", a "relativistic Hamiltonian system", an "evolution Hamiltonian system" and an "evolution Lagrangian system". These will be the prototypes for the generalizations Let = (nab)

in §§ 4 and 6.

x = (xO,xl,x2,x 3) = (x a) be the metric tensor of

denote a point in Minkowski space M

of signature

(i,-i,-I,-i),

The (nonlinear) Klein-Gordon equation for scalar fields Du = 0

u

M, and let

(a,b = O,1,2,3).

with mass

m

is

with ~2 u

(2.1)

Du = ~

ab

2 + m u + f(u) ~xasx b

where

f

is a smooth function of one variable and where

smooth function on

M.

(ab)

is the inverse matrix of

quently we shall use Einstein's summation convention. Let of

u

denotes the unknown

(nab) g

and here and subsebe an antiderivative

f.

2.1. Relativistic Lagrangian equations. Because time and space coordinates play the same role we call these equations relativistic. by (2.1) is a second-order differential vector bundle

F = Mx~ ÷ M.

(or "variational derivative")

D

The differential operator

of the first-order Lagrangian I

defined

operator on the sections of the trivial

is the opposite of the Euler-Lagrange

EL

D

ab ~ u

~u

L(u) = ~ ~

1

2 2

2 m u ~x a ~x b

- g(u).

L

on

differential F defined by

28

aL ~L EL(u) = ~ u - Da ~ u

In fact,

where

~u

Ua =

a respect

a

to

Therefore

x ,

so

-EL(u)

a necessary

2

the nonlinear

2.2. Relativistic

condition

Da

is the total derivative

for the

with

u

- = Du. ?xa?x b C integral IL(u)(x)dx

t o be e x t r e m a l

Hamiltonian

equations.

( h e r eJ

o v e r a s p a c e t i m e domain

dx = dx 0 A dx 1 A dx 2 A dx 3)

is

equation

Set

a

_ ~L _ qab ~u ~u a ~x b

is equivalent

and

~ = r~ 0 , I, 2 , ~ .

to the generalized H a m i l t o n i a n

equations

{

au

~H

~x a

s a

~x

for the Hamiltonian

by t h e i r

a

2

K l e i n - G o r d o n e q u a t i o n be s a t i s f i e d .

The nonlinear Klein-Gordon first-order

and

ab

= m u + f(u) + q

with fixed boundary conditions that

a

$x

values,

H(u,~)

1

au a

= ~

the equations

a

qab ~ ~

b

1

2

+ ~ m u

2

+ g(u). Replacing

aHa

become

?~

au ~x a

aH

3u

b nab

~a --

and

=

-

m

2

u

-

f(u),

Sx a whence ab ~2u

2 m u - f(u),

q ~xa~x b

i.e.j

the Klein-Gordon

equation.

We note that in this generalized

Hamiltonian

formalism

the number of momenta

a is equal nents of

to the number

of independent

u). In an invariant

Cartan form will replace

apart. We let

equations.

i,j = 1,2,3. Let

H(¢,~)

iHdx I A dx 2 A dx 3 ~ 3 x R x R ~ + R 3,

formulation

the Hamiltonian

2.3. Evolution H a m i l t o n i a n

~

(and not to the number of compo-

on fibered manifolds,

Here the time coordinate

and

the H a m i l t o n - P o i n c a r ~ -

function.

~

be scalar

functions

x

0

of

= t ]

plays a role 2

x ,x ,x

3

and let

] 2 1 "" ~ 3~> 1 = ~ v - ~ n lj ~ + ~ m2~ 2 + g(¢) = K(~) + 0(4). ~x 1 ~x ]

is a first-order

g.e.~

variables

the "cotangent

functional bundle"

on the sections

~ = (~,~)

of the trivial vector bundle

of ~3x~ __+ ~3

29

(see §6). This cotangent bundle possesses a Hamiltonian structure in §6. By definition, the Hamiltonian vector field tional

H

~

~

to be defined

associated with the func-

is

6H XH=~%7

~H ~H) = (~H =#(~' ~ ~-7'

The evolution Hamiltonian equation on

~3x~R~

6H ~7 ) '

÷ ~3,

can be written Be ~t

~H ~

3~ 3t

~H ~¢ '

and these first-order equations are again equivalent to the nonlinear Klein-Gordon equation. In fact, since 6H 3H 3H 6-~ = 3--~- Di - - = 3¢i

"" 32~ . m2¢ + f(~) + nl] 3xZZx ]

'

we obtain

~x °

=-r7

•

m2¢ _ f(¢)

•

9x13x ] 32¢ which imply

_

.. 32~

_

+

lj

.

• +

m2~

+

f(~)

=

0

which is the Klein-Gordon equation.

3x13x j

~x°~x °

2.4. Evolution Lagrangian equations. Here, as in 2.3, i,j = 1,2,3 and ~ and ~ ] ~2 1 "" 3~. 3¢ ! m2¢ 2 scalar functions of (xl,x2,x3). Let L(¢,~) = ~ + 2 nz3 " - 2 - g(¢) 3x I 3x j f ] K(~) - U(¢). The integral A dx 2 A dx 3 over a space domain is a first-order

are =

jLdx

functional on the sections the trivial vector bundle

w = (¢,~)

of

E3x~xR ÷ 3 ,

i.e.j the tangent bundle of

~3xR ÷ R 3. The Legendre transformation associated with

L

is the mapping from the tangent to the cotangent bundle of ~3xR ÷ R3, defined by 3L (x,#,~) ÷ (x,¢,~) where ~ = ~ = ~ . The inverse image under the Legendre trmnsformation of the Hamiltonian structure structure

~L

is the vector

~

on the cotangent bundle is a Hamiltonian

on the tangent bundle. The image under 3 B ~

3 - A T.

~L

of a l-form

Ad~ + Bd~

Setting [(%,~) = K(~) + U(~), and writing the evolution

30

Hamiltonian equation on

R3xRx~ ÷ ~3,

Dw ~--f = ~L 6f ~w

(w),

one again obtains equations which are equivalent to the nonlinear Klein-Gordon equation.

In fact,

6[2 .. ~25 ~ = m $ + f($) + n lj . . ~xi~x j

25

"

224

and

6[ ~ = @, so the equations are

m2~ - f(~)

Dxi~x j which imply that

~2~

+ ij

~x°~x °

~2

. + m25 + f(~) = O,

~xi~x j

the Klein-Gordon equation. This approach to the nonlinear Klein-Gordon equation is very close to that of Chernoff and Marsden

[6], pp.

17-19, where the Hamiltonian structure

~L

is viewed

as a weak symplectic structure on the tangent space to an infinite-dimensional

vector

space of fields.

3. FUNCTIONALS,

GENERALIZED VECTORS AND GENERALIZED FORMS ON FIBERED MANIFOLDS.

The case of infinite-dimensional tely analogous to the finite-dimensional that adequate generalizations

Lagrangian and Hamiltonian systems is complecase --at least formally-- on the condition

of the functions, vector fields and differential

over a manifold are introduced.

Call them respectively functionals,

fields and generalized forms defined over a fibered manifold

forms

generalized vector

~ : F --> M. This fibered

manifold plays the role of the velocity phase space in the Lagrangian formalism, while in the Hamiltonian formalism for field theory it is itself the cotangent bundle of a fibered manifold cations,

M

(see below), and it plays the role of the phase space. In the appli-

will be either the 4-dimensional

space, and the sections of

F

space-time or the 3-dimensional Euclidean

will be the physical fields. When the base manifold

degenerates to a point, the generalized objects over

~ : F ÷ {point}

reduce to the

usual objects over the manifold F. The basic definitions can be found in [19]. We recall the notations, with slight modifications.

Let

~ : F ÷ M

systems of interest will be defined. F

and

VF

TF

denotes the ~ertical bundle of

tangent spaces to the fibers. manifold

be the fibered manifold over which all the

~ : F ÷ M. If

p

VF

denotes the tangent bundle of the manifold ~ : F ÷ M,

i.e., the union of all the

is also called the tangent bundle of the fibered

denotes the projection of

VF

onto

F,

then

p : VF ÷ F

31

is a vector bundle.

Its dual is the union of the cotangent

This dual bundle w i ~ p l a y

an essential

the dual vector bundle by fibered manifold

(In section 6 we shall use the letter

F(F)

is the set of smooth sections

then

VF = F×F and V~F = F×F ~ M M On a given fibered manifold,

According

of

F. If

n-forms

to the needs of our discussion,

either Lagrangians on

F

by H(F)

the vector

space of Hamiltonians

on

in

An-ITeM,

formalism

An-]T~M,

element has been chosen on on

reason that whenever

M, an (n-l)-form

on

M

a divergence if it is the exterior differential u

of

dM

to

operators

in

AnT~M.

will be called

F, since the main

(§§ 5 and 6). We shall denote

operators on

F

on

M

with

with values

is orientable

can be identified F. A generalized

of a vector O-form

in

and a volume with a vector O-form 8,

H

is

i.e., for

is the exterior

= dM(B(u)),

differential

of forms on

M. We shall write

Functionals are equivalence classes of Hamiltonians modulo divergence. space of functionals

of the

with values F

M,

F,

H(u) where

F from

O-forms M

M, we shall rather call them vector O-forms on

each section

than

k.

which we could call generalized

for the obvious

rather

F. We shall also refer to a H a m i l t o n i a n

as a generalized O-form of order

but,

on

operators

In a similar fashion we shall consider differential values

p

is the dimension

Hamiltonians on

of this paper is upon the Hamiltonian

k

n

these differential

or, more frequently,

emphasis

of order

We denote

is a vector bundle over

operators

M, i.e., differential

on

F

~ : F ÷ M, where

M, we shall consider differential

the differential

to the fibers of F.

(~§6 and 7 )°

: V F ÷ F, and we call it the cotangent bundle of the

p

~ : F ÷ M.

p~.)

base manifold

spaces

role in what follows

on

F

will be denoted

F(F).

The equivalence

H = dMB. The vector

class of

H ~ F(F)

r

is denoted

IH e F(F). The equivalence

relation will be denoted by

A generalized vector field of order tor

X

of order

identity of for

x e M

XM(X)

k

from

F

F, which projects and

u e F(F),

on

F

is a differential

onto a vector

X(j~u)

c Tu(x)F.

field For

XM x c M

on

jkF

to

section operaTF

over the

M. Thus, by definition,

and any

u ~ F(F),

= (T~)(X(j~u)). The Lie derivative

section of

VF

along

u

Lxu

of a section

x

in

u

of

F

with respect

to

X

is the

field is called vertical if it takes values

in

VF. The

defined by (LxU)(X)

for

k

TF, i.e., a bundle map from

to

~.

= X(j~u)

- (rU)x(XM(X))

M.

A generalized vector set of vertical

generalized

vector

fields on

F

will be denoted by

V(F). Each

32

generalized vector max(k,]).

field

In fact,

L~ = LX. X Let

X

(xl,y~),

to the fibration.

X

i = l,...,n,

y~(xl,yB

=

Xl(x)u?(x), l

k

has a vertical representative generalized

a = 1,...,d,

~ ~ ) (xl,y ,yi,...,Yi(k)

Then

k, are local coordinates ya

of order

is the unique vertical

on

jkF.

~ B 'Yi ..... Yi(k)),

then

be l o c a l

, where

X

of order

vector field such that

coordinates

I(k)

on

F

adapted

is any m u l t i i n d e x

of length

If

X = X i ----v ~ + Ya -$- , with X i = X1(x j) and ~x I 9y~ a i B ~ B (6xU)(X) = Y (x ,u (x),ui(x) .... ,Ui(k)(X)) -

and therefore i a X = (Y~ - X yi ) ~Ya

Generalized AnT~M

0 ~ 1 ~ k

VF

@ ~(AnT~M)

along

is defined by

F

are differential

type. A generalized

maps each section

(jlu)~(V~jIF) of

]-forms on

of the following

u

of

~ : F ÷ M

which depends

u,

and

h(u

; v)(x)

operators

l-form

h

v = w o u, the differential

1 v ,), = ~(j u) ( -3x

and

n-form

%(u

; v)

be adapted coordinates

(xl,y ,yi,...,Yl(1),v Then

,vi,...,Vl(1))

~ a (xl,v a ,vi,...,Vl(1))

section w

of

VF

has components l(w) = %(u ; v)

where

h ,1~,...

values

in

AnT~M.

If

X

Hamiltonian

,h ~I ( 1 )

generalized

vector

]-form of order (k,O)

jkF + F

÷ F. The vector

V~F O ~ ( A n T ~ M )

will be denoted

denote by



of order

w

is a section

l(w) = l(u ; v) linearly on

(k,l)

on

with

F, and let on

V(jIF) ~ jIF.

(jlu)~(VjIF)

on

v. If

F×F M

÷ M. If the

then

operators

on

F

field of order

of order

m,

I o X

k

with

is a

sup(k,l+m).

We call a generalized

F

M,

in

with

I (u)v ~ + ll(u)v~ + ... + 1 I(I)~ ~ ~ a I ~ ~U)Vl(1)'

generalized l-form of order to

on

(k,l) of

u. If

depends

local coordinates

are local differential

is a vertical of order

l(u)

local coordinates

x ÷ (ua(x),v~(x)), =

with values

on the fibered manifold

be adapted

are adapted

VF

of order

only on the k-jet of

is a vector bundle, 1 is a differential operator n~ values in h T M, linear in the second argument• (xl,y ~)

on

F

onto a section

F

Let

on

in this paper by the Hamiltonian

In local coordinates,

(k,l)

is a m o r p h i s m

g(F).

1 = O. A s i d l e from

space of simple generalized

For

I ~ ~(F)

and

]-forms over

X s Y(F), we shall

I o X.

a simple generalized ~(u

s i d l e if

of fibered manifolds

]-form

; v) = ~ (u)v ~.

1

will be w r i t t e n

33

If

X = Ya 3 , then 3y ~

Remark

: Generalized

operators

(u)

l-forms of order

in the following

manifolds. A differential p : G + jIF

(jlu)~(G) ÷ M

D

(k,l)

sense. Let

D

from

0 ~ I ~ k and

of order

jkF ÷ jIF

maps each section

which depends

for

~ : F ÷ M

section operator

is a bundle map

jIF. Equivalently,

= ~ (u)Ya(u).

u

k to

of

~ : F + M

~ jIF

|-form on

to

is a

in

An-]T*M

F

onto a section

with

w

the generalized

(k,l)

is a differential

]-forms of order

vector l-forms on

and we call them

of

of

G = V~jIF @ AnT~M. (k,l)

F. A generalized

divergence if it is the exterior differential of a vector ]-form

for each section

Du

.

of order

P : G ÷ jIF

In a similar fashion we define with values

in

'

M

a generalized

with values

over the identity of

1°

.1 ] u

from

~ : F ÷ M

) G

t section operator

section

be fibered

u,

(jlu)*(G)

With this definition,

on

G * jIF

~ : F ÷ M

only on the k-jet of

are differential

p : G ÷ jIF

y,

on

F

]-form

i.e., if,

VF,

%(w) = dM(X(w)). We shall write

~ = dMX.

Two generalized

equivalent (denoted ~ ) if they are equal

l-forms are called

modulo divergence. If denotes

D

is a differential

is a vector bundle, If

(u~,v a)

VD

X

is a vertical (VDoX)(u)

for

u

in

the variables If lized

H

=

; v)

= VD(u;Xu)

indicated,

operator

to

~'

from

VF

of a section of

_ _

operator

~D

field,

VD ° X

to

VF'. When from

VF

to

VD F' F'.

VF,

~D v ~ + ~D ~.v a + ... + _ _ ~u ~ ~u~ z ~ a l Ul(k) vector

: F' ÷ M, then

~l(k)V

.

is the functional

defined by

= ~D (xu)a + ~3D~ Di(Xu)a + ... + 3D~ Di(k)(Xu )a, 3n a ~u.z 3UI(k)

Di,...,DI( k.

e.g.,

is a H a m i l t o n i a n

l-form of order

~ : F ÷ M

is identified with a differential

generalized

F(F), where

from

a differential

are the local components

VD(u

If

operator

its Fr~chet derivative,

(k,k).

denote

Di(Xu) of order

If

X

the total derivatives

(x) = ~

to

((Xu)a(x)).

k, its Fr~chet

is a vertical

with respect

derivative

VH

generalized vector

is a generafield, we shall

84

denote by X.H

the Hamiltonian

VH ~ X.

The vertical bracket of two vertical is

[XI,X2] V = VX I ° X 2 - VX 2 ° X 1

field

(see [18] and [17]).

which is again a vertical

It is a Lie algebra bracket on

4. NOETHER'S MAPPING FOR LAGRANGIAN The literature mostly from Matin Lagrangian

generalizing

Gel'fand and Dikii

theorem is very large. We shall draw

[331 where

Lagrangians.

of

calculus

to the case of several independent variables.

[40] [411 to the case of higher-order and [5] and

formulation

the formal variational

projectable vector fields on fibered manifolds, generalizing

related material

X2

V(F).

have found in [16] a very clear account of the general Noether

Trautman

and

generalized vector

[2311241 who give an invariant

and from Olver

[8] is extended

XI

SYSTEMS WITH SYMMETRY. Noether's

[271 and Kupershmidt

field theory,

generalized vector fields

of

We

theorem for ordinary

the classical papers of See

[39] for closely

[311 for up to-date surveys of related topics, with

references. We state and prove a form of Noether's generalized

infinitesimal

theorem are particular

symmetries

cases.

theorem that is applicable

of which the diverse classical

It is not surprising

that the introduction

generalized vector fields leads to a simple formulation whenever

a first-order

ordinary vector field

Lagrangian

L(u)

of Noether's

on some fields

u

N i = 8L (ya_~u~) + XiL contains the coefficients ~n? i constitute the components of the vertical representative X

4.1. Euler-Lagrange

when

X

Lagrange differential

~L) ~

denoted by

of

EL

work of Kupershmidt FL A

on

F

FL. The vector

L

be a Lagrangian

operator

of

of order

l-form

FL

of order

k

from

[27~ and

(2k,O).

i.e., a generalized

is the differential

6L

L

because

that

in [23] and FL

and not

in the particular

F

from the l-form in of

to the vertical

jet bundle of

should be called a Legendre

case of a first-order

and

with values

F, which is

[27~ and called there a Legendre SL

L

which is equivalent It follows

can be identified with the restriction

SL

AnT~M. The Euler-

(in a sense to be made precise)

denoted by

of

F

the

on a fibered manifold

to

l-form on

fields of the l-form defined on the infinite

We claim, however,

We now consider

[16]) that there exists a vector

generalized

or by

X, a generalized

k

F

l-form on

L. It is of order

(2k-l,k-1),

VL - EL

of

(also called the variational derivative of

L

[23] (see also

of order

T M, such that

VL

Ya - Xiu a. which l

transformations.

is the unique simple generalized

to the Fr~chet derivative

n-I

and Legendre

Let

: F ÷ M, i.e., a differential

of the Noether

is an ordinary vector field.

differentials

following general situation.

of the

theorem since

X = X i a . + ya ~ , the well-known expression ~x I Dya

I

of the

is invariant under the

current

vector field of order

to the

statements

transformation.

transformation

time-independent

Lagrangian

35

in classical mechanics transformation

on a configuration

between

TQ

space

Q,

FL

reduces to the usual Legendre

and

T Q, as we shall see shortly, while SL reduces to jl form on (~×Q). We shall touch upon this last point

the Hamilton-Poincar~-Cartan again in 4.11.

For each Lagrangian

4.2. Proposition.

simple generalized l-form

on

EL

(2k,O), which is equivalent to

k

on

X

on

(4.1)

F

there exists a unique

F, the Eu~er-Lagrange differential of

VL, and a vector l-form

a Legendre transformation associated with

(2k-l,k-1),

generalized vector field

FL

on

F

L, of order

of order

L, such that for each vertical

F,

VL o X = <EL,X> + dM(FLoX). The local existence

variations.

(EL)

Expressions k

of

EL

The local coordinate

(4.2)

and

of order

L

FL

is a classical

expression

for

k ~

1 ~L ~(-]) DI(1) ~ ~ l=u ui(1)

=

for

and

FL

in coordinates

for

~L

n

variational

arbitrary and

The global existence work of Kupershmidt. of the proof,

paper

derivatives

of the generalized

We summarize his argument

F

the construction

of

FL

k

l-forms

EL

and

n = 1 formulas

[]2].

FL follows from the

for the sake of clarity.

The details

calculus on the infinite

3.1 and 3.2). The proof of the

of the Lagrangian

in local coordinates,

and for

[32j, p. 238, while general

appear in [27] and

can be found in [2~ , (II.].3,

theorem is by induction on the degree

(-1 k ~L ) DI(k) ~uI(k)

k = 1,

using the full formalism of the differential

jet bundle of

of

is

D. ~L +...+ i ~ a u.1

~u ~

arbitrary can be found in Noether's

in terms of higher-order

EL

fact in the calculus

and is patterned

applying successive

after

integrations

by parts. If

k = i, then one defines

in the following way (cf. also section VL

v

of

VF

is a generalized

operator

of order

constant.

Since

FL

along

on

F

F

f

on

M,

- fVL(u;v).

Since

is a differential

FL(u,v,f)

is zero if

f

is a

f, it follows that there exists a vector

FL(u,v,f)

L

v

of

VP

along

l-form

u ~ F(F),

= dMf A FL(u ; v). By construction,

defines a generalized

l-form of order

(2,0) on

F

which

EL.

of order

first jet bundle generalized

f. Moreover,

= VL(u;fv) FL(u,v,f)

(l,O) such that for each section

; v)

operator

be a scalar function on M. For each

FL(u,v,f)

of order (],l),

is linear in

Assume that the proposition a Lagrangian

f

set

] with respect to

and for each function

we denote by

as the symbol of the differential

[11]). Let

u E F(F)

]-form on

FL

of order

VL(u ; v) - dMFL(u

FL

l-form

J|F

k

on

of

EL ]

holds for any Lagrangian

F. Then

L

can be regarded

F. By the induction assumption, on

jIF

of

order

(2k-2,0)

of order

k-l. Let

as a Lagrangian

Ll

L be on the

there exist both a

and a vector

l-form

FLI

on

36

jIF

of order

r of

VJ I

EL I

on

along

jIF

on

jIF

FL

from

F

(2k-3,k-2) z,

of

of

VL](z

order

order

VL

such that for each section ; r) = ELI(z

(2k-2,0)

(2k-l,]).

of order

(2k,O)

for each section

v

One applies

VF

set

FiL(u,v,f)

l-form

along

EIL(u In fact,

to

EIL

k = I. There

and a vector of

; r) + dMFLI(z

can be considered

in the case where

of

jIF

and each section

; r). The generalized

as a generalized

a construction

similar

exist both a generalized

FIL

on

F

of order

l-form

]-form

ElL

to that of

l-form

(2k-l,k-1)

EL

on

such that

u c F(F),

; v) = EL(u

= EIL(u

z

; v) + dM(F|L(u

; fv) - fEIL(u

; v)

; v)).

and define

F]L

by

~i L = dMf A FIL. FL 1 vector

can be regarded

]-form on

as a vector

l-form

FIL

on

F

of order

(2k-2,k-l).

The

F, FL = FIL + FIL

is of order

(2k-2,k-1)

and

FL

VL(u for each section

v

of

VF

If, in particular, l-form of order

over the identity FL(u

n ~

; v) =

of

F

(_i)i+I

~L

as defined

transformation coordinates

M

of

FL

I,

FL

follows.

is a simple vector

JIF + F to

V~F @ An-IT~M

in local coordinates

in

to

of order

FL

/~i dx

÷ F

is

means

MxQ, where

Q

and Sternberg

is a manifold

TM @ T Q * Q. If we let mechanics

to a bundle map from

given by the fiber derivative

element

" dx l

that

to L

with the Legendre

have pointed

of dimension

L

out,

d,

M = R, we recover

; when

TQ of

has been chosen on

is seen to coincide

[11]. As Goldschmidt

i in classical

reduces

and a volume

TM, and

FL

is

the case of

is a time-independent

T Q, the classical

whose expression

Legendre

in local

is FL = ((FL)~)

where

of o~de~

a bundle map from

is orientable with

bundle,

T M @ TQ ÷ Q

FL

F, and the proposition

i

is a trivial

Lagrangian,

of

~u?

can be identified

the Lagrangians

u

; v))

A. v~dx I A... A dx I A... A dx n, where

M, An-IT~M

a map from

; v) + dM(FL(u

is a Lagrangian

If, moreover,

F

; v) = EL(u

~ . e.j

,

is omitted.

transformation

satisfy

F, and the expression

i=I

when

EL

along a section L

(1,O) on

and

(q~,q~)

4.3. Lagrangians on Lagrangians.

are local

coodinates

with symmetry. Let

X

on

= (~L) ~ TQ.

We now define

be a generalized

~ = 1,2 ..... d, '

the action of generalized

vector

field

on

F,

XM

vector

fields

its projection

on M,

37

and

~

its vertical Since

k

on

a Lagrangian

with

F

representative.

values

X.u = lxU = X . u , Lie derivative

L of order k on F i s n A T M, and X a c t s on t h e

in

a n d on t h e

of

L

sections

with respect

(4.3)

r/ to

of X

AnT~M

a differential sections by

operator

u

of

F

of order

by

X.r/ = -IXMrl = - dMixNn,

the

is

X.L = VL o X + dM(iXML),

where

iXM

opposite

denotes

the interior

sign convention

When

X

product

was chosen.

is vertical,

by

X M.

See also

XM = O

and

(Cf. [18],

§§ 4 and 7 where

the

[16-J).

~ = X, so that

X.L = X.L = VL o X and therefore

this definition

In local coordinates, for

if

X = Xi ~

with the one introduced + Y~ ~ Dy~

, then

at the end of §3.

X = (Y~-XIy~) i

~ 3y~

and

L = £dx I A ... A dx n,

(4.4)

X.L = (~£ ~u ~

+

The equivalence of

is consistent

D.X i

Su? l

class

2£ + ... + - -

A ... A dx n .

ni.k.X-[ ) ~ + Di(Xi£))dxl

~U~(k)

of

X.L

is entirely

j X.L

=

defined

by the equivalence

class

L. In fact,

and

EL

depends

differential

only on the equivalence

of a divergence

We denote by Let

~tM

the flow of

d X.L = ~ be a relatively

(~t ° I)(u) preceding

class

=

J~t~

remark

of

L, since

the Euler-Lagrange

is zero. XM

and by

-I u = ~t o u o ~tM" The Lie derivative

Pt"

Let

VL o X = J<EL,X>

compact

L(~ t ° u)

L

with respect

the flow of

to

X

X.

satisfies

(~tM L(~t" u))[t=O

open subset and

of

~t' if it exists,

(X.I)(u)

of

M. Let

I(u) = I J

= ~-~ ( ~ t ' I ) ( u ) l t = O ' l

L(u), It

follows

from the

that

(4.5)

(X.I)(u)

To facilitate other ways of writing

comparison X.L.

r = ~ J

(X.L)(u).

with other papers,

in particular

with

[33], we derive

38

Let

£

be a scalar differential operator on

the action of the k-th prolongation of

F

of order

k. Denote by

X --a generalized vector field on

on £ , considered as a scalar-valued function on

prkx./

jkF--

jkF• The following basic relation

holds, prkx.£ = V£ o X + XM.I

(4.6) The proof is straightforward. Assume that L = In

where

1

who designate

M

is orientable and let

be a volume element on

is a scalar differential operator of order

is defined by

div XM

k

the "Lagrangian density" often refer to

L = In

The function

~

on 1

M. Then

F. (The authors as the Lagrangian.)

dM(i~in) = (diVDXM)~. In local coordinates, if M

n = dx I A ... A dx n,

divnX N = ~i X~.

The relation (4.7)

dM(IXM(I~)) = (divl]XM)/q + (XM.1)~

is easily proved since it is an invariant formulation of the local coordinate relation Di(Xi/) = (Dil)X i + /(DiXi ). It follows from the definition of

X.L

(4.3) and from the relations (4.6)

and (4.7) that (4.8)

X.(I~) = (prkx.£)~ + (div XM)/n,

which is the usual expression of 4.4. Definition• Let

Lagrangian on

X

X.L. (Cf. e.g.,

X

leaves

generalized infinitesimal symmetry of B

on

F

[16] and [33~ .)

be a generalized vector field on

F. We say that

vector O-form

[20],

L

L

F, and let

L

be a

invar~ant modulo divergence (or is a

modulo divergence) if there exists a

such that

(4.9)

X.L = dMBWhen

X

is vertical, this condition reduces to VL o X = dM~ •

This condition (4.9) means that the equivalence class of

X.L, i.e., the func-

r

tional

IX.L, is zero. In fact, in view of (4.5), this condition expresses the infi¥

nitesimal invariance of the associated action integral with fixed boundary conditions. For this reason such an

X

is called in [33j and elsewhere a symmetry of the

variational problem defined by 4.5. Lemma. Let

X. Then

X

X

leaves

L.

be the vertical representative of the generalized vector field L

invariant modulo divergence if and only if

invariant modulo divergence. More precisely,

X.L = dM~

X

leaves

if and only if

L

X.L = d @ ,

39

with

~ = 6 - i X LM

In fact, by formula X.L = dMB

(4.3),

X.L = VL o ~ + dM(ixL)~

A conservation ~

of the Euler-Lagrange

equation

of

such that for any solution

u

(4.]0)

EL(u) M

on

F

We first state a version it is simpler and because

= O,

of Noether's

the general

L,

of (4.10),

theorem for vertical

theorem appears

dM(W(u))

symmetries

vertical generalized infinitesimal symmetry of

L

= O.

because

as its corollary.

(Lagrangian Noether theorem for vertical symmetries). Let

4.6. Proposition.

SO

X.L = d M ( 8 - i x M L), proving the lemma.

if and only if

is a vector O-form

= X.L + dM(i~L)'M

X

be a

modulo divergence, that satisfies

VL o X = dMB. Then wx=FLoX-B

is a conservation law of the Eu$er-Lagrange equation, Proof.

By assumption,

VL o X = X.L = dM~. By (4.1),

EL(u) since

= O. X

is vertical

<EL,X > = VL o X - d (FL o X) M

whence ~EL,X > = -dM(FL The conclusion

o X - B).

follows.

4.7. Corollary.

(Lagrangian Noether theorem). Let

X

symmetry of

modulo divergence, that satisfies

X.L = dMB. Then

L

~X = FL o X + i

be a generalized infinitesimal

L - 6 XM

is a conservation law of the Eu~er-Lagrange equation,

EL(u)

= O.

%

Proof.

By lemma 4.5

tion 4.6,

,

X.L = dMg

<EL,X > = -dM(FL

If, in particular,

implies

Following

if

<EL,X >

proposition

Gel'fand

field

X

X.L = O, an associated

and Dikii

~ = B - ikLM follows.

conservation

[8] and Olver

With this v o c a b u l a r y

(cf. Olver

[33])

By proposi-

law is

= FL o ~ + ixML. [33] we say that a vertical

is a characteristic of a conservation

is a divergence. to obtain

with

o X + iXML -B)- The conclusion

Mx

ralized vector

X.L = dMB

law of equation

we can reformulate

gene(4.10)

the preceding

40

(Lagrangian Noether theorem with converse). If

4.8. Proposition.

invariant modulo divergence, then the vertical representative characteristic of a conservation law of ralized vector field then

X

Proof.

leaves

L

~

X

X

leaves

of

X

L

is the

= O. Conversely, if a vertical gene-

EL(u)

is the characteristic of a conservation law of

EL(u) = O,

invariant modulo divergence.

Since for a vertical

generalized

vector

field

X,

<EL,X >

and

VL o ~

are

%

equivalent

modulo divergence,

divergence.

Therefore

if and only if

X

X

conservation

is a divergence

is the characteristic

leaves

This propositon

<EL,X >

L

invariant

is important

of a conservation

it as a divergence.

VL o ~

law of

is a

EL(u)

= O,

modulo divergence.

in practice

because,

law associated with a given symmetry

try to express

if and only if

Noether's

in order to determine

X, one may consider

theorem guarantees

a

<EL,~ >

and

that this may be

accomplished. It is clear that X.L = dMB , which

~X

following equivalence

relation

Two vector O-forms where

~

class of

It now appears is exact, Hn-I(M)

~VX]

= O

of conservation

Noether mapping. 4.9.

Conserved

manifold

~

and

(Cf.

if

of

X

L

= O,

(and not on X +

B)- Therefore

[VX]

EL(u)

and applications. O-form

NX

Let on F

for each solution be the generalized

u

n

L

is of order

(4.11)

N i = ~l

formula, l~.l,

at

least [12].)

M

space of generalized

that can be called

the

EL(u)

TM

the symmetry

on the orientable

associated with

X. It satisfies

= O. in

TM

associated

with

8.

has components

o ~X) l" + I X l _ B i "

(y~_XJua)

when

NX

in

l,

~u~ r

[32],[2],[3],[4]

on

space of equivalence

be a volume element

O-form with values

N i = (~

a classical

of

L = £dx | i ... A dx n, then

If, in particular,

from the vector

w i t h values

(Nx(U))

B

(n-1)-form

in the case where

to the vector

= O. It is this mapping

div

Let

the

4.6.)

is called a conserved current associated with

a Lagrangian

such that

:

i.e., if each closed

WX

If

~

we introduce

are equivalent if ~ - ~' = dMS, n-2 with values in i T M. Let [~] denote

modulo divergence

laws of

M. The generalized

= O

but also upon

~'

F

Hn-I(M)

[43],[45]

currents

X

X. Therefore

~.

only on

symmetries

by

on vector O-forms F,

O-form on

that,

depends

determined

we obtain a linear mapping

infinitesimal classes

on

is a generalized

the equivalence

depends not only upon

is not uniquely

+ ~X i _ B i,

3 X

i s an o r d i n a r y

vector

field

on

F.

(See

e.g.,

41

A.

Assume

a = 0,1,2,3, I) Consider

M = N x N 3 is Minkowski space, with coordinates O x = t. Assume for simplicity that B = 0.

X (a) = ~ . Let 3x a

energy-momentum

if

that

i = 1,2,3,

tensor and

N

. Then

x(a) = N(a)

Nb (a)

uOa - "~)q = J( ~(~U~ g.

iN~0)n

(x a) = (xO,xi),

the components

are

is the energy which

of the

is conserved

ill_l__= O. 3x0

2) Consider

a vector

field

X

such that

X 0 = O. Then

NO = ~l 3u;

(ya_XJu~) ]

and N i = 1X i + 3L (ya_xJu?) 3u~ J 1 the components

B.

of the conserved

~en

M = N

current

(with coordinate

F = NxQ * ~, we recover

In this case the conservation

laws

Let

2)

If

vector 3L

X = ~---If ~x 0 "

X = ya Z

3~ a

ya + 3 L

(4.11)

31 = O, the energy, 3x 0

is a vertical

3Y a 46 = O)

3~ ~ 3@

then

= t)

(y~_xO~)

field on the configuration

3q ~

0

(or conserved

formula

NO = ~

x

derived

and

in [1], p. 479.

F

is the trivial bundle,

classical mechanics with configuration

the case of

ties. With the usual notations,

1)

(Tx,Tx)

,

currents)

reduces

_N O = 31 3~ ~

field on

space

Q, such that

N O : 3£

are just conserved

Q.

quanti-

to

+ gX ° _ B °.

vector

'

space

ya

qe - £, is conserved.

F, for instance

(i.e.,

X.L = 0

is conserved,

a time-independent

the elementary

Noether

3~ ~

theorem. 4.10. Example. field

Setting

-

u

a with

We apply Noether's

2 3 X = x - - 3x 1 3u

3x a

1 3 x -3x 2 we

obtain

theorem to the Lagrangian

leaves

L

invariant

from

(4.11)

the

since

conserved

X.L

current

L

of 2.1.

vanishes

N =

The vector

identically.

(NO,N1,N2,N3j ~

42

NO

u0(x|u2

=

_

x

2 u I)

N I = -ul(x|u 2 - x2u|)

+ Lx 2

N 2 = -u2(xlu2

- Lx ]

- x2u|)

N 3 = -u3(xlu 2 - X2Ul ). 4.11. and

On the Hamilton-Poincar6-Cartan X

a generalized

vector

form.

field on

F,

For

L

a Lagrangian

FL o ~ + ix L

of order

is a vector

k

on

O-form

F,

on

F

0L

is

i'i

which depends

linearly

a linear mapping the linear

on

X. We denote

from the linear

space of the vector

9L

the Hamilton-Poincar@-Cartan

The mapping defined

in

infinite

0L

is closely

[23]. Essentially,

prolongation

of

(which can be defined i T(dy ~) = Yidx . For a first-order OL o X

form

on

F

to

o

on

of degree of

for each section

intrinsically)

Lagrangian

SL

product

u

n

SL of

on

J°°F

with the F

and

x s H,

: (@LoX)(u)(x)

satisfies

T(dx l) = dx l,

F = ~×Q ÷ ~, the local

coordinate

expression

is

coincides

(y~_xO~) +

By formulas field

xOL,

with the usual Hamilton-Poincar6-Cartan

@ = ~L ~q (4.1) and

(4.3)

(dq~_~dt)

the following

form on

J]F,

+ Ldt.

relation

holds

for any generalized

X,

(4.13)

X.L = <EL,X > + dM(@LoX).

is an alternate

implies

fields

L.

to the form

~ SL)(JxU) pr X

~L

This

of

is the interior

X. More precisely,

T

vector

vector

F, satisfying

related

0L o X

T(i

so @L

on

by definition,

@e = FL o X + i X e. M

We call

of

@L o X. Thus,

space of all generalized

O-forms

(4.12)

where

it by

the following

Poincar6-Cartan

version

of (4.1) which

version

form of

e.

of Noether's

(Cf.

appears theorem

frequently expressed

[34], [7], [35], [137] and also

in the literature,

in terms

and

of the Hamilton-

[15] §21.)

43

For each generalized

4.12. Proposition.

divergence

that satisfies

conservation

law of

symmetries

laws of

From formula

mapping of

L

EL(u) = O (4.13)

equation

(cf. also

@L

field

gL o X - B

is a

space of equivalence X ÷ u

infinitesimal classes

of

[VX] = [@L ° X - B].

of

F

safisfies

the

= O

X. form of the Euler-Lagrange

equation

In fact we claim that in the case of a Lagrangian

(4.14)

for each vector and Sternberg

(n-l)-forms

to the vectors

tangent

on to

M

field tangent

to

jIF

reduces

[11]. Once the vectors

on

M

of

to the

are

by means of a volume element n, the restriction

is the n-form on JIF denoted by @ in ~11], .l = (ixO)(JxU), for X a vector field tangent to J]F.

(@L o X)(u)(x)

(dM(OL o X) - X.L)(u)(x)

whose expression

F,

invariant modulo

if and only if

(4.]4) Hamilton's

(3.7) of Goldschmidt

more precisely, Also

vector

[23] theorem 4.2).

identified with the of

to the vector

it is clear that a section

EL(u) = O

[1]], we call

I, the condition

condition

L

space of generalized

(dM(@L o X) - X.L)(u)

for each generalized

order

from the vector

is the linear mapping

(4.14)

Following

leaving

X.L = dM6, the vector O-form on

modulo divergence

Euler-Lagrange

x

EL(u) = O.

Therefore Noether's

conservation

vector field

J]F

= (ixdg)(JxU),

in local coordinates

so (4.14)

reduces

to

u (ixd@)

= 0,

is

9u ~

3H

~x z

9~ r

~H

~u a

~x r

where

i

= (FL)~~ _- __~L , H = __~L u~ - L = i 9u.

~- - L. This is the form into which

~u.

1

1 a

the Klein-Gordon

equation was cast in 2.2 with the notations

a

u

=

u

and

~

a =

~

,

a = O,1,2,3.

5. THE M O M E N T U M MAPPING FOR HAMILTONIAN 5.1. H a m i l t o n i a n

structures.

are generalizations

Fibered manifolds

of Poisson manifolds.

as a fibered manifold a single point.

SYSTEMS WITH SYMMETRY.

with H a m i l t o n i a n

In this particular

with the space of functionals

on

Any Poisson manifold

structure

case,

structures

in general,

of order

0

~ : F ÷ {point},

structures F

[23], [19]

can be regarded

whose base manifold

the space of functions

the Poiss0n bracket defined by the Poisson Hamiltonian

with Hamiltonian

on

is a Lie algebra with respect

structure.

For fibered manifolds

it is the space of functionals

is

F, which coincides

with

which is a Lie

to

44

algebra with respect to the Poisson bracket defined by the Hamiltonian structure. Other examples of Hamiltonian structures of order symplectic structures,

0

are vector bundles with

the cotangent bundle of a bundle

(to be described in §6), and

the tangent bundle of a Riemannian bundle. Hamiltonian structures of order i or more on

~×~ ÷ ~

Gardner's

have been described in !27] and [24]. The most famous of these is d -dx "

Let

~ : F ÷ M

be a fibered manifold. A linear map

the vector space of simple generalized generalized vector fields on i) for each ii) by

l,V

in

F

l-forms on

@ : g(F) ÷ V(F)

is called a HamiZton~an structure on

2(F), ~ -

from

to the vector space of vertical

(antis~nmetry~,

~ : F + M

if

and

Jacobi's identity is satisfied for the Poisson bracket of functionals defined {jH,

K} =

{H,K)

where

{H,K} =

The Hamiltonian structure generalized order

]-forms of order

@

(h,O)

. K, for

H

and

K

is said to be of order k into the vertical

in

H(F).

if it maps the simple

generalized vector fields of

h+k. Given a Hamiltonian

H

and is denoted

(system of partial differential equations)

Hamiltonian system associated with Hamiltonian vector field.

u t = XH(U)

is called

XH. The evolution is called the

H. Two equivalent Hamiltonians define the same

It follows from (ii) that for two Hamiltonians

(5.1)

X{H,K}

H

and

K,

= [XH'XK] V

can be interpreted as a vector field on the infinite-dimensional

of the fibered manifold

~H @ ~u

H, the vertical generalized vector field

the Hamiltonian vector field associated with equation

XH

F

~ : F ÷ M, whence the name of

Hamiltonian system" for the evolution equation

space of sections

infinite-dimensional

u t = XH(U).

(See 7.20 for the example

of the Korteweg-de Vries equation.) 5.2. Lie algebra actions. A Lie group action on a fibered manifold whose differential is a Lie algebra action by vertical generalized vector fields would be an action on a manifold of sections of the fibered manifold. not possess flows, even locally, actions. Let

g

it is more

Since generalized vector fields need

appropriate to consider only Lie algebra

be a Lie algebra. We assume that

g

is finite-dimensional but this

restriction could eventually be lifted. An action of : F ÷ M from

g

(by vertical generalized vector fields) to

V(F), which we denote by

into vertical brackets

in

(X,x) s g×M ÷ xF(u)(x) s F

V(F)

X

in

g

on the fibered manifold a linear map

X ÷ X F, taking Lie algbera brackets in

such that, for each section

is differentiable.

Hamiltonian if each generalized vector field each

g

is, by definition,

there exists a Hamiltonian

The action of XF

JX

u

of

F, the map

g

on

F

is called

is globally hamiltonian, on

F

g

such that

i.e., for

6J X XT = @ ~ ,

and the

45

mapping

X ÷ Jx

is linear.

The mapping the action of

g

is called a comomentumdensity mapping for l X ~ g ÷ J IJ X e F(F) is called a comomentum

~ : X c g ÷ JX e H(F)~ on

F. The mapping

I~ J

mapping. For

u

in

F(u), we set

g -valued Hamiltonian where ~

on

J(u)(X)

= Jx(U). The map

is the trivial vector bundle

momentum density mapping. Let

IJ

M × g

--+ M.

F,

note that

IJ

can be regarded as the g~-valued



for those sections in

J

class of

u

of

F

differential

is a

to

AnT~M @ gM

is called a J. The g~-valued g

on

operator on

F. We

F

defined by

= iM Jx(U)dx

[or which integration

over

M

of

Jx(U)

makes sense

g.

The action of equivariant

F

IJ, is called a momentum mapping for the action of

on

X

The may

be the equivalence

functional

for all

J : u ÷ J(u)

i.e., a differential operator from

F,

g

on

is called strongly Hamiltonian if there exists an

F

J, i.e., one such that

momentum mapping

JJ[x,Y]

5

:

{Jx'JY }

'

for all

X

and

Y

in

g.

Remarks. I. If

M

is orientable

density mapping M×g

÷ M. (If

morphism

J k

is the order of

jkF * M×g ~

M

J, then

J

over the identity of

a g -valued function on 2. When

and a volume element on

M

has been chosen,

can be identified with a differential

a momentum

~ : F ÷ M

to

can be identified with a fiber bundle

M.) Thus

J

maps a section of

F

into

M.

degenerates

a g -valued function on

operator from

M

to a point,

a section of

is just an element

in

F

is just a point in

F

and

g . In this case the momentum

density mapping and the momentum mapping both coincide with the usual momentum mapping from the Poisson manifold 5.3. Hamiltonian vertical

X

the functional

i.e.,

to

g .

systems with symmetry.

generalized vector field

other terms,

moreover

F

X

X

is a generalized JX.H

H

4.4, that a

invariant modulo divergence

infinitesimal

symmetry of

is zero, and that, by definition,

H

IX.H =

(or, in

modulo divergence) VH o X =

6K X = XK = % ~-UU' then

X.H =

XK.H :

of a Lie algebra

g

on

H F

{H,~}

= -

if

<EH,X >. If

is the globally Hamiltonian vector field derived from a Hamiltonian

We say that a Hamiltonian X ÷ XF

We recall from §4, definition

leaves

K,

XH.K.

is invariant modulo divergence under the action if each

XF

leaves

H

invariant modulo divergence.

46

We also say, in this case, that

Let

5.4. Proposition.

H

X = ~

is a sy~etry

9

Lie algebra of

H.

be a gZobally Hamiltonian vector field. If

invariant modulo divergence,

then

[XH,XK] V = O. If, morGover,

~

X

= 0

leaves

implies

L ~ O, then the converse holds. Proof. The proposition

follows from the formula

{H,K} ~ 0

X{H,K } = 0. Conversely, if r {H,K} ~ O, that is JX.H = O.

[XH,XK] V = 0 Let

implies X

and

Y

flows of

X

and

Y, if they exist,

By proposition divergence, Y

be two vertical

XL = 0

generalized vector

XK.H ~ O, then

implies

L ~ O, then

fields. We say that the dy-

system u t = X(u) is invariant under Y if [X,Y~v = O. This means that the

namical

if

(5.1). In fact, if

and therefore

5.4, if

Y

then the Hamiltonian

is the Hamiltonian

a conserved quantity for

Let

5.5. Proposition.

avz open interval d

u t = XH(U)

u t = XH(u)

symmetry of

H

is invariant under

for

modulo

Y. Moreover,

K, then

K

is

in the following sense.

be a globally Hamiltonian vector field leaving

u(t)

I C~,

system

vector field associated with a Hamiltonian

~

moduZo divergence. If

commute.

is a globally Hamiltonian

is a solution of the Hamiltonian system

u

t

H

invariant

= XH(U)

on

t ~ I, the equivalence class of the Hamiltonian

is zero.

d--~ K(u(t))

Proof. ~ d K(u(t))

= (VKoXH)(u(t))

and this Hamiltonian

~

is equivalent

~u,XH (u(t)) = - ~-~u,~K (u(t)) ~ - (VHoXK)(U(t)) to zero by assumption.

It follows from this proposition

that if a Hamiltonian is invariant under the

Hamiltonian action of a Lie algebra, the momentum is conserved in the following sense. 5.6. Corollary.

Let a H~iltonian H be invariant modulo divergence under a Hamiltonian

action

of a Lie algebra

X + XF

solution of the Hamiltonian system

9

on

~ : F + M. Let

u t = XH(U)

~

~H . = @-~-uu

on an open interval

u(t)

If

I C~,

is

for

a

t ~

t

the g~-valued functional

IJ

is conserved,

the equivalence class of the Hamiltonian

i.e., for each

d

~-~ Jx(U(t))

X

in

9

and

t

in

I,

is zer¢.

This corollary expresses the fact that for those sections u of F such that l it exists, the integral JX (u(t))dx is conserved as u(t) evolves according to

JM

the Hamiltonian

equation

exists a conserved "charge".

This corollary

5.7. Remark.

u t = XH(U).

integral

Thus, for each element of a basis of

(over the "space" variables

g, there

M), i.e., a conserved

is the infinite-dimensional Harsiltonian Noether ~heorem.

In [34], 01ver generalized

of an evolution equation

in

the notion of an absolute

(a simple generalized

l-form

a

integral

is an absolute

invariant

integral

47 invariant

of

u = X(u) if X leaves a invariant). For a one-to-one H a m i l t o n i a n t ~, he observed that for every symmetry X of H, the generalized ]-form

structure = ~-Ix

is an absolute

conversely,

if ~

integral

is an absolute

tutes a further generalization tries

X

of

H

invariant integral

of the evolution

invariant,

of the Hamiltonian

~

equation

u t = XH(U) , and

is a symmetry.

This consti-

Noether theorem. Only those symme6K X = # ~ u ' give rise to a conserved

which are globally Hamiltonian,

IK.

functional

6. THE M O M E N T U M M A P P I N G IN F I E L D THEORY. We shall show that the "cotangent p : E ÷ M

possesses

means of the differential equivariant action on

bundle

and let

of a fibered manifo]d. and the Liouville

p : E ÷ M. There

mapping

We must

vertical

of order

(defined by

V E

lifted from an

is a natural

V~E

is equal

On the fibered m a n i f o l d

first define the cotangent p : E ÷ M

projection

is a fibered manifold which we call the

to

T~E,

o : V~E ÷ M

|-form in the sense of Libermann as follows.

spaces

p : V E ÷ E, and the

M

is a point

the cotangent

there exists

bundle

(and only in this of the manifold

a canonical

which we call the Liouville vertical form of the cotangent

V~(V~E) ÷ V E, defined

bundle

be a fibered

be the union of the cotangent

p : E ~ M. If the base manifold

the total space

vertical

O

and we shall derive the

form. Let

= L J T~(Ex ) xaM

d = p o p : V E ÷ M

cotangent bundle of

0

form)

for any Lie algebra action on

V~E = [ ~ T~(Epz) zeE

to the fibers of

case),

of a fibered manifold

structure

of the Liouville vertical

momentum mapping

of a fibered manifold

composite

V~E

E.

6.1. Cotangent

manifold

bundle"

a canonical[ exact H a m i l t o n i a n

bundle

vertical

E.

l-form

V E. It is a

[25], i.e., a section of the vector bundle

For each

x s M, let

e

be the Liouville

]-form

x

Tm(Ex ). Let with to

Y

be a vertical

o~ = x. By definition,

tangent vector Y

field to

is tangent

a : V~E ÷ M. Let

to the fiber of

V E

V~E,

~

through

~,

T~(E ). Let x

(~) In local coordinates VE

P>

E ~0

= . to the double

fibration

> M, 0 = ~ de e

If

i B Y = Y (x ,# , ~ )

•

+ Y (xi,¢6,~)

~

<e,y>(xi,¢6,~)

It is clear that if

M

is a point,

, then = ~ ya(xi,~,~B).

the Liouville vertical

form

e

of

i.e.,

48

: V E ÷ M

reduces to the Liouville form of

of the Liouville

The differential of the vertical V E

T E, so

0 is indeed a g e n e r a l i z a t i o n

form in mechanics. l-form

8

is the v e r t i c a l 2-form

do

on

defined by d0(Y,Z) = Y. - Z. -

for any v e r t i c a l vector fields Let

~

Y

and

Z.

= - dO. In adapted local doordinates, m = d~ a A dv •

6.2. The canonical P o i s s o n structure of an exact regular Poisson structure on of the fibration

~(~I,Y) =

tangent v e c t o r on

for each v e r t i c a l

V E, there is a unique v e r t i c a l vectors,

such that

]-form

= A

on

V E

for any

~

on

V E.

1

and

The m a p p i n g (or, equivalently, 2d (where

d

~

frQm the

l-form

tangent vector

o : V E ÷ M. Thus fiber with ~0 = ~

(V~E)x

V E

~

satisfies

~

A = [A,@0]

: V~E ÷ M

there ~I

Y. Given any l-form 1

such

1

on

to the v e r t i c a l

Y. We set

~I = ~I v

V~E

V E

of constant rank

O : E + M). In fact, the symplectic o : V E ÷ M, ~

i.e.,

the cotangent

is the v e r t i c a l bundle of

is a " s y m p l e c t i c fibered m a n i f o l d "

is a symplectic m a n i f o l d ,

x. The Poisson structure

o

to the tangent vectors of

x e M, because the image of

V E

on

w h i c h we denote by

is a Poisson structure on

is the d i m e n s i o n of the fibers of

T ~ ( E x ), for

defines

by

leaves of this P o i s s o n structure are the fibers of bundles

1

o : V~E ÷ M

iv, the r e s t r i c t i o n of

l-forms of

the 2-tensor A)

~ = -dO

whose symplectic leaves are the fibers

for each v e r t i c a l vector

and we define a 2-tensor

l-forms

V E

~ : V~E + M. For any v e r t i c a l

exists a unique v e r t i c a l that

V~E. The v e r t i c a l 2-form

in the sense that each

the symplectie structure v a r y i n g smoothly

is exact because the L i o u v i l l e v e c t o r field where

[,

]

denotes the S c h o u t e n - N i j e n h u i s

bracket. This exact, regular Poisson structure on the cotangent bundle of a fibration O : E + M

is a p a r t i c u l a r case of the exact, regular P o i s s o n structure defined b y

L i c h n e r o w i c z on the cotangent bundle of any foliation The Poisson bracket

{f,g}

[26].

of two smooth functions

f

and

g

on

V E

is,

by definition, {f,g} = - < # d f , d g > = - A(df,dg). In adapted local c o o r d i n a t e s then

iv = %ad~a+%ad~a,

and

(x I , ~ , H a)

on

V~E,

if

% = %idxZ+% d ~ + % ~ d ~ a ,

49

If moreover

= ~idx i + ~ d% ~ + ~ed~

= = ~>

A(~,~)

and

{f,g} = ~f

~g

6.3. The canonical assume that structure

M @

~f

Hamiltonian

V E

defines

structure with

~

of order

of order

of order Then

k

(k,O). We set from

a Hamiltonian

~O(~)

is a section of

~ F(V~E) + @(IO({)) order

k

0

V~E.

on

on

to

Xa

IO

along

if

is a H a m i l t o n i a n 6H

IO(~) = %

~

_

~

6H

be a section o f

generalized

generalized

vector

field on

( . if

Y'@ = O.

and let

X

be the globally Hamiltonian vector field

6J X

on

V E, ~ ~ ,

field on Proof.

V*E

where

JX = ~ O

when

w

and

is

~

are

be two vertical g e n e r a l i z e d vector fields on

E .

respectively along the same section of and

X

E.

{Jx,Jy} % J [ X , Y ] v

Proof. and

X

let

X($(t))Jt=O '

in the sense that

6.7. Proposition. Let Then

X ~ U(E),

E

3X B 3q~ PB -3p-

are local canonical coordinates on

More generally, given VE

then

We s h a l l Y

are

introduce

of order

local

at most

= ~a(VXoy-VYoX)~ n = (~

SY_~ X B _

8X-~ y B + a ~B

~Y~

= 64~ 6L ya " Let g

denotes the scalar product

@0

(7.4)

~

and if

X

. ~.

. Xa. ~

is of order

gaB(~-~(g~pXY~P) $ XY ~P(~gBo +--

O,

~

~g

Y ag ~X - g (gyp ~

= Xa 3 ~a

SgY~P))

Xy

BP),P 3~----~ ~,a "

+

Special cases of this situation are : a) When the base manifold tensor local

g = (gab) , coordinates

M

is a point,

VE = TE

and

, q•~ )

(q

K

~

~g~P

XK = X~ - - - ga$(xY ~qa ~qy

lifting

~

of

X

to

TE,

infinitesimal isometry P~(q) = g

on

) q

T g

infinitesimal isometry

is an

X

gyp ~

rential of the isomorphism from X

is the function on

~X Y. -0

+

is a Riemannian manifold with metric

~1 g~BqaqB " Any v e c t o r

is

globally Hamiltonian vector field

v

E

of

TE

whose expression in

field

X

on

E

T~E. The vector field on .----~, is the image of ~q

to

TE

(E,g),

X

lifts

to

a

TE, under the diffe-

defined by the metric. If, moreover, then

~

~ = X~ ~ + -~X - a q.B ~q~ ~q8

coincides with the canonical

.----~. By corollary 7.8, for each ~q

X, and for each geodesic

q(t)

of

(E,g), the momentum

is conserved.

b) More generally, let

p : E + M

vector field

can be lifted in a natural way to a vertical vector field

on

VE

X

on

E

defined by

be a Riemannian bundle• Any ordinary vertical

d ~(w) = ]-~ X(y(t))it=O,, where

an infinitesimal isometry on the fibers of natural prolongation of

X

to

VE

w

E, then

= (t____~) dy dt it=O e VE. If

X = ~.

X = Y~(xi,y B) ~ ~y~ '

X = Y~ ~ + ~ya z B ~ ~y~ ~yB ~z ~

follows from the definition of an infinitesima] c) Let

(M,g)

In other words, the

energy of

E

E.

E

is defined by the metric

E = TM,

w = (~,~) isometry

X

(M,g).

Computing

TM, and

X

X =

E

could

XM

on

M

defines a

is such that for each section

P~(w) = P~(~,~) = -g q

E = TM. Assume now that and

S, j

E = TM. More generally,

g. Each vector field on

X(~) = -[XM,~]. Therefore

is a pair of sections of of

, and the equality

on

is a Riemannian vector bundle and the kinetic

vertical generalized vector field of

(xl,y~,z a)

isometry.

be a Riemannian manifold and let

be any tensor bundle over

is

is then Hamiltonian with respect to the

Hamiltonian structure defined by the metric. In local coordinates, VE, if

X

• , and using

~,3

XM

(7.3),

where

is an infinitesimal one finds

that

81

~(w)

: ~K

~

(w) = (-[XM,%],-[XM,~]).

natural lifting of

X

from

TM

to

As explained in (b),

~

is actually the

V(TM) : TM × TM. Furthermore, since

XM

is an

M infinitesimal

isometry

I

of

M,

i

~

" "

v

[K(W) = ~ gn = ~ gij *~'3~ _

leaves

EK = K E

for each infinitesimal isometry

according to

fact

~

XM,

U(+)

be a potential invariant under

~.e., VU(%,~XM,+]) = O, for all

L = K-U. Then, by corollary 7.8, the momentum

j(%,9) = -

In

gq

IP L

% s F(TM).

defined by

is conserved when

w = (%,~)

evolves

(E).

The application of the above Lagrangian/Hamiltonian Noether theorem to an invariant Lagrangian on the tangent bundle of a Riemannian manifold yields lwai's theorem 2.3 [14], up to boundary conditions. 7.10. The Legendre-Cauchy transformation.

~EL(0) w t = #L(0)(~w )(w)

EL(u) = 0

Cauchy

Evolution Lagrangian

transf.

vF(0) + ~(o)

Relativistic Noether mapping

Lagrangian

Relativistic Legendre transformation

Noether mapping X + v X = 0L o X

L on

F ÷ M

X + V x = FL°X+ixML

L (0)

~L (O)

FL

Momentum mapping L = X(0)> Px <eL(O)'

Evolution Legendre

I

transformation

Hamilton-Poincar~-

Evolution

Cartan form

Hamiltonian

@L

on

HL(O) on

r ! V~F(O) + ~(o)

F ÷ M

i ~HL(O) VX, (dM(@L.X)-X.L)(u)

=

and (XK" K)<W) : (V[ K o XK)(¢,~ ) : -gij~iEXM,~JJn

_12 [(XM) gn = O. More generally, let

Let

invariant.

- 0

~t " ~ (~-U---)(0

Momentum mapping JX =

on

V~F (0)

on

VF (O~

O

L

~ ~X = PX = Jx o T(S)z] @k , the so called 6£]~tuYLe

v(jk-l)jk- valued l-form

k , which verifies:

A (local) section certain

a

V(J k) . In general,

will be denoted by

q:Z--+S

of

s:X---+J k

of

Pk

is holonomic

p) if and only if: s*0 k = 0

(see e.g.

(that is,

~=3

.k

s

for a

[8]).

The local expression of the structure form is

i l~l I~I

;

Sv z A ~hi = ~yh ~

, if

I~I ~< InI

Hence, i h wj ® dy~ = A* o ~L + LD(2)~L = h,j ~ 181=o gBj =

~ h,i,j

i ~ (Ahi fi.)co. 0 dy~ + [~i=o l~k0.

A simple

calculation

distribution

P(s).

due

to W i g n e r

Indeed

gives

some

for a r a n d o m

sequence

E ds/0 (s)

P(0 ( s ) .

indications the

about

following

the

rela-

tion holds:

(2)

P(s)ds

P(0 (s) s

and

:

P(I

is the p r o b a b i l i t y P(I ( d s / 0 E S)

interval

ds

~%(x)dx

= r(s)ds

at

s

contains

are

considered:

that

is the one

the

spacing

conditional

level

with

no

is l a r g e r

probability

levels

that

in the

than the

interval

s.

Two

cases

(2a)

i)

r(s)

: const

(2b)

ii)

r(s)

: ~s

= I/S

which

are

between vels

equivalent

levels

(ii).

to the a s s u m p t i o n

(i) or t h a t

Since

from

there

that

there

is a l i n e a r

is no

repulsion

interaction of a d j a c e n t

le-

(2) we h a v e S

(3)

P(S)

Poisson well

= const

law follows

known

Wigner

r(s)

e- J0

from assumption

formula

r(x)dx

(i) w h i l e

,

from

assumption

(ii)

the

follows: ~s 2

(4)

P(s)

The

The m a i n

point

there

are n o t

(i) a n d

(ii).

in

is t h a t

the a s s u m p t i o n s (ii)

there

cannot

tests.)

certainly

is a s u r p r i s i n g

results.

(The l a r g e

to e x p e r i m e n t a l cal

432

in s u c h

for

tal

e

is d e t e r m i n e d

on

hand

~s 2~ 2

constant

ments s

-

Moreover for

of e x p r e s s i o n states

a fairly

plausibility

the

large

= 1.

P(x)dx

convincing

of e x c i t e d

provides

~

that

be c o r r e c t

agreement

number

measurement

a way

linear s.

dependence

O n the o t h e r

(4) w i t h

of n u c l e i

good basis

argu-

experimenaccessible

for

statisti-

90

In fig. ground isospin bution taken

I we p l o t

state

region

T

[8].

is q u i t e

the e n e r g y

for

states

It is seen good.

This

as an i n d i c a t i o n

that

spacing

the

the

strong

same

distribution

spin

agreement

phenomenon

of

4 0 1 ~

level

with

known

J ,

with

~

the W i g n e r

as level

correlation

in the

parity

among

and distri-

repulsion

was

levels.

Nuclear Data Table Fig.

I

0V 0

1

2

3

4

S/D

Nearest neighbour spacing distribution (taken f r o m ref. in the g r o u n d state d o m a i n , for s p a c i n g s b e t w e e n s t a t e s the same (J~, T) .

On the numbers

contrary,

( J ~ ,T)

appears

from

between

states

fig.

if we c o n s i d e r

together, 2. This

the

was

all

Poisson

taken

of d i f f e r e n t

states

as an

(J ~ ,T)

of d i f f e r e n t

distribution

are

indication

quantum

follows that

8) of

as

it

spacings

uncorrelated.

60 40

Nuclear Data Table xed jT[

Fig.

2

20

1

2

3

4

S/D Spacing (J~, T)

Analogous Hamiltonian

distribution

results

is k n o w n

model.

In Fig.

values

of a shell

as

for Fig.

are o b t a i n e d and

3 we p l o t model

compute the taken

if one

the

level from

I but

irregardless

assumes

eigenvalues spacing

ref.

[8].

of

that

the n u c l e a r

using

the n u c l e a r

distribution

for the

shell eigen-

91

50 V

, ~

0

Fig.

I

2

cal

results

cated

[9,10],

systems

it is t o o tions

lyses of

(N+n) , able

detailed Still,

individual

levels

of h e a v y

N

level

nuclei

highly point ture

of view, is w a s h e d

parity cal

level.

excited

remain

theory

may

assuming

good.

result

of

structure

is e x p e c t e d

of

to be u n d e r s t o o d

statistical

the

state

ture

ticles

are

to d e f i n e which

mechanics,

nucleus

interacting

and

laws

numbers an

in a n y

What

in w h i c h

as a " b l a c k b o x " according

of

of the

to u n k n o w n

precise

interaction

way are

observations

to n u m b e r

whether

all

but

which

system

laws.

The

equally

struc-

spin

and

not predict it w i l l of

level

is too

compli-

is a n e w k i n d

itself.

n o t of We p i c -

number

problem of

descri-

the

knowledge

a large

an e n s e m b l e

opposite

be a s t a t i s t i -

will

exact

or

the

shell

than

required

in w h i c h

of

as far as the

will

nucleus

we renounce

states

It is i m p r o b -

irregularity

is h e r e

of the n a t u r e

that other

theory

in

such ana-

diametrically

nucleus,

of

[10]:

and collective

inquire

inquiry

statistical

the d e g r e e

106 .

the

by

give precise N

be p u s h e d to

from

in a n y o n e

to o c c u r

in a m a t h e m a t i c a l l y

all possible

The

of

hypothesis

such

in d e t a i l .

of a s y s t e m b u t

a complex

of

levels

appearance

that

reasonable

as a w o r k i n g

be the g e n e r a l

cated

can ever

be u n d e r s t o o d

levels.

sequence

For example,

structure

freedom,

success

which

region

develocompli-

the equa-

excited

from number

shell

t h a t no q u a n t u m

The

of e n e r g y

the d e t a i l e d

numbers

stated

beyond

go.

levels

on

As

low-lying

of the o r d e r

based

It is t h e r e f o r e

out and

the

of

to i n t e g r a t e

impressive

a point

usefully

of

integer

quantum

states

of

(RMT)

for v e r y

of d e g r e e s

matrices.

had

and numeri-

Theory

that

in the n e u t r o n - c a p t u r e

a stretch

is an

number

have

come

cannot

assignments

individual-particle millionth

structure

levels

Matrix

meaningless

large

must

experimental

idea was

a large

analyses

there

concerning

where

that

with

Random

main

and practically

theoretical

the

information

b y the

or to d i a g o n a l i z e

nuclei.

of

of the a b o v e

[7]. T h e i r

s u c h as t h o s e

"The r e c e n t interpreting

provided and

difficult

of m o t i o n

complex

interpretation

has been

[6],

3

3

A qualitative

ped by

Nearest neighbour spacing distribut i o n for a s e c t i o n of 50 l e v e l s t a k e n from shell model eigenvalues. The s m o o t h c u r v e is the W i g n e r s u r m i s e . (taken f r o m ref. 8) .

then

systems

probable".

of p a r -

in

is

92

As a m a t t e r joint

linear

tian matrix of

systems

large,

of

fact,

a quantum

in H i l b e r t

infinitely

can

but

of

operator

therefore

finite,

rank.

many

system

is d e s c r i b e d

space which

dimensions.

m a y be

The

by a self-ad-

thought

above

as a

mentioned

be r e a l i z e d

with

The p r o b l e m

is h o w to c h a r a c t e r i z e

an e n s e m b l e

hermi-

ensemble

of m a t r i c e s

of

such

ensemble.

Actually

we are

w e are p r e t e n d i n g tonians.

This

terested

are

required

here

which

to d e s c r i b e

is m e a n i n g f u l the

same

priate

limit

(e.g.

condition average

values:

the

the m a i n

ing d i s t r i b u t i o n along

several

invariance

properties

o n the m a t r i c e s .

example

requires

lar,

the

very

useful

In the

are

the

under

fact

presentation

we have

[4]

leads

that

GOE

accurately

which also by

of

the

that

this

m a y be n u c l e a r the

the

energy

complex

GOE.

This

levels

data

real

are

same w h e n

as

and

this spac-

computed

by a v e r a g i n g

by

the m a t r i x .

One

depend

which

o n the

shown

3.

to the c o m interaction

therefore

should

rere-

fits q u i t e

I and

expects

the m a t r i x

distribution

underlying

(4) o b t a i n e d atoms

level and

is r e l a t e d

atoms

for

revealed

It has b e e n

figs.

to the

excited

of

Their

[4] w h i c h

agreement

and not

fig.

not

con-

In p a r t i c u -

has

of t h e b a s i s

must

by

certain

invariance

symmetric

variables.

shown

ionized

impose

properties

distribution

of h i g h l y

and

this to the

level

symmetric.

real,

results

satisfactory

is c o n f i r m e d

The

of m a t r i c e s

transformation

or e l e c t r o m a g n e t i c .

of n e u t r a l

is

dis-

close

computed

and

(GOE)

random

to r e a l i z e

considered

spectra

the

to c o m p u t e

and rotational

statistical

the physical

to t h e W i g n e r

system

the

or w h e n

are

ensemble the

the e x p e r i m e n t a l

We r e m a r k plexity

to be

in-

appro-

Under

very

method.

of the H a m i l t o n i a n

Gaussian

taken

--> ~)

easier

are what

in some

-->0.

are

Hamil-

we

state.

a similarity

that

out

GOE the m a t r i c e s

independent

is i n v a r i a n t flects

quantities

and

"self-averaging"

to w h i c h

the m a t r i x

nucleus

the m a t r i c e s

orthogonal

so-called

the e n s e m b l e

Time-reversal

in u n d e r s t a n d i n g

distributions. elements

that

Gaussian

of

More

the e n s e m b l e s

turns

at the g r o u n d

the

in g e n e r a l

of

of a g i v e n

ditions

systems.

according

over

are

for e x a m p l e

an e n s e m b l e

The

latter

levels

precisely,

of t h e s e

advantage

different

all

dimension

values

over

system

in w h i c h

property,

quantities

of a s i n g l e

if the p r o p e r t i e s

property,

as the

"typical"

constitute

them by averaging

only

is a t e c h n i c a l

of r e l e v a n t

in the p r o p e r t i e s

for a l m o s t

is an e r g o d i c - l i k e

persion

over

interested

that

be d e s c r i b e d

by u s i n g

the

in the r a r e - r e g i o n

atomic

(11).

93 I00

i

I

80

i

i

rWignerbution

60

Fig.

N e a r e s t n e i g h b o u r spacing d i s t r i b u t i o n of atomic e n e r g y levels in the rareearth region (taken from ref. 11) .

4

40 20 0

J

I

.0

i

Polyatomic in w h i c h ported,

i

1.0

~

I

2.0

I

3.0

molecules

.0

also

display

the

12.1<E means

0,

Indeed the

,

so far the [18]

More where

stronger levels

suggestion

we o b t a i n

RMT,

preh

is

local

and weaker goes

to

corresponds

if we c h a n g e so-called

oc = ( I + q ) 8

in-

o n the

for s t o c h a s t i z a t i o n

levels.

1+q

by

system.

= ~xqe - B x 1 + q

6=

the

distribution

results

[19] P(x)

that

definitively

between

Zaslavsky's

of a d j a c e n t r(x)

should

correlation

on com-

in p r e s e n c e

arguments

classical as

freedom

values.

settle

const/h

disappears.

repulsion

relation

distribution

idea

limit

to

analytical

of

spacing

and numerical

corresponding P(s) ~ s

striking the

parameter

accurate

distribution

predicts

repulsion

quite

the d i s t r i -

results

t h a t w e are

adequately

experimental

of the

The

as a t w o - d e g r e e s

suggest

some

in d e s c r i b i n g

or e x p e r i m e n t a l

It is i n d e e d

both

spacing

In the

to a n o n l i n e a r

may

by different

and therefore

a n d the

objects

describes

sufficiently

degree

successfully

spectrum,

On the c o n t r a r y ,

entropy.

instability

zero

hand,

Zaslavsky

the m e t r i c

model

or m o l e c u l e s ,

not

the

stochasticity cisely,

very

characterized

question.

dicate

GOE proved

atoms

On the o t h e r

above

2.5

such disparate

universal

systems

2.0

shell

a parameter-free for

i .5

the

Brody

95

For

the h i s t o g r a m

agreement

with

is r e p r e s e n t e d P(s) the

with

dicate

that

line.

A numerical h

of b i l l i a r d s

system

or numerical

of

by computing

with

slightly

system

[22]

one

line),

~

It h a s ment,

that

agrees

the

been

the W i g n e r

that

Every

that

in-

of

o n the m e t r i c

q

by Robnik q

more

with

[21]

h,

their

en-

on a class

contrary

to

to h a v e

by Brody

[19]

The

in the

two

(e.g.,

shown by Berry Hamiltonians surmise

entropy

statistics.

the

the

space

linear

re-

for e v e r y

any predelevel

re-

goes back of

to

symmetric

eigenvalues

for m a t r i c e s

of r a n k

is a

2 it is a

"unlikely".

[23],

P(s) ~ s

(q=1).

general

same

be v a l i d

argument

degenerate

is v e r y

the

with

However,

fact that

for e x a m -

billiards

et al.,

Hamiltonian

property.

with

predictions,

a better

[4] c a n n o t

can

experimental

dispersing

approximately

distribution.

on the

accurate

in o r d e r

a

statements

with

can be obtained,

several

construct

degeneracy

much more

via

a geometrical

as

s -->

A similar

ensembles

that

argu-

0 , and this

argument

allows

GOE exhibit

to

level

re-

precisely:

statistical

such that

elements

phenomenon.

with

fact

surmise

formula

set of m a t r i c e s

already

Theorem: N,

is b a s e d

that

More

matrix

[20]agrees the a u t h o r s

no definite

This

and with

by W i g n e r

spacing

means

pulsion.

rank

and

for

spacings

can always

that

to o b t a i n

as r e m a r k e d

given

for generic

with

[24]

of

of W i g n e r

is n e e d e d .

sizes

the

of c o d i m e n s i o n

which

indicate

to b e a " t y p i c a l "

and

real matrices, manifold

it

of

to t h e

made

an i n c r e a s e

In o r d e r

hand,

levels

seems

Landau

be d u e

recently

agreement

spacings

all

spectrum

pulsion

if h e r e

a better and

computation

oscillators

even

on the d e p e n d e n c e

found

different

other

since

termined

may

the e i g e n v a l u e s

and collecting

of

the

results.

number

On the

q ~ I

considerations

ple,

pulsion

of M o r s e

q = 0.8+.I

has been

a n d he h a s

concerning

a larger

a numerical

gives

q = 0.71

prediction.

The above be m a d e

Also

distribution

fit g i v e s

chaotic.

computation

of the

Zaslavsky's

for

completely

A best

model

with

the r e a s o n

is n o t

tropy

show

a full

distribution

(5) the B r o d y

data.

for a t w o - d i m e n s i o n a l Brody

model

h

in Fig.

experimental

the

ensemble

of r e a l

joint probability

is a b s o l u t e l y

continuous,

symmetrical

matrices

distribution

function

exhibits

level

the

of f

of

repulsion

96

Proof:

The linear space

T]G(N)

of real symmetric

matrices

is naturally a m a n i f o l d with a single chart modelled In this space, matrices with two eigenvalues a shell around the submanifold shell gives the probability Let

x.

1

E.

i=I, ..... N(N+I)/2

i=I, .... ,N

the c o r r e s p o n d i n g

coordinates

represented

of eigenvectors

N form

weight of this

of small level spacings.

the matrix elements eigenvalues.

and

A . 1

We introduce

by the eigenvalues

of each matrix.

of rank

~N(N+I)/2

close to degeneration

The statistical

of o c c u r r e n c e

on

a new set of

and by independent

More precisely,

we consider

functions

the

application: (6)

@ : [-I,1] N(N-I)/2

x~ N

> TIG(N)

defined by 2&t

~(01 .... 0N(N_I)/2 where

~

provides

a smooth mapping

the space of orthogonal It is possible of coordinates, eigenvalues forming

Ai;

(4). This fact,

moreover,

previous

(7)

eigenvalues

J(~)

of this change

defined change I to I

over the

of degeneration is useful

is readily evaluated:

of degree

of in per-

N(N-I)/2 li

=

it must

in the variables

~ j' i#j,

in virtue of the

then

= F(01 ..... 0N(N-I)/2) W~ C TIG(N) with

Pui ' ~ In the new coordinates ~,

defined being not the submanifold

it must vanish when

lj

0(N),

N.

far from being bothersome,

polynomial

Define now the set

over

near the submanifold.

considerations:

J(~)

[-I,1] N(N-I)/2

to show that this is not a properly

integrations

homogeneous

from

of rank

but only outside

The Jacobian be a

matrices

the transformation

entire manifold,

N

} 2Lf..... ~N)=~(01 .... 0N(N_I)/2) ( " . . ) ~ - I ( 0 i )

II

; We

i0 K 9--E--Vol =6-->0

6-->0 : lira 6__>0

-

'

/'d ~ -.

This implies of spacing

{ G i

~i

P(s)

i

[24] we

Aj)G( A i, ~j)

=

1

~i + 6) - G( ~i, ~. - [ )I < + -'

l

9 Vol( ~f )< ~ -lim [I 9--~-a-->o between levels =

properties

f~i+6 ~I _ ~i9~ 16O~i-6d~j { A ij d

that s

regularity

% vol({s) 9--~

= 0(s)

i.e. the probability

density

98

3.

How

Random

An

is t h e

important

more

or

this

respect,

less

Poisson

point

degree

been

taken the

presence

as

an

level of

prising

since

ready

stressed of

the

objective

This

theory

theory

randomness that

of

certain

Let

[25],

us

mensurate

does

consider

not

sides.

The

rearranging

we o b t a i n

the

neracies,

from

by

E N. formula

A

is

the

Therefore, perty

that

I lim ~

A beautiful shows

that

area

the

the

to

to

of

the

sequence

argument

by of

the

point

works

of

it c a n

a billiard in

sequences denote

the

somehow

randomness

we

of [26]

As

need

to

we

sur-

clas-

have

a precise

view and

aldefi-

and

of a l g o r i t h m i c [27].

an e f f e c t i v e

sometimes

test

enable

us

for to

say

sequence in t h i s

know

units,

with

incom-

are

(~ > I) --

'

we

in a r e c t a n g l e

suitable

2

that

E

in i n c r e a s i n g o r d e r , n,m c a s e t h e r e are no dege(asymptotically)

N

billiard.

of

spacings

N 4~ E s, i= I 1 A

statistics

a

has

it an o p e r a t i o n a l

in g e n e r a l ,

Since

4~ EN,,, ~--

where

random

systems.

that

double

Weyl's

result

should

In to

random.

+ n

sequence

as

lead

conclusions

associate

is

the

example

the

above

integrable

yet,

not

2

the

give

provide,

: am

This

systems

and

eigenvalues,

m,n

is t h e

eigenvalues.

systems

behave

chaotic

point

adhere

are

for

to

"random" we

of

spacings.

levels

of

naturally

sequences,

sequences

E

after

Here

the

central

of

the

found

order

originated

given

(10)

and,

the

integrable

literature

spacings.

we

more

and

concept

meaning.

complexity

among

would

sequence

that for

that

in t h e

in the

[14]

typical

speaking,

systems

discussed

= e -s

indication

one

chaotic

by

P(s)

correlations

sical

Eigenvalues?

randomness

shown

repulsion

Qualitatively

of

extensively

of

it w a s

distribution

while

nition

Sequence

= E N - EN_ I

has

the

pro-

by n u m e r i c a l

computations,

-

[14], the

sN

supported sequence

sn

should

yield

a Poisson

99

distribution. as

This

in a P o i s s o n

tribution

suggests

process.

alone

is not

detailed

information

spacings

are needed.

Indeed, rithmic

neralized the

to w i t h i n gram

shown

for

bers

Em, n

broader

the

class

~

which

of

and

in i n c r e a s i n g

The

shows as

thus

much

spacings

then

has

systems,

that

dismore

different

to be q u i t e

Nth

at r a n d o m

spacing

question:

the

log

in two second

order

of

seem

gives

come

of the

between

integrable

which

be d i v i d e d

K K(N).

levels

this

sequence

asymptotically

can

0 ~ m,n

to a n s w e r

proof,

precision

increases

algorithm

Em, n

that The

that

the k n o w l e d g e

the c o r r e l a t i o n

of an a l g o r i t h m

a given

lenght

The

[25].

to a m u c h

construction

sufficient

concerning

we have

complexity

the p o s s i b i l i t y

Of course,

zero

algo-

easily

ge-

is b a s e d

on

eigenvalue

the

required

EN pro-

N.

steps. step

The

first

rearranges

obtaining

the

provides

the

K(N)

string

num-

E'.n

(n ~ K 2 (N)) .

The

number

K(N)

must L

Ej

The length

1) 2)

length needed

to

the n u m b e r

eigenvalue

can be any

K(N)

[~/~N ] + I,

~N just


of

r

L(5,~L) [~] ~

following

in the

one.

which

then

isomorphism

WH

f = W(f) .

the n a t u r a l

quantization

quantization

quantization

of the o p e r a t o r

5' ~ S' ,

The Weyl

Weyl

f(x,y) ~ (y) dy.

[17,

section,

of d i s t r i b u t i o n s

this

= f

L(S, S')

in the n e x t

Theorem

The

the

(2. I) is not

topology

more

{ ~

is an a l g e -

onto

L(S,S') [~].

generalization

described

by

the

of the

following

theorem.

Theorem

2.4:

Let

tum operators mial, Weyl

This of

then

papers

type

of B E R E Z I N

W(f) ,

WH(f)

: W(f)

that

WH(f)

(2.3).

This

i ~x L 2(R) .

WH(f)

the p o s i t i o n If

coincides

and m o m e n -

f(q,p) with

is a p o l y n o the g e n e r a l

i.e. L(5, S').

can be d e f i n e d

result

and K L A U D E R

I

P

5 c

the q u a n t i z a t i o n

states G

and

on the d o m a i n

quantization

theorem

the

Q = x

has b e e n

[3, 4,

12,

by

proved

13].

integral in the

transforms

"classical"

112

III.

Quasi-~-Algebra

First -algebra papers

we

shall

L(I,S' ) [17,

L(S,S')

18,

give

we have

D

~ S

the r i g g e d

Hilbert

space

,

properties

introduced domains

~c

~ = L2(RI).

of

the q u a s i - ~

and d i s c u s s e d

in e a r l i e r

H

By c a n o n i c a l

imbedding

we get

S [t] c H c S'[t'].

(3.1) F ~ S'

, ~ e ~

product

in

tinuous

mappings

L(5,5') .

H .

we w r i t e

,

Consequently, of

With

5

L+(5)

(3.2)

resp.

spaces

S'

into

= L(5) n L(S')

is the

fined

on

S



(E,F),

= J W(x)

If the

E = F'

and

F 6 S'(R v)

of the G e l f a n d - D u n f o r d

dependence

[3,

holds

is the m a x i m a l

related

to be a

(w(f),p)

linear

the

shows

at(S)

[19].

- Let

F,

Let

for e v e r y

the

see

3.4.

is said

(3.8)

Remark:

(3.7)

right-hand

g (H)

consequence

space

p C F.

the

relation

of u n b o u n d e d

as

W(f)~

L(S,S').

convex

W(f)

If also

3'~ 2' ,

(3.5)

integration

Definition

the

~

algebra

important is the

dq dp

the q u a s i - ~ - a l g e b r a

domains

predual

ii)

generalizes

o I.

space

An

[21].

of the

is

Hilbert

~ (q,p)

fundamental

dependence

the W ~ - t h e o r y

more

6 of

L(5,5')

relation

W~-algebra

bras.

f(q,p)

is a w e l l - k n o w n

is d e n s e

continuous

2.3.

= f

be the W e y l

f & ~2 respect

(3.8) This

details, in the

(1.3).

the

continuity

is a c o n s e q u e n c e is s a t i s f i e d

since

the

following

operators W(q,p)

to the d u a l

of

for

continuous

theorem

(1.2)

.

and

is an o p e r a t o r pair

(L(~,£') ,

115

Therefore (3.9)

W(f) exists

Proof: tion,

the Weyl integral

in

j

=

W(q,p)

L(5,5')

dq dp

and is equal to the Weyl quantization

We have to show that i.e.

f(q,p)

W(q,p)

tr W(q,p) ~ ~ 3 2

is a

for every

(2.5) .

L(5, S')-valued test func-

~ ~ o I (~) .

For

~

= ~ (x)

we get i (3.10)

If

(W(q,p)~) (x) = e ~ qp e iqx ~ (x+p)

f (x,y)

is the kernel of

then as a consequence

~ ,

i.e.

(~)(x)

of (3.10) the kernel of

= ~ p (x,y)@(y) W(q,p)~

dy

,

is

I e

qP

eiqX

~

(x+p,y) .

Therefore, I

(3.11) Since

tr W ( q , p ) ~ p (x,y)~ ~2

tr W(q,p)~ ¢~2 defined

(Lemma 3.3),

e iqx

from

as a function of

Furthermore, in

S'

(3.9] yields

IV.

f

p (x+p,x)

and

f--> W(f) in

(3.11) we get immediately

(q,p). Thus the integral

is continuous

from

(3.9) depends continuously

(3.9)

hc.lds for

the Weyl q u a n t i z a t i o n

f--> W(f)

~

(3.9)

is well-

to

Since

but is

f c3 [24, 22, 23], the integral

in

f 6 ~2

f.

L(S,~' ) ~

for every

on

. []

section we have seen that the Weyl quantization

is a w e l l - d e f i n e d

one-to-one

linear and continuous m a p p i n g

l

I

5 2

to the q u a s i - Z - a l g e b r a

are linearly of

that

Twisted Product In the foregoing

from

dx

in the sense of the above definition.

also the integral dense

= e ~ qp

L(S,S')

W(g) 6 L+(S) We write

isomorphic to then

h = fog

~'2" W(f)

L(S,Z~).

Since

~ 2

and

L(~,S')

we can pull back the q u a s i - Z - a l g e b r a This means that if W(g)

and call

is defined h

W(f) E L(S,S')

structure

and

(3.2) and equal to a

the twisted product of

f

and

W(h). g.

116

A formal

(4.1)

calculation

yields

i S (f.g) (q,p)= ~ f(q+ql,p+pl)

[2, 25] 2i(qlP2-q2Pl)

g(q+q2,p+p2) e

dPldP2dqldq2 ?

Lemma

4.1.

- If we d e f i n e

a topological with

respect

i) f --> g,f on

~'2

ii)

Proof:

The

i)

involution

is g i v e n

W(f),

(Gf) (x,y)

Since

in

functions,

not

In fact, A'B

is not

restricted

if

If

B 6L($,~')

to an o p e r a t o r

We h a v e

First, dure,

going

in

maps of

How can

Q2:

Is

into

to i',

to a n s w e r

QI:

Av

there

to e x t e n d

of o p e r a t o r s , product

Let

two b o u n d e d

V

a general V

operators

A

has

of

on

H =L2(RI),

this m u l t i p l i c a t i o n (see

procedure

C = AV B

pairs

4.~.

L(S,~')

be a l i n e a r

and

then

of L e m m a

of

in c o r r e s -

to g e n e r a l

L($,S') I but

structure

and

the m u l t i -

to e x t e n d

space

with

a natural maps

(3.5)).

5

the

~ c V c ~' .

extension to

AV

~'

be d e f i n e d ?

that

exist

product.

by

two q u e s t i o n s

A V. B c L ( ~ , S ' ) ,

let us r e m a r k since

~

V

rises

twisted

to d e s c r i b e

L(5,1'-) .

twisted

is d e f i n e d

classes

AB ~

algebraic

mappings

(G[) (x,y).

are

and

linear

= f(q,p)

of the

f --> f+

assumptions

A,B ~ L(S~')

by the

N o w we are multiplication

the

(3.5)

i.e.

f+ (q,p)

fk

becomes

of d e f i n i t i o n

,

as c o n t i n u o u s

the q u e s t i o n

by the

is w e l l - d e f i n e d

covered

=

to l a r g e r

to e x t e n d

fog

(29,F)

is the k e r n e l of the o p e r a t o r + W(f) and f r o m (2.3) one can

of

manner

then

sense

the d e f i n i t i o n

iGf) (x,y)

natural

that

by

involution

is the k e r n e l

L(5,~')

with

than

defined

(Gf) (x,y)

see t h a t

In a v e r y

then

else

ii) . The

in the product

are

g ~ F.

= W(f) +.

pondence

, f --> fog

to show

immediately

twisted

for e v e r y

W(f +)

plication

quasi-~-algebra to the

is n o t h i n g

It r e m a i n s

F = W-I(L+(~)) ,

AV

i.e.

is

cannot

operators

AV B

continuous

be d e f i n e d

A ~ L(S,S'),

by the which

from

~

closure

are not

to

proce-

closable

117

as u n c o n t i n u o u s

operators

TO a n s w e r V,

5c

First

some

convex

which

(4.2)

nuous

of

.

such

that

~ [t~]

system

of

on

J" ,

suppose

Definition

i) t V

t~

following on

or its

4.3.

space

i))

A-B

is d e f i n e d

- Let

a class

of d o m a i n s

5 c V c S',

the

we p u t

strongest

locally

B ~ L V is c o n t i n u o u s

of

product

for w h i c h Lv

that

LV< L(S,v[tV]).

we d e n o t e

the w e a k e s t

operator

A ~ ~

i.e.

t~

locally

becomes

is d e f i n e d

by the

contifol-

for

all

A ~

space

~cV

c~ r ,

conditions

the V'

V,

t~

for a c e r t a i n

= v[tV] '

an

hold:

completion

A,B

is c a l l e d

of

~ c L +(~) ,

and

S

is a F r ~ c h e t

of o p e r a t o r s

space.

such

that

the

V

be an

F-domain. operators

The

extension

by

A V.

def.

AVB

If

of an A£i V

By

LV

of

Sit V]

Ael V , B ~

we d e n o t e

£V,

to

to V

then

the

~I, i.e. (see Def[ the p r o d u c t

by

A.B

Lemma

S ~

is a f f i r m a t i v e .

we d e n o t e

(4.4)

x

every

of the c o n t i n u o u s

4.2,

(A,B) cl v

t~

with

of p a i r s

(Q2i

= L(s[tV]'~r)"

The

S

dual

the c l a s s

Definition

operators

in

is s e p a r a t i n g .

--- ][t~],

to q u e s t i o n

Remark:

that

- A linear

if the

V[t V]

linear

such

By

II~PII A = I I A ~ i l

coincides

N o w we d e f i n e

~V

V,

denote

operator

~ ~ L2(RI),

that

V =

answer

1.c.t.,

such

4.2.

F-domain,

ii)

to d e f i n e

space

we

every

dense

seminorms

t~,

always

tV

~

domains.

vector

By

L+(S).

into

of

(4.3) We

every

: = strongest

topology

lowing

going

semiregular

B S c V]

on V,

be a s u b s e t

convex

For

are

the d o m a i n

v[tV] , i.e. tV

A

(QI) we

definitions.

into

5' w i t h

we call

B ~ L(S,~'),

topology

[t]

Let

question

V c D',

L v = {B;

in

A,B there

is d e f i n e d exists

only

for

an F-domain

V,

such such

pairs

(A,B)

of

that

.

4.4.

- Let

V

be an F - d o m a i n .

If

f e 5~ , g G V',

then

by

118

Tf ~ g ~ V

: i(V,S')

Tf ~ g ~

= f.

morphism

we d e n o t e The map

the l i n e a r o p e r a t o r

T

can be e x t e n d e d

T : 3' ~ V' --> L(V, S'),

is i s o m o r p h i c

to

i (~,V),

s v : L(3,v)

(4.5)

quite

to a l i n e a r

analogously

iso-

V $ £'

i.e.

--- v 6 S'

£ v : fi(V S') ~ S' ~ v' The L e m m a § 41,

states w e l l - k n o w n

§ 44])

having

N O W let

properties

in m i n d that

and

~'

® S'

and

6 £'~

V'~5"

be the two o p e r a t o r s

A,B.

T h e n the k e r n e l

of

by

(4.5)

A(x,y)

~

of t e n s o r p r o d u c t s

well-defined

(4.6)

(A~B) (x,y)

in the f o l l o w i n g bilinear

sense.

For

($~)

x

topologies dense

(6~S) of

be c o n t i n u e d

to

$' $ V'

in b o t h

5 " ~ 5' .

A(x,z) A,B

B(z,y)

e ~2,

the

A B

11,

spaces.

B(x,y) ~ V ~ £ ' C of

[14,

~'~'

is in c o n s e q u e n c e

dz integral

in

(4.6)

defines

mapping

A,B--> of

: f

are n u c l e a r

(see

spaces

SA,B(X,y) S'$ $' , ~

j

which

and

~' ~ V'

to a b i l i n e a r

A(x,z)

is c o n t i n u o u s

V $ 5" ~ S ~

, V ~ S' ,

mapping

T h u s we have p r o v e d

B(z,y)

of

with respect

to the

Since

is

the b i l i n e a r (~'$ V')

the f o l l o w i n g

x

dz

~ ~ ~

mapping

(V ~5')

SA, B

can

to

lemma.

/

Lemma

4.5.

respect

- ( S 2' S 2 )

B = B(x,y) C $ 2 tiplication i(x,y) ~

5' ~

g i v e n by

A , B --> A ~ B V',

multiplication (3.35)).

Let

comes

V

continuous

l o g o u s l y V~

convex quasi-*-algebra

of k e r n e l s

Let

A=A(x,y)

V be a:l F - d o m a i n .

can be e x t e n d e d

convex

be a l i n e a r

of

(4.6).

a canonical

in a l o c a l l y

locally

A~B

by c o n t i n u i t y

~ 52

with ,

The multo k e r n e l s

B ( x , y ) e V ~ S'.

N o w are g o i n g to d e s c r i b e

the w e a k e s t

is a l o c a l l y

to the m u l t i p l i c a t i o n

convex ~ 0 [ ~ v]

is d e f i n e d

subspace

topology into

procedure

for the e x t e n s i o n

quasi-~-algebra of ~ , ~ ~ V

on Of.0,

0~[6]

m ~0.

such that

for e v e r y

by the c o n t i n u i t y

(O~[~] , 50)

of

By { v we d e n o t e A --> AB

B 6 V •

A --> BA

of the (see

from

Quite

beana-

119

~0 [~]

to

06 [6]

for every

D e f i n i t i o n 4.6.

B ~ ~.

- A linear subspace

convex q u a s i - { - a l g e b r a

0~, 0 ( D ~

/

~0'

of a complete

locally

is called left regular,

if the following p r o p e r t i e s hold. i)

The topology

~ v

is stronger than

~

and c o o r d i n a t e d to it,

i.e.

_ ~v] ii)

~J~

~[~]

c

also is stronger than

~

and c o o r d i n a t e d to it, and

v = ~0[°{] The linear subspace

LJ , 06 D4)D0(0

is then a right regular

linear space. A pair

(&),@)

topologies

of linear subspaces of

[W , [~

satisfying

O( with the c o r r e s p o n d i n g

i) and ii)

is called a regular

pair.

Let

(~,V)

be a regular pair,

nets with ~v -lim of

A

and

B

A

= A

and

A(O,

A'B : lim

of ~ x g

B

and = B.

{A

] , ~B ] C~ 0

Then the m u l t i p l i c a t i o n

can be d e f i n e d by the c a n o n i c a l e x t e n s i o n

(4.7)

The product

B~ V,

~{ -lim

A'B, A ~ , in ~ .

B~ U,

A

B = lim

defined by

A B

(4.7)

is a b i l i n e a r m a p p i n g

With respect to this partial m u l t i p l i c a t i o n

comes a partial ~ - a l g e b r a

T h e o r e m 4.7.

in the sense of

- i) If

If O : ~' ~ V'

and

V is an F - d o m a i n ~ : V ~ S'

~

be-

[1] .

then

(Def. 4.3), ~ C (~,d)

V C~.'

is a regular pair

t

in the q u a s i - n - a l g e b r a multiplication

A.B

( ~2' ~ 2 )

of integral operators.

Lemma 4.5 and by the c a n o n i c a l e x t e n s i o n

(4.7)

ii)

(see Lemma 2.1),

If we put

(QT, UT)

OT

The

defined by the e x t e n s i o n by c o n t i n u i t y in

= ~-I~

,

VT

= ~-I~

coincide.

is a regular pair in the q u a s i - ~ - a l g e b r a

(~2,~)

then (see

Lemma 4.1) with respect to the twisted multiplication.

Proof:

i)

is a s t r a i g h t f o r w a r d c o n s e q u e n c e of Lemma 4.4 and the duality

relation between ii)

Since

~

V'

and

V.

is a t o p o l o g i c a l

We omit d e t a i l e d arguments. i s o m o r p h i s m of the q u a s i - S - a l g e b r a

120 !

(52, ~ (~,

with

[)

quence

with of

the k e r n e l the

multiplication

twisted

multiplication

result

Theorem tended

(4.1);

[22,23,24],

fog can be w e l l - d e f i n e d ,

Schmidt

onto

the q u a s i - X - a l g e b r a

ii)

is a d i r e c t

conse-

i).

It is a c l a s s i c a l tion

(4.6)

operators. 4.7,

But

ii).

In

to the case

[11]

the

if f,g e L 2. T h e n

(L2,L 2)

that

that

the

is not

W(f),

a regular

twisted

twisted

pair

multiplication

f is a m e a s u r e

and

W(g)

multiplicaare H i l b e r t -

in the

sense

f~g has

g a continuous

been

of ex-

function.

REFERENCES

[1]

A n t o i n e , J.-P., K a r w o w s k i , W.: P a r t i a l * - a l g e b r a of c l o s e d lin e a r o p e r a t o r s in H i l b e r t space. B i e l e f e l d , Prepr. ZiF, P r o j e c t No. 2 (1984)

[2]

Bayen, F., Falot, M., F r o n s d a l , D.: Ann. Phys. 111 (1978) 11]

[3]

B e r e z i n , F.A.: L o n d o n 1966

The m e t h o d

[4]

B e r e z i n , F.A., (Hungary) 1970

Shubin,

[5]

Bourbaki, Act. Sci.

[6]

Daubechies,

I. : JMP

[73

Daubechies,

I.,

Grossmann,

A.:

JMP

[8]

Daubechies,

I.,

Grossmann,

A.,

Reignier,

[9]

H~rmander,

of

M.A.:

C.,

Lichnerowicz,

A.,

Sternheimer,

second

quantization,

New

York,

Colloquia

Soc.

Janos

Bolai

N.: E l 6 m e n t s de m a t h ~ m a t i q u e , L i v r e VI, I n t 6 g r a t i o n , et Ind. Nr. 1175 u. 1244, P a r i s 1952, 1956

L.: B.:

24

Comm. C.R.

(1982)

Pure

AppI.

Kammerer,

[11]

Kastler,

D.:

[12]

Klauder,

J.R.:

JMP

[13]

Klauder,

J.R.,

McKenna,

[14]

K~the, G.: T o p o l o g i s c h e l i n e a r e H e i d e l b e r g , N e w Y o r k 1966, 1979

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Kuang

CMP

Liu:

Acad.

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[10]

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Math.

Sc.

I (1965)

JMP

4

17

2]

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Paris

(]980)

2080

J.:

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3_~2 (]979)

295

(1982)

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239

359

317

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(1976)

1055,

4

JMP

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859

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1058,

88 II,

5

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177

121

E16]

Lassner,

[17]

Lassner, G.: Q u a s i - u n i f o r m topologies on local observables, in: M a t h e m a t i c a l aspects of q u a n t u m field theory I, A c t a Univ. W r a t i s l a v i e n s i s No. 519, W r o c l a w 1979

[18]

Lassner, G.: Wiss. Z. Karl-Marx-Univ., R. 29, 4 (1980) 409

[19]

Lassner, G.: A l g e b r a s of u n b o u n d e d operators mics, Physica A, Vol. 124 (to appear)

[2o]

Lassner,

G., Timmermann,

[21]

Lassner,

G.A.:

[22]

Loupias, 39

S., Miracle-Sole,

[23]

Pool,

[24]

Segal,

[25]

Voros, A.: J. Funet.

G.: Rep. Math.

Phys.

W.: Rep. Math.

Rep. Math.

J°: IMP 7 (1966) I.E.: Math.

3 (1972)

Phys.

18

S.: Ann.

279

Leipzig,

and q u a n t u m dyna-

Phys.

(1980) Inst.

13

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3 (1972)

295

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66

Scand.

Math.-Naturw.

(]978).

6 (1967)

G E O M E T R Y OF D Y N A M I C A L

SYSTEMS WITH TIME-DEPENDENT

CONSTRAINTS AND TIME-DEPENDENT HAMILTONIANS: AN A P P R O A C H T O W A R D S Q U A N T I Z A T I O N

Andre Lichnerowicz

C o l l ~ g e de France Paris,

France

In the a t t e m p t to u n d e r s t a n d more c l e a r l y the r e l a t i o n s h i p b e t w e e n classical and q u a n t u m m e c h a n i c s ,

certain authors

(Flato, L i c h n e r o w i c z ,

S t e r n h e i m e r and al.) i n s p i r e d by the W e y l - W i g n e r q u a n t i z a t i o n have viewed quantization

as a d e f o r m a t i o n

on the space of f u n c t i o n s the phase

of the f o l l o w i n g two s t r u c t u r e s

( c o r r e s p o n d i n g to c l a s s i c a l

observables)

on

space:

- the a s s o c i a t i v e

a l g e b r a d e f i n e d by the o r d i n a r y p r o d u c t of f u n c t i o n s

- the Lie a l g e b r a d e f i n e d by the P o i s s o n bracket. In these attempts, and c o n s e q u e n t l y

the m e c h a n i c a l

systems c o n s i d e r e d w e r e a u t o n o m o u s ,

their H a m i l t o n i a n s w e r e

time-independent.

It is so

p o s s i b l e to make a d i r e c t use of the full r i c h n e s s of the s y m p l e c t i c geometry.

In c o l l a b o r a t i o n w i t h Hamoui,

to non a u t o n o m o u s

systems with time-dependent constraints

tonians explicitly time-dependent. have r e c e i v e d r e l a t i v e l y study

is n e c e s s a r y

for example, fields.

little

The p r o b l e m to q u a n t i z e

systematic

attention;

and H a m i l such s y s t e m s

although

for an i m p o r t a n t n u m b e r of p h y s i c a l p r o b l e m s

to lasers or to the

oscillators

its related,

interactions with electromagnetic

Also certain natural problems

dent h a r m o n i c

we have e x t e n d e d those a t t e m p t s

lead to the

[3, 4], o t h e r s

study of t i m e - d e p e n -

result from time-dependent

123

boundary

conditions

I will tems. the

present

I present

terms

and

of

a

manifold

introduces

The

t of

as

or

quantum

the

small

reflects

up

I - The

state

W

W

the

manifold

terms

for

of

star-proas

Hamiltonian

corresponds

in

example

our

is o d d to

this

of

a 2-tensor

of

of

t

the of

the

The by

in

to

the

the

distinction

between can

be

respectively

classical

the

one

roles;

approximations

corresponding

as w e l l

variable

as

the

(t+ ~ ) .

formalism.

geometric terms

framework,

of

its

parameter

system

is g i v e n an

always The such

the

a

we

first

deformation,

being

study

we

~ = {/2i

with

consider

.

that

dt

~ 0

manifold global

satisfying:

geometric

and

is d e s c r i b e d

everywhere. of

of

codimension t.

C~. by

proved

manifold, I

We

see

The We

a distin-

I have

a canonical

coordinate

constraints differen-

structure.

paracompact

time

(],t)

time-dependent

(2n+])-dimensional

interesting

classical

aregular P o i s s o n by

by

connected,

a structure

2n

r.

different

parametrizes

The

Usual

played

Thus

corresponding

which

time.

way.

variable.

playing time

r ,

and

role

admitting

is g i v e n

T ,

approach

a dynamical

t ~ N

rank

our

time

a canonical m a n i f o l d

supposed

then

the

and

deformation

= C~(W;R) .

admits

foliation

of

of

a dynamical

later,

freedom

function

a structure its

and

other

appropriate

as

W

is

= N(W)

guished that

of

the

of

out

the

space

manifold

N

motions

them,

appears to more

(see

in

a geometric

cases of

and

space

n degrees

manifold set

is d i r e c t

sys-

with

approach

applicable,

t

of

coherence the

mechanics,

tiable

an

directly

in a n a t u r a l

variations

mechanics

state

role

point

the

setting

The

such

analysis

a doubling

while

limiting

quantum

and

mechanics

of

mechanics

canonical

role

is m a d e

results

classical

a)

quantum

usual

denoted

the

system,

times

considered

This

and

since,

is n o t

the

of

on

times"

plays

two

After

notion

plays the

evolution,

to b i g

the

define

star-products

interpretation

quantum

that

is b a s e d

"two

time

control

the

manifold

we

the

[12].

approach

these

note

on which

situation

the

We

using

classical

and

symplectic

dimensional.

This

between

conventional

[7]).

on

approach

intrinsic

the

Groot

ducts

here a coherent approach to the evolution q u a n t u m

our

correspondence

precise

de

[5].

such that

that is that ~[

is

124

(I-I) in

terms

of

Schouten

brackets

(I-2)

defines The

nothing

the

leafs of

the

group

of

the

should

be

completed

but

the

(I-3) i(

. ) is

(W,~,t,E)

is

considered

b)

Consider

the

a W

In

our

the

{ x a]

=

such

that

Such

canonical

by

the

(I-6)

same

(

) of

symplectic

denote

rank

and

is

by

E

the

Lie

is

structure.

such

that

[12]

derivative~

state

manifold

t

the

The

structure

2n

such

(a,b:0,

~

admit =

space

(M,F).

projection

2-tensor

that

of

the

I,..

only

said

as

be

= q

a canonical of

: t

x

corresponding

][ a n d notations.

E

~

and

define Introduce

-

A

of 2n;

=

field

W

a 2-tensor

+-/t

the

and

vector

on

W

= q

by

the

and

2-tensor

We

set

: W x e

Z

./~ = "Z.AE

(W,~,t,E). if w e

Denote

by

on

we [8].

I ..... n ; [ = ~ + n )

X

W

case,

x°

let

= Po ~/~x °=

a vector

on

mani-

I

= p~

manifold

defines

charts d =

for

Mechanics

projection.

the

E

components

~

chart

Classical

(2n+2)-dimensional

of

atlases

=

is

the

by

a canonical

constraints

non-vanishing ~

(M,F)

is

such

2 n ; h = 1 .....

~

to

W

If

Introduce

W --> ~,

of

(W,A,t)

manifold).

I

notations

the

canonical

the

on

case,

coordinate

elements

~

are

the

The

and

hypersur-

system

= 0

there

W be

W.

~(E)A

time-independent

usual

of

1

the

field

has

X

: W-->

the

system

E~

Consider

of

a vector

canonical

E

symplectic of

a product

general

the

by

representation

~ / Q t.

{x ~ = t,xh]

a chart

as

dynamical

(I-5)

d)

and

of

(I-4)

obtain

t : const

transformations

system.

field

(called that

:

2n-dimensional

~

by

product

geometric

and

automorphisms

dt

inner

= M x ~

vector

say

C)

the

a 2-tensor

fold

the

dynamical

manifold

W

given

canonical

i(E)

where

11],

= 0

group

structure

Et

0

[10,

[A,t]

faces.

This

:

[J[,l]

the o denoted

125

Take U.

for We

W

a canonical

obtain

(A,B,...

for

= 0,

admits

W

chart

a chart

0,I,...,2n

the

Ix ~ = P ~ ' ~xAl[~ :

, i = 0,~)

nonvanishing

xO

{xl of

= qo,

= Pi

x [ : q~]

; xi:qi

domain

U

=

of

domain

O,x~;xO,x

= U x ~

such

that

components

Zoo= I o o = i The is

tensor a

_%

symplectic

ponding

Pi

ticular

-Po

(2-I)

on

°

If

+ E dx~A

conjugate

to

s ~ I,

If

~

is

the

(W,A)

corres-

i

with

+ dp~Adq

~

respect

conjugate

to

to

qO

F;

in p a r -

= t.

manifold and Z

= Z

where

Z

a vector field on

through

x~M

(y(s))

I

is

(y(o)

an

1,...,m)

is

open

is

real

a chart

M.

a smooth = x

curve

, ¥(s)

interval of

M

An

satisfying

= y)

centered

of

integral

domain

at U,

the we

orihave

U

is w e l l - k n o w n

that,

for

s

that

with,

the

y if

s,

s'

(2-3)

are

: f

Therefore

the Z)

= exp(s

formula x.

Let

s

(x)

of

Z

a flow

;s')

(2-2)

is

small, = f(x;s+s')

often

be

an

u

: f~

written

element

of

under N(M).

O

we

define

: f(x;s)

sufficiently

u

curves

small,

f(f(x;s)

y

= za(y(s))

integral

sufficiently

(2-2)

If

Therefore

flows

of

(a =

= 0.

chart

q

and

{ ya 1

= J[ .

dx ~ : d P o A dq °

d(ya(y(s)))/ds It

~,i

curves

orbit)

[~,k]

considered

definitiontcanonically

dy(s)/ds

all

that

the

is t b y

m-dLmensional

(or a n

satisfies

such in

is c a n o n i c a l l y

L e t M be an

gin.

2n+2)

have

dx°Adx

2 - Integral

curve

we

~

Thus

rank

manifold

2-form,

(I-7)

for

of

set S

S

u

: u O

o O

f S

the

form

f

S

so

126

it f o l l o w s

from

the p r o p e r t i e s

(2-4) and

dUs/dS

takes

the

initial

of the Lie

:

~(Z)

value

u

derivative

that

us

satisfies

us

at

s = 0. T h e r e f o r e

the e v o l u t i o n

in

O

s

of e a c h

flow

function

of the

u s solution

integral

3 - Classical

curves

orbits

Let

(W,A,t,E)

and

(W,J[)

of

of

(2-4)

of the d y n a m i c a l

be the

state

is s t r i c t l y

connected

with

the

Z.

space

system

of the

considered

dynamical

system

rv

tially an

the

functions

inverse

by

u;

we

H ~

system.

that

image

that

that

Dynamics

N(W)

This

, the

symplectic

are e l e m e n t s

~'u

say a l s o

a) C l a s s i c a l tion

associated

of the d y n a m i c a l

(W,~,t,E)

and

Denote

on the

by

state

lu,v]

space

U of this

translated

in f u n c t i o n

and

space.

of the

For

a function

(abusing

is i n d e p e n d e n t

on

W

state

u

essenadmits

the n o t a t i o n )

of

space

Po"

by a g i v e n

Hamiltonian

func-

of the

a vector

are d e s c r i b e d , time

u , v ~ N)

introduce

we c o n s i d e r

[],H]

system

(where

Such

(time-dependent)

YH = E +

Here

denote

on the

determines

The motions

main

function

classical

Hamiltonian

N(W).

also

is d e t e r m i n e d

(3-I)

YH"

of

we w i l l

this

manifold.

c(t),

state

t,

by the

integral

the

Poisson

bracket

a canonical

a motion

on the

the

chart

(qO

above

statement

space

curves

of

i(.~)(du^dv)

: t, q

,p~)of

do-

can be

by

(3-2)

dq°(c(t))/dt

: I

and (3-3) that

b)

dq~(c(t))/dt are

usual

Introduce

(W,.A.). (3-4)

Hamilton's

the P o i s s o n

We r e m a r k

that, {Po,U

= ~H,q~l (c(t)) equations

bracket { according

]~ :

of

dp~(c(t))/dt motion

, I of the to

~u/ ~t

= IH,p~\ (c(t))

the

symplectic

definition

of ~-,

(u 6 N ( W ) )

manifold we h a v e

127

The

hamiltonian

vector

field

of

(W,]t)

corresponding

to

(Po+H) ( N ( W )

is (3-5)

and

=

such,

t h a t r r ~ Y H = YH

Therefore

the p r o j e c t i o n

integral

curves

of

Y

with by

~

and

H

(3-6)

c)

Po

the

of

the

It c a n be e a s i l y respect

to

verified

t,

(3.7) Inversely

(3-7)

Hamilton's

the o r b i t s

du/dt

:

determines

can

9 H/ ~ q °

if

along

8H/

of be

YH

~t. are

the

completed

by

,

u ¢ N(W), of

yO = _

curves

equations

= -

that,

component

integral

Hamilton's dPo/dt

with

÷ Hj

YH'

its t o t a l is g i v e n

derivative

by

g u / ~ t + { H(t),u(t)] .

the orbits

and

thus

is e q u i v a l e n t

to

to

the

equations.

N

If

u ¢ N(~)

of

YH'

its t o t a l

is g i v e n

derivative

If

du/dt

u : u,

is e q u a l

4 - The

a)

(3-8)

+ H = const.

Suppose

by means

to

one

of t i m e

can

dy(~)/d~

~

If w e

the p o i n t

x

energy

u : Po along

orbits

+ H,

(3-8)

gives

the o r b i t s ,

up to an a d d i t i v e

the c o n s i d e r e d

: YH{y(~))

equivalent

of

take

say that,

we

Po

constant.

u ~

: x,

curves

of

YH

y(~)

: y)

= I equations.

of e l e m e n t s

W duT/d~

integral satisfying

see t h a t

to H a m i l t o n ' s

family

y(V)

(y(0)

dt/d V is t h u s

along

+ H, u ~

. We obtain

coordinates,

of the o n e - p a r a m e t e r

(4-3)

~ Po

roughly

of the

(4-2) (4-I)

:

(3-7).

that we parametrize

in c a n o n i c a l

t,

variable

of a p a r a m e t e r

(4-I) and,

reduces Thus

to t h e n e g a t i v e

change

respect

by

(3-8)

Po

with

:

,~ (yH) u T

of

The N(W)

evolution

in

satisfying

at

128

and

taking

flow we

the

fT

of

value

the

u°

for

integral

T = 0

curves

of

is

strictly

connected

YH"

For

sufficiently

a

with

the

small

~

,

have uz~

: fg-u o : u o

o f~,

with f(f(x;Z),[') It

follows

from

: f ( x ; Z + Z')

(4-2)

(4-4)

t(y)

Introduce

a canonical

main

U.

On

this

described

on

U,

with

b)

(4-4)

We

view

and

adopt to

value

u

we

we

have

for

f

( (x,

of

orbits

= 0

= t,

x ~,

x[ ] _

x

x~,x ~

:

notations,

suitable

in

but

the

on

and

.

following to

u~

subject

the

The

space

flow

with

fr

a do-

can

be

by

+ Z ; ~')

t(x) ; ~ + I~')

a different

introduced

space.

of

to

: f(x,

part

that

a phase

family

of

~ , ~'

similar

Hamiltonian

Z

+

t(x) ; r ) , t(x)

a one-parameter at

set

evident

systematically

that

now

domain

t(x)

{ x~

with

time-independent only

chart

=

In

elements

satisfy

the

in

this of

point

the

vein

N(W)

case we

a

consider

taking

differential

of of

the

equation

O

(4-3) , l e t (4-5)

du~/d~

: 9u~/St

+{~,u~]

du~/dr

=

u~] ~

or

(4-6)

The the of one

introduced global (4-5) is

roles ter) (4-5)

coordinate the

led

to

(a r o l e denoted can

Dynamics.

functions,

be

t

function the of

of

appears.

For

in

introduction

respectively considered

of

the

t

N(W),

two

and main

depend

u ° = t,

agreement

coordinate by

as

+ H,

elements

(t + Z)

geometric

{Po

time and Z .

with

we

upon

x ~ W

obtain

(4-4).

as We

having

a role

evolution

In

intrinsic

this

of

context,

equation

of

solution

see

variables

where

how

different

the

parameequation

Classical

129

5 - Tangential

star-products

a) L e t

be a r e g u l a r

(W~i)

dimension power

q.

series

(W,A)

If in

u~vv

where

the

satisfying

canonical

E(N;M)

denote

~ r=]

operators

conditions

into if

of d i m e n s i o n

E(N;~) in

the

N.

m

a n d co-

space

of

A star-product

given

yanishing

formal on

by

of

u ~

E(N;9) r

on the

constants

and

[6]

bracket

extension

is s y m m e t r i c

by

manifold

9 r cr(u,v)

(Poisson

x E(N;~)

manifold

coefficients

+

following

the

C

: uv

= [u,v}

Poisson

N x N --> E(N}~)

the

2

3

map

are bilinear

[1(u,v)

we

with

Cr

1

Poisson

N = N(W) 9 ~ C

is a b i l i n e a r

(5-I)

on a regular

on

(W,A)) ,

v

as a b i l i n e a r

map

from

is a s s o c i a t i v e

is e v e n ,

and antisym~etric

if

r is odd.

--r

We have also

according

[13,

to t h e s e

u ~

Introduce

(s = I .... plectic

q

on

W

foliation. domain

Cr(r21)

contain

arguments. symplectic condition

Come

back

invariant

an a t l a s

(on the

star-products

We

only

leaf

of

see

state

space

E

induces

proved

sym-

star-product

of t h e

a general on

if,

of the

a star-product

under

its

on each

cohomological

(W,~).

Dynamics

(W,A,t)

and denoted

so to

~hl...h

star-products

of o u r

(W,~,t,E)

on

and

expressions

derivatives

tangential

of Q u a n t u m

systematically under

I have

of

local

u

r xS,xh~

(W,/t)

to

the

star-product

(W,A).

v = v±~

is a t a n g e n t i a l

chart,

the t a n g e n t i a l

equation

to the

(5-I)

:

{ xal

adapted

say t h a t

the e x i s t e n c e

u r.

: u

of c h a r t s

of a n a d a p t e d

A tangential

formal

introduced

I : I X~u

: h = q + I .... m)

for e a c h

6 - The

[6]

14])

(5-2)

b)

assumptions

by ~

a given ,

dynamical tangential

that will

induce

system.

We

star-product the quanti-

zation.

a) We formal

set

N c = NC(w)

power

one-parameter

series family

in

= C

(W;C)

~ 6 u C

C

and denote with

by

E(NC;v)

coefficients

of e l e m e n t s

of

E(NC;~)

in

N c.

the

s p a c e of

Consider

satisfying

the

a

130

differential

equation

deduced

du~

(6-I)

u K

duction value

b y the

on u°

that

(i/i)

that

~

b y the

nical

symplectic

X¢

from that

product

(6-2)

(I)

The

du~/d%"

~

= I/2i.

solution for

of

u°

on

admits

(M,F)

and

the

(6-])

t,

value

of

shows

by

taking

we obtain

We h a v e

~ = M x R2 (x° = Po'

and a Moyal manifolds

~(2) by

(6-I)

u ~

in-

a given the

and the x°

space

= qO)

a natural

constraints.

and we choose R2

admits

~

(2)

dea cano-

[15].

star-product

on

(W,AJ

v = u }~

v

the

The

~

de-

star-product

a tangential

star-

c a n be w r i t t e n

= ( i / } i ) 2 V [Po+H,uz]~,~ ~%-powers

2n

star-product

has

We a d o p t

(W,~,t)

equation

time-indeFendent

: i/~{(Po+H)~

,v

Introduce

One

of d i m e n s i o n

coordinates

(I) a n d

.

system

manifold

structure

induces

~

and where

the d i s t i n g u i s h e d

product:

is a u n i q u e

symplectic

~

(6-])

of ~

2 ¢ Z .

canonical

of t h e

in

In p a r t i c u l a r ,

our dynamical

fined

[H, u~]~ I

Moyal

there

= 0.

a star-product

duced

It a p p e a r s

a symplectic

product

by deformation

by antisymmetrization

so-called

~

t +

Suppose

We h a v e

r at

solution

2v~--

is o b t a i n e d

= E 9 r U(r ) U "

suggested

b)

- (ilm

[ , ]k

(4-5)

.guc +

dt

where

from

u - 6 - UT~,,,(Po+H) t .

tv

u (~')p

of a f u n c t i o n

u

(~(~)p=~(~)p-1~

~)

set

(6-3)

Exp~

(~ s) : ~

(sP/p!) ~(*)P

(s 6C)

p=0 If

u = u,

the r i g h t

denote

the

left member

taking

the

value

ur where

the

7 - The

to

~

right

now

the

of

(6-3)

by Exp~(u at

~

= 0

s).

member

solution

is e f f e c t i v e l y

of

of

Po

(6-I)

a n d we or

(6-2)

formally

u ° /.~ Exp ~

(- ~ (Po + H)'~)

independent

of

Po"

of an o b s e r v a b l e

the p o i n t

value

is i n d e p e n d e n t The

can b e w r i t t e n

= Exp~ (~ (Po + H)~)A~

spectrum

Consider

u°

member

M/2i

of v i e w of the m a t h e m a t i c a l ([I,

2]).

analysis

and give

131

a)

Suppose

first

constraints. fine

that

Exp9

((i/~)u

neighborhood

s),

for

fixed,

Exp~ ((i/~)u

where t.

b)

to

the

l

depending

It f o l l o w s

Come

back

a state

the

relations

moreover If

such

are

from

space

and w h e r e

Lemma upon

case,

it f o l l o w s

If are

p~

- Substitute

the

define

can de-

in a c o m p l e x M.

case

where,

Fourier-Dirichlet

with

t

being

expansion

on

M

spectrum)

(~6

the

Nc

~ ~A(t)~

spectrum

of

the

9~ ~

~

are

such

depend

define

states. from

The

time-dependent

star-product

eigenvalues

0 for e a c h

u

t)

for e a c h

the

that

Suppose

pA (t) ~ o

only

upon

t

(~-

l) (t)

# 0

spectrum

relations

constraints.

~

of

(7-2)

are

for

each

satisfied~

for e a c h

u

We

that

and

the

t.

~A

characterize

the

(7-2)

: 0

for

to u e N

spectrum

A + k and

~

s

on

(6-3)

~ # ~'

PI ~ Pl= PI

u : E ~ PA

to p r o v e

t. The

A ~

t

system

two d i f f e r e n t

pl*

the

(i/~) I s ~l

a tangential

where

the n o n n o r m a l i z e d

It is easy

upon

with

(7-2)

is the

for

that

(7-I)

the e i g e n v a l u e s

for

spectrum;

a unique

time-independent

such

and

discontinuous

E e m

to a d y n a m i c a l

have

t

has

:

t

we c o n s i d e r

(purely

Exp~((i/~)us)

admits

as a d i s t r i b u t i o n

that

s)

s

system u eN(W)

for a f i x e d

simplicity

respect

(7-I)

dynamical functions

of the o r i g i n ,

Suppose

with

our

We c o n s i d e r

of

the

(u+k)

the c o r r e s p o n d i n g

is n o n - d e g e n e r a t e d ,

the

element can be

(u+k) & N, w h e r e

deduced

states

spectrum

f r o m the

remain

of

k depends spectrum

only

of u by

unchanged.

u 6N

is real

of Q u a n t u m

Dynamics

= 9uz/dt

i + ~ ( H ~, u z

and

the

~A

realvalued.

c)

In this

to

(6-2)

(7-3)

context

can

the m a i n

equation

(corresponding

be w r i t t e n

dur/d u

:

~ur/dt

+

[H,uz]~,

- u z ;% H)

132

8 - The

Let

evolution

u r

If

~

of the

be a s o l u t i o n

spectrum

of

is an e i g e n v a l u e

(7-3)

taking

the

corresponding

value

to the

u°

at

[ = 0.

eigenprojector

~r

'

we

have

(8-1)

u~ ~

It f o l l o w s

(8-2)

We

from

~ ~

(8-2)

It f o l l o w s

similarly

the

=

~r (t) :

Theorem by

(9-I)

We

~Ac/2t

- The

x~/d~pr

left ~ - p r o d u c t

by

{du~/d~) ~

pZ

the

pu

: ~Z ~

(8-I)

=

(d

: ~

with

respect

~ (9u~/dt) ~ ~

vanishs

and

there

according

to

to

t

.

is a f u n c t i o n

(7-3).

Therefore

~

of

t6 ~

the

spectrum

such

that

We h a v e

spectrum

of

ut

is d e d u c e d

from

from

of the

of

u°

(8-2)

U~ * (dpr/d r

-

and

eigenprojectors

from

9~r/~t)

the

+

above

result

[H,ur], ~

tgl

: [(t+E)(d~r/dr

set

(9-2)

(d~/d~)

~

by d i f f e r e n t i a t i o n

/~ t) £~

+ Zr

A (t+i).

evolution

It f o l l o w s

to T

by

from

member

A (t+Z).

k (t) ->

9 - The

respect

by d i f f e r e n c e

right

d~/dV

~ p~

(d p r / d ~ )

(~

But

with

=

+ u¢ ~

(d 9.~/d~)

We o b t a i n

p~ k u ~

by d i f f e r e n t i a t i o n

(dub/aT)

deduce

p~=

¢1

= d~_~/d~- ~pz/~t - [H,pr]9

- ~p~/~t).

133

(9-I)

a n d the

where

we have

tion

for

~l ,

u r

similar

= 0

relation

u~,{~

=

suppressed

the

and

we have

Z l{A

right

the

Theorem

f r o m the

- Each

lemma

of

eigenstate

~

(9-3)

d~z/dI~

=

§ 7 that

of

of

from

= ~z/~t

(9-2)

qA

and

uz ~ ~ %

= 0.

is an eigenfunceigenvalue (7-3) = 0

that a n d thus

We h a v e

satisfies

+

~ ~

o

~

u Z

to

For a d i f f e r e n t

We o b t a i n

~ It f o l l o w s

(t+~)

It f o l l o w s

A ~l j ~ k = 0.

then

~

:

eigenvalue.

= 0.

thus

,u=

argument

studied

£~, k ~ K

and

~

reduce

the

dynamical

equation

[H, ~z]~,

10 - I n v a r i a n t s

Let

f ~N(W)

be a f u n c t i o n

(10-I) The

~f/2t

function

fr

= f

a function

is

be an e i g e n v a l u e

~ (t)

f

for e a c h

said

~

we

is n e c e s s a r i l y

according

to be an

to the

have

theorem

- The

independent

spectrum

and e a c h

Interesting

invariants

The e q u a t i o n

[3].

9~

The

= -

for

equation

states is then

+

from

Therefore

r •

Such

quantum theorem

each

system. of

§ 8

eigenvalue p

of

satisfies,

equation

= 0

invariant

of a q u a n t u m

eigenstate the

the

eigenstate

invariance

[H,p]~

for

for e a c h

the c o n s i d e r e d

= ~ (t).

of e a c h

is an

system

is time-

invariant.

time-dependent

harmonic

oscil-

let

(i/i) ( H ~ p in t h i s described the

of

(7-3)

it f o l l o w s

§ 9, the

(10-2)

of

corresponding

are k n o w n

as the e x t e n s i o n ,

equation

of this

/~t

f;

corresponding

lators

= 0

invariant

~ (t+z)

of

the e q u a t i o n

a solution

9~/~t

Corollary

mann

[H,f]~

of

constant.

(10-2)

appear

+

is then

Let that

satisfying

- ~ ~

H)

framework, by

equation

of the q u a n t u m

invariants (9-3).

p .

The

yon Neu-

generalization

134

11 - I n t e r p r e t a t i o n

for the d y n a m i c a l

systems

with

time-independent

constraints

If

a)

u~

, Vr

are

Suppose

and

that

solutions

our

of

(7-3),

it is the

dynamical

system

admits

a time-dependent

denote

by

.q

Hamiltonian.

the v o l u m e

element :

We

suppose

that

intersection leaf

of

our

More

generally

from a Moyal

Lemma

- If

v.

(11-I)

the q u a n t i t i e s

(11-2)

b)

only

upon

The m e a s u r e d

state

p

=

~(t,~) above

=

~M

(11-I)

if

t

that

to the

holds

the

time

corresponding

for c o n v e n i e n t

for a s t a r - p r o d u c t

uz

urn


vector

idea

no outgoing group. with

and

role

ones.

t-->

both

formally

that

are

Moreover

S

the m a t r i x

results

are

field

+~,-

theory,

quite

mathematically

fixed,

if t h e r e

In p r a c t i c e

empirical

of r e l a t i v i s t i c

- ~

special

THEORY

putative

in a c c o r d a n c e

of

the

S

with

with

S

at t h i s

the

particles

with

elements

determined

limit

and physically.

no i n c o m i n g commutes

the

stage,-

S

leaves

intuitive

there will

action

required

the u s e

of

of

be

of the L o r e n t z for c o r r e l a t i o n

the

formal

ex-

pression

(##)

S = I + ~ gn(-i)n(n!)jO(tl,t 2 .... tn)Hi(tl)Hi(t 2) .HI(t )dr dt " "" n 1 2 " " n: I

where

the integration is o v e r

itself,

and

increasing write

S

8 ( t l , t 2 , . . . , t n) order

is

]

the

Practical

matrix

elements

interpretable ones

between

in

K

terms

of

and

of c l a s s

analysis simplest

and

being has

the

form

have

been

sharp-time

of

of

over

~

states

S I,

of

expressions

with

factors

finite

vectors

K,

that

in

can be correlated an e f f e c t i v e apparent

might

be u s e d

need to

such.

where densely

s u c h as

support

in

of M i n k o w s k i

and the

Hi(x)f(x)d4x,

correlated

are

expression, S = Z n:0 g n S n'

in d e r i v i n g

However,

renormalization

of c o m p a c t

tj this g:

all

product

the e x i s t e n c e

space with

from constant

succeeded

direct

treatment

the

constant

to o b s e r v a t i o n .

constant

and

I when

Apart

as m a n y - p a r t i c l e

subject

C ~

as

of M i n k o w s k i

To e x p l o r e

S I.

integration

likelihood

support,

[7], g

be

coupling

the

Expressions

with

is d e f i n e d

in the c o u p l i n g

the

formulation

against

of c o m p a c t

the

physically

for an i n f i n i t e argue

series

d4x ,

product

otherwise.

theoretical

that may

mathematical

0

first-order

Hi(x)

space.

with

a n d as

as a p o w e r

and consider this

the n - f o l d

.dtn,

#

f

is

C~

defined

and

operators

Hi(x,t)g(x)d3 x

on space h a v e b e e n

corre-

146

fated with

continuous

itself

appeared

has

Now development case

of the

of a c o n f o r m a l l y

because

of

the

sesquilinear development

sentially that

f o r m on

of t h i s

space

cludes

D ~ ( H 0) if p e r i o d i c

time,

which

shows

that

the u s e

of

The

the

corresponding

the

conformal

actions,

~, which

appears

a function

on

M0

self-adjoint,

More M,

whom

and

generalizing

M 0.

Thus

for e x a m p l e ,

live

on the t w o

choice takes

among the

under

form

of

conformally

M

are

Functions

(or g e n e r a l i z e d

which

metrics For of

only

for

on the

M.

Fourier)

the

operator

basic

of

space-

argument

without

the w a v e

by

Lie

itself, cover. groups,

on t h e m h a v e expansion

solutions

all

of t h e s e

~ = 0

coefficients

(or

group) of S3

implies

that

fields

finite

locally

correspondingly

into

the

of of

frame

and

context)

although

All

all

antisymmetry S]

extend

M

~

Einstein

on

from

is

equations

equation

the

maps

It

integral.

the c o n f o r m a l

theory,

such

permits

compactification

to the q u a n t u m M

As

theory.

expansion

wave of

inter-

its t r a n s f e r

of the

in a p a r t i c u l a r

~4

interaction

ones.

Fourier

invariant

permuted

the

connection

after

This

the a n t i p o d a l

2-fold

compact

defined

in-

On the o t h e r

in c u r r e n t

L,

conformal

the

to live o n

live

SI

in M i n k o w s k i

is s i m p l y

self-adjointness

f r o m the c l a s s i c a l

themselves

of

for n o n - d e r i v a t i v e

the c o n f o r m a l

solutions

product

c a n be c o n s i d e r e d

U(2).

the

S I x S 3.

SI

expression

on

of the

fold cover,

the d i r e c t

unlikely in M i n -

connection),

is i n e s s e n t i a l ,

e q u a t i o n , w e have a c l a s s

covers

the E i n s t e i n

(which e x t e n d s HI

the

for

all c o n f o r m a l l y

for a n y g i v e n

on f i n i t e

quite

domain

introduced

this

is in f a c t e s -

singularities.

are

and

that

directly

the

the p h y s i c a l l y

that

to a f u n c t i o n

live

which

It s e e m s

that

work

energy;

shows

the c o n f o r m a l

true

in the

is a c o n t i n u o u s

[9]

be e s t a b l i s h e d

of sign,

to be e s t a b l i s h e d

specifically,

SI

is a s e l f - a d j o i n t

fundamental

therefore

expansion

that

in the c i t e d

connection.

to i n c l u d e

quite

significant

a Fourier

SI

K,

of v i e w

from a matter

appear

it is a f o r m a l l y

point

in the e x p r e s s i o n

apart

in

D ~(H) .

SI

existence.

shown

Poulsen

infra-red

conditions

from a practical

integrand

Lagranian

of the

boundary

not

[8]. B u t

is the E i n s t e i n

paraphrasing

it is p r o b a b l y

because

hand,

H

operator

could

has

ensue,-

by N.S.

on the d o m a i n

merely

D ~ (H 0) mathematical

untreated

that

where

character

(without since

on

connection"

field,-

initiated

a self-adjoint

self-adjoint

particularly

forms

dubious

divergences

D ~(H) ,

of a m e t h o d

results

kowski

"conformal invariant

infra-red

form represents

to

sesquilinear

to be of q u i t e

covers

isomorphic

to

a Peter-Weyl of

irreducible

147

unitary G

(necessarily

in q u e s t i o n .

tions

on the

The

finite-dimensional)

Such

coefficients

Lagrangian

the

formally k

fk

L

form

thus

to c h o o s e

situation with Thus

L

yet

The

and

exists

the

same

however, in

M

the

that

K

with

in

such

comparable

the

lends

that

The

index

identifiable

Ak

to that

support

unitary

are

The m a t h e m a t i c a l

= A_k-

long

self-adjointness

some

fide

Ak

of g e n e r a l i t y

operators

and

field

that

in

M0

of

known

the

to the hope

operator

on

k

the

of the

d4u

does

equivalence

M,

in

in

leading

that

K,

S

when

of

g

existence ~

M0 of

applies

satisfying

(where

M,

that

the

The

by a

correspon-

suitable

compactness the

Of course,

4 g0

dif-

as a c o n s e -

with

that

~

of the L a g r a n g i a n

are q u i t e

densities

latter the

and

as the

limit

inte-

Fourier

different,

g4 .

of

and

In a sense of that

vanishes.

essentially

is u n i t a r i l y

in

be

equation

M

identical

insure

can be r e g a r d e d S3

case

g4 d4u.

M

in

is e s s e n t i a l ) ,

[10]

former. on

should

the w a v e

g = 0

In the g4 one

locally

= $I x S 3 f

field,

g

~

that

It r e s u l t s

Lagrangian

not

M.

are

and of

in

field

in

analogous

g4

the

the c u r v a t u r e

M0

on

regularity

M0

satisfying

by a c o n s t a n t

g04 d 4 x

implies

of a s c a l a r

of the

equation

invariance.

situation

in

g0

and

local

g0

is true the

and

]M 0 the

case,

function

M0, 4 g0 d4x

of

when

same

. loss

= fk"

this

in

Quantization tized

product

the

are n a t u r a l l y

operator

of c o n f o r m a l

expansion

inner

and

f-k

to the w a v e

bounded

normalization,

gral

such

is the m u l t i p l e

the w a v e

4-forms

SI x S3

indicated,

it is no e s s e n t i a l

regularity

field

M0,

analog

smooth 4 density g0

quence

as

Ak

nontrivial

The

on

fixed

ding

func-

expansion

coefficients

to be a b o n a

simplest

the p r o p e r from

group

formulated.

= 0

fers

L2(G)

and

Together

A 0 = -S I

representative. ~0g0

in

to be the

of

a formal

as a formal

define

a level

be f o u n d

suitably

fk that

fields.

coefficient may

the

densely

has

stochastic

given

has

multi-index

is then

closed

for

fk (x) '

a basis

operators,

is a d i s c r e t e

here

of the

basis

group.

L = Ek Ak where

representations

f o r m an o r t h o n o r m a l

change

equivalent

these

to t h a t

to the L a g r a n g i a n

results. in

densities.

The q u a n -

S I x S 3,

and

In all p r o b a -

148

bility

the a n a l o g s

valued

distributions

M0,

which

mutually domain

again

D ~ ( H 0) ,

smoothing.

that

the

for

are

adjoint,-

time

to the o p e r a t o r

closed

densely

expansion defined

on the d o m a i n 4 to w h i c h ~0 need conformal

over

nuous s e s q u i l i n e a r

coefficients

Fourier

but

The

integral

the

f o r m on

time

D_ 0o(H) ;

:~(x)

but

than

can p r o b a b l y

slice

in

this

as o p e r a t o r -

M0

on

and appropriately

rather

be a p p l i c a b l e

connection

a finite

exist

operators

D ~(H), not

Ak

of the L a g r a n g i a n

the

even

larger

after

space-

be u s e d to s h o w 4 @0 is a c o n t i -

of

integral,

formally

: d4x

t~ almost

certainly

of t h i s more

type

have

been

regular

case

in w h i c h

posed has

has no n o n v a n i s h i n g

in space.

the

on the e x p l i c i t

This may invariant natural,

be c o n s t r u e d

@4

quantized

rigorously

nevertheless temporal

quite

direction,

H0 +

:@0(x,t0)4:

j

~(H)

that

tended form

over

is t h e n

be a f i r s t quantum

group

haps

thereby,

study

of

that

generated

a basis

as a p o s s i b l e

than

semigroup

time

Parenthetically,

S-matrix

S

im-

S

of

finite

the

space,

to with

is q u i t e

is

finite

time

t 2.

f o r m on

integral

is ex-

the r e s u l t i n g

regarding with

would

for c o n s t r u c t i v e 4 dimensions, the

new approaches.

directly

there

with

but

semigroup

as n a t u r a l ,

in j o i n t w o r k

n 9ptimism

is a

semiboundedness

used

dimensions

for w o r k i n g

K,

for the

sesquilinear

such

of d e a l i n g

there

rather

and per-

Preliminary Stephen

these

terms

Paneitz as

the g r o u p

rather

terms

only

by the h a m i l t o n i a n .

the

the u s e f u l n e s s

as

the m e t h o d s

as p o s s i b l e

for g u a r d e d

or

that

that when well

S

that on

to a l a t e r

b y the h a m i l t o n i a n

terms

still

been

I' is e s s e n t i a l l y

implementation

but

+ ~as

technique

times,

Results

if the c o n f o r m a l l y

sense

Establishment

method

that

as a c o n t i n u o u s

in 2 s p a c e - t i m e

generated

all that

it is p o s s i b l e

extending

this

in the have

in the

tI

below, to

as p r o d u c t i v e

well

confirms

- ~

the h i g h e r - o r d e r

has p r o v i d e d

unitary

exists

towards

theory

clear

than

from

and proved conditions

over

indication exists

for e x a m p l e ,

in its d o m a i n .

D ~ (H) .

no u n i t a r y

f r o m one

semibounded.

step

field

it is n o t

as an

is n o t b o u n d e d time

integral

domain

existent

d3x

boundary

K

of an o p e r a t o r

theory

possibly

propagation

In t h i s

the

regularity

in

formulated

periodic

Nevertheless,

considerable

self-adjoint

rigorously

vector

of

study the

of the h i g h e r - o r d e r conformal

connection

not

for the t r e a t m e n t

149

of m a s s l e s s gests

problems

that a q u a n t u m

than M i n k o w s k i

in n o n l i n e a r field

space will be more

interesting

inasmuch

fundamental

and a p p r o p r i a t e

space global

originated structure

ultraviolet mental

empirically a global formal riant

observed and

is more

should

field

and

treated

time

But the

seems

is more

in the

change

the

to the

S-matrix,

funda-

although

is t h e o r e t i c a l l y

element

in a m a n i f e s t l y

sugrather

than M i n k o w s k i

intervals,

to a central

This

and changes

so be m a t e r i a l

theory.

over m i c r o s c o p i c in its r e l a t i o n

cosmos

not f u n d a m e n t a l l y

of the theory,

simply

physics

considerations,

but

cosmos

and convergent.

the u n i v e r s a l

for t h e o r e t i c a l

of q u a n t u m

field theory,

on the u n i v e r s a l

coherent

idea that

of space-time

structure

object,

group

as the

in c o s m o l o g i c a l

convergence

relativistic

theory b a s e d

of the con-

conformally

inva-

formalism.

REFERENCES

[i]

J. Schwinger (1958), (Dover, N e w York)

[2]

P.A.M. Dirac (1958), "Principles of q u a n t u m mechanics", ed. (Oxford U n i v e r s i t y Press), et seq.

[3]

S.M. Paneitz and I.E. Segal (1983), " S e l f - a d j o i n t n e s s of the Fourier e x p a n s i o n of q u a n t i z e d i n t e r a c t i o n field Lagrangian", P r o c . N a t . A c a d . Sci. USA 80, 4595-4598

[4]

G.C.

[5]

I.E. Segal (1970), "Nonlinear functions of weak p r o c e s s e s I"; J o u r . F u n c t . A n a l . 4, 404-456, and (1970), II, ibid. 6, 29-75

[6]

I.E. Segal (1970), " C o n s t r u c t i o n of n o n l i n e a r q u a n t u m processes, I", A n n . M a t h . 9 2 , 4 6 2 - 4 8 1 , and (1971) II, Invent.Math. 14,211-242

[7]

L. Garding and A.S. W i g h t m a n (1964), "Fields as o p e r a t o r - v a l u e d d i s t r i b u t i o n s in r e l a t i v i s t i c q u a n t u m t h e o r y " ; A r k . F y s . 2 8 , 1 2 9 - 1 8 4

[8]

I.E. Segal (1970), "Local n o n c o m m u t a t i v e analysis" in Problems in Analysis, ed. R.C. Gunning, Princ.Univ. Press, I~I-130

[9]

N.S. Poulsen (1972), "on C ' - v e c t o r s and i n t e r t w i n i n g b i l i n e a r forms for r e p r e s e n t a t i o n s of Lie groups", Jour.Funct. Anal. 9, 87-120

[10]

Wick

(1950),

"Selected

Phys.Rev.

80,

papers

on Q u a n t u m

Electrodynamics"

4th

268-272

S.M. Paneitz and I.E. Segal (1982), "Analysis in space-time bundles", I: J o u r . F u n c t . A n a l . 47, 78-142 and II, ibid 49, 335-4]4.

152

CURVATURE

FORMS

WITH

SINGULARITIES

CHARACTERISTIC

Akira

Department

0.

Asada

of M a t h e m a t i c s ,

Matsumoto,

AND NON-INTEGRAL

CLASSES

Nagano

Shinshu Pref.,

University

Japan

INTRODUCTION

The

purpose

of s i n g u l a r

of t h i s

gauge

paper

fields

is to g i v e

(curvature

characteristic

classes.

Such

of n o n - a b e l i a n

harmonic

integrals

of m e r o n s

Let

forms

a formulation (cf.

formulation

singularities)

m a y be r e g a r d e d

[8])

and

relates

and

their

as a t h e o r y

to the

theory

([9],[12]).

M be a s m o o t h m a n i f o l d ,

Then we consider

the

following

G=GL(n,C)

sheaves

the

sheaf

of g e r m s

of c o n s t a n t

Gd:

the

sheaf

of g e r m s

of

~I:

the

sheaf

of g e r m s

of m a t r i x

d@

Since

+ eAe

= 0

a matrix

over

valued

if d e + ~ ^ e = O ,

smooth

the g e n e r a l

over

Gt:

and only

a mathematical with

linear

group.

M

G-valued

G-valued valued

maps

maps

o v e r M.

o v e r M.

l-forms e such that

M.

l - f o r m 8 c a n be

setting

t(g)=g-ldg,

locally we get

written the

as g - l d g

following

if

exact

153

sequence

of

(non-abelian)

(I)

0 NOTE

sheaves i

> Gt

I: For an a r b i t r a r y

can d e f i n e

the same

exponential

map,

sheaves

f > p/~1

> Gd

o v e r M.

If exp(~)

is exact.

Here

hold

for t h e s e

NOTE

> U(n) t

> U(n) d

/ I is the sheaf of g e r m s

But

in the h o l o m o r p h i c

category.

manifold,

exp m e a n s

skew symmetric of this p a p e r

we do not state

we can d e f i n e Gto

the

> 0

Most results

T h e y are d e n o t e d

~ , we

(I). For e x a m p l e ,

of a H e r m i t i a n

for s i m p l i c i t y ,

2: If M is a c o m p l e x

as

Y>/I

l - f o r m 8 s u c h that d 8 + 8 ^ 8 = 0 . sheaves.

= G, w h e r e

sequence

the s e q u e n c e

matrix valued

0

Lie g r o u p G w i t h the Lie a l g e b r a

we get the same e x a c t

0

>

them.

the same

and ~ L

sheaves

instead

of

G d and ~ I .

Our f o r m u l a t i o n induced

is b a s e d from

on the f o l l o w i n g

exact

sequence

mology

sets

(2)

0 _ _ > H 0 ( M , G t ) i >H0(M,Gd)_~_~ >H0(M, ~%1) 6 >HI(M,Gt)

first

6 t e r m s of this e x a c t

sequence

together

But the last 3 t e r m s

mology

sets HI(M, ~ I ) , H 2 ( M , G t ) and H 2 ( M , G d ) , s e e m to be

We k n o w that a 0 - d i m e n s i o n a l global

classes

sections;

of)

HI (M,Gt)

G-bundles

Hom(~I(M),G), ~I(M),

the

the

set of

fundamental

the set of g l o b a l

integrable

set of a sheaf

G is r e g a r d e d

of M,

of)

is a F u c h s

6(e)

type e q u a t i o n is g i v e n by

identified

with of

H0(M, ~ I) is

o v e r M and the e q u a t i o n

dE + Fe : 0 , ~ ~ H0(M, ~ql)

Here

(equivalence

representations

in G. By d e f i n i t i o n ,

connections

is the set

to be a d i s c r e t e

H I ( M , G t ) is a l s o classes

[8],[10],

of the c o h o new.

and H I ( M , G d ) are the sets of

(equivalence

group

(cf.

the d e f i n i t i o n s

cohomology

over M, w h e r e

or a Lie group, r e s p e c t i v e l y .

group

with

had b e e n k n o w n

[11]).

of

>

~ >H I (M , ~ I ) 6 >H2(M,Gt)--~-->H2(M,Gd '~ )

i >HI (M,Gd)

The

of c o h o -

(I)

o v e r M w i t h the m o n o d r o m y

, representation

6(8)

.

154

6(e)

=

and regarded

{ h u h v - 1 1 ~ H I (M,G t)

, eIU =

~(h U)

to be an element of Hom(VI(M),G).

been shown that tr(8^.~.^G)

is a closed

hu-Idhu

=

,

In this case,

it has

form over M for any p and set-

ting BP(e) : the de Rham class of

we see that BP(8) 4 H2p-I(M,C)

is a monodromy p r e s e r v i n g

variant of the equation

dF+FS=0

global

BP(e)

solution over M,

denote by e p 6 H2p-I(M,Z) BP(e)

([3], cf.

the

homology nition

of the cohomology

homolgy H

&

of the map

1

1

2-dimensional ([5])

and others,

from the c o r r e s p o n d i n g

(I). For example,

set defined by

HI(M, ~&lI) must be regarded as the set of of)

singular

of characteristic

HI(M, ~7 I) are given in 2.. The definition a natural forms

(cf.

extension

of the definition

[6]). If an element

its characteristic

class

situation

for the elements

of the characteristic

class together of

class

from a G-bundle But at this

meaning of H2(M,Gt ) .

0

in the following

0

0

0 --> ~ t --> ~ d --> ¢I --> 0 0 --> It

--> ~d

--> ;I 0 ,-->

0-->Z

-->Z

-->0

f 0

0

is [

,

~ . The 6-image of an

its singularities.

(I) is imbedded

diagram

if we define

H2(M,Gt ) may be

of Chern class by curvature

is the Chern class of

we do not know any geometric

our co-

by the sheaves

(some equivalence

of HI(M, ~q I) comes

element of H I ( M , K ~ I) must evaluate

If G:GL(I,C)=C ~,

classes

(M,G t ) co-

(I).

gauge fields over M. The details of this

with the definition

2

our defi-

In fact,

are not defined absolutely

but defined by the sequence

and

~

non-abelian

2(M,Gt) using the sequence 0 - - > G t - - > G ~ - - > o ' ~ l ~ - - > 0 ,

different

then

":H (M, ~1~ )-->H

from these definitions.

sets H2(M,Gt ) , etc.,

Gt, etc.,

in-

for any p. In fact,

of H~(G,Z),

sets HI(M, y~1) , H2(M,G.)

sets had been defined by Dedecker different

deformation

([3]).

in I.. We note that although

is slightly

class

(2p-1)-th generator

H2(M,Gd ) together with the definition are given

,

[7]). If this equation has a

is an integral

is equal to F*(e P) if e:F-IdF The definitions

(-I)P-I tr(e~--2p-1--n ....... ^8) (2 F ) p

commutative

stage,

155

Here diagram,

@I

means

we can

the sheaf of germs of c l o s e d

rewrite

(2)

as

l-forms

0__>H 0 (M,C~)__>H 0(M,C#d)_>H

0 (M,¢I)

H p+I (M,Z)

The to take

corresponding the

) = H I (M,C)

= H P ( M , C { d)

, H2(M,C)

, p = 1,2

commutative

following

0 -->

diagram

= H I (M,¢ I)

0

Gt

> Gd

d e are

sheaf

of g e r m s

the m a p s

given

>

>

de(f)=

1__j____(exp(f))-Id(exp(f))

>~I

> 0

of m a t r i x

as

some

smooth

maps

the

problems

kernel

sheaves.

and we o n l y

_

I

e-mT~v~fd(e2~Vr~f

But

treat

to h a n d l e

the p r o b l e m

this

diagram,

to r e g a r d

to r e f i n e

l-dimensional

this

there

H I (M, ~ I )

HI (M, ~ I) as the

set of

singular

gauge

fields.

•

The

) ,

object.

we e x p r e s s

But

over

2~g:7

2-dimensional

In 2.,

valued

,

2~v~0d-->}d

Here

this

.

expt expI2

exp

Using

form 0

M,

M.

6 >H I (M,C{)__>

- - > H 2 (M, Z) - - > H 2 (M, C ) - - > H 2 (M, C ~ ) - - > H 3 (M, Z) H0(M,¢I)/dH0(M,Cd

over

follows

formulation, non-abelian

we n e e d Poincare

a non-abelian lemma

is the

/

Polncare fact

that

lemma. e is lo-

cally

w r i t t e n f - l d f if and o n l y if d e + e ~ e : 0 . The 2 - d i m e n s i o n a l non. a b e l i a n P o z n c a r e l e m m a seems to take the f o l l o w i n g form: L e t ~ be a /

matrix

valued

if

O

for

some

2-form.

satisfies e. At

Then

~

the B i a n c h i

least

in the

is l o c a l l y identity,

real

written

that

analytic

as d e + e r e

is d ~ = [ ~

category,

if a n d o n l y

,e] = ~ A e

it seems

- e~

that

the

156

following also holds: If ~ satisfies the Blanch± identity, ~ is locally written as P FQ, where E is a (finitely) many valued l-form such that d ~ = 7 ~ 7 = 0 and PQ = I. But at this stage, these are only conjectures. These formulations and thcir relation with Yang-Mills equations are stated in 3.. The above formulation starts from the differential operator d. But from the point of view of connections of differential operators ([I], [2]), such formulation is possible starting from an arbitrary differential operator. This is stated in 4..

I.

DEFINITIONS OF HI (M,D~I) , H2(M,Gt ) and H2(M,Gd )

f 3 As usual, for a locally finite covering t]~ = ~ Uil. of M, we denote by cP(hA,F) the set of p-eochains with coefficients in F defined by ~ . Here F is a sheaf over M. We set CI( ~ ' G ~ ) a

=

Igij / gii

= e, the identity map, gij-gji-

II ,~ is

t or d,

C1a(~' ~1)= {6Oij/ &)ij : ~(gij)' {gij } &Cl} ( ~a' G d ) DEFINITION: We define the map 6:C1a(t~,~1)-->C 2 t~, ~I) and the set zl (h~, ~I) by

6(Q)ij k = U j k -

60ik + gkj63ijgjk,

6Oij = ~ (gij) ,

z 1(t~,fn I) = I~ 18(u) : 0 I NOTE: { gij I is not determined uniquely by f4 1g01ig~ I" The condition 6(&)) = 0 means 60_._.-i3_.i.+gi_ 63: .g C3(~,Gd)

and the

(8[ c) i0iii2i3 -I = gi0ilCili2i3gi0il

-I ci0ili3Ci0i2i 3

C = { Ci0ili2] ( C 2 ( ~ , G t )

,

[ :

Z2(~,Gt ) = { c 6 C 2 ( ~ , G t ) I 5{c=e LEMMA 2: (3) (4)

= {gij]

,

{ gij] { CIa(~,Gd ) for some

be in CI(0~'Gd ) ' a

6~ (6[) = e , if 6 ~(C 2 ( ~ , G t)

I ~ @ C a ( ~ , G d) I

Then we have

,

-I 6 C 2 ([](,Gt) , if 6[c = e gi0i ICil i2i3gi0il LEMMA 3:

(5)

Let ~

-I ci0ili 2

If 6{c = e and a = [aij I ~ C I (]J~,Gt) satisfies

(gi0i1-1ci0ili2) (gi0i2ai2i3gi0i2 -I)

i) = (gili2ai2i3gili2 -

(' 'gl0il -Ici0ili2)

' { : Igijl '

then, setting (6)

c'

i0ili 2

-I e , -I = a. g. . a. . a, 1011 i011 ili2gi0il 10ili 2 ioi 2

we have 8 ~ , (c') = e.

158 NOTE:

If c = 6[ , the first equality of (5) always holds.

DEFINITION: cohomologous

Let c and c' be in Z2(t~,Gt ) . Then we call c and c'

if there exists a = laijl 6 CI([~,Gt ) such that a satisfies

(5) for [ , 6[ c = e, and c' is given by By lemma 3, the cohomologous

(6).

relation

is an equivalence

on Z 2(t~,G t) . We denote by H 2 ( ~ , G t) the quotient this relation. [,r'

If ~

= {. Vj lj ~ J _ I

relation

set of Z 2 ( ~ , G t) by

is a refinement of [~= ~{ U i { i ~ I I

and

: J-->I are the maps such that V.] C U r(j)~Ur,(j ) , we set -I aj0j I

c ~ ( j 0) r'(j0) r'(jl )c T(j 0) %-(ji ) T'(jl)

Then a = laijl 4 C ] (]{,G t) gives the equivalence between ~{(c) and T~'(c) by (4). Hence we can define the limit set of H2(M,Gt ) in H2(Oi, Gt ) with respect to ~

.

Take c and a from C2( O~,Gd ) and C1([]Y,Gd);respectively,

we can

define H2(M,Gd ) in the same way as H2(M,Gt ) . By lemma I and

(3), if

~ = IOijl

is in Z 2 ( 0 ~ , ~ ] ) ,

set

Q ij = ~ (gij)' 6 { is in Z 2 ( ~ , G . ) and its class in H2(b~,Gt ) is determined by the class of ~ in H I ~ [ ~ , ~ I ) . Hence we can define the map 6:H] (M,~I)-->H2(M,Gt) . On the other hand,

the map

i~ :H2(M,Gt)-->H2(M,Gd ) is naturally defined. without the assumption THEOREM

I:

Then,

since

(3) holds

6 {6 C2(t~,Gt ) , we obtain

The following

sequence

is exact

HI(M,ctli~>HIIM,Gdlf~>HIIM,~ZII6 >n2(M,Gt) ¥

i

2.

>H 2 (M, Gd)

CHARACTERISTIC

CLASSES

LEMMA

&] =

4:

Let

6(g)ijk

~ (~) be in Z 2 ( b ~ , ~ I ) . Then we have

163ki6(g) ijk : t0ki

'

~ =

{gij 1

159

COROLLARY: exists

Let

a collection

~

and

~

be the same as above.

of matrix valued

l-forms {8il

Then there

such that

-I (7)

~ ij = 8j - gij DEFINITION:

cohomology

We call {Sil

8igij a connection

class of ~ . The curvature

form

form of ~

~

: { ~il

or < ~ > , of {Sil

the is de-

fined by (8)

0 Connection

form {8i[

i = de i + ei^8 i

forms of < ~ > are not unique.

of ~ , any other connection

the following

lemma

But fixing a connection

forms of < ~ >

are determined

by

5. (

LEMMA another

5:

(i) If ~8i~

connection

is a connection

(ii)

~j

If { 8 ~

~' = { ~'ij I is given by setting

= gij -I ~ igij

is a connection 6:

its curvature

Let form.

leil

(9) Hence

~j

form of

~

= I J~jl

and

o'

is determined

: g i j - 1 ~ igij,

c------p --~ tr( ~ i ^ . . . ~ i

form of

~(hi))hi -I

be a connection

By definition,

form of and ~ =

[~il

2p-form over M

by < ~)>.

we have . [.0i'8i] . .

d ~i

) is a closed

tr(~' i.... A ~ ' i ) = t r ( ~ i A . . . A ~ i

if 8' i=Si + ~ i '

~(gij )

is

~(hj)+gij -I ~ (hi)g ij )h i -I' then

2p-form

If (9'i=hi(8 i- ~(hi)) hi-1 , we have Hence

~ij

Then tr( ~.~7.. z p. ~ ~i ) is a closed

and its de Rham class PROOF:

'

is a connection

O i'j = h i ( ~ i j -

8' i = h i (8 i -

LEMMA

t0 and I 8'i%

form of tJ , then

e' i = e i + ~ i '

8 '= [e'il

form of

~ j=gij -I ~ igij,

@i

8 1 - 8.1 ~ 1.

over M.

~' z'=de''+8''^el ,i=hi~'h'-ll i l

) in this case.

On the other hand,

set ~ i'= d 8 '.+8',^8'. i 1 l' then we get

160

tr(~i~

....

) : tr(~i^

Because t r ( ~ i ^ ~i )=tr([ ~ i,Si]):0, have the lemma by len~a 5. NOTE

1:

This proof

2-forms{ ~il

satisfies

also

.... ~ i ) + exact

form.

[~ i,ei]: ~ i ^ e + e i ^ ~i"

shows that a collection

Hence we

of matrix valued

(9), then t r ( ~ i ..... ~ i ) is a closed

form over

M.

NOTE

2:

If

~

= {~ij I

is in Z] (t~,}9~I) , then we have

•

tr( ~ i 0 i l A ~ i i 1 2 ~

..

.A ~ i p _ l i p

where cP is the sheaf of germs of closed homology

class

an element

of this cocycle

Since

p-forms

is determined

of HP(M,¢P)=H2P(M,C)

de Rham class

for < ~ >

for this element

the ring of even degree

forms

is commutative, ~)

their de Rham classes

are determined

by < ~ > .

We denote

by c P ( < ~ > )

of d e t ( I + t / 2 ~ / ~ )

< ~ >. The total Chern class THEOREM is the p-th

2:

(cf.

(complex)

[6])

c() (i)

EXAMPLE space,

I:

cP()

,

1]

=

is defined

~*([),

I

] 0

0

~m

~ij

by E cP(< w > ) . P

[ is a G-bundle,

cP()

~ . . is a Hermitian skew z] is a real class for any p.

Let M b~CP m, the m-dimensional

AI ~ij

6 shows

of { .

If each

l-form,

lemma

forms over M and

it the p-th Chern class of

of < ~ >

{~I ..... ~m ~ a set of complex

60,

the co-

the de Rham class of the p-th

If < ~ > :

Chern class

matrix valued

are closed

and call

(ii) symmetric

over M. Since

by < ~ >, we can associate

for any p. The corresponding

of d e t ( I + t / 2 ~

DEFINITION:

,

is tr( ~ i ~ ...A ~ i ) .

that the coefficients

coefficient

) ~ zP(o~,¢ p)

numbers,

complex

and let

projective

161

m (I+ A i t ) = I+~It

+'''+

~m

tm

'

=

~ij

dz. ~ z

dz. 1 z.

3

i

i=1

Then

~

: fl60ij~ ]defines

c

m E ~ ep p:0 P

=

EXAMPLE defined

2:

uij AO

of H I (CP m, m I)

r-- P--m , eP = eu...ue,

(cf.

for a c o m p l e x

(gij)

Then

an e l e m e n t

[9])

Let

number

= A

~ (gij)

and

that

e is the g e n e r a t o r

{gijl £ HI (M,Gd) A

such

be

such

of H ~ ( c p m , z ) .

t h a t {gij ~}

is

satisfies

, gij

8igij

= ej

,

= f (gij) : ej - gij leigij

( ZI(0[,~

I) and

c()

= E APcP(). P

3.

2-DIMENSIONAL

The

NON-ABELIAN

constructions

elements

of H I ( M , ~

I)

of c o n n e c t i o n in 2. we may

sional

de R h a m

theory

blem).

In this

section,

O~ I be the

Let 2

the map

given

(and

sheaf

the

}~

image

0

10)

NOTE:

2-forms

over

and

curvature

theory

must

informations

of

forms

as an e x t e n s i o n

smooth

be

for

the

2-dimen-

the Y a n g - M i l l s

for this

matrix

of

pro-

situation.

valued

l-forms

over

by (e) : de

sheaf

--> m I -->

If the

introduction)

forms

some

of g e r m s

THEORY

regard

its H o d g e

we add

2

and

DE R H A M

of

+ (9,,.(9 ,

I

~i

by

~2

2-dimensional

is true,

2

. Then

>~2

__>

non-abelian

~I 2 is the

M for w h i c h

~

sheaf

the B i a n c h i

we h a v e

exact

sequence

0

Poincare

of g e r m s

identity

the

lemma

of m a t r i x

holds

(for

(stated in the valued

some matrix valued

l-forms).

Let valued by

h be a s m o o t h

I- and

G-valued

function,

2-form, r e s p e c t i v e l y .

Then

e and to d e f i n e

~

are

the m a t r i x

h-actions

for e and

162

h(e)

= h(e - ~ ( h ) ) h -I

we can give Gd-actions

(11)

on ~ I ,

~2(h(e))

(10)

h(@)

= h @ h -I

~ I and ~/~ 2. Since we have

= h(~2(e))

,

is also the exact sequence as Gd-sheaves.

(11)'

£2(~h)

= h(~2(8))

By definition,

(11) also shows that

if and only if [ f (h),e]

= 0, eh:heh -I

Therefore,

holds if and only if h is a constant map. to define Gt-actions on ~ I and ~I by e h, (10) is exact if

we regard

~I

~I

and

6h=h(e)

~I

to be Gt-sheaves

and ~ 2

Considering actions, the 0-dimensional and j9~2 must take the forms

I

eolij : cj - g j i ( e * l '

f=

to be a Gd-sheaf.

coboundary maps on #}%1,

{ gijt

I'

= t or d

Using these coboundary maps with [ ~ C ! ( ~ , G ~ ) , we get the 0-dimensional Un 2 cohomology sets H 0 ( M , ~ I ) d , H0(M, ~I) d and HV(M, ~ ). Similarly, the 0-dimensional

cohomology

sets of M with coefficients

obtained to use these coboundary maps with by H0(M, }~I) t and H0(M, ~I) t. By the definitions of G~-actions O commutative d i a g r a m , w h e r e B 1 (M, y~l) 0-->H 0 (M, ~ I )d

in k}~I and

31

[ g cl ( ~ , G t) are denoted

and lemma 4, we have the following is

defined

as

usual

i >HO(M' ~1 )d ~,2 >H 0 (M ,m2 )

T

(12)

0__>BI(M,~GI) In this diagram,

i >ZI(M, g~1)

>HI(M, ~11)

>0

we set dR(HI (M, ~#~I)) = H ~ R ( M , ~ 2 ) . This is the set of

singular gauge fields over M. If the 2-dimensional

non-abelian Poincare

lemma would be true, we have 0 HdR(M, ~2)

= H 0 (M, ~2)

In fact, by note I at the end of the proof of lemma 6, we can define Chern classes for the elements of H0(M, ~2) . Similarly, dR(ZI(M, J171)) = 0 I) = HdR(M'~ d must be the set of singular gauge potentials over M.

163 On the other hand,

if

~ 4

C1((.~, C~1), we

6~ ~ijk : ~ jk- ~ik+gjk -160 ijgjk' Z 1 ( ~ , ~1) As in I., we call

set

~ : {gij] ~ cIa ( ~,%1

= { C9(C I (0~, ~I) [ 6[cO : 0

for some

Co , co' ( ZI(0~, ~I^) cohomologous

h = [hil ~ C0(O(,Gd ) and e = {ei< £ cU(0~, ~I)

r }

if there exists

such that

6(g) ijk-lei 6 (g)ijk = 8i ' 6[ ~ = 0, ~'lj = hj( ~ij

- @j + gij -18igij )hj

Using this relation, usual.

But

we can define the cohomology

this set vanishes by

The discussions (12)'

-I

set HI(M, ~I)

(a modification

in 2. shows that the sequence 2

0__>H0(M, y~1)t i >H0(M, ~ 1 ) t

6 >HI(M,~/~I) z

0 (M, 2) >HdR ~

>HI(M,~ I) = {0{

is exact.

Using these sequences,

following

2-step problem:

the Yang-Mills

problem splits into the

(i) To get the Hodge theory in (12)'

sional non-abelian

/

•

Pozncare

lemma is true,

Let M be compact and

@

a

valued 2-form in H 0 ( M , ~ 2} such that then

~

(12). If the 2-dimen-

it seems that the following

Hermitian

locally as P EQ, where

NOTE: In the holomorphic exact sequence 0--> ~ I i~ >

skew symmetric matrix

~ j=gij - I ~ igij ' /gijI6 C~ (~,U (n) d )

is a solution of this l-st stage problem

expressed

>

,

(ii) To get the Hodge theory in(the upper line of) also holds:

as

of) lemma 4.

~ is a harmonic category, 2 92 > ~

if and only if

~

is

form.

similarly as (10), we get the -->0. By this sequence, we

get the exact sequence

0 - > . °IM,

1

It->.

6 >H I (M, ~ I

I }

similar to

(12) '. But H I (M, ~ I

iu~

is

(< qO>)

the

obstruction

>H I (M, ~Ic0 )

>

,

) does not vanish in general and class

to

have

for

< q~> a h o l o m o r p h i c

con-

164

nection

(cf.

[4]).

characteristic (cf. note

If M is a c o m p a c t

classes

on HI(M, 6qI

2 at the e n d of the p r o o f

to H P ' P ( M , C ) .

On the o t h e r

characteristic

classes

hand,

by u s i n g

K~hler

manifold,

) using

tr(iJ

of l e m m a

if i *

we can d e f i n e

( ~ ) .... A i ~

6). T h e s e

classes

the

(~)) belong

( < ~ >) : 0, we can d e f i n e

holomorphic

curvature

forms.

T h e y be-

long to H 2 p ' 0 ( M , C ) .

4.

THE G E N E R A L

DIFFERENTIAL

OPERATORS

L e t E I and E 2 be c o m p l e x C~(M,E2 ) a differential G-valued

D g = g-1 (D ® IH)g m e a n s

By d e f i n i t i o n , (14)

It is k n o w n

~D(g)

the o p e r a t o r

>

H:C n, a s m o o t h

on C m ( U , E i ~ H),

i=I,2,

: Dg - D @

g i v e n by

JH

((D ~ I H ) g ) u = ( D ~

IH) (gu).

we get =

( ~D(g))f

A G-valued

that t h e r e

by D and a c t i n g

c(D)-class

+

~D(f)

map g is c a l l e d

to be of c ( D ) - c l a s s

if

}711 _ _ > D1

7:

the sheaf and

set

D2

functions

of g e r m s ~ D(Gd)=

such t h a t

element

such that

of c ( D ) - c l a s s

~ I D. By

G-valued

to Gc(D2 ) , t h e r e

c

diagram

DI

i >

if r ( D ) f : 0 .

functions

(14), we h a v e

the f o l l o w i n g

0 - - > G c (D 2 ) - > G d

g is a

of g is a s o l u t i o n

f on U to be c ( D ) - c l a s s

If Gc(DI ) is e q u a l

~1

valued

if e a c h m a t r i x

We call a f u n c t i o n

We d e n o t e

LEMMA

is a s y s t e m of differential operators r(D) de-

on s c a l a r

map if and o n l y

([3]).

o v e r M by Gc(D)

j:

operator

setting

= 0.

termined

r(D)

IH)g,

?D(gf) DEFINITION:

~m(g)

o v e r M, D : C ~ ( M , E I )

set

(13) (D ®

bundles

o v e r M. Then,

m a p g on U acts as a l i n e a r

and we can

Here

vector

operator

~,fl

D2

>0

is a b i j e c t i o n is c o m m u t a t i v e .

of

165

By

(14) and lemma 7, we have THEOREM 3:

(i) There is the following exact sequence

0__>H0(M,Gc(D))__>H0(M,Gd) -->H I (M,G d) ~

~D >H0(M,DtD ) 6 >H I (M,Gc (D) )__ >

>H I (M, ~9~ID) 6 >H 2(M,G e ( D ) ) _ > H 2 ( M , G d )

(ii) The cohomology

sets H0(M, ~ I D) and HI(M, ~I D) are determined

by Gc(D). That is, if Ge(DI ) is equal to Gc(D2 ) , there are bijections j~ :Hi(M,Gc(DI))-->Hi(M,Gc(D2)), NOTE:

i = 0,1.

As in I., the oohomology

absolutely by Gc(D).

set H2(M,Gc(D))

is not determined

Its definition depends on the sequence 0-->Ge(D)-->

-->Gd-->~ D--> 0. EXAMPLE: If M is a complex manifold and D is [ , then Gc(D) is G ~ , the sheaf of germs of holomorphic G-valued maps over M and ~ ~ (g) is the

(0,1)-type form g - l [ g- In this case,

germs of matrix valued In general•

(0,1)-type

setting

ferential operators

~I

is the sheaf of

forms O such that

~e+e^8

= 0.

k-1 ~ E 1 ® H,E 2 ~9 H the sheaf of germs of dif-

from C~(U,EI ® H) to C~(U,E 2 ~ H) with the order at

most k-1

~ I D is a subsheaf of ~ k-1 if ord D=k We can ' E ~ H,E 2 ~ H " define the cohomology set H I" ~k-11 similarly as H I (M, 91). (M, ]3EI (9 H,E 2 ® H )

Since this set vanishes, ~ D ([) ' ~ ~ HI (M'Gd)'has a trivialization in k-1 H I (M, ~ El ~ H,E2 ® H ) . This trivialization is the connection of D with respect to

[ ([I]•[2]).

If D=D I is imbedded in the sequence DI (I 5)

C m(M,EI)

D2 >C~(M,E 2)

>C ~(M,E 3)

Gc(DI ) : Gc(D2 ) , ord D I : ord D 2 take the connection el= lel,il

and 692= /02,il

j (t0) 6 Z I (M, ~ID2) , we define the curvature

,

,

of 6o 6 ZI(M • ~ I DI ) and (operator)

166

(16)

~i

= (D2 ®

IH)@1,i

+ @2,i(D1

= (D 2 (9 1H + e2, i) (D I ® NOTE

I:

If

(15)

ting D.,8 = Dj ~ 3 j

is a d i f f e r e n t i a l

is a d i f f e r e n t i a l

complex

IH)

+ e2,iel,i

I H + (9I,i ) - (D 2 (9 1 H) (D I (9 1 H)

I H + ej, i , j=I,2,

DI,81 ~ [)-->Cm(M,E2

Cm(M,EI

~

complex

the

D2,e 2 ® { )-->C~(M,E3

if and only

(f),

and ~ = ~ D I

if

~ ~)

~ ( e 1 , @ 2) is equal

to 0.

NOTE 2: If El=E2, set E3=E I and D2=D I in (15), we define vature o p e r a t o r of e, a c o n n e c t i o n of ~ , to be ~ (e,8). By using to the one (general)

curvature

operators,

If Cc(D), the f o l l o w i n g

the

C p'D

sheaf

is a subsheaf

over M and each

of germs

dD

of C p, the sheaf

d D is a l-st order

Hi(M, ~/~ID), i=0,I,

as follows:

i=0,I,

of c ( D ) - c l a s s

dD >cI,D - >c2,D

characteristic

and

constructions

because

we have no

functions

over M, has

resolution

then we can define duction

similar

complicated

the cur-

lemma.

0 - - > C c ( D ) - - i >C d where

we can give

in 3.. But they are much more Poincar~

set-

sequence

as the e l e m e n t s

of germs

differential classes

Using

in 2., we can define

> ...

the

operator

same m e t h o d

p-forms

(cf.

for the elements

characteristic

of H°dd(M,Cc(D))

of smooth

[3]),

of

as in the

classes

intro-

on H i ( M , ~ I d D )

and H e v e n ( M , C c ( D ) ) .

Then,

,

since

Cc(D) = Cc(dD ) by assumption, we get Gc(D)=Gc(dD). Therefore, there are b i j e c t i o n s j e :HI(M, ~/[ID)-->Hi(M,)gYIdD) , i=0,I, by lemma 7. Then we set BP(e)

= BP(j~(e))

cP()

the

(H2p-I(M,Cc(D)),

= cP(j~())

8 (H0(M, ~ I D)

6 H2P(M,Cc(D)),

EXAMPLE: If M is a c o m p l e x sheaf of germs of h o l o m o r p h i c

,

~ H I ( M , ~ I

D)

m a n i f o l d and D= 9, Cc(D) is C ~ f u n c t i o n s over M, and the above

lution is the D o l b e a u l d t complex. Hence P c ( < ~ > ) is in H 2P(M,C ~ ), r e s p e c t i v e l y .

BP(e)

is in H 2 p - I ( M , C ~ )

In this

case,

denote

by

, resoand

~P'q

167 the p r o j e c t i o n on the

(p,q)-type part, we then have the following cormmta-

tive d i a g r a m with exact lines and columns. 0 0 0

f

f f o,1 0-->m ~ -->~ i_ _ > ~

t

t

O-->G~

>

>d d - -

;G t

0 Especially,

>0

~=1

>0

>0

=

O-->G t

I

0

if M is a compact K i h l e r manifold,

class and c P ( < ~ >) is a ( O , ~ - t y p e

class.

BP(@)

is a

(0,2p-1)-type

In this case, we also have

the f o l l o w i n g formulas 770,2p-I(BP(e))

~0'2P(cP())

: BP(it0,1(8))

,

: cP(7[0'I~ ()) , eHI(M, ~/[I)

As a special case of this last equality, 7TO'2P(cP(~))

, ~ 6 H0(M, ~ I)

: cP(~

~

(~))

,

we get ~6 HI(M,G d)

REFERENCES

El]

Andersson, S.I.: Vector bundle c o n n e c t i o n s and lifting of p a r t i a l d i f f e r e n t i a l operators, D i f f e r e n t i a l G e o m e t r i c methods in M a t h e m a t i c a l Physics, Clausthal, 1980, Lecture Notes in Math., 905, 119-132, Berlin, 1982

[2]

Asada, A.: C o n n e c t i o n of d i f f e r e n t i a l operators, Shinshu Univ., 13 (1978), 87-102

[3]

Asada, A.: Flat c o n n e c t i o n s of d i f f e r e n t i a l o p e r a t o r s and odd d i m e n s i o n a l c h a r a c t e r i s t i c classes, J . F a c . S e i . S h i n s h u Univ., 17 (1982), 1-30

[4]

Atiyah, M.F.: C o m p l e x a n a l y t i c connections T r a n s . A m e r . M a t h . Soc., 85 (1957), 181-207

[5]

Dedecker, P.: Sur la c o h o m o l o g i e non abellenne, Math., 12 (1960), 231-251, 15 (1963), 84-93

[6]

Dupont, J.L.: C u r v a t u r e and C h a r a c t e r i s t i c Classes, Notes in Math., 640, Berlin, 1978

J.Fac. Sci.

in fibre bundles,

I,II, Canad.J.

Lecture

168

[7]

Flaschka, H.-Newell, A.C.: Monodromy- and s p e c t r u m - p r e s e r v i n g deformations, I, Commun. Math. Phys., 76 (1980), 65-116

[8]

Gaveau,

B.:

2 e series [9]

Integrals harmoniques

106

(1982),

f

,

non abel±ennes,

Bull. Soc.math.,

113-169

Manin, Yu.I.: Gauge fields and cohomology of analytic sheaves, Twistor Geometry and Non-Linear Systems, Primosko, 1980, Lecture Notes in Math., 970, 43-52, Berlin, 1982

[10]

Oniscik, Doklady,

[11]

Oniscik, A.: Connections with zero curvature Sov. Math.Doklady, 5 (1964), 1654-1657

[12]

Schaposnik, F.A.-Solomin, J.E.: Gauge field singularities noninteger topological charge, J.Math. Phys., 20 (1979), 2110-2114.

A.: On the c l a s s i f i c a t i o n 2 (1961), 1561-1564

of fibre spaces,

Sov. Math.

and de Rham theorem, and

YANG-MILLS ASPECTS OF POINCARE GAUGE THEORIES

J.D. Hennig

Institut f~r Theoretische Physik Technische Universit~t Clausthal Clausthal, Germany F.R.

I. Introduction

Apart

from

the

of the quantization classical

General

occurrence

of

singularities

and

the

outstanding

solution

problem there might be seen at least two reasons to modify

Relativity

(GR) by incorporating

structure

elements

of Yang-

Mills gauge theories (YMT), possibly within the wider framework of supergravity: - With

the

exception

of

gravity

all

known

fundamental

interaction

types

seem

to fit in the general YM scheme of 'internal' symmetry groups. - There

are

classical

striking (i.e.

parallels

non

between

quantized)

YM

the

step

theory

and

by the

step

construction

transition

from

of

a

Special

Relativity (SR) to GR ; in particular we mention: GR

YMT - Invariance lagrangian 'internal'

of

a

with

certain respect

symmetry

Lie

Invariance of SR with respect to

matter to

an

group

G

the 'external' Poincar6 group.

of 'global'gauge transformations. - Covariant

I

formulation

of

the

I -

Covariant

formulation

theory by introducing 'compensating'

introducing

i)

potential fields

coefficients ~

the

of SR by

Levi-Civita

o~

~

with zero field

~ o ~ and

strength

of the flat metric

ii) ~-orthonormal tetrads o

>

.~ Consequence:

Invariance

under

"=~ 'local' gauge transformations.

L

Consequence:

£

6L

Diffeomorphism cova-

riance.

- Generalization to potential fields

- Einstein's

with non zero field strength.

(EP) :

L

~Z

equivalence

Generalization

to

principle Lorentz

metrics g with non zero curvature.

170 According aspects

in

to these parallels

(gauge)

theories

following different trary

holonomic

being

the

i,k,.,

lagrangian

to

connected

types of gravitational

and

field

(contributes

there has been a continuous development of YM

of gravity

the right

potentials

orthonorma]

(matter hand

the introduction

(/~, ~ ,..

anholonomic

lagrangian)

side,

with

i.e.

denoting arbi-

indices ;

which yields

to the

of the

~{

( ~

)

the left hand side

'sources'

of the potentials)

of the field equations via the usual variational processes): potentials field lagr. Einstein 1915

g~

tetrad-

e~

field equations

~'

formalism~

YM 1954

1956

ECKS

e

f "~ '

1980

.

, ~

.

,1

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

R?

{-~"

.

.

.

.

-

.

.

,~ .

.

.

-

~ : ( y A ,D]., y~ ,e/') Q'~,,-O'~ ~'

' Qd'#-

W

~)", ~ "

PGTL

~ L ~ M variational the

~

of

The

course

procedure

following

correspondence between the several steps of construction in PGT L and PGTA:

direct

181 PG L

~

,

I

.

~-m( ~ A , ~ , =:

Z~

.

.

.

]

.

.

A .

.

,

~

,

.

.

.

.

.

.

.

D~yA)(de t ~ ()d~x ~

=.. ~7~( ~ A , d A~(P)

D

which i s

Ra~g(~

.... ~ )

=

~(}

.... ~)

g ~G

:= r'o£ ~ fa rD#~ ~b~

(A.2.2)

~p

---+

(~,D ~)

The partial derivative of ~

)

~ c-A~(P,F, ~),

v~

is said to be scalar, if

(A.2.1)

is defined by

~

L(p; %(p) . . . . ~ ( p ) )

~ ,..,

and a l a g r a n g i a n

ii)

depending on the 'fields'

~ o ( ~ . . . . S~ ) , f o r t h e mapping ( A . 2 . 1 ) and, i n p a r t i c u l a r ,

reduced to pairs of the form for all

P

:

~

in (A.2.1) with respect to the q-form

, where the mapping

A~(P,F , ~) ~ .. ~ A~(P,H,-C)

--~

A~?(P)

is given by

9~ For s c a l a r iii) := ~o

~

i ~k o

the n-q-forms ~

(~ .... ~)

Given 1 - p a r a m e t e r f a m i l i e s ,

O( := Vdo , the variation

~Y a r e of t y p e (F*, ~ * ) .

{#e}cA~(e,F,f) ¢9~( ~ . . . . 9")

.... :=

{%} ~

• a~(e,U,r)

~ (~, . . . . * ' )Io

,

186

of

~ ( ~ .... ~ )

contraction

iv)

is

yields

understood

Given

[~}

we get for scalar

3~

as

and

=

d~

=

A 9 ~~

~

usual

and

connection

~ (~,D~)

and similarly for scalar (A.2.5)

¢~

+

.. +

where

forms

d~

{~E~

@~^

~¢ ~

, where

:= d ~ A

'

fA

~'~ := ~"

with

, on

P

the scalar horizontal n-form

~ (D co)

~D ~

+ d(fco ^

9 k~

A.3. Invariance~ covariance and symmetries i)

Let

autvP field hy A

the

AutvP

Lie

X E autvP

:

P --~ ~

symmetry A AutvP

denote

algebra

of

the

group

then determines ,

vertical

automorphisms

automorphisms

an Ad-equivariant

of

P

of

P

; each

(~ -independent)

and

vector

function

p ~ - ~ ~(X~)

( infinitesimal

symmetry , reap.)

( X e autvP , reap.) with

ii)

of

infinitesimal

A lagrangian

~ ( ~ .... ~ )

~*~ on

of a connection

= ~ P

( LK~

~

on

P

is an

= 0 , reap.).

is said to be

passively gauge invariant if

~ ( ~ .... ~ )

actively

y (~ .... ~)

~(~.~

.... ~ . ~ )

and

~ (~ .... ~)

~* y(~,~

.... K , ~ )

,

gauge invariant if

gauge covariant

if

for all

~ ~AutvP

If

of

two

these

( ~ ,.., ~ )

( ~.

~* ~(~

.... W )

,

:= < * -4).

conditions

are

gauge invariant.

fulfilled Passive

(and

hence

all

gauge invariance

of

them),

is equivalent

we

call

to the

scalar property of The bundle version of Noether's theorem for internal symmetries now reads: (A.3.I) Proposition

Let of

{ ~ f } c AutvP ~o

with induced

& A~(P,F, ~)

be a l-parameter X ~ autvP

family of symmetries

, and let

~ (~ ,D ~)

, be gauge covariant with respect to

,

{~fff

and scalar. Then the Noether current (A.3.2) is an invariant horizontal (A.3.3)

1-form on P and conserved, i.e.

d ~<j~j$ for #

=

0

fulfilling the Euler equation

, D D__~

= (_1) 9 ~_~

187

Moreover, D ~ 3. ~~

for gauge invariant = 0

,

is tensorial •~~, ~

~

of type

= h x . ^ j ~~

~ ( + ,D~ )

fulfilling (~,Ad

' since

we get the 'covariant conservation

the Euler ~)

Lx~

.

The two kinds

= -h X A

3. ~ #

equ., where

law'

_ , ( ~ ' ( ¢ ) W~g ~ )

:=

of currents

are related

through

~I(%)

B. Affine connections i)

We

and affine

denote

by

LM = k] L ~ M

and

frames of the n-dimensional

AM

=

manifold

k] A × M

M

the bundles

with the natural

of linear

right

group

actions (B.I)

~

:

LM × GL(n,~R)

---> LM

,

(e,g)

~-:~

(B.2)

~

:

AM kGA(n,~)

---~ AM

,

((p,e);(t,g))

w--~

(

te

:=

t~ e~

,

eg

g = (g'k) e GL(n,[) LM

,

:=

(e~g~)

, e = (el)

As

connection

a

~(n,~)

=

4~@

{'~],

:

LM

{ ~;~}

- - ~ AM

,

are

1-forms

' ~

@'

k

~

constituting

Similarly in

group,

(B.3,B.4)

iii)

the

M ).

to

@; of

p

=

(p£)

product

¢

of

TM TM

, and

~

generalized

affine

, has a decomposition

~ ~ ~; of

the

a

~"

and

~ £ (n,~)

natural

embedding

~;

@

of

If LM

into

0

be

i

~,

::

+ part

~

being

on

LM

('soldering

form on

LM

, hence a linear connection on

subbundles

replaced

by

A~M c AM

the

form') and the

of the

and

generators

'linear' M .

J M ¢LM

, where

of

Lorentz

~;~

the

ke

~

~dx ~

~(e~) Moreover,

,

@ = ($;) :

@; =: e ~ =: e ~ d x ~

0

TM - - , TM

We call

+

l-form

'translational'

in (B.4) a local section

forms

~'

+

, the

on Lorentz

fields')

definition

determines (B.5)

l-form

= ~<j¢

For ~

('tetrad

endomorphism on

have

AM

bases

~ ~:

LM

a connection

for connections

2(~;~)
n.

¢,

operator

its e l e m e n t s

matrix

£

a given

For

c11

period

n.

is the

left

follows

from

range

[-M',M]

if cij

our purpose

we w i l l

assume

Let g . c . d

(M,n)

= N,

g.c.d

We

is

(M',n)

zero

= N',

n I • N = n and n I . N = n. The difference operator is called regular if the N quantities

ci,i+ M

for

I ( i ( N,

and

the

same

e.

The

with

all

.....

different

for the N'

c l+(n1-])M,i+niM

from

quantities

of P. V a n

periodic

a diagonal

Moerbeke

difference perodic

Jacobian

J(~),

where

det I ~ ~

- zId I

a n d D. M u m f o r d

operator

is the

~ Q( ~ ,z)

and different

operator,

Riemann

= 0. T h e

genus

space

in o u r

use

exists

algebra

a Lie

is l y i n g

K.A.S.

will

Hamiltonians Liegroup (-P'P)

be the

of -

decomposition

asserts

on the

G(~-P)

on

and hence

~

that

the

~

(N+N')

+

that

any

conjugation point

by the

is g i v e n

of

space

P . The

Killing

P

identifies

equations.

(K.A.S.)

~

(-P'P)

by

of the

curve [9]

2

for

of

~

of the

=~-P

of

(~) d e f i n e

the c o a d j o i n t

~/-P,

the

dual

to do If t h e r e

such that

~

pl,

the

commuting

action

f o r m K 0 is of m a x i m a l with

[I].

+ ~P, ~

Poisson

In o r d e r

theorem

submanifold

elements

Poisson

P±

given

construction

f o r m of the

the Kostant-Adler-Symes

in a c o a d ( ~ / - P ) - i n v a r i a n t

theorem

other

2

determining the H a m i l t o n i a n

t h i s we w i l l

~x

approach

asserts modulo

to a r e g u l a r

surface

(n-l) (M+M')

step

(vM.M)

R corresponds,

g = The n e x t

from each

Ci_ ( n ~ _ l ) M , , i _ n 6 .M,

difference

~

zero

[9]

Ci_M, , i_2M . . . . . .

1,i-M'

theorem

regular

are

. Ci+M,i+2M

rank of

of the on v ~ -p

240

For

any

~

ference

g ~,

operators

the v M . M

H ~ ~(~)

is a l i n e a r

flow

on

of the

difference

operators

In o r d e r and

to m e e t

the

lowing

two

types

are

considered

follow

afterwards, this

torus

both

allows

an

and hence,

isospeotral

the

Kac-Moody

structure

integration

requirements, of

the

through

deformation

algebra

while

the

in t e r m s

namely

operators,

that

by d i a g o n a l

transformation

us to c o n s i d e r

the

decompo-

regularity

of m e r o m o r p h i c

only

of

one

algebra

we w i l l

We r e m e m b e r

up to c o n j u g a t i o n the

surface)

dif-

J(~).

of d e c o m p o s i t i o n .

from

isospectral

set of J(~) . A n y H a m i l t o n i a n

that

action

allows

regularity

rators

gation,

coadjoint

complex

of

[9].

we have

the

position

consists

on P , d e t e r m i n e s

construction

determines

on the

~A

to an open

J (~)

sition

functions

through

corresponds

with

In this

orbit

(determining the same R i e m a n n

theorem,

veetorfield, which

the

the

type

decom-

consider

the

difference

matrices.

flows

the

ope-

As w i l l

under

of L a g r a n g e

fol-

conjudecomposi-

tion.

Let

h be a g i v e n

subalgebra

decomposition of b-

gonal

construction

representation

matrices

of

trace

of the

~

=

with

A 0 the

both

with

We

_f_ + b

define

(b) T o d a

the

the

operator

algebra

we will

choose

h is r e p r e s e n t e d

+

+

~'+ and

of a n t i s y m m e t r i c

following type

type

intersections

i = 1,2.

b +) the B o r e l

by d i a -

~

2 = A0+

matrices

and

~° i T }ti E gi gi ' i=I ~ the t r a n s p o s e d ,

to h.

(a) L a g r a n g e

The

(resp.

zero.

, ~ I = n

subalgebra

respect

difference

in w h i c h

-

Let

of g, b-

the n e g a t i v e (resp. p o s i t i v e ) r o o t s p a c e + (resp. n ) the m a x i m a l n i l p o t e n t s u b a l g e b r a

b+).

the

the m a t r i x

subalgebra

upon

of g and n-

(resp.

For

Cartan

constructed

with

decompositions: ~

=

d+

~

=

d

~ (-P'P)

will

J~ +

~

I 2

be d e n o t e d

by

J~-P

and

~ P i'

241 Proposition The

1.1.

following

[4]

projections

(a) for the L a g r a n g e p.i

f

(b)

p.i

morphisms.

type

$i Ai)

P

> ¢,

J{ pl I

for the

J~2

Poisson

- - > IP-1±1

P E i:O

=

are

Toda

=

•

PE

P

p-ql ( F i + 1 ~ i) ~1

i=O

type

-->J('

p-1 E P i=0 %p-I± q

P

along

their

are

taken

(~1+1Zi[.

complement

+

~i+1 Z-l)

~-Pm

1

follows

It

from

this

proposition

> ~ p+1 ~- -

f o r m an define

l'

inverse an

i

sequence

limit

i

of P o i s s o n

Poisson

s p a c e s J%_ we w i l l i

= together

with

the

coad(~f)-invariant P

( ....

use

]-p'

along

- -

spaces.

For

~-p+l .....

1

:

us to

~-1'~0

)

'

~ =

E ~-i A . Let E ~ i be a i=0 We d e f i n e g r a d = -PJz i ° K01 ~ d, w i t h

@f . F r o m

any H. 6 ~ (~) , the

allows

i

the

K.A.S.

theorem

xi (I)

This

the c o o r d i n a t e s

representation submanifold.

the p r o j e c t i o n

sequences

space

,lim~P ~ =~ i On b o t h

the

pZ _ _ > $ { p - l l -->J~

limit

inverse

that

flow

[grad Hi,~]IV

is d e f i n e d

.

by

we o b t a i n

[4]:

"

242

(2)

Let

Dt.

be the

derivative

along

the

Hamiltonian

vectorfield

cor-

1

responding (a) D r .

]

to H i . Then

Bj,~ .....

Lagrange

difference

type

operator

multiplication

by

k

This

in the c h o i c e

Some

trivial

maximal

If

choice

avoids

on the

O{

ponds tion

to an o p e n

"finite

P

dition

zone"

has

on

g r a d Hi]

have

of

(as a d i f f e r e n c e

the

:

~).

to be m a d e

corresponding for a n y p.

leaves

This

the

shift

bilinear

may

regular

yields

to e n s u r e

the

points

the

provided

that

K_I

operator.

coadjoint of J(~)

integration.

the

Riemann

{~ w h i c h

operator)

~]0

satisfy

from

the

and The

surface

the

right.

the

x 0

. 0

X

.

regularity

One

easily

X

I and

•

•

'X

°

~

applica-

:

•

are

is r e g u l a r .

:

.

the corres-

solutions

•

: "~ 0 :

be

is of

Then

action,

0

X

also

in-

form K_I=Res~=0K.

classes:

:

Because

flows

algebras.

difference

subset

~

zero.

by

of e l e m e n t s

following

to be

this

, determined

theorem

RI

R2

~

= 0

Hi, ~ .....

~z 6 ~ P

to a r e g u l a r

~

solutions

consist

we d e f i n e

, with

different

truncated

through

of the v M . M

M'

of the

adaptations

rank

[ ~ P corresponds

orbit

~ -I. < ~

is a d ( ~ ) - i n v a r i a n t ,

absorbed

the

Hi,

~) = D t . Q r ( Q ~ g r a d J

decomposition

by

variant•

3) Let

(~) one

I-2.

I) For the

2)

any H i , H j ~ ~

g r a d Hi - Dr, g r a d Hj + [ g r a d 1

(b) D t . Q r ( ~ x g r a d ± Remark

for

X

confinds

243

....

0

0

0

R3

: '~0

x x " " " x 0

li

6

=

and

X

~-I

:

Q

X

. x 0 0

"0 . . . .

R4

x x 0

0

=

: 90

and

. • 0

~-I • 0

0 0

X

x 0

For

the

Toda types regularity from the right will

regularity• the

. 0

X

It is e a s i l y

Lagrange

type

seen

is o p e n

that

and

the

dense

. . . .

X

automatically

set of r e g u l a r

in

~

if

~

imply

elements

is r e g u l a r

for

from

the right.

2.

INTEGRABLE

To d e s c r i b e

SYSTEMS

KdV-type

OF

a Hamiltonian

vectorfield

X H • , of

~

~ i' a l o n g

the

1

integral

curves

mic m o m e n t u m jet b u n d l e integral

operator into

~

curves

of

The to the

existence

following

a generalisation

Lemma Let

of a n o t h e r

Hamiltonian

[4].

This

sending

vector

operator

integral

field,

is a map

curves

of the

we n e e d

f r o m an

a holono-

infinite

jet b u n d l e

into

the

XH0 of h o l o n o m i c

theorem,

momentum

depends

of a lemma

due

on the

operators, following

to G. W i l s o n

which lemma,

is c r u c i a l which

is

[10].

2.1.

G(~ 0) be

f r o m the

the

inverse

of C ~ - m a p s

of ~

inverse limit

limit,

algebra

(with v a r i a b l e

simply ~ 0. Let x)

into

connected,

Lie

group

constructed

~ C G ( ~ 0) , J(R,@) be the ~ and

e ~ g a regular

jetbundle

element•

244

Then

there

(I)

Dx©

(2)

~0

(3)

Qi (~)

exists

:

a unique

[?tn~e~-1,~]

map

,

>~0

cO : J(R¢)

such

that

n > 0

= ~e~-1

-j :

E E. j =0 13

for

a given

The

lemma

follows

ker

ad(e)

part

Because Cartan

e is

of

are

Definition

J be

: J

decomposition

in g of

the

equation

Dx~-%~

:

regular

and

2.2.

a jet

[_~-1

h are

~-I~)0¢

found

bundle

and

] {Ci!

, Ci ~

that

for

The

holonomic

integral into

if on

evolution

an

the

components , while

inverse

i

and

the

of the

$-I

¢

in the

components

on

limit

Poisson

member

space.

Then

if , a set

of

C~-integrable

: O I C ' is a s m o o t h

. ~ being

, the

~f

the

the

Hamiltonian

o is h o l o n o m i c S

momentum

curves

part

inverse

projection

map

with

limit of

the

P.D.E's

values

sequence inverse

in a and limit.

I

operator

S c j,

Im ad(e)

equation.

operator

dimensional

~ H i & ~(~x) to o.

momentum

the

the

~-10~] .

Qi(¢-1~)

Ci+ I , i ( ~

~°ICi

equation

from

~

each

= Olci_

The

= e,

by

j = I ..... ~ }

[4]

finite

(2)

into

. Dx ~ + ~ n e ,

given

is a m o m e n t u m

such

[ Eij I i = 2 ..... n,

the

directly

> ~ (I)

constants

from

subalgebra

Imad(e)

Let

set

of XH0 equations

vector

with

fields

Hamiltonian

XH. are i

H 0 on

tangent

a constraint

: DxO

: OXHo

operator

sends

and

pulls

over

S.

back

integrable all

the

sections

Hamiltonian

of

S into

vector

the

fields

245

Theorem Let

Ec

right, :

2.3. ~ z. be a c o a d ( ~ ) - i n v a r i a n t s u b m a n i f o l d w h i c h is r e g u l a r at the 1 a n d Jr the jet b u n d l e of C ~ - m a p s of ~ into ~ , w i t h t a r g e t m a p

JF

>

P

"

Then (I) ~

: S C J

with S

quadratic

: ~ grad 0 and

(2) The

>

~

l Hamiltonian

H0 = v

the

PC

grad

constants

, a holonomic H

(~(~) o such t h a t

H0, Ek 1

momentum

and

constraint

v

is u n i q u e l y

Res A :0 Ak Qi (v)

:

P.D.E's

D t.~ = ~ XH , Hj ~ ~ (~) are ] 3 c o o r d i n a t e s of J F

bundle (3) E a c h

evolution

equation

Dt

V

Moreover the solutions

Hj + [ ~ g r a d

following

of the

Rest: 0 x k[Dt

Qr(

equation defined

equations

above

equation:

~grad

H0,V)

are

algebraic

in the

on

H0, ~4~grad Hj] satisfied

: DxQr( ~Z

by

o

= 9 ~ X H . is e q u i v a l e n t ]

]

D t . ~ W g r a d H0 - D x ~ g r a d ]

operator

S with

= 0.

identically

grad

jet

on t h e

Hj,V)],for

each

k.

] We w i l l tion.

call

The

entirely

upon

Kac M o o d y into above

above

lemma.

the

main

For

such

theorem

of

that

n classes

elements

of the

grad

H 0 takes

the

of

then

the

equa-

and

reposes

the

one

needs

a

corresponding

f o r m ~e~ -I of

regularity

follows

above

lengthy

condition

same

lines

as

the

and the

[4].

2.4.

(I) The b u n d l e grad (2) The

J c

reduces

identically

to the b u n d l e

over

the

domain

of

H 0 in equations

responds

D t.~ = 9~ X H. d e p e n d 3 ] to the c h o i c e of a s p e c i f i c

convenient

for the L a g r a n g e

type

grad

factors

F ~I

can

of the

for the

but

the

because proof

laws

difficult

each

taking

is p o s s i b l e The

conservation

is not

isomorphism

submanifold This

the

theorem

H 0 is q u a d r a t i c .

of

Remark

the

a new

equations

of the

algebra

lemma.

because proof

these

proof

H 0 IF

be w r i t t e n

as

through

on

orbit

systems •

the c o n s t a n t s

This

in

P.

In p r a c t i c e

to c h o o s e implies

Ek.l This

H 0 such

that

~

corit is

that

grad

H0

246 a ~

grad

H 0 : PK

(~-I

+

A s0)

I (3) As

a consequence

above

theorem,

of

the

equivalence

it f o l l o w s

that

the

of the e q u a t i o n s evolution

in

(3) of the

equation~Dt

u = m~XH 3

are

evolution

(4) The

equations

conservation

laws

of the e q u a t i o n s .

in the

may

They

target

be u s e d

form

the

variables

to give

PKI

(~-I

a variational

link w i t h

the G e l ' f a n d

3 + A ~0 ) .

formulation - Dikii

approach.

Examples

2.5.

IOo -? < :) -e

(I) Let

PKI

R2 for

(a_1

+ A a 0)

g = si(2) . The

8I ( 2 f x x x Because diagonal

0

0

evolution

+ A

equation

, which

is of c l a s s

for H I is ft =

_ fxfxfx ) "

the

difference

matrix,

operator

we m a y

is d e f i n e d

gives

for ~ =

c

~,

up to c o n j u g a t i o n

by a

o}

choose

= This

ef

_fl

u . ~ -I

Dx ( ~ , ~ .~-1) = [DxT ,h -1 + ~PK1 (a-1 + A ~0)'~ -1'

~ "u~'-l]

or

If: Introduction

Ii ii I

-fx/2

of the n e w

target

coordinate

v = fx y i e l d s

the M_KdV

equation '

I

V : -- ~ This of

transformation

class

R4 a n d

[5]. A f u r t h e r by B.

Miura

Kupershmidt

transformation.)

is the

systems

XXX

-

3v

X

.

v 2)

link

between

the

defined

by V.G.

Drinfel'd

type

and G.

(2v

transformation Wilson

[8].

systems

yields

(See a l s o

the [5]

of L a g r a n g e

and V.V. system for this

type

Sokolov defined

247

(2) The

corresponding

operator

DxV

Q

Toda

type

satisfying

the

[ ~

RI.

need

mx%

=

AND

Similar

constructions

for

the

of c l a s s

+ fxfxfx)] "

EQUATIONS

DETERMINED

BY

one

....

any

~

holonomic H0,9]

: J [ x ~ ~

+ ~, 9 % r a d

H0

momentum

and

~ ~ G.

~'F~-I

~-I,~]

. This

which

on the

Lagrange

differ

only

type

systems

in t e c h n i c a l

classes. easily Xln

+ 2 ~0 ) = 0

~

[9Wgrad

operator.

concentrate

other RI

=

[mx~..~ -I

we w i l l

case

Dx

to be a h o l o n o m i c

0 x12 PK(e_I

TYPE

operator

section

can be made In the

by H] b e c o m e s

as above,

satisfying

is a m o m e n t u m

equation

doesn't

of c l a s s

a submanifold

: J?

~.~ q-1

satisfying clearly

]

/

ef

+ e-2f) fx + 4I (2fxxx

OF L A G R A N G E

~

HAMILTONIANS

~ C ~be

operator Then

SYSTEMS

•

0

determined

by the m o m e n t u m

:I

+

0 -f

ft : - 2I [3 (e2f

is o b t a i n e d

o

+

equation

R2

equation

0

and

3.

of c l a s s

• Xn-1, n 0

finds

that

Ia Y!I Y{ll

0 0 a2

0

" "an

de-

248

where

a~,...,a n are

all

constants

determing

the

Fd:~

submanifold

-I

Let

al

0" . . 0

0 " e

=

0 0

0 a n

then

all

a.

are

different

and

E a

1

(xij,Yk£)

of C ~ - f u n c t i o n s

: JV

; i,j,k,~

of [

a fixed

system

into

>

We bundle

next

P

by

and

the

The

(B)

~0

= BeB

decomposition

with

of

Let

J

g

~

= ~

s y s t e m is d e f i n e d -I and DxU

by

=

o

(y)

: Jg

with

by

jet

JV with

bundle

V and

coordinates the

space

target

map

will

two

target

be

= R e s A = 0 .A p,

:

(-p,0)

+ p

called

,

holonomic

~

(~) .

Let

: JG

JG be >G a n d

p : ~

the

jet

define

:

~

(-p,-1)

+ Q

operator

H0 =

, Q =

~:JG-->P ~ ,

X ~0

~

(0,p)

s y s t e m is d e f i n e d b y t h e h o l o n o m i c momentum operator z >Q , o 0 = e w i t h S : y = o_i, t h e c o n s t r a i n t equation

satisfies

the

equation

the

transformation

=

property

[o~grad

one

H0,m]

finds

the

the

(1,p)

momentum

, u~grad

K([,[).

operator

systems.

map

(-P'P)

the

H0(~)

momentum

b e t h e j e t b u n d l e o f m a p s in C ~ ( ~ , g ) , w i t h t a r g e t m a p g >g a n d d e f i n e t h e f o l l o w i n g decomposition of ~ (-P'P)

DxO

Using

H0

[ A BeB-I,~]

(-p,p)

which

vectorspace

is d e n o t e d

and

holonomic

following

in C ~ ( ~ , G ) ,

(-p,p)

The

the

.!~ (-p'p)

the

Hamiltonian

introduce

of maps

following

: J

V defines

p consider

determined

: JV

y

The

E (I ..... n)

>V.

For The

= 0. l

i <j,k>2

(on S)

following

theorem.

and

249 Theorem

3.1.

Let

B = ~.~

, with

the

systems

(~),

transformations (~) a n d

~ ~ G(b+),

(B) a n d

~ £ G(n-)

(y) are r e l a t e d

with contact

inverse

and ~

a diagonal

matrix.

Then

to e a c h o t h e r by B ~ c k l u n d -

as follows:

(B)

with

Dxdt,St-I + y-IDx~/

Dt ~,A-I+ - 1

Dt l~

1

(B) and

=

-A(pn+9_I)A -1

= _A(pn+ ~;_i_1)~-1

1

(y)

~ . o / [ -1 = B-I~.B with D A . A -1 X

Dt. • -

+ B-1D

X

B = -A.Y..A.

+ B-IDt. B = - A o _ i _ 1 i[-I

A -I

1

(~) a n d

-1

i

(y) -I 7

-v-7

: °

with

v = - g - I D x 7 + 7 -1(pn + ~-I )

°-i-1

= _~

-IDt

.? + ~-1 (pn+ ~_i_1) ¢ 1

The p r o j e c t i o n S respect

are taken a l o n g b

to the q u a d r a t i c

and the d e r i v a t i v e s

hamiltonians

this t h e o r e m

is t h a t the q u a d r a t i c

the m o m e n t u m

operator.

the b a s e

s p a c e of the

(the i n d e x d e n o t e s

Absorbing jet b u n d l e

partial

Hamiltonians

the d i a g o n a l

tonians

are

conclusion

of

sense d e t e r m i n e

in B and e x t e n d i n g

we are a l l o w e d

to w r i t e

derivation): >p~

P = BeB -I - i=IE Bti_1

differential

in some

matrix

JG into J(~P,G)

: J (RP,G)

The p a r t i a l

are t a k e n w i t h

H i . An i m p o r t a n t

equations

determined

-I

-i

B

by the q u a d r a t i c

Hamil-

250

Btm_1

Dt

B -I)

= Dt

Formal allow

extension us

to

(B t m

r

to p

= + ~

and

finite

zone

equation

Cr

construct

8 -I ) , I ~< r,

m

( p-1

r-1 different

coad(~)-invariant

solution

for

several

truncations

well

known

field

equations.

(I)

o-fields If

we

[3] .

impose

the

Dxb

r

: ~-r

:

Dxb-r

: b-r-1'

we

find

[Z0,b_r]

: Dt

bo r

This

may

number

of

surface are

be

done

for

freedom.

has

any

One

infinite

with

finite

easily genus.

may

if

that

This

follows

one

seen This

solutions

remark

genus.

coad(~)-invariant

nevertheless

r defining

adds

that may

~ = Z _

the

as

i -I

•

• ~

Riemann

' i6~.

reduces

achieved

increasing

condition

: ~-r-1+i

surface

an

corresponding

from

~-r+1

this

be

the

with

to

follows:

C

r It is

which

a surfaces

let

Then

DxU 0 : D x b 0

5t The

bar

~-p-1

refers

= 0,

to

the

1

~0

: Dt

evolution

p-1 E i=0 The

equation finite

~0

- Dt

fields

i-I

bO

determined

by

~.

Imposing

one obtains

DxU_p

with

1

-p-1 genus.

[b0,Z_p]

-D t ± U0

= 0 truncates

~x ~ p the

Lie

algebra

determining

a curve

251

(2)

Imposing

Cr

the

equation

: 8tr-1

together

8-I

with

the

y i e l d s the self

- iStr+1

complex

dual

(3)

Imposing

Cr

: Btr-1

B-I

with

Bogomolny

the

iBtr+l

variables

equations

Remark

3.2.

]

systems

G. W i l s o n many

2

are

T

The

morphism

on J

(B)

son m o r p h i s m

systems

= t r + 1 - i t r + 3,

B-I)~

(By

non

• B -I r+3

y = tr+1+itr+ 3

Riemann

ral one

a B~cklund

the

curve

RI are

linear

described

P -i E ~-i A i=0 > (¥) and h e n c e

solution,

+ iB t

yield

the

B-I)9

of c l a s s the

interchanging

needs

B -I

: 8tr+1

z = tr+itr+2,

to s y s t e m s

corresponding

of a g i v e n

g

containing

conjugated

: ~ =

B-I

B-I) z :

described [10],

others.

The m a p

y

[7]

[7]

(B z

Wilson

fields

(By

• B -I r+3

the e q u a t i o n

together

The

_ iB t

z = t r + it r+2 ,

Yang-Mills

B-I) z :

B-I

: Btr+1

variables

(local)

(B z

B-1

in

systems

given

[8] by B. K u p e r s h m i d t

of c l a s s

by

equation

and

a n d G.

Rn.

P -i E ~-p+i A is a P o i s s o n i=0 Ad(8) o T : (B) >(B) is a P o i s >o :

flows

but

not

which

when

preserves

necessarily

transformation

specially

the

Schr~dinger

the

the

genus

the curve.

to c o n s t r u c t

a solution

genus

curve

of

the

of the

In g e n e out

is not

preserved.

From

the

representation

(B) one

obtains

the

following

geometrical

interpretation. Define

g(p)

(a) ~ p+1 tures.

P i E gi I equipped i=0 = 0 and (b) ~ p+1 : I. =

The

first

is the

Kac

Moody

with

two Lie

algebra

This

defines

two Hamiltonian

structure

described

structures

above

strucand

252

grad

-I : K 0 ~ d.

ad-invariant

The

second

bilinear

structure

K 0

. 0 O

0 0

.

0 0

(with

2e s t r u c t u r e tur~

one

Jacobi

Lie

: I).

One

le s t r u c t u r e

defining

constructs

connected the

Z p+1

for the

group

fields

comes

f r o m the

the

with

along

.

0

verifies

there

same

easily

0

....

0 K 0

tonian

which

O K

. . . . .

on g(P)

is the one

form

for e a c h

vectorfield.

a left

algebra the

that

is a q u a d r a t i c From

invariant

geodesics

of

this

struc-

= I).

this

Hamil-

for the

metric

g(P)(IP+1

quadratic

Hamiltonian second

on the

simply

The e q u a t i o n s

are

metric.

REFERENCES

[1]

M. p.

[2]

I.V.

Cherednik,

[3]

D.V.

Chudnovsky,

[4]

P. D h o o g e , integrable

[5]

V.G.

Drinfel'd,

[6]

B.A.

Dubrovin,

[7]

P. F o r g a c s , Z. H o r v a t h , L. T h e o r y " , Ed. N.S. C r a i g i e ,

[8]

B.A.

Kupershmidt,

[9]

H.P.

McKean,

A d l e r , P. 318, 1980

van Moerbeke,

Physica G.V.

B~cklund systems,

[11]

G.

Wilson,

Ergod.

Math.

Z. P h y s i k

Surveys,

Inv. Inv. Acta

Dyn.

C.

5, p.

Lie

36;2,

Math. Math. Math.,

Syst.

267 a n d

1981

SSSR.

p.

62,

11,

p.

30, Vol.

I, p.

55,

algebras

P a l l a , in " M o n o p o l e s P. G o d d a r d , W. N a h m ,

D. M u m f o r d , and

306,

38, p.

Dokl.Akad. Nauk.

Moerbeke,

Th.

in M a t h .

on K a c - M o o d y

Sokolov,

G. W i l s o n ,

P. v a n

van Moerbeke,

I, p.

Chudnovsky,

Russian

P.

3D,

equations preprint

V.V.

[lO]

Advances

1981

in Q u a n t u m F i e l d p. 21, 1981 1981

217,

143,

361,

and

258,p.11,1981

403,

p.

1980

p.

1981.

1975 93,

1979

NON-LINEAR DIMENSIONAL

TECHNIQUES

GRASSMANNIAN

Allan

Institute

Suppose

of L a u s a n n e ,

z is a c o m p l e x

E 2 = {Xl,X2]

variables

x+

. Then,

Physics

Switzerland

defined

on the

2-dimensional

z is a f u n c t i o n

euclidean

of the

complex

= x I ~ ix 2

z : z[x+,x_)

the p u r p o s e

structure

of c o n s i d e r i n g

it is n e c e s s a r y

the c o m p a c t f i e d

E 2,

i.e.

(I .2)

in a f i e l d

to f i n d

theory

S 2.

language

applications the

i.e.

complicated

extremal solutions

models

to r e q u i r e on

S[z]

for many

maps,

field

equivalently,

(1.1)

or

MODELS

INTRODUCTION

plane

For

TWO

Din

for T h e o r e t i c a l

University

I.

M.

IN SIGMA

to the

non-linear

a non-trivial

in a d d i t i o n

If one

is g i v e n

: / d 2 x .~[z(x)]

an e u c l i d e a n

(semi-classical points.

with

In o t h e r equations

differential

that

on

functional

,

action

one

of m o t i o n

equations

z is d e f i n e d

an e n e r g y

then

approximation, words

topological

has

to

which

[I].

it is of

interest

WKB methods, look

etc.)

for h a r m o n i c

in g e n e r a l

are

254

Few of

non-trivial harmonic

ques, will

which here

examples

maps.

There

allow

to

consider

class

of

theories

monic

maps

are

exist,

find

at

a rather [2]

known

however,

least

have

to

f is a h o l o m o r p h i c

non-linear

of

a given

the

quite

property

special

technitheory.

I

interesting)

that

the

har-

form

function

(1.4)

f

alternatively,

lutions

z can

rations

acting

The

of

classification

z : z(f)

where

or,

of

nevertheless

remarkable

are

(I.3)

a complete

solutions

(but

the

(1.2

admit

a number

some

special

which

corresponding

which

be

field

in t e r m s

on h o l o m o r p h i c

sigma

z

f(x+)

anti-holomorphic

classified

Grassmannian

matrix

an

:

model

(I ~ m

< n)

function. of

objects

G(m,n)

,

In o t h e r

certain

explicit

words

the

non-linear

soope-

f.

can

fulfilling

be

the

defined

in t e r m s

of an n x m

constraint

+

(1.5) and

with

an

~

(: L a g r a n g i a n ) ~

where

Db~ also

=

~

n~n-abelian

~

U(m)

z 4U(n+m)/U(n)

As

~ - ~z + ~

have

a special

I

(~.2}

defined

in

(1.6)

x+ we

z z

z.

= Tr(D

z)+m

Introducing

= 2Tr([D+z)+D+z

+

transformations:

by

z

derivatives

with

(D_z)+D_z] . ~

z -->zU

or

I that

G(1,n)

action

solutions

is

in o t h e r

respect

to

invariant

under

words

x U(m) .

case

we

have

complete

classification

[3],[4].

Explicitly,

(an n - d i m e n s i o n a l

for

of

for

=

finite

any

complex

that

m

solution

vector,

as

z(x+,x_) z)

and

= CP n-1 has

there

an

for been

exists

integer

which

a

found. an

f = f (x+)

k 6 [0,n-l]

such

A(k)

Z

(1.7)

z -i~(kll

(I 81

~(k) = ~kf _ ~k f~.(f, ~+ f

"

To

+

prove

too next

that

z given

complicated section

but

I will

by

the

+

(1.7)

and

(1.8)

completeness

describe

how

the

'"

"''

are

proof

is

~k-1 f] +

really less

generalization

harmonic trivial of

(1.7)

maps [3]. and

is n o t

In t h e (1.8)

255

works the

in t h e

general

procedure

2.

GENERIC

The

has

case

not

of G ( m , n )

yet

been

where

however

demonstrated

the

completeness

of

[5-9].

SOLUTIONS

G(m,n)

equation

(2.1)

D+D_z

It is c o n v e n i e n t

of

motion

+ z(D_z)+D_z

to r e w r i t e

to be

solved

is

: 0

this

equation

using

the

projector

the

following

(an n x n m a t r i x ) +

(2.2)

P = z z

in t h e

following

simple

(2.3) We

form [ 3+

get

a class

of

generic

[7], [9] ~_

P,

P]

solutions

= 0 of

(2.3)

by

construc-

tion: Let

fi

= fi

Choose Then

(x+),

integers

i = 1,...,m

ki,

will

3+

be

1 fi

a basis

= 1,..,n

and

for

'

~i

C n.

that

define

the

n-component

k I ~ k 2 ~...~

vectors.

km and

Ek i = n.

the

conventionally, vectors

(2.6)

g are

=

vectors,

in

some

order,

by

gs,

{ gl ..... g~ 1

H 0 = ~ and chosen

~+H 8 c HB+ m

Gram-Schmidt

these

, i = I .... m

subspaces H8

also,

which

= 0 .... ki-1

Denote

(2.5)

By

holomorphic

such

in g e n e r a l

(2.4)

and

be

i=1,...,m

must

H B : C n for be

such

8 > n.

The

order

the

vectors

in

that

, B = I ..... n

orthonormalization

one

next

constructs

el,...,e n (2.7)

e B : e 3 / [e~( eB

The

statement

defined lently

by

the

(2.3).

is

then

that

orthonormal

the

= gB

- g8 ~ H ~ - I

Grassmannians

vectors

z (B)

p

8 = I ,..!

e S .... e s + m _ I s o l v e

(2.1)

n-m+1 or equiva-

256

The

proof

and

denote

of this

z (B) . A l s o onto

define

projector

Q to be the

a holomorphic

plane

its a d j o i n t

that

~_QQ

P 8+P

+

=

i.e.

To

equation

prove

lutions finite tence

the

one

simple

But

for

the

the

(2.6)

= 0 and

follows

simply

that

~+Q maps

H~_ I is a l s o =

a holo-

9+Q

and

~+ 0 _ Q

above

-

~+ ~ _ Q

procedure

an a r b i t r a r y

2_pP

that

2+p

+

fl,..,fm

of G(1,n)

for

solution

0+PP

2_P]

related

P of

and

to P via

it is h o w e v e r

not

and

P ~+P

of motion

are

at

least

can be w r i t t e n

(2.3)

show

with

the

(2.7).

so-

exisWhile

so in g e n e r a l .

h o w e v e r f r o m the r e q u i r e m e n t + P = zz . A f i n i t e S i m p l i e s (under D+z

= 0

constructing

emerges

equation

with

is f u l f i l l e d .

of the

vectors

that

instanton

together

so P # + Q

to

P+Q p r o j e c t s

to an

tells

since

Q ~+Q

P =

(2.3)

with

case

solution

assumptions) since

m - P 3+ ~

start

since

= 0, w h i c h

and

B

: 0

It n o w

S = 2f d 2 x T r [

conclusion

of a g i v e n ness

= 0.

completeness

of h o l o m o r p h i c works

9_Q

of m o t i o n

should

action

~+Q,

a specific

corresponding

Finally

P +

?+ ~_m the

this

?_Q.

~+Q

(2.9)

=

i.e.

corresponds

condition

~+Q

Choose

on HS_ I. N o w

(i.e.

~ _ ( P + Q ) (P+Q) The

(P+Q)

~_P

also

in C n

that

simple:

{ e~ .... e B + m _ i I

= 0. T h e r e f o r e

~_QP

(2.8) and

plane

~_(P+Q)P=0.

H ~ + m _ I such

morphic

B+m-1

we have

= 0 implies

into

is r e l a t i v e l y on

of G ( B + m - l , n ) ) , QP

statement

by P the p r o j e c t o r

of f i n i t e reasonable

0(I/%xI)

for

One

action smooth-

Ixl ~ ~.

as a c o n s e r v e d

current

equation (2.10)

~ +(P ~_P)

it f o l l o w s

that

there

exists

~_PP

is p r e c i s e l y

trouble phic ses

is that

plane of

These

special

stronger (2.12)

equation

it does

as b e f o r e .

solutions

morphic

the

for w h i c h

than

9_Q

(2.8)

is not

if we

in the p r o o f

(2.6): ~+HBC

HB+ m,

above,

Q is a p r o j e c t o r

that

but

the

on a h o l o m o r -

to c o n s t r u c t

special

clas-

so.

start

and proceed

Q such

= 0

used

that

: 0

n x n matrix

it is p o s s i b l e

this

arise

fl,..,fm,

requirement

+

follow

In fact

solutions

vectors

not

9_(~+PP)

a selfadjoint

(2.11) This

-

as

from a number in

(2.4)

and

m' (2.5)

< m of h o l o but

with

a

257

A special sarily

t y p e o f solutions

adjacent)

consecutive

vectors

vectors

is t h e n

e B from

and also

given

(2.7)

the

by a choice

such

"holes"

that

of m

(not n e c e s -

the p a t c h

in b e t w e e n ,

length

all h a v e

of

a length

m'

3.

FERMIONIC

The theory

SOLUTIONS

Grassmannian

involving

supersymmetric completely only

model

fermions

CP n-1

[10] b u t

be p a r t i a l l y

model

c a n be g e n e r a l i z e d

by

supersymmetrization.

the

for the

done

(1.6) ~

solution

structure

supersymmetric

m a y n o t be c o m p l e t e

[8],[11].

volves

a quartic

selfinteraction

solutions

in the

necessitates [10],

fermionic

an

sense

of

full

the b o n a fermi

consider

the

simpler

could

of

of the p u r e l y

looking

fide

the

this

supersymmetric

and

of

case

of the

bosonic theory

in-

for c l a s s i c a l

equations

fields

course

of m o t i o n s

as C - n u m b e r

fields

[11].

I will Dirac

however equation

here for

only

r~ in the b a c k g r o u n d

(3.1)

with

solving

interpretation

The

In the

c a n be d i s e n t a n g l e d

G(m,n)

in so far as the p i c t u r e

theory

to a n o n - l i n e a r

~y

the

metry)

additional

-

zz

y

orthogonality

o n the n x m m a t r i x

~

problem

of

of a b o s o n i c

solving

solution

the

z :

: 0

constraint

(following

from

supersym-

: +

(3.2)

z

:

y

0 +

Denoting

the c h i r a l

component

of

~

by

~

(3.1)

can be written

+

(3.3)

D+ ~--

=

z

2+

I

where

~

are

+

a solution different a given

m x m matrix

representation

B, the

(3.4) i.e.,

some

of the g e n e r i c

valued

functions.

considered

c a n be

found

Let

following

in the

us t a k e

(2.7).

following

way:

z to be

A slightly Define,

for

vectors

~ , = g~ the

type

-

g~

$ HB- I

g's

are now only

subspace

HB_I~

The vectors

z as the

old e

, ~ = B,..,B+m-1,

,

~

= B ,.., B+m-1

orthogonalized

with

respect

to t h e

fixed

A

b y the ~ (3.5)

by

e~ n e v e r t h e l e s s since

z we h a v e z = z M

-1/2

define

denoting

the

same

Grassmannian

the n x m m a t r i x

formed

258

where ^

(3.6) is a p o s i t i v e It

+

M : z

is n o w

definite

easy

to

m x m matrix.

verify

(3.7)

that

D M I/2

: D M -I/2

: 0

+

But

then

one

can

write ~ M~

(3.8)

y +

where

+

0

fulfils

z + 0-

= 0 and

the

+

(3.9)

~+

where

the

covariant

derivatives. projector

: z

derivatives

Consider

first

on H S _ I . T h e n

it

equation +

0

(3.10)

D+ h a v e

the

(3.9)

The

where

proof

is e a s y

h + is an

follows

by

to

+

fulfilled

with

substituted

for

show

h+(x

arbitrary

using

been

equation

0 + : P~-I

solves x_.

I/2

: ¢

0 + and

ordinary by

PS-I

the

that

)

m x m matrix

~+PB-I

by

denote

= - P ~+P

depending

such

that

only

on

(3.9)

is

2.

is of

+

b

= - z

~+P

h

Similarly (3.11)

¢

solves

4.

the

second

ACTION

For interest Q = 2~/

the

AND

equation

(4.1)

of

TOPOLOGICAL

purely

to e v a l u a t e d2xq

(I - P B + m - 1 ) (3.9).

solutions

explicity

the

number)

discussed

action

given

: 2[ (D+z] * D+z

~

(x+)

CHARGE

bosonic

(winding

h

in

in S e c t i o n

S and terms

topological of

the

it

charge

densities

(1.6)

+ (D_z) tD_ z]

and (4.2)

q : 2[(D+z)+D

respectively

[8].

found

G(1,n)

in t h e

It t u r n s case

out [3]

+

z -

that

have

{D - z ) + D the

_

z]

remarkably

a rather

logic

simple

formulas

generalization

for

m>1

259

Let

us

consider z

given

by

: eB,

(2.7).

(4.4) one

z = z(B) :

that

rewriting

(4.2) +

: 2 Tr[(~+z)

q(B)

to a CP n-1

e B + I .... e B + m _ 1 )

Then

q(B)

sees

ding

solution

(s)

(4.3) as

the

is a s u m q(B)

topological

as

~+z - ( ~ _ z ) + ~_z] B+m-1 E qi of m t e r m s e a c h c o r r e s p o n i=B for w h i c h qi = 2~+ 9 _ l o g l~il 2

:

charge

Thus

(4.5)

where

To

M is d e f i n e d

find

like

(4.6)

expression

in

(4.5)

=

relates

. Using

written

(4.6) as

Tr( ~

But a p p l y i n g (4.7)

is

twice

~_P'P'

where

the

(4.6)

is

simply P'

last just

Tr

P')+(

: -

~+

-

z(B+m)) + D _

two =

(z

M

different

the

)

+

and

the Z

of

following

simple

identity:

(~+m)

solutions

relation ~+P

also

. To p r o v e

m+z

it

follows

that

+ Q)

~_p

p -

is a c o n s e q u e n c e

of

:

which

the

-

projector

~ +m

while-the

P'P') .

P P ~_P

in t e r m s

(eB+m,..~eB+2m_1)

(~+m)

P Q_P

~_(P

density

establish

~

(2.8)

equality Tr

Tr ~ n

(4.6)

it

projectors

(~+m)

HB_ I =

{ e I .... eB_iI

of

the

action

first

z (B+m)

to c o n s i d e r

the

= Tr(D

"norms"

: z

of

may

z (~)

+

P = z(8) (z(B)) + , P'

side

B+m-1 Z ,{n;eil 2 : 2 ~ + ~ _ i:B

for

one

(e B ,.. , e B + m _ I) a n d

is c o n v e n i e n t

~_

(3.6).

Tr(D+z(~))%D

equation

z (B)

: 2 ~+

by

a suitable

quantities

This

B+m-1 ~' qi i=B

q(B):

z

the

right-hand

~_p p2

left-hand side

can

be

: - p ~_p

= p.

is p r e c i s e l y

Q on

Thus

equal

the

to the

RHS

of

left-hand

side.

From

the

definitions

that

the

action

cal

charge

(4.8) where

~

(4.1)

density

~

and =

(4.2)

~(B)

together

can

with

be w r i t t e n

(4.6)

it n o w

as a s u m

of

follows

topologi-

densities: (B)

= q(B)

~ = B(mod

m)4

+ 2q (B-m)

+...+

[1,2 .... m }

. If

2q (~)

+ 4Tr(D_z(~))+D_z(~)

~ = I the

last

term

of

the

RHS

of

260

(4.8)

is

zero

c a n be u s e d of a l o w e r (4.9) But

since

z [I]

is an i n s t a n t o n .

to s h o w

that

the

dimensional

same

Grassmannian

4Tr(D_z(~))+D_z(~)

since

given

this

by the

E qi i=1 U s i n g the

term

z is an

z =

argument

charge

proving

(4.6)

~ ; I can be w r i t t e n

(el,..,e

in t e r m s

_ I) as

: 4Tr(D+z)+D+z

instanton~D_z

topological

The

for

•

: 0, and

(4.9)

is t h e r e f o r e

simply

density

2

(B) general

(4.10)

The

~

integrated

Qi

degree

in the

~ 0),

The

Y1 of

CP n-1

the

between

¥0

:

n o w be e v a l u a t e d

to the As

(B)

charge

(4.11)

can

expression

defined

action

expressed

and

B+m-1 E qi i:B

:

1 /

8-I E i:I

+ 2

we thus

find

qi

d2x 2 a + a -

2~

IeiI

at

this

infinity,

in x+ of

in terms

and

}eil ~

is g i v e n

{x I

out

for

r

Ix I ~ ~0 .

of a d i f f e r e n c e

:

Yi

to be e q u a l

Yi

Yi-1

(with

[3]

charge type

turns

in terms

two p o l y n o m i a l s ,

of e i.

following

~nlSil2

integration

degree

topological

by the

for q

density

by p a r t i a l

case

degrees

(4.5)

of a g i v e n

of

solution

z (B)

is t h e r e f o r e

formulas

S (B)

: 2 ~ ( ¥ 8 + m _ I + yB_I )

Q(B)

= ¥B+m-1

(4.12)

Another the

interesting

stability

of a s o l u t i o n is g i v e n

question

under

small

z then

the

- YB-I

concerning

the

fluctuations fluctuation

generic

[3],[8]. of

solutions

concerns

If ~ is a p e r t u r b a t i o n

the a c t i o n

to

second

order

in

by

(4.13)

6S

= 4 /

d2x

V(~)

where +

(4.14)

As

V(~):Tr(D_~)+D_~-Tr

for the CP n-1

like

solution

case

with

D+

~+~(D_z)+D_z-Tr[z

it is n o w e a s y z # 0, the

to see

special

+

D_~+~

that,

+

D_z~

+

[z D _ ~ + ~

+

D_z] .

for n o n - i n s t a n t o n

fluctuations

261 ¢+ are

£ D+z

( £ constant)

therefore

saddle

special

solutions

display

a large

solutions a negative

5.

THE

RIEMANN-HILBERT

The

technique

of c e r t a i n picture

ture m a y case)

and

in

In fact

for

action

(3.11)

all

in the p r e v i o u s equations

solution

sections

was

manifold

seen

since

turn

for

to give

in q u e s t i o n .

the

out

to fer-

to p r o -

finding a rather

Although

of c o u r s e

interest

to c o m p a r e

it w i t h

other

for

this

soluextenpic-

the CP n-1

non-linear

techniques

problem.

of g e n e r a l

of w h i c h

equations

the

(4.14).

(except

approach

solutions

so for

PROBLEM

applied

same

Such

it is p o s s i b l e

the

incomplete

Another

bility

(3.10)

V(¢). is a l s o

to be

to the

in t e r m s

This

2..

modes

inserted

non-linear

out

it is of

in S e c t i o n

(3.8),

when

of the

turn

applied

[12]

by

V(¢)

a negative

the a c t i o n .

of n e g a t i v e

mion

sive

given

produce

of

discussed

class

duce

tions

will

points

for

the

the

~+ ~

interest

is the

equations

linear

Riemann-Hilbert

of m o t i o n

(2.3)

arise

technique as c o m p a t i -

system

2 : I+Z

[ ~+P'P]

_

[ ~

T

(5.1)

where

~

2

(x, ~ ) is an n x n m a t r i x

additional

Solutions

complex

to

(5.1)

parameter

can be

2~

found

p,p]

valued

function

depending

on the

[13].

explicitly

in terms

of the p r o j e c t o r s

PB: (5.2)

~ 8 : I +

Alternatively factor)

this

4~ (i_i)2

PB-I

+

~

can be r e e x p r e s s e d

2

(PB+m-1

- P~-1 )

(up to an o v e r a l l

A -dependent

by ~g

= I

41 (A+I)2

(I - PB)

2 ;t+1 (PB+m-I

- Pb-I )

(5.3)

2 = 1 + A--~ This

shows

equivalent

how

the pole

ways.

PJ]-I structure

2 ;,.+1 of

(1 - P 8 ) ~B

can m a n i f e s t

itself

in v a r i o u s

262 The

~

B g i v e n in (5.2) is r e l a t e d in a s i m p l e w a y to the f e r m i o n i c + (6) ~-- in the z b a c k g r o u n d f o u n d in S e c t i o n 3. E x p l i c i t l y

solutions

we

have +

(5.4)

+

~

y6H

:

2 where

H+ -

H- =

(1 - P s + m _ 1 )

It w o u l d

to o t h e r

PB-I

2

interest

so as to e x t e n d

non-linear

is less

h + M+I/2

and

h- N -I/2

s e e m to be of

lationships

fold

(~-I) (I+I)

to u n d e r s t a n d

the u s e

equations

where

of

better

this

kind

the R i e m a n n - H i l b e r t

the k n o w l e d g e

of re-

technique

of the

solution

mani-

complete.

REFERENCES [I]

J. E e l l s ,

[2]

H. E i c h e n h e r r , M. F o r g e r , Nucl. Phys. C o m m . Math. Phys. 82 (1981) 227, A.J. 82B (1979) 239

B 1 5 5 (1979) MacFarlane,

[3]

A.M. A.M.

B174 (1980) 397 95B (1980) 419

[4]

V.

[5]

A.M.

[6]

J. R a m a n a t h a n , C h i c a g o Univ. p r e p r i n t (1982) S. E r d e m , J. W o o d , Univ. of L e e d s p r e p r i n t no. 9

Din, Din,

Glaser, Din,

L.

Lemaire,

W.J. W.J. R. W.J.

Bull.

Zakrzewski, Zakrzewski, Stora,

London

Nucl.Phys. Phys.Lett.

Zakrzewski,

L e t t . M a t h . Phys.

A.M. Din, to a p p e a r

W.J. Z a k r z e w s k i , L a u s a n n e in L e t t . M a t h . Phys.

[8]

A.M.

W.J.

[9]

R.

Sasaki,

Zakrzewski,

Hiroshima

Soc.

I O0, I (1978) 381 a n d Phys. Lett.

unpublished

[7]

Din,

Math.

Univ.

CERN

Univ.

preprint

preprint

RRK

5,

(1981)

(1982)

preprint

TH

3746

83-4

553

(1983)

(1983)

(1983)

[10]

A.M. Din, J. L u k i e r s k i , (1982) 157

[11]

K. F u j i i , T. K o i k a w a , R. S a s a k i , H i r o s h i m a Univ. p r e p r i n t R R K 8 3 - 1 5 (1983), K. F u j i i , R. S a s a k i , H i r o s h i m a Univ. p r e p r i n t 8 3 - 1 8 (1983)

[12]

V.E.

[13]

A.M. Din, Z. H o r v a t h , W.J. Z a k r z e w s k i , (1983) to a p p e a r in N u c l . P h y s . B.

Zakharov

a n d A.V.

W.J.

Zakrzewski,

Mihailov,

Nucl.Phys.

Soy. Phys.

JETP

Univ.

47,

B194

1017

of D u r h a m ,

(1978) preprint

A GEOMETRICAL UMBILICAL

OBSTRUCTION

TO THE E X I S T E N C E

COMPLEMENTARY

A.M.

FOLIATIONS

Naveira

Departamento

y Topologla

de M a t e m ~ t i c a s

Burjasot,

Valencia,

interesting

aspects

Spain

INTRODUCTION

Among

the most

a differentiable topological cation J~

MANIFOLDS

Rocamora

de G e o m e t r l a

Facultad

0.

- A.H.

OF TWO T O T A L L Y

IN C O M P A C T

,

tensor

we can point

or g e o m e t r i c a l .

of the p o s s i b l e according

In

to the b e h a v i o u r

families

interesting,

leaves.

in this paper.

and

in this due

p o s e d by two m u t u a l l y umbilical

[11] the

almost-product

defining the structure

One of the larly

manifold,

Corollary

4.1.

a geometrical

curvature

TQ~(~)

scalar

curvature

This p a p e r

manifold,

~

determined

is c o m p o s e d

by the v e r t i c a l

with

leaves,

I,

I.

totally result

to the e x i s t e n c e

(~

operator

In

of J~) .

as the main

and

and h o r i z o n t a l

sections.

(I, 1)-

is the one com-

umbilical

= J~9~r-

manifold

the

to be p a r t i c u -

foliations

obstruction

totally

the Hodge d u a l i t y

of four

connection

properties,

can be c o n s i d e r e d

of the global

(P b e i n g

appears

complementary

with

gives a c l a s s i f i -

~7 p,

that

of

either

on a R i e m a n n i a n

~7 the L e v i - C i v i t a

foliations

scalar

author

of the tenser

to its g e o m e t r i c a l orthogonal

This p r o v i d e s

compact

first

classification

structure

its o b s t r u c t i o n s ,

strutures

of two c o m p l e m e n t a r y

oriented

of a g e o m e t r i c a l out

in terms

being ~

an the

distributions.

following

Mat-

264

sushima valued

[7] and

Eells-Lemaire

in a v e c t o r

bundle

[3] we e x p o s e

that

in p a r t i c u l a r ,

the W e i t z e n b 6 c k

we

concepts

revise

some

structures,

and

in p a r t i c u l a r ,

pe F 2 a c c o r d i n g characteristic

In 3., we particular

will

formula

to the n o t a t i o n and

connection

deduce

Theorem

some

3.5.,

for

geometrical of the in

in the

rest

l-forms.

[11],

forms

in 2.,

of the a l m o s t - p r o d u c t

umbilical as well

of the

of the paper;

Analogously,

properties

totally

its

the p r o p e r t i e s

be u s e d

foliations

as t h o s e

or of ty-

concerning

the

curvature.

results

based

from which

the

on the p r e v i o u s

geometrical

sections,

consequences

in

studied

in 4. follow.

The trical

I.

manifold

objects

WEITZENBOCK'S

Let

(~,g)

tor b u n d l e DM

~

will

considered

be a s s u m e d

throughout

FORMULA

FOR

I-FORMS

be an n - d i m e n s i o n a l

over J~

with

to be c o n n e c t e d

the p a p e r

a metric

will

VALUED

and

C

all

manifold

a covariant

the

geo-

.

IN A V E C T O R

Riemannian

< , >,

and

be

BUNDLE

and

~

a vec-

differentiation

satisfying

We d e n o t e

by A

P(E,J~)

the

vector

space

of

~ -valued

p-forms

on J~. It is a w e l l dD

:AP([

known

,~)--->IP+1

fact

that

( ~ ,v~),

where

E i<j

M i ~ ~(j~),

The

covariant

lued p-form

exterior

(p:0,1 .... ),

(dDo) (MI,M 2 ..... Mp+1 ) = +

the

P+J 2 i=I

i+I (-])

differential

is g i v e n

by the

operator formula

^ DM (O (M I ..... M i ..... Mp+1) ) +

( - 1 ) i + J o ( [ M i , M j ] ,M I ..... Mj

....

Mj ..... Mp+ I)

i=l ..... p+l.

derivative

DM8

of

~ 6 /~ p ( ~

,J~)

is the ~

satisfying P

(DMS) (M I ..... M p ) = D M ( @ ( M I ..... M p ) ) -

E 8 ( M I ..... ~MMi ..... Mp) i:I

-va-

265

where

~7

isthe

Levi-Civita

PROPOSITION

1.1.

connection

of

(9 { ~ P (

~ ,J~)

- Let

(~,g). and M I ..... Mp+ 16 /

(J~),

then (dD(9) (MI ..... Mp+I)

=

p+i E i=I

)i+I

e

(--I

(D M

i (9)

A M i .....

M p + 1)

(M I

A . . . .. .M. .i,.

Mp+

i)

Proof: p+1

i+I

(-I)

w (DM.(9) 1

i:I

(M 1 . . . . .

p+1 = i=IE (-I) i+I DM I ((9(M ±

+

=

..... Mi, .... Mp+1 ) ) +

p+1 p+1 )i A Z E (-I (9(M 1 ..... M i ..... ~ZMiM j ..... Mp+ I )

i:i j=1 j#i NOW I p+1

p+1 E i:I j:1 j~1

i(9

. .... V M Mj . . . . Mp+I) = 1 : E ((-I)i(9(Mi ..... ~i ..... VM.Mj ..... Mp+I) i<j 1

(-I)

,

(MI

,

.... Mi

+ (-I J(9(M I ..... VM M i ..... Mj ..... Mp+1)) =

which

implies

E (-I)i+J(9(VMIM~ i<j .

:

~ M Mi'MI ..... AMi ..... Mj i) J ..... Mp+

the r e s u l t .

The c o v a r i a n t tensor

-

+

derivative

field of type

(O,p+1)

~(9

defined

of

(9 e A P (

by the

[ , J K)

is an

~-valued

identification

(~e) (}41 . . . . . Mp,M) : (D~M8) (MI . . . . . Mp). Now, rator Let

by u s i n g

the c o v a r i a n t

derivative,

we can i n t r o d u c e

6 D : A p ( [ , ~ t ) - - - > A p-I ( [ , J < ) , (p>O) , in the f o l l o w i n g x ~

and let

{ e i ..... e n }

(6D~ )x(Ul ..... Up_l)= u I ..... Up_ I ~ T x J ~ If

~

is a forms

basis

of

TxJ~

n E ( < k ~ ) (ek,u I ..... Up_ I) k=1

• [ -valued

It is well k n o w n differential

-

be an o r t h o n o r m a l

the ope-

way:

that

O-form

we can define

the L a p l a c i a n

is given by D = d D 6D + 6D d D

operator

6D~ = O. ~D

on

k-valued

266

DEFINITION metrics

1.2. - Let $ , ~I' ~2

be vector bundles on J~ with

< , > , < , >I and < , >2 respectively

i) We define the following metric on the dual

~

of

~

,

m

< ~,~>~ where

[[kl

(x) =

k:1,.

E k=1

Z~( ~ k )b( [ k )

,m is an orthonormal .

X

ii) The metric on the tensor product < ~I ~

~I ~

~2' ~I (9 ~ 2 > x : < ~I' Y I > I

This induces a product THEOREM

basis of

,

1.3.

~2 is given by

< ~2' ~2>2

in ~ I A ~2"

(WEITZENBOCK'S

FORMULA)

[7]. - Let e be an [ -valued

l-form. Then < •2 ~ , ~ > where

~

= ~I

~ + + A

is the Laplacian operator of the Riemannian manifold

(J{,g)

and A is a function on J~ defined by A(x): E - E RD(e ,ei,~(ej),e(ei)) i i,j J with [ e I ..... en~

an orthonormal

basis of TxV~,

TxJ~ defined by the Ricci tensor o f J ~ RD(M,N,~,~)

= , W M , N 6 ~(J() , ~,~ 6 / ( 6 ) .

Proof: Let x be a point in H a n d T x ~ . Choose E I ..... E n 6~(J~) i,k = 1,...,n. Then

[ e] ..... en I an orthonormal such that Ell

: e i and

(~EkEi) x : 0,

x

(6DdDe) (el)=- kE (~ekdDe) (ek,e i) : - kE Dek((dDe) (Ek,Ei)) = - kE Dek((DEke) (E i) - (DEiS) (Ek)

+ kZ mekDEi(e(Ek))

basis of

:

: - kE De~((DEk~) (El) +

- kE e(~Tek~7 E1 Ek)

since 0 : (Deke) ( Dez' Ek) : Dek(8( ~Ez.E k))

e( V e k D E l 'Ek) "

On the other hand 6De = - E gkt(D~ E e) (Ek) where kpt t

(gkt) is the

267

inverse

matrix

of

(g(Ek,Et)) , and thus we have

(dD6Do) (e i) = D e

(6D@)

: _ ~ k,t

i

(e

ig

kt

¥

) (Det@) (e k) -

¥

- ~ 6ktD ((DEt~) (E k)) k,t ei = - ~D e D E (O(Ek)) k i k

= - ED ((DEkO) (E k)) k ei

:

+ E (9( D e ~ E k E k ) . k i

Therefore,

(Zl De) (el)

: kE (DekDE± (e(E k) ) - D e i D E k ( @ ( E k) ))

- 8(E(~zek~TEk i Ek - iTei EkEk)) and since

- k~ D e k ( ( D E k @) (Ei))

[Ek,Ei] x = 0,

(x)

= -

= z

E ~(ek,ei,8(ek),fig(ei)) i,k

:

+ E i

-

E = i,k ek ¥ = A(x) - E i,k k

-

It is easy

-

to see that

E

If the m a n i f o l d inner

product

(8,~)

(~,g) of two

: f(J)

is compact ~-valued

(e,~) =]4. and it is well PROPOSITION of d D, i.e.

known 1.4.

for this

inner

(x) + (x)

and oriented,

p-forms

we can define

the

as

<e,,~> ~ 1 product,

[3] - The o p e r a t o r

IdDe,~l = le,6D'II,Ve~APIf , ~ I ,

6D

that the following result is the adjoint

IcAP+~I f ,~I

hblds,

operator

268

Proof: Given

x C v~

we c a n c h o o s e

and

~ e I ..... enl

E I ..... E n 6 ~ ( J ~ )

(

=

- < e , 6 D i > ) (x)

as

[

i 1.

n _ Ip! ii .... Z

in t he

Theorem

< (dDe)(ei~ ip+l=l

. . . .

basis

of T x ~

1.3.

=

n ~

1 (p+l

an o r t h o n o r m a l

....

ei

),~(ei~... p+l

=1<e(eil ..... eip)' ( s D ~ )

,e i

)>p+1

(ell ..... e i p )> =

' p I

n p+1 ~ )> + E < Z (-I)k+1(D e e) (eil ..,e i ), ~(eil . . , e , (p+ 1).fii ,... ,ip+1:ik=1 ik ' " " "'elk' p+1 ' 3_p+ 1 n <e (eil ..... e i ), Z (De %)(es,eil ..... e i )> : P,f i I ,.. ,ip:1 p s=1 s p n E

+ I_

n

n

I

<s=IE (Dese) (eil ..... eip) , ~ (es,eil ..... e ip) > +

P] i I , .... ip=1 n 1 ~ + P! i 1 . . . . .

n =

<e(e ip+l=l n

1

~

~!

s=l

n ..... e i ), Z (De ~) (es,eil ..... e i )> = p s:1 s p ±1

z

z 1 , .... ip:1

( + P P

+ <e(eil ..... e i ), D e (~(Es,Eil ..... E i )>) : p s p :

n E s=1

es(1 )

n E i 1 .... ,ip=l

) : p p

= ( d i v i"I) (x)

being

M =

Now,

n E j:1

the

f E 6 ~ (v~) , w h e r e J ~ n I •= Z <e(Eil ..... E i ), ~(Ej,Ei] ..... E i )>. f3 p,I i 1 , . . . . i p = l p p result

Clearly, (~D

e,(9)

follows

~ e & AP( =

(dDe,dDe)

from

the

Green's

C ,v~) +

( 8 D (9,

6 D (9)

Theorem.

269

RIEMANNIAN

2.

ALMOST-PRODUCT

CURVATURE

OF T H E I R

A Riemannian (J~,g)

on

J~

= I and

A Riemannian

lues

of P,

In turn

-I,

whose

associated

~

to

will

(J~,g)

We w i l l

is t o t a l l y

geodesic,

- In any

[11]

(1,1)

two m u t u a l l y

to the

vertical

eigenva-

and h o r i z o n t a l . manifold

a Riemannian

a com-

almost-product

distributions

are

~

almost-product

and

~m

structure

a foliation

minimal

of

~

of ~

~

on a R i e m a n n i a n

or t o t a l l y

are

totally

umbilical

geodesic,

if

minimal

, respectively.

Riemannian

almost-product

manifold

we have : g((~LP)N,M)

g((~LP)M,N)

+ g((~LP)Pm,PN)

= 0

~ ~(J{) is o b v i o u s .

in

[11]

almost-product

by

algebraic

by d e c o m p o s i t i o n

condition

dimensions

of

(~,g,P),

0(p)

of t h e s e

classes

are

each

~

36 d i f f e r e n t one

P. This

of w h i c h

as the

tensors

tensor

2.2.),

under

x 0(q) , w h e r e / are

and given

~ in

classes

was

of o r d e r

y defined the

of Rie-

is c h a r a c t e r i z e d

classification

of c o v a r i a n t

(lemma

of the d i s t r i b u t i o n s

one

there

on

space

properties

= g((~LP)M,N),

group

that

manifolds,

of the

algebraic

¥(L,M,N)

of e v e r y

of type

determines

Riemannian

manifolds

2.2.

It is shown

tural

where

6 ~(~)

on a R i e m a n n i a n

say that

submanifolds

mannian

same

P

called

and h o r i z o n t a l

umbilical

The p r o o f

the

(J~,g,P)

field

, corresponding

, and hence,

be c a l l e d

integral

ii)

some

and ~

determines

i) g ( ( ~ L P ) M , N )

VL,M,N

/M,N

structure

~

~I

2.1.

the m a x i m a l

LEMMA

#

.

DEFINITION

or t o t a l l y

P is a t e n s o r

respectively ~

vertical

this

(~,g,P)

and

distribution

manifold

is a t r i p l e t

: g(M,N),

almost-product

respectively;

all

g(PM,PN)

a distribution

structure

and

(F2,F 2)

CONNECTION

manifold

manifold

distributions

I and

plementary

OF T Y P E

satisfying,

p2

complementary

CHARACTERISTIC

almost-product

is a R i e m a n n i a n

defined

MANIFOLDS

obtained 3 that

have

by

action

p and q are

of the the

struc-

respective

. Some

non-trivial

[8].

The a l g e b r a i c

examples conditions

270

on

~

P defining

and horizontal the most can

the c l a s s e s

distributions

interesting

point

by geometrical are

classes

conditions

interpreted

of Riemannian

in

o n the

[5]. F o r

vertical

instance,

almost-product

among

manifolds

we

geodesic

fo-

out:

i) ~

and

ii) ~

and

~

(or one

~

(or at

of

them)

least

are

one of

foliations. them)

are

totally

liations.

Remark.

- If

~

zhe m a n i f o l d iii)

~ was

and

~

is l o c a l l y

are

totally

is a f o l i a t i o n

with

mainly

by R e i n h a r t

Naveira,

studied

[10],

geodesic

foliations,

then

product. almost-fibered

and Vidal-

metric.

[13],

Vidal

This

structure

[14], M o n t e s i n o s ,

Costa,

[17],

among

[9],

other

references. iv)

~

and

~

(or at

least

one

of them)

~

(or at

least

one of

are m i n i m a l

foliations,

are

umbilical

[16] . v)

~

and

foliations,

The m a i n

object

product

structures

totally

umbilical

It is w e l l only

each

of t h i s w o r k

known

that

is the

2.3.

is a t o t a l l y

is a r e a l

denotes of

[5].

the

a

=

E a=1

reference

( VE of

almost-product

a

of the R i e m a n n i a n are

integrable

P)E a

'

is t o t a l l y

function

linear

foliation

E

.9

structure

almost-

and have

a = 1 , . . . ,p

J~ , ~(x),

if a n d such that

attaching

to

on

~

on a R i e m a n n i a n

if a n d o n l y ,

I/

A,B

associated being

a

on

umbilical

transformation

- A distribution

umbilical

I

P is the

~

-~B X lying

( ~zAP)B : ~ g ( A , B ) ~ where

study

distributions

a foliation

~Zthere ~x

the c o m p o n e n t

PROPOSITION (~,g)

both

leaves.

~(x) I, w h e r e B6 ~

totally

[9].

for w h i c h

if f o r e a c h x 6

x :

them)

manifold

if ~)

to

a local

and orthonormal

271

Proof. If B,C 4 ~

Suppose,

and

X & ~,

first,

that

~

is a t o t a l l y

umbilical

foliation,

then

we have g( ( V B P ) C , X )

: 2

A (X)g(B,C)

hence

g(~$ ,x) = 2p k(x) and

given

therefore

g((~Bp)c,x)

= !g(~,x)g(B,c)

Conversally,

since

A,Be ~

, then

g(dxB,C) And

If is,

we

I : ~

~

get

is a f o l i a t i o n

be

both

totally

2.4.

said

Next

we d e f i n e

the

on a R i e m a n n i a n

2.5.

(F2,F 2)

It is e a s y DP

= 0

- The

manifold

to

manifold,

it is o b v i o u s

that

almost-product

manifold

if the d i s t r i b u t i o n s

~

(~,g,P) and

~

are

see

and we

connection study

of a R i e m a n n i a n

its c u r v a t u r e

in the

alcase

(F2,F2).

characteristic

(~,g,P)

DMN = ~ M N + ½PlUMP)N, and

that

Moreover,

curvature.

characteristic

(~,g,P)

is of type

almost-product

we h a v e

foliations.

manifold

DEFINITION

~ A,B ~ ~ a foliation.

by t a k i n g

its m e a n

to be of type

(~,g,P)

is

,X)

A Riemannian

-

umbilical

most-product

~

,X).

up to a c o n s t a n t ,

will

= ( ~BP)A,

, and

the r e s u l t

g(~

DEFINITION

that

( ~AP)B

[A,B]~ ~

I ~ g(B,C)~ = ~g(

then

l(X)

P

that

connection

is d e f i n e d

of a R i e m a n n i a n

by

/M,Ne ~ ) . this

is a c o n n e c t i o n

satisfying

Dg = 0

272 PROPOSITION nifold.

2.6.

- Let

If R D and R denote

(~,g,P)

be a R i e m a n n i a n

the curvature

tensors

almost-product

of D and

~

ma-

respective-

ly, then RD(L,M,N,O)

~R(L,M,N,O)+

R(L,M,PN,PO)+

~g( ( ~ M P ) N , ( ~ L P ) O )

~g((~7LP)N,

VL,M,N,O

,

6

(~MP)O)

#((~) .

Proof. By using

the d e f i n i t i o n

g(DmDLN,O)

of D and the p r o p e r t i e s

= Mg (DLN,O) Mg(~TLN,O)

g (DLN, DMO)

i

: g( ~ M ~TLN,O)+ I - ~g((VMP)

we can write,

- g( ~7LN , ~ M O)

I

I (

- ~g( P(gLP)N,~TMO)--4g

~Ig ( ( ~ M P ) ( ~ L P ) N , O )

1

-

(~Lm)N,(g~)O)

=

I (~TM( ~LP)N,I:O)_ + 2--g

(m • L N) ,0) - lg(( V L P ) N ,

1

~P

:

+ 1Mg(P(~LP)N,O)

- ~g( ~ L N , P ( ~ M m ) O )

of

1 (

: 2 g( ~7M ~TLN'O) + -9.( 2 ~M ~TLPN'PO) + ~

( ~TMP)O ) =

(~TLP)N' (~TMP)O) "

Therefore,

RD(L,M,N,O) = g { D [ L , M ] N - DLDMN + DMDLN,O)

={.g(

[7[L,M]N,O)

1

:

1

+ 2~g( ~/[L,M]PN, PO) - 2--9-(~'LVMN,O)

-

4~g(( ~TMP)N' (~LP)O) + 2--g 1 ( PrM VLN,O) +

+ ~(

~7.M [TLPN,PO ) + 4~:/( 1 ( VLP)N, (~7"MP)O) =

= 1R(L,M,N,O)+ 1R(L,M, PN,PO)+ 4-,¢~ ~xtO

which

-

V2

gives

Yij

'

dxldx3

"

:

[~I

(~)

+

2

62

(~)

]

d~Td

~

+ dz 2

(lO) O

: - I/V

(g] (~), g2(~)

c a n be e l i m i n a t e d

Therefore,

complete

(11)

the

dS 2 = - [ 6 1 ( x + i y )

metric

+ E2(X

by a c o n f o r m a l

in the c a n o n i c a l

- i y ) ] d t 2 + 2 dt d Z

transformation). form

(7)

is

+ dx 2 + dy 2

289

The

signature

the

coordinates

The Wick to

maps

t = i~

t = iF

On the

this

other

hand,

into

If

61(~)

and

is of P e t r o v

~2(~)

associated

o models.

~ ~ N

are

,

(~-

aq)

(~-

bq)

associated

The

for t h e s e

this

condition

takes

solution

associated

solutions

solution

(11)

On the o t h e r

gives

@

:

Yij

Note

change

[10]

by

(GI)2

o

of the

instantons

self-dual

solutions

( o i , o 2 , o 3)

satisfies

lowest +

-+++

++++

euclidean

(o2)2

_

( 3)2

equations

~ ~(~) .

action

are

with

signature)

signature)

the

non-

signature.

metric

regime.

that

is n o t a

complete

(++++)

= 1I

(9), w h o s e

Note

equations,

instantons

equations

(i.e. (i.e.

+ iy)

of E i n s t e i n ' s

Gravitational

which This

The

does

metric

curvature.

~ I = 0 10)

[which

or

~ 2 = 0,

is flat implies

of T a u b - N U T that

with

V + i¢ = ~ ( x

a solution

(eq.

are

i.e.,

solutions

the

field

equa-

of p a r a m e t r i z a t i o n

where

the E i n s t e i n

~(I/V)

a metric

is s e l f - d u a l .

if

where

o

signature

of t w o d i m e n s i o n a l

function,

f o r m of the C a u c h y - R i e m a n n

(Riemann)

hand,

the

(3,1)

with

of the E i n s t e i n

°~ 9v°Y : 0

is a L o r e n t z i a n

metric

This

of

(2,2)

(multi)-instanton

o model

+ 3)-I

in the e u c l i d e a n ,

has n o n - s e l f d u a l

~x

(]I),

instanton.

singular

spatial

the

is g i v e n

metric

gravitational

the

solutions

By the

¢ = G2(oi

,

z = i

signature

solutions

the o c o n f i g u r a t i o n s

class.

of the

(y = - i t ) ,

~ = a rational

of the

is e q u i v a l e n t

signature

t = iy

is r e a l

the p o s i t i o n s

E~v & ~ y

defines

(o I + 03)-I

with

represent

instantons

that

homotopic

tion

(11)

of h o l o m o r p h i c choice

of

but with

solution

.....

a0, ..... aq

exist

is the

.....

V =

not

solution

case the

rotations

to be E u c l i d e a n .

dt 2 - 2 dt d

a 0)

condition

general

the

one

(~ - b 0)

sense

in e a c h

[2(x+t)]

class

~o ~ This

+

This

in t h i s

rotation real

Wick

signature

not change

(~-

centered.

in the

does

the

the m e t r i c

o model,

the

In p a r t i c u l a r , =

gives

AZ)

the W i c k

type.

to the

By appropriate

to get

a different

dS 2 = d x 2 - d t 2 + [ 6 1 ( x - t )

tions

(3,1).

, z = i ~ (which

, Az-->i

solution

=

is

it is n o t p o s s i b l e

rotation

setting

metric.

of t h i s m e t r i c

in t h i s

and

then ~

in t h i s

and

V = ~ i¢,

the

three-

V

the

rela-

case

type.

The

curvature

case,

the

metric

satisfy ~2(I/V) of t h i s

is c o m p l e x

: 0]. solution with

signa-

290

ture

(3,1)

or r e a l w i t h

signature

(3,1)

Note

the m e t r i c

that

or

n o t of T a u b - N U T Pohlmeyer's

symmetric which

types

The

solutions one

also

depend

where

V ~ V (~,~,{)

real

picture in

is

solutions,

Sine-Gordon holds

In t h a t

type

between

the

or to

for a x i -

case,

a n d on L i o u v i l l e

of the c o n n e c t i o n

metrics solutions

these

different

[I].

vector

in a s i m p l e

is p r e s e n t .

way

In t h i s

to the

case

~ ]

case when and

~ 2

, i.e.

2 + 2 d{d(

the W i c k

solutions

tional

plane

called

pp

parallel

with

waves. wave

+ 2 d~d~

satisfies

~

~

V(~,~,[)

= 0,

i.e.

In t h i s

density

to be V =

(z-t)

signature

choice

solutions

we

(plane

can o b t a i n

directly

61(~,

z-t)

fronted

(]2)

c a n be g e n e r a l i z e d ~ 2 V = 4 ~ g for

matter

F(F)

z-t)

a sub-class

interpreted

= ~2(~,

of

as g r a v i t a -

z-t)

gravitational

in the

space.

The

to the

+ E i(~,

are

z-t)

holomorphic

+ in

case

9 positive solutions

of the E i n s t e i n - M a x w e l l

gives waves

the

so

with

C 0 F(~, ~ (~)

(12)

equations z-t)

when matter

definite

is p r e -

representing

c a n be g e n e r a l i z e d

by t a k i n g

F(~,

z-t)

and depend

where

arbitrarily

the

func-

on

.

generalization

Killing

vectors

(13)

ds 2 = V d {

where

(3,1) The

t = i ~

case

solutions

[ i(~,

tions

rotation

rays).

solutions

sent.

The

with

61(~, {) + E2(~, {)

If we m a k e

the

metric

of the o m o d e l

reduction

equations.

transcendents

{ = z + i~

ds 2 = v d {

The

is no r e a l

symmetric

to the

This

c a n be g e n e r a l i z e d

(12)

V =

there

instantons

[11]

hold.

the E i n s t e i n

Killing

on

but

non-axially

not

is g i v e n

(11)

(null)

to the

these

does

of

A clear

solutions

only

(2,2)

of the ~ m o d e l

on P a i n l e v 6

found. of

For

equation

solutions

depend

c a n be

associated

type.

reduction

the L i o u v i l l e

signature

(4,0).

of the

is g i v e n

solutions

(12)

to the

case

by

2 * 2 d{d[

+ 2 d~d~

+ Gd~

2

when

there

are no

291

with

~"t A1 + ~{" B1 -

~.~ A 2 ~[ B 2 = 0

1

1

1

1

2

oiI ~ A 2 + ~- B 1 ~ A 2 + ~-[B 1~? + B 2] 9,~ A 2 : 0 2.

~

~B2

+ ~ AI ~# B2 + ~ [ A I ~

Different

types of new solutions,

solutions

of the

metrics

(complex)

The solutions

(~,y)

(13) have

(anti)

14)

~

~

gauge

:

-

where

real)

has as

having real,

G

non-zero

By Wick rotation imaginary

consig-

there is no

of the coordinates

( $=

it

and

real one with signature

in general non-selfdual

(2,2).

curvature.

equations

[

euclidean invariant

space without formulation

[13] for the self dual Yang-Mills

( ~ ,~ , ~ , ~)

The Killing equations

of a Killing vector)

and

to be purely

self dual Einstein

±n four dimensional a manifestly Yang

V

the metric maps onto a different

The solutions

by

[12]. These

This allows us to assign a (++++)

For both

(3,1) signature.

or by specializing

For the

symmetry.

(in particular

eigenvalues.

to these metrics.

real solution with

parametrized

(K {, K ~, K {, K Z) = 0.

(13) are complex

stant and positive

sub-classes

equation are given in

(K i being the_ components_

the only solution

y = iY)

including

Liouville

do not exhibit any space-time

Ki; ]. + K.3;i = 0

nature

+ A2] ~F B2 = 0

any Killing vector,

analogous

field.

we give

to that given by

In complex

space

292 ~'-2 ¢' =X1 +

eq.

(14)

iX 2

can be w r i t t e n

,

~"2 "~ =

X3 + ix 4

as

(15)

where

R bv

are the f i e l d

strengths,

Fb

= G~; F

0

are the g a u g e p o t e n t i a l s

(GT)~ = 6 ~ 6 9o

are the g e n e r a t o r s

(Christoffel

of the GL(4

I °g I0 (complex)

tensor.

T-F/[,/~

Since

Here

0

connections),

Lie a l g e b r a

IS

and

the m e t r i c

is the 2 x 2 n u ~ m a t r i x .

/- P-( /~, / ~ )

are c u r v a t u r e l e s s ,

F

= j-1

j

/-

=

we can c h o o s e

a gauge

J

in

which

(16)

~

~+-~7~ j + Em)

its

is d e r i v e d

We d e f i n e

Any

to

by W a n g

"chunk-wise"

BASIC

manifold

idea

character

a local

condition

equivalent

2.

4, the

global

as an m - d i m e n s i o n a l

boundary) physical

which space

compact

manifests S,

which

differentiable

itself

through

con-

is an n - d i m e n s i o n a l

manifold.

configuration

K

is, b y

definition,

an e m b e d d i n g

K : B --> S of class The

ck(0

< k < + ~).

set

QB of

all

possible

configurations

sional

manifold

[7] w h i c h

sequel,

where

configuration

there

tion

space from

{~:

S ~

of a g i v e n shall

call

is no ~ m b i g u i t y ,

we

S} body

the

B

is an

infinite

configuration

shall

use

space

the n o t i o n

Q

dimen(in the for

the

space).

An e l e m e n t

away

we

:

has

6K

of the

the p h y s i c a l

tangent meaning

bundle

space

of a v i r t u a l

TQ

of the

displacement

configurameasured

the c o n f i g u r a t i o n K =~QI6K),

where

~Q

: TQ

an e l e m e n t

: TQ --> Q

is the

can be r e i n t e r p r e t e d 6K

of

the b o d y

the d i a g r a m

into

the

tangent

tangent (see

bundle

for e x a m p l e

projection. [8])

Indeed,

such

as a m a p p i n g

: B --> TS bundle

TS

of the p h y s i c a l

space

such

that

302

TS

B commutes, at

where

a given

a vector

~ S

is

the

configuration 6~(b) E TS

Intuitively,

we

natural

K ,

at

>

to

a point

conceive

projection

point

K (b)

in t h e

physical

of

a force

linearly

on

virtual

displacements.

a force

f

is

an e l e m e n t

A

force

where

f

that

is

the

natural

displacement

thus

with

i< =

~Q(f)

at ~Q

the

virtual

work

W

of

f W

where It




is w o r t h

denotes

6

is no

natural

such

~ can

A global behaviour

the

mentioning

placement

of

=

on

6


evaluation

map.

although,

as

with

representation

body

an

on

of

same

(f)

associated

elastic

the

the

= T ~ Q The

use

of

the

constitutive

term

law

configuration local

in t h e

ticular

As the

forces comes

~

but

sense

that

is m e a n t

not

also

characteristics

material

of

"global"

in w h i c h

its

only

to

the

action

on

it

involves

the

of

it

as

(such

stress

operator

the ~

a virtual entire

the

6 ~

value

of

generality depends

on

of the

displacement rather its

this entire

is n o n -

than

jet m a p

any

par-

at a

point).

a force

f

may

configuration, and

the

rather

the

"internal"

artifical.

in c e r t a i n

cases

distinction forces

In a n y

between

dictated

case

be

we

say

by

given

as

a smooth

"prescribed" the

that

function

(external)

constitutive a configuration

laws

be-

303

~

is a s t a t e

of e q u i l i b r i u m

if a n d o n l y

if

~(~o ) + f = 0 where

the

force

f

satisfies ~Q(f)

If

f is g i v e n

equation the

may

state

as a s m o o t h

be c o n c e i v e d

If the 8

one-form

and

CONSTITUTIVE

(global)

is e x a c t ,

8

the

SYMMETRIES

the e q u i l i b r i u m

problem

to be

solved

for

~

of

symmetry

if

{

of

exists

the m a t e r i a l

a real

scalar

is c a l l e d

global-

function.

the c o n f i g u r a t i o n the c o n s t i t u t i v e

on

the p u l l - b a c k

space law

Q

[

is,

by defini-

if

~,

:

as a o n e - f o r m

denotes

if t h e r e

= d~,

AND LOCALITY

~

star

value

i.e.

~

is its e n e r g y

(3.11 namely,

of c o n f i g u r a t i o n ,

as a b o u n d a r y

such that

A diffeomorphism a

function

[

: Q -> R

ly h y p e r e l a s t i c

tion,

~o

of e q u i l i b r i u m .

function

3.

=

also

Q

is i n v a r i a n t

operator,

under

so t h a t m o r e

~ .

In Eq.

explicitely

(3.1) we c a n

write (3.2)

< 6(~(X)),

for all derived

6 < map

It is n o t

and

for c o r r e s p o n d i n g

of ~

hard

to

see t h a t

forms

a group

of a c o n s t i t u t i v e

interest.

viously

under

Let

induces

B:

feomorphism

law

where

TQ -> T Q

T~

~ o

on

Q

and

which

:

we c a l l

Two particular

diffeomorphism

~ B

on

is the

Gs

the

g~lobal s y m m e t r y

sub-groups

of

of the b o d y . Q

GQ

are

It o b -

by

= ~oB o

of the p h y s i c a l

by

'~o(ic) GB

law,

be a d i f f e o m o r p h i s m

a diffeomorphism

sub-groups

6~>

set

~ .

B -> B

~S(~)

(3.5) The

the

composition

a unique

(3.4) Similarly,

< ,

> : < ~(%),

.

group of

T ~(6~

=

Co< of

GQ

defined

by

space

S

induces

a dif-

304 (3.6)

G B :f ~

d Diff. Ql

~ 6GQ

and 2//= ~ B

for some Bc- Diff. B [

and (3.7)

GS = I

~

will be called,

Diff-QI

~6

respectively,

groups of the c o n s t i t u t i v e

Let

VC B

GQ and

~=

the material and spatial global symmetry

be an open set in

in Eq.

B

and denote by

GQ(V)

and we call Let

= {~ GQ(V)

V x denote

6 g V'

GQ(V)

any virtual

then the group

say, w h i c h will

GQ

will

include more

law under the smaller

We may write

6 Diff. Q I < ~ , the global

6 KV

If we now should restrict the

p r e s e r v i n g the c o n s t i t u t i v e

class of virtual displacements. (3.8)

V.

(3.2} to such

in general be e x p a n d e d to a group diffeomorphisms ~

S]

law

d i s p l a c e m e n t with compact support in "test functions"

~ o for some o ~Diff.

6

6 g V > = < ~,

symmetry group of

~

6 ~Vt

~V > for each relative to

VCB.

now the family of n e i g h b o r h o o d s of a material point

We define the global symmetry group

GQ(X)

of

~

relative to

X

X6B. as the

union (3.9)

Thus

GQ(X)

GQ(X)

=



VB(b)

differential

us that P is locally

displacement

(3.14)

on a n e i g h b o r h o o d

that due to the a s s u m p t i o n

tells

configuration.

we could 6 ~>

P in

displacement

becomes

Theorem

of virtual

action

X ~ B on the w h o l e Otherwise

to realize

P: TQ -->~(B)

by the d e r i v a t i v e s

The

virtual

the

(3.13)

= I

where

on

generated

[10]).

at any given p o i n t local

action

write

VB(b)

the

operator

V B is a v o l u m e

is such

306

that

there

exists

displacement

a mapping

6~

(3.15)

P(6~)

and where induced

As rials,

jk 6 K

by

of o r d e r

(X)

is t h e

6<Eck(B,TS),

an e x a m p l e i.e.,

we c o n s i d e r

materials

introduced

with

B

such that

for e a c h v i r t u a l

X6 B

of the k - t h say

means

that

that

jet f i b r e

bundle

the m a t e r i a l

jk(B,TS)

is j e t - l o c a l

for h y p e r e l a s t i c

potential

e n d of S e c t i o n

the v i r t u a l

6~ > : 6~

2.

@

such

that

In g e n e r a l ,

mate~

= d8

for a g l o b a l -

if t h e r e ~

(~)

work

is g i v e n

b y the F r e c h e t

differential

We

say t h a t

a hyperelastic

material

exists

a scalar

function

6 ~.

(3.17)

jet-locality

we h a v e

= ~ point 6 ~(X))

t h e n we

material,

8

= m(jk

section

ly h y p e r e l a s t i c

of

jkTQ

k.

as a l r e a d y

which

~:

and each material

:

has

B x Q -> R

such that (3.18)

e

where

vB

Therefore, given

is

a volume

the

element

hyperelastic

local

potential

~

B. action

P

is

at

each

point

X~ B

(X)

~(X,K

then we

say t h a t

defined

the m a t e r i a l

previously

in

(3.21)

: 6~

) = ~(X,

material

The

work

virtual

point

jk W(X))

is j e t - l o c a l

(3.15)

has

a(j k 6 ~ (X))

for a g i v e n

(~(X,~),X)

is s u c h t h a t

(3.20)

(3.221

on

local

P ( 6 Z)

If the

is as

~VB

by

(3.19)

a

P JB

=

of o r d e r

k

as t h e m a p p i n g

a form : 8 ~ (~(X,j k < ( X ) ) , X )

X 6 B.

is n o w g i v e n

by a first

variation,

which

in c o m p o n e n t s

follows



:

k ~ j:0

Q~

?

6( < i '~ I " ' ' ~ 3

,

~

~j) I"'"

VB

307

4.

MATERIAL

UNIFORMITY

Roughly

speaking,

a body

made

of the

same m a t e r i a l .

this

way

formulating

of

presupposes

locality.

of the

into

make

body

this

say that

the

if there

exists

In the

the

The

and

precise

point

Yg B

context

idea, then

by

X

however,

:

B

is that

the

the

following

B

such

are theory,

since

of m o v i n g

local

isomorphic

of

global

is p r o b l e m a t i c

for

suggesting

is m a t e r i a l l y

if its p o i n t s

of a c o m p l e t e l y

checking

a diffeomorphism

(4.1)

uniform

idea of u n i f o r m i t y

key

another

idea m o r e

is m a t e r i a l l y

it

one p i e c e

response.

We

definition:

with

the

we

point

X 6 B,

that

B(Y)

and (4.2)

~B

where

~B

ficult

is a d i f f e o m o r p h i s m

to s h o w

We are

compactness

larly

of

when

the

of g e n e r a l i t y ,

For

this

the

say that

materially

two m a t e r i a l

(4.3) ~ &Q,

We have

now

: TyB

global

L~Xy

uniformity,

uniformity

of

defined

uniformity

tisfying

where

X

and

(4.1)

The

by

(4.3).

implies and

converse,

B

the

(4.2)

however,

Clearly,

by

as

effect.

we can follows.

a material

are

isomorphism

LXy. for a l e t - l o c a l

for e a c h

pair

(4.1)

and

for t h e s e

one,

needs

hand,

such

an

to this

that

uniformity

satisfying

local

level

= L XY m(j 16 re(Y)

requires

TB I TyB

of

exists

such

particu-

At this

isomorphism

Y say,

if there

-> TxB

other

unrea-

body,

can be m a d e

I, on the

required

physically

same.

the

of

its p o i n t s

of the

S

are

definition.

if all

global

is i n d u c e d

which

some

dif-

[5] d e f i n i t i o n

because

imply

statement

of m a t e r i a l

B ~ Diff.

standard

that

It is not

above

if and o n l y

of m a t e r i a l

points,

8.

of a t r u l y

of o r d e r

if a n d o n l y

two n o t i o n s

the e x i s t e n c e

(4.3).

and

B

m(6(j1.~C~Lxy))

for all

bal

of

definition

isomorphic

may

layer

no p r e c i s e

LXy

the

definition

material

standard

Note

by

of the

the

uniform

"outer"

dimensions however,

generated

to a d o p t

isomorphic.

in an

a jet-local

rephrase We

B,

Q

transitivity

is m a t e r i a l l y

materially

behaviour

of

and

in a p o s i t i o n

a body

pairwise

sonable

reflexivity

now

uniformity: are

6 GQ(Y)

since

be

type

and

material

true~

in

B

the

local

of m a t e r i a l s ,

for d i f f e o m o r p h i s m

is a local not

of p o i n t s (4.2)

material:

B

isomorphism

a locally

uniform

glosaof

308

material

needs not be g l o b a l l y

is n o t

guaranteed

Therefore, sary and

the

remainder

sufficient

to be g l o b a l l y

near

of

frames

[cf.

on

B

Sternberg

if a n d o n l y

that

pg

[11]].

isomorphisms

such

is d e v o t e d

for a l o c a l l y

s u c h that,

manifold

local

X c B

We

for a n y p o i n t

show

LXy

first

gives

and a linear LXX

that

rise

~

(4.5)

X

(gij)

corresponding

form a sub-group

ar f r a m e s

in

TB

isomorphisms

material

(e I .... ,em)

G

obtained

m ( E i:i

g ( G local

on the b o d y Given

in

a mate-

TxB,

for

(gij)£GL(M,R)

m Z eigim) i=i

eigil,...,

to all

p

li-

i.e.

of G L ( m , R ) . from

g =

if

of all

uniform.

a matrix

is a s u b of all

and any matrix

the c o l l e c t i o n

p :

G ~

to a G - s t r u c t u r e

exists

(Lxxe I .... , L x x e m ) :

where

if a n d o n l y

is l o c a l l y

frame

there

B,

pe ~

in

L X X p = pg

matrices

of n e c e s -

jet-local

of the m a n i f o l d

is c o n t a i n e d

(4.4)

at

isomorphism.

to a s t u d y

uniform

manifold ~

if the m a t e r i a l

symmetry

The

of d i f f e o m o r p h i s m

local material

section

is a s u b m a n i f o l d

material

rial point

of

on a m-dimensional

the p o i n t

each

of t h i s

conditions

GL(m,R) ,

ge GL(m,R)

as the e x i s t e n c e

uniform.

A G-structure group

uniform

b y the e x i s t e n c e

local Let

b y the

material

~

symmetries

be the c o l ~ c t i o n

action

of all

L y x , that is a frame q at Y B is c o n t a ~ e d ~

local

of l i n e material

~ if and o n ~

if

there exists a l ~ a l material isonmrphismLyx such that q = Lyx p. It is easy to verify n o w t h a t ~

is a G-structure on B if and only i f t h e m a t e r i a l

Let

now

us c o n s i d e r

consists

of the

a triclinic

for s i m p l i c i t y

identity

crystal.

and applying

Starting

local material

trary m a t e r i a l

point,

B

for e a c h

in

s u c h that, p

there

at

exists

smoothly

The

(4.6)

Y.

on

matrix

Note

with

that

a unique

G =

local

This

case when

corresponds

the b a s i s

isomorphisms

we o b t a i n Y ~ B

a special

only.

smooth

([I(Y)

p =

{ identity}

where

vector

fields

material

means

isomorphism

the g r o u p

Y

Lyx

b CC 13 m

functions

[ ~ i' ~j]

=

E

k:l

on

B

Ck

are d e f i n e d

~j fk

unique

for e a c h which

Y.

structure

at

by

to X

is an a r b i -

~I ..... ~ m

is the that,

G

for e x a m p l e

( e l , . . . , e m)

Lye,

..... ~ m ( Y ) )

is locally uniform.

on

frame Y~ B

depends

309

In this p a r t i c u l a r cal u n i f o r m i t y structure

G

here,

is of finite

of order

the

of our next

Sternberg

[11]]

that

if and only

lo-

if the

B. can be reduced

of p r o l o n g a t i o n s ,

[11] but this

we have one

assumption

paper

shown

structure

provided

to the that

is s a t i s f i e d

to be

functions

locally

uniform

special

the group

in all the

ck

this

is g l o b a l l y

uniform

is locally

homogeneous.

Global

uniform

jet-local

it is n e c e s s a r y

on B.

homogeneous.

This

material

but not every uniformity

leads which

local

now a special

for a locally

us to the concluis also

globally

appears

ma-

sufficient

It will be a task

is in fact a c r i t e r i o n

stage b e t w e e n

and

. One can c o n s i d e r

C kij z]vanish

jet-local

homogeneous

an i n t e r m e d i a t e

for a locally uniform

functions

to show that

sion that a locally

that

to be g l o b a l l y

structure

uniform material

materials

[c.f.

uniformity

in elasticity.

to have c o n s t a n t case when

by means

type

considered

Therefore, terial

to global

ck are c o n s t a n t on 13 of an a r b i t r a r y group G

case

discussed

cases

it can be shown

functions

The general case

case

is e q u i v a l e n t

locally

uniform material

to be for jet-local

uniformity

and

local

homogeneity.

REFERENCES

[I]

Epstein, M. and Segev, R., " D i f f e r e n t i a b l e M a n i f o l d s and the P r i n c i p l e of V i r t u a l Work in C o n t i n u u m M e c h a n i c s " , J.Math.Phys. 21(5), 1980, 1243-1245

[2]

Segev, R. and Epstein, M., "Some G e o m e t r i c a l A s p e c t s of C o n t i n u u m M e c h a n i c s " , D e p a r t m e n t a l Report No. 153, Dept. of Mech. Engg., U n i v e r s i t y of Calgary, March, 1980

[3]

Segev, R., " D i f f e r e n t i a b l e M a n i f o l d s and Some Basic N o t i o n s of C o n t i n u u m M e c h a n i c s " , Ph.D. Thesis, Dept. of Mech. Engg., Univ e r s i t y of Calgary, May, 1981

[4]

Segev, R. and Epstein, M., "The P r i n c i p l e C o n t i n u u m Dynamics", 1981 (unpublished)

[5]

Noll, Arch.

[6]

Wang, C.-C., "On the G e o m e t r i c S t r u c t u r e s of Simple Bodies, a M a t h e m a t i c a l F o u n d a t i o n for the T h e o r y of C o n t i n u o u s D i s t r i b u t i o n s of D i s l o c a t i o n s " , Arch. Rat. Mech. Anal. 27, 1967, 33-94

[7]

Michor, London,

[8]

Ebin, D.G. and Marsden, J., M o t i o n of an I n c o m p r e s s i b l e 102-163

of Virtual

W., " M a t e r i a l l y U n i f o r m Simple Bodies Rat. Mech. Anal. 27, 1967, 1-32

R.W., 1980

"Manifolds

of D i f f e r e n t i a b l e

"Groups Fluid",

with

Work

and

Inhomogeneities",

Mappings",

Shiva,

of D i f f e o m o r p h i s m s and the A n n a l s . M a t h . , 92, 1970,

310

[9]

Noll, W., "A M a t h e m a t i c a l Theory of the M e c h a n i c a l B e h a v o i r of C o n t i n u o u s Media", Arch. Rat. Mech. Anal. 2, 1958, 197-226

[I0]

Kahn, D.W., "Introduction to Global Analysis", New York, 1980

[11]

Sternberg, S., "Lectures on D i f f e r e n t i a l Geometry", Hall, E n g l e w o o d Cliffs, New Jersey, ]964.

A c a d e m i c Press,

Prentice-

DIFFERENTIAL THE

GEOMETRICAL

THEORY

APPROACH

OF A M O R P H O U S

TO

SOLIDS

R. K e r n e r

Departement Universite 4, P l a c e

I.

The u n d e r s t a n d i n g difficult

75005

it h a s

network notion gress

widely

Curie

Paris,

qualitatively,

more

seriously

In w h a t

FRANCE

shall

give

follows,

coming

we

hints

The m a i n

from quite

the to

concerning goal

distant

such

physical

paper

random

way,

some

structures

enabling

a pronot

us to t r e a t

systems.

ourselves

to the d e s c r i p t i o n

number

N c at the e n d of this

the g e n e r a l i z a t i o n

of t h i s

have been such

lack

Zachariasen

continuous

there

coordination

3, o n l y

by

to be a

of the

it is too v a g u e

to d e s c r i b e

restrict

with

equal

of

work

however,

Recently

proved

because

so-called

in a q u a n t i t a t i v e

shall

model

neighbors)

some

N c ~ 3.

also

solids

probably

fundamental the

in a t t e m p t

thermodynamics

of a t w o - d i m e n s i o n a l b e r of c l o s e d

the

that

constructively.

but

the

Since

most

modelisation,

[2],[3],[4],[5]

only

or g l a s s y

with,

admitted

is the p a p e r

to w o r k w i t h made

to d e a l

description.

been

(CRN)

of a m o r p h o u s

problem

of an a d e q u a t e

and

Jussieu,

et M a r i e

INTRODUCTION

very

[I]

de M e c a n i q u e ,

Pierre

to t h r e e

is to e x p l a i n

theories,

how

s u c h as g a u g e

(the n u m paper

we

dimensions the

ideas

fields

or

312

gravitation, least

may

the m a i n

Before

be h e l p f u l

directions

explaining

set up the p r o b l e m consider to be and

a model

identical

atoms,

lent

silicon

of b o u n d s lized

bonds

atom

in w h i c h

allow

them

solid,

each

closest

to

approximation

i.e.,

the

atoms

to.

We

roughly

equilateral

atom being

p l a c e d in a vortex i n w h i c h

We

shall

call

its

three

closest

by t h r e e

unit

set of all

kl,k2,k 3

in

cell

information and

elementary

~2,

and b o n d s

we

then

elementary

cells

meeting

at the

(ki-2) ~ ki four (Fig. to

,

tripod

in i n t e g e r

These

perfect

"criystalline

all in

polygons

(k2-2)~ k~-~ +

numbers:

configurations"

(6,6,6), nets There

which

sides

the

that

at P. is

We b e l i e v e

of f r e e d o m

of

to d e t e r m i n e the p o l y g o n s cells

iden-

is so b e c a u s e

are

non-perfect

polygons.

networks.

The p r o b l e m to

form

, and

(4,8,8),

if

angles

lots

And

(3,12,12).

regular

(not all exists,

is to k n o w why,

crystalline

are o n l y

and c o r r e s p o n d

of o t h e r

there

there

(4,6,12),

known

are n o n - h o m o g e n e o u s

and w i t h

prefers

This

are w e l l

of r a n d o m Nature

of

elementary

(k3-2)~ k3 2V

identical),

circumstances,

~2.

P

(kl,k2,k3) , the t h r e e

an

infinity

the

at

meeting

sufficient ask

visualized

are

homogeneous

configurations

i.e.

if we

together

atoms.

degrees

is q u i t e

example,

solutions

(ki-2)~ kI +

therefore

semi-regular

four

For

(k1+k2+k3-5)

internal

so d e f i n e d

o u t of p e r f e c t

central

solutions I).

only

cell

number

the net h o m o g e n e o u s ,

we h a v e

P

(N c = 3).

bonds,

whose

and

is m a d e

an a t o m

covalent

out

each

meet

the p o l y g o n s

be p e r f e c t then

polygons,

polygons

elementary

of b o n d s

as the

constituted

polygons

random

cell

are

three

our

each

other

to the

completely

tical,

P

to be

pattern

each

nets

at

in the

network.

supposed

whatever

from

three

it c o n t a i n s

contained

norma-

are

perfect)

call

to t h r e e

tetra-va-

and

tripod

shall

a bond each

to be c o n s t a n t

three

the

such

supposed

unoriented

length

always

and

are

The

forces

far

atoms

shall

neighbors).

resulting

belonging

contains

closest

necessarily

neighbors

respectively,

the

tripods

call

vectors

atoms

If an e l e m e n t a r y

that

(but not

via

let us

We

the b o n d s

dimensions

speaking,

that

all

is l i n k e d

to be as

suppose

all

(in three

four

of convex,

with

the

interatomic

tend

fruitful,

in w h i c h

is s u p p o s e d

The

quite

at

be a t t a c k e d .

context.

atom

its

found

and d e f i n i n g

could

and p h y s i c a l

silicon),

neighbors

is l i n k e d

and central,

in place,

we

in an a m o r p h o u s

I for c o n v e n i e n c e .

repulsive is put

and

called

insight

the p r o b l e m

analogies

of a c o v a l e n t

in f i r s t

to

the

a new

which

in its g e o m e t r i c a l

(like

equivalent,

other

in f i n d i n g

along

(regular)

or

the c e l l s

of course, under

some

configura-

313

X > x

(6,6,6)

(4,8,8) Fig.

>-
>-< (3,12,12) (4,6,12) T h e f o u r r e g u l a r honmgeneous p l a n e . The e l e m e n t a r y c e l l s (3,12,12) .

tions,

while

introduce a least

in o t h e r

kinematics

action

configurations

2.

tion fact,

conditions and dynamics

principle, are m o s t

KINEMATICS:

or

DESCRIPTION

FIBER

The

considerations

of the

random

if we c o u l d

(xi,Yi) ~ ~2,

it p r e f e r s of

random

such networks

its a n a l o g ,

likely

OF D O U B L E

above

t r i - c o o r d i n a t e l a t t i c e s o n the are: (6,6,6) , (4,8,8) , (4,6,12) ,

which

networks. in o r d e r

will

decide

We h a v e

to

to f i n d o u t what

kind

of

to a p p e a r .

OF A C O N T I N U O U S

RANDOM

NETWORK

l e a d us q u i t e

naturally

IN T E R M S

BUNDLE

network determine

i = I,...,N,

in t e r m s

of

the e x a c t taking

into

fiber

bundles.

position account

of e a c h the

to the d e s c r i p As a m a t t e r

of

atom,

constraints

for the

314

closest

neighbors

the w h o l e result

should

because

be

would

not

it m a k e s

no s e n s e

to

tion"

"absolute

to s p e a k spect

of

with

words,

speak

the r e l a t i v e

to the b u l k

for all.

respect of

atom,

atoms

should can

themselves

displacement

of d i s t a n t

atoms

Like

"absolute

of a n y

other

(like

their

position",

or c h a n g e the

only

the

upon rela-

Relativity,

"absolute

it w i l l

make

in d i r e c t i o n

distant

~2,

closest

in G e n e r a l

whereas

of

the

not depend

feel

and

then

hand,

to the r i g i d m o t i o n s

neighbors.

about

of polygons,

On t h e

the n e t w o r k

between

so close)

velocity"

the c o n v e x i t y

once

individual

and directions

(and m a y be o t h e r

or

: I) a n d fixed

properties

In o t h e r

distances

be

invariant

the e s s e n t i a l

the o b s e r v e r . tive

(distance

network

stars

direcsense

with

re-

in G e n e r a l

Relativity).

In the v e r y sest n e i g h b o r s

first

should

d o m of an e l e m e n t a r y which

cannot

distance greater

than

are

space

of the

open

atoms

~

in o r d e r

degrees

of

as f o l l o w s :

bonds,

to

in the p l a n e

the

and

degrees

of

given

by t h r e e

unit

polygons.

tripods

denote

to c h o o s e These

(@i,~2)

to k e e p

can not

tripod,

the a n g l e s

conditions

cut

on Fig.

vectors a minimal

Because

a representative

displayed

free-

f o r m an a n g l e

unoriented,

of an e l e m e n t a r y

~i,~2,~3

"

in o r d e r

concave

the

freedom let

the c l o -

internal

are

interaction

1, a n d w h i c h

to a v o i d

it is s u f f i c i e n t

cI 4 c~2 4 ~3 ~ ~ set

which

between

b y the

to e a c h o t h e r ,

equal

indistinguishable

rametrized

with

tripod,

too c l o s e

between

bonds

three

be

approximation, be p a r a m e t r i z e d

out

the

all

the

internal

E,

c a n be p a -

between

the

parametrization a quadrilateral

2.

~2 Fig.

~,

2

,"

i

The manifold representing the i n t e r n a l d e g r e e s of f r e e d o m of an e l e m e n t a r y tripod. The point A corresp o n d s to the c o n f i g u r a t i o n ~ i : ~ 2 : ~ 3 = 2 ~ / 3 , the e d g e A B corresponds e d g e BC

r//I

'~,,

pI/ i

%\%

/s

k\ /3

2 /3

to ~ i = ~ 2 , t h e

to ~3 = ~ ,

C D to

~i = ~ / 3 ,

D A to

~2=~3

and

the e d g e the e d g e

315

It has only

an u n f a i t h f u l

onto of

to be u n d e r l i n e d

~2. ~2,

this

can be

by A B C D

they

been

have

ginal

plet the

only,

We can [2

of

the

a structure with

the

fact

that

group

abelian E

A net having

N

more

the

information

the

as

corresponds whose i.e.

we

tripod,

of

should or at

internal tral

ters

~2 or

the

~3'

to e a c h

of

of

upper

will

this

we m u s t

if t h r e e

reduces

have

the

F,

the

clo-

structure (kl,k2,k 3) polygons

triplet angle

is o r d e r e d , ~I of the

kl,k2,k 3 correspond This

which the

meet

shape

leaves to the

of the

k-gon

at a v o r t e x

(kl-3)+(k2-3)+(k3-3)-3 the p o s s i b i l i t y

still

correspond

an e q u i l a t e r a l

of polygons'

des-

together

with

of t h r e e

in w h i c h

polygons

E.

is some

fiber

numbers

fixed.

k I + k 2 + k 3 ~12.

for the n u m b e r

has

at the

freedom,

cell

in

can not

There

its b o n d s

triplet

bundle:

to an a t o m

integer

~i,~2

space

the

of this

a second

freedom

is b e c a u s e

be o n l y

of course,

bound

of

an e l e m e n t a r y

freedom;

there

This,

degrees

for

but

a section

by

is f o u n d

unordered

of p o l y g o n s

fixed;

of

angles.

a tripod

polygons.

k l , k 2 , k 3. This

the k l - g o n

+ k 2 + k 3 - 12)

degrees

degrees

identi-

in i n t r o d u c i n g

~2+6),

them

n e x t the fiber f, w h i c h

internal

the

the b a s i s

section of

such

formed

of

from

equal

definition.

corresponds

triplet

A

tri-

it is d i f f e o m o r p h i c

the

introduce

shape,

two

from

with

(~i+~,

closed

we

is r e s p e c t i v e l y

positions

has b e e n

way,

form

DA c o m e

different

locally

to e a c h

P(IR2,E)

to e a c h

know whether

k l , k 2 , k 3 at will: also

of

sides

is given, left.

in

introduce

parameters

tripod

internal shape

(k]

therefore,

A point

not

such

complicated.

by a d i s c r e t e

attribute

needed;

foliation:

six d i f f e r e n t us w i t h

we

the p o i n t

f r o m one

difficulties

->

set

f r o m the o r i -

CD and

three

complicates

will

We

that

the

tripod,

example,

come

P(~2,E) some

so in an a r b i t r a r y

a space

number

~2

of a p a r t i c u l a r

sest n e i g h b o r s . of a s i n g u l a r

a boundary

tripods

follows.

tripod

between

bundle

of

containing

set q u i t e

We h a v e

of a s u b s e t

coming

BC,

the q u a d r a n g l e

be r e p r e s e n t e d in

if we do

a net:

defined with

should

AB,

triplets

S I x S I : (~i,~2)

have

For

is

its p r o j e c t i o n

point

~1=~2=~3=~comes__

it is o b v i o u s

torus

does

points

Of course, cribe

here,

each

(~i,~2,~3).

inside

E.

of

topology

classes

on the e d g e s

a fiber

fiber

a kind

figure

of an e l e m e n t a r y

equivalence

of the w h o l e

construct

typical

shape

permutations

topology

already

and

although

equivalent

the p o i n t s

set,

see on this

to the

follows:

situated

of three

we

a unique

of a n g l e s

six p o s s i b l e the

of this

configuration

the p o i n t s

makes

as

what

equivalent

as some

triplets

to the

finally,

fication This

seen

represents

identification

angles,

is not

obtained

unordered

corresponding

that

representation

Its t o p o l o g y

delimited

here

free

cen-

has

k-3

whose parame-

in c h o o s i n g

In a r e a l i s t i c sides

should

model

be giv~%~

316

in o r d e r should

to

include

go up to k

all

the

( 12,

l

regular

in the

homogeneous

simplified

lattices

version

known,

it is e n o u g h

we to stop

at k I = 8. The sheafs

second

which

are

Given

of a t r i p o d

two p o i n t s

(kl,k2,k3);

(kl,k2,k3)

is a s i n g u l a r

to p a r t i c u l a r

deformation

any

of g i v e n

F

correspond

a continuous tinuously

fiber

in s h e a f s

on the

oriented in

E

triplets

the

all

to

to d i f f e r e n t

sheafs

with

the

(kl,k2,k3).

it is p o s s i b l e

corresponding

contrary,

containing

By

join

con-

orderings

different

numbers

disjoint.

a continuous

random

network

it as a g l o b a l

discrete

section

commuting

the two

canonical

with

foliation

of

with

N

: 3 we can r e p r e s e n t c fiber b u n d l e P 2 ( P ( [ 2 , E ) ,F) ,

the d o u b l e

projections

~2

and

~I

defined

a very

delicate

na-

turally:

P2 (P (R2'E}'F)

(2-i) In the play

double

between

what

for e x a m p l e , ~i (x,u) closest fiber

if we

= x,

only

very

tripods

E

polygons

as we can

possible

sections

point

of

~I.

a point able

are

them

is a t r i v i a l

to w h a t

point way.

~I

A lift

is an a s s i g n m e n t

of

~(t)

If a c u r v e

is g i v e n

~(t)

•

x

all

the

with

happens

to e a c h

sections ~2 in

in w h i c h

the

tripods

t,

as w e l l

as the

the

inverse

is needed:

having

the

in c l a s s i c a l

on

mechanics;

in mind.

space

with

from

a geometrical

is time

represented

value

space

the p r o j e c t i o n of

"

at e a c h

of

and

restrictions

t,

it d e f i n e s

~

a relation

between

~,

~3,

(t,~)->t.

we can d e f i n e the v e l o c i t y ,~ dv accelaration a(t) = d--t" In real

problems

second

three

net

is the c o n f i g u r a t i o n

x ~3

three

to the

situation,

it; that

in the

discrete

kinematical

we have

of the

inter-

over such

a point

of a g i v e n

real

The b a s i s

~I

u ~ E

belonging

can be r e g a r d e d

there

bundle

points

coincide

in R 2. T h e s e

view in the f o l l o w i n g of

F

analogy

also

vertices

to the in

fibers

the p o s i t i o n

Among

the

correspond

similar

point

x.

in the

a point

choosing

of o t h e r

onto

the p o l y g o n s

and

to d e f i n e

at

project

see

x ~ ~2

(Xl,X2,X3} ;

of a m a s s i v e

At e a c h

the w h o l e

and

the m e c h a n i c a l

Mechanics

by

x,

ones

and

let us r e c a l l

exists

space

to m e e t i n g

which

special

in

there

the p o s i t i o n s

adjacent

P2(P(~2,E),F)

so d e f i n e d in the base

we are

of

defines

polygons

choose

then

neighbors F

bundle

happens

]/1 > ~2 ....

__~2 > p ( R 2,E)

a curve

~(t)-

dx dt mechanical ~ and

317

we

search

~(t).

as a d o u b l e

We o p e r a t e

fiber

bundle,

(the c o n f i g u r a t i o n cities).

A second

onto

x ~{3:

~I

kinematics

tell on the

for w h i c h

~(t)a=

starting ~(t),

from

x(t),

space),

dt

'

the

curve

fiber

systems

can be g i v e n

by

(the s p a c e

from

exists

us t h a t

they depend

of v e l o -

(t,x) .

The

in

(~I x ~3)

one

and only

higher

determined

~3

(~I x ~{3) x ~{3

of c u r v e s

tell

bundle

correspond

there

c a n be u n i q u e l y

can be v i e w e d

fiber

~3

now,

infinity

dynamics

which

first

shall

(t,x(t)) ,

In c o n s e r v a t i v e

a n d an a r b i t r a r y

the

second

(t,~,~)

among

same c u r v e dx ~d(t) . The

~ -

space,

~{I ,

c a n be d e f i n e d

point

us t h a t

t.

a n d the

projection

to e a c h

projecting

in the p h a s e

the b a s i s

one

derivatives,

as f u n c t i o n s only on

a differential

x ~3

x

of

and

v

system

dx

d-~ : A(~,v) (2-2) dv dt The

dynamics

kinematics curves

of the

tell

for w h i c h

or r a t h e r

with

fied with

E,

rules

to

the

to t h e

k i n d of t r i p o d s

vicinity

A(x,v)

and

have

E,

= v.

v

space

are

~2,

with (2-2),

to be e n c o u n t e r e d

and what

sufficient

We hope

to d e t e r m i n e

are

will the

F.

t

be

to e s t a b l i s h

rules

will

in the v i c i n i t y will

are

tell

the n a t u r e

some

of a g i v e n

rules of the

~2,

identi-

us w h a t

be e n c o u n t e r e d

these

the

with

space will

We h a v e

that

B(v,x) , the

admissible

identify x

these

k i n d of p o l y g o n s cell.

function

which

analogy

of

equations

in the

curves

Our

subset

of an e l e m e n t a r y

of m o t i o n " )

contained

the o n l y

a discrete

analogous

belonging

system are

us t h a t

B(x,v)

tripod

in the

("equations resulting

lattice.

All ticle

we needed

was

the

i has

situations

~

(X,~)

dt

Thus

from

; for

any

a function

we

will

be

similar

First

of all,

shall

encounter

roughly

contained dering

of c l a s s i c a l

mechanics

of a p a r -

interval

[tl,t2]

this

action

tI

reasoning

lattices.

be

case

integral:

on

IR3 x ~3

we c a n

deduce

the

(~(t) ,v(t)) .

Our

will

in the

t

to be m i n i m a l .

curves

dom

to k n o w

f o r m of the a c t i o n

similar,

in some

of the m e a n

relation

applied of

at d i f f e r e n t

and that

integral

when

the n o t i o n

over

points

the e s s e n t i a l

the basis

between

to the

homogeneity

[2

the p o i n t s

of

continuous

requires the b a s i s

information this in

E

that

amounts

ranthe

~2

should

be

to c o n s i -

a n d the p o i n t s

in

318

F.

For

example,

characterized ~1

=

~2

which

=

makes

bution

~2

Next

unique

we

should should

useful

way

compact whose

(kl,k2,k3)

and

gauge

obtain tell

basis

l-form

~

over

shall

of m i n i m a l field

is the

the p o i n t s

space-time along

the

with

vector

field

subgroup

then

G,

(2-3)

~X

field

and vertical (2-4) The

part

~

~

is c a l l e d

The

over

curvature

(2- 6 )

G

gauge

in the P

to

Lie

internal G,

of the

fiber

on

variaaction.

theories.

the a c t i o n

in a

symme-

usually bundle

P(V4,G)

trans-

is a l e f t - i n v a r i a n t

algebra

generated

the

group

acts

over

our

the g a u g e

The

field

F,

minimize

define

as a f i b e r

group

The

values

E

Lie

P c a n be d e c o m p o s e d

= d ~(hor

action

V 4. The

some

averaging

by any

~G;

if

X

one-parameter

now

into

a horizontal

, X,

differential

(X,Y)

is t a k e n

over

distri-

after

integral.

by

F

perfect

from

they

to

three

from

comes

[7],

E

= a d ( - X)

X = hor X + ver

covariant

(2-5)

X

recall

[6],

fibers.

with

configurations

curvature

which

P(V4,G)

vector

in § I a r e

is the a v e r a g e

in E,

function

are d e s c r i b e d

a left-invariant of

some

us w h i c h

we

semi-simple,

lating

Any

mentioned

(6,6,6)

lattice

the p o i n t s

formulation

in t e r m s

a cell

in a r a n d o m over

analogy

geometrical

of the

to t h i s p o i n t

is i m p o r t a n t

principle

In t h e i r

tries

correspond

of c e l l s

tional

lattices

b y f i x i n g one a n d o n l y o n e p o i n t in E, e.g. 2~ and defining a constant mapping from

What

the b a s i s

homogeneous

= ~-- '

~3

hexagons.

the p e r f e c t

so t h a t

of ~

X,

,

integral

/

defined

hor Y)

or the g a u g e

of

the

~ (X)

=

~(ver

X)

= 0

as I + ~

= d ~(X,Y)

field

tensor.

gauge

theories

[ ~ ( X ) , ~ (Y)] ~ G

is t a k e n

as

£ A * ~ dp P (V 4 ,G)

The

important

thing

is t h a t

therefore,

we c a n w r i t e

(2-7)

]

~ A~

~

is i n v a r i a n t

dp

: VG /

P (V 4 ,G) where

VG

is the t o t a l

along

(d]t~)A ~ V4

volume

of the

group

G.

the

fibers

(d t r Y ) d 4 x

of

P;

is

319

The

invariance

variational

In our case

case

in the

first

tion

bundle,

another

to the

second

The c u r v a t u r e

2-form

of

(2-9)

second

in

P

to a

two c o n n e c t i o n s ,

bundle.

Let

by

P2(P(V,GI),G2)

X = hor2X

= 0, being

us d e n o t e

~,

and

by

A.

the

Now

and v e r t i c a l

+ ver2X,

i(ver2X)

In the

P 2 ( P ( ~ 2 , E ) ,F).

P ( V , G I)

its h o r i z o n t a l

A,

A

to d e f i n e

bundle

into

A(hor2X)

bundle,

has

in the

bundle

connection

(2-8)

problem

fiber

one

first

can be d e c o m p o s e d

respect

one the

connec-

any v e c t o r

parts

with

with

= A(X)

horizontal,

F = D A = d A ( h o r 2, hor 2)

we can h o w e v e r it onto

decompose

P(V,GI) ,

to ~

: if

then

[ = hor I [

X

into

a horizontal its

+ ver 7 [ ,

then

form

(2-71)

B

second

part (2-6) ~ ~

@

splits

=

now

~(ver I ~)

therefore ,

is i d e n t i f i e d

splits

from

P2'

vertical

after

parts

d~2(X) 6 TP(V,GI) ;

,

B = A o hor I

integrand

and

projecting

with

respect

let d~2(X)

= f,

with

~ (hot I ~ ) = 0

The c o n n e c t i o n

vector

horizontal

is A - h o r i z o n t a l ,

(2-70)

The

a double

bundles

in the

in the

a variational

V 4.

fiber

]-form

l-form P2

us to r e d u c e

in

we have

of p r i n c i p a l

connection

in

enables

problem

into

into

=

{

two

invariant

parts

~ = A over 1 by p h y s i c i s t s

three

as the

Higgs

field.

The

parts:

( ~ O h O r l ) / k ~ ( ~ o h o r I)

+ 2(~

hor])A ~ (~Over

I) +

(2-12) (~overl) which

are

identified

Lagrangian

of the

field,

finally,

and

variational the p u r e etc.

as the L a g r a n g i a n

interaction

between

the p o t e n t i a l

principle

gauge

A~ (~°verl)

gives

rise

configurations,

of the

the p u r e gauge

of the p u r e to the m i n i m a

e.g.

the

stable

gauge

field

Higgs

field,

and

the

field.

which

are

Yang-Mills

the

Higgs

This

new

impossible

in

monopoles,

[8] , [9] .

There construction

is one which

radical

difference

is a d a p t e d

between

this

to the d e s c r i p t i o n

approach

and our

of the a m o r p h o u s

solids.

320

In our

case

the h o m o g e n e i t y

in a v a r i a t i o n a l the

first

citly

principle

fiber

on the

bundle.

not

in

space,

the b a s i s

words,

~2,

/

the b a s i s

over

In o t h e r

coordinates

(2-13)

concerns

we have

L(x,e,f)

space,

as n o t h i n g

being

the

" v o l u m e " "of the

in a d i s c r e t e in the on

F,

in

Although

dP2

For

E

that

task.

space

its g l o b a l

tifications

that

automorphisms

shapes

of

F

trivial

mation

Physical

forces

are lead

by a solid, of a t o m s

group

the m e a n

acting

value

co-

and

as a f u n c t i o n

E

trivial

if t h e r e

of

to be

help

us

of the

lattice.

these

same

of P(~2,E)

contrary,

an

the the

us a c l u e

three

conas

adjacent

infinitesimal

de-

such a connection:

in c e n t r a l

tripod's

polygons

that will

a minimal

w e can n o t

construct

find

properties.

of

available

time

we can

R2:

iden-

group

between

give

on t h e

adjacent

Although

of the The

connection

in d e f i n i n g

at the

explicitly,

E.

On the

a change

on

to a d d i t i v e

possible

undergoes

fibers.

acting

information

it s h o u l d

the

our

'

of

The

imposed

in a c e l l

group

at all b e c a u s e

~2.

trivial:

upon

~2 + C2)

is no

is s t i l l

to d e f i n e

equivalent

complicated.

provoking

possessing

of the

at the b o r d e r s

and repulsive,

its c u r v a t u r e of

has

to a d e f o r m a t i o n

surrounding

algebra

a variati-

it e x p l i c i t l y

and transitively

(al + 61'

points

considerations

maximal,

we c o n s t r u c t

is no p r e f e r e n c e

is n o t

tripod

central

surface

o f the

connection

there

movement

if the c e n t r a l

their

some

only

to do

is l o c a l l y

is n o t place

at d i f f e r e n t

infinitesimal

keep

de df

the n u m b e r

under

we are y e t u n a b l e

the L i e

because

P2(P(~2,E),F)

should

of all,

is m u c h m o r e

to w h a t

shape

occupied

N,

should

bundle,

..... >

take

(flat),

in

if the

by

concern

now how

of t r i p o d s

the p o l y g o n s ,

of t r i p o d s

formation.

in

expli-

MODEL

fiber

that

topology

should

nection

polygons

will

effectively

(~I'~2)

concerning

a fiber

depend

L(e,f)

space

is i n v a r i a n t

clear

act

clear

(3-I)

be

L

SIMPLE

First

should

the c o n f i g u r a t i o n

should

configuration

principle

OF THE

it is q u i t e

however,

seek

E x F

if

in o u r d o u b l e

difficult

groups

over

should

= V }

c a n be r e p l a c e d

it is q u i t e

principle

a very Lie

shall

E.

CONSTRUCTION

onal

V

Moreover,

the v a r i a t i o n a l

ordinates

3.

version

lattice.

but

we

to w r i t e

P 2 ( P I ( ~ 2 , E ) ,F) V

what

the

In o u r

action

deforthis integrand

simplified

model

321

we

shall

first

neglect

the

third

terms

and

reasonable

to a s s u m e

Ut

: ~

interaction of that

+ ~

term,

(2-12). the

+

For

energy

2 =

keeping

central

only

of a t r i p o d

21 + ~

+

the a n a l o g s

and repulsive

(2~

of the

forces

is p r o p o r t i o n a l

-~I-

it is to

s2 )2 =

(3-2)

= 2 a~

The m i n i m u m log of the

The take

first

last

into

of

term,

should

Ut

term

account

to a c e l l

+ 2 ~

is a t t a i n e d

corresponding

the

fact

then

Up =

cause

is the

surface

per

atom

~

being

it d e p e n d s and will should at

vary

least when

amount

I ~

this

(2-12).

field potential,

of the p o l y g o n s

the

over

belonging

of

contribution

equal

to

l

i-th polygon

T -> 0,

ki

U = Ut +

~ U

would

calculus of

However,

of

it is d i v i d e d

by

k

one

of the

to m a k e

cells

bei

adjacent

tripods.

The

total

po-

are

the d i s t r i b u t i o n

strength

by a function

sense,

such

of c e l l s

shall

average

which

is o c c u p i e d

over

E

of

U. the e n o r m o u s

only,

computing

results

of the a v e r a g e d i.e.

the

should

(3-4)

configurations,

involve

reasonable

in s h a p e

is v e r y

while

re-

values

of the

tripod

angles

suppose

to an a v e r a g e

that

the

shape,

and

homogeneous.

the p o i n t s

b y an a t o m

will

forces

expression

any conceivable

substitution

close

The

to the m i n i m a

is b e y o n d some

two c o n t r i b u t i o n s ~

interatomic

the p r e f e r r e d

an e x p r e s s i o n

fiber

of

to a n o t h e r .

lattice,

can expect

first

of the

properties

element

and

all quite

As w e

P

correspond

such

freedom

s u c h an e x p r e s s i o n

In o r d e r

to the

the w h o l e

of f r e e d o m

elementary

will

Sk 1

as a r e l a t i v e

degrees

point

surface

on the p h y s i c a l

placing

that

3 E i:I

from one chemical

of d e g r e e s

possibility.

~I"

the

to t a k e

equally

defined

of c o u r s e

be a v e r a g e d

Of c o u r s e

to the H i g g s

of

is them:

(3-4) with

lagrangian)

'

it c o n t r i b u t e s

tential

that

= ~2 - 2 ~3 . T h i s is the ana-

al

field

+ 4ff2

be m a x i m a l .

(3-3)

Sk. 1

when

(pure Y a n g - M i l l s

It is r e a s o n a b l e

where

+ 2 c{i~ 2 - 4 r[~ I - 4 ~ 2

in

(a v o r t e x

~2, of t h e

a n d as o v e r lattice)

each

there

such is

322 one e l e m e n t a r y trical cell.

mean

cell,

value

we shall d e f i n e

In the case of the r e g u l a r

identical,

and it is e n o u g h

such cell.

For e x a m p l e ,

the f o l l o w i n g

the a v e r a g e

of all the p o l y g o n

angles

angle

~

homogeneous

elementary

lattices all the c e l l s are

to take the m e a n g e o m e t r i c a l

for the l a t t i c e s

as a g e o m e -

in an a v e r a g e

mentiones

value

in one

in § I, we o b t a i n

values:

(6, 6, 6)

: ~ =

2~ (~--) = 120 °

(4, 8, 8)

: ~ :

[(~)4

,3~,16] ~--~

(4, 6 , 12)

: ~

[(~)4--

2~ 6 (~--)

(3,12,12)

: ~ =

[(~)3~

,5~24] (~--)

=

1/20

:

124°30 '

~,5V, - ~ 12]

1/22 =

128 o20!

1/27

In a s t a t i s t i c a l

approach,

we s h o u l d c o n s i d e r

in w h i c h o n l y the p r o b a b i l i t y

of f i n d i n g

denote

Pk"

these probabilites

If we d e n o t e mentary

cell,

by

by

Pk

:

the p r o b a b i l i t y

a k-gon

an a m o r p h o u s

of f i n d i n g

a k-gon

we

in an ele-

k Pk Pk - EjPj

(and for

Nc = 3

we have a l w a y s

If all the p o l y g o n s pute

~

in the

E]Pj

= 6).

lattice were perfect,

klP

The m a i n d i f f e r e n c e as we b e l i e v e , polygons

by the v a l u e s

T, the

polygons

continuities

the l i q u i d

to p e r f e c t ; of

Pk'

etc.;

farther

at the p h a s e

I/EkiPk

i (kn-2)i knPkn

i

and the a m o r p h o u s

in a l i q u i d we m a y c o n s i d e r

can n o t c h a n g e

may change

continuous

between

the p r e s s u r e Pk s

i kI

the f o l l o w i n g :

are v e r y c l o s e

(3-6)

rature

then we can com-

as I, ( k i _ 2 ) ,

state

lattice

in a net is given;

then we h a v e

(3-5)

via

135o30 '

moreover,

which after

adjust

transition

the a n g l e s

on w i t h c o n s t a n t

transition

is d e t e r m i n e d

themselves

the p h a s e

anymore,

then. (But its d e r i v a t i v e

~

Pk'S.

m i g h t be, m i g h t not.)

the

solid

then

to the t e m p e into s o l i d

of the t r i p o d s Whatever Pk'S

is,

that all the

and

the dis-

have to be

323

I Fig. 5

Fig 4

FiB. 6

Fis. 7

Fig. 3: The curves r e p r e s e n t i n g U as the function of ~ for fixed P6" The lower curves w i t h one minimum only (at ~ : 2 ~ / 3 ) corresp o n d to the values of the p a r a m e t e r ~ b e l o w the critical one, for A big e n o u g h (upper curves) the minima appear at two other d i f f e r e n t angles, c o r r e s p o n d i n g to the amorphous configurations, the c r y s t a l l i n e c o n f i g u r a t i o n has greater energy then.

¢

Fig. 4: The curves r e p r e s e n t i n g the free energy F as function of P6 (for the liquid), a) W h e n A is low enough, b e l o w some temp e r a t u r e the m i n i m a l value of F is always at P6 = i (crystallization), b) W h e n A is big enough, even b e l o w the critical temp e r a t u r e the m i n i m u m of F appears at P6 ~ i (amorphous solid).

324

It is quite easy to find the simple f u n c t i o n s haviour

of U t

and

Up.

For

(3-7) has

proached

its m i n i m u m by

polygon's

the f o l l o w i n g

S k -> A k

Its m a x i m u m constants

in o r d e r surface

From

to m a k e

remain

the b e -

the e x p r e s s i o n

which

surfaces

displays

shall

, Ak

this

Here

c a n be ap-

a maximum

at p e r f e c t

-

(k-1)~)]

are

the n o r m a l i z i n g

expression

are

with

the v a l u e s

the p e r f e c t

of

some

of

the

in w h i c h

the

A 7 = 0,664.

discuss

only pentagons,

flat,

(k-2) r k

coincide

~ = ~k"

is i n d e p e n d e n t ,

globally

~ + sin((k-2)~

~k -

A 6 = 0,5,

n o w o n we

contains

P6

sin

for

when

: A 5 = 0,362,

lattice

The p o l y g o n s

expression

[(k-l)

is a t t a i n e d

polygon's

only

imitating

angle

(3-8)

Ak'S

~

to t a k e

3~ 2 - 4 ~

~ - 2 3~

at

of

it is e n o u g h

U t ->

which

have

Ut

the

hexagons

whereas

so t h a t

simplest

P5

P5

and heptagons,

= P7'

: P7

model,

=

because

~(I-P6)"

our When

in s u c h

a case

lattice

has

P6

= I,

we

by external

h e x a g o n a l l a t t i c e , w h i c h m a y n e v e r t h e l e s s be d e f o r m e d 2m stress (~# ~-) ; if P6 # I, it is an a m o r p h o u s l a t t i c e

(especially

if P6

the p u r e l y

The now,

full

in o u r

(3-9)

is c l o s e r

expression

crude

is the m e a n

ves

U(s)

played minimum

grows

as

+ ~

internal

3. T h e r e

l o n g as

bigger;

I/3 t h a n

the m e a n

to

I).

potential

energy

per atom will

k

then

7 AkP k E k k=5

energy

per

to d i f f e r e n t is a l w a y s is b e l o w

new minima

[(k-l)

atom at values

an e x t r e m u m

some

critical

appear

sin

~ + sin((k-2)~-(k-1)c()]

zero of

temperature.

A

for

The

cur-

and

P6 are dis2~ ~ - 3 , w h i c h is a

value,

and a maximum

for c o r r e s p o n d i n g

when

to n o n - c r y s t a l -

line configurations.

In a l i q u i d , simplified

model

(3-10)

which

~

yields

however,

we c a n put,

=

be

approximation,

corresponding

on Fig.

to

for

U = 3~ 2 - 4 ~

This

1

to

3~ 5P5 [ (~-)

P6

and

in f i r s t 2~ 6P6 (~--)

~

depend

on e a c h o t h e r ,

approximation, 5~ 7P7] (7--)

[10]

I 5P5+6P6+7P7

in o u r

325 28.1

(3-11)

Now, P6

when

= I.

the py

Log

~ ~

0(P6(I,~ At

minimum averaged

- P6

varies

very

slightly,

temperature

T

of

free

F = U - TS.

the

over

the

energy

cells

the

should

in o u r

model

have

the

S : -

at

enough,

the

enough; we

the

curves

minimal

beyond

120 °

when

correspond

configurational

to

entro-

form

Pk

F ( P 6)

value

some

Tc

a liquid.

I-P 6 Log(~)

(I-P 6)

(Fig.

4) w e

is o b t a i n e d the

When

is b i g

see

Log

that

P6

when

~

is

small

for

P = I f o r the T small .... ~F 6 twlrn~ = 0) is no m o r e at P6 =

minimum

A

- P6

I,

t h e a b s o l u t e m i n i m u m of F ~F is f o u n d s o m e w h e r e b e t w e e n 0 a n d I f o r P6' ~ p - 0 e n a b l e s us to f i n d 6 the temperature dependence of P6" The critical temperature of t h e 9F p h a s e t r a n s i t i o n m a y be f o u n d t h e n w h e n b o t h % P6 0 and 92 F 0. 2 -

get

The

to

should

is

(3-13)

Looking

121 ° 52'

equilibrium

S : - E Pk L o g k

which

from

finite

(3-12)

and

- 2.25P 6

37

enough,

-

9P 6

It h a s

to be

underlined

tion

liquid-amorphous

and

Cv

tion

in a c l a s s i c a l

ters

have

at

this

been

describe

the

should

average

cell

and

to

not

transition rather

4.

the

mean

described

which

we

over

by

model

though

there not

is

elementary taken

rather

to of

kinetic

energy

per

of

our

described

here

an o r d i n a r y all

cells, in

number

obtain

that

the

the

in t e r m s

we

a phase

is a d i s c o n t i n u i t y

describe

reason

usually

transition

CONCLUDING

hope,

The

correspond

divided

and

it d o e s

sense.

quantities

smooth

The we

point,

averaged

perature

that

solid,

the

and

degrees

of

atom,

etc.

parameters

in o r d i n a r y

phase

therefore

kinetic

transi-

e.g.

of

Therefore

would

parame-

do n o t

energy

freedom

transi-

density

essential

thermodynamics; mean

of

our of

tem-

an

this

cell

a sharp

correspond

to

a

parameters.

REMARKS

we

presented

could

) be

think

it

done.

here

is o n l y

It r e l i e s

is u s e f u l

to

upon

underline

a sketch some again

very

of w h a t strong

clearly.

should

(and

assumptions,

326

Our g e o m e t r i c a l as the basic physically. tent,

U

then

it w o u l d mean

atom

theorem

kinetic

stored

that

the e l e m e n t a r y of the

turns

energy

of a cell

which

in a cell

already

to

In this

to the p o t e n t i a l

- 5);

(3-9),

energy

the mean n u m b e r

it is quite

of the degrees

of f r e e d o m

these

to the cell

is equal

to

Therefore,

the r e l a t i o n

temperature

and the mean v a r i a b l e

can be d e t e r m i n e d

the

e n e r g y per

show that the number

(Ej2pj+3) .

solid,

case,

the mean p o t e n t i a l

(3 E k2Pk

justified

to some ex-

or in the a m o r p h o u s

to the cells.

is given by the e x p r e s s i o n

is equal

are c o n s i d e r e d

be also

out to be a d e q u a t e

should be equal

We have

cells

space must

in a liquid,

could be e x t e n d e d

in this cell.

in a cell,

of atoms

in w h i c h (points)

If such a d e s c r i p t i o n

the virial mean

image

constituents

atoms

easy

to

contribute between

the

from the e q u i p a r t i -

tion of the e n e r g y (Ej2pj kT 2

(4-I)

+ 3)

- -

The close,

dependence

with

phase

strong

formulation as the

space.

to compute

assumption

of the

invariant

its phase

space

to compare

the

subset

of the m i n i m a l

given by goes

be m a x i m a l i z e d

we have

(3-12).

unless

the t h e r m o d y n a m i c a l an excess

perimentally. of the model.

in the

P6=I

of the p r o b a b i l i t i e s the volume.

defined

in our model.

the

In order

noticing

has been

possibility

Pk'

since

we

sense,

should cor-

in o r d e r

to take

entropy

corresponds

in the c l a s s i c a l

but also

into

part

that even when

to the global This

Pk'S

to e x p a n d

spaces

"configurational"

solid can not be c o n s i d e r e d

which

is another

the

space,

phase

easy

lattice,

In p r i n c i p l e

in d i f f e r e n t

It is w o r t h

equilibrium

of entropy

This

liquid phase

Pk'S.

in the

it is quite of the

in a given phase

to 0, this c o n t r i b u t i o n

fact that the a m o r p h o u s

system occupies

are fixed,

In

can be con-

s y s t e m not only tries

volumes

maximizes

the

the entropy.

the entropy

of f r e e d o m

by a d j u s t i n g

repartitions

this p h e n o m e n o n ,

the entropy,

the

volume

itself

configuration

Fk

However,

invariant

to d i f f e r e n t

to tell w h i c h

sesses

or at least be very

concerns

mechanics

degrees

therefore,

the m a x i m a l

changes

known

space.

changing,

to occupy

temperature

of the

of internal

be able

account

coincide,

we have made

statistical

volume

the phase

are c o n s t a n t l y

responding

should

out of the c o n d i t i o n

If all the p r o b a b i l i t i e s

the number

and to define

in order

obtained

F.

The other

ceived

U

5)

thus o b t a i n e d

the r e l a t i o n

free energy

a usual

:

(3Ek2Pk-

of

the

will

not

to the well as b e i n g

in

the glass pos-

long ago m e a s u r e d

for the e x p e r i m e n t a l

ex-

check

327

However,

it is clear

ment can be imagined For the d i s c u s s i o n sider

only

no more

lenghts

flat,

and there

in ~2.

fiber)

has the d i m e n s i o n

analog

of the e l e m e n t a r y

same central

from 6 to

12

the d i a m o n d

There cell,

the

atom,

is no more

lattice

dimensions,

cell we propose

giving

is very

is the

h e xa g o n s

are

(the first

complicated.

of bonds

of these m i n i m a l

with

their e q u i l i b r i u m

set of all

couples

con-

of the lattice

of the t e t r a p o d s

its t o p o l o g y

any c o n s t a n t

is c h a r a c t e r i z e d

The

the minimal

originating

polygons

at

can vary

in an e l e m e n t a r y

cell

of

angles

of bonds

expressions.

Also

between

results

soon

but also by the

in an e l e m e n t a r y

and the r e l a t i v e c i n d e p e n d e n t number

angular

the planes

are e s s e n t i a l

On the other

tentative

of p o l y g o n s

the only mean

variable

defined

by the

for the r e s u l t i n g

hand,

tripod will be d e t e r m i n e d

elsewhere

number

not only by the N

polygons,

a bond.

the dihedral couples

Some appear

space

dimensions.

we should

lattice).

around

energy

in three

the experi-

to three

for example,

relation

12 identical

of p o l y g o n s

mentary

lattices

the number

(there are

of d i f f e r e n t

ficient:

5, and

with

the model

I. But the p o l y g o n s

six i n d e p e n d e n t

frequencies

pendent

to

The c o n f i g u r a t i o n a l

by the

comparison

silicon,

is no simple

like

spanned

(Nc=4)

normalized

angle,

polygons

serious

of the a m o r p h o u s

the t e t r a - c o o r d i n a t e

the bonds'

the

that any

if we g e n e r a l i z e

the four

concerning

6 inde-

density

solid angles

by the o r d i n a r y

is insuf-

and the

of the ele-

and d i h e d r a l

the t h r e e - d i m e n s i o n a l

angles.

model

will

[11].

REFERENCES

[1]

Zachariasen

[2]

Rivier

N.,

Duffy

D.M.,

[3]

Xleman

M.,

Sadoc

J.F.,

[4]

Dzyaloshinskii

W.H.,

J.Chem. Phys.,

I.Ye.,

Vol.

J. Physique, J. P h y s i q u e Volovik

[5]

Kerner

R.,

Kerner

R., Ann. Inst. H. Poincar~,

[7]

Trautman

[8]

Forgacs

[9]

Kerner

R.,

Journ.Math. Phys.,

[10] [11]

Kerner

R.,

Phys.Rev.

Dos

Phil.

A.,

Magazine

Rep.Math. Phys.,

P., M a n t o n N.S.,

Santos,

D.M.,

I,

(1935) (1982)

40, p.

569

(1979)

39, p.

2, p.

151

(1983)

(3), p.

143

(1968)

(1970)

Comm. Math. Phys.,

B, 28,

162 293

J. Physique,

B, 47, No. 9

p.

Lett.,

G.E.,

[6]

3, p. 43,

24,

2, p.

356

p.

5756

(1983)

J. Physique,

Aug.

1984

72,

I, p.

(1982)

15

693

(1978)

THE

ISING

MODEL

ON F I N I T E L Y

AND THE

M. Dipartimento

di F i s i c a

Among nal

the

di F i s i c a

several

Ising Model,

one

for g e n e r a l i z a t i o n The

latter

situations morphic Let

us

is d e f i n e d

first

graph

review

the

schematically.

Let

have

the

some

Italy

solution

finitely

far-reaching

Italy

of the

2-dimensio-

promising

so-called

implemented over

di P a v i a ,

and

Pfaffian

in p a r t i c u l a r

a finite

presented

properties

lattice group

that

suitable

method

[I].

in t h o s e L0

iso-

GO .

s u c h an a s s u m p t i o n

presentation

G O : ~ 2 ;

2c

of - ~

[4],

Se, Z I ( S c ) ,

over

F({tk])

Here

so t h a t

H] (S c)

of

element

theorem

~ 1(Sc)

(8)

extension

(multiplicative)

of S c

the

finite

ex-

of g e n u s group

to the

of map-

group

Sc

which

Tc,

(finite)

isomorphic

preserve

On the o t h e r

handles

the

surface

hand,

isotopy Sc

(or

is iso-

and

~ 0 Diff+(Tc )

classes

of o r i e n t a t i o n

preserving

diffeomorphisms

.

general

denotes in the

the form

regular

representation

of

G, A

can be w r i t t e n

in

332

n

(11)

A :

where

the

depend

Zk(+)

coefficients

no a m b i g u i t y the

same n o t a t i o n

two

alternative

If

R ( a k -1)

Zk(-) ({tkl) ,

of

arises

with

be

+ z (-)

Zk(+)

We h a v e could

R(ak)

on t h e p r e s e n t a t i o n

- since G

E { k=1

GO

and

- we d e s i g n a t e d

as for t h o s e

ways

and

Zk(-):

the h o m o l o g y the

of

of

Zk(-)({tkl)

Sc

only;

generating

and

symbols

of

GO.

of r e d u c i n g

(8) to an a l g o r i t h m

which

solved.

D(J) (g),

g~ G, denotes

of d i m e n s i o n

/J/,

the

recalling

(12)

J-th

irreducible

representation

of

G,

that

R(G)

:

~

[J] D (J) (G)

J we

can write

(13)

free

- 8f = In 2 +

where the

the

A

sum

Thus,

(J)

is the

is o v e r

all

on the o n e

n E k=1

J-th the

hand,

the

irreps

of

the

finite

set of

O n the o t h e r

energy

G,

hand,

per

f

in the

2-2c-I uk + - NO

in c o s h

irreducible

form

E Tr J

block-diagonal

in A (J)

component

of

Ai

irreps. we are

and

site

faced

with

the a l g o r i t h m

finite from

determinants (8),

in F(~tk~ ) = -2 -2c-I

E

(14)

the p r o b l e m

is r e d u c e d d e t A (J)

of c o n s t r u c t i n g

to the for

r

calculation

all

of

J's.

(11) n E

I

p [ IT

E

~

zk

(~i)

]

Z

x Tr [RIak111 ~ ( a k Pl I P Notice

that

if w e d e n o t e

to the e l e m e n t luate

g = ak

those

a non-vanishing In f a c t o n c e The p r o b l e m

words

the

latter

of d e c i d i n g

G = T . . . . > T . . . . > K .... > I

there

is a h o m o m o r p h i s m

tion

L

riant,

of

simple

since

morphism

of

is a g a i n group

of

0 T

up

and

since

realizable group

fills

K

as a g r o u p

of of

G, Sc

one

surface

by

c a n be of

by

sending

as well)

thought

to the

auto-

of as a f i n i t e

isometries

can

is G - i n v a -

L.

acting on

to the q u o t i e n t

which

any collec-

(such a c o l l e c t i o n

up the

is a s s o c i a t e d

of d i f f e o m o r p h i s m s

defined

Sc

GL

Fuchsian,

as

to

by conjugation

induced

OT,

is d e f i n e d

T

filling

the o r b i t

for the p r e s e n t a t i o n

group

from

curves

T

such a Fuchsian As

(in f a c t

the

identify preserve

space

[8]

K

a cut

sub-

H(2)/T~Sc ,

G/K. with

the

sub-

system.

The

latter

follows.

Let

Cp ;p r . : I ' .... c ]

Tc~

:

be a set of d i s j o i n t

cycles

on

f

Se

=

is

t

en

a

c-p

nctered

sphere.

335

An

isotopy

then

the

class

group

Denoting

by

exchange

between

i)

Q 6 G

G

ii)

~c where

~n

by

@

£

group

The

elements

and

2c-I

to

system.

K

is

we have:

K.

--->

~

the b r a i d

gs 6 G

is a cut

sequence

of p e r m u t a t i o n s

belonging

Thus

cycles,

Q

denotes

the

ters

of

an e x a c t

~CpI

C's and r e v e r s e s t h e i r o r i e n t a t i o n s . P G s u p p o r t e d l o c a l l y by an h o m o l o g y

intersecting

exists

of

the

element

is g e n e r a t e d

There

iii)

= I .... ,c>

permutes

group

of

are

~ /c ---> 0

--->

n

on

represented

[ QP;p

4 ~ ] u K

the -I

relations

of

G

are

that

since

is f i n i t e l y

n

strands,

and

~n

objects. by w o r d s are

Ws

indeed

generated

whose

elements

by w o r d s

let-

of

of the

K.

form

Wsg s There

follows

all

the

relations

of

Sc

of g e n u s

Finally - due

to

tained Thus the

there

- the

F({tk~)

2c-I"

the

n

and

Let

us t h e n

thought distinct

n

of d e g r e e

K

plectic

can

- recently

p(n)

K

of

on

G

Moreover,

subsurfaces

which

and which

representation function,

characters

of

derive,

induction,

by

in c o n c l u s i o n , group

~c

some

has

can be obof

K.

depending

on

[7]. from

those

of the b a s i c

of

properties

~n"

fundamental

in a plane.

(n))

as the

obtain

matrices,

seems

group If

theory

,

of the

p(n)

~ i ( P (n))

space

space

is the

of u n o r d e r e d

space

of p o l y -

namely induced

of

~n

the m a t r i c e s by c i r c u i t s

present

by Sato, - between

= 0 , i > I

of h y p e r e l l i p t i c

a representation

to b r i d g e

established

field

the

of

points

product

matrix

as an a u t o m o r p h i c

of as the

of the c u r v e s

somewhat

supported

G.

n,

in fact

integral

homologies

quantum

of

so is

representation

of a w r e a t h

and

recall,

~n~1(P

thinking

relations

(finite)

of the b r a i d

can be

one

This

of

from

matrix

the

representations

(19)

n

from

of

nomials

a finite

can be w r i t t e n

and d e f i n i t i o n s

sets

follow

presented,

2.

structure

induction

representations

In turn

G

at m o s t

exists

(18)

by

of

K

Jimbo

of d e g r e e

group

of a u t o m o r p h i s m s

in the

approach

with

and M i w a

[9]

the p r o b l e m

curves in the

coefficient

of

sym-

of the plane.

the c o n n e c t i o n in t h e i r

of e v a l u a t i n g

the

holonomic 2-point

336

Green's

function

isomonodromy n

has

of the

(n-l)

form

for

the

2-dimensional

generators

Si,i

(20)

s i si+ I sz = si+1

interesting

1

link

s

]

= s

of the

to the pointed Work

latter

(euclidean), generalized out

]

between

Yang-Baxter-Zamolodchikow

algebra

and

the

Schlesinger

and

(n-l) (n-2)/2

relations

[10]:

s

mulation

model

= I ..... n-1

(19)

An

Ising

problem.

s

,

i

the

i-j

> 2

s.1 Sl+1

relations

factorization within

whereby

the the

1 _< i i

(n-2)

of the b r a i d

group

equations,

scheme

leading

of an i n f i n i t e

connected

Roger-Ramanujan

'

has b e e n

the

to the

for-

dimensional

combinatorics

identities,

and

Lie

is r e c o n d u c t e d recently

[11].

is in p r o g r e s s

along

these

lines.

REFERENCES

[I]

F. M.

[2]

M. R a s e t t i , in " S e l e c t e d T o p i c s in S t a t i s t i c a l M e c h a n i c s " , N.N. B o g o l u b o v jr. and V.N. P l e c h k o Eds., J.I. N.R. Publ., D u b n a 1981, p a g e 181 M. R a s e t t i , in "Group T h e o r e t i c a l M e t h o d s in P h y s i c s " , M. S e r d a r o g l u and E. I n 6 n H Eds., S p r i n g e r - V e r l a g , B e r l i n 1983, p a g e 181

[3]

M.E.

Fisher,

[4]

P.W.

Kasteleyn,

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