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 --
the space of linear
subspace (defining the infinitesimal representation). somewhat,
(s : , s i-)
it: may be shown
that, for any p,
and ( s j+ , s j-) , up t o a s c a l a r
Theorem 2.
9ij
Extending the results of [7] Pij depends only on the spins
factor.
Moreover we have
and
Is7
equals zero unless
Is~-stla Naturally relations
such a
these
between
= ~
constraints
on
the generators
remaining constraints on
the
s]l
@ij are
of the SL(2,C)
= ~
(1)
equivalent
to
the
commutation
and H(2) subgroups of
~0"
The
@ due to commutativity of {dP(0$F); F @ H(2) } have been
computed using explicit matrix elements for standardized
@ij's
whose correspond-
ing spins satisfy (i); however, these are less succinctly presented. representation is indecomposable
Recall that a
if it is not a nontrivial direct sum.
By ad hoc
and additional argument we have Theorem
3.
Any indecomposable
trivial
on
the
conjugation, of dimensions sequences
of
translations
or anti-duality, 4,5,6,
representation of the Poincare group that is non-
and
of
dimension
~ 9
is,
up to duality,
complex-
linearly equivalent to a unique such representation
or 7, or either
spins characterizing
of two 8-dimensional representations.
their
restrictions
to the homogeneous
The
Lorentz
group are,
respectively,
1
i 1 (0,0) + (~,~),
(½,0) + (0,½),
1
l
1 1 (0,0) + (~,~) + (0,0),
1
(0,i) + (~,~), (0,0) + (i,0) + (~,~), and (½,0) + (1,½).
Furthermore, sentations of
there are only finitely many inequivalent
~0 of dimension smaller than 24, and a continuum of inequivalent 24-
dimensional representations. of
~0
able
indecomposable repre-
A complete determination of all such representations
(or at least of the simplest, prototypical group SL(2,R) ~ R 2) is not deriv-
by standard
group-theoretical
methods
at present,
but appears within reach.
Decomposition of Corresponding Induced Fields Before As
noted
• .. + Vn
induction
in
[8],
to G,
way
formal weight)
to each V3
Now define indecomposable)
such that P
wz = w3 + 1
be
special
if
the
is
a
finite
second-order
weights
RIxsL(2,C) subgroup but
set.
This
follows
subgroup,
For example,
~.
V1 +
wj (conDenote the
wj (determined by
@.
of
G
w=w n if p is
induced
from
p'
,
@' = id on the translation subgroup, In fact, the set of special conformal
from the value of
the suitably
normal-
Casimir of ~ acting on fields induced from an irreducible rep-
resentation of P of spins (s+,s_), lation
must be extended to the decomposition
whenever Pij ~ 0.
representation
is not equivalent to that induced from
ized
~0 given
also by @ .
the system of conforma] to
@' = p on the
weights
of
to do this,
under SL(2,C) as in Theorem i, is by assignment of a constant
resulting representation of
where
the representations
one convenient
which
value is
conformal weight w, and trivial on the trans-
w(w-4) + 2s+(s++l)
+ 2s_(s_+l)
[4, Part III].
the spannor fields are induced from an indecomposable represen-
tation of P denoted
(w+l,(½,0)) + (w,(0,½)).
Then evidently
w
is special when
(w+l)(w-3) = w(w-4), i.e. w = 3/2, recovering the conventional canonical dimensions 3/2 and 5/2 in a more rigid and structured context. twining operator from Ind @'
Dirac operator, which becomes singular when for the
For generic
w
the inter-
to Ind @ is a differential operator, essentially the w=3/2.
The special conformal weights
'wrenched' scalar and vector fields induced from the 5- and 6-dimensional
indecomposable representations of
P
are familiar integral weights.
REFERENCES i.
I.E.
Segal,
"Mathematical
Cosmology
and
Extragalactic
Astronomy,"
Press, New York, 1976. 2. I.E. Segal, Comment on paper of L. Wormald, J. Rel. Gray., in press.
Academic
3.
I . E . Segal,
A class of operator algebras determined by groups,
Duke Math. J.
18 (1951), pp. 22]-265. 4.
S.M.
Paneitz
and
I.E.
Segal,
Analysis
in space-time
II, J. Func. Anal. 47 (]982) pp. 78-142 and 49 (1982)
bundles,
pp. 335-414;
Parts
I and
S.M. Paneitz,
Part III, ~. Eunc. Anal., in press. 5.
I.E.
Segal,
Covariant
Chronogeometry
and Extreme Distances
III, Int. J. Th.
Phys. 21 (1982), pp. 851-869. 6.
I.E. Segal, Chronometric cosmology and fundamental
fermions,
Proc. Nat. Acad.
Sci. USA 79 (December 1982), pp. 7961-2. 7.
S.M. Paneitz, All linear representations of the Poincare group up to dimension
8, Ann. Inst. H. Poincare (Th. Phys.), in press. 8.
G. Mack and A. Salam,
Finite-component
field representations
of the conformal
group, Ann. Phys. 53 (1969), pp. 174-202.
Comment Steven
of the e d i t o r s : M.
PANEITZ
considered
two
topics
for his
talk
at this
conference: "Sharp
Asymptotics
Equations
and
of
the
Solutions
Conformal
to the Y a n g - M i l l s
Connection"
and "Indecomposable of the
finite
Poincar6
He d e c i d e d
to r e p o r t
manuscript
of the
on the
second
Group first
one,
dimensional and
associated
one.
we add
Representations Fields".
As we h a v e
this
to the
only
a written
proceedings.
THE ENERGY
MOMENTUM
MAPPING
OF T H E L A G R A N G E
TOP
R.Cushman Mathematics
Institute Utrecht,
, Rijksuniversiteit
H. Mathematisches
CONSTRUCTION
and
a constant leaving
Euler ment
angles, called
The group
SO(3)
the m o t i o n
the L a g r a n g e
see
[I] o r
of
of the
its c a n o n i c a l
which
<,>
tensor
of
of top,
space
top
Phase
form
at t h e
> ~
axis
a Lie [3],
top
space,
body of m a s s
The p r o b l e m using
theoretic
identity
is the
SO(3), space
bundle
=- d@.
inner
: X,Y
fixed.
treatment
group
cotangent ~
rigid
its c e n t e r
treat-
[9]).
the L a g r a n g e
is the
on
a classic
give
is a n o n d e g e n e r a t e
: so(3)xso(3)
is a s y m m e t r i c acting
symmetry
(cf.
~3.
symplectic
is the t a n g e n t
of i n e r t i a
approach
rotations
MAPPING
For
[2]. We w i l l
of all p o s i t i o n s
and momenta
top.
Bonn,
F.R.
force
its b o d y
o f the
the L i e - P o i s s o n
space
with
so(3), ment
system
of a l l p r o p e r
positions
Germany
vertical g r a v i t a t i o n a l
the b a s e p o i n t
is to d e s c r i b e
Universitit
OF T H E E N E R G Y - M O M E N T U M
As a p h y s i c a l with
Kn~rrer
Institut,
Bonn,
I.
Utrecht,
the N e t h e r l a n d s
the L i e
of all
T~SO(3)
On t h e L i e e ~ SO(3),
product
> k(IX
,Y)
of
algebra
the m o -
13
where
k
there
is the K i l l i n g
is a k - o r t h o g o n a l
diagonal,
that
is,
is a s y m m e t r i c
f o r m on basis
IE i = I i E
rigid
body,
i
the
SO(3)
standard
given
is the
the m o m e n t TSO(3)
: 12
tr%vialization
k
of
so(3)
i=I,2,3.
is p o s i t i v e such
Because
definite,
that
is
I
the L a g r a n g e
top
that
> 0
and
of the
tangent
of
> TSO(3)
at e of
inertia
left
tensor
13 > 11
tangent
bundle
TSO(3)
of
: A,X
> TeLAX
translation
to a left
LA
invariant
: XA
by
,
A,
inner
we e x t e n d
product
<,>
on
by d e f i n i n g
<XA,YA> Using
Since
by the m a p p i n g
SO(3)xso(3)
which
for
we a s s u m e
I] Using
so(3).
[ EI,E2,E3 ~ _
the
invariant
identification metric,
=- d~
of
phase
where
~
= <X,Y> TSO(3)
space
is the
. and
becomes
pull
back
T~SO(3)
TSO(3) of the
given
with
by this
symplectic
canonical
left
form
l-form
8
on
TeSO(3).
The
Hamiltonian
and p o t e n t i a l
V
H
of the
energy
Lagrange
top
is the
sum of k i n e t i c
K
with
K
: TSO
3) --> IR : X A -->
V
: S0(3)
I/2
<XA,XA >
and > IR : A
> k(Ad
_IE3 , ~ E 3) A
where Next begin
by
we
show
finding
that
the
Lagrange
two c o n s e r v e d
S I = [ B e SO(3) i Ad B E 3 = E31 TSO(3)
is a s y m m e t r y
S
I
x TSO(3)
Physically, symmetry Therefore
the
axis
of the
-->
left
SI
of the
top.
its m o m e n t u m
top,
TSO(3)
mapping
is c o m p l e t e l y momenta.
Then
SI
which
is g i v e n
: B,X A -->
action The
top
angular
action
integrable.
We
on the
left
on
by
TALBX A = XBA
.
to r o t a t i o n
leaves
> 0.
Let
acting
corresponds
left
~
~
and
about H
the b o d y
invariant.
14
J• (where
RA
vector
field
nent
is r i g h t
of the
axis.
XH
which
about
the
translation
angular the
the r i g h t
SI
axis.
follows
that
AdBI
variant
under
the r i g h t
preserved
Jr
: TSO(3)
is an i n t e g r a l the
total
action,
Jr
we have
shown
H,J~,J r
is
The
~
[m.
for all
of t h e s e
to find
the
TSO(3)
TARB-I
and
symmetry
is a s y m m e t r y
XA
of
.
to r o t a t i o n the
B ~ S I.
: X A -->
of the
definition
Hence
<,>
of
and
top
SI V
it
are
in-
-
Classically, Since of
is, we
top
is the
vertical
is i n v a r i a n t
r Hamiltonian
bracket have
of any
recovered
is L i o u v i l l e
mapping
of ~
Jr J
the
Poisson
momentum
-1(E,a,b)
on
is the c o m p o -
the b o d y
a c t i o n of SI Consequently, ~ and H are I S action. T h e r e f o r e its m o m e n t u m m a p p i n g
the q u a l i t a t i v e
topology
necessary
corresponds
1 1 = 12
(H(XA),
properties which
fibers
and
image,
critical
of
the
field
two of the
of
left Xj
.
SI Thus
functions
the
integrable.
top
~
(XA),
the
flow
is i n v a r i a n t
to see h o w t h e y points
component
under
vector
of the L a g r a n g e
: TSO(3)-->~ 3 : Xn -->
To d e s c r i b e
the
action
The L a g r a n g e
energy
~/~
the that
SI
of the H a m i l t o n i a n Ji
top a l o n g
: XAB-->
momentum.
that
zero,
integral
Classically,
TSO(3)
--> ~
X H.
H.
of the of
is an i n t e g r a l
Theorem:
fiber
of
angular
is an
action
From
=IAd B
by the r i g h t
to
by
x S ] -->
vertical
:X A -->
by A),
momentum
right
is g i v e n
TSO(3)
Physically,
--> ~
corresponding
total
Similarly,
the top,
: TSO(3)
and
is
Jr(XA)) of under
X H,
we
X H.
fit t o g e t h e r , critical
study To f i n d it is
values
of
the
15
2.
REDUCTION
In this to r e d u c e
TO ONE
section
defined
We b e g i n
no f i x e d smooth
points
(where
manifold,
for
so(3)'
of
and
M
Pa
=
fiber
the
Here
induced
a second
Pa
: Pa --> ~
we h a v e
orbit
called
see
is a s m o o t h Since
SI j~1
: YA -->
(AdA-I
is i n v a r i a n t
is a s i n g l e
M
the
first
S I -->
Pa
for
and
SI
I k(X,X)
hence
under orbit.
the
~3
has
is a
space.
To
(X,Y')
left
SI
the
image
Pa
we use
for all the
: x :
the
= ,
I/2 tr XY
,
first
reduced
is
+ ~ k ( x , E 3)
(AdBX,AdBY') .
B ~ S
I
once
identification
(xl,x2,x 3 - - > X
properties:
between
= a ]
is
: (X,Y') , B -->
=IAd B
for
Hamiltonian
I/2 k(I -I Y',Y')
action
AdBI
following
I). an i s o m e t r y (x,y)
which
(a)
Since
=
again.
0
~x 3 x 2
x3
0 -x I xI 0
it is
0R3, (,)) = k(X,Y)
To c o n s t r u c t
map
-x 2
on
=
: I, k(X,Y')
is a m o d e l
reduced
: (X,Y')-->
> so(3)
the
every
action
phase
E3'IY)
I
has
free-
that
to
SI
~3
which
Since
manifold,
this
reduced
of
variety.
[3].
M = $I\
first
process
degree
semialgebraic
space
the
reduction
to a one
action,
action.
the
~
right
used
model
top
is
On
: Pax
I
(a)
SI
I (X,Y') 6 s o ( 3 ) x s o ( 3 ) '
space.
Ha
of
S
J~,
so(3)xso(3)'
is d i f f e o m o r p h i c
phase
and
note
left
: I.so(3)cgl(3))
each
~ /j~1 (a)
Pa
of
left
~ 1(a),
: TSO(3)-->
action
the
-
on
symplectic
a model
the
the M a r s d e n - W e i n s t e i n
on a two d i m e n s i o n a l
value
under
OF F R E E D O M
of the L a g r a n g e
by r e d u c i n g
is a r e g u l a r
is i n v a r i a n t
find
we use
the H a m i l t o n i a n
dom Hamiltonian
value
DEGREE
and
(so (3) ,k)
is the E u c l i d e a n
where inner
product
16
2). an
isomorphism
where
x
3). e q u i v a r i a n t ~3
and
of L i e
is the with
the
Therefore
where
a second
Pa
=
-I
y'
= Iy
model
[ (x,y')E
and
product
respect
adjoint
Ad A X : AXA
algebras
vector
for
for
~3
to
action
the
the
A£
[X,Y]=XY
orthogonal that
reduced
first
reduced
Hamiltonian
: I,
Since
on
Pa
The
: Pa - - > [R : ( x , y ' ) - - >
reduced
right
SI
action
(x,y')
Ie.
S I -->
Pa Pa
on
becomes
space
is
= a]
= I.e.
for
i = 1,2,3,
i i
,2
,2
~i1(Y I
on
Ax
is
1
Ha
- YX,
action
is,
phase
l
the
[,])
S0(3).
first
: I(R~.
(so(3),
and
so(3) ,
x ~R3' [ (x,x)
~3'
and
~3
usual
on
all
(IR3,x) on
+ Yl
1
'2
)+ ~ 3
Y3
+ ~
x3
is
: (x,y') , B - - > ( B x , B y ' )
where sin t 0
Next ~Iw
we
leaves
Weinstein
reduce
out
the
invariant
the
symplectic
reduction
theorem
Jr,a which not
is
act
critical and
has
induced freely values only
by
on
that
Pa"
closed
the
the
and
of
orbits,
SI
on
as
momentum
fixed
its
~
Pa
.
invariant
Jr"
see
I
P
a space
To c o n s t r u c t [4].
A
short
polynomials
IT I = x 3
=/12+y'2 0
, , 773 = x 2 Y I - X l Y 2 d
]74 = Y3
If5 :
that
Marsden-
correspond
phase
manifold.
theory,
the
Unfortunately
leaves
reduced
by
Note
mapping
>-Y3
~
Pa .
given
~(e3,ae3)
Since
r,a the s e c o n d
on
momentum
map
points
J
is a t o p o l o g i c a l use
action
form
has
SO(3) I t6 ~R
> gR : (x,y')
right Its
~
right
: Pa
b = ±a
P a , b = j-1 r , a (_b)/S I model for Pa,b we shows
cos t 0
x2 X2 I+ 2 ~ 0 l
7T6 = x 2 Y 1 + x 2 Y 2
I
'~ to
does the
invariant
an e x p l i c i t calculation
17 generate under
the a l g e b r a
the d i a g o n a l
generate
of p o l y n o m i a l s
action
the a l g e b r a
of
on
SI
~3 x ~3'
which
The p o l y n o m i a l s
of p o l y n o m i a l s
w h i c h are
are
~iIPa
invariant i=I,...,6
invariant u n d e r
~.
There-
fore the r e l a t i o n s 77 5 +77" 21 = I, if4 = b
(the f i r s t t h r e e
7T2 3 +7T62 : w h i c h are space P
satisfied
Pa,b"
]76 + ITI TE4 = a,
by all the
Eliminating
0
= F(
~3
= 0
other words, on
~
plane.
region Pa,b
2
:
7T 3
and
7T
+
the fiber of 7 r
and for
SI
~2,
and
while
and
shows
)2
-
the p r o j e c t i o n
points.
for to
map.
or
in the s h a d e d
or
when
-1
i
(-I,0,0) j r e s p e c t i v e l y ;
~2,
has
S I cone-like
is s m o o t h e v e r y w h e r e
else.
a # + b
The
I: P r o j e c t i o n
on
fold c u r v e
-I
projection.
a = b % 0
region
jection.
-I
a
-b
4
]
~
~3 = 0
a : b = 0
v
is
of the Hatched
is image of
Pa,b under
I I
vertices second
plane. D a r k c u r v e
1 ~I
b#~a,
b = -a}0; P a , b
of P a , b
,
P a , b ; in
Over e a c h p o i n t
Therefore
b = a%0
]Ton
edge of
Fig.
I I
phase
that
7T~)
T/2(1-
over e a c h p o i n t
(1,0,0)
is h o m e o m o r p h i c (-1,0,0)
Tr 1
i s the a p p a r e n t
while at
~6
/.
is two d i s t i n c t to
second reduced
and
of the p r o j e c t i o n
is a point;
with vertex
b=a=01P0,0
(7,0,0)
(a-b
Pal- u n d e r
curve
~
the
~4' ~ 5 by
17-/II ~
The h e a v y
is d i f f e o m o r p h i c
is a cone on
at
17 3 )
the fold c u r v e
the fiber of
j-1 (-b)) r,a
define
define
set d e f i n e d
is a s k e t c h of the image of
1
the
~iIPa,
the v a r i a b l e s
7T 1 , T/"2 ,
7T2 -> 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} <
~,
19
Since
is strictly convex,
ha, b
cal point of
Ha, b
on
Pa,b
(~0,0)
is the only nonsingular
and it corresponds
criti-
to the absolute mini-
mum value E(a,b) of
Ha, b.
~-I
It follows
(E(a,b)a,b)
from the reduction
is diffeomorphic
these relative equilibria regular precession, moves
b2
I
= E'(a,b) +
of
H
process of
2. that the fiber
to a two torus
correspond
to motions
that is, as the top spins,
in a circle about the vertical
T 2.
Physically of the top called
its body symmetry
axis.
We now study sections of the graph of the function planes through the
Eiaxis.
Suppose
I~/ % I.
Then the following
argument
strictly
increasing
Ib/>/b/ ,
then
as
is tangent t o
on
h~
a=(b
(-I, I) .
is greater
where
shows that
Ib{ increases.
h~>h b
axis
Let
4
E'(a,b)
with
is fixed and
Eqb)
= Eq~b,b)
is
h b : hgb, b = b2hw~0 If
Therefore
the value
than the value
E'(~) where
E'(b) where
~
is
!
tangent to
h b.
Since
h b = b2h~
which does not depend on Because
ff 01(eb,b)
of
independent E'(a,b),
ment holds
of
E'(b)~cb 2 b.
except
increases
from above by as
a=~b,
is a s y m p t o t i c a l l y
the singular
critical
Eliminating
I - 2i I ]72 + ~
(a-b~) 2 -
where a=~b
convex.
points of
must work algebraically. /
is
-2[I1,.
Ibl increases. hl
takes constant
of the graph A similar argu-
b=0.
E I with the planes
E
~I'
section
section of the graph of
I
~
as
/b/--> ~ for some positive
Thus every plane
for the plane
Next we analyze
and the slope of
~(4b,b)
is bounded
its minimum value, c
b,
~2
a=+b.
Ha, b
and the
In this case we
from
Tf I
7T2(I-~ ~) = 0,
l fr ~ [ ~ l ,
7r 2
,which is
>_ o
Pa,b(] ~ff3 = 0 ]
gives f(~1 ) : (~-g~1) (I- rr~)-(a-b~1)2 where
~ = 2I I E'
7F0 = (TT ,7[2,0)
and
B : 2Ii~
> 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+c<) 2 : 6 ( a ) .
Since
A_ 6 = and
fl>O,
real we
6
roots
obtain
has
if
two
~
the
double
and
roots
~#-B~
B=const.
P2
8B(~+~)2(~-~)
slice
if a n d
and of
one
~ f
only
if
double
a:B,
real
sketched
root
in fig.
the locus
PI Q a n d
P2
, which form
unstable ning
ing,
but
given
~ = 2IIE in f i g u r e
fiber
minima
of
of
is c a l l e d
an
Thus
its
the the
where axis,
thread motion
is
a
gives
2
to t h e
figure
the
arc
thread,
the
critical
the
these top
is
energy
of E J @
stable.
is
Finally
values
the
Over
points
that
is,
S I. H e r e
also that
the
the
discri-
open
of
the
the on
PI
fiber
top
to
spin-
parabola
top
a = -b
is
to
of
correspond
the
the
arcs
to
joining
values
points
suppose
of
correspond
parabola
momentum
sleeping,
is v e r t i c a l .
fiber
of
slice
2(b) . O v e r
these
Over
which
the
i~
= ~ + ~
is T 2 s i n c e
Ha, b. the
Fig. 2(b): a:b cross section of image of energy momentum mapping. Light curve joining PI to P2 is the thread of criticaI values corresponding to sleeping motion of the top .
2(a)
E~
S I . Physically
motions
about
outside
mapping
P2 Q t h e
nonsing~lar
~J
if ~ : - B .
2(a).
[-~
Q
Fig. 2(a) : a:b, 8:const. slice of discriminant locus ~f=0. Tangents the same at P. and P2; tangents distlnct at Q .
minant
simple
PI
Q~B
Applying
four
is
sleep-
, then
21
f(Tfl)
Now
~i=-I
then
g
=
(I+~i) [(~-B~I) (1-~1)-a2(1+Tr1) ] :
is a d o u b l e
has
a double
g Since
~>-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), <@l,V > ~ - <@~,~>
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
<(SJ)(u),X>
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
<~K > <~H _ > ~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
= <(Ox)~,
i a (x ,¢ ,~ )
adapted
y >. 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.<0,Z > - Z.<0,Y > - <0, [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~
= <~t,~> = ~>
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
(<)(x) and
o : V~E ÷ M.
vector
field
~I
structure
(~)d~ ~ + l~(~)d~
,
(<)
~
of of order
then
.
operators
0H
~,a ~ /H
and
K
on
V~E.)
If, in particular
is
the functional
vector
ct
field theory. fields
from
V E. We denote by
E
~ ~ F(V~E).
to
<8,Y >
defined by
x e M
l-form on
section operator
is a H a m i l t o n i a n
~
o f two f u n c t i o n a l s
formula from classical
6.4. Lifting of vertical
for
o : V E ÷ M
local coordinates,
which is the familiar
vertical
~
~,~
c~
V E
on
on V E, then
(~7~)(~) = ~
In adapted
O
is a differential
generalized
~ s ~(V~E) * @% ~ Y(V~E)
The P o i s s o n b r a c k e t
of order
M. The Poisson
6, the mapping
are local scalar differential H
V E. We shall h e n c e f o r t h
be a simple generalized
is a vertical
local coordinates,
and
6H = ~-~ , where
l
V~(V~E) ÷ V~E. Let
(~)(~)=~a(<) (Here
of
structure
where
V~(V~E)
s F(V(V~E))
V~E, and
In adapted
I =lOn,
o : V~E ÷ M
O
a fixed volume element on
in the sense of 5.1 in the following way. Let V E
~ a
~g
is orientable on
,
= <(8x)%(x) , Y(~)(x) > ~(x)
V E. Let
Y
be a
the H a m i l t o n i a n
on
50
In adapted local coordinates local components
(~(x)
(x))
if
<@,Y>(~)(x) Given of
such that
~ = dx I A ... A dx n
Y = ya(~,~)
9
+ Ya(¢,~)
and
~
~(x)
has
, then
= ~ a ( x ) Y a ( ¢ , ~ ) ( x ) d x I A ... A dx n.
X ~ ~'(E), we say that
Y ¢ V(V~E)
projects
onto
X
if, for each section
V E,
( T p ) ( Y ( ~ ) ) = X(p o ~ ) ,
i.e.,
Y = Xa(¢)
$_
+ Ya(¢,~) ~
X. For e a c h s e c t i o n and
X(po~)(x)
C
is in
of
V E
. Let
and f o r each
T(po~)(x)(Ex).
<e,Y>(<)(x)
so the H a m i l t o n i a n Y or
<@,Y>
along the fibers <@,X >. Thus Let
lized
~
of
on
vector
on
in
M,
: <<(x),X(po<)(x)>
V E
Jx(~) = <0,X>(~)
fields
x
and let
depends only on
and we d e n o t e
= <@,X>(¢,~)
2-form on
V E. Then
Y g V(V E)
~(x)
is
in
project
onto
T(po~)(x)(Ex )
Then
p : V E + E),
be a vertical
X c V(E)
is
X
this
(and not on the components
H a m i l t o n i a n on
= <~,X(¢)>n
V E. Let
~(Y,Z)
n(x),
Y
and
V E
by
of
JX
= ~axa(t)q. Z
be two vertical
t h e H a m i l t o n i a n on
V E
genera-
d e f i n e d by
~(Y,Z) (¢) (x) = w ( x ) ( Y ( ~ ) ( x ) , Z ( ~ ) ( x ) ) n ( x ) . If
w = -dO,
then by the definition
of the canonical
~(~,Z) for
% g g(V~E)
and
of
Q
Y
leaves
6.5. Proposition.
~
of
V E,
= <~,Z >
by a vertical
the generalized
l-form on
(y.o)(z) We say that
structure
Z E V(V~E).
The Lie derivative is, by definition,
Hamiltonian
Let
@
=
invariant
X ~ V(E)
generalized V~E,
vector
field
Y
onVE
Y'@, defined by
dO(y,z) + ~. ~ - <@,Y>,Z > . if
Y'@ = O.
and let
X
be the globally Hamiltonian vector field
6J X
on
V E, ~ ~ ,
field on Proof.
V*E
where
JX = <@'X>" Then
which projects onto
It is clear that
X
X
projects
X
is the unique vertical generalized vector
and which leaves onto
X. Moreover
@
invariant.
k
leaves
@
invariant
since
51
6J X <6J x dO(i,Z) = ~(X,Z) = ~(~ ~-~-,Z) = 6-~--,Z> and
-
therefore
X'O = O.
In adapted local coordinates (6.1) and
6J X <6%-$ ,e> ^ = <~--~-,z> , and
if
X(%) = X~(%) ~
JX($,~) = ~ Xa(¢)~, 6J x = ~--
k([)
6J x ~
6J x So
~(~,~)
(6.2) k
The unicity of
xa(~) ~
=
6(x~ ) a
with the stated properties follows from formula (6.2).
6.6. Remarks. When the base manifold
M
is a point, formula (6.2) reduces to the
classical formula for the lift of a vector field from X(q,p) = Xa ~ bqa where
(qa,p)
defined by
X(w) = ~d
the adjoint of sections of
X
VE
and
V E
X
where
to
T E,
T~E.
be the natural lifting of d$(t) dt
Y
to
k
It=O = w ¢ F(VE). Then
<[,X(w) > + <X(~),w > ~ 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($(t))Jt=O '
in the sense that
6.7. Proposition. Let Then
X ~ U(E),
E
3X B 3q~ PB -3p-
are local canonical coordinates on
More generally, given VE
then
We s h a l l Y
are
introduce
of order
local
at most
= ~a(VXoy-VYoX)~ n = (~
SY_~ X B _
8X-~ y B + a ~B
~Y~
<6Jy ^ 6(~aXa) --6~ 'X> = ( 6 5 B ~
__
+
= (~a ~$B~XaYB-Di (~a i~i ) YB
~y~
V E. A s s u m e t h a t
D.yB
X
~ X~ Dl(k)yB a ~$1(k)
Di(k)XB)~ ,
~$1(k)
{Jx'JY } = - X'Jy = - VJy °~ % -
va ~ B
~X a ~
~ya
~@6
_
on
k.
J[X,Y] V
_
i a (x ,~ ,~ )
coordinates
~y~)
XB+D i (Ta ~#~
X6
_
..
yB
6(~aYa)
65 B
XB)n
~ Xa ) yB " + (-')k Dl(k)(~a "---~----~$1(k)
~ya
"'" - (-l)k Dl(k) (~a ~ l ( k )
)
XB)r1"
52
J[X,Y]v {Jx,Jy}
In the difference
Xa B
~
appear the terms
a DI(h)YB - (-l)h DI(h)(~a ~X_~_~__)y6
$¢I(h) for on
el(h)
I ~ h $ k. Each of these terms is a divergence, as can be shown by induction h. It follows that
J[X,Y]v
is equivalent to
6.8. Lifted Lie algebra action on each Lie algebra action on V E
E
V E
{Jx,Jy}.
and its momentum mapping. We shall prove that
lifts to a strongly Hamiltonian Lie algebra action on
for whose momentum mapping we shall give an explicit formula.
6.9. Proposition. Let manifold
X * ~
be an action of a Lie aZgebra
o : E ÷ M. There exists a u n i q ~
action of
g
on
g
on the fibered
o : V E ÷ M, called
the lifted action, which projects onto the given action and which leaves The lifted action,
X ÷ ~
8
invariant.
is strongZy Hamiltonian and its e q u i v a ~ a n t comomentum
density mapping is : X ~ g ÷ Jx = J
(xE)
= <@,X E> c H(V~E).
A 6 J(xE) Proof. We set, for each X in g, ~E = X E = # 6~ 5J ~ ( ~XE, YE~ V) [X'y]E = = ~ 6~ . On the other hand, By proposition 6.7,
{J(xE) , J(yE)}
and
J
xE,yE]v)
onto
XE
g
on
X
and
[xE,yE]v
Y
~E
leaves
@
.g,
6J X 6Jw [~ ~ ' '~ 6~-£]V"
r^E ^E7 = IX ,Y JV" Therefore
o : V~E ÷ M. It is clear from proposition 6.5 that
and that each
in
are equivalent and therefore
(E
have the same variational derivative. Thus an action of
. For
X ÷ ~E ~E
is
projects
invariant. The unicity of the lifted action
also follows from proposition 6.5. By construction,
~
is a comomentum density
mapping for the lifted action. The equivariance of J (i.e., the equivariance of r the associated momentum mapping, }J) follows from proposition 6.7 and the fact that
[x,YElv by the definition of a Lie algebra action. In adapted local coordinates
JX(¢,~) The formula
JX = <e'xE>
=
(xZ,¢~,x)
<8,xE(¢)>q
obviously
on =
V~E, set
xE(¢) = Xa(~) ~
. Then
~ X~'(¢)~.
generalizes
the expression
for the
comomentum mapping of the lifted action on a cotangent bundle (cf. [I] 4.2.11). The
53 % J : X ~: g ÷ JX = s H(V*E)
comomentum density mapping for the lifted action,
f,~ : X c g ÷ JJx r jJ = I <0,~>
gives rise to a comomentum mapping
ponding momentum density mapping is the J(~) c AnT*M 0 gM
the equivariant
The i n t e g r a l s arising (cf.
)M ~a x a ( ~ ) n
[13] [6~). The f a c t
with
H
is
J : ~ ~ F(V~E) ÷
JJ
= ~x~(~)n.
: ~ c F(V~E) ÷
are the usual expressions
J(~)
that
these
integrals
Hamiltonian
be a Hamiltonian
on
with
transformation
are conserved is actually
Noether
c 9M
for the conserved quantities
o f a H a m i l t o n i a n u n d e r an i n f i n i t e s i m a l
of the infinite-dimensional H
= Jx(~) = <e,xE(¢)>
momentum m a p p i n g i s
from t h e i n v a r i a n c e
Let
g -valued Hamiltonian
with
<J(~),x> Finally,
s F(V*E). The corres-
theorem,
a special
as the next proposition
a : V E ÷ M. The Hamiltonian
case shows.
system associated
6H @ ~ (~)"
~t =
In adapted local coordinates
= (~,~)
where
3t
the system can be written
~H 6~
(6.3) 3~a 3t Let
invariant
X + ~
be an action of a Lie algebra
modulo d i v e r g e n c e
under the lifted action
Let
6.10. Proposition.
acts on
@H ~
p : E ÷ M
under
X ~ ~E,
H
X ÷ XE
if
H
i.e., for each
be a Hamiltonian on
and t h a t
g
on
E. We say that
is invariant X E g,
H
is
modulo d i v e r g e n c e
--~'H ~ O
or
{H,J X} ~ O.
V E. Assume that a Lie algebra
g
H i s i n v a r i a n t moduZo d i v e r g e n c e under t h i s a c t i o n
(i.e., under the lifted action). Then the momentum mapping
IJ
is conserved.
With
the p r e c e d i n g n o t a t i o n s ,
dat(j~a(t)x~(*(t))n) when Proof.
(7 (t),ta(t))
satisfy (6.3).
Since the lifted action is Hamiltonian
this proposition 6.11. Example.
= 0
with momentum mapping
follows directly form corollary
Consider
the vector field
X = x 2 -~-
generalized vector field
~
5.6.
3x 1 field defines a vertical
J =
x 1 -~3x 2
on
~3. This vector
X = (x I~2_x2~i ) ~3
on
E = ~3xI{~3.
54
The lifted vertical generalized vector field on (6.4) ~6
~H
V E,
3
is .
(~(xl~2_X2~l)) = Dl(~X2 ) - D2(~x I) = -xl~2 + X2~l , formula (6.4)
follows directly from formula (6.2). For on
~3~RxR~ ÷
+ (xlv2 - x 2~1 ) ~
X = (xl+2 - X2@l ) ~
In fact, since
H
V~E =
X'H = VH o X = ~
~H
A + ~
X = A ~-~ + B ~ , ~H
B + --
by (6.4) and for the evolution Hamiltonian
and a Hamiltonian
DiA. We compute H
X'H
for the
X
given
of the Klein-Gordon equation given in
2.3. We find that
X'H = (m2%+f(~))(x[~2-X2%l)+v(xlv2-x2 ~l)+~l(%2+x1~12-X2%ll)+%2(x ] ~22-%1 -x 2 ~12 )
+#3(x1~23-x2~13 ) = -Dl(X2H(~,~))+D2(xIH(~,~)),
where
H(~,~)
leaves
H
is the Hamiltonian defined in 2.3. Since
X-H
Ks a divergence,
invariant modulo divergence.
By the Hamiltonian Noether theorem we know that the functional r f JJx = j~(xl*2 - x2*] )dxl k dx 2 A dx 3 is conserved. The other two infinitesimal generators of the rotation group
S0(3)
also yield
conserved functionals, and the momentum mapping for the rotation group is the angular momentum
(~,~) ÷
We compare
J~(~,~) = J~(~ A grad ....~)dx JX
A dx 2 A dx 3.
with the conserved current obtained in example 4.10. Setting
= Duo~L = Uo' the conserved quantity
JX
becomes the time component
called the conserved density) of the conservation law
v~
(usually
~X" This fact is general, as
we shall see in proposition 7.16.
7. NOTES ON THE LEGENDRE TRANSFORMATIONS.
classical Legendre transf~mation associated with a time-
In mechanics, the independent Lagrangian FL
of
L
with configuration space
Q
is the fiber derivative
L : TQ ÷ ~. It is the purpose of this section to show that the Legendre
transformation admits three types of generalizations in the infinite-dimensional case : the relativistic Legendre transformations,
the evolution Legendre transformations,
and the Legendre-Cauchy transformations. 7.1. Relativistic Legendre transformations. Consider a field theory where the fields are sections of a fibered manifold of order
k
on
~ : F ÷ M, which can be described by a Lagrangian
F. We call such a Lagrangian a relativistic Lagrangian. In §4 we
55
have considered the classical Lagrangian
the vector
Legendre
on
l-forms
FL
transformation
on
F
of order
when
L
is a time-independent
~ : ~ × Q ÷ ~. To distinguish
in_t~hi~ section we shall call the vector
transformations associated with
(2k-l,k-1)
which reduce to first-order
them from the other objects l-forms
L. We recall
FL
to be discussed
relativistic Legendre
the
that %
(i) the Noether mapping is w r i t t e n generalized
vector field on
F
which leaves
(ii) once the H a m i l t o n - P o i n c a r ~ - C a r t a n Noether mapping
is written
X ÷
a manifold
n+|
denote
Let
i.e.,
u(xO,x l) ~L
@L
is defined
in terms of
that the base manifold
be the coordinate
(ii)
on
k
such that
L
depends
is a
~L, the
for each m u l t i i n d e x
M
~ × M (O)
~
and (xi),
derivatives
to the time
I(r) = (io,il,...,i n)
will
on
on the time, of a section
i.e.,
x O = t,
such that
is
a Cauchy
L
does not depend explicitly
] with respect
~ : F÷ M M (O)
i = 1,...,n,
M (O). We consider Lagrangians (i) L
of
where
there will correspond
on
only upon the partial
which are at most of order
= O
~u?
and
n. To each such splitting, xO = t
of order
~L = O, and ~x O
invariant,
We now assume
a system of local coordinates
: F ÷ ~ × M (O)
L
and that it can be split into a product
of dimension
transformation.
form
X
[VX] = [@L o X].
7.2. The Cauchy transformations. is of dimension
[v X ] = [FL o X + i XM L], where
X ÷
i O ~ 2.
l(r) We call such a Lagrangian
admissible. Any time-independent
first-order
Lagrangian
is
admissible. (For local splittings, Let
p : F (O) ÷ M (O)
the inclusion
I : x ¢ M (0) ÷
are of the form
~(xl),
on
k
F
that of
of order L(O)(w)(x)
VF (O)
one could introduce be the pullback
= L(u)(O,x)
along the section
defines for
F (0)
a Lagrangian
u(O,x)
of
w(x)
Conversely,
assuming
that the partial
there corresponds
F (0) = ~~u -[
formation because the splitting in
on
F
sections
on
VF (O)
and
to
M(O) under
p : F (O) ÷ M (0) Lagrangian of order
w e F(vF(O)),
k
L such
a section
such that
(t,x) It=O
on
equations
condition,
(7.1) have a unique soto a Lagrangian
F. We call the mapping
to the Lagrangians
M = ~ × M (O)
of
Each admissible
L (O)
differential
a unique Lagrangian
Lagrangians
a Cauchy hypersurface
~ : F ÷ ~ × M (O)
u c F(F)
for all times for a given initial
from the admissible
of
~ = 1,...,d.
x c M (O),
(7.1)
lution defined
I~F
(O,x) c ~ × M (O). Locally,
i = ],...,n,
uniquely
local Cauchy transformations.)
corresponds
M. In local coordinates,
on
VF (O)
where
of
L ÷ L (O)
the
to the choice of
w = (~,~)
L (O)
Cauchy transM (0)
as
#(x i) = u(O,x i)
56
• = ~8u (t,xl)[t=O. ' 4(x i)
and
i O = O,
Let
I(r) : (iO,ll, " " "''' in), with
UI(r)(O'x i ) = @J(r) (xl),
~i(r)(O, x i ) = 4K(r_]) (xl),
where
In the case of mechanics, M (0)
reduces
to a point
transformation itself,
and
J(r) = (i] .... ,i n ). If
K(r-1)
= (i] , .... in).
F = ~XQ + [
fields are sections
on
first-order
transformation.
corresponding
of the m o m e n t u m mapping with respect
L
is a simple generalized
evolution Legendre transformation Legendre defined
transformation.
of
VE
tional derivative 6L %-~
of
field on
along of
are
LI~
element on the orientable VE
defines
]-form
LI$
of
into
evaluated
at
(X~,Y~),
are the values
structure
of
V E.
~L
on
VE, which we call the
L, which generalizes
for
If
Y
derivative
the classical 6L 6~-
of
L,
of L to the vector space 6L w e F(VE), ~-~ (w) is the varia-
(x))
local coordinates,
is a vertical
~L
generalized
~
l-form
from the sections %L
the
generalized
6L > = 64~ 6L ya " Let <%-~4,Y
then
M. The simple
÷ ~(x) = ($~(x),~
the
p : E ÷ M. Given a Lagrangian
w. In adapted
~ = 1,...,d.
manifold
field theory,
and we have shown in §6 how the
o : V E ÷ M. In local coordinates,
w(x) = ($~(x),4a(x))
L(q,q),
be the restriction
a mapping which we still denote by
to the sections
Q,
of
with
~ ~ F(E);
),
on
of the Hamiltonian
bundle
associated
: let
64 with components
VE
p : E * M
~, where
(O,
manifold,
VF (0) = TQ. The Cauchy
Lagrangian
is the partial variational
in the following way
of sections
ponents
~L
i O = I,
to the canonical H a m i l t o n i a n
T : VE ~ M, the vertical
VE, there
I. If
is the configuration
and
In nonrelativistic
to symmetries
We now consider on
Q
or
VF (0) = TQ.
of a fibered manifold
quantities
where
p : F (O) = Q ÷ {point}
as a function
7.3. The evolution Legendre
conserved
where
maps a time-independent
considered
iO : 0
com-
vector
be a volume 6L %L = %-~ of
on
T : VE ÷ M
is described by
where
~a(x)q = 6L (xi,$B(xj),$~(x j) ..... @B(xj),4~(xj ) .... ). @L induces a mapping, also 65 ~ denoted ~L, from the generalized O-forms on V E to the generalized O-forms on VE, and defined by VE
to
V E
if
}L(H)(w) 6L %-~
case we say that Lagrangians
L
L, if
LI$
2)
O,
ordinary
is of order
any Lagrangian
from a first-order E
is of order is an
Note that i.e.,
}L
only defines
is an ordinary
Lagrangian.
a mapping
l-form on
We list examples
VE.
from In this
of ordinary
:
I)
3) L, if
= H(%L(w)).
L (O)
on
Lagrangian
is a Riemannian
O
for each
t e F(E)
E = VF (O) ÷ M (O) on
F ~ ~xM(O),
and
3L 84
6L 64
is of order
0,
obtained by the Cauchy transformation such that
bundle with metric
tensor
3L ~-~
is of order
g, and
L
O.
is the kinetic
57
energy
K
of
element on
and
E
M. If
(g~8)
w = (~a,@a),
Thus
~
l
defined by
then
K(w) = ~ g(w,w)q, where
are the components
K(w) = K ( ~ , ~ )
= gag~ 6, More g e n e r a l l y ,
energy by a potential section of
VE
along
~
~
~
Lagrangian
o~s~ o~ ~ o ~ n f o a = on
a function on
the morphism
VF (O) = TQ, reduce
to the same morpbism of
of a Lagrangian
~L" For example, and
if
L
~L (0)
on
E
VE
is
TQ
into
mapping from
F(VE)
~L ~71~
@L
VE. The ~ g ~
the pullback under
to
L
l-form
on
-d@L(#L%,Y)
be of order
0
VE
considered T~Q
as
defined
O-form
~ 8 ~
L
Al of
L
is the equivalence
on
VE
is the class
L(w) = K(w) - U(~), then
L
VE
to
l-form on
V~E
d~a.
is nondegenerate
for each
from
of the generalized
if
condition
~L for
defines a one-to-one L
to be nondegenerate
~. An ordinary Lagrangian
L
is nondegerate
V~E. we define ~L
to be the Hamiltonian
obtained as the image of the canonical Hamiltonlan
structure
~L -l. -d@(#~,Y)
= <%,Y>,
for
= <~,y>
for
~ e ~(VE),
following formula holds. Let
~ ~ ~(V~E), Y s V(VE).
% = (A d ~ + B
Y e V(V~E),
#L ~ = BB
~
it follows that
In adapted local coordinates,
dp~)q e ~(VE).
Then
~P
(7.2)
L
@. In adapted local coordinates
is ordinary and nondegenerate,
under Since
of
~L
F(V~E). A necessary
is a diffeomorphism If
is the generalized
is a Riemannian bundle and
We say that a Lagrangian
V~E
is ordinary.
first-order
L (0)
w g F(VE). The ~
@L ~ ~L
structure
is a
EL(W) = K(w) + U(~).
defined by the Liouville vertical
on
where
w
is one-to-one.
is a time-independent
on
for each
~L = AL - L
We denote by
eL
~L : VE ÷ V~E
L
O-form
if
and therefore
and
generalized
is that
6L ~K ~-~ = ~-~
FL
AL(W) = <~L(w),w >
of
E
L(w) = K(w) - U(~), where
when
defined by
~L
on
= ~K =g~6~6q a~ from t h e k i n e t i c
a differing
F = ~×Q ÷ ~, both
The ~ o ~ o ~
I
only,
= (~K) ~-~
and (~K)
~L (q,q) ÷ (q, - - (q,q)). ~q
locally by
A L = 2K
= ~1 g a B ( ~ ) ~ a C g n ,
~ g F(E), is such that
We note that for such a Lagrangian
is the Riemannian volume
of the metric tensor along the fibers,
any L a g r a n g i a n
function of
D
(--~
--Z))
the
58
For example, @L
if
E
is a Riemannian
= O K = g~B*Bd¢ ~
(7.3)
bundle
and for a simple
~ ~
The following
and
generalized
- U(~),
then
VE, ~ = (A d#a+B
d~a)n,
~gB~ _ ~ g ~ o ) ~ )
lemma is obvious.
on
H L = E L ° (~L) -I
= K(w)
l-form on
9____ _ gaS(As+gPYBp
To each nondegenerate Lagrangian
7.4. Lemma.
L(w)
V~E. If
~
L
on
VE
is associated a Hamiltonian
is ordinary, the evolution Hamiltonian equation
(with respect to the Hamiltonian structure
~L) on
VE,
6E L wt = ~L ~ (w),
(E)
and the evolution Hamiltonian equation on
V E,
S HL
(H)
~t = ~ ~
(~)'
are equivalent. 7.5. Remarks. theory, while and
E
Let
F(E) F(VE)
be the trivial
or some subspace = F(E)xF(E),
vector
of it plays
the tangent
Hamiltonian
weak symplectic
structure
[6]. In the same way, Lebesgue
measure
described
structure
on
on
set ~n,
F(V~E)
K(w)
O
of
on
space,
is the phase
space
space.
The
gives rise to the canonical
for
~ E F(E),
structure
to the weak sy~lectic
field
phase
in the sense of Chernoff
12 = ~ ~
= K(~,~)
In a scalar
is the velocity
F(E),
V E
= T~(F(E)),
then the Hamiltonian
above gives rise
F(E),
space
of order
~nx~ ÷ n.
the role of the configuration
space of
F(V E) = F(E)x(F(E)) ~, the cotangent
canonical
bundle
~K
and let q
of order
structure on
and Marsden
O
be the on
VE
F(E)xF(E)
f w((~|'@I)'(~2'~2)) described partial
in
[6] and used there to investigate
differential
wave equations. tensor bundle Riemannian
= J~n (~I~2 - +2~I )n
equations.
In fac~ over
vector on
M
let
K
VE
Hamiltonian
structure
[[4],
in a natural
which ~K
on
the conserved
Iwai applies
be a Riemannian
(or spinor bundle
bundle
Lagrangian
M
In
induces
if
M
manifold
and let
way and therefore from
there VE
of scalar
treatment
Then
is a kinetic to
to vector
p : E ÷ M
is a spin manifold).
an isomorphism
VE. See 7.9.
quantities
a similar
V~E,
be any E
is a
energy
and a
59
7.6. Liftings of vector fields to Z E V(VE)
project onto the same
VE. We first observe X E V(E), then
their common value by
<0L,X>. For each
local coordinates,
X = Xa ~
L
if
Y
and
<~L'Y> = <0L,Z> , and we denote
X s V(E)
P~(w) = 6 L
(cf. 6.4) that if
PXL = <@L 'X>" In adapted
we set
(w) X~(~). It is clear that
if
is ordinary, PXL = jX o ¢L.
If
L
is ordinary and nondegenerate,
we can consider the globally Hamiltonian
vector field (with respect to the Hamiltonian X
and
structure
@L )
on
VE, depending on
L, 6p L ~L = ~L 6w x
So,
~• = Xa 9#a ~ _ (6___6%y(~BXB) + XS(~Trxg~8 ~))~8 ~a~vX ~@~ ~ . It is clear that the vector
fields
%
to
and it leaves
VE
on
VE
and
X
on
OL
divergence if and only if propositions
are
invariant.
In addition,
X
HL
leaves
L
vanishes and
equation
Thus
~
<
is a lifting of
leaves
EL
X
invariant modulo
invariant modulo divergence.
be an ordinary nondegenerate Lagrangian on
is invariant modulo divergence under
~L ~-w
tL-related.
Applying
5.4 and 5.5, we obtain
7.7. Proposition. Let
[L
V~E
PX =
<0L'X>
VE. Aesume that
X . Then the vertical bracket of
<
and
is a conserved integral for the evolution
(E) .
This proposition should be compared to [6]. It can be generalized
to Lie
algebra actions, as in corollary 5.6 and proposition 6.10. 7.8. Corollary. Let
[L
X ~ g ÷ ~
be a Lie algebra action on
~ V(E)
is invariant modulo divergence under the lifted action
Then the momentum when
w(t)
[jPL is conserved, i . e . , for each
is a solution of
X
in
E. Assume that
X c ~ ÷ (XE) L c V(VE).
g,
(j~t a Px(w(t)) T.
= 0
([).
Because of the role played by the Lagrangian
L
and because equation
([)
is in Hamiltonian form, this corollary is a Lagrangian/Hamiltonian Noether theorem. 7.9. Example. Assume that energy
K, or
L = K-U
where
PKL(w) = pK(w) = g~8*aXB(~)n in the fibers of
p : E ÷ M U
is a Riemannian bundle and
is a function of
= <~,X(~)>g~
p : VE + E, and
where
<
~ '
L
is the kinetic
only. Then >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~(~,~) = -<~,[XM,~]>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
<@,[XM,~]>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 = <0'x(O)>
%
62 The diagonal arrow in the above diagram, which we call the Legendre-Cauchy transfor-
mation, is obtained by composing the Cauchy transformation with an evolution Legendre transformation from
VF (0)
to
V~F (O).
Given an admissible relativistic Lagrangian is nondegenerate,
the evolution Hamiltonian
L
HL(O)
on
v : F ÷ M
such that
V~F (0) + M (0)
on
L (0)
has the folio-
wing property which generalizes the equivalence in classical mechanics of the EulerLagrange equation for a time-independent,
first-order, nondegenerate Lagrangian and
Hamilton's equations for the associated Hamiltonian on
7.11. Proposition. Let L
be an admissible Lagrangian on
is nondegenerate and let
L (0)
Hamiltonian associated with EL(u) = 0
T~Q.
EL(O)
and
7 : F ÷ ~×M (0)
such that
be the energy density and the
HL(O)
L (0). Then the relativistic Euler-Lagrange equation
is locally equivalent to the evolution Hamiltonian system on
V~F (0)
6H L (0)
(H)
~
Ct = #
If, moreover,
is ordinary, then
L (0)
evolution Hamiltonian system on
(~)" is also locally equivalent to the
EL(u) = 0
VF (0)
6EL(O) (E)
w t = #e(O) ~ - ~ - -
Proof. If
L (O)
is ordinary and nondegenerate
(E) and (H) are equivalent by lemma 7.4.
We shall use local coordinates to prove that, if order
k
such that
L (0)
admissible Lagrangian 6L
= ~L
6~
~u ~
6L
~L
6~ ~
~u~
_ D. ~ L
L
is nondegenerate, of order
k
_ D. ~La
L
is an admissible Lagrangian of
EL(u) = 0
is equivalent to (H). For an
we shall set, with a slight abuse of notation ,
+ ... + (_1)k Dl(k ) ~L --~
1 au I
(w).
where
l(k) = (O,i] , ... ,in) ,
and
~UI(k) + "'" + (-l)k-] Dj(k-]) ~L
1 3Uo, i
where
J(K-I) = (i],...,in).
3Uo,J(k_|)
Then (EL(u))
6L
DO(%
6u~ Since
L (0)
is nondegenerate,
moreover, the relation
6L (0)
"
6L (0) ~ }~ B
) + ~L
6~
~+~
is necessarily of order
(x1,~8,...,~l(2k),~B) = ~
0
and,
can be solved uniquely to
63
yield
$~ = @~(xi,¢5,¢~,...,$1(2k),~S8 ) = ~($,~). By definition, - L(O)($~,~S(~,~))
~(~,~) Let if
L
us first
does
variables
not
assume
depend
together,
6H
L
" that
on mixed
and
derivatives if
~
6~
involving
the
is first-order.
3L(O) ~ S
time
is satisfied and
space
Then
~(~,~),
3~ B
)
+ ... + (-I)2k Di(2k)(~ B - 351(2k)
3L (0) 35 B
$*_~) 3¢~ + D i ( ~ ~(0) 35~
3L (0)
3L (O) D. - + ...
L(O) I~
i 351
(_] 2k 9L(O) Z@S -) ) Dl(2k)(3~6 3~i(2k)
_
depends upon
~y,~y,i,...,~y,i(r). However, since
3L (0) + (-1) k DI(k) 3~i(k)
is of order
is an admissible Lagrangian of order L (O)
assumption
~_i!)
3~
More generally, when
involved since
O. This
~L (0)
6L (0)
where we have set
L
- Di(~B
6L (0)
L
is of order
partial
in particular
6H (O) 3~ ~ L ($,~) = ~6 -
if
L(O) I
(O)
= -
HL(O)($,w) =
k-1 > O
--this will be the case
k > | --, the computations are somewhat
@S,~, "'''~I(k-1)' S while ~
is of order
O
in
B ~I(r)
depends upon
~, the following
identities hold : ( ~(DI(r)~B) D~,J(r) 3 Since ~ For i n s t a n n e ,
and
D. 1
I
3rBm if J(r) = l(r) 3w O otherwise .
commute, expressions such as
3(DI(s)@S) 3~ ~,J(r)
can also be simplified.
64
2¢ 6 Dk ~ a
if
i=j
Dj ~ 6
if
i =k
~(Djk~B)
O
otherwise.
Using these identities and Leibniz's rule we obtain the same result, ~HL(O)
a
and
~ Let us consider a one-parameter ~(t,x) = (¢a(t,x),~a(t,x))
such that
~H (0) L
5L (O)
5¢ a
6¢ a
family of sections of ¢~(O,x) = ¢~(x)
V~E, is
E (O,x) = ~ a (x). Then
a solution of (H) if and only if (7.4)
Ct(t,x) = ~a(t,x)
(7.5)
where
~B(t,x)
since
u~ = ~ t
5L -
-
5L(O)
E~(t,x)
(~B(t,x),~B(t,x))
= ~6(~Y(t,x),E
(t,x)). Setting u(xO,x i) = (~(t,xi)) X 5L(O ) 5L by equation (7.4), (¢B,~6) = ----~(u), and E 5¢ a 5~
(u), so that the system of equations
+ 5L -Do(SL--L--5~a)(u) ~(u)6¢ = O,
we see that
6L (O)(¢B,~B) 5~ a
(7.4) and (7.5) is equivalent
to
i.e., EL(u) = O.
7.12. Remark. In §2 we treated the example of the relativistic Lagrangian
L = LKG
of the nonlinear Klein-Gordon equation. More specifically, in 2.4 we obtained the energy [ = EKG of this Lagrangian and we proved that the evolution Hamiltonian equation
([KG)
introduced an (H)
is equivalent ¢o the nonlinear Klein-Gordon equation.
ad hoc Hamiltonian
H
also turned out to be equivalent
to the nonlinear Klein-Gordon equation. This
fact now appears as a special case of proposition nothing but the Hamiltonian 7.13. Examples. Let (i)
Let
F = ~x~
EKG o (~L(O)] " KG "-I d ÷ ~xE
7.9 since the Hamiltonian
associated with
H
is
LKG.
with the usual coordinates
(t,x,ua).
L(u) = P(x,u,ut) + Q(x,U,Ux,Uxx,...).
6L (0) Then
In 2.3 we
for which the evolution Hamiltonian equation
L(O)(¢,~)
: P(x,¢,~) + Q(x,¢,¢x,~x x .... ) , - 5~ ~
~L (0) -
~
-
=
-
~P ~a
(x,¢,~)
=
a
:
65
~2L(O) and
L (0)
is ordinary.
Moreover
L (O)
is nondegenerate
if and only if
is
an invertible matrix. [ = [L(O)
= ~a~a-L(O)(¢,~)
= *a SP-~ ( x , ¢ , ~ )
- P(x,¢,~)
i[_~ = ~B 32p
6[
By formula
(7.2),
([)
~P
- Q(x,¢,¢x,¢xx) •
Dp
+ ~B 32p
3p
46 ~2p
is the system
a = ~a
~t
32p
3P
or
+ 6Q
32p
B
As an example,
] ~2~2 = ~
if
$2p
]
B
~B 3 2 p +
~g = O
~p
+ ~Q
which is precisely
2 2
l
2
1
2
L = ~ u u t + uu t + ~ u t + ~ Ux,
EL(u) = O.
then
! 2 I 2 +7 - ~ ~x' f
J Ct
and (E)
= ~ ( 2+i)~ t = _ ~ 2
i.e.,
(u2+l)utt + uu~ + u
(ii)
Let
L(O)(~,~)
L(u)
xx
is equivalent
be an admissible Lagrangian of order 3 on
= R(x,~,~x,~xx,~xxx,~,~x,~xx).
O
~xx
L(O)(~,~)
~x
so that
operators of order
EL(u)
0
~L (O) ~B 6~ - DxxA-DxB + ~
in
in
~
= O,
= O.
~L (o) 6~ I~ to be of order and
to
_ ~xx
~
for each
F. Then
It is easy to find necessary conditions
~.
R
= A~xx + B~x + C
and of order 3 in
~C ~B ~C ~x + ~--~ = DxxA - --~x + --~
for
must depend at most linearly on where ~, and
A,B
and
C
are differential
~A = O. Then
which is an operator of order
O
in
66
and of order 5 in
t.
As an example , if 6L (0) 64 - ~x
and
example
L = Utx x + xutUtx + UUxUt, then
- ~ = ~" Thus
L(O I~
L (O)
is nondegenerate
is of order 2. Since
H(~,~x,¢xx,¢xxx,~,~x,~xx)
~ = ~x
L (O) = ~xx + X ~ x
but not ordinary.
+ ~x ~ In this
- ~ ' we can compute explicitly
= ~(~,~x,~)-h(O)(~,#x,~(~,~x,~),~x(~,~x,~xx,~,~x),
~xx(~,+x'¢xx'+xxx,~,~x,~xx)" @H ~ = ~ and
We verify that
6H 2 2 6--$ = - ~x ~ + ~%x + ~ }xx
7.14. Symmetries and conservation on
~ : F ÷ ~xM (O)
~ s F(F (O))
lift
k(O)
of
X (0) and
X (O)
with Hamiltonian
X
be a vertical generalized vector field
i.e., there exists a vertical genera-
which is time-independent,
lized vector field each
laws. Let
on
F (0)
u c F(F) to
J
V~F (O)
satisfying such that
6L (0) 64
which is indeed
(x(O)(%))(x i) = (X(u))(O,x i)
for
~(x i) = u(O,xi). Recall from §6 that the
is the globally Hamiltonian vector field on
= <8,X (0)>, where
0
V~F (O)
is the Liouville vertical form of
X (O) V~F (0). We shall abreviate
Let each
L
Jx(O)
~ s F (F(O) ). Since
w = FL o X + iXML
be the time-component 7.15. Proposition.
for each
L
L (0) = <eL(O),X(O)> Px(O)
In fact,
~u~
L
of
of
is of order
X M = D O . By definition ~x
PXL "
0
for
X, where (Xu) a = -u~.
EL(u) = O, and let
~.
the generalized O-form on
VF (0)
L(O) I~
defined by
is of order 1~v 0
L (0).
being first-order in the time derivatives,
(i~wO)(~,¢)
(See [13] and 4.9.)
F
to
it is invariant under
be an associated conservation law for
(u) u~ + L(u)
t~ O
on
L(O) I~
For an admissible Lagrangian such that
t z F(F(O)),
Thus
and
is time-independent,
opposite of the ~nergy density of
O(u ) = _ ~L
JX'
be an admissible Lagrangian such that
is the vertical representative Let
to
and
DL (0) = - _ _ (t,~)~a
0 ) is the component N(O
+ L(O)(%,~)
= - [L(O)(¢,~).
of the energy-momentum
tensor.
0
is the
0
87 7.16. Proposition. Let
L
be an admissible Lagrangian such that
ate . If the time-independent vertical generalized vector field L
invariant modulo divergence, then the lifting
HL(O)
k (0)
of
X (O)
L(O) x
on
to
is nondegenerF
leaves
V~F (0)
leaves
invariant modulo divergence. If L (0) is ~rdinary, the liftin~ ~(0) = ~(0) of L (0)
X (O) to VF (O) leaves [L~) invo~iant modulo divergence. If, moreover, order
0
for each
@ c F(F(O)), then the time-component PXL = <SL(O) ,x(O)>
~X" the momentum density
Jx = <0'x(O)>
on
V~F (0)
on
VF (O~
O
L
~ ~X = PX = Jx o <6L ~u 'x> ~ 0
First we note that
(DoX~)
~X (0)~ ~8
It=0
~H ~n (0) ~-~ = ( - - ,~a) expression for <6H ~(0)> ~,A
of a conservation law
and the momentum density
~iB
%L(O)•
and we show that this implies that
6L <~-~,X> ~ 6L 6~ ~ X~ + 66L~
~X (0)~ +
~
is of
satisfy
(7.6)
Proof. We assume that
~
L(O) I~
(DoX~)
B SX (0)~ 9i + "'" + 8
and
<~H~,~(O)> ~ O.
6L ~ ~it= 0 = ~
,
~B On the other hand I(k)"
~(k)
(see the p r o o f of p r o p o s i t i o n 7 . 1 1 ) . T h e r e f o r e , u s i n g the
~(0)
g i v e n by formula (6.2) we o b t a i n
6L (0) 6(x(O)aTr~ ) x(O)~ _ ~B Since 6~ ~ ~+B "
~B 6 ( x ( O ) ~ ) 6%8
) ~ (~X (0)~ ~B + ~8X (0)~ ~B +...+ ~X (O)a ~B ~%B ~%~ i ~--~l(k) l(k)
divergence, the conclusion follows.
is always a
(The terms that occur in this difference are of
the same type as those encountered in the proof of proposition 6.7.) The result on the invarianee of coordinates, L
[L(O)
follows from 7.6. The equality (7.6) is proved using local
v OX = ~L ~
X~,
O _ ~L (O) x(O)~
I vX
~
e
= PX' where we have used the fact that
is only first-order in time-derivat{=~es. In sum, assume
L(O) I%
is of order
L 0
is admissible, L (O) is ordinary and nondegenerate, and
for each
~ c F(F(O)). If
X
is a symmetry of
L, then
68 0
of
• 0
6(~ w X) ~w
L(O)
is a symmetry of
EL(O)
4((,~ VX ) o ~..--6¢
and
~L(O)-I) is a symmetry
H (o)" L
We note that the proposition Lagrangian
depends on higher-order
variables.
For example,
X = Xa ~
, then
Du o
(7.6) are not valid if the
mixed derivatives
involving the time and space
L = L(x,u,ui,...,Uo,Uo, i)
wO = (3#_ O
Du a
PX = (~La - D .
if
7.15 and formula
D. ~L J ~Uo, j
on
)X a + ~ L D.X a a j J ' ~Uo,
F = ~Rn~R d ÷ ~
and
while
3L__~__) Xa J ~u a . It=O O,j
We have shown how the conserved densities symmetries
of a system can be obtained
formalism with an (n+1)-dimensional as the time-components
associated with the infinitesimal
in two ways
: In the relativistic
laws
(vector O-forms, i.e., vectors on the
depending on the fields).
In the Hamiltonian
in which time plays a role apart from that of the space variables,
7.17. Generalized Legendre transformation
transformations.
in field theory. However, structure
with a Hamiltonian
evolution Legendre
3-space)
are obtained as values of the momentum mapping.
with its canonical Hamiltonian o : G ÷ M (0)
theory,
the fields are
base (e.g., Euclidean
of ~ibered manifolds with an n-dimensional
and the conserved densities
Lagrangian
(e.g., Minkowski space) they are obtained
base
of the conservation
base manifold With coefficients
sections
n
transformation
We have just described
the Legendre-Cauchy
in more general situations where is replaced by an arbitrary
structure
~G
V~F (0)
fibered manifold
in the sense of §5, the notion of
associated with a nondegenerate
Lagrangian
admits
the following generalization. 7.]8. Definition.
and let on
F
~G
and
Let
be a Hamiltonian structure on L (0)
the Lagrangian on
transformation. Let G. We also denote by G
~ ~
G. Let
L
r : VF (0) + M (0)
the mapping defined by
L
~
be two fibered manifolds,
be an admissible Lagrangian obtained from VF (0)
L
by the Cauchy
to the sections of
from the generalized
VF (0) . We say that
~
O-forms on
is a G-Legendre transforma-
if
is injective and the energy density
a)
~
b)
the Euler-Lagrange equation
Hamiltonian system on
o : G ÷ M (O)
be a mapping from the sections of
to the generalized O-forms on
tion associated with
(H)
and
~ : F ÷ ~ x M (O)
EL(u) = 0
G,
EL(O)
is in the image of
is locally equivalent to the evolution
~H
vt
~G
6v
~, and
L(O) (v),
69
where
HL(O)
= t
-I
and is called the Hamiltonian associated with the
(E (O))
L Lagrangian
L.
It follows such that
L (0)
field theory
from proposition
7.]] that if
L
is nondegenerate,
the evolution
Legendre
is a special
(This generalization mations where
which are of a different G
is not
7.19. Example Let
o : G = ~x~ + ~
generalized
(of a certain
~G : ~(G) + V(G)
involving
h g ~(G)
~L (0)
of
G = V~F (O).
Legendre
transfor-
transformation,
Hamiltonian
x. Generalized
fields
on
G
is Gardner's
v E F(G),
of the function
with base
by
(~Gh)(v)
%(v).
Let
structure.
~ = M iO)~ " O-forms,
all reduce
k) on scalar-valued
d #IG = dxx
and
Gardner's
vector bundle with base
vector
order
defined by
the total derivative VF (O) = ExEx~
transformation
be the trivial
]-forms and vertical
for
where
the relativistic
We now study a G-Legendre
on the base being denoted
tial operators
Explicitly,
nature.)
transformation
transformation
to include
Lagrangian
V~F (O)
: Legendre
the coordinate
(cf.
case of a G-Legendre
is not intended
is an admissible
and fiber
simple
~,
generalized
to scalar differen-
functions,
The map
Hamiltonian
structure
= Dx(%(v)),
where
~ : F = ~mRx~ ÷ ~x~
Dx
on
G.
denotes
as above.
Then
IR. A Lagrangian L on F is said to be of evolution type i L = ~ UxU t + B(u) where B(u) is a time-independent
[36]) if it is of the form
Lagrangian
of order
is always
admissible
Let
¢
k
involving
and
only the space derivatives
EL(O)(#,~)
be the mapping
of
u. Such a Lagrangian
= -B(¢).
from the sections
defined by
(9,4) E F(VF (O)) ÷ v = Cx g F(G).
mapping
from the generalized
of
VF (O)
to the sections
This mapping
induces
of
G
an injective F
~
defined by
E(¢,~)
= (¢H)(~,~)
We now prove ~B D--~ = O then
that,
(a necessary ~
if
O-forms
L
the definition
B,
6B ' under = -Utx + ~-uu
tion
EL(u)
is written
same energy
O-forms
on
~
VF tO)
of variable
with
type on
F
such that
to be in the image L. In fact,
Therefore
of
set
6B "'' = ~-~ " Since u x = v, the Euler-Lagrange
t
L = ~ UxU t + C(x,u)u t + B(X,Ux,Uxx,°..) differential
¢),
~G -~ ~H = _D x ~-~ 6B . But , b y
equa-
6H vt = ~G ~-vv "
of the type
and Euler-Lagrange
EL(O)
associated
H(v) = -B(v).
the change
of evolution
for
6~ ~B D2 ~B -Dx ~-v = -Dx --~¢x + x - -~ x x
EL(u)
A Lagrangian
condition
transformation
= -B(~x ). Then
= O
to the generalized
is a Lagrangian
EL(O)(¢, ~) = -B(¢) of
G
= H(¢ ). x
and sufficient
is a G-Legendre
on
as the Lagrangian
of evolution
has the type
70
]
UxU t + B(X,Ux,Uxx .... ). Therefore
~
is a G-Legendre transformation associated
with any Lagrangian of this type. Conversely,
any
first-order
Lagrangian for which
is a G-Legendre transformation is of the above type. There also exist higher-order Lagrangians for which is of order
O, then
B(X,Ux,Uxx,...) that
~ L
with
is a G-Legendre transfor.mation. If, for example, is necessarily
B
linear in
ut, L = A(x,U,Ux,Uxx,...)u t +
arbitrary and additional
conditions on
~6u B " Such higher-order Lagrangians exist,
EL(u) = -Utx +
L(O) I~
A
which ensure
e.g.,
one can choose
l L = ~ (Ux+nxx)U t + B(X,Ux,Uxx .... ).
7.20. The Korteweg-de Vries equation as a Hamiltonian and as a Lagrangian system. We recall that the Korteweg-de Vries equation is a Hamiltonian system on G = ~R×[R + £R d with respect to Gardner's Hamiltonian structure, #G = d-~x" Consider the Hamiltonian on
G,
I 3 ] 2 H(v) = ~ v - ~ v x. Then
to the Hamiltonian structure
6H 6v
I 2 2 v + Vxx. The Hamiltonian system (with respect
#G ) associated with
Korteweg-de Vries equation,
H
i.e.,
the
v t = vv x + Vxxx.
If we now consider the Lagrangian of order 2 on ] ] 3 ] L(u) = ~ UxU t - (~ u x - ~ u 2 ), we find that
Euler-Lagrange equation of
~H vt = Dx ~ v '
is
L
is
F,
EL(u) = -uxt +u x u xx +u xxxx . Thus the
Uxt = UxUxx + Uxxxx, which is the Korteweg-de Vries
equation for
v = u . In fact, L is the Lagrangian of evolution type such that x is associated to L be means of the Legendre transformation #. The fact that the potential equation of a Hamiltonian evolution equation
respect to Gardner's Hamiltonian structure) noted by several authors
(see
H
(with
is an Euler-Lagrange equation has been
[35] [36] E42]) in relation with the study of the
Korteweg-de Vries equation. And in [42] Tu called the mapping
v ÷ -u
a generalized
x
Legendre transformation,
a claim which we have justified here.
Conclusion. We have discussed the various Legendre transformations relevant to classical field theory and we have introduced generalized Legendre transformations which relate Lagrangians folds with base
(on
M (O)
~ : F ~ ~ x M (O)) to Hamiltonians on arbitrary fibered maniwhich possess a Hamiltonian structure,
in such a way that an
Euler-Lagrange equation is transformed into an evolution Hamiltonian equation. all cases, the role of evolution equations is fundamental.
In
This is easily explained
because the natural generalization in infinitely many dimensions of the Hamiltonian systems on a Poisson manifold,
i.e., the
infinite dimensional Mamiltonian systems on
a fibered manifold with a Hamiltonian structure, are indeed evolution equations. This in turn explains why it is only the admissible Lagrangians that can admit Legendre transformations
in this sense. If an Euler-Lagrange equation is to be replaced by an
evolution equation in which the parameter is the time, the Lagrangian must be firstorder in the time derivatives.
If it is not, more machinery must be brought in,
71
~.e.j higher-order tangent bundles to fibered manifolds. It might be interesting to explore the possible consequences of this approach for quantum field theory, and it could be worth-while to find the generalized Legendre transformations corresponding to various Hamiltonian structures.
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[I]
R. Abraham and J.E. Marsden, Foundations of Mechanics, 2 ed., Benjamin, Reading, Mass., 1978.
~]
V. Aldaya and J.A. de Azc~rraga, Variational principles on r-th order jets of fiber bundles in field theory, J. Math. Phys. 19 (1978) 1869-1875.
[3]
V. Aldaya and J.A. de Azc~rraga, Geometric formulation of classical mechanics and field theory, Riv. Nuovo Cimento 3 (10) (]980).
[4]
R.L. Anderson and N.H. Ibragimov, Lie-B~cklund Transformations in Applications, SIAM, Philadelphia 1979.
~]
S. Benenti, M. Francaviglia and A. Lichnerowicz, eds., Modern developments in analytical mechanics, Proc. IUTAM - ISIMM Symposium (Torino 1982), Suppl. Atti Acad. Sc. Tor~no, Turin, 1983.
~
P.R. Chernoff and J.E. Marsden, Properties of infinite-dimensional Hamiltonian systems, Lect. Notes Math. 425, Springer-Verlag, Berlin ]974.
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[8]
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[10~
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~II~
H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier, Grenoble 23 (1973) 203-267.
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C. Itzykson and J.B. Zuber, Quantum field theory, McGraw Hill, New York 1980.
[14]
T. Iwai, Symmetry of vector wave equations dealt with in Hamiltonian formalism, Tensor N.S. 35 (1981) 205-215.
E15~
J. Kijowski and W.M. Tulczyjew, A symplectic framework for field theories, Lect. Notes Physics 107, Springer-Verlag, Berlin ]979.
[I~
I. Kol~, Lie derivatives and higher order Lagrangians, in Proc. Conf. Diff. Geom. and Appl. (Prague 1980), Univ. Karlova, Prague ]981.
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[18]
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~45~
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AN AXIOMATIC
CHARACTERIZATION
OF THE POINCARE-CARTAN
FOR SECOND ORDER VARIATIONAL
FORM
PROBLEMS
Ja~b~le Mu~oz Masqu~ University
of Salamanca
(Spain)
i. Introduction.
As is well known, manifold
p:Y---+X
the first order Variational
developed by means of the Poincar~-Cartan [~).In
these cases
(r = 1
or
to each r-order variational to characterize
methods,
n = i)
[~,
[~,
~])
in a canonical way
j2r-1
which allows us
problem under consideration.
form; however,
through different
to each variational
problem of
in the general case this form is not
depends on a linear connection on the base manifold
for every variational
choose, among the Poincar~-Cartan
X .
problem of second order it is really possible to
forms associated
in a canonical way, as in the classical
to the problem,
cases ([6~,
The aim of this work is to characterize associated
on
have recently proved,
that it is always possible to associate
Nevertheless,
@
are
are [2], ~ ] ,
sections and also to study the geometric aspects of "the
arbitrary order a Poincar~-Cartan unique and essentially
to associate
fibered
dim X = n = 1
form (standard references
it is possible
for the variational
([i],
Calculus when
problem an ordinary n-form
the critical
manifold of solutions" Several authors
Calculus on an arbitrary
and the higher order Variational
one of such forms
[7]).
axiomatically
this canonical form
to each second order Lagrangian density and to use these axioms for
studying their functorial properties.
2. Preliminaries.
In what follows, we shall consider a surjective manifold
X
dim Y = n + m
of dimension
n
. The k-jet bundle of local sections of
with canonical projections
pk:Jk---+Y
Fibered charts for the submersion 1 Siam l~I S k
submersion
oriented by a volume element p
, pk:Jk---+X
p
are denoted by jk
i .k y~(Jx s) = D (Yi ° s)(x)
where
D~ =
~1~1 ~x~1
... 8X~nn
and
is denoted by
and
. Induced fibered charts in the k-jet space i , with Y0 = Y i ; that is:
= (~1'''''~n) g ~ n
p:Y--+X
on a
~ . We also write jk=jk(y/X )
~hk:Jh---+J k
(xj,Yi) are
for
h%k
, I4 j~ n ,
denoted by
,
is a multi-index.
(xj,y~)
.
75
The v e ~ c ~
subbundfe of
on
jk
X
given an arbitrary fibered manifold
, we write
T(Z/S) = Ker[T(Z)
There exists on
foJTm of order
jk
s
q*
> 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
Oi~ stands for the ordinary 1-form
multi-index
~
~Y~i
'
Oi~ = dy~i _ ~ Yc~+(j)idxj , and
(j)
is the
(j) = (0 ..... 1 ..... 0) .
(J In the sequel, we shall denote by
Dk, j
the vector field in
jk
defined by the
formula i ~ Dk,j = ~x---7+ ~ ~ Y~+(j) i 3 i l~l
(dxj)
, (dy)l~l=k
isjSn.
are a local basis of
(~--~)~y~~ ~
are a local basis of the subbundle
Definition: A vector field
of ord~
, (e
,
k
endomorphism
D
in
with
(--~)~y~i~i= k ,
Ker O k
is an infin~e6im~ ¢0mtac,t ~ta~6fo~m~tion
jk
if for every linear connection A
T*(J k)
V
of the vertical vector bundle
on
V(J k-l)
v(jk-l)jk
there exists an such that:
where the Lie derivative is taken with respect to the connection
LD @k = A ° 0 k ,
V .
It is easily checked that if the previous condition is fulfilled for one connection,
it is also verified for any other.
Proposition:
For every vector field
unique infinitesimal Furthermore,
Let
the map
D = ~ u. ~ . j 3 3
D
contact transformation D ~--+D(k )
+ ~ vi ~ i
in the manifold D(k )
of order
Y k
there exists a projectable to
D .
is an injection of Lie algebras.
be the local expression of
D , with
uj,vigC°°(Y) .
76
Then, one has
8 '7 i 3 D(k ) = [ uj ~ + ~ ,~ v a J 3 i lal =o 3y z where the functions
(2.1)
v
i
are determined by the following recurrence relations:
=v.
v o
;
~C~(d I~1)
v
1
i Dk,l v i+ ( l ) = Dk, £ v ei - ~. Y~+(J) 3
(2.2)
We only shall use these formulas explicitly vector field
D
is p-projectable.
one may consult
'
uj
,
/~1
in the case
k=2
and when the
For a more detailed statement of these results,
~].
3. Legendre and Poincar~-Cartan
forms in second order Variational Calculus.
~k,k_i:Jk---+J k-l
As is well known, the projection on the vector bundle
skT*(X)@ jk_IV(Y)
is an affine bundle modelled
• Thus, there exists an exact sequence of
vector bundles:
0 ---+ skT.(X) @ jk v (y)
Pk
(~k,k-l)*,
> T(J k)
> T(jk-1)j k ..> 0
or in other words, T(jk/j k-l) = skT*(X) OjkV(Y)
In particular,
P2
is determined
P2(dxj @ dx.j @ ~ 3 )
•
in local coordinates by the formulas:
i3 SY(jj)
,
, if
~2[(dxj ® d X k + d X k O dx.)3 ® ~ ~ ] = ~Y(jk) i~ where in both cases
to
S2T*(X) ® j2V(Y)
S2T(X) @ j2V*(Y)
(3.1)
,
(jk) = (j) + (k) .
Now, let us consider a differentiable dL
j
defines a
function
section
L
F = d L o ~2
on
j2 . The restriction of
of the dual vector bundle
, whose local expression is
F = X
X
i j,k sjk
~L
i 3Y(jk)
(~--e 3x. ]
~
3--~ ¢ dYi) '
77
gjk=l
, if
j=k
; Ejk = I/2 , if
Definition:
Let
on
X . We call the
Lw
the
covariant by
i
j #k
.
be a differentiable
function on
jz
and
co a volume element
second ord~ momen~2m fo~m associated to the Lagrangian density
T(X) 8 2V*(Y) - v a l u e d (n-l)-form A obtained by contracting the first J index of co with the first contravariant index of the section F determined
L , as explained
above;
that is,
A = c~(o~ By choosing
the coordinates
(x.)3
F)
0
.
in such a way that
co= d x i ^ "'" ^ d x n
, locally
one has :
(3.2)
A = ~
~
(_l)j-1
i j,k
SL
gjk
i 3Y(jk)
3
coj @ - - ® ~Xk
dYi '
A co. = d x A ... A dx. A ...A dx 3 I 3 n
where
Since
T(JI/Y) = T * ( X )
exists a canonical
@jIV(Y)
epimorphism
of
is a vector subbundle
V*(JI)~---L+T(X)
Given a V*(Jl)-vaiued defined by:
~ =~o
Theorem V*(J I)
j3
(a) q
Lw
is horizontal
v(J 3)
~ , we denote by
be a Lagrangian
(n-l)-form on
~
T(j1/X) = V ( J I) , there
@jIV*(Y)---+0
~ . With these notations,
i: Let
-valued
form
of
vector bundles
on
j3
~
the
density on
form
j2 . There exists a unique
which fulfills
X ; that is,
T(X) @jiV*(Y)-valued
we have:
iDa= 0
the following
for every vector
conditions: field
D
in
.
(b) ~ = A , where
A
is the 2 nd order m o m e n t u m
(c) The exterior differential
form associated
of the ordinary n-form
to
Lw .
@ = e 2 ^ ~ + Lco
satisfies
a
decomposition
where
~'
is a
V*(Y)j3-valued
n-form and
~
a
Homj3(V(j2),V*(J1))-valued
(n-l)-form. and
~
are the
Legrendre and PoincoA~-C~t~an fo~ms, respectively, associated
to the variational
problem defined by
Ar~y
verifying
n-form
~'
Lw . The forms
~'
condition c)will be called an
and
N
are not unique.
E,~£~t-Lagrange fo~m~
78
associated to this variational problem.
Proof: First we shall prove that the conditions (a), (b), (c) determine in a unique way the
~
form (and therefore they also determine the
@
form).
From condition (a), locally one has 1
(3.3)
~ =
~ i,j
1
~
fi i og co~ 0 dy~
Io~1=o
for suitable functions
@ = ,
I z,j . .
fZ.gC°°(J3) , and from (b) it follows that: ~J
(3.4)
i j-1 f(k)j = (-I) gjk
Calculating
d@
I fi i lal=o aj eaAwJ + tw ,
in the local basis
DL i 8Y(jk)
(dxj) , (dy:)i~i= 4 , (8)l~i< 4 of
T*(J 4)
and by imposing condition (c), one obtains: i
.
.
(_i) j fi. ~J 0i+(j) A e + i,j
IC~I=o
'J 2
+X
I
i,j
~L 0i --:-
i Iod=o 3y~
for a certain
(n - I) - form
i ~,
^
IGI=o
+
~'
This equation implies that the coefficient of left hand
0i A W
for
I~1 = 1 l
I
or
2
on the
side must be zero. Hence,
(3.5)
SL + [ (_])j fi. = 0 . i ~+(j) =(k£) og 8Y (k£)
(3.6)
~L + [ (_l)j( D 4,j f(k)j i i ) + (-l)k fiok = 0 ~Y(k) J From (3.4) it follows that (3.5) is automatically verified, and from (3.6) one
obtains: fi = i ok (-l)k-
(3.7)
because
(~L/$Y~jk))
( ~L ) ~L + ~ (_l)k i gjk D3,j ~ ' ~Y(k) 3 ~Y(jk)
is a differentiable function on
j2
o
Equations (3.4) and (3.7) determine completely the Legrendre form. By substituting (3.4) and (3.7) in the local expression of differentiating, we get:
@
and by
7£
= .
o^
CO- ~. dfoijA COj
i So, it suffices
~, = $.
(3.9)
q = -
for obtaining q
( Va)
by
(~a)
co - I. dfolj ^~j J
(3.8),
~f~k) i h ® dY(k)) $ h j co. j ® (dy~
~ {~I~<2
y$
. Moreover,
- i32(supp L), as follows
is isomorphic
the valued forms associated
, {supp qa }
Thus one can define
~Y~
it follows
,
the supports of the forms
Y
~nx]Rm----+~n j2(Va/U a)
to the function
L a = ~a L
that the collections
the
~
form is horizontal
d@a = 0i^~'a verifies
p~C~(X)
02A (~3AQa)
+
conditions
Remarks. In fact,
if
on
(a),
L = Z L
i) The forms L~ = L'~'
. Then:
a
, one has a
~ = ~ ~ = E A = A • It is also obvious a a a a
X • Finally, imply that
(b),
by formulas
finite.
~ = aZ ~ a , ~' = a~ ~'a ' q = ~ qa " Since
A = ~ A , from where a a
, and
. If one denotes
of supports
a
F = ~ F ; hence a a
d8
equations =
~'
such that the submersion
projection
to the cover
are locally
~,
from (3.4) and (3.7).
to the canonical
(3.9), respectively,
{supp ~a } , {supp ~;}
IBIs<2
O dy i
be a partition of unity subordinate
~a ' ~'a ' qa
(3.3),
i h,j
be an open cover of the manifold
P : V a - - - + U a = P ( V a) let
~L
~ ~ h,i j,k
in
O(k ) A
to take:
d @ = 0 1 A ~' + G 2 A (03 ^q)
are contained
Let
~ i,k
locally
(3.8)
and
-
3
01A~ '
+
@a = 0 2 ^ ~a +
LaCO
02A(~3AQ)
Hence,
.
that
(c) in the theorem.
A , ~
one has
and
@
only depend on the Lagrangian
L' = p-l[ , ~' = p~
density
for a n o w h e r e - v a n i s h i n g
V' = p-IF, A' = c~(~' @ V') = ci(~ @ F) = A • Hence,
~' = ~
Lco •
function and
®, = @. 2) The
q
Notation: @L
form may be taken to be horizontal
Once the volume element
the forms associated
Corollary:
Let
~'
w
to the function
, ~'
density Lco . For every section
of
[ by means of the previous
p:Y---+X
lj3s
X .
has been fixed, w e shall denote by
be two Euler-Lagrange s
on
forms associated
, one has:
lj~s
~L '
theorem.
to the Lagrangian
80
Proof:
From
d@ = 0 i A g '
(*)
+ 02 A (83 A~)
01 A (Q'--~') Thus,
for every vector
field
+ 82 A (83 A T )
02A ~03A (~-- gl')]
=
D
= 81 A~'
on
it follows
.
j3 , one has:
o [ 'lj s
-
=o
,
because in (*) the second member contains double products of structure Since
D
is an arbitrary vector
Remark. critical
field, necessarily:
~'
.
j3s
Having arrived at this point, one may develop,
sections
for a second order variational
First characterization.
A section
~'
that:
s
of
lj~s
problem.
p
- ~'
forms as factors. = 0 .
lj~s
as usual,
the theory of
In particular,
is critical
one obtains:
if and only if
=0
is fulfilled.
Second characterization.
A section
s
of
(iDd@)[j3s is fulfilled
for every vector
4. Functoriality Noether
of Legendre
invariants
field
D
in
p
is critical
if and only if
= 0
j3 .
and Poincar~-Cartan
forms.
for second order variational
Infinitesimal
symmetries
and
problems.
The m a i n result of this section is the following:
Theorem 2: Let
D
be a p-projectable
an arbitrary derivation V(JI)j2
such that:
transformation LD(2)(L~)
law in
V(J l) . We denote by
11)(2)0 2 = A o 0 2
associated
to
of
D
on
X . Then,
i) ~L'
= A* ° ~L + L D ( 2 ) Q
2) ®L'
=
LD(2)
@
•
, where
D , and let
= L'~ ," or in other words,
projection
vector field in the manifold
L •
L'
L' = D ( 2 ) L
one has:
D(2 )
A
Y
and
the unique endomorphism
is the infinitesimal
be the Lagrangian + L(div D')
contact
defined by
, where
D'
is the
V of
81 Proof: By taking Lie derivatives, we have:
LD(2)®/ = LD(2) (@2 A~L) + [ ' w =
d[LD(2)®L]= for certain valued forms
~'
@ 2 ^ ( A * ° ~L +
LD(e)~L) + f'~ '
LD(2)(d@L) : 91A~' + G2A (03A~)
~
,
as follows from the characterization of the
infinitesimal contact transformations by means of structure forms. Thus, the form
A*° ~L + LD(z)~L - fulfills conditions (a) and (c) in the
previous theorem, because Now we prove that chosen.
is
pc-projectable.
A*o ~L + LD(2)~L
In fact, if V'
V~(2)-VD(2) = B
D(z )
does not depend on the derivation law
is another derivation law in
is an endomorphism of
V(J~)j2
V(J I) , the difference
and one has:
LD(2 ) , @2= LD(2)O2 + B o 02 = (A+ B)o @ 2
' LD(2)~L
=
LD(2)~L
-
B*o
~i
Hence, (A+ B)*o ~L + LD(2)~[ = A* o ~L + LD(2)~L "
But condition (b) in Theorem 1 may be checked locally; we can therefore use the flat derivation law for doing so. In this case, if 2 j is the local expression of
A =
J
i lel=o
~ 3v z
D'2"k ) , we have: hi h ~ ~ A ~ dy~ O , h,i I~I=o I~I=o ~yZ
where A hi ~
: 0
, if
I~I > 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~k
i
(fh ~oj) ®d h
h,j I~=o ~D(2)
~J
Y~ "
82
In particular
h
g(k)j
=
Z
i (av(/)
i,£
i
+D(
h
%u/ + ~ ( _ l ) / _ j _ 1
~u. jh ~xl f(k)Z '
3Y(k)
where we have used DUg LD(2) c°j : ~ ' ~ J
: l~ ~
h + 1Z ~ _ ~ j - e - ~
~u1 ~xg ~e "
Moreover, from formula (2.2) we obtain $v{£) h ~Y(k)
Suk Sv i -6hi ~--~£+ 6k£ ~Yh
and so, (*)
h
h
~Uk
g(k)j =D(2)f(k)j- ~ ~
h
f(l) j
+
~u. i h -8ul - + ~ (_l)Z-j-1 J fh axz ( k ) l ~ ~hYh f(k) j +f(k)j ~ 9xI /
~v.i
"
On the other hand, by differentiating the idenLity
LD(2)@L = 82 A (A* o ~L + L D 2)~L ) q- L'co = ~ ( h,j
i h 86h ACO. g6J J + L'co 161=o
,
one obtains: 81^~,+82^
(03^~)
=
h,j
By applying successively
i
I~I =o
iD
h h h {d g~jA 8~A0J.+ (--I)j h ^~}+dL' A~ . ] g~j 8~+(j)
,...,i D 3,1
, i 3,n
(--$--) to
this equation, one gets:
~Y(jk) h (-l)J g(j)j + ~ SL' = 0 ~Y(jj)
(_i) j
~L' h + (_l)k h g(k)j g(j)k + - -h SY(jk)
But it follows from (*) by using (3.4) that: the previous equations therefore imply that h
"
- 0
,
h (-l)J g(k~i
~L'
g(k)j = (-l)J-~ Ejk ~ h Y(jk)
j#k.
=
(_l)k
h g(j)k ' and
83
Definition: A p-projectable vector field
S~g/J~y
D
for the variational problem defined by
LD(2)(Lw)
in Lw
Y
is called an
inf£~6~g~a~
if
: 0
is fulfilled.
Corollary: A p-projectable vector field
D
is an infinitesimal symmetry for
L~
if and only if:
LD(2)~ i = 0 Corollary: If
D
is an infinitesimal symmetry of the Lagrangian density
for each critical section
s
[w
,
we have
d[iD(e)®L] lj3s = 0 Thus, each infinitesimal symmetry
D
defines a function
fD (s) = [iD(2)@LIIj3s from the set of critical sections into the vector space on the manifold
s
of closed (n-l)-forms
X .
Proof: The result follows from section
n-i ZX
LD(2)® L = iD(2)d@ L + diD(2)® L = 0 , because the
is critical.
REFERENCES
[i]
M. FERRARIS, Fibered connections and global Poinnar6-Cartan forms in higher-order Calculus of Variations. To appear in the Proceedings of the Conference on Diff. Geom. and its Applications, Nov6 Mesto na Morave, Czechoslovakia, 1983.
[~
P. GARCIA, The Poincar6-Cartan Invariant in the Calculus of Variations, Symposia Math., 14, Academic Press (London, 1974), pp. 219-246.
[31
P. GARCIA, J. MUNOZ, On the Geometrical Structure of higher order Variational Calculus, Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, 1982, Volume I - Geometrical Dynamics, pp. 127-147.
[~
H. GOLDSCHMIDT, S. STERNBERG, The Hamilton-Cartan Formalism in the Calculus of Variations, Ann. Inst. Fourier, 23, pp. 203-267 (1073).
[~
M. HORAK, I. K O L ~ , On the higher order Poincar6-Cartan Forms, ~SAV Brno, (preprint, 1982).
[4
D. KRUPKA, Some Geometric Aspects of Variational Problems in Fibered Manifolds, Folia Fac. Sci. Nat. UJEP Brunensis XIV, pp. 1-65 (1973).
84
E~
D. KRUPKA, Lepagean Forms in Higher Order Variational Theory, Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, 1982, Volume I - Geometrical Dynamics, pp. 197-238.
E~
J. MUNOZ, Formes de structure et transformations infinit~simales de contact d'ordre sup~rieur, C.R. Acad. Sc. Paris, t. 298, S~rie I, n ° 8, pp. 185-188 (1984).
E9]
So STERNBERG, Some Preliminary Remarks on the Formal Variational Calculus of Gel'land and Dikii, Lect. Not. in Math., 676, Springer-Verlag (Berlin, 1978), pp. 399-407.
ENERGY
LEVEL
CHAOS
DISTRIBUTIONS
IN Q U A N T U M
Giulio
Dipartimento
MECHANICS
Casati
di Fisica Via
20133
AND
dell'Universit~,
Celoria Milano,
16 Italy
I. I n t r o d u c t i o n
The
prediction
of
the q u a l i t a t i v e
dynamical
laws
discovery
of c l a s s i c a l
a big progress with
systems
chaotic even
whose
motion
real
future,
selves
to the
and discuss
The
of m o t i o n .
direction
has b e e n m a d e
of the
equation.
an
notion
systems
experiments
consideration general
role.
As
chaotic
needs begin
the
like
last
governed
with
In t h i s
respect,
[I] a n d w i l l
paper
finite
in t e r m s
of c l a s s i c a l
feasible
in c o n n e c t i o n
years
to q u a n t u m
to u n d e r s t a n d
to be c l a r i f i e d .
In the p r e s e n t
known,
thirty
of s y s t e m s turning
influence
to be
given
after Newton's
in the
motion
we w o u l d
with
it is n o w w i d e l y
is n o w
of c o n s e r v a t i v e ,
features
classical
of c h a o t i c
of p h y s i c s
interest
behaviour
important
their
The
In p a r t i c u l a r
on q u a n t u m
some of
so-called
laws.
qualitative
laboratory
in the
properties
problems
equations
deterministic
Schr~dinger
of m o t i o n
of the m a i n
in t h i s
the d i s c o v e r y
by purely
of
is o n e
features
we
limit
and bounded the
play,
oursystems
statistical
limit.
motion
of c l a s s i c a l
deterministic
systems
is
87
by now firmly stability relevant cessary
grounded.
of orbits theorem
and
[2] states
sufficient
truly r a n d o m
despite
for almost
with
therefore
finite
of the
trajectory
and only
Turning of finite, derstanding
greatly
Is it p o s s i b l e (QCS)
properties
that
improve
By this
of m e a s u r e -
complexity
the
for these meaning
are
and
systems
the notion
(being e x p o n e n t i a l l y
the d i s c r e t e n e s s
of chaotic
by
"complexity"
significant.
the main o b s t a c l e
of important
behaviour
questions
our k n o w l e d g e
to c h a r a c t e r i z e by the p r o p e r t i e s
of the
spectrum
on the way of unin q u a n t u m mecha-
remains
of the general
open whose behaviour
of
to introduce
to c h a r a c t e r i z e
c) The m e t r i c
Can the
whose
classical
of e i g e n v a l u e
sequences
entropy
in q u a n t u m m e c h a n i c s
limit
is
or by the
somehow
a notion
of com-
QCS?
measures,
in a sense,
and the rate of a p p r o a c h
racterize
systems
of e i g e n f u n c t i o n s ?
Is it p o s s i b l e plexity
d)
places
is a ne-
systems.
chaotic
chaos
mechanics,
A
to be
are c h a r a c t e r i z e d
which measures
methods
the m a n i f e s t a t i o n s
answer w o u l d
b)
systems
in-
and random.
this p r o p e r t y
It is clear
motion
of these words.
algorithmic
looses any p h y s i c al
To this end a number
quantum
chaotic
a quantity
now to q u a n t u m bounded
instability
any sequence
has p o s i t i v e
statistical
exponential
of the motion.
local
contradiction
possessing
entropy,
local
for the d e t e r m i n i s t i c
trajectories,
precision
s y s t e m as a whole.
unstable)
a)
all
are
of the s p e c t r u m
exponential
the seeming
systems
metric
of a single
nics.
that
it is u n p r e d i c t a b l e ,
Dynamical positive
features
condition
we mean that ments
Its main
and c o n t i n u i t y
the degree
to equilibrium.
of c l a s s i c a l
Do this notion
cha-
even the q u a n t u m m o t i o n ?
instabilityltypica]
for c l a s s i c a l
chaotic
motionjalso
appear
in q u a n t u m m e c h a n i c s ? e) The answer
to the p r e v i o u s
to c h a r a c t e r i z e quantum
QCS.
observables?
Which
questions are the
may a l l o w the p o s s i b i l i t y
implications
on the b e h a v i o u r
of
88
2.
The Level
We b e g i n tion which classical we are dual
characterize
systems
not
Distribution
by looking for some general property of the energy level
may
levels
systems
of e x a c t
states
of e x a c t
positions
a large
A first This
number
is g i v e n
is n o t
A
shape
is the
and
of t h e b o u n d a r y ,
the billiard,
does
One m u s t
then
the
fluctuations
one
usually
defined
the
of D y s o n
in s u c h a w a y
fifties
properties neutron and
[2,5] the
not
related
assumption
energy
data
resolution
tribution
were
the
or
spacing
was
observed
~ (E).
properties:
density
the d y n a m i c a l
of
for
levels
the
and
of d e g r e e s
of c h a o s
of r e s o n a n c e P(s)
Already
statistical
possible,
More
s later,
strong
b y the
nucleus was
when
number
led to
yield
As a matter well
a better
deviations
scarce
complexity
should
quite
small
a of
with
the
neutron
from Poisson of
on
vague
and cer-
of c o m p l i c a c y
agree
in
formed
quite
freedom
= !e-S/~
to
spacing
the
of
levels
is
the
s+ds.
or a l g o r i t h m i c notion
these
and the
that
"complicated"
intuitive
available.
caused
s
to d e s c r i b e
term
seemed
of
describe
distribution
s u c h as a c o m p o u n d
spacings
the
Among
P(s)
spacings
between
which
~ (E).
is the p r o b a b i l i t y
the
while
properties
properties
density
distribution
introduced
times
distribution then
density
(I).
[4]. The
lies
the position
became
in
in s y s t e m s
of the b i l l i a r d ,
local
to the h i g h n u m b e r
of
Indeed
the k n o w -
as the k n o w -
level
average
the m e a n
P(s)ds
However,
distribution
experimental
level
to the n o t i o n
that
the P o i s s o n
fine
systems
those
connected
we n o w have.
Poisson fact
At
sense
to the c h a o t i c
characterizes
levels
quantity
same
smoothed
in e x p r e s s i o n
and Mehta
of c o m p l i c a t e d
capture.
simply
tainly
this
is the
around
that
of t w o n e i g h b o u r i n g
the
molecules,
the p a r t i c l e s
the p e r i m e t e r
at m o r e
levels
considers
/k3-statistics
indivi-
freedom.
related
L
enter
look of
of the
properties.
(3)
which
not
are m e a n i n g f u l ,
L_A_)
(A
area
of
for e x a m p l e ,
formula
[{E) -- ~i
where
of
interest
billiards,
b y the W e y l
(1)
s
of
quantity, h o w e v e r ,
two-dimensional
in t h e
and velocities
distribu-
corresponding
determination
or e x c i t e d
is m e a n i n g l e s s
the
methods
statistical
nuclei
of d e g r e e s
quantity
exact
in t h e i r
s u c h as h e a v y
with
statistical
in the
only
ledge
In a n a l o g y
only
here
but
ledge
with
QCS.
for w h i c h
interested
energy
complex
Spacing,
dis-
spacings.
89
At
this
point,
energy
cannot
bution
of
Wigner
[6], L a n d a u
be t r e a t e d
spacings
and
as a r a n d o m
cannot
hold
Smorodinsky variable
since
P(s)
[7] r e m a r k e d
and that
should
that
Poisson
approach
the
distri-
zero
as
s-->0.
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<13.6
features
[12].
Fig.
(5),
is re-
eV. Fig.
N e a r e s t n e i g h b o u r spacing dist r i b u t i o n for the 505 c a l c u l a t e d e n e r g y levels of C2H ~. The
5 r=l
h y s t o g r a m is c o m p a r e d with the Br o d y d i s t r i b u t i o n (5) with q=0.71 (full line) and with the W i g n e r d i s t r i b u t i o n (dashed line).
2O .
.
.
.
n,
2
i
4
As we h a v e troduction
6
already
of RMT was
large number
8
S/D
mentioned,
the notion
of degrees
of freedom.
usually
referred
the p r e d i c t i o n
of RMT,
dispersive
[13,17],
they have p o s i t i v e
since
classical
motion
computations tribution
is very
in very
Sinai's
billiards.
idea w h i c h
"complicacy" Now we have
to as
"chaos"7
plane metric
"complicated"
of e i g e n v a l u e s
illustration
the main
of
of complexity,
good
same
the n e a r e s t - n e i g h b o u r spacing h i s t o g r a m for 505 level + refers to the C2H 4 energy levels in the interval
a well then,
is p r o v i d e d
with
by Fig.
led to the related defined
in-
to the notion
in order
to test
billiards
were m a i n l y
studied
entropy
and therefore
their
or chaotic.
for these b i l l i a r d s
good a g r e e m e n t
mainly
the W i g n e r (6) w h i c h
Indeed
numerical
led to a spacing surmise. refers
dis-
A fairly
to d e s y m m e t r i z e d
94
. . . . .
0.9 0.8 0.7
\
Sinai's
\\
r . l ~ k r. b i l l i a r d \
IJ"l
IAkl
Nearest neighbour spacing distribution for d e s y m m e t r i z e d Sinai's billiards (tak e n f r o m ref. 16).
-1
............
° o
0
The
0.5
1.0
fact
that
bution
P(s)
for
chaotic
billiard,
plex of
nuclei,
some
available
are
that
property. theory,
repulsion.
linear
(5)
(2b)
is t h a t
weaker h -->
into
P(s)
~
the
larger
h
time
= xq
depend
~1 F ( 2 + q i
1+q"
1
s --> 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
i<~j
( ~i -
2~j)
= f& TAI G ( N ) L
s.t. A has two
- • I < E~ . i
3
this set is easily defined by
= 6i<j { (0'~) ~ [-q'1]N(N-1)/2
x~N s't"
'li- A'' ~
97 Let ~(x i) the absolutely matrix elements. Then:
Vol(~
) ~
N
continuous
joint distribution
function
of
Vol('~J)
z
i<j
Vol(~iJ) : /~j /(xI0,~))r(0,z) ~<sIT(~-~s)d~a0 : (8)
= ~
d?~ d0 f(x(0,~))F(0) ~ (Ag - )~s ) : . . . . . Z<s
;ti- Ajl < £
Ai+ £ -~
,_~
]
l
1
where
(9)
G( li' Aj) ~ ] d O
f(x(0,1))F(O)
dl
( ~'k
2~I) "
k
I
over i and
j
are
now
~
~lj
_ I
9 -
d 2~i/
~-[
d ;%i(;5 i- A,j)G( 2~i, }tj) .
1
Since G( Ai' gj) obtain:
has the required
lim I @ (w~J) lim 6-->0 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
<
square
at the ~
2 +
of
(I +
s
with
K(N)
side
to e n s u r e
above
is,
that
J ~ N
needed
for
with
the
basically,
the
that
the
triangle
we
.
denotes see
one
in fig.
curve
Therefore
[ ..... ]
required
to c o m p u t e
2k .
illustrated
K(N).
Therefore, 2~)]
the a c c u r a c y
precision
is
such
where
dashed
(I+ ~ ) . [ ~
EN
integer
the
looking
number
finding
inside ~
for
a way
K(N)
the
K(N)
in such
specify:
irrational
of
= Ej
of the p r o g r a m
the
One w a y
be c h o s e n
H = EN we m a y the
that possible
7. lies
completely
assume
integral
[ ~ N choice
part.
]2 < 2N of
K(N)
By
whence will
be
100
m
[~/-~N ] +I
I
I
I
I
I
I
I
I
I
I
'
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
~
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
÷
+
+
+
+
+
+
+
+
%
+
+
+
+
+
+
+
+
+
+
+
+
~
+
I
I
I
II
1
L + + +~ - 7 ~ +
Fi~. 7
t~-~-~N ] -
~
"~
+
+
+
I0
I
J
I
:
:
+
~,
v
[Q--~N ] I
Illustrating
As to the
second
point,
the c h o i c e
the
error
of K(N)
in c o m p u t i n g
E
is m,n
6E
= 6 ~ m 2.
m,n
Since 6~
m 2 ~ K2(N)
one
<
-2N
,~ K 2 (N)
This /~
shows
the n u m b e r
minimum
program
increases is n o t
that
not
length
faster
in
to c o m p u t e ~
must
required
that
that
N
log2N ,
m,n
eigenvalues
increase
to c o m p u t e
const,
6E
as in N. N
and
<
/%
with
precision
Therefore,
eigenvalues the
if
sequence
the
within of e i g e n v a l u e s
random.
Our
construction
integrable ficient
systems.
in o r d e r
(i) H(II, .... Im) (ii)
in o r d e r
of d i g i t s
has
the
ratio
seems
easily
In p a r t i c u l a r ,
that
the
same
is a s m o o t h
of the v o l u m e
the h y p e r s u r f a c e
H = E
generalizable the
conclusion function VE
class
conditions
seem
of suf-
holds:
of a l g e b r a i c
of the
to the v o l u m e
to a w i d e r
following
smallest SE
growth
at i n f i n i t y ,
hypercube
of the r e g i o n
containing
where
101
H K E,
is b o u n d e d .
We h a v e
no
idea
as to w h a t
the e i g e n v a l u e s
of n o n
little
at p r e s e n t
of
is k n o w n
systems.
may
yield
about
Nevertheless,
a model
the
about since
for c h a o t i c
complexity
of
for c o m p a r i s o n
tained
integrable
Let
us c o n s i d e r
be
said about
systems. the
as w e h a v e
behaviour,
for the,
the c o m p l e x i t y
As a matter
spectral
features
seen,
of
of t h i s Matrix
useful
a Random
albeit
of fact,
Random
it s e e m s
the e i g e n v a l u e s
to get a t e r m for an
could
integrable
class Theory
to e n q u i r e
Matrix,
particular,
of
very
in o r d e r
result
ob-
billiard.
for e x a m p l e
the O r t h o g o n a l
Circular
Ensemble
w h i c h c o n s i s t s of s y m m e t r i c u n i t a r y m a t r i c e s of r a n k N, whose N e i g e n v a l u e s are d i s t r i b u t e d on the u n i t c i r c l e a c c o r d i n g to the d e n s i t y
W N ( e I ...... e N)
= CN
le
~-
ie. i@ K 3 _ e
I~j{K{N with
CN
when
the
a normalization
constant.
WN
8i
spaced
therefore
distributed
are
the
equally
(4). O n a c c o u n t
configuration, ing
½NInN
one
8
might
and
has
a max±mum
of the r e g u l a r i t y
conjecture
is a s o m e w h a t
that
ordered
the
string.
are
WN0=CNe
very
of t h i s string
On the
regularly "most
likely"
obtained contrary
by orderwe h a v e
l
shown
that
Proof:
Suppose
spacing 6=I).
I
string
to
given
length
N,
them,
that
we g e t
We a r e
N-->
N,
~
the
can
~N
complexity
number
of
are m e a s u r e d
strings
6
a string
going
in u n i t s
(in the of
N
sequel
integer
to i n v e s t i g a t e
number
of the a v e r a g e we
take
numbers,
rang-
the c o m p l e x i t y
of
of n o n d e c r e a s i n g numbers
from
I
strings to
N
- ( 2 N - i)! ~rfN - I)!
of one
with
~N
form by using
O(log2~
The
complexity.
.
total
one
(11)
The maximum
positive
an a p p r o x i m a t i o n
N.
for
has
81' ..... 8N
with
By o r d e r i n g
For of
such a string
that
2~/N
ing f r o m this
[25]
such
N)
string
: O
complexity
is,
asymptotically,
(N)
not
exceeding
log2 ~ N - I
is
is:
102
2-1+I
~N'
associated the max.
prob.
Suppose (~NON find
and the statistical with such strings
in
[ N
i.e.,
for large
N,
of the same order, ¥
oN
in
~ N
where
the o v e r w h e l m i n g
as
N--> ~
we can actually
find.
,
for
oN
is
/~N
we
N-->
l=iN=(1-y)l°g2
exponentially
will give strings of complexity
This
~N
~NYON-->0
Then assuming
~ 2 J~N Y o N --> 0
meaning that,
~ [l+i
y,0
by definition).
~N(IN)
weight of the matrices bN
of one string.
that we can find ~ I
is
as
N -->
majority
greater than
of matrices ¥ l°g2
~N
as strings of m a x i m u m complexity.
For any given string
Xl, .... ,x N
we have
Prob
{x I ...... XN]
: NI
<
dO I .... < I
dON W(01...O N) H
01 ~ 02 < ... g 0 N where
~i
are intervals
in
(0,2~)
Therefore,
of width 2 ~ / N .
we may assume
(12)
o N ~ NI
2~ N ½NInN (~--) CNe
Using the known expression
for
C2m
I
C2m - 2 4m m! in inequality N--> ~
~m
(12), and relation
provided
that
0<¥<
¼
(11), we find that
2e
Thus we have shown that most matrices with a s y m p t o t i c a l l y
maxima]
of
in
systems,
is that in order to operate chaotic
systems,
according
~N
for
have eigenvalues
the idea that statistical
the eigenvalues of a random matrix reproduce of chaotic
ON-->0
complexity.
If now one is willing to accept of eigenvalues
~
log 2 -~-
somehow analogous
properties properties
the sense of our whole discussion
a distinction
between
to the different
integrable
degrees
and
of randomness
103
of their chaotic grable
eigenvalues, systems
one must
as b e i n g
systems.
The
such d i s t i n c t i o n
between
levels
by t h e
generically
important
is p o s s i b l e
An interesting,
consider
or the
at all,
of
random
remains
at least
"rigidity"
of c l a s s i c a l l y
than those
open,
of inte-
whether
any
generically.
test to measure
so-called
~ 3-statistics
more
question
empirical
the e i g e n v a l u e s
long range
of the
correlations
spectrum,
is
given
[28] x+L
(13)
~ 3 (L'x)
I Min = L a,b
/
[N(E)
- aE - b]2dE
X
which
is a m e a s u r e
density
N(E)
of c o n s t a n t an
level
"unfolding"
sequence
will
(one can always
then
be i n d e p e n d e n t ~3(L)
regular,
of the
straight-
[12,16,19]).
invariant,
sequences
the most
fluctuations
density
procedure
is t r a n s l a t i o n a l
random
of the
from a b e s t - f i t t i n g
reduce
of
x.
cumulative the a s s u m p t i o n
to this case via
If we assume
the average
= L/15
equidistant
staircase line under
that
the s p e c t r u m
/~3(L)
over the
It can be shown
[28] that
(strong
sequence,
fluctuations), ~3(L)
= 1/12
while
spectral for for
(maximum rigi-
dity).
It is i n t e r e s t i n g ly
(74) with in
to n o t i c e
that
for the GOE one has a s y m p t o t i c a l -
[28]
a standard
(14)
deviation
indicates
of
~ 3(L)
the p r e d i c t i o n s
[16] of
~
based
a very
with
with
9]2
~3(n)
3
the GOE.
of
[inL - 0,0687]
+ 0,11.
strong
on e x p e r i m e n t a l of GOE
for Sinai's
The
rigidity
(fig.
logarithmic
of the
data reveals 8). Also
billiard
gives
A measure
a surprising
a numerical the
dependence
spectrum.
agreement
computation
same good a g r e e m e n t
104
~3 0.3
rig.
0.2
0.I
• Experiment (Nuclei)
5
The h i g h
10
random
not
~3(L)
[29]
that
this
of
levels
case
sequence
seen
system
/~
(fig. levels
(10).
as r a n d o m
in p a r t i c u l a r ,
systems
levels
from
(10).
9)
This
only
the c o m p u t a t i o n
indicating
in a c c o r d a n c e
over
small
Poisson
to
conclusion
It has b e e n
It is i n t e r e s t i n g
to the
reinforced
contrary
systems.
for e x a m p l e
between
sequence
behave
between
integrable
~3(L)~
of the
of c h a o t i c
in i n t e g r a b l e
as c a n be
simple
t h i s gives rise,
spectrum
correlations
correlations
complexity
and
25
L
of the
strong
correct
in t h i s
long range
zero
of
for the
that
and
20
appearence
is h o w e v e r of
15
rigidity
the c o n v i c t i o n the
A v e r a g e v a l u e of ~ 3 as a f u n c t i o n of L for nuclear energy levels (taken f r o m ref. 16) .
8
shown
strong with
rigidity
the
to n o t i c e
energy
intervals
distribution
of
spacings.
4.
Conclusions
We h a v e of energy
levels
properties here
discussed
squared
a theorem n-th
distribution of QCS
are
of q u a n t u m
the
as
34] .
[30]states
n--> ~
relevance
as c o r r e l a t i o n
,
that,
u~(x) thus
It h a s
limit.
exhibit
tends,
been
Gaussian
of the fluctuation
on the b e h a v i o u r functions,
which
we
are
that
the c h a o t i c
also
weakly,
not
considered
relevant.
billiards,
the
to the u n i f o r m
the e i g e n f u n c t i o n s
suggested
that
[31]
the
fluctuations.
properties
of q u a n t i t i e s
refer
with
We h a v e
for e r g o d i c
indicating also
of the d i s t r i b u t i o n
in c o n n e c t i o n
classical
the e i g e n f u n c t i o n s
to
of Q C S
properties
systems
eigenfunction
and eigenfunctions such
of
due
delocalized.
eigenfunctions
For
general
of the c o r r e s p o n d i n g
the p r o p e r t i e s
Indeed
some
to
some
of e i g e n v a l u e s of p h y s i c a l
recent
papers
interest [32-
105
~3(L) 2
4 / • /
L/15
0/ or or • / o~/ */
1.5
~r Q,
Fi~.
/
9
/
r~ r:
r /o / / • / / •
0.5
/ /o
r ro
1" 0
;
Graph
We c l o s e
by
patterns
systems, signal
and some
440
of the p h a s e
showing
tion
1;o
the
averaged
square-root
stressing
are
that
obtained
feature
~3(L)
dependence
for L ~
data
numerical on n u c l e i ,
of q u a n t u m
L
on L.
the r e m a r k a b l e
f r o m GOE,
from experimental general
900
1,000
Here
fact,
the
[ NI
same
computations atoms
mechanics.
fluctuaon c h a o t i c
and m o l e c u l e s
may
106
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Shnirelman,
[31]
B.V.
Chirikov,
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4
3-MIR E d i t i o n s
Italy
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and Control
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Vol.
14
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602
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181
communication
I. G u a r n e r i and G. Mantica, in "Chaotic B e h a v i o u r Systems", e d i t e d by G. Casati (Plenum, New York),
QUASI-W-ALGEBRAS GENERAL
WEYL
G. Lassner,
AND
QUANTIZATION
G.A.
Lassner
Naturwissenschaftlich-Theoretisches Karl-Marx-Universit~t,
I.
Leipzig,
DDR
Introduction
The e x t e n s i o n distributions 21,
Zentrum
25].
elosab l e
has
of the Weyl
found an i n c r e a s i n g
For d i s t r i b u t i o n s operators,
of the H i l b e r t the forms,
but
space.
only
Since
~;
chosen
topology
is the dual t
In a series quasi-}-algebra systematically. correspondence this
general
pers
[6, 10,
of the Weyl
In
of
~
[16-19]
and
years
longer
on a dense
regularity
operators
of
[6-]0,
lead to domain
properties
from
D
of
to
to an a p p r o p r i a t e
Weyl
correspondence
15].
In all these play
the t o p o l o g i c a l
its s u b a l g e b r a s
[21] we had used
In this paper
forms
some
with r e s p e c t
for all d i s t r i b u t i o n s
quantization
in recent
does not
classes
~ .
of papers L(~, ~)
to q u a d r a t i c one assumes
to c o n t i n u o u s
space
on
to d i f f e r e n t
interest
the q u a n t i z a t i o n
they c o r r e s p o n d
where
quantization
them already of
~
have been
of the
investigated
to define
Special
the Weyl
properties
investigated
condiderations a crucial
properties
have been
of
in recent pa-
topological
properties
role.
we shall show how the systematic use o~ the topological
109
p r o p e r t i e s of the q u a s i - ~ - a l g e b r a of the q u a n t i z a t i o n procedure.
L(~, S')
leads to a s y s t e m a t i z a t i o n
With that as background,
cation of certain classes of pairs of o p e r a t o r s of
the m u l t i p l i -
L(2, 3')
and the
c o r r e s p o n d i n g twisted product of their symbols are defined.
For the d e f i n i t i o n of the general Weyl q u a n t i z a t i o n there exist two approaches. The first goes back to the d i s c o v e r y of H e i s e n b e r q that to the p o s i t i o n and m o m e n t u m c o - o r d i n a t e s operators
Q
[Q,P]
(~ = I).
= iI
and
P,
q,p
c o r r e s p o n d the symmetrical
w h i c h must satisfy the c o m m u t a t i o n relations We suppose that
Q
and
tors on a dense domain of d e f i n i t i o n
~
w h i c h remains
Q D c D and
PD ~D.
WH
are symmetrical operainvariant,
This leads to the f o l l o w i n g problem.
set of all p o l y n o m i a l s ping
P
f(q,p)
in
p
and
q,
Let
~
i.e. be the
then we ask for a map-
(quantization on the basis of the H e i s e n b e r g c o m m u t a t i o n re-
lations), w h i c h a s s o c i a t e s to every p o l y n o m i a l an o p e r a t o r
WH(f)
such
that
HI.
WH(f)
is a linear m a p p i n g from
defined on
D
P
into the set of operators
,
(1.1) H2.
( Q+pp) n =
, ~
E n ~+k=n ~
WH(qZpk) l ~
are real variables and
The second a p p r o a c h to the q u a n t i z a t i o n tation of the
CCR.
(1.2)
n
~
k
,
integer.
starts w i t h the Weyl r e p r e s e n -
Let W(q,p)
i = e iqQ e ipP e 2-qp
be the unitary Weyl o p e r a t o r s and (1.3)
[(q,p)
the F o u r i e r t r a n s f o r m of
(1.4) Since (q,p),
W(f) W(q,p)
= j
f.
=
f
e -i(qu + pv)
The Weyl q u a n t i z a t i o n
W(q,p)
are u n i t a r y operators,
the integral
(1.4)
f(u,v)
~(q,p)
dq dp
du dv
is d e f i n e d by
.
w h i c h c o n t i n u o u s l y depend on
is w e l l - d e f i n e d
in the Riemann sense,
if
110 is s u f f i c i e n t l y tizations
WH
smooth (1.1)
classes
of f u n c t i o n s •
bounded
functions.
which
relates
[9, 21,
25],
to e v e r y which
In S e c t i o n
definition the
matical
3 we
i.e.
II.
of o n l y (but
show that
Weyl
and
paper
and
the
only
W(f)
both
for for
by the g e n e r a l
next
quan-
different smooth
and
quantization
a linear
operator
section.
integral
W(f).W(g)
(1.4)
is w e l l - d e f i n e d
distribution
for m o r e
are d e s c r i b e d
we r e s t r i c t
f e S2 .
general
pairs
in S e c t i o n
~
: 3(RI),
tempered
By
5 It]
L(S, S')
in
The
of
f,g
4 in a s y s t e -
product
5' ~
~'
that
S'(R I)
we d e n o t e
The
generalization
to
linear
This
Schwartz
their space
space
spaces
strong of all
of
test
topologies continuous
is i s o m o r p h i c
to the
t
funcand
operatensor-
[14]
every
S')
=~ ._q' "~
operator
.,¢'
A eL(S,
$')
can be w r i t t e n
as an
operator
(2.2)
(A~) (x) the k e r n e l
following
be the with
the
$' [t'] .
L(~, means
variable.
(q,p) 6 R 2 ,
of D i s t r i b u t i o n s
~'=
(2.1)
integral
to the case
is s t r a i g h t f o r w a r d .
distributions
into
ourselves
and momentum
dimensions
Quantization
of
Lemma
Lemma
(2.3)
Therefore,
first
f(q,p) E S' 2
for e v e r y
product
one p o s i t i o n
t'
where
covered
in the
L(5, $')
finite)
tors
which
defined
way.
Let tions
are
distribution
twisted
at i n f i n i t y .
are
for p o l y n o m i a l s
cases
of the p r o d u c t
related
(1.4)
we d e s c r i b e
In the w h o l e
higher
decreasing W
WH(f)
Both
the q u a s i - * - a l g e b r a
and
and
and
=
f
is a d i s t r i b u t i o n
A(x,y)
(Y) dy
A(x,y) e
S' ~
S' ~ S 2
We have
[21].
2.1:
The
integral
g =
(Gf) (x,y)
is a h o m o m o r p h i s m
transform _ 2 9I
f -->
/f(x2~
f -->
Gf
Gg p)
of
3
e ip(x-y) onto
dp S'.
the
111
(2.4)
On the b a s i s arbitrary
The
inverse
f =
(G-Ig) (q,p)
of this
lemma
distribution
Definition W(f)
= ~
mapping
2.2.
~
For
is a l i n e a r
f(x,y)
:
f e-ip~
=
we can
f(q'P)
is g i v e n
define ~2
by
g(q+~/2,
the W e y l
q-~/2)d~
quantization
.
of an
[21]]. I
f = f(q'P) operator
~ ~2
of
(Gf) (x,y) 6
the Weyl
L(5, S')
~+®5 ~
quantization
given
by the k e r n e l
: S2
(2.5 W(f) }
f (q,p)
The
is c a l l e d
±somorphism
uniform once space
From
on
and L e m m a
braic
2.3:
and
so d e f i n e d
Heisenberg
symbol
only
52
2.1
an a l g e b r a i c 18,
and
t'
we get
topological
the
19],
If
we
shall
the
describe
topology
on the
~2[t'].
f ~ W(f)
I
~ 2It']
is the n a t u r a l sense
denotes
theorem
f --> 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
fined
in
L(3,5').
are
the
the
scalar
L(J)
linear
diagram
of con-
subspaces
of
[18]
~"
closable
the p r o p e r t y
, ~ > :
get
resp.
L(S,S' )
set of all with
generalizes
L(S)
itself
we
L+(S)
L+(S)
which
the
L(S')
By
Integra !
fundamental
we h a v e
for g e n e r a l
Here
For
some
which
19]
and W e y l
# >, If
#,I~
unbounded
A,A~5
m
,
a
~
A
-operation
then
A ~ L+(S)
operators
in
H
de-
S.
A
is the
A-->A
is de-
restriction
of
+ (taken
in
~ )
Every
to the
space
transformations bounded logies
of
domain
(3.2)
5 ,
considered
can be e q u i p p e d
convergences
[14].
corresponding
to
i.e.
with
Then
A
= A
as a space the
we get
natural
the
of c o n t i n u o u s topology
following
linear
of u n i f o r m l y
diagram
of
topo-
(3.2)
z &
(3.3)
[~
means described case duct
D
in =
that
[17,
5
tors
[
is s t r o n g e r
for g e n e r a l
~ .
These
topologies
of d e f i n i t i o n
topologies
(3.3)
~
are
.
are
For
tensor
the pro-
We h a v e
3.1. are
than
domains
we are c o n s i d e r i n g , the
topologies.
Lemma
18]
Us
- The
topological
isomorphic
to the
spaces
following
(3.2)
of c o n t i n u o u s
complete
tensor
opera-
products,
113
where
E ~ F
denotes
L(S) [r~]
(3.4)
Z
~ ~ ~'
L(j, S') [Z5] Furthermore
Let ~0
us c h o o s e = L+(S)
sense
of
the
then [19],
(I) Or=
: S'
~
S'
L+([) [r~]
[
~'~
abbreviation
~[[]
(3) An
Now be the 5 c L2
linear
respect and
leaves
let
o I (5)
[20].
We have
Lemma
3.2.
-
the b i l i n e a r
~
(3.6)
.
If we put
C~ : L(S,5')
and
quasi-~-algebra
dual
convex
space
with
A-->BA,
in the
a distinguished
A-->AB
on ~
for
multiplications
~
0
are
defined
every
as
B e ~0
becomes
"
an a l g e -
~0-modu]e.
A-->A +
is d e f i n e d
on
~ ,
which
is c o n t i n u o u s
invariant.
0
: {~ ~ L + (5) ,A ~ B operators
(L(~,S'), form
coincides
L(~,S') [~]
and
associated
also
for with
is a dual
= tr A p .
and
The
nuclear
GI(5))
(A,p)
strong
pair.
It also
holds
~
is the
spaces
[14]:
~ $ S'
transformations
is an
of n u c l e a r
to that
product
~0"
to these
~
and
L(5,$')
= Us
tensor
= S'~ S
is a t o p o l o g i c a l
subspace
involution
ideal
~' ~
multiplications
continuous
bra
~
is a l o c a l l y
linear
(2) The p a r t i a l
With
L(S/) [r S]
,
( O~[~], O~0)
projective
i.e.
dense
(3.5)
the c o m p l e t e
The
topology
with
the M a c k e y
a1(S) [B ~]
the d e n s e
pair
with
topology
the M a c k e y
strong
A,B ~ L + (S)]
all
in
are
respect
r : r5
topology 01(5)
,
domain
to
in
with
respect
is d e n o t e d
topology.
Therefore
reflexive.
by both
Especially,
it
that
L(S,S' ) [~]
Lemma
3.3.
- i)
: 01 (S) [B ~] '
If
function
of
morphism
between
the
ii)
For
W(f). ~
~ 2"
~ ~ a1(S),
The
is n u c l e a r
then
correspondence locally
and
convex
W(f) e L(5,S')
its
symbol
~ (q,p)
~ <--> ~(q,p) spaces and
is an iso-
o1(S) [8"] p c oi(~)
is a
the
and
~ 2 "
operator
114
(3.7)
tr W ( f ) . ~
Proof. then
i) (3.7)
a1(S) the
is T h e o r e m
The
in
(3.6)
~ (H)
In the
to the
case
same
sense
ralized
v. N e u m a n n
general
~
,
ralized
to
W(a)
dual
pair
side
fact
4, 24].
(3.7)
But
and
the p r e d u a l
that
operators
,
of the of
to q u a s i - ~ - a l g e -
W~-algebra
on the
is the m a x i m a l
dense
of
Theorem
generalization
leads
L(S,S')
since
as a c o n s e q u e n c e
of
that
already
to the
of the
theory
domain
fact
that
every
for W ~ - a l g e b r a s ,
f(x)
space
functional
But we on
Theorem
p
~c
~.
geneFor
W~-algebra
which
3.5.
when
the
has
a
can be g e n e -
shall
avoid seen
- Let
function
[5(VI, here
of with
locally
into
with
E.
respect
3 ( R V)
to
for e v e r y
then
by the
relation
dx
functional
on
side
then
of
§ I, 4)].
such
explicitly
W(q,p)
of the Rv
F
defined
on
is c o n t i n u o u s .
right-hand
can be
of
= gw,p(x)~
the G D F - p r o p e r t y ,
theorem
space
function
f(x)
linear
(3.8)
on the
valued
o1(S)).
(W(x),p)
is d e f i n e d
has
transform
test
(W(x) ,~)
of
F
dual
a mapping
be a d i s t r i b u t i o n ,
f
the F o u r i e r test
if
dx c E
side
be the
W(x)
(E-valued)
def.
p ~ F,
right-hand
F = o1(S).
x -->
(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
[15]
Kuang
CMP
Liu:
Acad.
1453
[10]
Chi
Math.
Sc.
I (1965)
JMP
4
17
2]
Math.
Paris
(]980)
2080
J.:
JMP 2 4
3_~2 (]979)
295
(1982)
(1982)
239
359
317
14
(1963) J.:
(1976)
1055,
4
JMP
(1965)
859
6
(1963)
R~ume
I,
1058,
88 II,
5
(1964)
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
(1963)
Anal. 29,
H. Poincar6
31
104-132
3 (1972)
295
495
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<
0~
[
on
star-productS.
, v~
,.. are
t,
=
of the o b s e r v a b l e
by an e l e m e n t
asymptotic (M,F)
deduced
We have
solutions
of
(7-3),
jM
(u~ 9~ v ~ ) i
< ~
...
If
product.
u z
(ur ~ Pz ) ~
and
~z
of
N
u 6 N in this
at time
t
framework
for a
[I,
2] is
JM (u~ p)
the e x t e n s i o n
to the M o y a l
lemma.
for e a c h
by
value
being
is real.
constraints
§ 6, b and
by
restrictions
similar
~(t) :
ponding
given
of
(t+T).
described
formula
u(t)
and
JM
(11-3) this
that
from
holds
defined
t,r
depend
or
notations
we have
It is the c a s e
product
v~.
[u'z]2 ~ : 0
we a s s u m e
on
u r~
(Fn/n~)
is such
of the
is c o m p a c t ,
fM
~ n
for
time-independent
the
(M,F)
(2 ~ ~ )
supports
(1 1-1 ) conditions
of
star-product
of the
u, v 4 N
We use
same
of the c l a s s i c a l
We n o t e are
depends
that
it f o l l o w s
solutions only
upon
Wigner
of
(7-3)
(t + z),
formula
from the
corres-
(11-I)
that
integral
according
to the
135
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[I]
F. Bayen, M. Flato, C. Fronsdal, A. L i c h n e r o w i c z Sternheimer, Ann. Phys. 111 (1978), p 61-65
[2]
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[3]
H.R. Lewis, J. Math. and W.B. Riesenfeld,
[4]
P. Camiz, A. Gerardi, C. Marchioro, E. P r e s u t t i and E. Scacciatelli, J. Math. Phys. 12 (1971), p. 2040-2043
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[6]
A. L i c h n e r o w i c z , p. 157-209
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S. de Groot, La t r a n s f o r m a t i o n de Weyl et la f o n c t i o n de Wigner: une forme a l t e r n a t i v e de la m e c a n ± q u e quantique, Presses Univ. de M o n t r e a l (1974)
Ann.
and D.
Lecture
Notes
Phys. 9 (1968), p 1976-1986, H.R. Lewis J. Math. Phys. 10 (1969), p , 1 4 5 8 - 1 4 7 3
M. Felix,
Inst.
E. Fijalkow,
Fourier
Grenoble
J. Math.
32
Phys.
22
(1982),
I
[8]
R. A b r a h a m and J. Marsden, B e n j a m i n (1978)
[9]
C. Godbillon, H e r m a n n Paris
Geometrle (1969)
Foundations
of M e c h a n i c s
(2nd edit.)
Diff~rentielle
et M e c a n l q u e
Analytique,
[10]
J.A. Roma
[11]
A. Nijenhuis,
[12]
A. L i c h n e r o w i c z , J. Diff. Geom. des t r a n s f o r m a t i o n s canoniques, (1979), p. I05-135
[13]
O.M. N e r o s l a v s k y (1981), p.71-75
[14]
P. Lecomte and M. de Wild, S t a r - p r o d u c t s Lett. in Math. Phys. (to appear)
[15]
J.E. H.J.
[16]
H. Basart and A. L i c h n e r o w i c z , (1982), p, 681-685
[17]
C. yon W e s t e n h o l z , D i f f e r e n t i a l Forms N o r t h - H o l l a n d , A m s t e r d a m (1981).
Schouten, (1954)
Conv.
Int.
Geom.
Indag.
Math.
17
and A.T.
Diff.
(1955),
Padova
p~ 390-403
12 (1977), Bull. Soc.
Vlassov,
1953 C r e m o n ~ s e
C.R.
4
Acad.
Acad.
Sci.
Paris
on c o t a n g e n t
Moyal, Proc. Cambr. Phil. Soc. 45 (1949), G r o e n e w o l d Physica 12 (1946), p. 405-410 C.R.
I
.
292
I
p. 253-300, g e o m e t r l e Math. B e l g i q u e 31
Sci.
bundles,
p. 99,
see also
Paris
295 I
in M a t h e m a t i c a l
Physics,
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 S-MATRIX
IN 4 - D I M E N S I O N A L I.E.
Segal
M.I.T. C a m b r i d g e , MA 02139,
I.
USA
INTRODUCTION
T h e r e are a v a r i e t y of a p p r o a c h e s one to be c o n s i d e r e d here f o r m a l i s m of Dirac,
to q u a n t u m
is a m a t h e m a t i c a l
H e i s e n b e r g and Pauli,
r i z e d by the f u n d a m e n t a l torvalued distributions order
PERTURBATIVE
SPACE-TIME
and others.
role given to n o n l i n e a r (or other
in the c o u p l i n g c o n s t a n t
field theory.
develooment
generalized
The
f r o m the
1920s
It is c h a r a c t e -
o p e r a t i o n s on o p e r a -
functions).
To first
(or c o e f f i c i e n t of the n o n l i n e a r
differential
m a l i s m gave finite r e s u l t s
that w e r e e x p l a n a t o r y of basic p h y s i c a l
phenomena.
e q u a t i o n of motion)
term
in the f u n d a m e n t a l p a r t i a l
this
Thus the e m i s s i o n and a b s o r p t i o n of light by atoms,
was t r e a t e d q u a s i - p h e n o m e n o l o g i c a l l y Heisenberg-Schr~dinger
for-
which
in the Bohr m o d e l and in the early
q u a n t u m m e c h a n i c s was not m e r e l y d e s c r i b e d but
explained.
However, perturbative
to 2nd and all h i g h e r orders, analysis
appeared
Tomonaga renormalization
infinite,
t h e o r y circa
has b e e n e m p i r i c a l l y e f f e c t i v e
however,
1950.
in d e r i v i n g
w h i c h had a l r e a d y b e e n m e a s u r e d ,
the r e s u l t s of a formal
prior
to the F e y n m a n - S c h w i n g e r -
The r e n o r m a l i z a t i o n several
for the most part.
it r e m a i n s very c o m p l i c a t e d ,
theory
subtle effects, Mathematically,
and at best a t t a i n s
ries in the c o u p l i n g c o n s t a n t of d u b i o u s c o n v e r g e n c e .
a power
Schwinger
c o n c l u d e d that the t h e o r y c o u l d not be fully r a t i o n a l i z e d
se-
[I]
f r o m basic
137
principles
within
in it.
Dirac
zation
less
the
[2] has
framework indicated
than totally
As a c o n s e q u e n c e state
of affairs,
idea of h a v i n g
valent
hamiltonian
tradional tained
a specific
plectic
proach,
of the
(i.e.
unquantized) Paneitz
Thus
thereof.
Even
selectivity subject
progress
fairly ones. ging
nique
regularity
recent work
case
theory
to be finite space),
while
connection
(in the
is central.
field theory
some of the
to me.
space-time
has made
quantization
approach;
are k n o w n
quite
the
to e x i s t under
unlike
to indicate
relatively
the q u a n t i z e d
that an e n c o u r a -
in 4 d i m e n s i o n s
is possible,
field t h e o r y
operators
in H i l b e r t
that will
asymptotics
in Paneitz'
case
space,
lecture.
expansion
a self-adjoint it is the
is seen to be finite,-
ideas,
and quite in
Yang-Mills
In the q u a n t i z e d that will be
operator
integrated in both
re-
The tech-
also be n e c e s s a r y
of the c l a s s i c a l
in the S - m a t r i x
sense of b e i n g
that
on space-
or some g e n e r a l i z a t i o n
Paneitz.
in the c l a s s i c a l
space)
and p a r t l y
subject.
Stephen
that will be s u m m a r i z e d term
latter
quantum
of the temporal
leading
of c l a s s i c a l
with
connection
re-
this ap-
of c o n s t r u c t i v e
for g e n e r a l i z e d
the c o n f o r m a l
it is the
Minkowski
in c o l l a b o r a t i o n
involved
as an in-
on some a s p e c t s
equations
theory
sym-
on w h i c h
speak about
space,
importance
wave
This
are d i s t r i b u t i o n s
circumstances,
a viable
equations.
of q u a n t u m
symplectic
of the
operators
concentrate
I want p r i m a r i l y
a conjunction
treatment
equations
here
step towards
essentially, the
via the
of n o n l i n e a r
and c o n t r o l l e d
involves
part
I will
from the that re-
for lack of time,
about
equi-
different
speak on the
in H i l b e r t
giving
consequence
the theory
partly
of 4 - d i m e n s i o n a l
except
solutions
general
presenting
and
involves
(or f o r m a l l y
such e q u a t i o n
I won't
with
speak
restricted
case
Nevertheless, first
I will
wave
differential
will
seem of f u n d a m e n t a l
The p h y s i c a l little
others
unsettled
the d e v e l o p m e n t
of a given
associated
are o p e r a t o r s
in this
A later
was
field.
equations,
is required,
that
classical
wave
involved
of r e n o r m a l i -
This
of m o t i o n
to n o n l i n e a r
variety
quantized
solutions
values
here.
quasi-manifold,
and p e r h a p s
the
time whose
theory was born.
and so is r a d i c a l l y
approach
is c l o s e l y
since
concepts
the theory
and p e r s i s t e n t l y
equation
of m o t i o n
solution
the p u t a t i v e
which
field
a specific
equation
finite-dimensional presented
imprecise
consideration
quantization
formulation
of this
structure),
line u n d e r
space-time
convincing.
axiomatic
up the
of the
that he finds
action
cases
shown
in H i l b e r t (over
the c o n f o r m a l
138
The present invariant the
quantum
isometry
cover
space
This
and
will
fields,
or
group
on a finite M 0.
results
tized
to give
equation
field.
makes
Our main
in familiar @
denote
then asserts,
conclusion
JM
compactificiation
and let
: ~(x)4:
of e o n f o r m a l l y
invariant
R I x S 3,
for a r e l a t i v e l y
an example
~ @ = 0,
class
to fields
Universe,
of the c o n f o r m a l generality
to a general
broadly,
of the E i n s t e i n
it may be well
the wave
apply
more
that M
under
are d e f i n e d
of M i n k o w s k i
abstract terms.
treatment,
Consider
then
the c o r r e s p o n d i n g in informal
quan-
terms,
that
d4x
0 is a s e l f - a d j o i n t generally, :@(x)4: space
the Fourier
relative
ficients.
tized
Apart
term @4
applies field
flavor,
foregoing
as the
Lagrangian
being
More
density
of M i n k o w s k i
one of the coef-
the e x p r e s s i o n
represents
of the S - m a t r i x
dimensions,
of an a r b i t r a r y
expression,
finite
part of
This p r e s c r i p t i o n being
for the order the case
designed
and a c o r r e s p o n d i n g
conformally
the
for the quan-
invariant
result
quantized
hilation
analysis,
RENORMALIZED
of this,
as the result
of e x p r e s s i n g
and a n n i h i l a t i o n
But
of these
the
the basis
treatment thus
are the
operators,
operators
in spite of this
Thus
in
(linear) @. as 3 and then
so that
the anni-
opportunistic natural
for the Feynman
of n o n l i n e a r
object,
diagram
quantized
fields
it is important.
OF FIELD O P E R A T O R S
of W i c k p r o d u c t s
to show their
@j
is in fact a m a t h e m a t i c a l l y
provides
PRODUCTS
the
ad hoc
standardization
of field operators. where
representation;
The r e f o r m u l a t i o n both
in a product
form a somewhat
a convenient
:@I@2...@n :,
act first.
or more broadly,
interaction
to p r o v i d e
of p r o d u c t s
the Wick p r o d u c t
independently
in its o r i g i n a l
of c r e a t i o n
the order
operators
appea r a n c e ,
field,
is d e f i n e d
combinations
rearranging
has
:@(x)4: r e p r e s e n t s what may be 4 , a c c o r d i n g to the p r e s c r i p t i o n
@(x)
basically
of the factors
of a boson
field operators,
serves
displayed
factor,
space.
as an e x t e n s i o n
expansion
in 4 s p a c e - t i m e
S-matrix
interaction
Universe
the e x p r e s s i o n
field Hilbert
[3].
of Wick.
2.
of the
from a c o n s t a n t
theory
In the
in the
expansion
in the p e r t u r b a t i v e
to the
described
and
in the q u a n t i z e d
to the E i n s t e i n
is self-adjoint,
leading
linear
operator
locality
in natural
and to provide
mathematical
an immediate
terms
generali-
139
zation
to p r o d u c t s
of
interacting
fields.
in the t r e a t m e n t o f h i g h e r - d i m e n s i o n a l may
represent
densely
distributions
defined
operators. domain
ones.
D
of r e g u l a r
specifically,
if
A
whose
They
A convenient
can
values
o n l y be
formulation vectors
But
one m u s t
space-times,
are b o n a
taken
is as
in the
any
fide
up,
that
at l e a s t
they
operators,
as a s p e c i e s
sesquilinear
underlying
is an o p e r a t o r ,
give
idea
of g e n e r a l i z e d
forms
Hilbert
the c o r r e s p o n d i n g
even
on a dense
space form
K.
More
a is d e f i n e d
by the e q u a t i o n
a(x,y)
Typically,
D
hamiltonian or
is c o n v e n i e n t l y H0
analytic,
as e.g.
the
or e n t i r e ,
the
time
to
field
But
here
the c o n f o r m a l
invariant
The p r o b l e m
is t h a t
2, one b e i n g priate
they
to the E i n s t e i n
differ
appear
these
only by
to be
is a l w a y s
~(x)
is a r e g u l a r
infinitely
Minkowski fra-red M 0.
but
so, pond
Although
in f a c t has
a discrete
a stronger
domain latter
that plague
M
are
replaced
that
(H0) ,
failure
invariant
energy
ranging
topology
than
of
invariant scalar
are
coercive
the
the D~
D~
(H0) ~
field
hamil-
denotes
the
of the
field
in-
theories
in
indeed
M0
and
canonically
They
group;
interval (H)
energy,
of all
Einstein
not.
which
measurability.
fields
the c o n f o r m a l
0.
locally
space,
Thus
H0
appro-
energy.
D ~(H)
to the
equivalent,
over
excludes
primarily
that
by the E i n s t e i n
where
operators of
but
being
physical
if
and clearly.
is a c o n s e q u e n c e
conformally
space-
latter.
although
fields).
respect
D~
generators
spectrum
natural
with
unitarily
and Einstein
spectrum
be
H 0.
explicit
sufficiently
f o r m on the d o m a i n
the c o n f o r m a l l y
a continuous
is n o t
on
from
Einstein
distinct,
of d i r e c t
energy
and must
more
the o t h e r the
free
and may
the
effectively
in the c u r v a t u r e
threshould
vectors
to d i f f e r e n t
has
terms
(for p o s i t i v e - e n e r g y
The
Universe
the M i n k o w s k i
H0
h~s
any
n o t on the
divergences
the E i n s t e i n
are q u i t e
map
hamiltonians,
short
sesquilinear
hamiltonian.
"free"
for
or
H 0,
for
be m a d e
Universe,
operators
larger
must
and
differentiable
H,
several
to
a smooth
to be t r e a t e d
the
dependence
topology
one,
purposes,
which
tonian
are
(or r e l a t i v i s t i c )
for m a t h e m a t i c a l
becomes
with
differentiable,
respect
relativistic
secondorder
far b e l o w
The M i n k o w s k i
in
there
with
a natural
are
D
infinitely
of the a p p a r e n t
connection
fields
the u s u a l
Mathematically
vectors,
with
x,y6
in a s s o c i a t i o n of
x-->~(x) , t h e n
forms,
conformally
defined
in s p i t e
itself,
sesquilinear
,
totality
etc.
then be L o r e n t z - i n v a r i a n t Even
=
correswhile
(0,~),
is c o n t a i n e d and basing
H in a n d
analysis
140
on the n a t u r a l cedures
Einstein
required
treatment
of c o n f o r m a l l y
Yang-Mills
To d e f i n e
tion; more
:@(x)P:
functions
f.
effect,
differential t i o n at the
for
but
one
space,
over
its use
equation time
first with
D(x,y)
[~(x),~(y)]
the
with
field
which <"@(x) Having larly
2 "
operator
be
fixed
v,v>,
where
defined defined
2
rely shown
K
v
powers
of the
to e x i s t with
in a n y scalar
as n a t u r a l
by the
when
are are
func-
taken. similar,
be d e s c r i b e d .
may
be,
its
The
right
equation
identity
operator.
unambiguously
so it b e c o m e s "@(x) 2"
is u n i q u e that
denotes
unknown
= 2@(x)D(x,y)
the
by r e q u i r i n g
of the
form
B y the
mathematical
it d o e s , -
dimensions,
side
(after
as a
but only
irreducibility
as
of
within
an a d d i t i v e
constant,
its v a c u u m
expectation
value
the v a c u u m p = 2,
defined
a definite
In fact,
dimensions.
"@(x) 2''
"@(x) 2''
is
differen-
side
side
case will
integra-
latter
pronounced
of the
left
class
the
The
evolutionary
two o p t i o n s
in 2 s p a c e - t i m e
in h i g h e r
by r e c u r s i o n
monomials
coincident
means
the r i g h t
on the
is e s s e n t i a l l y
in
"@(x) p''
the
defined
a
that
whatever
,@(y)]
exists
a true
of the
the
vector, higher
should powers
vanish. may
be
simi-
[5]:
["@(x)P",@(y)]
In fact,
distribution
a relatively
space-time
take
function),
there
operators,
may
and
regarding
on the v a l u e s
p = 2,
I being
equation
form
the
scalar
a test
sesquilinear
the
pro-
in the
in the M a x w e l l
space-time.
having
requires
only
should
is the
whether
time
treatments
case
: D(x,y)l,
self-adjoint a
for s p e c i a l
for an a p p r o p r i a t e
two o p t i o n s
forecloses
of n o t i o n
@(y)
foregoing
smoothing
s u c h as
or o v e r
the d e r v a t i v e ( s ) the
[@(x)
question
fields,
has
This
depends
that
simplicity
commutator
of the
need
divergences
as an o p e r a t o r - v a l u e d
Here
formulation.
technically
Taking
where
the
infra-red
as an o p e r a t o r
over
integration
equation
However,
eliminates known
invariant
f(x)dx
it c a n be e i t h e r
smoothing
and
e.g.
J :@(x)P:
regular,
tial
well
fields.
to d e f i n e of t e s t
structure
to n u l l i f y
free
field
field,
: p,,@(x)P-1,,m(x,y)
and
m a y be t r e a t e d
mathematical
the c o r r e s p o n d i n g
its d e r i v a t i v e s ,
objects,
Wick product.
and not me-
in a s i m i l a r which
are
way,
formally
and
141
Intuitively, terms
of d e g r e e
evaluate
x)
of
in the
to a t e s t
of o p e r a t o r s
case,
Lorentz) . Property
assertion
that
Thus
convenient a true
x,
for
the o r i g i n
would
of the W i c k
it has,
rather
case
of an i n t e r a c t i n g
and annihilation
be e x t e n d e d .
This bears
partial
field
operators,
here
it m a k e s
no
formally
that
to a f r e e power
on the m e a n i n g
differential
the
time.
sense at
field,
to
However,
product
to the
(in the B a i r e
as an i n t e n d e d attributes
extends
the r e p l a c e m e n t remains
of
coherent
with
the p h y s i c a l
field that
be a s y m p t o t i c
invariance
must
In t h i s w a y
to t h o s e
vanish
the
to
free
at all
"Heisenberg
Even
unitarily
assuming
equivalence
of the
concept,
at l a r g e
times
field
field
fields,
and
according
expectation
Wick in q u e s -
at a n y o t h e r
free
interacting
at l a r g e
by
its
concept-
physical
vacuum.
is m o r e o v e r to w h i c h
to the
values
field,
equivalent
corresponding
b y the p u t a t i v e
free
free
unitary
scattering,
of the
not
quantized
as the
of p o w e r s
vacuum
can
the
" : @ ( x ) 3 : ''
and covariant of
fundamental
n o t be m e a n i n g f u l
s h o u l d be a s y m p t o t i c the p h y s i c a l
as
directly
idea
no c o r r e s p o n d i n g
as e a r l i e r ,
the
would
of the
a local
are
definition
field was
via
the c h a r a c t e r i z a t i o n
then
by
~(x)
s u c h as
"@(t0,x)3"
equation
"@(x) p''
should
be e q u i v a l e n t
there
than
the
its t r a n s f o r m
ually,
implies
tO
and defining
properties
interacting
rather
time
commutation
This
in the m a s s i v e
,
" d e f i n e .... @(x) 3''
initial
the d i f f e r e n t i a l
with
or
remarkably,
of the
equation,
interacting,
(more e x a c t l y ,
tion),
of
in an a r b i t r a r i l y
of
and Wick's
@ . ,,@3,, = 0
being
its a v e r a g e
(von N e u m a n n )
(conformal,
a function
a local
neighborhood
the
y
:@(x)P:
function.
In the
nonlinear
@(y)
to that
is i n d e e d
by a small
invariance
is i n d e e d
rationalize
with)
and other
seeks
cutoff
(more e x a c t l y ,
(a) c l a s s i c a l l y
:~(x)P:
despite
it
supported
b y the (b)
if one
:@(x)P:
it is a f f i l i a t e d
standardization,
local
creation
of
that
locality, that
by constant
an u l t r a v i o l e t
attributes
(a)
sense
generated
neighborhood
@(x) P
"infinite",
(or i m p o s i n g
function
(more p r e c i s e l y ,
from
that are
are:
small
class).
differs p
The k e y
pth power ~(x) ,
respect
is in
ring
removed).
"true"
function with
than
them by continuity
is g r a d u a l l y as the
:@(x)P:
less
of
free
the field.
"@(x) p''
times,
and
so
times.
field"
@
c a n be n a t u r a l l y
formulated
142
as a s o l u t i o n
of the e q u a t i o n
(+)
[] @ + : @ 3
: 0 phys.
satisfying
the
usual
it is n o t k n o w n "practical"
whether
evidence
turbative
expansion
make
cally
trivial.
this
(+) m a y h a v e
of the
times
a unitary
interaction
representation
may be representable vector
space
powers
physical case
vacuum,
the c r u c i a l
Einstein
tions
and
large
Universe
of the
shown but
below
or p h y s i -
equation
but
it s u g g e s t s
propagation
over
not be represented
with
shortly,-
from time
- ~
on the
the m a t h e m a t i c a l
of
scalar
or
in the
to t i m e free
by
interaction
but
+
field
distances.
a spatial
space.
The
relative
cutoff
is a s e c o n d a r y by virtue
of h a v i n g
In the
existence
field,
[6]. A s p a t i a l
this
divergences)
of
cutoff
the h y p e r b o l i c i t y take
analog
is r e q u i r e d , field
in the
(not a f f e c t i n g
cutoff
2-dimensional
Heisenberg
is n e e d e d
issue
the
of r e n o r to the
thus
effect
of t h e
by virtue
exists
for e q u a -
form
l~ @ + RI x SI
that
not;
transformation
the p o s s i b i l i t y
not even
(++) in
been
space,
of the c o m p a c t n e s s
meaningless
terms
a well-
coupling
renormalization
in the H e i s e n b e r g
be d e a l t
the p r o p a g a t i o n
dimensions,
"ultraviolet"
at a r b i t r a r i l y
whether
(indeed
K.
has
equations
does
renor-
in the p e r -
any
constant
indicating
In o t h e r
includes
constant
without
mathematically
in t h i s h i g h
term
is f i n i t e
space)
from
coupling
here
coupling leading
coupling
of a s e l f - i n t e r a c t i n g
of M i n k o w s k i
of the
K, will
against
summarized
can p r o b a b l y
by a unitary
In 2 s p a c e - t i m e malized
in
latter
Among
an i n f i n i t e
the
it p r o b a b l y
space)
equation. prejudice
(see below)
is far
exist.
to t h i s implicit
infinite
in H i l b e r t
indeed may
transformation the
for an
dimensions
to be n e c e s s a r y ,
analysis
term either
(in M i n k o w s k i
representations,-
the
infinite
in i t s e l f
S-matrix
that
typically
S-matrix
An
a solution;
is an
specifically,
operator
leading This
a unitary
need
More
renormalization.
would
state
hand,
In 4 s p a c e - t i m e
a solution
there
on the g r o u n d s
the a p p a r e n t
is s p u r i o u s .
exists
appears
other
self-adjoint
constant
finite
primarily
On the
that
malization
that
there
relations.
physicists
renormalization
a dimension.
defined
commutation
theoretical
the p o s s i b i l i t y , constant
vac.
for a g e n e r a l
and have
p'(0)
= 0.
:P'(@) :phys. class
vac.
: 0
of p o l y n o m i a l s
It is k n o w n
that
this
p
that class
are b o u n d e d is n o n t r i v i a l
143
and open a given
in the
nomials.
but
p(@)
the e x i s t e n c e tion
In the
in the
of
similar
of the W e y l to t h o s e
terms
all
where
exists
there
even
integer.
in the
sense
of
of
such polya solu-
In m o r e
is k n o w n m a t h e m a t i c a l l y
these
matters
THE
must
from being
about
solutions
of e q u a -
representation" only
on
interacting
representation to t r e a t
in
of w h i c h
H0
can
but
used,
nonlinear for r e a s o n s
relations
quantum
mechanics.
in the
in p l a c e These
treatment but
of h i g h e r -
for b r e v i t y
this
theory
via
"practical"
is s i m p l e
remove
in p l a c e
of the v e c t o r
and
time
be u s e d
HI
function
are
the
far field.
"interaction
("unitarily", to the H e i s e n -
treatment
in the e s t i -
analysis.
f r o m the H e i s e n b e r g
to t h e
are
of p a r a m e t e r s .
"free"
formally
hamiltonian) self-adjoint
is c o m p l i c a t e d , generated
of t i m e
u(t)
by =
of
In p a r t i c u l a r ,
theory
by a perturbative
(the
variation
for the
diagrams.
field
exist
interacting
of the term)
Feynman
of v a r i a t i o n H0
above
is e q u i v a l e n t
use
in f r e e
field,
to t r e a t
the b a s i s
quantum
S-matrix
where
indicated
field
it f o r m s
monomials
free
a putative
or analogical
hamiltonian)
to f i r s t
of
ordered scalar
as
however
is by the m e t h o d
the
been
issues
of c o m m u t a t o r s
fields,
expansion,
of the
transformation
e-it(H0+HI)
"interaction"
of
the W e y l
the W i c k
Formally,
of the t h e o r e t i c a l
the
of
taken
here.
for p o w e r s
field
used
Formally,
instead
have
important
D ~(H),
in a n y e v e n t
quantum
it is c o m m o n l y
should
are
:@P:
powers
field.
and
Thus,
has been
of m a t h e m a t i c a l
powers
perturbative
dimensions,
in a p r o g r a m m a t i c
field,
license
REPRESENTATION
forms
Wick
[5]
the u s e
be o m i t t e d
a substitute
free-field
ones.
in e l e m e n t a r y
s u c h as the p o w e r s
sesquilinear
mation
require
relations
In 4 s p a c e - t i m e fields,
primary
S-matrix
INTERACTION
a certain
the a v o i d a n c e
of r e n o r m a l i z e d
relations
in the
and
relations
that
Weyl-like
order
(the
fields,
exposition,
f r o m the p r e s e n t
nonlinear
berg
known
such polynomials
it i n c l u d e s
an a r b i t r a r y
nothing
succinctness
of the H e i s e n b e r g
but
even
being
of H e i s e n b e r g
characterization
variants
The
n
dimensions
foregoing
sake
disjoint
as
of all
(++).
f o r the
3.
it is n o t
= @n,
2 space-time
space
it is n o t k n o w n w h e t h e r
In p a r t i c u l a r
tion when than
finite-dimensional
degree,
interaction
In o r d e r and
HI
operators
it m a y b e a d v a n t a g e o u s H0[
Specifically,
e-lt(H0+HI)u0
that
144
s a t i s f i e s the e q u a t i o n e i ( t - t 0 ) H 0 u(t) where differential
v' (t)
There than but
having
entirely
one
;
Hi(t)
operator
the p r a c t i c a l l y
made
it c o n v e r g e s
time.
=
satisfies
v(t)
that
of the
free
free
Thus
on the
- free,-
right
the
early
field, field
for the
vac.
~ : ~ 0 ( x , t 0 ) : d3x,
side,
interacting
where
~
the
field
interacting that
as
field
to--> - ~ , the
sub-
in the e x p r e s s i o n scalar
de-
case,
to c o n v e r g e
interacting
is the
assumptions
assumption
to v a l i d a t e
invariant
is s u p p o s e d
certain
The main
in f a c t
enough
complicated time-dependent,
it c a n be e x p r e s s e d
modulo
tO ,
so c l o s e l y
strongly
d3x
that
practice.
times
more
in b e i n g
fields,
in the c o n f o r m a l l y
:phys.
Hie-i(t-t0)H0
feature
physical
at v e r y
by a free
: ei(t-t0)H0
convenient
of k n o w n ,
to the
HI.
~ :~(x't0)
v(t)
introduce
representation
in t h e o r e t i c a l
is to the e f f e c t
stitution
is a f i x e d
for the H e i s e n b e r g
is a p p r o x i m a b l e
fining
tO
: -iHi(t)v
in t e r m s
generally
= -i(H0+Hi)u ,
equation
is n o w o n l y those
u'
to
and
the
free
field.
This
assumption
is n o m a t h e m a t i c a l to the
contrary. theory,-
a finite
result
Feynman tation
and
apart
when
forms
diagrams. is g i v e n
S(t,t')
taking
(formally)
exist, the
since
same in
K
quence
of t h e i r
a mathematical
that
S(t,t') ;
continuous
states
compu-
in t e r m s
unitary
of
represen-
operator
at t i m e
t,
S(t',t')
= I
no s u c h o p e r a t o r s
under but
under
spectrum.
it w o u l d
analytic
interaction
(symbolically)
that
is i n v a r i a n t
sense
to e x t r a c t
analysis
= -iHi(t)S(t,t') ,
is i n v a r i a n t for
explicit
there
indications
self-consistent
required
in the
into
are
and
given
equation
it is c l e a r
Hi(t)
to m a k e
t'
differential
invariance
vector
b y the
plausibility)
there
to a f o r m a l l y
evolution
at t i m e
(d/dt)S(t,t')
in f a c t
for p e r t u r b a t i v e
temporal
(formally)
on p h y s i c a l
and
the m a n i p u l a t i o n s
it is s o u g h t
states
space
it,
it l e a d s
from
the b a s i s
The
b y the
In M i n k o w s k i
entirely
for
Nevertheless
and neat
tations,-
is b a s e d
evidence
have
space
translations,
the v a c u u m space
Thus
if
to l e a v e
can actually
vector
translations, S(t,t') the
free
implying
is t h e
only
as a c o n s e -
did exist vacuum
in
vector
145
fixed, would
which
would
be a f i r s t
In the E i n s t e i n so t h a t
the
S(t,t') but
make
step
Universe,
foregoing
satisfying
there
nonsense
towards
of
the p h y s i c a l
establishing
space
argument
translation does
an a p p r o p r i a t e
are more
sophisticated
S-MATRIX
IN P E R T U R B A T I O N
not
interpretation,
mathematical has
rule
a discrete
out
spectrum,
the e x i s t e n c e
interpretation
reasons
and
triviality.
of the
of e q u a t i o n
to be d o u b t f u l
~),
about
such
existence.
4.
THE
In the c o n t e x t S(t,t') has
as
a very
the v a c u u m physical
t'-->
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<7 some
0
by
: e - 2 ~ /-//~f
~ 0 d and N~, d are
seems
0
exp(f)
remain
case
f
0
~ d is the
,
0
t and
,
in the n o n - a b e l i a n
>~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 <= 0 for some [g.. ~(C (L~,G_) such ±J ±~ ~3 ±3 3~r ~ I ~ 1319 a 0 that L0ij= ~(gij) . In the rest, if ~) ij] 6 Z (~, ~ ' ) , ~ (gij)= gOij means that { gij ] also satisfies g0ij-~ik+gkj &)ijgjk = 0. LEMMA I: Let g0 = { ~(gij)l ~ C I ( ~ , ~ if and only if [ gijl satisfies a {6(g)ijkl
:
I)
then '
Igijgjkgkil 6 C2(L~,Gt )
63 is in ZI(o~ ~I)
157
and
DEFINITION: Let 60 and co~ be in ZI (~, ~I) . Then we call co o)' cohomologous if there exists h = { hul E cO( [~,G d) such that 6(g)ij k g0'~ . =
13
~ (hi)6(g) ij k
hi(
LOij
-
,
~(hj)
+
gij - I ~ (hi) gij )hi-1
The quotient set of ZI(0~,~9~ I) by this relation is denoted by H I (~, ~I) . H I (M,~I) is defined as the limit of H I ( ~ , ~ I ) with respect
to t~. DEFINITION: set Z2(~,Gt ) by
We define the map 6[ : C 2 (O[,Gt)-->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(<~0>). 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~ ()) , <60> 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
~<(~ , D ~
'
~
.
Utiyam~ | e
matter lagr.
P2
d'KQ~e=9~s~; 0. ~
'
W~(~,9~, e)
-t~'
=
~1975 PGTA A~ 1980
~
t,
A
~ ~'
A~
A
~.
C I
Whereas
in Einstein's
and matter fields
to
= e~ ~ ~ K
anholonomic
~
coefficients
g~
served as potentials
(in order to get a cova-
and
the
replacement
of
g~
by
tetrad
This implied the separation of holonomic
('internal')
approach
A
•
were treated as GL(4,~)-tensors
e~W )"
to an equivalent this
A
~ . ( 9 ,D~- ,BY)
the introduction of spinors in GR required the generaliaation
S0(3,1)-tensors
(g~
=
theory the metric
~
riant formulation),
A
=-~
version
indices and of GR
automatically
]ed,
apart
from the extension
('tetradformakism'
are
internal
fields
/i/ ); matter
Lorentz
e~
('external') and
invariant.
to spinors,
lagrangians in The
first who
pointed out this analogy with YMT and gave an interpretation of the Levi-Civita LC~ connection coefficients ~ as YM like 'compensating potentials was Utiyama I, /2/ . The generalisation to arbitrary metric compatible ~x s by allowing for torsion
then
resulted
cients
Q ~
as
in a second
algebraic
field
functions
equation of
yielding
the canonical
the torsion coeffispin
tensor
s~.$'
- 'no torsion outside spinning matter' - (Kibble /3/, Sciama /4/ and essentially contained
already
in early
papers by E.Cartan
E_instein C_artan Kibble S_ciama theory).
/5/
, thus often referred
to as
The interpretation of the Poinear6 group
as a gauge group for gravitational theories, first indicated by Kibble and later developed
by Bregman
and Hayashi
/6/,
yon der Heyde /7/, Hehl /8/ and others,
led to the notion of Poincar6 gauge theory (PGT).
In PGT a further analogy with
171
YMT
was
realized via
including
terms
the
generalization
quadratic
in
curvature
of
~¢ =
and
both on the same footing ('propagating torsion') Modern
coordinate
in two steps. calculus' be
External
G-bundle
by
M
of linear
the
which
formulations frames
formalism
represents
of
on
field
PGT
M
entered
PGT
(and
YMT, CR)
from
the
the
space
internal
coordinate
time
turn out
degrees of freedom.
to as PGT L)
free
versions
this
of
However,
to be
the
l-forms
comparison
differences
M
to the
In the ease
is the
bundle
the potentials
(7~' of
,
~
the c a n o n z c a l
the type of
LM
e
.
and (linear) connection forms
structural
mainly
'Cartan's
manifold
"
then
of
.
have
(here referred
The
-
I~'~
lagrangians treatment
forms (cf. /9,10/) and internal (bundle) coordinates may
'lifting'
over
usual
formulations
to
which meant a
(space time) coordinates may be removed by using
of differential
avoided
of
free
R~
torsion
o~
e
l-forms
uo k • between
YMT
and
GR
also
shows
up some of
the main
which still are inherent Jn PGT L ; we mention the keywords
a) Internal symmetries versus external symmetries (acting on space time).
(i.i)
-
b) The role of the tetrad fie]ds as 'compensating'
-
c) The diffeomorphism group as the local gauze group of a suitable
potentials.
space time group. -
d) Local gauge in_variance versus diffeomorphism c ovariance.
-
e) Is there a modification of EP in complete formal analogy with the transition from zero to non-zero field strengths in YMT ~
The of
occurrence PGT L to
of
the
O'
and
bundle
AM
ua"~ as of
backs of connection forms on thus
giving
a
uniform
AM
15/).
The
aim of our
as
possible.
from
the
paper
(referred
PGT L
Poicar6
frames
then
group
M
of
both
(
@'
of
i
out
to
be
a broken
subgroup.
the
pull
: LM --9 AM ),
strictly in the sence of YMT (cf.
Lorentz
translation
are
gravitational
is to construct a consequent
its
the
, ~',
embedding
types
to as PGT A ) following
turns to
in PGT L suggests on
via the canonical
interpretation
by only one type of connections,
an affine version
potentials
affine
potentials
/11,12,13,14,
translation
of PGT L to
the general YM scheme as close PGT A, with Moreover,
symmetry PGT A gives
breaking answers
to the questions indicated in (1.1).
2. G-equivalence principle in classical YM theories
Consider certain
type
Minkowski
the
usual
of F-valued
space
or,
more
starting
point
( F=~ TM or {
TM
general, over
for
the
) matter a
flat
construction fields Lorentz
of
a YMT
, i.e.
( ~A ) , A=I .... m, manifold
(M,~)
a
over , with
172 equations of motion given by the Euler equations
(2.1)
#
~/**A
~+~
of a first order matter lagrangian {-induced volume 4-form ~ of an
= 0
~(
~A,d ~A ) d #A = ~
( +A ~.j~A)
~
on
M
and
~A
dx ~
).
r-dimensional Lie group
G
and a linear representation
(with
The existence ~
of
G
on
O
F leaving
{~
invariant
(2.2)
f~(~A,d~A)
O
o
= _~m(f(g); ~ a , ~ ( g ) ~ d # @)
('global' gauge invariance) then induces, for
~
,
g eG
,
fulfilling (2.1), conservation
of the Noether currents O
(2.3)
D ~ i¢,~ We
indicate
lagrangian
= 0
the
formalism
,
!~,~
translation on
bundles
:~
into
given
~9~,
fa~
coordinate in
free
Appendix
~ language
A and
then
using
the
introduce
the
'non-flat' theory (minimal coupling) via the notion of 'G-equivalence principle': 2.a. Flat theory The background structure of the flat theory is given by a trivial G-bundle :
~ --4 M
, the trivialization being fixed by a distinguished global section
: M --+ ~
We consider tensorial 0-forms
~
on
get the matter fields in (2.1) by the identifications be the flat connection on X ~ TM
, and
_~ , ~
o
(2.4) ~
lagrangian
induces
~
via
~ (&,~(X)) = 0
M
and
~
~
in (2.1) as a 4-form on
Ao
~(~
the
,~ ~ ) on
~
automorphism +E< on
and
gauge
invariant
free
matter
The corresponding Euler equs. on
~
are
free matter f i e l d s of type (F~ ~) .
E< a ~
induces a
~ ~ M ×g
then is a symmetry for
then reads
gauge invariant).
actively)
~ : M --9 ~
( ~ -dependent) i n f i n i t e s i m a l v e r t i c a l
via l e f t G-action (to be d i s t i n g u i s h e d from the
-independent fundamental vectorfields +E~
M
,
, respectively.
(see A.3.) via
the s o l u t i o n s of which we c a l l Each generator
('local
(passively
with arbitrary sections
Let
determined by
~___~( ~ ,~ ~&) now
of type (F, ~), hence
~A := (~ ~ ) A
its covariant derivatives on
The covariant expression of
~
~
+ E~
on ~ , induced via right G-action)
~o , which implies the existence of
r
conserved
173
currents, on
¢ ~
&(M)
(2.7)
.
2.b. G-equivalence Given co
expressed by
isomorphic , there
lagrangians
~(
(globally)
~
principle
,
G-bundle
~A
G-bundles
is a
P~
natural
~ SD~ ~ )
on
'isomorphic' ~
~ . ~ ,D 0<. ~ ) =
P~
~(#
M
, with
'isomorphy'
~f~ ( ~ ,D ~-)
there
:
over
of
P~ and
if
isomorphism
, Pz
notion
exists --~
a
P~
,D ~ )
on
Pa
strong
such
connection
between
gauge
. Both are called
(i.e.
that
forms
invariant
17a~ = ~
co
=
)
~~ ~
and
We apply an infinitesimal version thereof
to get the notion of 'G-equivalence principle': Let
~7
isomorphic
:
to
-valued
fundamental
x e U~
be an arbitrary
~ in 2.a.
l-form
= (Ad #'~) o o
E×(~,P)
P ---> M
with
g ~ G
,
P×
vectorfield
on
P
of
.
A connection
TM/P×
on
local
strong
For fixed
,
:
, i.e.
P
over
x
Z+
and
locally
x e M
. and
is a
R~
co x =
the Z-induced
we call a (maximal)
~ ~-~(U~)
v ~ T~/P~
P
Z ~ ~
f
class
--~-~(U~)
(of first order) over
,
~I.(v)
x e M
~
M
on
= Z
denoting the fibre over
equivalence
=
~.(v)
over
form co x
~O×(Z +)
isomorphisms
, an infinitesimal
(2.8)
G-bundle
x
c P
, if
~ ' ~
~ E.(P,P)
D
Hence the elements of
o
E (P,P)
For a fixed connection
.
~.
on
are of first order contact at each point ~
over
x
an infinitesimal
of
equivalence
P..
E (~.P)
thus determines a unique connection O
(2.9) on
P
~0 x := over
~.
~.
( ~ ,~ ~ )
on
equivalent over
x
~
o~
=
~×(} R e Ex(P,P ) and
~
~
and
if there is an infinitesimal
(2.10)
, ~
~ ~ E (P,P)
x
We call lagrangians
for each
,
, g
( ~.
to
~.
~
,~$)
&= ( ~ , D ~)
on
equivalence
E~(~.P)
(hence
=
K'~
, cO~ and
D(~¢)
(~ ~
P
infinitesimal
= ~(~)
with )
and
,Dd.~) ,
~o~ being the r e s t r i c t i o n s of
WP/e~ and r $ / F ~ ) . °
(2.11) Definition
Given (F, @0 by
a free matter
lagrangian
' a matter lagrangian
~-~ ( ~
,~ ~ )
via
constructed as follows:
~( ~(
~ ,~ 4) ~
,D ~ )
the G-equivalence
on on
principle
of type P
induced (G-EP)is
174
i)
We assume
that for each
x e M
an infinitesimal
equi-
O
valence ii)
E×(P,P)
is given.
The connections
w~
(2.9) are assumed determine
a
, induced by
differential
(global) connection form iii) We define
~(~,
~)
o
~e
Ah(P,F, y )
(The
definition
o~
and
to be C~-dependent 1-form
~
on
on
on
E (P,P) x
via
, i.e. they
P
and
hence a
local
strong
P
fibre by fibre through
a
,
f e Aa(P,F,~ )
is
independent
from
the
isomorphisms
: $~(~, over,
~)
~
in (2.11)
( ~,D ~)
e ~(u~)--~-~(U~)c~
i s gauge i n v a r i a n t
is infinitesimal
w i t h r e s p e c t to
E,(P,P)
(2.12)
Remark
If,
= ~(~
,D ~)
in
on
,
P
particular,
P = ~
in
and of type (F, f ) . O
e q u i v a l e n t to
x~U~
o
~(
~ ,D ~ )
(2.11),
~
hence coincides with the usual minimal coupling procedure.
More-
in each
we get
Then G-EP means simply the replacement of
.)
~(#
by
D
in
x e M
,D#) = ~
,
However, our aim here
is to use G-EP as a generic term which subsumes the notions of a) 'minimal coupling' in YMT and -
b) 'Einstein's equivalence principle' in GR.
We shall see in the affine version of PGT that, up to the constraint of vanishing torsion, b) in fact can be interpreted as G-EP for
G = T~
SO(3,1)
2.c. Field equations and conservation laws The application non-flat theory. P
over
M
-f~ = D e
of G-EP to the flat structures
and
G-EP
implies
, as tensorial
same
assumes
footing
the existence
invariance of of ~ =
~
=
8~
as
~
~
+
~ ~f
occurrence
~
P
of
non-vanishing field strengths
of type ( ~ , A d ) .
Thus,
in addition to
gets dynamical character and is treated in YMT on
, concerning
of a first
order
the
field
field
equations.
lagrangian
Accordingly, one
~ f ( ~ ,D ~ )
Gauge
implies independency of to because of the non tensorial quality
The independent ( ~ ,D ~ )
the
2-forms on
, also the gauge field the
in 2.a. yields the general
Here the geometrical background is realized through the G-bundle
+
~
variation
(D ~ )
, where
of the total
with respect
to
lagrangian ~
and
~ ( ~ ,D~ ,D~)
to then yields
~
:= =
175
(2.13)
~o Wt
Hence
$~
= 0
(2.14) (2.15)
gives the
matter (field) equations
D
(gauge) field equations
D
-
as equations on Moreover, $~
= LX~
,
gauge invariance ~
= L~
of
, ~
i
~
~
implies from (2.13) (with
, X e autvP ) the conservation
~ ~W~
d(Lx #
~
@ ~*
P
(2.16) for
and the
fulfilling
(2.14,2.15).
+ LX ~
S~
= LX~
= 0 ,
laws
~+ ~ ) = .a%-N-
0
These r e d u c e t o t h e u s u a l c o n s e r v a t i o n of
t h e Noether c u r r e n t s
(2.17) if -
j4, × .-
X
i s a symmetry of
~
= ~
and
X
is a fundamental
centre in The
relation
~
X = +E~
~
,
LZ~ = 0
in the flat
Ag~
, which e . g . i s t r u e i f
c a s e of 2 . a . ,
vector field on
P
or i f
corresponding
to an element of the
.
between
the
(G-invariant,
~-valued)
l-form
j~ix
on
P
and
the
j ~ , × = ha ^ 3
~ ,
'source term'
(2.1s)
J~ := - 5
in (2.15) where
(tensorial
h x := i x ~
Eventually, Dj~
=
0
3-form of t y p e ( ~ , A d ' ~ ) )
:
gauge if
P --+~
invariance
@ fulfils
3. The affine formulation
The
application
scheme outlined -
specification
to
is
given through
(cf. A.3.). of
g~
implies
the
'covariant
conservation
law'
(2.14).
of Poincar6 Gauge Theory
gauge
theories
of
gravity
of
the
general
classical
YM
in section 2 requires of the bundle
rily restricted particular
(+)a~
P
to the topology
over flat space time of
(M,~)
( M
~ ~ ) as a fixed background
the choice of the structure group
G
,
not necessastructure,
in
176
-
-
-
introduction
of matter
fields as tensorial O-forms on
the S0(3,1)
transformation
properties
of
'usual'
P
matter
such that at least fields are included,
description of the 'flat theory', i.e. fixing the free matter lagrangian on P , construction of the framework for the full theory by -
application of G-EP derivation of the field equations,
conservation
laws, etc. for the general
form of matter and field lagrangians. We treat these points in the indicated
order,
in particular
taking into account
the problems mentioned in (Ioi).
3.a. Affine matter fields Matter fields in SR are described by tensorial O-forms (~m, ~) lently, the
i)
on the SO(3,])-bundle
by sections in bundles
introduction
and
in
L° M
B
= (A)
of ~-orthonormal frames on
~ -associated
Appendix
~
the
to
LOM
) 2)
gravitational
of type
M
(equiva-
As pointed out in potentials
@£
, ~i
occurring in ECKS-theory and in PGT L correspond to connection forms on the bundle A°M
of affine ~-orthonormal frames on
M
A°M
, i.e. to choose
being the Poincar6 group
We
introduce
~
with
A°M
~
equivariant
:
(T~
functions
tangent bundle, T~
SO(3,])) x ( ~ ) ,
Y
on
(y, ~)
P
the bundle associated S0(3,1)
affine frame field
on
(p,e)
may
:
to
A°M
M - - ~ A°M
S0(3,1)
;
is deter-
--~
~R~x~ ~
~-~
(~y+t, ~(~)~)
seen
as
sections
the
'affine'
with respect to the usual affine Y(x)
p(x) + y~(x)e (x)
In particular, TM
of
in the
the canonical function
has components
~
= 0
with respect
(p=0,e)
(3.2) Remark
In the case of Minkowski space
cal interpretation
of the pair
'shifted'
via
1) 2)
be
IR~ ; hence we have
corresponding to the zero section of to
to
as ~ -equivariant
, where the a[fine representation
(t,~)
action of
(Y, ~ )
T~
L°M
through
(3.1)
The
G
the notion of affine matter fields
functions on
~4×iR~_valued
mined by
P = A°M
This suggests to pass from
on
M
a
(Y, 9")
diffeomorphism
(M,g)
there is a simple geometri-
, namely as the usual matter field depending
on
Y
:
For
each
x e M
where the complex case is included if we replace { ~ by its underlying [ z ~ More exactly, the existence of spin requires the replacement of L°M by a covering SL(2,{)-bundle of spin frames; to keep simple notations we don't discuss this case here, the transition to the specific covering structures - groups, connection forms, tensorial forms, etc. - being obvious, once the covering bundle is fixed.
177
the
identification
~y(X) e M
T~M
Then
phism
~
of
These
Y's
M
m ~
0 m x
,
~y : ~ --~ M
Y
such that
constitute an open neighbourhood , of
~(TM)
with
~
given by
TM
, as a group is abelian).
(Y, ~) ~ ~ y
hwr ~y
fold
~
; e.g.
Y
id~ =fi~
in the Whitney Ce-topo -
~¥~Diff ~ since
being a C ¢ homeomor-
ff(T~)
, the space of
Our interpretation of
(Y, ~)
:= # y ,
~
now is
,
corresponds to the matter field ~ actively transported on
This property of hood of
onto a point
, where
(3.3) i.e.
T~
fi = ~y
U¢ of
Y ~-~
phism (although not a local group isomorphism, sections of
•
is C ~ and, in particular, for each diffeomor-
there is a unique
logy, ~ ~ IN ~ ~ ]
Y(x)
maps
,
as a 'shifting field' (at least for
Y
~
via ~ ¥ .
in an open neighbour-
~ ) obviously generalizes to our case of an arbitrary flat Lorentz mani-
(M,~)
if one uses the exponential map corresponding to
In this sence diffeomorphisms on transformations
M
correspond to active 'translational'
AOM ~(p,e) F-~ (p+Y,e) ~ A~M
gauge
, which gives an answer to (l.l.c).
3.b. Affine matter lagrangians o~
Again,
consider
first
~-orthonormal coordinates riant)
_
matter
formally treating
i
case
(z ~)
lagrangian
4-form on M , we L ~( ~A ,~£ ~A)(de t
the
~
of
on
Minkowski
M
(M,g)
Given
in
a special relativistic (Poincar~ inva-
( ~A, ~A)
d 4 z~
introduce arbitrary ~ , )d~x~ e~ ( ~ 7 =:
and ~
space
of
type
coordinates eo~ ~ ~ , ~
(IR~, 9 )
as
a
(x ~) by writing o ; D~' ), thus =: e~
as separated 'internal' and 'external' indices.
Next, we define the notion of a 'free affine matter lagrangian' by interpreting ~
as a function invariant with respect to 'internal' Poincar6 transformations:
Take the single affine frame
(~,~)
the coordinate
given above,
$ = (~
I~)
system
(z ~)
on the (affine) space
Via the identifications
i.e.
M
corresponding to
~ ~ p ~-~ (0,..,0) siR ~ and
~ m T~
,
x ~ ~
, the frame
uniquely determines an affine ('holonomic') frame field as a section of denoted also by = ~
~;
(~,$)
, i.e.
for the components
We define
~
on
~
~(~A,~
(~)
~A)(de
This yields
of the canonical function
with respect to the section
,e ) (3.4)
~(z) = -zi~i(z)
(p,e)
~
~7(z)
(~,~) A~M
,
= ~ ? z ;=
(zero section).
by
:= t ~ ;)dYx~
=:
~(~A,[4~]-~
~A)(de t ~)d~x~=:
A
The obvious extension of the domain of morphisms of lar) yields
~
~
(see Remark (3.2) ; then
~(~A,d?A
,dy ])
, which is
to all (O~yl)
Y
corresponding to diffeo-
is guaranteed to be non singu-
178
-
a function of the affine matter fields
(Y, ~-)
and its first derivatives, O
given in the affine 'holonomic' frame fields of type
-
-
globally T 4 @ S 0 ( 3 , 1 ) Replacing
to
~
d
, we get
O
(p,e)
and
gauge invarJant. ^ ~,
in
¢ D_ corresponding
by the affine covarJant derivative
the locally
gauge
invariant
expression
for
~ _
on
~
The
latter finally allows the introduction of the free affine matter lagrangian as a T4~ $0(3,1) gauge invariant 4-form on (3.5)
C~
~(
9" , D ~
,DY)
:=
A°~
_~((0":::~)
with arbitrary g-orthonormal affine frame fields
to
the
isometrics
M --+ M
The definition of
case
of
flat
Lorentz
and the induced
~
A°M
on
, ( c ~ D ~ ) A , ( 6 - ~ D Y ) ~)
0-= (p,e)
This construction of the internal Poincar6 generalizes
(M,~)
"~
yields
bundle isomorphisms
of form
arbitrary
on
simply
use
Ao M
local
- - ~ A~M
we follow the general
G = T 4 @ S0(3,1)
This
A
~ ,D ~,DY)
an ~
A°M
in section 2 and apply G-EP with
A
~(
W~
E.g.,
then Js independent of the chosen isometrics.
To get the (non free) affine matter ]agrangian on YM scheme indicated
; cf. (2.5).
invariant lagrangian
manifolds
local
~
via
via
the replacement
('regular',
see
of
Appendix
~
B)
by the covariant generalized
derivative
affine
connection
field
lagrangian
A°M
3.c. Field equations in PGT A In contrast
to
the usual
case
of YMT
the
choice
of
the
in PGT A (and in PGT L) is not unique.
Because of the occurrence of equal external
and
and
internal
functions
index
of
lagrangians.
~y
, ~iy
= 0
an
~Y
) may
A
hA
m
be
~Zy
used
A2
~ (~,D~,D~,DY,D For
~
also
to
(higher
derivatives
construct
D ~ -dependent
interpretation
A
Y) of
the
=
m
A
A
Y~(~,D~,DY)
Y
are
invariant
('affine')
Euler
A
A
A
~
+ ~(D~,DY,D
Y)
equations
resulting
from
we remember the following facts:
The affine formalism was introduced via the canonical embedding to
of
Am
We thus get the total affine lagrangian A
(3.6)
types
redefine
the
tetrad
coefficients
e~f as connection
i
:
coefficients
L°M --q A°M ~kJ
As
a consequence and to be conform with the general YM scheme we had to accept the occurrence of additional degrees of freedom, namely -
We
the point fields
p
the shifting fields consider
both
as components of the affine frame fields, and Y
fields
as components of the affine matter fields. as
physically
irrelevant,
allowing
equivalent descriptions of the same physical configuration:
for
different,
but
179
A
'translational'
-
the affine matter fields
automorphism
-
the affine connection form
-
the affine frame fields
~y
:
(p,e) ~-~
(p+Y,e)
of
AOM
, mapping
O
subbundle
(Y, ~)
to
~
to
to ~y
the matter fields the connection form
~
-
the flat metric
to
~y~
~
~
on
Y = ~
From
$~
of
M
~ = 9y
, to
,
(cf. Remark (3.2)) shifting
~y~
~
on
~y~L°M
c LM
we deduce as equations on
equations
^ LA
A°M
S Y = 0
, and
the affine matter
A
D ~ A -
:=
, and
~y~
from the variational procedure,
= 0
(3.7)
, hence the Lorentz
= (Y,e)
{Y(M)} x ~ L~M c A°M
L°M c LM
Y
,
and
~y%(O,e)
to the equivalent metric
Consequently we exclude set
~
to
{O(M)} x ~ L°M c A°M
-
~y~
(p=O,e)
corresponds to the diffeomorphism
-
~ ¥ ~ ( Y , ~) = (Y, ~ )
~A
=
0
and the affine field equations
=
(3.8)
zy
A
A
•
=
with the constraint
y~ = ~ ~
-
=:
(O,e)
because of
To get the physical interpretation of the solutions
i~ ~
~
~
Appendix G~
we
= ~ B
M
and is
~ L~M
to the bundle
f~
there
~(~)
subbundle on
return
a
i~
L°M
c A°M
= ~o + ~
unique
vertical
The application ( LM
-t]
, where in the first equation the terms including
y~ vanish in each affine gauge of the form
(3.7,3.8)
~y~]
of
(~, ~ )
, i.e.
consider
~°"~'~.~ +
now maps
ioj) 0 , ~o
According
-6-® of L~M
of equs.
the pull backs
~d
automorphism
~
~~
~ LM
LM
such
to that
onto a Lorentz
, thus defining an (in general non flat) Lorentz metric
and a (linear)
connection
Riemann Cartan space time
~@
(M,g,~) LM
~
=: ~
on
L$ M
g
; this constitutes a
as the final solution of equs. (3.8).
+~-~- LM
L~M ~---
~-@~
AM
L~M ~ _ _ ~ A°M $
Analogously
the
'physical'
tensorial forms on (3.10) (g, ~ ) Remark
Remark given (3.2).
matter
solutions
are
derivation
of
given
by
~® ~ ~
=: ~
as
LSM
Each above
step has
in
the
a natural
In particular,
geometrical
consider
the
Riemann
interpretation
the transition
~ ~-~ D
Cartan
structure
consistent in
~
with
to get
180
the affine matter x e M
lagrangian via G-EP.
According
the choice of an infinitesimal
class of local vertical automorphisms
~
Since
way
of
each
A°M
class
~
induces
, the class
B×
of
local
in
an
E×
determines
which yields as
~,×
'deforming'
corresponding
obvious
:
B~
A °M
a
~
fulfilling
(local)
of
M
this means for each
as a first order contact co~ =
translational ~
with
~
4. ~
automorphism
) a first order contact fi×point
x
(cf.
coincides with that linear isomorphism
L~M --~ L~M
the metric
of
Ex
(for regular
diffeomorphisms
One easily verifies that
to (2.11)
equivalence
(3.2)).
TxM --, T~M
This justifies the interpretation of
into
g
through
~
in (3.9),
~"
being
the
'deformation potential'.
3.d. Comparison with PGT L Using the
the
canonical
relationship
With
the
free
~
o
e "~ ~( ~
J
matter
:=
~(~A,DwA,e:)
, ~) is
(the
rather
o
i*~Y
,
~.~(
on
~
L°M
to
recover
, and PGT A
(cf.
,~ ~ , ~ )
in
the
or
(2.5,3.5)), G-EP
the
defined
character
L ° M-version
of
PGT)
matter
by
lagrangian
of
the
transition
and
with
the
field
we get:
i*(Y,~)
$
=
= (0,¢)
~
~
,
AZo
i*DY ~°
,
= ~
±* ~(~,~,~) ^
=
i*D
,
Y = DO
$~(~,D~,e)
,
~2 ° ,DY
i*
straightforward
theory
0
i*Y = 0
(3.11)
now
o
coupling
obscured
0
is
lagrangian
minimal
~ F ( D ~ , @,D ~)
lagrangian
it
, as a lagrangian o
( ~,D~ ~
PGT L
(PGT L-)
$)
~,D~,
~-~
embedding
between
=
,
~ = pC
,
i*t i = t i
(i)
LA
:=
D ~
(ii)
c.K
:=
D .~0 ~,,~ ~
(iii)
P1
:=
D%~
,
,
where
(3.12)
are
the usual
PGT L equations
interpretation is of
the e}~'
same in
of as
the
69
as
-
0
+ 2 69c~ ~$, ~~ i 7
=
+ ~(};
the
for
of
tt
~@, , ~o
'deformation'
corresponds
formulation
- 2 Y ; l~A3 Y 3 ~~ 2 ~
-
of motion
defining
in PGT A and tetrad
%'~
to GR.
the
and L°M
standard
Thus
we
get
69
(ef.
=:
_s
=:
-t~
/16/).
---> 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 ' d ~FA ' ~ )
~
--,y
__~.( ~ A , d ~ A , d Y i) I
'
covariant formulation
~(T,B
y,
i I
l
cevariant formulation
l
G-EP
&)
~ minimal coupling
~I~
~/~ 2/~" ~ total lagrangian
,
G = T~e S0(3,1)
.,, ,, .,, ~"'( "r ' D "r ' DY )
~/~
total lagrangian
"I
o /
I matter + field equs. LA = 0
,
c;k= -s~ ~ ,
solutions on
(~)
,
(~)
l~ (
~)
,
,
aff. matter + field e q u s .
~I~ 12
^
p; = -t,, I
LOM
=
LA
I
~(
~
,
~c. ~
"
, = -s i
~
"~
,
p;
solutions
on
AeM
(~)
,
,
(~'~)
three
types of invariances
iI
(~'~)
o
(~')
(Y=~)
A
= -t;
: L°M --~ L~M (~)
,(~'~) ,
,
g,
I
Riemann C a r t a n structure on M
3.e. Conservation laws in PGT According Passive
to Appendix
and active
gauge
A we consider invariance
and
gauze covariance.
type is the notion of 'diffeomorphism covariance'
Related
in a YMT : to the last
in GR which can be reformulated
in bundle language as follows. Let
(N,g,~)
and
let
Aut ~ L N
LN
, i.e. automorphisms induced by diffeomorphisms of
case
of
GL(4,~)
be a Lorentz manifold with metric compatible connection ~ Aut
tensor
LN
denote
fields,
the
which
subgroup
are
of
equivariant
'natural' N
automorphisms
cO of
In contrast to the
functions
on
LN
, the
182
treatment of Lorentz tensor fields (defined on the Lorentz subbundle has to take into account the non invariance of Similarly
for spinor fields.
covariance' arbitrary L$ N
Thus,
on the fixed bundle
Lorentz
bundles
L~N
L~N
, we have to translate lagrangians given on
Lg~N c LN
,
g' ¢ g
Take a Lorentz gauge invariant lagrangian
~(~,D
LSN
(i.e. invariant under
g' = v.g
g' # g
on
~'(~,D~
Lorentz
invariance
covariance (3.14)
of
~
splitting
~
the
in
account
the
to
PGT
@
of type
v 6 AutvLN
gauge
of
~'
the
'translational'
of
use
usually
=
invariant
h*~(h,~,h,D~
correspondence
of
gauge
implied
in
in
~ Aut LN
lagrangian
.... h,e)
between
AOM
the
via
the
by diffeomorphism
affine
on
A°M
formulation
diffeomorphisms
(Remark
covariance
lagrangians
gauge covariance
laws
the
h
is equivalent to the condition
A°M does
notion
of
diffeomorphism
(3.2));
i.e.
with respect not
fields
take
~
into Thus,
covariance
cannot
be
In fact, there are no of
PGT.
simplicity we carry out the discussion of differential identities on and
Given
such that
, g e Aut~LN
to gauge covariance on
However,
x-dependence
identities
A °M
,
Diffeomorphism
v~h
of a Lorentz LSN
.... ~ )
corresponds
conservation
on
transformations
automorphisms.
derived
of
~
.... v,e)
into a product
covariance
remember
differential
additional
and
h 6 Aut L~N
we
covariance
implicit
from
LN
, one easily proafs:
( # ,D # .... @ )
'translational'
translational
on
L~'N vial
uniqueness
(3.14)
Diffeomorphism
for all
diffeomorphism
,
( ~ . ~, ~ . D ~ ..... K . @ )
Z~(~,D¢
and
~ .... ®)
v*~'(v,¢,v,D~
v ~ AutvLN
(3.16)
to
=:
= ~* ~'~
< ~ Aut ~ L N
~
Returning
O
:
in the sence of Appendix A.3.).
~ $ ' on
implies
invariant and
(3.15) Proposition
M
L@N
there is a (non unique)
.... e)
.... @)
each
L~N
of
to
defined on
then means by definition
~$(,,D~
leaving
in
AutvL~N N
We define the lagrangian
(3.13)
By
, into lagrangians
This can be done with the help of the canonical 1-form LN
on
AutnLN
in order to get a notion of 'diffeomorphism
a 'projection' of natural automorphisms of
a Lorentz metric
L~N c LN )
with respect to
covariance
in
Hence, L°M the
for
instead sence
of
condition (3.16). It on
L°M
is
straighforward
now
to
derive
the
following
(cf. /16,17/):
Lorentz gauge invariance -
a)
of
~(~,D~,
O)
-
b)
of
~ f (D ~, @ , D O)
implies
Ds,,~ =
2 ~t~7
implies
Dclk
2 @ ¢ ~ ^ P~7
=
differential
identities
183
Dif feomorphism covariance --
cl)
of
~.
gives
Dt'
c2) -
of
~f
gives
Dp~
d2)
~o~
Q
and
P#' + E R ~ ; ~ (9~/~ c$ ~
and
W~ m~, A
- D~' ~A.~
d,'~ @ ~
=
where
QJ
curvature
forms of
(3.12.i).
co
~f
, and for a), cl)
L.~g. k _ Dt "
connected
identities
t 2 , s~ ~ by
with
a)
and b) coincide
t~
~
, =
via the
For empty space (vanishing
conservation
- IRJe,,. o % . ~ 6)~
:
are the torsion and
has to fulfil the matter equations
and spin ten'sots
~x~9~t~)~ % L~ g~k
:=
,
,
_~iJ~ =: ~¥ R~e,,~ O ' A
The canonical energy-momentum
t,w,
s$
i)
,
and
~
A
,
=: Ai Q$,k 69'A O ~
(9 k
tj, + ~ RJe;k
"~Y{ = ~f, - Q~'~. e ~ ,
c2).
A
QJ{kO ~
t~. =
dl)
are
=
s ~4,
:=
t,~ ~ O ~ field
2 t,k% (p.:~%L,~%_~_~ ~A g D0 s, k = s ,.~,
,
equations
in
torsion and curvature)
~ (~'~'
The
also
cl) and
(3.12),
they reduce to the usual
laws of angular momentum and energy-momentum.
4. Concluding remarks
We have not touched question
of
lagrangian of
here
quantization ~f
(unphysical
singularities,
the whole
and
with
the
solutions
discussion
of
complex choice such
Newtonian
of problems
of as
ghosts
limit,
fibre
bundles
viewpoint,
we
structure
and
gave
elements
accepting a
consequently
coordinate
of Poincar6
free
gauge
the
with the
meaningful
tachyons, problem,
field
existence
teleparalle-
Instead, using the language general
formulation
theories.
or
Cauchy
lism /18/, etc.; see e.g. /8/ and refs. given there). of
connected
a physically
In
of
classical the
basic
particular,
Yang-Mills geometrical
we investigated
the questions listed in (i.I) : Thus, theory,
it turns out that gravitational
if at least formally
of Minkowski way
ground
metric as
relativity given
I)
starting
a
~
then
on
the
description
gauge
forms
~
of
= ( ~ ' , ~'~)
on
This
frame
symmetries
(affine)
of the geometry
eye" , F~; w
for horizontal forms
affine The
group
This is possible in a
a (non observable)
'arena'
invariance
Dynamics
potentials'
with
orthonormal
geometrical
'global' A°M
of connection
:= iekl
time
corresponding
underlying
imply
'gravitational
~k
space ,
rigid
as 4-forms
introduction usual
space time Poincar6
space as an internal gauge group (3.a.,3.b.).
natural
serving
theory may be seen upon as a YM gauge
one treats the (external)
come A°M
matter
flat
back-
bundle
A°M
of
special
lagrangians
into play
via
the
which replace the
structure
can be derived
184
from G-equivalence equivalence minimal as
a
coupling
natural
and
for
and
(Def.
on
to
A°M
comparison
(3.e.).
The
equations,
of
Lorentz
remove
subgroup,
a
PGT
reduction fields,
of
to
A°M
via
Y
treating
on of
uniquely
the
as
the
L ~M
translational
the metric
of Lorentz subbundles of
g
diffeomorphisms on
gauge
they allow
in/covariance
of the affine
the Poincar6
unphysical
of
field to
its
fields'
and
We thus end up with
¢)
g-orthonormal
components the
group
'Goldstone
'Higgs field'.
through
A°M
'fibre
information on the physical
of
c A°M
the
on
(global)
Moreover,
local
the derivation
breaking
Y
bundle
as
between
with
replacing
, introduced
(Remark (3.2)).
after
symmetry
, thereby
translations
covarianee
as the symmetry breaking
L~ M
giving
on
and hence,
formally [
fields'
time
they bear no additional
them
the zero section formulation
space
diffeomorphism
consideration
we
'shifting
of YMT
, show us the correspondence
On the other hand,
system under
to all kinds
express
'local fibre translations'
a
(2.11)), which reflects the idea of Einstein's
applies
procedure.
vehicle
translations' M
principle
principle
co
of
~
frames, , the
'deformation'
L~M
the
tetrad --9 L S M
LM
Appendix
A. Lagrangian densities on G-bundles
A.I. Notations i) (pseudo-) =
Consider
an
metric
g
~ d e t ( g p v )' ~ 4 A
basis on Let
%
{E ~}
U ¢ M :
of
n-dimensional, of
.. ^ ~
,
with cobasis
P --~ M
{ ~
~
--9 9 ~ A f 9
{fA}
,
{f~]
is denoted by A~(P)
vertival
be
G
M
; we set
P
~
:
F ), we write
.~I
~
=-~7~
,
with form
~>
of the Hodge star operator on
:
representation
G × F ---> F ~l :
~F
F ---> @ ~ ~ F of
~
,
on
fA
= K ~
;
(with basis
,
~---> 3~I~A
The contragredient
,
(M,g) F
--->F
{E~}
(E~,fA) ~-E ~ ~ f@ '
representation
9~
the
space
of JR-valued
X • TP ) q-forms on
~ e A h~(P,F,
M
volume
with fixed basis and cobasis
P
horizontal
and let
(i.e. horizontal and equivariant) q forms on Each
manifold
corresponding
.
of
or, equivalently,
for the corresponding
Let
of
connected
denoting a positively oriented coordinate
Given a linear representation
I
~
and
be a G-bundle over
the Lie algebra
and cobasis
of
and
(r,s)
{~I
' Z ' denotes the natural lift to ii)
oriented
signature
~)
then
has
a
local
(i.e.
Ag(P,F, ~) P
~ (..,X,..)
=
0
for
be the space of tensorial
of transformation type
representation
~
=
(F, f)
~ A (9 fA
'
185
A ~]%...~ ~
A
~=
~
.. & ~ / ¢ ~ 9
Moreover,
~
defines
tensorial
a
q-form
'( of type
(~
iii) on
P
~ F,Ad*®y)
If
D
and if
i.e.
D~A
= d~
is the covariant derivative belonging to a given connection ¢o
~
A~(P,F, ~)
= d~A + w ~ , ~ y ~
+ £~ o~bA c h iv)
~&
,
we have
~~
and
'
D~ D~
, with structure constants
For
4 e A~(P,F, ~ )
a vector
iv~
6 Agh4(P,F, ~ )
via
the form
= d~+
being an arbitrary vector field on
P
¢o a ~I(~) ~
A
= doa + ~~ ~oAad(~ )
field
c&
of
v
on
M
'
9&+~' (P,F, ~) i • e.
,
=
determines
on
P
i v ~ (X~ .... X9_4 ) := # (~,X~ .... X~_~)
,
which projects onto
v
, ~..~ = v
A.2. Lagrangians i)
A lagrangian density on
.., ~c-A~P(P,H,-c)
is a mapping
(A.2.1)
~
: A[tP,F,~)
( fulfilling
~ ( ~ . . . . ~F),
C" f u n c t i o n of
p
and
For s h o r t , we w r i t e ' on
km ( ~ , D ~ ) P
'
,
'
=
in
~ . . ~ A[(P,H,-U)
~
....
4 .... ~
~ 6 A~(P,F, ~ )
and all
~
:
, if
A~(P)
~--~
~(~
,
p e P
.... ~) , where
L
is a
there i s given a covariant d e r i v a t i v e
A~(P,F, ~ )
A9.4 × .. a ( P , F , y ) ---> 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,
,
@ = ($;) :
@; =: e ~ =: e ~ d x ~
0
TM - - , TM
We call
+
l-form
'translational'
in (B.4) a local section
forms
~'
+
, the
on Lorentz
fields')
definition
determines (B.5)
l-form
= ~<j¢
For ~
('tetrad
endomorphism on
have
AM
bases
~ ~:
LM
a connection
for connections
2(~;~) <
1-forms
a
,
is the fibre
on
denotes
=
on
same type as the canonical
gives
T~
(p+te,eg)
decomposition l~ ~
with
~
usual
(O,e]
.~
(B.4)
as
(t{)e AM
~
=
the
e ~-~
, we get a corresponding
~;~
=
(n,IR) valued
form
~.~
where
part
~
/19/, i.e. a connection
(B.3)
AM
t ).
, and
AM = TM × ~ L M
ii)
i
, e LM
eg
=
regular
~'~
each
v = ~f(v)$~
on
M --~LM M
, which
0
also
determines
~ - ~ v* = e£(v)$~
if this endomorphism
yields the local
In particular,
can be understood a
(i.e.
vertical
:
LM
--~
LM
,
is an automorphism.
(~)g
~ - + (el)g
bundle
a (l,1)-tensor
a vertical bundle automorphism T ~
~=
,
Then
@
188
g ~ GL(n,~)
, e~(e~) := ~ k
, which fulfills
(B.6)
Thus
"CO , ( 0 ) iv)
~o
we
have
objects on
LM
in (B.3) the
is called
following
and objects on
an affine connection
canonical AM
=
i-I
if
correspondences
~
= ~
between
in (B.4). (pairs
:
((l,l)-tensor, linear connection) @-9 generalized affine connection (B.7)
(~
@ AutyLM , linear connection) 6-~ regular general, aff. connection linear connection @-~affine connection
References
/i/
Weinberg, S.,
"Gravltation and Cosmology", Wiley, New York (1972)
/2/
Utiyama, R.,
/3/
Kibble, T.W.B.
/4/
Sciama, D.W., "Recent Developments in General Relativity", Pergamon, Oxford (1962), p.415
/5/
Cartan, E.,
/6/
Bregman, A., and Hayashi, K.,
/7/
yon der Heyde, P.,
/8/
Hehl, F.W., in "Cosmology and Gravitation", eds. P.G. Bergmann and V. de Sabbata, Plenum Press, New York (1980)
/9/
Trautman, A.,
/i0/
Straumann, N.,
/ii/
Petti, R.J.,
Gen.Rel.Grav. ~, 869 (1976)
/12/
Pilch, K.A.,
Lett.Math.Phys. i, 49 (1980)
/13/
Hennig, J.D., and Nitsch, J.,
/14/
Ivaaov, E.A., and Niederle, J.,
/15/
Drechsler, W.,
/16/
Kopczynski, W.,
/17/
Schweizer, M., in "Cosmology and Gravitation", eds. P.G. Bergmann and V. de Sabbata, Plenum Press, New York (1980)
/18/
M~ller-Hoissen, F., and Nitsch, J.,
/19/
Kobayashi, S., and Nomizu, K., Vol.l, Wiley, New York (1963)
Phys.Rev. I01, 1597 (1956) J.Math.Phys. 9, 212 (1961)
Ann.Sci.Ecole Norm. Sup. 40, 325 (1923) Ann.Phys. 75, 562 (1973)
Phys.Lett. A 58, 141 (1976)
Sympos.Math. 12, 139 (1973) Lect.Notes Phys. 150, Springer, Berlin (1981)
Gen.Rel.Grav. 13, 947 (1981) Phys.Rev. D 25, 976 (1982)
Ann. Inst.Henri Poincar6 A 37, 155 (1982) J.Phys. A 15, 493 (1982)
Phys.Rev. D 28, 718 (1983)
"Foundations of Differential Geometry"
of)
SUPERMANIFOLDS AND BEREZIN'S NEW INTEGRAL
Yuval Ne'eman
Tel Aviv University Tel Aviv, Israel
+
also on leave from the Physics Department, University of Texas, Austin, Texas
*
also Ministry of Science and Development, Jerusalem, Israel
190 1.
Introduction F e l i x Aleksandrovich Berezin (1931-1980) made important contributions to
d i f f e r e n t i a l geometry and i t s applications to r e l a t i v i s t i c quantum f i e l d theory. In p a r t i c u l a r , although there may have been some mathematical antecedents, he pioneered the use of Grassmann algebras as the key Building block for Fermion f i e l d s and in the generalized functional i n t e g r a t i o n . over anticommuting variables de ~ = o
,
Berezin's integral [1]
e~ ~ e ~ de B = 6 ~B
(I,I)
played an important role in the successful renormalization of the Yang-Mills f i e l d , which required anticommuting a u x i l i a r y ghost f i e l d s in the Feynman path integral as layed out by Faddeev and Popov.
In l a t e r work, he was the f i r s t
(with G.I.Kac) to introduce supergroups [2] and his c o n t r i b u t i o n was seminal for the b i r t h of supersymmetry.
With Leites, he defined supermanifolds [3] and the
corresponding Jacobian-like superdeterminant has been named the Berezinian. Developments in gauging supergroups
-
e s p e c i a l l y supergravity
important to perfect the a n a l y t i c a l t o o l k i t over supermanifolds.
make i t In the l a s t
paper Berezin wrote [4], j u s t before his recent t r a g i c death in a boating accident, he made several suggestions in this context.
I shall devote most of
the present a r t i c l e to these suggestions. Supermanifolds were introduced from two points of view.
De Witt [5] used
the d i r e c t approach, i . e . a manifold with commuting and anticommuting variables. The Salam-Strathdee superspace [6] r e a l i z a t i o n of the super-Poincar~ group followed this approach.
Berezin and Leites [3] and Kostant [7] , on the other
hand, took the algebraic geometry viewpoint, reproducing the supermanifold's geometry through the algebraic structure of i t s sheaf of functions. approach is sometimes named "graded manifolds".
This
M. Batchelor [8] showed the
equivalence of the two approaches and determined the global properties. A. Rogers [9] introduced a generalized norm and extended the realm of possible global and local structures, including n o n - t r i v i a l structures in the anticommuting piece.
191
2.
Generalized superdifferential We s h a l l
open-ball finite
now o u t l i n e
like
Let Ap/q (U) c(x)
Berezin's
domain i n r e a l
forms
new l o c a l
p-dimensional
formalism
space.
[4]
.
Let
U c Rp
be an
~(x), x~U is
A function
if
{f = f'
i.e.
exterior
= f"
:
...
= o
be the a b s t r a c t
I x c boundary o f U}
Grassmann a l g e b r a w i t h
the number o f " g e n e r a t o r s " are i n f i n i t e l y
oi = l " ' ' q ,
differentiable.
(2,1) q
the d i m e n s i o n a l i t y
with coefficients
Thus i f
c(x),
C(x,O) ~ A p / q ( U ) ,
x c U. it
of
A1 ,
The
can be w r i t t e n
as
C(X,G) = cO(x) + c~(x)O i + c2. " " ~3(x)Q1,og We can use a g e n e r a l i z e d zi = xi
,
z p+j = 6 j
coordinate
i = 1 ...
p
, j = I ...
q
+ ...
(2,2)
i n a superdomain
(2,3)
C(z) ~ Ap/q (U ; z) The g r a d i n g (modulo 2) ~(z I). = o m(z l ) = I
, ,
i = 1 ... p i = p + I ....
For a change o f v a r i a b l e a)
(2,4) p+q
zi = zi(v),
contravariant,
we i n t r o d u c e
w i t h an ~ v e r s i o n
dz i =
~dv k ( ~
~(dz i )
= ~(z i )
two t y p e s o f d i f f e r e n t i a l s ,
of statistics
(i.e.
bosons o r f e r m i o n s )
zi)
~vk b)
(2,5)
+ 1
covariant, without inverting v i = ~(~ z k) ~- z k
the s t a t i s t i c s ,
~vl m(~- z k) = ~(z k) Thus, z i and dz i z , dz, dz ) .
dz i
q e n e r a t e a Grassmann a l g e b r a
Elements o f t h i s
superdifferential
exterior
algebra of differential algebra
(2,6) A2p+q/2q+p(U x Rp x Rq •
a l g e b r a w i t h p o l y n o m i a l dependence upon dz, dz are
forms.
This i s i n f a c t
forms o v e r a m a n i f o l d
the " h i s t o r i c a l "
Grassmann
g i v e n by the a b s t r a c t
Grassmann
Ap/q(U).
w(z) i s a homogeneous form o f degree ( m , n ) , o r an (m,n) terms a r e o f degree
m
w = z dz I -dz I
in
dz I
and o f degree
n
in
~z I .
form i f The
all
(1,1)
its form (2,7)
192 is general c o o r d i n a t e i n v a r i a n t . We have thus used two kinds o f d i f f e r e n t i a l s , derivative
o p e r a t o r in C a r t a n ' s e x t e r i o r d = ~ dz i a
,
d
but o n l y
is an e x t e r i o r
a l g e b r a sense
d2 = o
(2,8)
gZ l
locally, dw = o It
÷
i s a graded d e r i v a t i o n ,
i.e.
d(Wl.W2) = dw I We denote a p,q form such t h a t
w
w = p(x,e)
w°
w
an i n t e q r a l
form i f
there exist
zi,e j ~ Ap,q(U;z)
only a n t i c o m m u t i n g d i f f e r e n t i a l s
dz p de I . . . ~0q
dz I . . .
(2,11)
(y,q)
(2,12)
w1
+
is again an i n t e g r a l
form, and
each a t l e a s t one commuting d i f f e r e n t i a l wo
(2.10)
dw2
as a form i n v o l v i n g
c o o r d i n a t e system
w = wo
where
w2 + ( - i ) ~ ( W l )
can be w r i t t e n
In an a r b i t r a r y
(2,9)
w = dw 1
i s connected to
(2,11)
wI dy
i s composed o f summands i n v o l v i n g or
dq .
through
wO = p o ( y , q ) d y l . . . d y p dr] I . . . dn q = = p((x(y,n), ~(z/o//y/q) dz ]
definition
D(x/~//y/q)
d y 1 . . . d y p dn I . . . ~ n q
i s the B e r e z i n i a n s u p e r d e t e r m i n a n t [3]
An i n t e g r a l for
e(y,n)))
form can be i n t e g r a t e d
and the c l a s s i c a l
in [1] thus r e l a t e s I #hi = o
This s u f f i c e s
,
.
with existing
Berezin i n t e g r a t i o n to the c o v a r i a n t
(2,13)
definitions:
(1,1)
f o r de.
differential
conventional
Berezin's do,
I n i dn j = ~ i j
(2,14)
to d e f i n e S Po(Y, n) dY I . . . d Y p dq I . . -
dq q
which does n o t depend on
dz, # z ,
we i n t r o d u c e a u x i l i a r y
the choice o f c o o r d i n a t e s . To d e t e r m i n e the i n t e g r a b i l i t y fermionic
variables
gij'
o f a form in
independent o f the odd g e n e r a t o r s
eA2p+q/2q+ p (U x Rp x Rq ; z, dz, dz)
e ,
d-e ,
dz
193
We r e p l a c e the even g e n e r a t o r s , do ] = ~ dx j ~ji (2,15) Tx I = ~ ~ji
~eJ
•
so t h a t w ( x , d z , d e / e , d o , dx) = f ( x / e , g )
f(xlo,g)
and i f f w
i s an i n t e g r a l If
w
dx I . . .
(2,16)
dx p dO 1 . . . do q
2,17)
= f(x/e) form.
i s an i n t e g r a l
form and
w = dw I , we have 2,18)
w = ~ dw I = o Two forms
wI
o f degree
complementary i f integral
form.
ml,n I
and
mI + m2 = p,
w2
o f degree
n I + n2 = q
m2,n 2
and t h e i r
are s a i d to be
product
Wl,W 2
i s an
Example:
w I = dx I de I + dx 2 #o I ; w2 = dz I Te I + dz I ~0 2 (2,19) w I • w2 = dx I dx 2 Te I #e 2 the i n t e g r a l
o f the p r o d u c t o f complementary forms i s w e l l - d e f i n e d .
(2,18) t o d e r i v e a f o r m u l a f o r the i n t e g r a t i o n
by p a r t s ,
o = ~ d(Wl.W2) = T dWl.W 2 + ( - 1 ) ~ ( w i )
3.
Integration
a sub-supermanifold.
fi(x,o)
(2,2G)
o f the concept o f complementary forms i s the i n t e g r a t i o n over ( f i s even, ¢odd) be given by the e q u a t i o n s Let M ~ A p,q(U) ; ~j(x,Q)
= o
= o
;
fi
' #j s
A p,q(U) (3.1)
l~j~q
issSp
~ Wld w2
over a s u b - s u p e r m a n i f o l d
One a p p l i c a t i o n
M
We can use
il
II _ ~
= )
,
rank
@=0
SXk
~el
e=o
We now c o n s t r u c t a form oM = 6 ( f l )
...6(f~)
6 (¢i)
...
6(¢ 9 ) df I . - .
df~ de I dp 2 . . . T¢~ (3.2)
194 M and does not depend upon ambiguities
which depends only on the supermanifold in the s e l e c t i o n of the functions to
CM"
Now l e t
~i ' Cj"
w
be a form complementary
The i n t e g r a l w := ~W~M
(3.3)
M
is thus well defined and serves as the d e f i n i t i o n o f i n t e g r a t i o n over a subsupermanifold.
Indeed, in the d e r i v a t i o n of Gravity and o f Supergravity as Gauge
Theories on a Soft Group Manifold [10,11] (see [12] f o r the most precise d e f i n i t i o n o f the S.G.M.) on the f u l l
we f i n d the action as an i n t e q r a l over space-time
d i m e n s i o n a l i t y o f the SGM : R10
In these cases, we noted the i m p l i c i t
quasi-dynamical r o l e [10] o f the c o n s t r a i n t
functions r e s t r i c t i n g
the i n t e g r a t i o n to
have in one gauge f o r
N=I S u p e r g r a v i t y ,
p(H 1'3) = 6(e 1) . . .
M1'3 r a t h e r than
f o r G r a v i t y , R10/4 f o r N=I Supergravity.
6(84 ) 6 (El2)
~1,3.
...
With the above formalism, we would
6(z34) dE12 . . . dE34 ~el . . . ~e4 (3.4)
where
em
is a Majorana 4-spinor and z [ i j ]
are the 6 Lorentz group v a r i a b l e s ,
and the action would be w r i t t e n as A =
lw p(M 1'3)
w = W w =
W~w~ Here
Rab
=
(x I
Ra b
~v
= I w H1,3
...
~ ec
o
x 4)
dx ~
^ dx v
^ ed
• Cabcd - 4 i ~ v
is the Lorentz curvature,
the t e t r a d f i e l d s and
~ dx ~
R~
~m the q r a v i t i n o .
" f a c t o r i z e d " expressions [10]
as
de I . . . de4
^¥5 ¥a
The 2-forms
in those curvatures.
^ ea
~ ^ ~
Rab
i n v o l v i n g only space-time
the r e s u l t of Spontaneous F i b r a t i o n [10,11] d~ 12 . . . d -34=
(3.5)
the Rarita-Schwinger curvature,
,
i.e.
a p p r o p r i a t e Cartan-Maurer f i b r a t i o n c o n d i t i o n s , in
^ dx ~
Rm are the z ~.
In G r a v i t y , t h i s is
the equations o f motion impose
i.e.
the vanishing o f components
In Supermravity. t h i s is not s u f f i c i e n t ,
are in the q u o t i e n t o f the Super-Poincar~ group by the Lorentz
group, and we require in a d d i t i o n a rheonomy [13] c o n d i t i o n . same as our (2.17).
This is in f a c t the
We can now understand one o f the most basic d i f f i c u l t i e s the search f o r a geometric d e r i v a t i o n of Supergravity. Lagrangian, the
ea
e differentials
would be o f the
thus y i e l d n o n - i n t e g r a b l e Lagrangians. appeared in a d i f f e r e n t form:
de
encountered in
In a g e o m e t r i c a l l y derived type, i . e .
commutative, and
To the workers in the f i e l d
the Berezin i n t e g r a l
this difficulty
(1.1) reduced the
195
d i m e n s i o n a l i t y , so t h a t the s u r v i v i n g expression would have to be a a p p r o p r i a t e terms a r i s e in the natural geometric expressions.
4-e term.
The problem was
resolved by e i t h e r the method in (3.5) or by using the "measure" i t s e l f Lagrangian in a constrained system [14] as in the S t r i n g . t h i s is in f a c t an expression in managed to evade the d i f f i c u l t y
do
, the Berezinian.
as
From (2.13) we see t h a t In both cases we have
caused bv the confusion between the geometric
derived through the e x t e r i o r calculus and the c o v a r i a n t
No
de
de
e n t e r i n g the Berezin
integral.
4.
T r e a t i n 9 commutative d i f f e r e n t i a l s
as v a r i a b l e s
Berezin's approach is to regard commutative d i f f e r e n t i a l s
as p l a i n commuting
variables, replacing d~i = ~i
(4.1)
~x i = ~i One may thus define p s e u d o - d i f f e r e n t i a l forms w(x,~,~/e,dx,d@) a A2p+q/2q+p(U @ Rq @ Rp) these can be r e w r i t t e n as d i f f e r e n t i a l A finite
forms i f
p s e u d o - d i f f e r e n t i a l form ( p . d . f )
go to zero with a l l t h e i r d e r i v a t i v e s for
they are polynomial in has the c o e f f i c i e n t s of
oi ,dx i ,~-ei
[ ~ I ~ I ~ I -~ ,x÷F .
I t is thus possible to extend the system by i n t r o d u c i n g o r d i n a r y d i f f e r e n t i a l s o f the a u x i l i a r y v a r i a b l e s boundary values.
Let
w
~,~
and d e f i n i n g i n t e g r a t i o n , using the above
be a f i n i t e
p s e u d o - d i f f e r e n t i a l form of maximal degree
w = Wo + w 1
(4.2)
w = p ( x , ~ , ~ / e ) dx I . . . dx p Te I . , . ~e q o wI
is o f lower degree.
The i n t e g r a l on
U is defined as f o l l o w s , (4.3)
~w = ~wo = ~p(x,~,~//e) dPx ~qe dP~ dq~ ~,~
where a l l
d~ , d~
=
-~
i n t e g r a t i o n s are taken from
not depend on the basis in With t h i s d e f i n i t i o n , differential
forms.
-~
to
+~
This i n t e g r a l does
Ap,q(U). it
is again possible to define complementary pseudo-
The r e s u l t w i l l
resemble ( 3 . 3 ) , with
w such a p . d . f . ,
and
196 PM the same as ( 3 . 2 ) . to
d~
and from
(2.20), i . e . 5.
~
Using e x t e r i o r d e r i v a t i o n ( j u s t as we used i t
to dE) on p . d . f . ,
to go from
we can reproduce the analog to (2.18) and
i n t e g r a t i o n by parts and a f u r t h e r g e n e r a l i z a t i o n of Stoke's formula.
Scalar products Berezin has provided [4] the necessary d e f i n i t i o n s f o r a scalar product,
possibly h e r m i t i a n
.
This involves metric tensors
grading given by the sum of the indices gradings. Two duals ( i n the s p i r i t ~w(z;t,s)
w(z;t',s)
, ~ = sdet11~iklj
i . e . an anticommuting d i f f e r e n t i a l v a r i a b l e , with
d-t'
the
~
(4.3).
i = p+1 . . . .
for
Similarly,
dt
s i = d-zI , i . e .
i = 1...p
ds I . . . dSpds p+m . .. ~Sp+q
becomes an a u x i l i a r y
commutative d i f f e r e n t i a l
changes o f v a r i a b l e s in
p.d.f..
6-operators,
with
* =* 6 , i ii
~ - like
which becomes
i = l...p,
~
ds i ;
the commutative for
~
i = p+l . . . .
v a r i a b l e of p+q,
si
is
v a r i a b l e , with a B e r e z i n - i n t e g r a b l e a n t i -
~s.
Like the e x t e r i o r d e r i v a t i v e 6l a n d
t i = dz i ,
(also anticommutative and
is a commutative d i f f e r e n t i a l
for
o-like
(5.1)
again anticommutative and i n t e g r a b l e as in
( 4 . 1 ) , with an ( i n t e g r a b l e ) anticommutative an anticommuting a u x i l i a r y
dt p+q
are the density superdeterminants;
p+q i t
v a r i a b l e of ( 4 . 1 ) , with
i
... dt p dtP+l..,
d-t i t s Berezin-type c o v a r i a n t d i f f e r e n t i a l
integrable); for
with t h e i r
o f Hodge's) are defined,
= o-½ ~ [ e x p ( i ~ t k g k l t k ' ) ]
g = sdetlbikll
on U,
All forms here are p . d . f .
~-½ IIexp(i~skbkl#')] w(z;t,s') ~ w ( z ; t , s ) = (g) Here
~ik
gik'
~
d,
the two d u a l i t y operations commute with a l l
I t is also possible to adapt two c l a s s i c a l
i = 1,2.
(5.2)
The s c a l a r product is given by (w
, w') =
f w ( * * w') 12
(5.3)
which takes the form (using (4.3)) (f,g) f,g
= f ( ~ [ e x p ( i s~)] f ( x , s )
g(x,~) ds d~) dx
the functions o f two real v a r i a b l e s .
To achieve p o s i t i v e d e f i n i t e n e s s we have
to use a n a l y t i c a l c o n t i n u a t i o n f o r t h e commutative v a r i a b l e s . (5.1) we can continue in
t,s
when these coincide with
~
or
Using the n o t a t i o n in ~ .
We imbed the
197 a c t i v e a n a l y t i c a l l y continued Grassmann manifold of forms (or a u x i l i a r y v a r i a b l e s )
(z;t,s)
Ac
~
A
2p+q/2q+p
(5.4)
(z;t,s,{,~)
3p+q/3q+p
where the l a r g e r manifold has an i n v o l u t i o n , and the v a r i a b l e s tI
II~ ik
i = 1...p,
t i = ~i,,
t i = (ti) *
for
i = p+1 . . . .
p+q
s~i = (si )*
for
i = 1.. .. p ,
s i = ~i x
si where
independent f o r
independent
for
i = p+l,
5.5)
t i = ~Ti
. . . p+q ,
( )* denotes complex conjugation.
(x/e)ll
,
s i = ~i
=
(w,w')
#(g 9) -½ dPx ~ q e #exp(-zt i gik
.w(x/e;t,s) w'(z/e;{,s)
and
I f the metric matrices l l g i k ( X / e ) l
are hermitian and become p o s i t i v e d e f i n i t e f o r {k _
e = o,
si ~ik . .~k ).
(s.6)
dP+qt dP+q{ dP+qs dP+qs
which in simple cases reduces to (f,g)
=T (#[exp(-s s)] f ( x , s )
g*(z,s) ds ds) dz
Such duals and products have been u t i l i z e d
(5.7)
in s u p e r g r a v i t y e t c . by Kallosh and
others.
6.
C h a r a c t e r i s t i c classes Lie superalgebra valued connections
and curvatures Aa(z;dz,~z)
Fae
and
m(Fa) = m(ea). superalgebra fields,
(with
ea
the a l g e b r a i c basis)
can be generalized to p s e u d o - d i f f e r e n t i a l forms
F~(z;dz;dz)~
where the gradings
~(A a) = ~(e a) + i
,
Such a system can be constructed, f o r instance, using the Lie su(2/1),
so t h a t some connections are odd one forms
as in the usual Yang-Mills case)
gauge f i e l d s )
Aaea
[15]
(even gauge
and some are even one forms
(with ghost
On a generalized supermanifold, one can p o s t u l a t e a Bianchi-
like identity dF = IF,A] where [ ] are the superalgebra Lie brackets. nomial
P(L)
(6.1) Taking an e v e n - c o e f f i c i e n t s p o l y -
on the Grassmann envelope of the superalgebra, i n v a r i a n t under the
a l g e b r a ' s action
( i n the a d j o i n t r e p r e s e n t a t i o n ) ,
we w r i t e a p . d . f .
198 (6.2)
w = P(F) closed by (6.1) dw = o In
SU(2/1)~
(6.3)
for instance, we can use the supertrace
w2 = P2(F) = s t r F2
,
dw2 = o
(6.4)
which is what we need to define Chern and Pontrjagin l i k e characteristic classes.
Bi bl i ography [1] Berezin, F.A., The Method of Second Quantization, Academic Press, [2]
New York-London, 228 pp. (1966). Berezin, F. A. and Kac, G. I . , Mat. Sbornik Eng. t r a n s l a t i o n : i i ,
[3]
8__22,pp. 124-130 (1970).
pp. 311-326 (1970).
Berezin, F. A. and Leites, 0., Soy. Math. Dokl. 16, pp. 1218-1222 (1975); Arnowitt, R., Nath, P. and Zumino, B., Phys. Lett. 56B, p.81 (19751.
[4] [5]
Berezin, F. A., report ITEP-71, Moscow (1979). De Witt, B., Bull. Amer. Phys. Soc. 20, NTO (1975)- D i f f e r e n t i a l Supergeometry, in p r i n t .
[6] [7]
Salam, A., and Strathdee, J., Nucl. Phys. B76, p. 477 (1974). Kostant, B., in D i f f . Geom. Math. in Math. Phys., K. Bleuler and A. Reetz eds., Lect. Notes in Maths 570, Springer Verlag, Berlin/Heidelberg/New York. pp. 177-306 (1977).
[8]
Bachelor, M., in Group Theoret. Meth. in Phys., W. Beigelbock, A. Bohm and E. Takasugi eds., Lect. Notes in Phys. 94, Springer Verlag, Berlin/Heidelberg/ New York, pp. 458-465 (1979).
[91
Rogers, A., J. Math. Phys., 2~i, pp. 1352-1365 (1980); 22, pp. 443-444 (1981); 22, pp. 939-945 (1981).
[10]
Ne'eman, Y. and Regoe, T., Phys. Lett. 74b, 54 (1978); Riv. Nuo. Cim., Ser. 3,_1, #5 (1978).
[11]
Thierry-Mieg, j . and Ne'eman, Y., Ann. of Phys. (N.Y.) 12__33,247 (1979).
[12]
Ne'eman, Y., Takasugi, E. and Thierry-Mieg, J., Phys. Rev., D22, 2371 (1980).
[13]
D'Adda, A., D'Auria, R., Fr6, P. and Regge, T., Riv. Nuo. Cim. Ser.3, 3, #6 (1980).
[14] [15]
Wess, J. and Zumino, B., Phys. Lett. 74B, 51 (1978). Ne'eman, Y., Phys. Lett. 81B, 190 (1979). Ne'eman, Y. and Thierry-Mieg, J., Proc. Nat. Acad. Sci., USA, 7_77_,720 (1980) and 799, 7060 (1982).
SPONTANEOUS
COMPACTIFICATION
S. Institut
fur
AND
FERMION
CHIRALITY
Randjbar-Daemi
Theoretische
Boltzmanngasse
Physik, 5,
Universit~t
Wien,
Wien
Austria
and Institut
fur
Theoretische
Sidlerstr.
I.
INTRODUCTION
ries.
A gauge
theory
gauge
fields,
Fermi
The
successful
a principal red
to
back,
as by
fibre the
the
G.
The
fermions
and
the
basic
them.
the
* Present This
gauge
address
report
P
the
This
fermions
and
therefore
are
supposed
of
the gauge
scalars.
whose
to
physics
three
scalars
are
to be
fields
models
there
of
shall
support more
field
group
manifold we
based
of b a s i c
G
normally
is w h a t
they
gauge
A gauge
one
base
are
types
structure
In p h y s i c s
P,
role
Bern,*
Switzerland
particle at m o s t
and
group.
section,
In n o n - a b e l i a n
between
fields
bundle
The
with
of
contains
connection.
field".
associated
Bern,
Universit~t
MOTIVATION
models
gauge
a local
racterizes "gauge
AND
5,
Physik,
the
sections linear
to
transmit
is a l s o
is u s u a l l y the
theoi.e.
l-form mean
the
the
cha-
the word
of v e c t o r
than
refer-
which
by
in
pull-
bundles
representations
fundamental
is
gauge
fields,
is a c o n n e c t i o n
employs
also
on
of
scalars
forces
between
self-interaction
fields.
;
is b a s e d
on
joint
work
with
Abdus
Salam
and
J.A.
Strathdee
200
The most
important
function
of the scalars
breaking,
a phenomenon
group
G
to one of its subgroups
making
the bosonic
solution fibre
which
type
opens
equations
consequence
scales
the way to the
of several
mass
This
partial
H.
is to induce
the r e d u c t i o n
This
happens
to admit
structure
dynamically
a suitable
of an a s s o c i a t e d
symmetry
of the
i.e.
by
solution,
bundle with
a
the
G/H.
energy
following
with
of m o t i o n
is a c r o s s - s e c t i o n
An important physical
identifiable
so called
scales
is not
of symmetry
and the vector
hierarchy
in the model
enumerate
resolution
of the
b) N o n - u n i q u e n e s s
of G1e.g.
c) A b s e n c e
S0(18),
of the
e)
problem.
Generation
To c l a r i f y
(d) and
but
of also
the o c c u r r e n c e
explained.
builders.
shall
In the
aim towards
a
couplings
in grand
unified
of scalars. models
G may be
etc.
of the g r a v i t a t i o n a l
d) C o m p l e x i t y
same effect
two only.
and n o n - g e o m e t r i c
S0(I0),
of model
others
a) N o n - u n i q u e n e s s
SU(5),
The
problem, n a m e l y
problem
several
last
is the a p p e a r a n c e
not yet n a t u r a l l y
the only u n s o l v e d
we shall
breaking
b o s o n masses.
fermionic
interactions
from the picture.
representation.
(e) we recall
that
in the
"standard
model"
G = SU(3) c X SU(2) L x U(1)y where
SU(3) c d e s c r i b e s
les and SU(2) L x U(1)y fied e l e c t r o - w e a k fermion
masses
Any = ~R handed
strong
At
~%'
Dirac
where
and out
high e n e r g i e s
them by two c o m p o n e n t
spinor
~R
projected
of the c o l o u r e d
~
~L
~
one
Weyl
may be d e c o m p o s e d are,
from
partic-
group of the w e l l - e s t a b l i s h e d
sufficiently
and r e p r e s e n t s
fermions
interaction
is the gauge
theory.
4-component +
the
respectively,
by the p r o j e c t i o n
uni-
ignores
the
spinors.
into a direct right
and
sum
left
operators
(I_+ y5 ) , i.e. I
(i)
Y
Now,
the c o m p l e x i t y
given tions of
~
+
R~ : ~(I+-x5)~ of the
the c o r r e s p o n d i n g G.
For instance,
¥5
:
fermionic ~R
and
consider
¥5'
2
Y5
= +I
representation ~L
belong
the e l e c t r o n
means
that
to d i f f e r e n t family.
for a
representa-
In the
standard
201
model
e L accompanies
SU(2) L
(i.e.
charge
of
I. W h i l e
non-trivially e R which tations
the
I = I/2),
neutrino
U(1)y
constitute This
the
with
of
of
to f o r m
a doublet
SU(3) c and
equalling
e = eR+e L b e l o n g
for all o t h e r
a Y-
transforms
to 2. T h u s
to c o m p l e x
fundamental
of
carries
SU(3) c x SU(2) L b u t
a Y-charge
observed
is true
~e
is a s i n g l e t
e R is a s i n g l e t
under
of G.
electron which
e L and
represen-
fermions
of the
model.
Finally onal
fact
plet
of
the g e n e r a t i o n
that
there
fermions.
are
problem several
For
example
model
observationally,
we can
same
G-representations,
has
simply
copies
in the
leptonic
identify
to do w i t h
of the
at
same
sector
least
the
observati-
massless of the
three
G-multi-
standard
copies
of the
namely
I
t
(2a)
I)~)e e
(~)~ L
(2b)
eR
Similar
structure
In the ken
~R may
standard
for granted.
4
L
L
TR
be w r i t t e n
model
this
It is not
for quarks.
rather
explained
bizarre
fermionic
pattern
by any m a t h e m a t i c a l
is ta-
or p h y s i c a l
principle.
One
of the
aims
cal e x p l a n a t i o n tegy
will
be t h a t
These
are
of
in a
sional
effective d
solution
them
original the
symmetries by
small
will
This
Z. H o r v a t h
group will et al
and
then
They
equations length
of B d w i t h
the o b s e r v e d in s e c t i o n
see
also
L.
Palla
spinor
to a 4 - d i m e n -
- by f i n d i n g a compact
at e n e r g i e s
be e x p l a i n e d [3],
reduced
a d so that
such
and
compactification
- to f o r m
for c o n s i d e r i n g
stra-
theories.
are m a d e
scale
topologi-
underlying
to Y a n g - M i l l s
by a d y n a m i c a l
be u n o b s e r v a b l e
motivation
isometry [I].
field
The
compactified
space-time theory
a differential
spectrum.
coupled
dimensions.
of the
an a p p r o p r i a t e l y
The
gested
of g r a v i t y
(space-like)
with
is to p r e s e n t fermion
spontaneously
low e n e r g y
the e x t r a
associated
talk
(4+d)-dimensional
propriate
sociate
this
peculiar
the
theories
fields
B d with
of
of this
an ap-
manifold
the
excitations I s m a l l e r t h a n -- . ad
a scheme
was
4-dimensional 2.
of
It was
[15],
that
to asgauge
however one m a y
sug-
202
invoke
the
differential
an e x p l a n a t i o n dimensions. the of
It is p r e c i s e l y
Yang-Mills the
in the
relevant
vanishing
values
of the
Schwinger)
operator
zero m o d e s It
gauge will
mensional lity
The
counting
large
group
struct
b y the
formations.
2.
Kaluza
of for
for a unified
In a m o r e This
simply
taining
then
by t h e s e
(or R a r i t a -
in the b a c k g r o u n d of
also
of an
zero modes.
of the
group
in
(4+d)-
the observed
explain
their
the m a i n
non-trivial zero m o d e s
features
example
we
4-di-
stabi-
shall
such
a
in s e c t i o n
shall
in b a c k g r o u n d s We
of
framework
illustrate
which
admit
a
not
attempt
to c o n -
of
outline
the
constraints
imposed
on o u r
invariance
also
under
concludes
local
supersymmetry
trans-
the p a p e r .
COMPACTIFICATION
some the
the
internal
appropriate first
"internal"
t i m e attempted
theory
of g r a v i t y
sense
writing
symmetries
one
down
Yang-Mills
starts a
space
dates
back
and electromagnetism
from
based
physics to
to f i n d a g e o m e t r i c a l
"physics"
(4+d)-dimensional
fields
of p a r t i c l e
on a L i e
I [ 4+d I/2 I S = ~ jd z g (- ~
19.21
setting
[I].
in 4 + d - d i m e n s i o n s .
action Group
integral
K and
I "~ -~MN + fermions R - ~ FMN F
+
con-
fermions,
i.e.
(3)
These
isometry
the L a g r a n g i a n
mathematical
transformations.
section
modern
gravity,
represented
model.
field
means
will
general
fermionic
i d e a of r e l a t i n g
to i s o m e t r i e s when
of
to i n c l u d e
in the n o n -
the D i r a c
to r e p r o d u c e
is to e x p l a i n
the
requirement
SPONTANEOUS
The
talk
4. we b r i e f l y
This
needs
find
in 4-
The n o n - t r i v i a l i t y
spectrum
choice
B d to
spectrum
deformations.
symmetry
a realistic
that
B d and
succeed
This
a relatively
of the of
classes
a rich
over
be r e f l e c t e d
representations
suitable
one will
one
Lagrangian.
known
afford
spectrum.
this
In s e c t i o n scheme
that by
by describing
3. By c o n s i d e r i n g the
can
bundles
fermionic that
B d will
space
to l i n e a r
space-time
a i m of
mechanism
field
continuous
reason
characteristic
in a c o m p a c t
fermionic
under
over
It is w e l l
belong
is h o p e d
dimensional
low energy
for t h i s
bundles
connections.
appropriate
of the v e c t o r
(4+d)-dimensional
vector
Yang-Mills
of B d.
topology
of the c o m p l i c a t e d
)
203
where
the
constants
respectively. dimensions. it c a n
be
The
Except
ways
or
for
assume
the
the
that
the
Now
from
Set
all
Find
part
(3)
B d by yb,~
der
equal the
makes
the
The
ansatz
for
the
Yang-Mills
of
up
If
"ground
state" of
the or
observed
of
ordinary to
the
naturalshall
continue
to
al-
in-
observable
[3]:
of
the
(4+d)-dimensional
a product
x m,
M 4 x B d,
this
should
m=0,],2,3
manifold
of
coordinate
and
those
chart
on
is a d a p t e d
have
be
in
accompanied
(3).
of
potential
transformation
physics
universe,
4-dimensional 10-parameter
with
i.e.
In the flat •
an-
invariant
un-
Then
is we
we
look
invariant always
gauge
identify
for
under
mean
group
this
then
a possible if t h e it
in-
K).
solution
it w i l l
group.
ground
space
is e i t h e r
following
Minkowski I
Polncare
a suitable
with
break
the
subgroup.
is m e a n i n g f u l
thereof.
the
theory,
to a s m a l l e r
solution
of
we
(4+d)-dimensional
this
by
ds 2 is
on M 4 x B d w h i c h a gauge
particle
theory
Assume
G 4 x G d in M 4 x B d.
equations
4-dimensional
approximation
identical
and
We
B d is a c o m p a c t
M 4 by
that
A
motions
gauge
the
of
part
ds 2 s h o u l d
S spontaneously
M 4 is t h e
origin
[2].
+ g~v(Y)dy~dy ~
for
of
with
(4+d)-dimensional
a good
because
integral
a 4-dimensional
and
on
assuming
invariance
Identification the
action
theories
(4+d)-manifold
potential
to a l o c a l
in a n a l o g y
symmetry
(4)
group
by
an
unspecifical
zero.
M 4 x B d we
Yang-Mills
variance
a
Then
iii)
to
space-time
ds 2 = g m n ( X ) d x m d x n
(where
of
bosonic
coordinates
structure
G x x Gd,
such
(length) d canonical
size.
the
= 1,...d.
a solution
(3)
and
~ is a c o n s t a n t .
compactification
of
small
a transitive
left
supergravity
spontaneous
fermions
denote
been
a length
4-dimensional
it.
which
product
(4)
satz
in
of
their
action.
for
equations
us
S has
kind
M 4 is a 4 - d i m e n s i o n a l
the
~
has
field
to
of
have
this
a solution
an a p p r o p r i a t e l y
dimension
possibilities
where
Let
the
fields
Rarita-Schwinger
S in
the idea of
i)
the
constant
logical
derived
ii)
b have
of
in h i g h e r - d i m e n s i o n a l
vestigate theory
all
fermionic
Dirac
ly a r i s e s
k and
Hence
we
flat
shall
space.
state
of
M 4 resembles
Then
Minkowski always
assume
G 4 will
be
204
It has b e e n such
shown
a compactifying
tains
sector
all
dual
s u b g r o u p of the o r i g i n a l
where
massive
B d and
tern
the m a s s i v e
the
the
follows:
There
will
of
always This
equals
The
number
unless
One
of
the b a c k g r o u n d
of the
features based
from
solution
which
gauge
the
final
the o f f - d i a g o n a l basis.
gMN(X,y)
are
Our shown
9y ~
the the
necessarily
is a
of m a s s -
they belong
be Y a n g - M i l l s
constructed
fields
cannot
to
gauge
on the
be p r e d i c t e d
under
some
distinguishes
the o r t h o d o x
compactification
com-
the
a prio-
supersymmetry
To c l a r i f y
ansatz
this
for the m e t r i c
4-dimensional components
Yang-Mills
of the m e t r i c
Kaluza-Klein
is the
structure
point we of the
recall
(4+d)-di-
potentials tensor
arise
written
B~ ( ( yx ))K {m
+ B( ~ ( x ) B y~ ( x ) K)~ ( y ) KmB ( y ) g ~ v
in
g~v (
of
g~v(Y )
~ : I .... , d i m Gd, continuous
4-dimensional
analysis
that
B~ (x).
pat-
=
= K~(y)
p~ementing B~(X)n
space
correct
B~n(X)K~(y)g~(Y) K@
the
One w r i t e s
gmn (x)
where
a d is
the n u m b e r
They will
is i n v a r i a n t
fields.
Kaluza-Klein
manifold
a coordinate
spin-zero
on s p o n t a n e o u s
spin-]
in the u s u a l
mensional
(5)
which
ad
In o r d i n a r y
Their
G d x K and
theory
I
to - -
[4].
the t h e o r y
solely
group.
4-dimensional
of m a s s l e s s
the m a s s l e s s
that
of t h i s
is a r e s i -
proportional
ones.
0.
invariant.
R. T h e r e f o r e
However, of
I and
where ~
solution
low e n e r g i e s .
dimension
on B d - c o n 2,
In g e n e r a l
2-field
graviton.
spins
about
background.
transformations
from
B d.
length
at
spin
the
masses
the m a s s l e s s
to the
effective
with
Planck
of
of G d x K,
have
invisible are
is the
representation
of a full
pactifying
are
oscillations expanded
fields
leaves
fields
he a s i n g l e
modes
adjoint
which
of the
small
harmonically
associated
modes
G d x ~.
spin-1
fields
ri,
modes
relevant
is as
singlet less
scale
of
multiplets K,
4-dimensional
to be of the o r d e r
energies
- when
irreducible
a d is a l e n g t h
assumed
spectrum
4-dimensional
They
The
into
the
solution
in its b o s o n i c fall
that
several
4-dimensional
isometric
Gd-Yang-Mills
spontaneously effective
are
the K i l l i n g
action
vectors
im-
of G d on B d a n d
fields.
compactified
Yang-Mills
fields
models cannot
have be
just
205
To o b t a i n
the e f f e c t i v e
neously c o m p a c t i f i e d (5) for Mills
the c o m p o n e n t s
group
contain must
of F ~n
transform
The
(3)).
exactly field
effective~
one
(massless)
should
write
of the K - Y a n g - M i l l s This
a f i e l d Ca(X)m w h i c h
a Gd Yang-Mills
B ~ (x) m
low-energy
background
ansatz
under
in the
fields.
for
same w a y
in a s p o n t a analogous
(K b e i n g
the K - p o t e n t i a l
an x-dependent•
in M 4. N o w o u r
theory
an a n s a t z
action
to
the Y a n g -
A(x,y)
will
of G d on B d
as B~(x) in eq. (5), i.e. as m c a n be f o r m u l a t e d as f o l l o w s :
result
4-dimensional
Gd-Yang-Mills
B(x)
are b o t h
field
is a c o n v e x
s u m of
a n d C ~(x) . m Geometrically
principal dicates as the
bundle
that
a certain
observable
From cessful
the
a n d C(x)
o v e r M 4. The
combination
Gd-Yang-Mills
aforementioned
compatifying
of B(x)
field
must
on the
of the b i l i n e a r a n d C(x)
should
same
Gd
spectrum
in-
be r e g a r d e d
in M 4.
description
scheme
connections
investigation
involve
it s h o u l d at l e a s t
be clear the
that
following
a sucthree
steps: i) F i n d i n g ii) iii) N o w we
shall
Investigating
briefly
the Y a n g - M i l l s
provided
b y the
desired its
Constructing
i) The m a i n of
the
on e a c h
under
equations
following
solution
stability.
an e f f e c t i v e
elaborate
problem
compactifying
(i)
on B d.
4-dimensional
one
is to In the
of t h e s e
theory.
points.
find G d invariant
solutions
interesting
these
are
integral
eq.
cases
theorem.
THEOREM: The Y a n g - M i l l s (3) a d m i t are
equations
of G d i n v a r i a n t
following
solutions
f r o m the a c t i o n
provided
the
following
conditions
satisfied: a
There
is a c l o s e d
to G d / H a n d b
H ~ K
c
H ~ K but
Under
condition
c)
the
subgroup
Riemannian
H of G d s u c h t h a t structure
B d is h o m e o m o r p h i c
of B d is G d i n v a r i a n t .
or H and K have
condition
f o r m of G d p u l l e d
the
back
b)
a common
the H - c o m p o n e n t
to B d s o l v e s
component
of
the
non-trivial of the
normal
canonical
the Y a n g - M i l l s
Maurer-Cartan
equations.
same M a u r e r - C a r t a n
form
subgroup.
Under
lying
in the
206
Lie
algebra
Mills
of the
equations.
of t h i s
With
these
number
As
that
stable. fects
subgroup
is a s o l u t i o n
0(5)-invariant
solutions of the
the E i n s t e i n
ii) known
normal
SU(2)
of t h e Y a n g -
instanton
solution
is
type.)
ly s o l v e a large
common
(The s i n g l e
of
It
standard
far as the
derived
solutions
equation
f r o m eq.
on B d one can easi-
(3).
T h u s we p o s s e s s
[6].
stability
is c o n c e r n e d
we k n o w v e r y
little.
5-dimensional
Kaluza-Klein
ground
is c l a s s i c a l l y
is h o w e v e r
[7].
Yang-Mills
equations
unstable
In 6 - d i m e n s i o n a l
under
quantum
state
mechanical
Einstein-Maxwell
theory
It is
tunneling
where
the
ef-
extra
two
•
dimensions
compactify
to f o r m an S 2 it has b e e n
x SU(2)
x U(1)-invariant
lution,
however,
ect.
is c l a s s i c a l l y
a spin-zero against
The
only
turbations
an
iii)
K : SU(2)
extra
tunneling
effects
example
which
0(5)-invariant
due
stable
from
(ii)
the possible
dimensions
instanton
effective
in the
in 8 - d i m e n s i o n s
is s e e n
with
to the n o n - t r i v i a l i t y
[10].
above
the
d is e x p e c t e d
to be
so-
O(n)
develop
of the K = U(1)
stable
under
case
small
compactified
on S 4. H e r e
again
instanton
theory
background
has
requirement
per-
to
the'stability
bundle
[10].
also been
in 6 - d i m e n s i o n s
for K c o n s i d e r a b l y . limited
same
K - U(n),
parturbations
to be
stability
candidates
The
K = SU(2)
4-dimensional
[8],
[8].
e.g.
is u n k n o w n .
of the
K = U(1)
stable
stability
is k n o w n
model
the P o l n c a r e
small
mechanical
only
can restrict
the the
The m a s s l e s s
constructed
As
because
that
group,
[9]. A l s o
other
is p r e s u m a b l y
gauge
mode
is an 8 - d i m e n s i o n a l
M 4 x S 4 with
is classical]y
in a l a r g e r
unstable,
tachyonic
quantum
solution
embedded
/
shown
and
in p r i n c i p l e The number
by s u p e r s y m m e t r y
of
require-
ments.
For
the k n o w n
been possible
as M 4. N o r m a l l y tion
turns
(ii)
and
out
(iii)
higher
to o b t a i n the
dimensional
compactifying
4-dimensional
to be an a n t i - d e have
been
carried
supergravity solutions
submanifold
Sitter out,
models
with
containing
space
[11].
partially,
it h a s
not
a Minkowski
Nevertheless
for the
space
the t i m e
direc-
the
/
supergravity. results
are
Here
hard
due
to the
to i n t e r p r e t
lack [12].
of P o l n c a r e
invariance
steps
11-dimensional in M 4 the
207
3.
FERMIONS
AND THEIR
As mentioned
STABLE
in the
introduction
r e n t i a l topological e x p l a n a t i o n sentations fiber
bundles
For are
of the
the
Dirac
the
~
of c l a r i t y o f the gauge
shall
assume
(7b)
F~
=
distinguish
two
S0(1,3+d).
where
y
=
invariant
¥J
(8a)
~
(8b)
I
~
L
point
representations
The m a s s
the
fermions some
in eq.
linear
(3)
repre-
e A is an o r t h o n o r m a l respectively
denote
basis
the
in M 4 x B d-
and consider
the
following
realiza-
~ = 5 ,... , 4 + d 4- a n d d - d i m e n s i o n a l of the
SO(1,3+d)
Dirac
invariant
representations
representation
of
is S O ( d ) - i n v a r i a n t
operator.
matrices
respective-
matrix,
of the
tangent
F-matrices
that
branch
into
two
I
?'
:
+
I
Y5 = - I
~'
=
-
I
= + I, m u s t x SO(d
Y5
=
+
to e m p h a s i s of
and
It is c l e a r
SO(I,3)
{R
The main
that
supporting
space
group
we have
Y5 x ~'
,
say
repreof
--- F4+d
5¥6...Y4+d
subgroup
fermionic topology
P A
inequivalent
chirality
SO(1,3+d), the
the
= FoE1
F= r
the
diffe-
a : 0,1,2,3
In the a b o v e
(7d)
of the
invoking
a n d AA,
¥5 x y~'
¥~ a r e
f
WA
I
two e i g e n v a l u e s
(7c)
find a natural
Thus
d is e v e n
matrices
= Ya x
Ya a n d
K.
connections
that
Pa
The
SO(I,3+d)
of M 4 x Bd,
(7a)
ly.
by
= p A ( e A + O A + A A) w h e r e
space
of the D i r a c
where
complexity
models
let us a s s u m e
group
group
and Yang-Mills
We
to
B d-
= PAP A_
tangent
Riemann
tion
sake
of the
in w h i c h of
over
we hope
of the
4-dimensional
spinors
sentation
CHIRALITY
is the u s u a l
an i r r e d u c i b l e irreducible
SO(I,3) spinor
pieces
of
under
,
is t h a t
~
SO(d).
operator
Y5
takes
the
form
L and
~R
belong
to i n e q u i v a l e n t
208
(9)
~L
where
~
i#a~a
operates
in the
operator
corresponding
fermions
in t h e
can
in g e n e r a l
For Mills
~
upon which labels
maps
general
expand
if for
ponding
some values
and
formula
~R
and
D n uniquely.
Singer.
and Yang-
of the
index
To c l a r i f y
this
set of b a s i s
func-
~L"
Here
n stands
The
Dirac
operator
i e. .
of n the D nR ( o r
D Ln (or D ) s h o u l d
(or ~ R ( X ) )
of t h i s m o d e
be
D L)
a zero m o d e
~L
(x,y) R
will
satz
(x) D L R
R
in M 4.
We c l o s e
section
by considering
in eq.
(3).
be M 4 x S 4 for the m e t r i c
cribe
an 0 ( 5 ) - i n v a r i a n t the e q u a t i o n
single
of m o t i o n
I ~2 ~ = 4
(12) where
R B are
orthonormal stanton
field
invariant an
the
frame
frame
dimensional
The to p i e c e s
according
n
coefficient
following
~ L(X)
compactifying
instanton
in S 4. W i t h
solution
of S O ( I , 7 ) .
Ricci
(y) d e s this
tensor
SO(I,7)
assume
N
S4 relative
on M 4 x S 4 . N o w is the
that
~
an-
to an
structure
supports
a
inis
let group (2 t + 1 ) -
of K = SU(2).
on S 4 i n v o l v e s [14]
of
F ~ the c o m p o n e n t s of t h e aD 4 on S . The w h o l e s o l u t i o n
representation
to the c h a i n
solution A
R2~ 2 6 8 ~B
of the
irreducible
Let
to
of P O l•n c a r e. x 0(5)
of
example.
(y), w h e r e
of M 8. We a l s o
expansion
the
a suitable
~ ~B with = F~B to the same f r a m e
spinor
unitary
harmonic
Then
R~B
components
bundle
. The
of M 8 a n d A = d y ~ A
reduces
:
the a c t i o n
irreducible
of the
~
cortes-
(y)
and F 2
relative
under
of
then the
n
~L
fermion
d = 4 a n d K = SU(2) will
: E n
be a m a s s l e s s
this
is m i s s i n g
in the e x p a n s i o n n
(11)
be
such modes
Riemannian
complete
[13]
to m a s s l e s s of
of A t i y a h
arguments.
the
of t h i s
n ~ n X~ n D L ..... > D R ..... > D L
(10) Now
the
rise
number
Gd-invariant
characterize
n--> n DL<__D L
give
The
theorem
theoretical
we c a n
which
will
index
n and D L n respectively by D R
on G d / H
= #~
the
Gd/H with
group
The e i g e n v e c t o r s
space-time.
can by-pass
simple
space.
eigenvalue
from
space
one
by using
for a set of
internal zero
be d e r i v e d
a quotient
let us d e n o t e tions
to
+ h.c.
four d i m e n s i o n a l
connections
theorem
~R
its d e c o m p o s i t i o n
in-
209 SO(1,7)
x SU(2)
---> SL(2,C)
x SU(2) B x SU(2) A x SU(2) K
---> SL(2,{)
x SU(2) 8 x S U ( 2 ) A + K
(13)
Here
SU(2) K d e n o t e s
gent
space
algebra
group
the g a u g e
0(4)
of SU(2) A.
gauge
of S 4. The i n s t a n t o n
transformations.
The
tangent
Finally,
space
spinor
of SO(I,7)
rotations
~L
With respect
~(2,1,
2t + I)
'
are
by
irreducible
with
harmonics.
The
With respect
into two p i e c e s
SU(2) B x SU(2) A x SU(2) K c o n t e n t
subgroup
are a c c o m p a n i e d
in S0(5)
is 8 - d i m e n s i o n a l .
to lie in the
to the d i a g o n a l
the p i e c e s w h i c h
x SU(2) B x SU(2) A it b r a n c h e s
(14a)
~L
to
and
~R"
is
~R
~
(1,2,
2t + I)
to SU(2) B x S U ( 2 ) A + K we have
(14b)
~L
~ (2, 2t + I)
(14c)
YR
~ (I, 2t) •
TO f a c i l i t a t e
(I, 2t + 2)
the e x p a n s i o n
the n o t a t i o n
we r e - e x p r e s s
[ml,m 2] of G e l ' l a n d
(15a)
~L
~ [t +1
(15b)
~R
~
The
is s u p p o s e d
with respect
to SU(2) B x S U ( 2 ) A + K can be e x p a n d e d
irreducible SL(2,C)
K and SU(2) B x SU(2) A is the tan-
It is i n v a r i a n t
of SU(2) A x SU(2) K in w h i c h
respect
group
are
(16a)
@
[t+~
labelled
(nl,n 2) and the e x p a n s i o n (nl,n 2) s u b j e c t
these
SO(4)
representations
in
Zeltin
, t-~]
[t-~I , t-~)
SO(5)-harmonics
and
, t+~]
by a p a i r of i n t e g e r s
of an SO(4)
piece
(or half
[ml,m 2] m u s t
integers)
include
all
to the i n e q u a l i t i e s n I 2 m I ~ n 2 21m21
The d i m e n s i o n a l i t y
of the S O ( 5 ) - r e p r e s e n t a t i o n
characterized
by
(nl,n 2)
is g i v e n by I d ( n l , n 2) = ~ (n1+n2+2) (nl-n2+1) (2ni+3) (2n2+I)
(16b)
It is e a s y to v e r i f y both
~L
and
~R
that the o n l y h a r m o n i c
which
fails to a p p e a r
in
is
(17)
n I : n2 = t
which
is p r e s e n t
plet.
Its d i m e n s i o n a l i t y
only
in ~ R "
This m u s t
is g i v e n by
I 2 constitute
the z e r o - m o d e
multi-
210
1 - ~
d(t which
agrees
This taining of
an
with
example
appropriate example in w h i c h
To
group
cently
been
there
are By
in e a c h tions
gauge
reproduce
M 4 the
several
defined
families
of
[16].
on
spaces
them.
The
interesting
CONCLUSIONS
AND
The
motivation
initial
electromagnetic gravity
plained
here
boundaries
factory
starting to be
compactification [6].
The
spectrum also
the
guarantee
It of h i g h e r mensions nors,
gauge of
forces.
(i.e.
He
for
be
fields
fields,
indeed
fermions This
CP 3
,
of
one
The
An of
has
in re-
dimensions.
Then S 2 CP 2 x ,
,
the
can
Dirac
operator connec-
obtain
number
characteristic
was
attacked
the
We
of of
of
The
believe
fields)
bosonic
to be
scalars
merit
of p r o v i d i n g
of
all
in M 4.
the
of
supergravity
several
families B d.
and
or
gravitational starting
from
the
scheme
ex-
that
within
the
dimensional cannot field fields gauge
be
theory.
Spi-
necessary fields
a rationale
non-trivial
theory
a satis-
for
for
[3], the
topology
may
compactification.
these
fields
theories.
a supergravity
[17]
Their
by
view
4-dimensional
case.
the
problem
a higher
to o t h e r
fermions
scalars
unify
point
understanding
chosen
that
to
this
this
a realistic
stability
exists
ten S6
=
real.
SO(10)-Yang-Mills
modes.
Euler
Kaluza
in a n y
have
massless
dimensional there
Bd
are
[15].
(8).
suitably. from
ob-
representations
observed
spectrum
that
zero
no c o u p l i n g s
point
the
is
From
day
introduced may
e g. •
the
SO(5)
in
the
chosen
anti-Kaluza-Klein.
present
is r e m a r k a b l e
vector
of
rather
the
of
appropriate
the
of
of
non-triviality
dimensions
complex
starts
B d,
the
OUTLOOK
is
gravity
have
with
5-dimensions.
of p u r e
nors
with
in
of
for
result
4.
and
identical
be
studies
fermionic
out
to
constructed
He
possibility
of
compactified
spectrum
B d must
turns
pure
been
space
he
the
belong
Witten
K =SO(10)
these
to be
has
possibilities
choosing of
E.
the
representations
elaborate
the
by
theorem.
in g e n e r a l
over
the
index
+ I)
a consequence
fermions
group
the
K and done
one
all
massless
the
that
in M 4 is bundle
however,
(t + I ) ( 2 t
of
illustrates
vector
Kaluza-Klein
2 = ] t
prediction
fermions
In t h i s
etc.
the
massless
example the
1 , t - 5)
For
Lagrangian
a second
are
rank
essential
instance containing
[2]
ingredients in
10-di-
gravity,
anti-symmetric
spi-
tensor field.
211
However, symmetry° with
the
by
model this
of
of
such
strongly not
constrained
admit
of
any
flat
Minkowski
such
a
solution
the
a symmetry
breaking
term
a term
will
find
space
by
local
super-
compactifying
as
the
supersymmetry [18].
justification
solution
observed
low
has
energy
to be
It
is h o p e d
from
quantum
bro-
that
the
mechani-
considerations.
In a d d i t i o n
to p r o v i d i n g
compactification
the
transformations dimensions.
There
exists
[19].
As
dent
a unique
pointed
out
that
seven
is t h e
the
group
SU(3)
paper
it
is a l s o
from
It
introduced eq.
(3).
in the The
a freedom. only
an
which
also
space-time
hopefully
Another gauge
to
some
on
that
effective
the
For
instance
it h a s
smallest
constructed
such
originate
the
in o r d e r
the
in
extra
dimension
4-dimensions. 11-dimensions
is n o t
an
acci-
a manifold
group.
fermions
the
to
In t h e
can
be
same
ob-
from been
of
constant
the
flat
to be
action
integral
given
so
do n o t
far
classical
~
permit
equations
mechanical
by such
are
equations
-term.
dimensions
the
and
requirement
shown
group
M 4 to be
had
quantum
a suitable
number
gauge
for
cosmological
however
K does
10-dimensions
that
models
group
anomalies.
chiral
2 in
for
isometry
of
possible
it
dimensions
(4+d)-dimensional
contain
restriction
probably
its
realistic
adjustable
argued
approximation
of
for
supersymmetry
number
highest
Lagrangian
of as
the
necessary
local
supergravity.
original
be
no
on
spin
[20],
number
remarked
an
in
fields
under
is the
a maximum
x U(1)
that
supergravity
It c a n
will
minimum
be
bound
supergravity
x SU(2)
argued
upper
Witten
11-dimensional
should
Minkowski
E.
invariance
eleven
and
simple by
admit
tained
that
supersyn~etry
non-gravitational of
a strict
is b e l i e v e d
with
the
requirement
imposes
It
consistent
the
is t o o
it d o e s
To p r o d u c e
addition
inclusion cal
to
4-dimensional
space-time. ken
the
Due
of
by
E.
Witten
K for
an
anomaly
the
choice
freedom
[16] free
of
from
that
in
theory
is
SO(10).
SO any
of
better -
maybe
far the
no
completely
problems
understanding through
questions.
some
satisfactory
alluded of new
the
to
in
the
structure
conceptual
solution
has
introduction. of
higher
inputs
been It
proposed
is h o p e d
dimensional
- will
shed
to that
a
theories
a light
on
these
212
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KI
(192])
For a m o d e r n t r e a t m e n t of d i f f e r e n t aspects of K a l u z a - K l e i n theories see B. De Witt, "Dynamical Theory of Groups and Fields", Gordon ~ Breach, New York 1965 R. Kerner, Ann. Inst. H. P o i n c a r 6 9 (1968) 143, ibid, 34 (1981) 437 and CERN p r e p r i n t TH.3669 (1983)~ Y.M. Cho, J.Math. Phys. 16 (1975) 2029, L.N. Chang, K.I. Macrae and F. Mansouri, Phys. Rev. D13 (1976) 235, R. Casalbouni, G. Domokos and S. K o v e s i - D o m o k o s , Phys. Rev. D17 (1978) 2048, C. Orzalesi, Fortschr. der Phys. 29 (1981) 413, W. Mecklenburg, Phys. Rev. D21 (1980) 2]49, R. Percacci and S. R a n d j b a r - D a e m i , J.Math. Phys. 24 (1983) 807, C. Wetterich, Preprint U n i v e r s i t y of Bern BUTP-83/16, L. Palla, CERN p r e p r i n t TH.3614 (1983), T. A p p e l q u i s t and A. Chodos, Yale P r e p r i n t 83-05, T. Appelquist, A, Chodos and E. Myers, Yale P r e p r i n t 83-04, F. M a n s o u r i and L. Witten, P r e p r i n t U C T P - 1 0 1 / 8 3 [2]
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S. K o b a y a s h i and K. Nomizu, F o u n d a t i o n s (Int. Science, New York 1969 Vol. II) ,
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S. R a n d j b a r - D a e m i
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Salam and J.A.
Strathdee,
Nucl. Phys.
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Salam and J.A.
Strathdee,
Phys.Lett.
[10]
S. R a n d j b a r - D a e m i ,
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Salam
Strathdee,
to be published,
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[12]
Duff and C.W. Pope in " S u p e r g r a v i t y Taylor and P. van N i e ~ w e n h u i z e n ,
and R. Percacci,
Nucl. Phys.
B195
(1982)
82",
Ed.
105,
S. Ferrara,
of D i f f e r e n t i a l
Phys.Lett.
117B
Geometry,
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41,
481,
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294,
Phys. Lett. 127B
(1983)
124B 47,
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Salam,
J.A.
Strathdee,
[14]
S. R a n d j b a r - D a e m i , print IC/83/75,
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L. Palla, P r o c e e d i n g s of the 19th I n t e r n a t i o n a l High E n e r g y Physics, Tokyo {1978) p. 629,
[16]
E. Witten, Lecture P r e p r i n t 1983,
[17]
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[18]
S. R a n d j b a r - D a e m i , 124B (1983) 349,
[19]
E. Cremmer,
[20]
E. Witten,
Abdus
Ann. Phys.
Salam
at Shelter
and S. Weinberg,
Abdus
B. Julia Nucl. Phys.
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Island
S a l a m and J.A.
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II Conference,
University
and J. Scherk,
141
76B
Phys. Lett.
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469,
OFF-SHELL
EXTENDED
EXTENDED
SUPERGRAVITY
IN
SUPERSPACE
Alice
Department
Rogers
of M a t h e m a t i c s
King's
College
Strand London
I.
INTRODUCTION
cific
physical
In t h i s
of
spaces
have
come
seminar
I shall
problem
with
of the
commuting
to be c a l l e d
and
values
even part
in t h e
rather
than
acquiring tain
formulae
siders
Ixl
As well
(IYl)
are
the
developed
extra
:= X Y -
0 or
as s u p e r m a n i f o l d s
algebra
of
to t h e s e
from
geometry
an
These
(m,n)
of a G r a s s m a n n
that
the
usual with
algebra,
for
take
algebra
on the manifold,
structures,
factors"~
spaces
dimensional
m of w h i c h
functions
the G r a s s m a n n
"grading
of v e c t o r
to a spe-
coordinates,
It is f o u n d
over
application differential
coordinates.
briefly,
+ n local
the
inherited
Lie bracket
[X Y]
m
itself.) carry
requiring
a graded
(1.1) where
calculus
a Z2 grading
recently
a n d n in the o d d p a r t extend
the m a n i f o l d
tions of t e n s o r
be c o n s i d e r i n g
anticommuting
is a s p a c e w i t h
(Some d e f i n i t i o n s
2LS
supermanifolds;
supermanifold
[I].
WC2R
manipulaobjects and
instance,
cer-
one con-
fields
(-I)~XI IYIyx
I according there
are
as to w h e t h e r super
Lie
X(Y)
groups,
is e v e n
which
or odd.
are groups
215
which
are a l s o
supermanifolds
tions)
[2]. The
infinitesimal
a graded
Lie
the
fields
into
fermi
generators
the
spacetime
vary
is to
which
that
the u l t i m a t e
there
that
are
could
is t h a t
two
among of
make
gravity,
to
the
theory;
include
the
describes
satisfy
their
shell
ry,
struct
an
field
form,
work
field
spinorial
N = 8 supergra-
of the
hope.
four
One
a finite
fun-
is t h a t
it
theories
f r o m the
phenomenon,
various
(which c a n be
quantum
unified
field
theory.
The other
of N = 8 s u p e r g r a v i t y
of the w e a k ,
motivated
is to
"off-shell"
are
(There
b y the
strong
say,
and
and elec-
are
Lorentz
of the
with
only
all
theory
alternative
the
graded
if the
characterization However the
the u s u a l
symmetry (that
is to
of o n quanti-
say,
theovalid
to c o n -
schemes,
in the
Lie
fields
of the
desirable
quantization
and working
to
N ~ 3 are only
closes
it is c l e a r l y
symmetry
to k n o w w h e t h e r
a hard question
in a f o r m w h e r e
section.)
thus
desire
has p r o v e d
theories
to e x p l o i t
version
configurations)
this
(An a l t e r n a t i v e
in the n e x t
if t h e y
manifest
a
supergravity
supersymmetric
arising
transformations
equations.
is g i v e n
such versions.
sacrificing
that
infinitesimal
techniques,
require
for all
as
the
theories
zation
at p r e s e n t
"on-shell" of
has
ge-
[5].
supergravity
in
this
symmetries
answer
algebra
this
of
symmetries
is f i n i t e ;
known
theory
a possible
additional
or n o t N = 8 s u p e r g r a v i t y because
behind
odd
a super-
of o d d
the m a x i m a l
infinities
and hence
interactions
seminar
the
words
viable
to the n u m b e r
calculations
"local"
on s u p e r s y m m e t r i c
are e i g h t
that
of t w o
and one
in o t h e r
unified
is
and
transformations
restrictions
N = 8 supergravity
the v a r i o u s
enough
reasons
in q u a n t u m
supersymmetry
the c o m m u t a t o r
there
of t w o
supersymmetry
transformation,
physical
out
a
under
bose
commutator the
if the
gravity,
It is t h o u g h t
including
even
N = I to 8 a c c o r d i n g
interactions
This
form
is a p p l i e d ,
transform
graded
so t h a t
to a n o t h e r )
are m a n y
it t u r n s
geometry
If the
theory.
reason
the
is s y m m e t r i c
provide
[4])
tromagnetic
versa}
intertwined.
point
cancellations
large
opera-
if it is s y m m e t r i c
generators
generator,
might
theory
are
are
this
odd
in the
found
and
and vice
theory
There
forces;
fields
whose
includes
has been
explained
group
coordinate
damental
are
to w h i c h
to a g e n e r a l
labelled
theory
on g r o u p
c a n be o d d or even,
supersymmetric
theory
[3], a n d
generators
there
the
theory.
theories,
vity
fields
spacetime
leads
theories
Lie
symmetry
supersymmetric gravity
is c a l l e d
is a t r a n s l a t i o n
say,
f r o m one
nerators
to the p h y s i c s
theory
of a s u p e r
odd
(that
now
field
action
conditions
which
algebra.
Turning quantum
(with s u i t a b l e generators,
such
lightcone
216
gauge
[6].)
Before to be
described
between first by
outlining
and
of
for
of
N
two
N to
extra
In
section
= 1 theory, higher
same
From are
used
dimension
the
of
structure
will
4 the which
while using
for
of
make
fact
Poincar~
group
finition
of
that to
the
and
down
action;
how-
spacetime
trans-
this
"superspace"~ odd
which and
is
an
dimensions
has
for
formulation
superspace,
N
been
cor-
developed
supergravity of
but
theory
they
[10].
supergravity
is
particular
higher
beyond
5 describes extra
the in
the
of
the
how
emphasis
Section
one
content
four
spacetime
superspace
same
3 con-
involves
is t a k e n
desirable
The
superspace
2 contains
physical
that
the
dimensions.
the
N.
usual
extended
some
section
features
approach
may
choice
of
these
"measure"
to
have
N.
work.
of
of
The
view, author
is w a i t i n g the
laborious
OF N
an N
odd
rather
heavy-handed
believes
to be
that
discovered,
checking
an
techniques
underiying and
that
this
redundant.
: ] SUPERGRAVITY
: I supersymmetry
for
to
differ
in t h i s
requiring
nature
N
with
over
dimensions
point
include
manner:
= 2 supergravity~
section
FORMULATION
superspace
expressed
writing
is
following
carried
further
each
much
the
of
the
handling
[7,9]
incorporates
by
this
a simple
A SUPERSPACE
The
bosonic
mathematical
in m u c h
in
to be
identical,
is d e t e r m i n e d
structure
2.
N,
out:
representations,
generators,
in the
on-shell
is
in a m a n n e r
dimensions the
of
connection
sections.
theory
with
dimensions.
the
the
theories
were
fields,
including
superspace
- I supergravity
of
versions
superspace
the
is o r g a n i Z e d
versions
two
"superfield",
supersymmetry
of
theories
invariance
a supermanifold
local)
the
pointed
supersymmetry
constructing
supersymmetry
subsequent
features
extend
odd
seminar
a review
of
the
to
be
various
proving
off-shell
paper,
[8]
the
between
for
in o n e
description
in
The
shell
fields spacetime
to
for
interplay tool
this should
supergravity
rules
explicitly
the
rigid(not
included
the
then
of
component
A detailed
off
and
a useful
responding
of
of
[7]
transformation
because
tains
rest
supersymmetric
extension
both
in the
constructing
supermanifolds
formations,
on
in d e t a i l
to
and
a lagrangian
many
approach
supergravity
stating
ever,
the
spinorial
group
is an e x t e n s i o n
generator
= I supergravity
to be
motivates
of the
a supermanifold
the dewith
217
4 real z
M
=
even
and
2 odd
( x m , s b , 8 -~)
corresponding
to t h e
it m i g h t
natural
seem
complex
dimensions;
(m = 0 , 1 , 2 , 3 , metric
and
to p u t
local
~ = 1,2) ; t h e r e connection
a metric
on
simpler
to
use
a tighter
structure,
of
the
cotangent
bundle.
In f a c t ,
cotangent
bundle
Zumino
with
the
[10,12],
preferred
the
coframes EA
with
E A transforming
lowing
=
[known
(E a , E a
under
as
infinitesimal
is
severe
due
to an
While
in f a c t re-
to Wess
SI(2,C)
bundle
denoted
SI(2,C)
rotations
in t h e
fol-
6E A = E B LB A Lab
= - Lba
(2.2)
LaB
= LBa
aB all
other
LA
B
are
A connection curvature
are
I = ~
LaB
~ is d e f i n e d
in t h e
usual
one
has
Bianchi
(TA
:= ~I E B E c T C B A D
identities , R
(2.4)
IAB C
:= AiB C
(2 • 5)
A gB C
(D A R B C
A~C
denotes
additional
accompanied to t h e
representation
A theorem [2.2),
Two shell" certain
due
the
to D r a g o n
surprising
cyclic that
the [13]
[2.5)
things
N = I supergravity torsions
to be
which
change);
in c o m p o n e n t
and
form
read
sum
the
Lie
about
(that
the
group
algebra
an
of
that,
such
of
a cyclic
two
SI(2,C)
odd
defined
of
the
a superspace
"off-shell"
are merely
sum with
indices
C D on
consequence
be p r o d u c e d
the
say,
indices
because
automatic
can
while
is to
interchange
states
theory
= 0,
: 0
also
are
zero,
torsion
m : D~
+ T A B E REC)
of
identities
bundle~
( R A B c D - D A T B C D - T A B E T E C D)
a graded
a sign
this
as
:= ~I E B E C RCM)
convention
by
on
way
TA = DE A
and
(oab)a B Lab
zero•
one-form
defined
(2.3)
the
approach
is r e d u c e d
vielbein)
it
, E~)
where
Here
relativity. [11]
to a m o r e
the
denoted
structure
manner
[2.1)
and
general
corresponding in
are
additional
superspace
duction and
of
coordinates is
is
RABCD by
refer
(2.2).
representation of
that by
version
(2.4).
the
"on-
constraining can
be
given
218
by a combination action Both
which
these
facts
supergravity motion,
invariant
under
from
follows sidering of
the
the
are
by
usually
if one
looks
involved~ have
[x]
derived
to g i v e
equations
of
clo-
of
mo-
which
such a theory
torsion
is n o t
constraints
does
track
an
of
algebra
equations
How
at the c o r r e c t
to k e e p
theory
f r o m an a c t i o n
on the
such a theory
it is u s e f u l
one must
their
with
superspace.
supersymmetry
a commutator
obey
constraints
consequences
torsions
with
a local
transformation.)
imposing
but
their
with
fields
the
"on-shell"
fields,
under
infinitesimally) is w h e n
the c o n s t r a i n t s ,
dimension
other
together
over
By an
a set of
each
a supersymmetry
start,
the v a r i o u s
rect
among
constraints
function
explanation.
of m o t i o n
merely
throught
torsion
of a s i m p l e
informally
that
equations
c a n be c r e a t e d
stringent
further
(defined
on-shell,
tionS(the
dent
need
transform
transformation
is
less
integral
one m e a n s
which
sing only
of
is the
emerge.
evi-
and
In c o n -
of the d i m e n s i o n s
the p h y s i c a l f i e l d s t h e i r c o r I = ~ (where w e a r e w o r k i n g
= I and
[8]
in u n i t s w i t h ~ = c = 7). T h i s m e a n s t h a t t o r s i o n s h a v e d i m e n s i o n I I 0, 5' I or I~. In a s c e n d i n g o r d e r of d i m e n s i o n s , the c o n s t r a i n t s w h i c h give
the o n - s h e l l
Dim (2.6)
The
only
constrained that
0
TaB
Dim
Tbc d =
I
torsions,
torsions
and
a spin
panding
the
vielbein
(2.7)
their
it m a y be are
field
Tab ¥
seen
, Tab Y
severely
denoted
in a c o o r d i n a t e
conjugates).
f r o m the
, are n o t Bianchi
restricted).
field ~m ~
(expressed These
are
explicitly
identities
(2.4)
The p h y s i c a l in v i e r b e i n obtained
fields
form
by ex-
basis
coordinates
such that when
~ = 0
the m a t r i x Em a Em ~ Em ~ )
•
d
E A = dz M E M A
and choosing
(2 8)
[10]
= 0
complex
gravitational
3/2
are
: - i oa~
T bY = T bY
remaining
the
theory
T ~ ¥ : T ~~ = 0
with
are
d
= T g¥
(together
as e m a)
part)
: 0, T ~
T bd
these
theory
d
I Dim ~
(although
in f a c t
of the
N = I supergravity
EMA =
E
a E ~ ~ E ~.~
E~ a E.~ ~ E.~
(and x h a s
no n i l p o t e n t
219
takes
the
special
form
em (2.9)
EMA(~,0)
' ~m 8
=
I i a(~) where also
and --~)m~ a r e the s p i n m the c o n n e c t i o n c o m p o n e n t ~ m
time while
the c o n n e c t i o n
transformations superspace free
approach
coordinates spacetimes mation
rules
= DM(A an
other
By c o n s i d e r i n g
the
by the
constraints
obvious
approach,
ly w i t h
the c o m p o n e n t s
finds
their
[10]
complex
then
in
coordinate-
transparent
over
when
non-trivial
[1,15].)
determined
general
that
and
field
In o r d e r first
fields be d o n e
of
The
by
transfor-
[16],
coordinate
their
transforma-
in v a r i o u s
ways;
the o f f - s h e l l
described
constraint
T bY
equations.
off-shell
by H o w e , = 0
implied
are
(It is s i m p l e r ,
identities
the
spin
rather
imposed
if n o t than
the
direct-
A s an e x a m p l e ,
Stelle
(and c o m p l e x
into
~ field
- the the
the
geometry.
no
constraints
longer
For N = I supergravity done
version;
by W e s s
this
Townsend
conjugate)
and
is the
N = I supergravity and
[18]; are
equation.
transformation
torsion
identities
originally
an a l t e r n a t i v e
of the n e w - m i n i m a l
[17]
equations.
are
which
connection.)
theory,
the B i a n c h i
it w a s
these
Vielbein
in s u p e r s p a c e
are b o t h b u i l t
until
that
field
I bcd , I by6, I ~bY3.6 together with
to g i v e
"on-shell"
field
and
is the
finds the
Bianchi
identities
to o b t a i n
one
components.
the
combine
theory
and
the v i e l b e i n
equation
h e r e we d e s c r i b e
version
identities,
torsion
of
the
be r e l a x e d
to o b e y
the o n - s h e l l
on the c o n n e c t i o n
conjugates
laws
the
are
(A m o r e
is m o r e
authors
of
transformations
superspace
several
group
6 B TB M A
to w o r k w i t h
is the m e a n i n g the
of
on the
This
West
of
and
in s p a c e -
allowed
a n d ~M"
the p h y s i c s
theory +
Bianchi
restrictions
but
by
The
coordinate
f o r m of E M A
infinitesimal
ingredient
b y the
must
is zero.
respectively,
connection
in s u p e r s p a c e .
The
one
~b
construction
of the p h y s i c a l
6 A parametrizes
is the o r d i n a r y
[14] b u t
the
3 spin ~ fields
2 and
of g e n e r a l
this
considered
6EMA
where
group
is p o s s i b l e
has b e e n
6. &
component
preserve
are u s e d }
(2.10)
tion
is the
which
(~)
0
r~
o
a
e
(~)' ~ m
force this
the can
Zumino
[12]
superspace of
Sohnius
in t h i s
dropped,
and
version
a n d an
220
additional incl~ded through
U(1)
(acting
in the the
identities) their
spinor group
consequences
the B i a n c h i to s a t i s f y
on
structure
indices
of t h e s e one
equations
reduced
finds
that
of m o t i o n .
a zero-dimensional
scalar
is a c r o s s - s e c t i o n
of an a s s o c i a t e d
This
gives
derived
from
that the
E denotes
the
E d4~ d2e d2~ gravitational
as r e q u i r e d .
to the
Wess
and
3.
is the
ON-SHELL
the
which
are
~2
[12])
used
applied
(This
by Brink
and Howe
SU(2)
bundle
(3.1)
forced of
(A " s u p e r f i e l d " of
Vielbeins.)
dynamics
is n o w
(m a,
infinitesimal following
this
IN N O R M A L
odd
version The
manner 6E B = E A L A B
K
is
on the t o r will
of E M A,
then
(due to
in s e c t i o n
AND
EXTENDED
supersymmetry
formulation
explicitly of o n - s h e l l
coframe
bundle
in a
generators;
(4,8)
dimensio-
m = 0,1,2,3, by H o w e
~ = 1,2,
[20] ; it is
N = 8 supergravity is r e d u c e d
to an
denoted
, i = 1,2.
SU(2)
4
SUPERSPACE
~)
, ~ = 1,2 ~
and
technique
described
(xm,e~ ,8-~ t) ,
vielbein
SL(2,C)
of the
are
[19]),
(so
S is d i m e n s i o n l e s s ,
constraints
components
spinorial
zM
'
thus
(EM A)
of N = 2 s u p e r g r a v i t y .
is p r e s e n t e d
Ea i
element and
to the
the
superspace
[21]).
d28 matrix
volume
subject
action
two
with
a = 0,1,2,3
in the
longer
the e x i s t e n c e
the
of the
to a c h i e v e
superspace
EA =
Under
infers
d2~
= 0; d e t a i l s
coordinates
formulation
of the
d4~
restrict
TabY
has
with
superspace
~
no
to the b u n d l e
= ]/ 1 6 ~ G ) ,
N = 2 SUPERGRAVITY
nal
SL(2,C)
([
to the
the o r i g i n a l
given
is
following
(by c o n s i d e r i n g
are
version~
indices)
bundle)
C on s u p e r s p a c e .
C E
in t u r n
equation
N = 2 supergravity
a truncation
one
off-shell
action,
this motivates
t = 1,2.
fields
Also
superspace
constant,
field
Zumino
they
coframe
constraints
the
bundle
superdeterminant
Varying
sioncomponents
where
of the
S -
lead
not o n v e c t o r
the a c t i o n
(2.11) where
superfield
the k i n e m a t i c s
but
of the r e d u c e d
rotations
the v i e l b e i n
transform
221
where
Lab
= - Lba
~!~ : ~ a3 Li
(3.2)
~ + ~ 3
= 0
•
~
p
~J : 6 i ~ al =
all
has
a connection
the
torsion
{3.3)
other
LAB
are
C(
(°ab)aB
zero.
the
ij T~Bd
Dim 0
As
(3.4)
Dim~
in the c a s e
theory.
These
= 0, T i ~Bjd
two-component
according
spinor
to the
used
are
rule
in
brackets
extracted
when
8 = 0
(and x has
a n d the c o n n e c t i o n
notation
Also
by once
in v i e r b e i n
has b e e n u s e d for the v e c t o r i n d e x b = o BB Vb " F u l l d e t a i l s of the con-
curved
again
no n i l p o t e n t
components
of N = 2 s u p e r g r a v i t y
special
are
[ik] : -L E B M ( ~ )
brackets
antisymmetrization.)
is n o w
the
form),
two
The p h y s i c a l
using part)
a spin
a special 2 field
°hi
0
o
spin ~3 f i e l d s
f o r m of the v i e l b e i n
is
EM A
=
0
o1 6.
~t
~i
/
in s p a c e -
system
where,
components
form.
em a i~ and
~m
symmetrization theory
a coordinate the v i e l b e i n
~ M A B take are
denote
e m
(3.6)
of
conjugates).
Vb<->VB~
[20].
time
fields
one
conjugates).
(and c o m p l e x
square
constraints
components
= 63°~Bd
Ti k Ti k ~b# <-> ~BBy
ventions
on some
All torsions zero except
(3.5)
(Here
of N = 1 s u p e r g r a v i t y ,
constraints
All torsions zero.
Dim I
field
Lab
with
(and c o m p l e x I
1
= - LaB
in the b u n d l e
defining
~J C(
I
~
L&~ and
+ ~
]
LaB
and
]
1
L
b,
L i. ~
EM A
The p h y s i c a l
(the g r a v i t a t i o n a l a spin
I field~
222
Here
the
spin
spin
one
field
when
8 = 0
strength
(and x h a s
on the c o n n e c t i o n ; ~m~8'
#m~
in o r d i n a r y
in the
(3.7)
and
one
finds
equations
value
Conditions
are
connection
components
to
the
take the
since
[ij]
from the
(3.3)
f o r m of the L o r e n t z i ~m j are pure
(3.4)
(3.5))
c a n be e v a l u a t e d
f r o m the i R A B j is zero.
from
[ij]
D i M(~8)
Bianchi
imposed
components
(as m a y be d e d u c e d
strenght
~ i
:
usual
SU(2)
also
identities
that
the
fields
obey
their
of m o t i o n .
There
bosonic
field
same w a y as b e f o r e ; the ik] of M ~ ) (and M ( B ¥ ) [ik])
SL(2,C)
while
zero)
in the
part).
the c o n s t r a i n t s
spin
6M(~)
gravity,
the
space,
set to
identities
Variations
Again
f r o m the
no n i l p o t e n t
as b e f o r e ,
(and u s u a l l y
Hianchi
comes
appear
a n d ~ m ab are r e q u i r e d
connection gauge
3 spin ~ fields
2 and
is an a l t e r n a t i v e
employing
superspace
an e x t e n d e d
coordinate
y[tU]
version
superspace
(~[tu])
with
(t, u = 1,2)
of o n - s h e l l an a d d i t i o n a l
and
N = 2 supercomplex
corresponding
"sehr-
vielbein" EA : with
the E [ij]
scalars
~ , ~i (E a ' E i
under
(3.8) this
means
the c a s e w h e n
SU(N) Howe
SL(2,C)
6E[iJ]
(In f a c t not
or
U(N).)
This
and Lindstr~m
where
the
Carton
extra
mensions,
The
[22]
although
Dim
0
(3.9)
also
the
full
All
scalars
have
was
I
All
SU(2)
as
in
that
torsions
there
zero
torsions
zero.
in t h e
retains
is s o m e w h a t
s e t of o n - s h e l l
but
this
is
considering
developed
of the t h e o r y .
[20])
aspect
SU(2),
N when
by
for N = 8 s u p e r g r a v i t y ,
meaning
are
context The
extra
of t h e
explicit
bosonic
di-
vestigial.
new components
constraints
are
except
T i" = - io ~d , T ~ r ~ : - 2i £ ~ ~Bjd [pqj
Dim ~
under
originally
[23]
a natural
E 7 symmetry
mean
under
or for h i g h e r
superspace
geometric
dimensions
to be c o n s i d e r e d ;
are
to U(2),
of the
transforming
= 2E [ik L k 2]
(given b y H o w e the
and
and by Kallosh
dimensions
to N = 2
extra
they
k i n d of
representation
truncation
(3.10)
that
we extend
, E[iJ] , ~ [ i j ] )
ij 6[pq]
of t o r s i o n
223
Dim
I
All
torsions
Ti
(3.11)
zero
except
k
[ik]
~ B~ i :
i ~B
M(B+)
[Pg] (From here
onwards,
omitted,
a n d may
Bianchi
identities
fied ry,
without and
further
D[iJ]M(B~)
To e x t r a c t procedure
the
[kl]
theory
part)
any
~MA:
and
the
Here
A
connectlon
formations theory
on the of the
fact
that
approach
does
not
using ferent
0
6t~
0
0
0
&~t
0
0
0
0
0
0
~i
@m'
a spacetime approach
considered
to that
seem
Hassoun,
are
play
and
Taylor,
Restuccia
satistheo-
one
finds
A m [ij ]
0
0
0
0 0 ~[tu] o [ij]/
spin
no p a r t
while
I field,
in the
by Howe
the
dimensions
seems
[22],
are
zero;
theory,
essential.
in s u p e r s p a c e
and H a s s o u n , [24].
is
zero.
of the trans-
spacetime
and L i n d s t r ~ m
superfields
to t h e o f f - s h e l l
and T a y l o r
~[tu] because
supersymmetry
for o b t a i n i n g
of all
the
(and x has
form
in the N = ] t h e o r y
of e x t r a
are
in s u p e r s p a c e
as b e f o r e
the
procedure
be
of the
of the
@ = y = 0
0
described
as
use
theory
[ij] 6[tu]
for
they
in p a r t i c u l a r
[ij] Am
to be a p p l i c a b l e
"slice"
to the
that
superfields
when
the
&i
@bt
the y - d e r i v a t i v e s
by R e s t u c c i a
and A m e r i g h i ,
they
The
the
~m
will
components
= 0
takes
s Ymi
torsion,
is d i f f e r e n t
the
from
em
theory.
finds
on the
[kl]
conjugates
more
superfields,
sehr-vielbein
components
one
in w h i c h ,
[ij] , Am[ij ] are p o t e n t i a l s
m constraints
uses
new
= D[ij]M(B~)
a
(3.13)
but
conditions
coordinates
the
to c o m p l e x
are n o w m a n y
in s p a c e - t i m e
is to c h o o s e
no n i l p o t e n t
There
to be c o n s i d e r e d ,
introducing
[Pq]
references
be a s s u m e d . )
imposing
without
(3.12)
explicit
M
Restuccia
which this where A dif-
has b e e n
and Taylor
224
4.
OFF-SHELL
In t h i s off-shell same
N
are
= 2 SUPERGRAVITY
section
a full
superspace
two
reasons
the
geometric
considerations
which
the
dimensions
the
extra
perspace the
N
"measure"
for d4x
= I superspace
There
are
two
torsion
constraints
then
action
an
The
which
the
in
group
this
off-shell
of
the
= -
up
the the
reduced
[25])
which in
is of
other
same
action
of
uses
the
section course
to
as
3.
the
introduce
is t h a t
dimension
the that
suof
is p o s s i b l e .
off-shell
theory,
are
dynamics
first
unrestricted,
the and
is c o n s t r u c t e d .
bundle
restriction
except
version
Lindstr6m
The
coframe
the
one
fields
constraints
zero
and
of
correct
removes
torsions
the
kind
so t h a t
torsion
T i" ~Bjd
same
the
merely
All
has
setting
generates
Howe
theory.
a new
described
a superspace;
dy dy
thus
of
in
version
inspired
relaxed
The 0
such
on-shell
d4@
stages
factor,
tion(3.2). Dim
and
are
structure
by a U(1)
d4@
briefly
on-shell
using
SUPERSPACE
is g i v e n
(described
as
for
IN E X T E N D E D
description
= 2 supergravity
extended
There
N
is a g a i n
Li,
l
extended
= 0 of
equa-
are
for
6~ 3 °~d
i
(4.1)
T~ B [pq] I
Dim
= - 2i
All
torsions
Ti{
~k
~ ~B 8 [[p@ ij]
zero.
= Ti~,c P q
(It
is
sufficient
to
require
most
general
that
: Ti~bd
(4.2) =
T i . yk
~Sj =
T.
T i [pq]
= [Pq]
~ :
[rs] 0.)
~lC
(4.3)Dim Using
the
possible treated the
I Bianchi for in
second
dimension information sion
d
Tbc identities
the
torsion
increasing set
of
is
components
order
Bianchi
identities
are
learned are
and
(2.4),
one
curvature of
may
dimension these
find
the
components~
dimension.
identities of
from
zero.
= 0.
Even
the
form
identities
in e x t e n d e d
are
superspace
(2.5) is r e d u n d a n t [26]. T h e I ~ (an e x a m p l e is I ai S j ky [ p q ] ) ;
identities
since
all
dimension
lowest
no new 1
~ tor -
225
In c o n s i d e r i n g venient to divide the between
the d i m e n s i o n identities
the n u m b e r
identities
fall
of u p p e r
into four
I and h i g h e r
into g r o u p s
and lower U(2)
separate
I
: 6)
Ijl
[Pq]
[rs]
group
2
= 4
i I~j
[pq]
[rs]
I~ c
[Pq],
3
I~
I~
4
J
[Pq]
k6i l ~Bjyl'
Z] = 0
i i. aBj
•' ' I~[pa]
C [pq] '
involve
I identity yields
the t o r s i o n s
give no f u r t h e r
'
i i, [pq]c ~Bj
no n e w
I~ '
[pq]c
information.
T [ij] [kl] and T i [pq]yk
" t o r s i o n s ) . F r o m I i]' Bc[k~] one l e a r n s i i. ~Bj [pq] [rs] one l e a r n s that Ti[pq]@ identities
[rs]
i i" d @Bjc '
ii , [rs] liJ @Bj [pq] ' ~flc[pq] The g r o u p
I
'
[pq] [rs]
[rs]'
group
the d i m e n s i o n
[Pq]c
ij ol IaByk
i ~Bc d ' I~iBj i
indices;
iijk 61 @By
'
iijk6 @Byl'
= 2
it is con -
to the d i f f e r e n c e
groups:
group
group
identities
according
The g r o u p
2 identities
(and the d i m e n s i o n
~ that T c [ij] [kl] yk is zero.
zero
is zero, w h i l e
The r e m a i n i n g
group
from 2
information.
The g r o u p 3 i d e n t i t i e s i n v o l v e the t o r s i o n s T i yk, T i [ p q ] ¥ b ~ k ' [pq]d [pq] yk Tc ' T-~i ' T[rs] [tu] [vw] , Tbc [Pq] and the c u r v a t u r e s ij ij i]' k R By6, R B~8_ and R ~ 1 " The i d e n t i t i e s are b e s t h a n d l e d by c o n s i d e r i n g the
irreducible
representations
the m e t h o d
of Grimm,
dering
identities,
the
Wess and
of SI(2,C) Zumino
one m i g h t
~ U(2),
[27]. For
expect
in an e x t e n s i o n
instance,
T~bYk-
to c o n t a i n
c!ik)
~ [ik]
before
of
consi-
t e r m s of the
form A ik
[ik]
ik ~B
In fact one
finds
'
that o n l y
N[ik] (~B)'
terms
R
(ik)
,
v[ik]
.
lik] lik] of the f o r m M ~+) and N ~B) can occur,
226
and
the
group
ayy
TV%
'i
3 torsions
= - 8il
T i [P q]
+
N [pq] (¥6)
afC~y
f~8 + d
IcN( aB )
Bj
a
curvatures
NliJl
-?I~
[Pq] 6 6• = 2 i ( c
and
3
'
have
, Iijl
the
form
)
M [pgj (~5)
6¥6)
i
B =
3
(4.4) [P q]
= 8i(6~y~(~y) (By) M [ ~ ] + $ By N [pq]
T[iJ]
[mn]
: 0
[kl] ij
RaBy6 ij
=
(8-2c)
o
~[ij] L~(y6) $ aB ~ [i~]
Rijk tN[ik] aB i = 8, (aB) Hence will
c and show
zation
d are
them
which
of
will
be
, T c and
and
GaB
T
[rsl
.
YY [Pq] Ri
(4.5)
this
stage
dimensional
A similar
k are
analysis
identities
arbitrary
4 identities,
which
61
and
normali-
T[rs]
' that
[pq]c the o n l y
involve
the
the
curvatures
non-zero
components
also
will
can
6aY)
+ ik HB~
set
be
and
i 6 Y k
traceless)
s[rs] V[pq]
cca6 + Gsicc
:
h and
group
HB~ ~ is h e r m i t i a n
: - 6i Gy~
identities
dimensional
8 is an
later.
. 6B ~ + h GB~
3c ~(6~6~
Ri k ~Bj 1 At
and
- 3(Gyi
~Bjy6
(higher
factor
are
(6 Ga~
is h e r m i t i a n
the
the
[rs] T[Pq]yk ' R ia Bj k i' s h o w s
curvature
T i 'Y : i i ~BBk 6k (where
of
numbers
while
convenient
[pq]
61)
real
zero),
analysis
Ti ab
torsion
arbitrary
to be
A similar torsions R ai B'j y 6
' [jk] 6~ + N(aB)
.i y)6 j
,k 6i
k i
- 46j61)+ arbitrary them
k H BI real
both
applied
to
to
3
numbers,
but
again
higher
zero.
the
dimension
3/2
identities.
227
The
superfields
mension
which
occur
in the
torsions
and
curvatures
of this
are all c o v a r i a n t d e r i v a t i v e s of the d i m e n s i o n I fields; 3 i i ~ s u p e r f i e l d s are ~ ~ + ) ~ & and "~ ~(B¥)' w i t h
di-
the
dimension
DT k N[iJ]
:
[i
6~]
[i [ij]
i]
[i
i]
= 2
(4.6)
i The
non-zero
i 3
dimension
.
i
~ torsion
i
TBByyg
:
8~og
By
Ri , ~BBy6
i = - i(4~B(y6)
R i ..-
:
and
curvature
components
are:
i
4 ~ ~y~g{By)+$gBy(7~l fyg
+
i (~B(~y)&
E ~B +
t
+ ~;
fBg)
+~i B6
~(~)
~By )
(4.7) i , i
i
i
Ri k
The
dimension
equations need
Sohnius
3 ~ identities
(4.4)
to be
i
and
k of
considered
[28]
that
ly s a t i s f i e d ,
the
of any
determine.)
The
solved
superfield
without
that
show
that
zero.
61 )
6
the The
of d i m e n s i o n
that
is that
).
coefficients remaining 2.
(It has b e e n
identities
the
are
commutator
which
the
c and
d of
identities
then
shown
which by
automatical-
of two c o v a r i a n t
torsion
and c u r v a t u r e
2 s u p e r f i e l d s are in t u r n c o v a r i a n t d e r i v a I and ~3 s u p e r f i e l d s ; the d i m e n s i o n 2 i d e n t i t i e s further
D&
means
+ 12
dimension
of course
(4.8)
This
Ol
are
those
k
dimension
tives of the d i m e n s i o n be
also (4.5)
higher
provided
derivatives
can
are
,,k
(as in
~&
restriction
: 0 ,
[18]) G ~(x,0)
m~
and
on the
superfields
apart
from
H ~& j i H ~Bj .i (x,a)
can be r e g a r d e d
as
228
the
(supercovariantized)
fields.
Apart
rivatives
of
We thus
from the
field
(4.8)
fields,
have
strengths of a n t i s y m m e t r i c
there and
are no r e s t r i c t i o n s
the
a complete
fields
off-shell
are
thus
geometry
tensor
on the
gauge
spacetime
de-
"off-shell".
for N = 2 s u p e r g r a v i t y .
Because D k N [ij] ¥ (~) (4.9) the
on shell
to c o n s t r u c t G ~ = 0. As zero
which
real
scalar
are
between
the
zero
covariant
derivatives
(4.11)
N[( ~]) ij
following
C is the
Howe,
matrix
Berezin
by v a r y i n g because than
the
(EMA) ; the
fields
field
the u s u a l
first
(that
the
is,
other
N aB [ij]
G ~) can be e x p r e s s e d of the and
relationship
G~,
find
we
a su-
and as co-
(4.9)
that
it is
D~])C scalar
superfield
and
= 0
Stelle
and
Townsend
[18],
it is n a t u r a l
superfield
E is the
integration The
to
action;
here of
of L a g r a n g e
by W e s s
and
Zumino
,
introduced
superdeterminant
is then
field
the c o m p o n e n t s
technique
developed
scalar and
[29].
in the
C and
d 4 ~ d 4 e d 4 ~ dy d~
c E
constant
prescription
the
use
method
Dj]_ B)u
zero-dimensional
gravitational
vielbein the
D [i (~
:
S : 1 2
is the
of N ~B [ij]
identifying
all
is
action
(4.13)
where
require
step
equation
I Dj~ D S j - DB j D~) C = ~(
D[ai
the
and
The n e x t
field
G ~ and
version
because
zero-dimensional
(4.12) Then,
on-shell
of N lij] sB)
will
that
zero.
to the
C so that
c~
consider
this
C such
In fact,
(4.10)
C is a real
set G ~ to leading
(2.11),
in the
of C.
to c h o o s e
if we
field
derivatives
derivatives
[ij] = D~k N(~B)
in s u p e r s p a c e
in the N = ] case
the c o v a r i a n t
where
6~ ]
GB)+
is o b t a i n e d
an a c t i o n
variant
possible
ml~
theory
dimension
perfields all
i
and
= 0
performed
equations one
are
encounters
(EM A)
are
and
according then
we
sehrto
obtained
a difficulty
constrained~
multipliers, [12]
above, of the
follow
subsequently
rather the used
229
by Howe,
Stelle
and T o w n s e n d
[18], of e x p l i c i t l y
evaluating
variations
in S.
To do this,
following
[72],
let
(4.14)
HA B
:= E A M 6 EMB
(4.15)
~ AB C
:= 8 #AB C
(4.16) (where
Then IAl
6 E = E HA
is the G r a s s m a n n
(4.17) where
6S =
degree f
6C is the i n c r e m e n t
A
(-I)
of A)
(6C + H A A
IAI
and
(-I)
im
C)E dz
of C.
Now
(4.18)
+
TABD HDC - HBD TDcA - (-1) IBllCJ HcD TDBA
of m o t i o n
finds all v a l u e s
straints cause
TAB C = D A HBC
(-I) IN IBI DB HA C
- ( ~ B C A - (-I) IBIICI£ CB A)
The e q u a t i o n s if one
6
variations
here;
and e q u a t i o n
in the t o r s i o n
of f i n d i n g
to the a c t i o n
of H and 6C w h i c h
on the t o r s i o n ,
full d e t a i l s included
corresponding
the r e s u l t
are c o n s i s t e n t
(4.18)
allows
are e x p r e s s e d
solutions
for H, ~
is two i n d e p e n d e n t
families
= 6 @ g %~
a n d the o t h e r HAA(_I) IA; :
(4.20)
where V ~
(m k D ~ ( i D BJ) L d (ji) k
+ D~k D(~ D B J ) L ~k
(ij))
6C and L ~(ij) k
: - HAA(-I)(AI
are a r b i t r a r y
with
the con-
one to do this be-
in t e r m s of H and ~
~AAI-I) Im = (D~ D~i - D~i D~)V ~ 6c
can be f o u n d
. The
and 6C are too long to be
first g i v i n g
(4.19)
(4.13)
superfields.
of s o l u t i o n s ,
the
230
Substituting field
these
values
c~
(4.22)
In f a c t
(4.22)
Up tended
to
addition
OFF-SHELL
the
to
necessary.
of
the
The
name
described
6C
into
(4.17)
gives
the
cause
they
not
to be
described
reasons
for
and
to be
ment and the gant
geometric
[31]
uses
dimensions
here.
how
they
the
correct
may
be
with
version
various
"central
from
of
commute
two
and with
version
(in will
version
Sohnius all
supersymmetry
The
ex-
authors
charges"
an e x t e n d e d
Lopusanski
extra
first,
an
with
for
of
some
the
extra
to
relax
[3]
other
generators
N = 2 supergra-
N
= 3 version
even
the
are
two
on-shell N
(both
N
= 2) even
= 3 fails,
and
g i v e the superspace volume eleof
an
action.
= 3 supergravity correspond
of p o s s i b l e (up to
element
dimensions,
There
dimensions
which
N
N
(as f o r
construction
higher
volume
dimensions.
again
on-shell
table
extra
superspace
the
generators
for to
attempt
the
In t h e
include
odd
dimensions
following
chosen
again
unextended
extra
group
dimension
rotations).
include
dimension
The
U(2) we
in an
the
odd
above)
required.
N supersymmetries)
comes
Haag,
generators.
section
formulation
extra
to
of
possible
correct
charge" due
and
which
this:
secondly
as
covariant
called
commutator
we
torsions
appears
theory
constructed;
generators
commute
doing
been
the
in t h i s
radically,
supergravity
(as e x p l a i n e d
includes transformations of t h i s s o r t g e n e r a t e d by tu y (although they are not truly central, be-
dimension
more
fully
invariance
translation
above
on-shell
even
in
the
(4.21)
the
symmetries,
"central
bosonic do
~ 3 has
algebras
occur
to
N
coordinate
extra
odd)~
and
while
gives
no o f f - s h e l l
extra
further
and
(in a d d i t i o n
but
A
= 3 SUPERGRAVITY
with
super-Poincar6 includes
(4.21),
time
that
general
be
but,
(-I)
= 0
which
N
present
noticed
generators
from
condition
supergravity
have
vity
follows
the
TOWARDS
which
A
D~( i D~3) D k C = 0
and
is e x a c t l y
[30]
HA
equations
(4.21)
5.
of
dz.
to
extra the
due the
8)
ele-
to A w a d a
extra
dimensions
maximal
An
to
odd shows give
231
TABLE
N x
m
.
(dlmenslon
I) I ~)
oUt(dimension
y[tU] (dimension 1,
I
2
3
4
8
4
4
4
4
8
12
16
32
I
3
6
28
16
224
complex) X ~ [tuv] ( d i m e n s i o n
I 5)
-
4
v [rstu] ( d i m e n s i o n
I)
-
- I
2
2
dimension
In t h i s N
of dz
section
we
propose
= 3 supergravity
ordinates m = 0,..., this
et, ~
x m, 3;
extended
on
components
superspace
...,
4;
is r e d u c e d
t,u,v
to
an
= I,...,
SL(2,C)
70
2
of
3.
the
torsion
I, w h i c h
has
in co-
with
coframe
bundle
(real
2
~ [tuv]
The
x U(3)
of
table
~t ' Y [tu] ' Y- [ t u ] , X ~ [ t u v ] ' X
~ = I,
superspace
constraints
in t h e
(complex)
bundle
with
on
"zuviel-
b e in" EA =
The
action
(E a,
of
E~,
SL(2,C)
x U(3)
(5.1) where
E[iJ] , E[ij] , ~
~i,
on
6 EA the
LB
A
are
as
the
[ijk] ' ~ [ijk] )"
zuvielbein
is d e f i n e d
by
= E B LBA
in e q u a t i o n s
(3.2),
with
the
new
non-zero
B[imn] L~[ijk]
(5.2)
B
= L d
(together The
with
proposed
Dimension
0
the
=
llmn] 6 ijk]
complex
[i ~mn] l [ i -jk]
conjugate)
constraints
on
All
are
torsions
~ + 66
(5.3)
Ti . ~Bjd
(5.4)
' T 13" 8[pq]
(5.5)
Ti [pq] ~S[jkl]
the
torsions
zero
will
now
except
= - °~8d : - 2i ~ B
6[iJ] Pq
= 6i $'~B 6
[ipq] jkl
exp
be
given.
addition
232
Here
L is a z e r o - d i m e n s i o n a l I ~
Dimension
All
T
(5.6)
torsions
are
[mnp] y
1 ~[ijk]~
T[ijk]
(5.7)
real
scalar
zero
except 6[mnp] ijk
i = - Q8 ~ ¥
1 B ~ [mnp]
=
field.
1 g~@ QB
6[ijk] mnp
[ijk] E
(5.8)
Tijk
(5.9)
1 [mnp] 1 ~, 6 mnp T~[ijk]8 y = - QB ~¥ [ijk]
(5.10
T[rs] [ijk] = _ w[ijk] d [pq] ~
(5.11
T~ [ijk]
6rs [pq]
[pqr] = _ W~ [iron] 8[ijk]
[iron] [pqr] g /
with D ia L : Q~i (5.12
and No h i g h e r
D!ijk]L
= W! ijk]
dimensional
(5.13)
Preliminary
constraints
c
T a b
calculations
define
off-shell
theory.
If exp
L = 0, the
[21]
or Howe
the e x t r a sonic is
dimensions
J
exp
The aim N)
are
field
only
supergravity
quantization by Howe, feature
are
required,
apart
from
suggest
on-shell
Lindstr~m
optional of the
in p r o g r e s s
to
these
rules
theory
[22]
X~
[tu]
in v i e w
that
constraints
(in the
and K a l l o s h
[stu]
The
dimension
the
and
action
action
with
the bo-
to c o n s i d e r
superspace
this
and
is r e g a i n e d ,
decoupled
obvious
of
see w h e t h e r
f o r m of B r i n k [23])
completely
extras.
on the
for the N = 3 s u p e r g r a v i t y
measure;
leads
to the
equations.
ideas
is not
.
transformation
dimension
y
L dz,
calculations desired
and
spinorial
L
= 0
torsion
Howe
exp
in this
section
to d e v e l o p theories,
techniques
Stelle
and
but
to do
introduced
Townsend
of the m e t h o d
are
still
an o f f - s h e l l
proposed
by G r i s a r u
and
further
is the use
stage~
of N = 3
so in a f o r m w h e r e
[4] may be here
at a t e n t a t i v e formulation
the
Siegel
exploited. of a c t i o n s
but
the
(and h i g h e r
superfield
[31]
and
used
An e s s e n t i a l which
are
233
integrals cause
over the whole
of the choice
of superspace;
of extra
such actions
are p o s s i b l e
be-
dimensions.
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[I]
F.A. Berezin and D.A. Leites, Sov. Math. Dokl. 16, (1975) 1218, B. Kostant, "Graded Manifolds, graded Lie theory, and p r e q u a n t i z a tion" in D i f f e r e n t i a l G e o m e t r i c M e t h o d s in M a t h e m a t i c a l Physics, p r o c e e d i n g s of the S y m p o s i u m held in July 1975, Lecture Notes in M a t h e m a t i c s 570 (Springer-Verlag, 1977), M. Batchelor, Trans. Am. Math. Soc. 258, (1980) 257, A. Rogers, Journ. Math. Phys. 21 (i980) 1352,
[2]
A. Rogers,
[3]
R. Haag, 257,
[4]
It is only p o s s i b l e to give a s e l e c t i o n of results, J. Wess and B. Zumino, Phys. Lett. 49B (1974) 52, P. Howe, K. Stelle and P. Townsend, "Miraculous U l t r a v i o l e t Canc e l l a t i o n s made Manifest" ICTP/82-83/20, to appear in Nucl.Phys. B, J. Iliopoulos and B~ Zumino, Nucl.Phys. B76 (1974) 310, S. Ferrara, J. Iliopoulos and B. Zumino, Nucl.Phys. B77 (1974)413, M.T. Grisaru, W. Siegel and M. Rocek, Nucl.Phys. B159 (1979) 429, M.F. Sohnius and P.C. West, Phys.Lett. 100B (1981) 245,
[5]
J. Ellis, M.K. Gaillard, L. M a i a n i and B. Zumino in " U n i f i c a t i o n of the F u n d a m e n t a l P a r t i c l e Interactions", eds. S. Ferrara, J. Ellis and P. Van N i e u w e n h u i z e n , P l e n u m Press, N e w York, p. 69,
[6]
S. M a n d e l s t a m in Proc. 21st I n t e r n a t i o n a l Conf. on High E n e r g y Physics, eds. P. Petiau and M. Porneuf, Journal de Physique, C o o l o g u e C3 supp. au No. 12 (1982) p. 331, L. Brink, O. L i n d g r e n and B. Nilsson, U n i v e r s i t y of Texas Preprint UTTG-1-82,
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D.V. V o l k o v and V.P. Akulov, Phys. Lett 46B (1973) J. Wess and B. Zumino, Nucl.Phys. B70 (1974) 39,
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S. Ferrara, D.Z. F r e e d m a n and P. Van N i e u w e n h u i z e n , D13 (1976) 3214, S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335,
Phys.
[9]
Abdus
477,
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Lopuszanski
22
(1981)
939,
and M. Sohnius,
Salam and J. Strathdee,
Nucl.Phys.
Nucl.Phys.
B76
B88,
(1975)
109,
(1974)
Rev.
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J. Wess
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R. A r n o w i t t and P. Nath, R i e m a n n i a n G e o m e t r y in Spaces w i t h Grassm a n n - c o o r d i n a t e s , Proc. Conf. The Riddle of Gravitation, on the o c c a s i o n of the 60th b i r t h d a y of Peter G. Bergmann, Syracuse N.Y., 1975) General R e l a t i v i t y and G r a v i t a t i o n 7 (1976) 89,
and B.
Zumino,
Phys.Lett.
66B
(1977)
361,
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J. Wess
[13]
N. Dragon,
Z. Phys.
[14]
A. Rogers,
London
[15]
L. Bonora,
P. Pasti
[16]
J. Wess
[17]
M.
[18]
P. Howe,
[19]
F.A. B e r e z i n R. Arnowitt,
[2O]
P. Howe,
Nucl. Phys.
[21]
L. B r i n k
and P. Howe,
[22]
P.
[23]
P. Kallosh,
[24]
A. R e s t u c c i a and J.G. Taylor, J.Phys.A: Math. Gen. 16 (1983) 4097, J. Hassoun, A. R e s t u c c i a and J.G. Taylor, S u p e r f i e l d A c t i o n s for N = 2 d e g e n e r a t e charges, King's College p r e p r i n t June ]983,
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A. Rogers,
[26]
J.G.
[27]
R. Grimm,
[28]
M. Sohnius in "Superspace and S u p er g r a v i t y " , and M. Rocek, C.U.P. (1981),
[29]
F.A. Berezin, The M e t h o d New York - L o n d o n 1966,
[30]
B. DeWit and S. Ferrara, Phys.Lett. 813 (79) 317, M. Sohnius, K.S. Stelle and P.C. West, Phys.Lett. 92B (1980) 123, J.G. Taylor, Phys. Lett. I05B (1981)429, Ibid. 434, I07B (1981), V.O. R i v e l l e s and J.G. Taylor, Phys.Lett. I04B (1981) 131,
[31]
M. Awada, gravity",
[32]
and B.
and B.
Sohnius
Howe
M.T.
Zumino,
Phys.Lett.
C2
(1979)
University
Thesis Journ. 79B
Phys. Lett.
51,
(1981), Math. Phys.
(1978)
I053
and P. Townsend,
23
lec.
3199
(1982)
394,
(1981)
353,
Phys.Lett.
I073
(1981)
420,
Private
J. Wess
495,
Phys. Lett.
given
Phys. Lett.
883
(]979)
Nucl.Phys.
at
1981
1323
3181
Trieste
(]983)
268, (1981)
Workshop
487, on supergravity,
333,
communication,
and B.
"The g e o m e t r y
Zumino,
Nucl.Phys.
of Second
of the
3152
(1979)
eds.
Quantization,
S.W.
and W. Siegel,
255, Hawking
Academic
I
Press,
Spin 7 p a r t i c l e in N = 3 superRev. D,
ICTP/82-83/18 to appear in-Phys.
Grisaru
(1982)839,
and D.A. Leites, Sov. Math. Dokl. 16 (1975) 1218, P. Nath and B. Zumino, S u p e r f i e l d Densities,
and U. Lindstr~m,
Taylor,
Ph.D.
Phys. Lett.
and P. West,
(1978)
29,
and M. Tonin,
Zumino,
K. Stelle
74B
Nucl.Phys.
B187
(1981)
149.
236
COMPLETELY
INTEGRABLE
ISOSPECTRAL
SYSTEMS
PERIODIC
P.F. Kath.
chiral
"finite
fields
I.V.
the
and
surfaces "finite
Universiteit
Leuven
Mathematics
Moerbeke
the
linearisation
Riemannian metrical
Central Mumford
that
a set
ponds face
to ~
to [9]
of
a set . Any
Kac-Moody-Lie tors
the
Euclidean
Yang-Mills
algebraic
way
This
of
Dubrovin,
classical
[I]
approach
on r e g u l a r regular
is
the
the
points
difference
operator
algebra
via
isospectral
class
its
and
systems
of
P.
operators.
the
will matrix
is an o r b i t
remarkable found
on
by
McKean
Novikov
the
of
results
[6]
Jacobian
-
and of
a differential
van
This
Moerbeke theorem
difference
Jacobian be
described
8-functions
S.P.
of
are
principle
a
geo-
constructions.
theorem
periodic of
the
construction
different
isospectral
~
equations
difference
regular
of
KdV-type
Matveev
invariant
fields
means
with
V.B.
these
by
together
mechanical
suggests
relating
our
result
solutions
B.A. of
of
relativistically
dual
surface approach
of
in a p u r e [2].
[9],
solutions
self
zone"
van
D.
zone"
Cherednic
Riemann on
TO
OPERATORS
INTRODUCTION
The
by
RELATED
DIFFERENCE
Dhooghe
Dept.
0.
OF K d V - T Y P E
REGULAR
operators
J(~)
considered
of as
representation. in t h e
and
asserts
orthogonal
corres-
a Riemann an For K
element some to
surof
a
opera-
a subalge-
237
bra
K of
~
under
N to K in Z
. In t h i s
mine
a completely
[I].
The
Poisson
The written
case
structure
vector
field
of a m o m e n t u m
ference
operator.
is i n d i c a t e d
consider
a n d the
Toda
two
type
(N)-invariant
solutions a finite b y B.A.
determined
For are
algebra
The
linear
[6] are
flows
first
The
among
type
an o p e n
on the
"finite
of the
subset
of
set of r e g u l a r zone"
one
we
a finite its C ~-
points
solutions
type
finds
operator).
determines
and dense
is
of
described
them.
decomposition
is d e s c r i b e d
G, a n d the
systems
by
dif-
the L a g r a n g e
is i m p o s e d ,
the p e r i o d
of t h e s e
of the
a n d P. v a n M o e r b e k e
i ; namely
is
is r e a l i z e d
s u c h an o p e r a t o r
symbol
condition
(n e q u a l s
has
Jacobian.
of
field
of a g i v e n
second
on JG'
on Jg,
two c o n j u g a t e the
the
jet b u n d l e
jet b u n d l e
representations of c u r v e s
of c u r v e s
in
in the
g.
f o r m u l a t i o n is u s u a l l y u s e d to d e s c r i b e c o m p l e t e l y i n t e g s y s t e m s of K d V - t y p e , as for e x a m p l e in the w o r k of B. K u p e r -
grable
schmidt,
J
G. W i l s o n
When well
The
group
regularity
vector
This
detero n J(~)
on K ± m * N .
curves
of
on the
of M. A d l e r
on K ~
linearized
system.
existence
sublagebra
structure
integral
imposed
truncation
are
if a H a m i l t o n i a n the
[4]. The
which
functions
flows
integrable
n classes
by
the L a g r a n g e
given.
the Lie Lie
subsystem,
Dubrovin
of t h i s
the
contains
dimensional
along
of d e c o m p o s i t i o n
When
that each
dimensional
appear
condition
types
type.
whose
coad(N)-orbit
by the w o r k
Each
coad
is t h e
operator
of a c o m p l e m e n t a r y
ad(~)-invariant
system
of K d V - t y p e
by a regularity
will
action
of a p a r a m e t e r
ensured
As
the
integrable
equations
in t e r m s
Hamiltonian means
the c o a d j o i n t
known
Yang-Mills pressed
the
flows
field
not obtained
a n d V.G.
are written
the
on JG'
on t h e
KdV-type
infinite
constraint
equations
are
constraint
equations
building
Sokolov
[5].
truncations
o-fields,
[3] [7] . The
is v e r y w e l l
the
determine
self
fields
possible
dual
are
ex-
within
this
here.
equations,
jet b u n d l e .
imposed.
V.V.
different
equations
formulation
n o t be p u r s u e d
to the
Drinfel'd,
the p r i n c i p a l
Bogomolny
a bundle
but will
In c o n t r a s t
[10]
equations,a.o,
fields,
locally;
framework
[8]
But
u p an
the
They
field
appear
equations
when
there
is an
infinite
inverse
limit
of
sets
are
additional set of of
such
solutions
238
determined examined.
by
Since
whether the neral
finite
dimensional
the c o n s t r a i n t
algebraic
Jacobians.
equations
solutions
give
any
The
inverse
are n o t
limit
unique
information
is n o t
it is n o t
about
the m o r e
clear ge-
solutions.
The in t u r n
JG formulation
allows
a second
to f i n d a g e o m e t r i c a l
as J a c o b i
fields
Lie
with
group
determines
along
Lie
the
interpretation
geodesics
algebra
9
igi,
Hamiltonian
of a l e f t
i ~ ~m,
structure
of the e q u a t i o n s
invariant
for
which
metric
on a
s o m e m.
The above results generalise to any semi-simple Lie algebra using the theorems of M. Adler and P. van Moerbeke [I] .
I.
KAC-MOODY
ALGEBRA
DECOMPOSITION
AND
REGULAR
PERIODIC
DIFFERENCE
OPERATORS
sider
Let
g be the L i e
the
following
algebra
{m
=
The
algebra
namely
(resp.
the p o s i t i v e
obtained
a basis
is o b t a i n e d
~+)
by
powers
of a d - i n v a r i a n t
functions
determines
p,q
for the
induction
of
set of
function
functions
Qi determines
~ i 6 g, m
i e~.
functions
Sl(n) . We c o n -
) . The b r a c k e t s
on b o t h
write
on g of d e g r e e
by Qi,k will form which
Qi'
sides,
~ p for powers
on t h e s e
on g. L e t
i. The
in
algebras
i=2,...,n
be
set of t h e s e
functions
: Res £ : 0 A k - i
i-linear
~
]
of the n e g a t i v e
a set of a d ( ~ ) - i n v a r i a n t
an
of
We w i l l
subalgebra
of A
generated
group
arbitrary
by truncation
~ 0,
the b r a c k e t s
Qi,k (~)
The
a n d G the Lie
algebra
~ i ~il
= 0 = A p+i,
( - ~ 'P) , ~ _ ( r e s p .
are
. E
~ (-q'P)
~ -q-i
sl(n)
Kac-Moody-Lie
on
~
by
Qi ( ~ A )
be d e n o t e d we will
by
~ (~) . E a c h
denote
b y the
same
symbol.
Let K denote ciated fined
the K i l l i n g
quadratic
function
f o r m o n g.
We w i l l
assume
on g. T h e
induced
Killing
t h a t Q2
is t h e
form on
~
by K 0 = Res£
By constructing
an
infinite
=0 ~ - ] K
dimensional
matrix
representation
asso-
is d e -
of
239
(-q'P)
as f o l l o w s
6o
any
~ 6
~
represent upper cij
(-q'P)
corner
M'
R by
of a g i v e n
= Ci+n,j+ n
outside
defines
the m a t r i x
. The
the b a n d
a difference
r o w of
-M'
~ i-j
> 0, M > 0 a n d M + M'
(cij),
R with
where
$ 0" The p e r i o d i c i t y
R has ( M.
> 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<2
p-1 /-
p
f
z P.
.
The p r o j e c t i o n s
({iz i .+ ~iz_~) .
>('
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
[<0 el .~
=
0
~0 =
The
evolution
CONJUGATED QUADRATIC
Let
9
9 =
the
In this
tails
> ~
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
~ <~,e> + + A
is the Laplacian operator of the Riemannian manifold
(J{,g)
and A is a function on J~ defined by A(x): E <8(S(ei)) , e(ei)> - 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,~,~)
= <~,N]~
S the endomorphism of
that is, S(ei)= E Skl e k, k
and
-DM(DN~)+DN(DM~) ,~>, 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 <(ZlDe)(ei), i
@(ei)>
E ~(ek,ei,8(ek),fig(ei)) i,k
:
+ E <8(S(ei)) ,8(ei)> i
-
E = i,k ek ¥ = A(x) - E i,k k
-
It is easy
-
to see that
E
(Ei) , 8(ei)>
If the m a n i f o l d inner
product
(8,~)
(~,g) of two
: f(J<8,8>)
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
<(9(Eil ..... E i ),~(Es,Eil ..... E i )>) : 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-
II
'IN,
2.7.
- If
(F2,F2),
then
i) RD (A,B,C,D)
-
:
-
1%P/O/. (~,g,P)
R(A,B,C,D)+
is a R i e m a n n i a n
almost-product
~II ~vII 2RI (A,B,C,D) 4p 2 1 c~ ,/6 4--~ g( ,X)R I (A,~ ,B,C)
ii)
RD(A,X,B,C)
: R(A,X,B,C)+
iii)
RD(x,Y,A,B)
= R(X,Y,A,B)
: 0
iv)
RD(A,B,X,Y)
= R(A,B,X,Y)
= 0
mani-
273
v) RD(A,X,Y,Z) vi) ~A,B,C,D6
RD(A,X,Y,b/)
1 ( 2~,A)RI(S~,X,y,z 4--~g
= R(A,X,Y,b/)+
1 ii J6[i 2 RI (X Y,Z,W) ~q2
~ ,X,Y,Z,W6 ~
respectively,
The proof
, where
)
p and q are the d i m e n s i o n s
of
and
and
R I (L,M,N,O)
i
= R(A,X,Y,Z)+
: g(L,N)g(M,O)-
follows
easily
g(L,O)g(M,N)
from P r o p o s i t i o n s
FL,M,N,Oe 2.3.
~(~).
and 2.6.
In fact,
RD(A,B,C,D) = R(A,B,C,D)+ ~(~7AP)C ,(~BP)D)- -gl 4 ((~BP)C,(~7AP)D)
=
= R(A,B,C,D)+ I[~II2 (g(A,C)g(B,D)- g(B,C)g(A,D)) 4p2 ii
I RD(A,X,B,C) = R(A,X,B,C)+ ~
g(A,B)g(~
I - 4-p g(A'C)g((~7~ )B'sD) I = R(A,X,B,C) + ~
L~ '
(V~)c)
-
=
g(A,B)g(( V ~ ) ~ ~ ,C) -
I - 4-p g(A,C)g(B, ( V ~ ) < v ) : I = R(A,X,B,C)+ ~-~ g(X,~ ~) (g(A,B)g(~,C)
-
- g(A,C)g(B,s ~)). iii
4(RD(x,Y,A,B) - R(X,Y,A,B)) : g((C7~)A,(~7~)B) - g((~7~)A),(~7~)B)
:
= g((V~) ( VXPm,B)- g(A, (V~) (VxP)B) = I I = ~ G(Y, (~7~)A)g(~ $6 ,B)- ~ G(Y, (~7~)B)g(A,~ ~ ) = =
i__ 2 g(Y'X)g(~ 'A)g(~A? ,B)- ~ g(Y,X)g(~ ~ ,B)g(A,~ J() = 0. q
O. Gil Medrano manifolds
of type
q
proved
in
[4] that
in the R i e m a n n i a n
almost-product
(F2,F2)
R(X,Y,A,B)
= 0
VX,Y~
The proofs
of parts
iv),
2
, A,B6
~.
v) and vi)
are a n a l o g o u s
to those
of iii),
ii) and i), respectively. Obviously, have
in any R i e m a n n i a n
almost-product
manifold
(~,g,P)
we
274
RD(L,M,A,X) : RD(L,M,X,A) : 0
AN APPLICATION
3.
RIEMANNIAN
The
OF T H E W E I T Z E N B O C K
ALMOST-PRODUCT
characteristic (~,g,P)
dD6 D and
~ D, as d e s c r i b e d
Riemannian
valued
in the
induces
PROPOSITION fold
of t y p e
i) ii) iii)
the in
almost-product tangent
MANIFOLDS
connection
manifold
the
~ L , M d ~ ( ~ ) ,Ac- 0 , X G ~{
vector
3. 1. - If
FORMULA
IN THE
OF T Y P E
(F2,F 2)
D of a R i e m a n n i a n
covariant
structure bundle
the
~ and
tensor
field
can be c o n s i d e r e d
T(~)
(v~,g,P)
almost-product
differential
I. M o r e o v e r ,
COMPACT
the
operators
P defining
as a l - f o r m
.
is a R i e m a n n i a n
almost-product
(F2,F2) , t h e n
: 2~I II~I12 + ~I l{aX/12 off 2 + I
= -4I //
<6Dp,6DP>
: ~1 ( II ~ / / 2
//<1211 2 +
II
cx
il
2
Proof. Let Lw a n d
[Eal P a=1 , j respectively w
IE u I pu:p+1
be l o c a l
orthonormal
n
i) =
g((~P) (Ei,E j) , (~P) (Ei,E j) ) : i,j=1 n
=
E
i,j=1
g((~E P)E i,(~E. P) (Ei) 3 O I
and since
(DE P ) E i : DE. PEi - P ~E. Ei : 2(~TE.P)Ei ' 3 O J 3 n 4~P,DP> = E g((~E.P)Ei,(TE.P)Ei) : i,j=1 l 3 n g((~7~p)Ea,(~7~p)Ea) + a,b=1 p n g((]7 E P)Ea,(~7 E P)E a) + u u a=1 u=p+1 p n E 7. g ( ( V E P)Eu' ( ~ E P)Eu) + a a a= I u:p+ I n u, 0 =1o+I
frames
for
mani-
275
II a~ll
=Z
2
I
+
P E
--
P
n E
q a=1
+!
p~. a=1
P
g(Eu,(~7 E P)Ea)g(a~,E a) + u
u=p+1
nE g(Ea, (~7E P)Eu)g(a¢ ,Eu)+ I /[ct'~(12 u=p+1 a
=2 llJ;I 2
+]2 ,11zll
P
2
n
ii) = ]I i,j=IE g((dDp) (Ei,Sj), (dDp) (Ei,E j) ) = n
~ ~ g ( (D E P) Ej _ (~E P) Ei' (DE P) Ej _ (DE P)Ei) : 2 i,j=1 i j i 3 I
I 4
Z
n
1
= ~
(g((~E P)Ej'(~7E P)Ej) - g(([z E P)Ej,([7 E P)Ei)) = i i i j
i,j=1 P
n
E a=1
E g((~z E P)Eu,(~z E P)E u) + u=p+1 a a
I P + ~ E a=1
n E g((~7 E P)Ea,(~z E P)E a) = u=p+1 u u
I
2
1 i1~11 2
4-~ n
iii)
<6'-'p,,~"p>
:
g( (~DE.P)Ei, (DE.P)E j) : i,j=1 i 3 P I £ g((~z m P)Ea,(V~P) ~ ) + a,b=1 a n
z
g ( ( V E P)Eu,( V F ~ ) E 1) :
u,~=p+1 COROLLARY fold of type
3.2.
u
- Let
(,1Dp,p) where
: p+l 4p
l/ ~ "° ll 2 ( vZ ) : /..ff. The proof
follows
DEFINITION
3.3.
one can define
(~,g,P)
(F2,F2) l o r i e n t e d
11 ~'11 2 (~)
from parts
+
T~
n E a,b=1
2)
almost-product
mani-
Then
Ii ~11 2 ( j . ( )
l, a ~r ll 2 ( v~) ii) and iii)
- In a R i e m a n n i a n p
II o11.. 2 + II c~ll
be a R i e m a n n i a n
and compact.
II ~ v ll 2:~1,
1 ~-(
= Jr{
in the proposition above.
almost-product
R(Ea,Eb,Ea,Eb)
n E R ( E u , E ~ , E u , E v) U,~=p+1
ll a ~ l/ 2~ l
manifold
(u~/,g,P)
276 ug~ :
p ~ a= I
where
~Eal
a= I ..... p and
for ~
and
~ , respectively.
It is o b v i o u s written
n E R ( E a , E u , E a , E u) u=p+ I
~ Eul
u=p+ I ..... n are
that the s c a l a r
curvature
local o r t h o n o r m a l
of
frames
can be
(~,g,P),
as =
PROPOSITION manifold
3.4.
of type
<~Dp,p)
+ 2 ~
~
- Let
(~,g,P)
+ ~ be a R i e m a n n i a n
almost-product
(F2,F2) . Then
= 2Z~
_ m-3 [i uii 2 _ q-3 ~ 2~ 2 4p 4q
Proof. From Theorem
1.3.
it f o l l o w s ~
Now,
n
:
from Proposition
+
n
E g(S(Ei) ,Ei) i=I
-
E i,j:1
R D (Ej ,E i ,PEj ,PE i)
2.7.
n E
RD (Ej ,Ei,PEj ,PEi ]
~9
:
+
i,j:1 + 7 J{ +
=
"~
+ z
JI~ff 2 4q 2 J{
[i vii2 4p 2
p E m I (~,Ea,%,E 2 a,b=1
n E R I (Eu, Eu,E~ ,Eu) u, O =19+I
p-1 + 4p
II ~Ull
2
q-1 + ~
The r e s u l t
follows immediately from part n and the fact that E g ( S ( E i ) , E i) = T i=I
+
:
i1~112 i) in P r o p o s i t i o n
3.1.
l
THEOREM f o l d of type
where
3.5.
- Let
(u/f,g,P) be a R i e m a n n i a n
(F2,F2),oriented
7/
(v~)
= P-14p
~
(~)
=
To p r o v e use C o r o l l a r y
this, 3.2.
and c o m p a c t ,
almost-product
mani-
then we h a v e
I[~'ll 2 (V~) + ~q-1 ll .~ll 2 (v~) Z
integrate
X I.
the f o r m u l a
given
in P r o p o s t i o n
3.4.
and
277
4.
GEOMETRICAL
Throughout
CONSEQUENCES
this
section
(~,g,P)
will
be
assumed
4.1.
- Let
(~,g,P)
be a R i e m a n n i a n
Z~I(~)<0,
then
(~,g,P)
is not
to be o r i e n t e d
and
compact.
COROLLARY fold.
fold
If
The p r o o f
follows
COROLLARY
4.2.
satisfying
vertical Then
one
mension
I or
- Let
Z2~
or the
the o t h e r
trivially
from Theorem
(//,g,P)
horizontal
one,
cannot
it is t o t a l l y
one
mani-
(F2,F2).
3.5.
be a R i e m a n n i a n
(u}i)K0 and w i t h
distribution
almost-product
of type
of the
is a t o t a l l y be of this
almost-product
mani-
two d i s t r i b u t i o n s , umbilical
kind,
unless
the
foliation. it has
di-
geodesic.
Proof. Let from
19
be a t o t a l l y
Theorem
3.5.
umbilical
P4pI I1~11 2 ( ~ ) and
if q#1,
then
~
if X , Y 6
The obtained be e a c h
of the
Z
fold
of type of
horizontal (v~,g,P)
4.3.
Wx,YE
if so,
too,
then
= o
Z and
of C o r o l l a r y the
therefore,
4.1.
hypothesis
(
~{
(Corollary
T#~(~t)<0(
is t o t a l l y
4.2. [~
geodesic.
resp.)
can be
(wzi)i0,
resp.)
K 0, resp.)
sectional
-
Let
(F2,F 2) w i t h
curvature
(~,g,P)
and
and
leave
vertical
is a l o c a l l y
follows
(non p o s i t i v e ,
be a R i e m a n n i a n
p and @ > I, and
the v e r t i c a l
The p r o o f
~
following:
< 0
Negative
q-1 }i ~// 2 + ~-~ (~)
~TxY ~ 2 ,
conclusion
COROLLARY
vatures
then
substituting
one
i) ii)
~
same by
If
= 0. H e n c e
(Vxp)Y = 0 where,
foliation.
we h a v e
horizontal
product
almost-product
satisfying
foliations,
respectively,
resp.).
that
the m e a n
restricted
have
divergence
fact
proven
manicur-
to e a c h zero.
Then
manifold.
immediately
from
the
by F.J.
Carreras
278
in
[I],
if
that
n Z u=p+1
in the
Riemannian
~ g(~7 E
~
,E u)
u
almost-product
p Z a:1
manifolds
of type
(F2,F2),
~ g( ~7E
~f
,E a)
: 0,
~
K 0 .
a
REFERENCES
[I]
C a r r e r a s , F.J.: " L i n e a r i n v a r i a n t s of m a n i f o l d s " , Math. P r o c . C a m b . Phil. Soc.,
[2]
Eells, J . , L e m a i r e , L. : "A report Math. Soc., 10 (1978) , 1-68
[3]
Eells, J., L e m a i r e , L.: " S e l e c t e d t o p i c s in h a r m o n i c m a p s " . C . B . M . S . R e g i o n a l c o n f e r e n c e s e r i e s of the N a t i o n a l S c i e n c e F o u n d a t i o n , (1981)
[4]
Gil M e d r a n o , O.: " C i e r t a s p r o p i e d a d e s g e o m e t r i c a s y t o p o l o g i c a s en a l g u n a s c l a s e s de v a r i e d a d e s r i e m a n n i a n a s c a s i - p r o d u c t o " , T e s i s d o c t o r a l . P u b l i c a c i o n e s del D e p a r t a m e n t o de G e o m e t r i a y T o p o l o g ~ a no. 7, U n i v e r s i d a d de V a l e n c i a , (1981)
[5]
Gil M e d r a n o , O.: "On the g e o m e t r i c p r o p e r t i e s almost-product s t r u c t u r e s " , R e n d i c o n t i Circ. (preprint)
[6]
K o b a y a s h i , S., Nomizu, K.: " F o u n d a t i o n s in d i f f e r e n t i a l I n t e r s c i e n c e Publ. vol. I, II, (1962) , (1969)
[7]
M s z d i d j o , s . Y.: " V e c t o r b u n d l e sions of R i e m a n n i a n m a n i f o l d s " ,
[8]
Miquel, folds",
[9]
M o n t e s i n o s , A.: "On c e r t a i n M i c h i g a n Math. J. 30 (1983)
Riemannian 91 (1982),
on h a r m o n i c
almost-product 99-106
maps".
Bull.
London
of some c l a s s e s of Mat. di P a l e r m o ,
geometry",
v a l u e d h a r m o n i c forms and i m m e r C s a k a J. Math. 8 (1971), 1-13
V.: "Some e x a m p l e s of R i e m a n n i a n P a c i f i c J. of Math. (preprint) classes
A.M.: " V a r i e d a d e s f o l i a d a s Math., 21 (1970] , 1-61
almost-product
of a l m o s t - p r o d u c t
structures",
[10]
Naveira, Collect.
[11]
N a v e i r a , A.M.: "A c l a s s i f i c a t i o n of R i e m a n n i a n a l m o s t - p r o d u c t m a n i f o l d s " , R e n d i C o n t i di Mat. Roma, (preprint)
[12]
Poor, A.W.: " D i f f e r e n t i a l Company, (1981)
[13]
R e i n h a r t , B.L.: A n n a l s of Math.
[14]
R e i n h a r t , B.L.: " F o l i a t i o n s and s e c o n d f u n d a m e n t a l forms", Proc. of the IV Intern. C o l l o q u i u m on Diff. Geom. S a n t i a g o de Ccmioostela, (1978), 246-253
geometric
"Foliated manifold 69 (1959), 119-132
con m e t r i c a
mani-
structures",
with
casi-fibrada".
McGraw-Hill
bundle-like
Book
metrics",
279 [15]
Reinhart, B.L.: der Mathematik,
"Differential G e o m e t r y of Foliations", S p r i n g e r - V e r l a g (1983)
Ergebnisse
[16]
Tachibana, S.: "On a c h a r a c t e r i z a t i o n of R i e m a n n i a n spaces admitting minimal subspaces c o m p l e m e n t a r y orthogonal", Nat. Sci. Rep. O c h a n o m i z u University, 22 (1971), 85-89
[17]
Vidal, E., Vidal-Costa, E.: "Special C o n n e c t i o n s and almost foliated metrics", J. Diff. Geom. 8 (1973), 297-304.
EINSTEIN
EQUATIONS NON-LINEAR SELF-DUAL
WITHOUT
YANG-MILLS
Norma
ER Departement
KILLING
SIGMA MODELS
VECTORS,
AND
THEORY
Sanchez
176-CNRS,
d~Astrophysique
Observatoire
Fondamentale,
de Meudonl
92190-Meudon,
FRANCE
Abstract:
We analyze dual Y a n g - M i l l s dual,
with
equations tons,
the c o n n e c t i o n theory
and without and at the
instantons
between
and G e n e r a l Killing
level
non-linear
Relativity
vectors),
both
of the d i f f e r e n t
and calorons)
of these
sigma models,
(self-dual
at the level
types
theories.
self-
and non
self-
of the
of solutions
(soli-
281
A fundamental
quantity
in Q u a n t u m
Field
Theory
is the f u n c t i o n a l
integral
Z =
which
enables
configurations classical nal
the c l a s s i c a l
of motion,
equations
of the time
axis
points
being
~[¢]
non-linear,
in the c o m p l e x e
regime
is the E u c l i d e a n
admits
the p a r t i t i o n T
is d e f i n e d
another
level,
t-plane,
fL
action.
-I
(kS)
path
integral
satisfy
which
In order
a Wick
to
rotation
gives
^
d [¢] e -I [~]
This p r o l o n g a t i o n
to the E u c l i d e a n
independent
interpretation
of a finite
temperature
system.
the
temperature
is p e r f o r m e d
theory,
solutions
In the Table
detailed
family
t=i-c,
of
saddle
of
Z as
In that case,
in the range
of solutions
thermal
the
alternative,
function
being
In the q u a n t u m
types
contributions
solutions
level,
on makes
the
functio-
only
~
< B
,
of the theory.
over
all the
In this case,
the
field c o n f i g u r a t i o n s
that
the c o n d i t i o n
but also tions.
the
are non-local.
integral,
0 < T~ =
i.e.
field
satisfying
the d o m i n a n t
At the q u a n t u m
of the path
all
by the c l a s s i c a l
of the action,
Z =
where
as a sum over
not n e c e s s a r i l y
and w e i g h t e d
of motion.
besides
the c o n v e r g e n c e
d[¢])
objects
semiclassical
stationary
equations
improve
quantum
(with m e a s u r e
I[¢] . At the
from the
point
¢
equations
action
come
us to c o n s t r u c t
I [¢] d [¢] e i
~¢
discussion properties
not only
at i m a g i n a r y
solutions time
I b e l o w we d i s p l a y in F i e l d of the
Theory
topological
instantons,
time
entirely
schematically
and their
Sitter
see ref.
are
important,
complex
the main
characteristics.
invariants
of the K e r r - N e w m a n - d e
of g r a v i t a t i o n a l
at real
and even
associated
soludifferent For
and T a u b - N U T - d e
[I].
a
to the Sitter
and persist in time)
which have f i n i t e energy
(Solitons in real time
SOLITONS
d=2 : VORTICES
d=3 : MONOPOLES
d=4 : I~ISTANTONS
(F = + *F)
I pl
Higgs f i e l d
[~]
.
Solutions of a non-Abelian [SU(2)] gauge theory coupled to a
I = 292
i~
is such that
gauge theory coupled to a Higgs
U(1)
~I~Im
at the spatial i n f i n i t y
[ ~].
Solutions of an Abelian field
(superconductivity model of Ginsburg-Landau).
that the f l u x of the magnetic f i e l d is quantized
I~I,= < ~>vacuum and ~00 =fA@d@. Topological invariant: an integer, degree of the map SI--> S1 and means
Example:
the zeroes of ~ ).
Topological invariant: an integer, the winding number, degree of the map $2-* S2. This is the magnetic charge (carried in
Example:
Euclidean action:
Topological invariant: an integer, the Pontrjagin number P = -(8~)-2 ~JFAF, degree of the map S3-~S3"
Example: Self-dual Yang-Mills theory
EUCLIDEAN (++++) signature regime
SOLUTIONSIN FIELD THEORY (d stands for the dimension of the Euclidean space)
LORENTZIAN (-+++) signature regime
TABLE I:
being periodic ( 0 ~ Topological invariant: d i f f e r e n t i a l winding number of the mapping.
(N on-
Selfdual
of e l e c t r i c type monopoles
(which may rotate)
B-I
L
f
for Schwarzschild
8~[M+M2(M2+12+Q2) - I / 2 ] for Kerr-Newman
8~M
covered by d i f f e r e n t "patches". O ~ 8 m n
This removes the Dirac string and horizon type s i n g u l a r i t i e s .
(Self-dual
Riemann-
curvature)
as magnetic type
gravitational monopoles
(the only solution is f l a t space).
-Asymptotically Euclidean (in 4 dimensions): are t r i v i a l
instantons. I = O.
sort of (twisted) vacuum state. Gravitational analogues of the Yang-Mills
-Asymptotically Locally Euclidean (in 4 dimensions): They describe some
Euclidean Action: ~ = 4~n 2
the gravitational f i e l d .
Topological invariant: "NUT' charge r e f l e c t i n g the magnetic aspects of
(n is the number of instantons).
-Asymptotically Locally Flat in 3 dimensions: The 3-spatial i n f i n i t y is
TAUB-NUT's:
TAUB-NUT's
related to the Euler number.
Euclidean Action: ~ = B2/4~, interpreted as the i n t r i n s i c entropy and
T
, ~)
of the manifold. ( ~ k ' ~k ) are Kruskal type coordinates and ( r
are of Schwarzschild type. This gives the temperature of the solution.
Riemann-
curvature)
~Ck* ± i ~ k = e2~B~T~ * + i ~) defining the maximal analytic extension
~B). They describe f i n i t e temperature states.
which may be considered
and by
Asymptotically Flat in 3 dimensions, the fourth imaginary time dimension,
BLACK-HOLES:
by
BLACK-HOLES which are the analogues
with (++++) signature (J~can be zero or non-zero).
~,~
INSTANTONS
of SOLITONS are provided
=-/~g~v
Complete non-singular solutions of the Einstein equations
GRAVITATIONAL
In GRAVITY, examples
284
We ask o u r s e l v e s types
if there
of solutions
lity p r o p e r t i e s logies
of these
and e x p l i c i t
physically (SDYM)
field
with
two K i l l i n g
with
the gauge
adjoint
problems
vectors
spherical
or axial
status
models tion
to the
[5,6].
to solve
The c o n n e c t i o n tions
possible
between
:
~0(~0,xil
~i
:
~i(xi
for
gMv
These
links
Wether
to the Toda
have
of the
been
and to the Ernst techniques
in the
subtle
configurations
a general
in the
equa-
developed of
[7].
equa-
feature,
space-time
of the E i n s t e i n
usesigma
problem
of the E i n s t e i n
is in fact
field
(in the
equation.
of t w o - d i m e n s i o n a l
structure
the
sigma mo-
SDYM e q u a t i o n
field
is e q u i v a l e n t
[3,4]
of syn~etries
under
static
and B i c k l u n d
a 3+I d e c o m p o s i t i o n
it is c h r o n o m e t r i c
equations
SDYM field
the n o n - l i n e a r
~0
I
[2].
) multi-monopole
is invariant
The
properties
sigma model
non-linear
For a clear picture
have been useful
of the p r e s e n c e
which
(I)
i.e.,
energy
to p e r f o r m
~9 = 0,
see
Yang-Mills
the g r a v i t a t i o n a l
(Bogomolny)
this
of ana-
describing
self-dual
as the Higgs
respectively.
the a l g e b r a i c
equation
i.e.
formulation).
connections
and the n o n - l i n e a r
irrespective
formulation,
integrability
Conversely,
stablished:
different
and i n t e g r a b i -
a number
equations
gravity",
is assumed,
these
structure
last years,
the m o n o p o l e
four-dimensional
(finite
been
identified
equation
the
the Ernst
generating
A0 gives
of these
ful for e x t e n d i n g
reduced
symmetry
chain or to the Ernst
In the
have
(Ernst
potential
between
non-linear
("R gauge")
"two-dimensional
representation)
present
theories.
in the Yang
and
connections
the n o n - l i n e a r
links b e t w e e n
unrelated
dels
exist
and b e t w e e n
[8].
It is
equations
transformation
{± : 1,2,3)
invariant
(CI) . The
are then p r o j e c t e d
four-dimensional
Einstein
into a t h r e e - d i m e n s i o n a l
space
of m e t r i c
hij in w h i c h ~
'l =
linear
(Cl) ~ 'l -
operators
conformal
the p r o j e c t e d
metric
replace
structure
new set of v a r i a b l e s
- goigoj/go0
are defined.
(g0i/g00) ~ 0
sigma model
expressing
: gij
lies
equations
(two kinds
¥ij
= V hij
[For instance,
the o r d i n a r y
in the e q u a t i o n s ~00
= 0,
of p o t e n t i a l s (instead of
the d e r i v a t i v e s
ones
~0i
= 0
V,~) hij
~ i ] . The non~0V
and
= 0.
By
in terms in terms
itself)
an
of a of the
0(1,2)
285 sigma
model
The
structure
(V,~)
one
by
The
o
for the
parametrization
V = I/(oi+o3), field
responding (++++)
lies
tential
@ = o2/(oi+o3),
where
in a t h r e e - d i m e n s i o n a l
an
~ + = V ! ~]-
with
sition one
respect
of
the
recovers
This
3+I
some
kind
aspects
~7
0(2,1)
~
model
i
decomposition of D u a l i t y
Yij
Ernst
in G r a v i t y ,
with
emerges.
(oi,o2,o 3 )
metric
¥ij'
equations
too b u t presence
ordinary
it g i v e s
field.
With
for a real of one
(3+I)
po-
Killing
covariant
the
cor-
with
ones
decompo-
two K i l l i n g
vectors,
formulation.
well
namely,
field.
[" # i' *~)o ] [ ~ i
the
and
is p a r t i c u l a r l y
gravitational
space
appears
become
= V+i~
( o i ) 2 + ( o 2 ) 2 - ( o 3 ) 2 = I.
in the
gravitation
the w e l l - k n o w n
~
standard
[For E i n s t e i n
1
~
to the m e t r i c
stationary
of the
..
In p a r t i c u l a r ,
v e c t o r ' t h e CI d e r i v a t i v e s ~i
potential to the
to Cl d e r i v a t i v e s
signature,
complex
is r e l a t e d
the
Note
=
S
i
suited
for r e c o g n i z i n g
"magnetic"
and
"electric"
that
°SO
*
~ k ] = 2Aik
0
where
~ 0 = 1/ ~
~-~0
=
~i
Aik
: 2
Ei
= -
i
V ~ g00
and
~i
~.
k @i
-
02
-
i
o
i@k )
(2)
Here, Ei
the
notation
describe
the m a g n e t i c
respectively. Aik of
.
1
is i n v a r i a n t . the
(eq.
stationary
I) are
1 The
~
l
are
changes
electric
under
Aik
and
Ei
case.
With
one
= X 0 +~(xj)
the
~ g0i/g00 parts
generalize
~J
is used.
of the
= X 9,
(eq.
the h o m o l o g o u s
vector,
the
under
Aik
and
gravitational
transformations
Killing
and
(V~i)
no
which
generalizes
symmetries
in the
the
Ernst
space-time
equation is g i v e n
quantities
transformations
which
to the c a s e by
field,
I) but
3
equation
there
~0
and
~
when
286
_
• ~7 2(V+i@)
1 ~[
~'
~7(V+i¢) ] 2
*
+
~Tk(
9 0 ~k)
+
(3)
+ 1 [~ 2V 2
Besides
0 ~k )2+(*~
the p o t e n t i a l s
(*~ 0 ~ k )
Here
and
Dik = 21
Dik
"3
21 DkDk + 41 D i k D ik = 0
~k ) (*~ k¢)]+
o
(V,¢)
this e q u a t i o n
which
vanishes
0Di~'
D =
'9
when
0
in
involves
other
stationarity
fields
is required.
lhl
The e q u a t i o n s I
Ak
(4)
V
~ (
~Tj (h ij D - D ij)
define
¢
and
The p r o j e c t e d
ik
:
(5)
corresponding
-
(Dj
~0Dik
(Dij+Aij)
j
(~
0 ~k ) = 0
to
+ Aj)
1
~ik
+ DDik
{
Ricci
In terms
vector
field,
'
(6a)
(6b)
1
+ 3 A i j A ~ + EiE k + 2 { V i E k
of
(V,¢)
~72~
(3)~
ik
and
_
(3)
tensor
~ik
=
0
calculated
the full
with
set of e q u a t i o n s
to
2 Ak Ak + ~ k Ek - EkEk + 2 ~ T k ( A k / ~ ) (3) ~ i k
+
+
h. z3
of one K i l l i n g
(5) reduces
is
- DijDJ
+ ~TkEi)
is the t h r e e - d i m e n s i o n a l
In the p r e s e n c e
= 0
.~
+ EiEk + 2 ( V i E k
to the metric
(3) and
- e ijk ~ j
equation
(3)~i k
respect
0 7k )
( 8 0 ~k ) .
+ 3 AijA
where
*~
~k @ +
1 (Re~)
¥ij
= V hij
(~z~)
I ( Viv 2V 2
2
VkV
= 0 +
: 0 +
~kEi } : 0
these
,
equations
~ = V + i¢
~7i¢ ~ k @)
= 0
read
287 In t h e
presence
in t h e
canonical
(7)
dS 2 =
where and
of
¥ij
In t h i s
I.
case
dxj
eqs.
(6)
R33
: A ~ -
~3
94¥
~4S S
: 0
These
(8c)
must
be
I - 2V 2
[ ~3 V ~4 V
_
[( V 4 V ) 2
situations
~2 3S -
compatible
(3)
Different
on
X3
and
X4
I 2 2] (Re£) [2,,(~3E) - ( ~ 4 £)
D 4Y + ~ ~
give
[( ~ 3 V ) 2 +
1 2V 2
only
+ $2(dx2)2]
a3 s
4Y + ~
_ 2 V21
(3)R44
i
~2 4 s : 0.
with
(~73¢)2
+
V3¢
~ 4 @]
+ ( V 4 0 ) 2]
described
by
the metric
(7)
are
axially symmetric stationary f i e l d s (for ~ = - I, I 2 3 4 x : t, x : ~ , k = p , x : z)
(ii)
cylindrical
(iv)
plane
non-axially : t,
For
waves
(for
waves
symmetric
~ = I, x (for
the
situations x4
and
the
I
(i) , (ii)
x4 = y
and
coordinates
= z , x
2u = 1,
stationary
x 2 = z, x 3 = x,
or
: 0
~ 3 Y - - -S
(i)
(iii)colliding
be written
2 23,4 S S
~4 s
~2
(3)R33
R34
A(dx4)2
a4s
~ 3y
S
(j : 2 , 3 , 4 )
eqs.
can
2 ~ + ~_3.s. 9 3 / + ~ 94s 94 r ~ sS S ~
+
3'; -
z3
g~
dx i dx j
(__~)~4] [
~4 s
_
me{ric
read
{3)R4 4 : ~, ~2 (3)R
-
depend
~ 2 + 2u(_~_)~ 3 4 -
-
(3)R34
S
a3s
[~3
the
+ rij V
[(dx3)2
and
(3)
(8b)
vectors,
~2dx2)2
: e2Y
V,~,¥
2
(8a)
Killing as
KV(dx I +
dxi
A = ~
two
form
=
~
, x
I 2 x = p , x = ~
fields
(for
3
= p , x
, x
3
4
= t)
= z , x
4
2~ =-I,
)
(iii) , are
2
of
S
is
taken
cylindrical
equal
type
either
[9].
All
to these
t)
288
situations which
correspond
are n o t
to
solutions
holomorphic
functions,
the C a u c h y - R i e m a n n
equations.
these
S
situations
For
the
are
of C a r t e s i a n
situation
morphic
is n o t
(iv),
functions,
i.e.
i.e.
This
and
¢
v=
x
8
and
the E r n s t ~
do n o t
to the
equation satisfy
fact
that
for
[8].
to a c o n s t a n t
corresponds
V
V
of
is a s s o c i a t e d
is e q u a l
This
= V + i¢
a constant
S
type.
~
a n d the
to s o l u t i o n s
~
coordinates
given
by h o l o -
satisfy
¢
y
(9)
The
holomorphy
neralized
to
ansatz
include
¢
where
W~
F r o m eqs.
2. 1
X + iy
=
(8b-c)
and
[I(~)
2y = in [
u
imposes complex
~--[E
=
=-~x ~
v
Y
1
(?)
,
E'
W ~
(4),
+
(from eq.
8a)
solutions,
for
2
S : const.
This
c a n be g e -
namely
(.~)] x - iy
=
¥
and
~o a
tO Z
one
gets
~2(@)
2
] + gl (~) + g2 (~)
~x ¢
-
Y
V2
9>,¢~ ~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 + E
and s u c h that
Therefore
the
(17)
g
j(j+
m ~T)
self-duality
r[ @[ (j-1 9{ j)
:0
eq.
(15) t a k e s
+ g~
- )~
is a 4 x 4 c o m p l e x
the form
(j-1 ~7 J)
non-sigular
=
0
matrix,
element
of the GL(4)
group.
Since
the
I
/Z s
follow
from a metric
they s a t i s f y
the s y m m e t r y
condition
293 d
i.e.
In t e r m s
of
J
this
condition
gives
(18)
=
In the FB ral
gauge
and
f~
J
as d e f i n e d
are
those
by eqs.
of the
(15),
type
the
~ ~B
non-zero
and
&~ ~
components
.
J
has
of
as g e n e -
solution
j =
t
0
gc~B
with
(19) and
~ det
through
= g = I
: ~
g~g
(the d e t e r m i n a n t
a conformal
the b a s i s
gag
g
transformation.
(~'~'f'~),
/-b, F ~
,
c.c.
c a n be a l w a y s
I is the
are
given
2 x 2
set e q u a l identity
r
'Z
(17)
reduces
For
hermitian
Yang-Mills defines
a K~hler
and
of a
(modified)
2n
g,
theory
null
n ~ 2,
to a 2 x 2 e q u a t i o n
matrices SU(n)
provides
dimensions.
general
gT ~ ~
(V,~,C)
namely
equation
for
is t h a t
the a d d i t i o n a l
type metric.
identity
g
this with
a K~hler
In o r d e r
_
_J
o .
.
of a
(modified) (19).
This
In 2 n d i m e n s i o n s ,
0 and
I are
type
to a l s o
and parametrize
metric
g
g
if eq.
allow
.
.
selfdual
condition
Even
In
g,
respectively,
theory.
matrix.
by
{
Eq.
to o n e
is n x n a n d eq. (14)
with
complex
in t e r m s
condition the n x n (20)
is m e a n i n g l e s s
vanishing
Ricci
solutions of t h r e e
is t h a t
for tensor
in
we c o n s i d e r
in
complex
functions
294 I V- 1 (21)
CV-1
g :
1TV-1 This c o r r e s p o n d s
V+
to the t r i a n g u l a r
r : (~/-~)-1
f1
0
I/
V
"/7CV-1
gauge
e
:
(¢V)
Hence
the l e n g t h e l e m e n t
is g i v e n by
(22)
ds 2 : V -I d { d [
(V+ 7 r C V - 1 ) d ~ d ~
Eq.
(20)
takes
*
g = re
1
C
0
V
-I
+CV-Id[ d~
+ ITV-Idfd~
the form
(23)
(V2+ /TC) 9~ (V -2 g{
and c o n d i t i o n s
(19)
C)
+
9 ~ (V -2 8 ~ C)
: 0
are
(24)
,
These equations
can be e x p r e s s e d
El
in terms
c.c.
of the c o m p l e x
functions
63 = V + iC
: V + i~ r
64
6 2 = V - i~ If with
C : T
and
(++++)
In this case
V
is real
signature
(i.e.
and eqs.
e = (~)-I
110
61 = (23) ~] V
~4'
are = r+
= V - iC
~2 : ~3 )'
(modified)
ds2
Yang's
is real
type e q u a t i o n s .
This can be c o n s i d e r e d
as the
295 "R gauge
If
parametrization"
C : ~
(24)
and
reduce
V
are
of the
real
(i.e.
selfdual
~I
gravitational
field.
~3: ~ 2 : E 4 ~ E ) ,
eqs.
(23) ,
this
is a four
to
(25)
and
(26)
, c.c.
Eq.
(25)
is a
dimensional vector
Note ~ the
the
ansatz
V -I (~,~,~)
selfdual
can be
Killing
and
generalized
vector.
This
one
{28)
ds 2 : v -I d f d [
with
Note
the
that
here
existence
of any K i l l i n g
eqs.
and
v
G
(23)
vector
(24)
(~).
for This
gives
being
solution
to give
- v~-~
= - "z %
arbitrary
(27)
- did R-d{d~ a non-selfdual to the
ansatz
solution~without any 2 Vz C =- ~-This
dfd~
+ v~-1
did
: F({,[,~I
Eln ~ , f , ; Z ~ functions
corresponds
Killing
vector
In ref.
[14] we h a v e
temperature
Killing
by
IT_I and
(null)
corresponds
is g i v e n
The
assume
metric
solution
F
equation.
-C=V= 7r satisfies b o t h
: 0
ds 2 : V - l d { d [
(27)
This
Ernst
We do not
field.
that ~
(modificd)
equation.
+
~,~,~
of the c o m p l e x
to a s s u m e
variables
F = constant
and one
indicated. (null)
[~] .
solutions,
generalized namely
these
solutions
"gravitational
to
include
calorons".
finite
These
solutions
296
satisfy
V(X,T)
and
We
8 -I
have
: V(2,
= kT
also
metrics
of
sources
with
The
is t h e
Taub-NUT
by
where
caloron type
equal
(~'~)
:
(~I ' X 2 ' X 3 ' ~ ) '
temperature.
obtained
multi-Taub-NUT
given
~+g)
solutions
and
from
masses
m
in
metric
has
one
the the
from
the
known
superposition Euclidean
(non-null)
multicenter
of
n
Schwarzschild
regime.
Killing
vector
and
is
[15]
as 2 = u - 1
(29)
(dx I +
2 + u d~,d~
~d~)
where
U =
(30)
nE i:I
~2 m 1
r o t 60 = g r a d
and
The
I +
0 ~
corresponding
(31)
u =
62 -
X
I
~
caloron
8m ~ I + ~-- k =E0
8m
m k=0
r i : I~
,
U,
i.e.
- ~iI
cO =
n E i=I
x
$
2m -r i
:
--z-{Y-'x3'x4~
I +cos
8i
sin
~. l
87Tin
solution
K 0(
is
~
1+cos sin
given 2r~k ~
) cos
8k ek
K0
by
( i- -
2~k~ (--T-)
cos
To)
2~k T
Here
U(~,t)
: U(p, r +
B)
P 2 _- (x 2 _ x 2 ( 0 ) ) 2
The 4~m 2
The
action for
caloron
of the
the
caloron
solution
+
(29)
(X 3 _
is
zero
X 3 (0) ) 2
whereas
that
is
multi-Taub-NUT.
solution
obtained
from
the
superposition
of
n
Schwarz-
297
schild
sources
equal
ds 2 = u d x ~
(32)
where
x ~
imaginary
We
with
have
ture
+
r
2
obtained
these
of
the
U
,
U
solutions
from
the
zero-temperature
meters
mi
all
equal
over
infinite
(-~
k
These
caloron
ones.
In t h e
holes,
known
time
T.
The
ties
in o r d e r
These of
the
on
the
field.
the
x
by
we
is g i v e n
by
of
ones
the
~ k
finite
the
tempera-
solutions
by
equal
at
as
x 4
(31).
caloron
i (0)
located
interpret
eq.
analogy The
instanton
poles
to
in
we
ones.
Yang-Mills
here,
setting and
=
by
Z(0)
an
of
known
are
the
ob-
para-
summing
up
+ kS,
finite
Euclidean [
components
Gravity
are
eliminating as
periodic
additional
parameter
are
the
not
we
analogues
of
of
singulari-
instantons. ~
In is n o t
functions ~
the the
the
functions
periodic
recover
as
coordinate
are
8--> ~
(black-
removable
gravitational
the
temperature
interpreted
components
limit
calorons
by
solutions
metric
contain In t h e
gravitational
metric
present
the
coordinate
arises
the
The
solutions
The U
define
from
solutions
periodic
is c y c l i c .
metrics
known
different
periodic
coordinate.
These the
are
, the
periodicity
caloron
~. to
metrics
Taub-NUT's)
a cyclic
all
of
regime
+ ~).
~
imaginary
the
and
number
and
is g i v e n
Yang-Mills
tained
an
Euclidean
2 2 2 = x2 + x3 + x4
coordinate
solutions
in the
I) dr 2 + d x . d x
(U -I
( x 2 , x 3 , x 4) , time
masses
with known
of
respect ones.
caloron
solutions
field.
REFERENCES
[I]
N. S a n c h e z , Proc. 2 n d M a r c e l G r o s s m a n n North-Holland, 501-518, (1982)
[2]
N. S a n c h e z , P r o c . "XII. I n t e r n a t i o n a l Theoretical M e t h o d s in P h y s i c s " , Lec. Springer-Verlag, (to a p p e a r )
[3]
K.
Pohlmeyer,
Comm.
Math.
Phys.
72,
Meeting,
ed.
R.
Ruffini,
Colloquium on G r o u p N o t e s in P h y s . , ed.
37-47
(1980)
298 [4]
L i n g - L i e Chau, 1574 (1981)
[5]
D. Maison,
[6]
V.A. B e l i n s k i and V.E. Zakharov, Zh. Eksp. Teor. 1955 (1978), (Sov. Phys. JETP 48, 985 (1978)
[7]
P. Forgacs, (1980)
Z. Horvath
and L. Palla,
[8]
N.
Phys.
26D,
[9]
B. Harrison,
M.K.
J. Math.
Sanchez,
Prasad
Phys.
Rev.
Phys.
Rev.
and A.
20,
871
2589
Lett.
Sinha,
Phys.
41,
1197
[ll]
K.
[12]
N. Sanchez,
[13]
C.N.
[14]
N.
[15]
G.W.
Yang,
Phys. Phys.
Sanchez, Gibbons
Phys.
Math.
Lett Rev.
125
(1983)
Lett.
38,
1377
125B,
Hawking,
Rev.
Fiz
75,
Lett.
45,
505
(1978) Zh.
4_66, 207
94A,
Lett.
and S.W.
Phys.
24D,
(1982)
A.A. B e l a v i n and A.M. Polyakov, Pis'ma 503 (1975) [JETP Lett° 22, 245 (1975)] Comm.
Rev.
(1979)
[10]
Pohlmeyer,
Phys.
403
Eksp.
Teor.
Fiz 22,
(1976)
(1977)
(1983)
Comm.
Math.
Phys.
66,
291
(1979).
L O C A L I T Y AND U N I F O R M I T Y
IN G L O B A L E L A S T I C I T Y
M a r c e l o E p s t e i n ~, M a r e k E l z a n o w s k i ~, J~drzej
Department ~
].
Department
of M e c h a n i c a l
Engineering,
of M a t h e m a t i c s
U n i v e r s i t y of C a l g a r y
and Statics,
U n i v e r s i t y of C a l g a r y
INTRODUCTION
This p a p e r
is the first of a series of papers
s e n t a t i o n of a c o v a r i a n t ral global
framework.
s t r e s s i n g the assumptions 2,3],
Sniatycki~
d e v o t e d to the pre-
f o r m u l a t i o n of E l a s t i c i t y w i t h i n the m o s t gene-
The basic p h i l o s o p h y of this a p p r o a c h c o n s i s t s
importance
of g e o m e t r i c
to a minimum.
objects
and k e e p i n g
Such an a p p r o a c h was a l r e a d y p u r s u e d
in
but there the q u e s t i o n of m a t e r i a l b e h a v i o r was left open.
p r e s e n t work,
on the other hand,
and p r o p e r t i e s a d o p t e d that
of c o n s t i t u t i v e
such n o t i o n s
laws,
the paper
introduced
3 is d e v o t e d to some
in
[I].
Section
of the m o s t g e n e r a l
[cf.
As preinvolved
4]
as s e l f - c o n t a i n e d as p o s s i b l e
2 some of the basic
symmetries
laws.
l e a v i n g the m u c h more
consideration
we shall r e v i e w in S e c t i o n
involving
u n i f o r m i t y can be
of the c o n s t i t u t i v e
v i o u s l y o n l y S t a t i c s will be d i s c u s s e d ,
In the hope of r e n d e r i n g
In the
and the point of v i e w will be
as l o c a l i t y and m a t e r i a l
for f u r t h e r
[I,
e m p h a s i s will be p l a c e d on d e f i n i t i o n s
r e g a r d e d as s y m m e t r y - l i k e p r o p e r t i e s
p r o b l e m s of D y n a m i c s
of
structural
geometric
objects
already
important definitions
global e l a s t i c
law and to the e l u c i d a t i o n of the n o t i o n of m a t e r i a l
constitutive
locality.
301
In S e c t i o n Because not
of
the
reduce
further
trivially
developed
specifically, may
not be
plicit
of
local
[6],
when
which
such
such
introduced
Noll's
to pass
local
by N o l l
[5] and More
for u n i f o r m i t y
global
chunk-wise
does
is local.
test
our
thoroughly.
uniformity
the m a t e r i a l
passes
enough
for
is i n v e s t i g a t e d
approach
counterpart
even
uniform
local
the
test.
An ex-
uniformity
to be
uniformity.
NOTIONS
a body
(possibly
B
with
in a g i v e n
differentiable
such
of u n i f o r m i t y
material
to s t a n d a r d
figurations (n>m)
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<
In o t h e r b 6B,
which
performs
mathematical
cotangent
words, assigns
space.
entity
the
6K
bundle
viewpoint T~Q
of
the
Q.
configuration
T Q.
base
~Q(6~)
Given
a force
configuration,
f
i.e.
and
a
such
that
,
is n a t u r a l l y
given
by
6~>
evaluation
map.
although,
as
with
representation
body
an
on
of
same
(f)
associated
elastic
the
the
=
that
be
the
the
projection
6 W
as
From
of
is a c o - v e c t o r
is a s s o c i a t e d
~Q
virtual
space,
TS.
material
such
configuration
of
each
work
then
S
of
a force
constitutive
law,
B,
form
is a o n e
remarked
a vector
above,
field f
as
completely ~
on
Q,
on
a virtual
dis-
< ( B ) C S,
there
a one-form
defining
on
the
~ (B).
material
viz
: Q -> 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)
=
<2 V{V
GQ(V) x
provides us with information about symmetries of a global
constitutive
law at a fixed point
X.
We show now that if this group
is large enough the m a t e r i a l can be c o n s i d e r e d to be germ-local precise
in the
sense that its response to virtual d i s p l a c e m e n t s with compact
support in some n e i g h b o r h o o d is u n a f f e c t e d by the c o n f i g u r a t i o n outside that neighborhood.
To make precise the notion of just how "big"
must be, we define the global identity group open set
VC B
IV
relative to the
as
L0 v : { where
LQ(V)
GQ(X)
Diff
v
denotes the r e s t r i c t i o n to
for each
V.
It is important to note that there exist n o n - t r i v i a l elements of
LQ(V) ,
such as the ones induced by c o m p o s i t i o n by any d i f f e o m o r p h i s m of
B
w h i c h reduces to the identity on By analogy to the point (3.11)
V.
(3.9) we now define the local identity group
X EB
as the union LQ(X)
:
\] V4V
LQ(V) x
LQ(X)
at
305
where
VX
is again
the
family
We are now in a p o s i t i o n state X
that
a body
is c o n t a i n e d
(3.12)
of n e i g h b o r h o o d s
to formalize
is ~ e r m - l o c a l
in its global
at X
Germ l o c a l i t y different common
is the
framework,
notion
was
group
X.
of locality.
identity
relative
to
group
X,
We at
i.e.
LQ(X) C GQ(X)
least
restrictive
introduced
of locality
our notion
if its local
symmetry
Germ L o c a l i t y ~
of the p o i n t
form of locality
by Noll
[9]. The
is what may be c a l l e d
and,
in a
stronger
and more
jet-locality
of order
n.
We shall law
5
space B
say that an elastic
is of local TQ
with
action
of virtual
exists
displacements
as d e f i n e d
by a c o n s t i t u t i v e
a linear m a p p i n g
to the
space
~ (B)
P
from the
of m - f o r m s
on
the p r o p e r t y
(3.13) and
material
if there
supp P(8~) ~
such that
supp
6Jc
for any given c o n f i g u r a t i o n
displacement
6 ~
the virtual
work of
m
and c o r r e s p o n d i n g
"internal
forces"
virtual
~ (~) on
6r
is gi v e n by (3.14)
< ~(~),
8 <>
= J
P(Sa)
B
Here we have
Note
ignored
that
evaluation
P
P(6~)
a possible
represents
contribution
a "density"
at a m a t e r i a l
point
from the b o u n d a r y
of work and,
represents
of
B.
so to speak,
the c o n t r i b u t i o n
the of
that p o i n t towards the total virtual work. The c o n d i t i o n (3.13) however, crucial to ensure that work is not a s s i g n e d to a p o i n t
is, unless
there
of it~
is a n o n - v a n i s h i n g
It is also linear
interesting
operator
B. C o n s e q u e n t l y
local
Peetre
< ~(m), element
=
trivially / B < ~(~),
on B such that
/B
a linear
may
(cf. Kahn
still
depend
If, however, always 6 ~>
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
<~,6 ~> 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
=
in the d i r e c t i o n potential
now
an e l a s t i c
at the
(3.16)
a local
-> ~ 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
<[(~I ~ < >
:
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 :
where
of
been
cases:
has
the
of
to be p a r t i c u l a r l y
generalization the m o d e l
Torino,
Universita
methods
3-dimensional
implies,
(1)
out
GROUPS
D'Ariano
Volta",
different
turns
to
Politecnico,
when
with
GO
"A.
GENERATED GROUP
Rasetti
del
G. Dipartimento
BRAID
ak r s
k : I ..... n[
denote
Rj
([akl) ; j : I ..... m>
the g e n e r a t o r s
of
GO
and
Rj
its d e f i n i n g
re-
lators. The M
Cayley
graph
C
of
be a t o p o l o g i c a l
each element IG01
gs E G O
is the
order
in o n e - t o - o n e Join
of
is c o n s t r u c t e d
(either
select GO;
correspondence
then pairs
nIG01. - f o l d
GO
space
of p o i n t s
set of e d g e s
a point
in s u c h with Ps,Pt,
defined
in the
a set or a m e t r i c Ps ~ M;
a way
that
the p o i n t s s,t as
space),
s = I .....
way. and
IGo(
the e l e m e n t s
,
of
in
M, w i t h
Let
for where GO
selected.
= I,..., IG01
follows.
following
a
are
329
If -I (2)
ak gs
then
the p o i n t s
edge
of t y p e
ginning
at
P~
exactly
gins
at e v e r y
The
with
Rj
M,
which
complete
symbols
ment
gs 6 G O
at
connected
by an
at
Pt'
be-
of the
n
types
be-
Ps" except
at the p o i n t s
Ps"
to
group
PI"
subwords
of
isomorphic
to
can be a s s o c i a t e d W
in the
is closed,
of the
form
coincides
with
the c o r r e s -
zero, W'
namely
= f f-l,
one of the
relators
in
can
be r e c o n s t r u c t e d
then
E w(k)
ge-
of o r i e n t e d
GO .
of m e a s u r e
of the p r e s e n t a t i o n
is c o n n e c t e d ,
M
by a p a t h
a path
subpaths
by the e d g e s
in
Any word
represented
identity
GO
~ = ~ I, w = I ..... 2nlG0) .
any p o i n t
If such
no
group
where
elements
(I).
given and
path-
the
in terms
as the
of the
relators
gene-
induced
by
C.
hence
L0
is c o n n e c t e d :
indeed
any e l e -
c a n be w r i t t e n I
(3)
of e a c h
it be
it r e p r e s e n t s
the
to be
C
PI ~
and has
induced
is a s s u m e d
GO, let
set of d e f i n i n g
C,
class
that
and e n d i n g
= a k ~ gs'
to the
correspond
graph
paths.
,
from
the w o r d
rating
in
~
is e q u i v a l e n t
closed L0
are
Ps
mutually
is u n i q u e l y
is c l o s e d
equivalence
Notice
conjugation
starting
word
the
Ps
at
edge
ends
gt (AIk)
[ a k~
of the
one
intersect
and
symbols
is a word,
Given
and
element
components f
=
(s,t,~)
If the p a t h
and
Ps"
oriented
do not
identity
in
at
Ps'
Pi
beginning
c o l l e c t i o n of all p o i n t s P s = I . . IG0~ (~Ik) s. . . . (gs,g t ), w h e r e the i n d e x w is i n d e e d a m u l t i -
a trivial
ponding
ending
point
: gt
Pt ' a n d
respectively and
E k)
nerating edges
and
gs
is the
w =
the
ak
one p o s i t i v e l y
C
and e d g e s
Modulo
Ps
the e d g e s
graph
index,
'
k,
Thus
Moreover
= gt
I 2
gs : ak I
ak2
As ... a k s
(k r ) and,
if
joins The
E
the
r
upon
the
of
writing
an I s i n g
model
then
[2]
the p a t h
is b a s e d
defined
over
Erl
er2
n
E0 s= I
E k= I
Jk
variables
o
lattice
0 gs
Og& ~ 2 '
akgs g ( GO
have
been
.Ers
on the p r o p e r t y
the
form
N
dynamical
(r-l,r, A r } ,
gs (°r PI to Ps ) . the P f a f f i a n m e t h o d
- for
in the
H = -
(4)
to
theory
the H a m i l t o n i a n
where
, wr =
w
r identity
general
that,
= E
explicitly
L0 -
330
labelled Jk
are
ted)
by
elements
of
the
coupling
constants
edge
IG01
is
of
L0,
indeed
(5) where
B
and
GO
we
have
the
number
Z =
E [Og;g &G0~
is t h e
of
inverse
z
[2 Yf
:
by
the
characteristic
than
of
set
sites
IG01 = N O of
points each
the
Ps
type
in o r d e r
L0;
of of
L0;
(unorien-
to r e m i n d
partition
that
function
e x p (-BH)
temperature,
n
(6)
instead
(cosh uk)]
is g i v e n NO
by
~(~tk~l_
k:1 where
(7)
Uk
and of
F([t~ the
edges
)
is t h e
number of
t k = tanh
loops
types
is c o m p u t e d
decorated
'
generating
of c l o s e d
different
F([tkl)
: 8Jk
function with
k,
one
according
[3] r i.e.
each
to
site
of
draw
the
following
L0
is b l o w n
Let
lattice~
L
is
Each
the
decorated
denotes
on
the
the
indeterminates
number
can
q
L
- in t h e
specified
n = 3 ( q - 2) points, where q L0, e a c h of c o o r d i n a t i o n 3. us n a m e
uk
of
sides
tk -
along
the
L 0. scheme. up
into
First
coordination
number
of
L0
a set
number
sites
is
of of
of
N = nq N O . site
with
of
L
is n o w
gs ~ G O a n d
F([tkl )
is
L,
weight
when
weight
tk
the
correspondence
with
a pair
(gs,i),
i c I a _ ~I ..... n q ~_
identical
I to
in o n e - t o - o n e
with
the
is g i v e n
(4q - 9)
dimer to
bonds
covering
generating
the bonds
in
ba6
added
LXL 0
L~L
0
by
function
of the
type
k,
for and
decoration
procedure. If
L
is a 3 - d i m e n s i o n a l
periodic ded
boundary
conditions
in a 2 - d i m e n s i o n a l ,
genus
c,
but
One
introduces
of
L,
not
corresponding
the
to a l l
- L
orientable,
in o n e
then
in w h i c h
lattice
a new bonds
of
genus
lattice are
possible
- or has
a 2-dimensional genus
closed
c ~
(c -
I)). which
orientations
(i.e.
surface
Lc,
restricted
I
to
Sc
is t h e the
one
L can of
by
be
with embed-
topological
22C-fold
subsets
required
endowed
of
the
covering
those
of
combinatorial
problem. It t u r n s
out
that
if
Lc
is h o m o g e n e o u s
under
the
group
L
G
which
is:
331
i)
locally the
the
central
of the r e l a t o r s Kasteleyn's ii)
globally of
Sc,
group h:
then
the
rated
module
:
satisfy
of
the
the with
the
group
set of
the
GO
respect
and
the N o e t h e r i a n
by the
ring
is the
antisymmetric
c,
Tr
(Ast = - Ats)
Pt,Ps & L c in
if the b o n d G
signature
theorem
group
at m o s t
a maximal
morphic
to the
T R
c
G
are
not
L c PtPs ~ L c
incidence
nearest
is the
image
b~ E L
PtPs e L c is the
of type
oriented
image
k in L.
bond
subgroup
sphere
isotopy
c
of
S c / Isot
thereof}.
with
description
PtPs ~ L c
of all
of an o r i e n t a b l e
is g l o b a l l y
= Homeo
MC(Sc)~
If
the
of h o m e o m o r p h i s m s
(10)
of
of the
determination
Thus
M C ( S c)
of
[5], group
to the
MC(Sc).
(9)
group
weighted
of a b o n d
if the b o n d
Itkl
fundamental
is e q u i v a l e n t
classes,
the
gene-
~ 2);
neighbours
is the
of the
the
abelian
homo-
G.
Baer-Nielsen
namely
is free
in A
if
sgn(t,s)
under
Sc,
ping
Se
is a f i n i t e l y
namely
sgn(t,s)
tensions
group
fundamental
homomorphism of
group
of a b o n d
By the
of
homology
of the
to the
the h o m o l o g y
= 2 -2c-I
sgn(t,s)
geneous
of
modification
requirements
group
under
where
by a d d i t i o n
consequent
images
fundamental
in F({tk~)
Lc,
obtained the
by a P f a f f i a n :
{o
of
Ats
over
is g i v e n
GO and
these
(namely
generators
A = { Ats ~
matrix
extension
--> ~ 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 =
the w o r d
~ ) ,
B e i nPg
for w h i c h
contribution
more
w _ ((p) ki,
...a k
Tr I R ',..(ki, W (p) ~i))I I .
that only
(15)
by
in
in the g
to the
in
(14) we h a v e regular
identity
r.h.s,
of
for a g r o u p
G
k = I ..... n ) R j (
ak
corresponding indeed
to e v a -
representation
is the
correspond
G
element
in
implies G
give
(14) .
to c l o s e d defined
paths,
now
by a given
) , j = I ..... d>
in
Le -
presentation
333
d > m, of
for an a r b i t r a r y
steps,
long
wether
standing
problem
[6].
In o u r
second
For both
one
can
alternative of
state
can be
mannian neous
for
of
metric
G
of
discrete the
finite
cover
say
Sc
Since
covering
Sc
the
can
also
isometry The
gether
with
of
Sc
a fixed
isomorphism)
Then
into
M C ( S c)
~1(Sc), The
~
action
isotopy
of d i s c r e t e
some
hand
the
there
the
is a n
compact
manifold
with
-I,
group
as w e l l
acts
L
and
H(2)/T,
as a h o m o g e -
of
G
of a p o i n t
itself.
,
cover T
The
as w e l l ,
P O l n c a r e, m e t r i c
where
subgroup
as a d i s c o n t i n u o u s
curvature
with
a Rie-
compact
is t h e o r b i t
negative
Fuchsian
the
space
of
in
uni-
so it is
,
H (2)
act b y
isometries,
is a s u b g r o u p
group
space
subset
~ 1(Sc).
of h y p e r b o l i c
~ 1(Sc)
to the
of
to
T
of the
them
of g e n u s
c,
of d i s c r e t e
Since
the
surfaces
Sc
(where
between
group
Sc
acts
to an a c t i o n
automorphism
of the
and can
hyperbolic be
problem
c the a c t i o n
is an i s o m e t r y
representations
finite
Sc(O)
the w o r d
two
to-
surfaces
respecting ~c
topo-
this
[7].
representations
of
of o u t e r
automorphisms
of
on
~c
of
by pulling
G
since
of
~1(Sc)
into
B,
back metrics,
the p o i n t s
of
c
of m e t r i c s .
and
of
c,
group
ly d i s c o n t i n u o u s
point
G.
properties
on b o t h
of the u n i v e r s a l
by
of as the
genus fixed
general
is a n o r m a l ,
endowed
Teichm~ller
descends
Every
for
.
to an a c t i o n
c.
disk
to the
is
classes
On the o t h e r
cends
constant
is i s o m o r p h i c
say
problem
word
up to c o n j u g a c y .
diffeomorphism
a n d the are
B
K
and
under
isomorphism
c a n be t h o u g h t
as D e h n ' s
H (2)
if t h e r e
is the
has a
is an i n d i s p e n s a b l e
the
curvature
fundamental
G
unit
is i s o m o r p h i c
genus
i (Sc)
of
of
G
number
o r not,
is k n o w n
of
discuss
where
translations
B
are e q u i v a l e n t
~c
The
be r e p r e s e n t e d
group
latter
logical
Gaussian
G/K,
has
to the o p e n
G
led to the w o r d
briefly
of a u t o m o r p h i s m s ,
isometric
and
of
irreps.
index.
of
thus
and how they bear
region
element
theory
of as a 2 - d i m e n s i o n a l
G,
subgroup
g e n e r a t o r s , in a f i n i t e
the p r e s e n t a t i o n
of c o n s t a n t
of
fundamental
versal
G,
in the identity
group
we will
of the
thought
space
W the
we are
solution,
Hereafter
the c o n s t r u c t i o n Sc
word
defines
in c o m b i n a t o r i a l
ways
ingredient.
W
of group
faithful.
structure
fixed
isometry
0;
on
K
of
be r e a l i z e d on a surface
point
of
of
Sc(°)
~
c
The q u o t i e n t
subgroup
therefore
group
~ i (Sc)
K
when
G,
o n the
S u c h an a c t i o n
space
on
~c'
of g e n u s
Indeed,
in the
~c.
c.
For
isotopy
space
has
of
on
des-
is p r o p e r -
is the m o d u l i
acting
space
the a c t i o n
as a s u b g r o u p
acting
to i t s e l f
.
acts and
a
isometrics
each class
let k~ K of
of
334
Notice could
that
isometry tity, of
such
generate of
S
The
the
same of
word
in
K
isotopic
have
class
which
to the
of
to Then
all
is a b s u r d . generated
K
represents
the
(hence
for any manifold
S
were
of
elements
two,
one
the o t h e r
trivial equal
:
to
word
to the
(S c)
of
by choosing
is i s o m o r p h i c
: Diff
exists
the
which
(16)
if t h e r e
inverse
- an
i d e n t i t y b u t n o t e q u a l to the i d e n (2) H commuting with every element
[cB kC
identity
the
to the
a lift
endpoints,
because
one with
on H ( 2 ) ) .
isometries
in e a c h
is u n i q u e ,
isotopic
(acting
group
metry
(o)
c would
which
~ 1(Sc)
have
isometry
- by composing
K
~ I (S c)
such a unique
itself,
in
should
K
iso-
because
any
is an i s o m e t r y
identity).
The map
.... > G
the q u e s t i o n
wether
or n o t
it is p o s s i b l e
C
to l i f t
G
back
tative
in
Diff
lifted
elements
is r e f e r r e d Now
the
isotopic
genus
is n o t
T
c
T,
to the
for if
G
0 T'
from homeomorphisms
to c h o s e
in
identity
G,
a single
so t h a t
is e q u a l
indeed
~
is c e r t a i n l y
K c G
is f i n i t e .
For
generated,
discrete
can
represen-
any w o r d
in the
to t h e
identity,
problem.
t h e n be v i e w e d
acting
of
namely
element
is a f i n i t e l y
cyclic.
of
(Sc) ,
lifting
problem
group
morphisms
Diff
for e a c h
to as the
of a r b i t r a r y
which
(S c)
lifting
Fuchsian
into
on the
space
solvable
for
finite
surfaces
genus,
c
the
subgroup
as the g r o u p
S
of
B,
of o u t e r
of r e p r e s e n t a t i o n s ,
auto-
induced
S C
The
center
of
is c y c l i c ) . responding
T
is t r i v i a l
Upon
defining
to the
exact
(17)
finite
the
centralizer
extension
T
of e v e r y
of
T
by
element K,
cor-
sequence:
I . . . . > 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)
which
the two
(18)
___>~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,
[5]
J. N i e l s e n , A c t a Math. 50, 189 (1927); 5_33, I (1929); 58, 87 (1931) ; 75, 23 (1943); H. Z i e s c h a n g , " F i n i t e G r o u p s of M a p p i n g C l a s s e s of S u r f a c e s " , S p r i n g e r L e c t u r e N o t e s in Math., No. 875, B e r l i n 1981
[6]
J.J. 1973
[7]
W.J. H a r v e y , Ed., " D i s c r e t e A c a d e m i c Press, L o n d o n 1977
[8]
W.P. T h u r s t o n , "A P r e s e n t a t i o n for the M a p p i n g C l a s s G r o u p of C l o s e d O r i e n t a b e l S u r f a c e s " a n d "On the G e o m e t r y and D y n a m i c s of Diffeomorphisms of S u r f a c e s " , p r e p r i n t s 1983, S. W o l p e r t , Ann. Math. 117, 207 (1983)
Lund, M. R a s e t t i and T. Regge, C o m m u n . M a t h . Phys. 51, R a s e t t i a n d T. Regge, R i v i s t a N u o v o Cim. 4, I (1981)
Rotman,
J.Math. Phys.
7,
J.Math. Phys.
"The
Theory
1776 4,
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(1966)
287
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of G r o u p s " ,
Groups
15
Allyn
and
Bacon
and A u t o m o r p h i c
Publ.,
Boston
Functions",
337
[9]
M. Sato, T. Miwa and M. Jimbo, Proc. Japan Acad. Sci. 53A, 6, ]47, 153, 183, (1977); Publ. RIMS Kyoto University, I_44, 223 (1978)~ I_~5, 201, 577, 871 (]979); 16, 531 (1980)
[I0]
E. Artin, Ann. Math.
[11]
M. Rasetti, in "Group T h e o r e t i c a l Methods in Physics", G. Denardo, L. Fonda and G.C. Ghirardi Eds., Springer-Verlag, Berlin 1984.
488, 101
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