Lecture Notes in
Physics
Edited by H. Araki, Kyoto, J.Ehlers, MLinchen,K. Hepp,ZSrich R. Kippenhahn,MSnchen,H.A. Weide...
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Lecture Notes in
Physics
Edited by H. Araki, Kyoto, J.Ehlers, MLinchen,K. Hepp,ZSrich R. Kippenhahn,MSnchen,H.A. WeidenmSIler, Heidelberg J. Wess, Karlsruheand J. Zittartz, K61n Managing Editor: W. BeiglbSck
280 Field Theory, Quantum Gravity and Strings II Proceedings of a Seminar Series Held at DAPHE, Observatoire de Meudon, and LPTHE, Universit~ Pierre et Marie Curie, Paris, Between October 1985 and October 1986
Edited by H.J. de Vega and N. S&nchez
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Editors H, J. de Vega Universit~ Pierre et Marie Curie, L.P.T.H.E. Tour 16, ler 6rage, 4, place Jussieu, F-75230 Paris Cedex, France N. S~.nchez Observatoire de Paris, Section d'Astrophysique de Meudon 5, place Jules Janssen, F-92195 Meudon Principal Cedex, France
ISBN 3-540-17925-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-17925-9 Springer-Verlag NewYork Berlin Heidelberg
This work is subject to copyright. All rights are reserved,whether the whole or part of the material is concerned, specif{callythe rights of translation,reprinting, re-useof illustrations,recitation, broadcasting, reproduction on microfilmsor in other ways, and storage in data banks. Duplication of this publicationor parts thereof is only permitted under the provisionsof the German Copyright Law of .September9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violationsfall under the prosecution act of the German Copyright Law. © Springer-VerlagBerlin Heidelberg 1987 Printed in Germany Printing: Druckhaus Beltz, Hemsbach/Bergstr.; Bookbinding: J. Sch~.fferGmbH & Co. KG., GrSnstadt 215313140-543210
PREFACE
This book contains the lectures delivered in the third year, Paris-Meudon
1985 - 1986, of the
Seminar Series.
A seminar series on current developments
in mathematical
physics was started in
the Paris region in October 1983. The seminars are held alternately at the DAPHEObservatoire
de Meudon and LPTHE-Universit&
encourage theoretical
physicists
ticians to meet regularly. October
1983 - October
Pierre et Marie Curie (Paris VI) to
of different
disciplines
and a number of mathema-
The seminars delivered in this series in the periods
1984 and October
published by Springer-Verlag
1984 - October 1985 have already been
as Lecture Notes in Physics,
volumes 226 and 246,
respectively.
The present volume "Field Theory, the lectures delivered up to October
Quantum Gravity and Strings,
topics of current interest in field and particle theory, mechanics.
Basic problems of string and superstrin~
porary perspective
and quantum field theoretical
cosmology are presented.
cosmology and statistical
theory are treated in a contem-
as well as string approaches
Recent progress on integrable
in two, four and more dimensions
II" accounts for
1986. This set of lectures contains selected
is reviewed.
It is a pleasure to thank all the speakers for their successful delivering comprehensive
and stimulating lectures.
for their interest and for their stimulating Scientific
Direction "Math~matiques
toire de Paris-Meudon
efforts in
We thank all the participants
discussions.
We particularly
- Physique de Base" of C.N.R.S.
for the financial
extend our appreciation
to
theories and related subjects
thank the
and the Observa-
support which made this series possible.
to Springer-Verlag
for their cooperation and efficiency in
publishing these proceedings.
CERN, Geneva, F e b r u a r y 1987
We
H. DE VEGA N. SANCHEZ
CONTENTS
P. Di Vecchia:
Covariant Quantization of the Bosonic String: Free Theory ....
I.A. Batalin and E.S. Fradkin:
Operatorial Quantization of Dynamical Systems
with Irreducible First and Second Class Constraints
.......................
11
Kaluza-Klein Approach to Superstrings ............................
19
Non Linear Effects in Quantum Gravity ..............................
41
Our Universe as an Attractor in a Superstring Model ...............
51
M.J. Duff:
I. Moss:
K. Maeda:
J. Audretsch:
B. Allen:
Mutually Interacting Quantum Fields in Curved Space-Times
.....
Gravitons in De Sitter Space ......................................
D.D. Harari:
R.S. Ward:
Effects of Graviton Production in inflationary Cosmology .......
Multi-Dimensional
N.J. Hitchin:
J. Isenberg:
J. bukierski:
82
97
106
Monopole and Vortex Scattering ................................
117
The Ambitwistor Program ........................................
125
Supersymmetric Extension of Twistor Formalism .................
137
M.A. Semenov-Tian-Shansky:
Supersymmetries of the Dyon ........................
Classical r-Matrices,
Groups and Dressing Transformations
M. Karowski:
68
integrable Systems .............................
E. D'Hoker and L. Vinet:
A.M.
I
Lax Equations,
156
Poisson Lie
.......................................
174
On Monte Carlo Simulations of Random Loops and Surfaces ........ 215
Nemirovsky:
Field Theoretic Methods in Critical Phenomena with
Boundaries ................................................................
229
COVARIANT BOSONIC
QUANTIZATION
STRING:
OF THE
FREE T H E O R Y
P. Di V e c c h i a Nordita,
Blegdamsvej
The b o s o n i c
string
S[x~(~,~),
g~B(~,~)]
t h a t is c l a s s i c a l l y variant
under
on
x~
6x ~ = E ~ ~
8~
a c t i o n I)
g~B~ x . ~ B x
to the N a m b u - G o t o
(I)
a c t i o n 2) and is in-
of the c o o r d i n a t e s induces
of the w o r l d
the f o l l o w i n g
trans-
gee
x~ + ~ y gab
s Y gy ~
+ ~ 8
are two a r b i t r a r y
The a c t i o n
by the f o l l o w i n g
A reparametrization
and
Copenhagen
= _ ~T ~dT ~d~ ~ 0
equivalent
= 87 ~ 6g~B where
is d e s c r i b e d
reparametrizations
s h e e t of the string. formations
17, D K - 2 1 0 0
sY 6
functions
(i) is in a d d i t i o n
(2) gay
of
also
T
and
invariant
~ . u n d e r Weyl
transfor-
mations : 6x ~ = 0 where
6g~8 = 2A(T,O)
A(Y,o) The
is an a r b i t r a r y
invariances
the c o m p o n e n t s However the q u a n t u m
the W e y l
of
(3)
(T,o).
(3) are s u f f i c i e n t
to g a u g e away
all
tensor.
invariance
c a n n o t be in g e n e r a l
maintained
in
theory.
Therefore formal
(2) and
of the m e t r i c
function
g~8
in the q u a n t i z a t i o n
gauge characterized
of
(i) we can o n l y fix the con-
by the f o l l o w i n g
c h o i c e of the m e t r i c
ten-
sor :
gas = P(~)
~B
;
~iI = - ~00 = 1
(4)
where
p(~) Since
critical
is an a r b i t r a r y in w h a t
dimension
choose
p(~)
D = 26
= 1
in
where
the
, where
gauge
second
obtained
ghost
coordinate
of
~
~ (T,o).
we will
the W e y l
the a c t i o n
J0
2--9
minant
dinate
however,
consider anomaly
only
the c a s e
of
v a n i s h e s 3) , we can
(4).
In the c o n f o r m a l
-
function
follows,
e
term
b
(i) b e c o m e s 4 ) :
B
(5)
is the c o n t r i b u t i o n
from having ca
and
fixed
of the F a d d e e v - P o p o v
the c o n f o r m a l
a symmetric
and
gauge.
traceless
deter-
It c o n t a i n s
antighost
a
coor-
b °B
The
conformal
We can
still
gauge.
They
functions
~
are
~(~)
-
=
c
0
+
~
not
Y
fix c o m p l e t e l y
that
transformations
leave
characterized
by two
the c o n d i t i o n :
eY = 0
(6)
+
;
~-
=
~
-+ o
~
;
+
=~
1 /~ ~
~ ~
+ ~
~
the e q u a t i o n s
the gauge.
in the c o n f o r m a l
coordinates 1
c
qeB
(4) does
transformations
satisfying
In the l i g h t - c o n e +
choice
gauge
the c o n f o r m a l
eB + 8Be~
E-
gauge
perform
(6) g e t
the
simple
(7)
/
form:
(8)
2 + ~- = 2 + s- = 0 implying
that
e+[e -]
It is easy formal
is o n l y
to c h e c k
transformations
conformal
fields
with
a function
that Lagrangian
provided conformal
that
x
U dimension
of (5)
~+[~-] is i n v a r i a n t
, b A
and equal
c
under
transform to
0,2
conas
and
-i
respectively. In a c o n f o r m a l (5) w h e r e venient
invariant
the o - v a r i a b l e
to use,
instead
theory
Z = e i(T+O) that
in e u c l i d e a n
as the o n e
described
v a r i e s in a f i n i t e d o m a i n + of ~- , the two v a r i a b l e s
(0,~)
by action it is con-
z = ei(T--O) space
(T ÷ iT)
become
one
(9) the c o m p l e x
conjugate
of the other. A conformal traceless
invariant
energy-momentum
theory tensor
is c h a r a c t e r i z e d with
only
two
by a conserved
independent
and
components
and
T(z)
T(z)
A conformal under
field
a conformal
¢
with
(A,Z)
dimension
transforms
as f o l l o w s
transformation:
[
] [
E'
']
(i0)
~z
For
the
sake
pendence z
can
also
The
¢(~)
=
in the
in m i n d
following
the de-
that w h a t e v e r
we
do w i t h
is o b t a i n e d
(OPE)
~/~ ~(~)- + z-~
tensor
of
by
T(z)
r e q u i r i n g the 5) with ~ :
~(~) + regular (z_~) 2
T(z)
following
terms
is a c o n f o r m a l
(ii)
tensor
with
A = 2 .
implies5) : T(z)
T(~)
additional
-
generators
the From
[L n
where
of
integral (12)
b = b zz
in terms
is d e f i n e d
it f o l l o w s
=
8x.~x
8 -- ~
t e r m can
the c o n f o r m a l of
terms
in g e n e r a l algebra.
be a d d e d
(12)
with-
The V i r a s o r o
T(z) :
(13)
(n-m)
1 + ~
b
they
a way
satisfy
c L n + m + ~-~ n(n2-1) z
and
z
that
dz ~-- = 1 .
the V i r a s o r o
algebra:
(14)
~n+m;0
the L a g r a n g i a n
corresponding
to
to:
' ~ ;
in such
that
of the v a r i a b l e s
N
c/______~2+ reg. (z_~)4
z n+l T(z)
is p r o p o r t i o n a l
L
T(~) + (z_~) 2
singular c - n u m b e r
can be c o n s t r u c t e d
, L m]
In terms
+ 2
the c l o s u r e
L n = % dz
where
8/~ T(~) z-~
more
out destroying
(5)
(i0)
expansion
energy-momentum
This
An
omit
be d o n e w i t h
product
T(z)
we w i l l
z , keeping
transformation
operator
The
of s i m p l i c i t y
on the v a r i a b l e
(b ~ c
~~z =
b zz
+ b
8c)
(15)
and
;
c = c
z
,
~ = c
z
(16)
Since
xU , b
and
-i
and
c
transform
respectively,
6L = ~[e(z)L] implying
that
The lowing
L
fields
with
A = 0,2
is a c o n f o r m a l
density:
(17)
the c o r r e s p o n d i n g
energy-momentum = TX(z)
that
+ ~[~(z)L]
transformations
T(z)
as c o n f o r m a l
it f o l l o w s
on
action
x
tensor
, b
is c o n f o r m a l
and
c
invariant.
are g e n e r a t e d
by the
fol-
:
+ Tg(z)
(18)
where
as
TX(z)
= - yl.{ .k~
xh] 2 :
Tg(z)
=
+ 2c'b
: cb'
it can be
They
:
seen by u s i n g
<xU(z)
x~(~)>
= - g~
1 = Z-~
allow
(19)
one
(20)
the
following
contraction
rules:
log(z-~)
(21)
(22)
to c o m p u t e
aiso
the O P E w i t h
two e n e r g y - m o m e n t u m
ten-
sors: D-26 T(z)
T(~)
~/~ =
implying
that
critical
dimension
the c - n u m b e r
As p r e v i o u s l y tities
that
depend
on
taining
for i n s t a n c e
the V i r a s o r o
(23)
algebra
we have In the
limited
case
is v a n i s h i n g
sets
of m u t u a l l y
string
to c o n v i n c e
=
analysis
that
at the
commuting
of the
string
to the q u a n -
string
depend
it is c o n v e n i e n t
of the end p o i n t s c ( z = e iT)
our
of a c l o s e d
for the q u a n t i t i e s
of an o p e n
that
It is e a s y
two
(z_~) 4
.
z .
everything
the p a r a m e t r i z a t i o n implies
of
D = 26
repeat
In the c a s e
+ 2 T(~_____J__} + (z_~) 2
explained
however
This
T(~) z-~
we
on
can z
Virasoro
algebras.
to r e q u i r e is l e f t
ob-
that
unchanged.
~(z=e i~)
oneself
that
In the t r e a t m e n t of the g h o s t we f o l l o w F r i e d a n , M a r t i n e c and S h e n k e r 6) .
for an o p e n
closely
string
we can
the a p p r o a c h
of
use all the p r e v i o u s
formulas with
z = e i~
In the following we
limit for s i m p l i c i t y our c o n s i d e r a t i o n s to this case. Having fixed the c o n f o r m a l gauge we have lost the general invariance
(2) keeping only the i n v a r i a n c e under c o n f o r m a l t r a n s f o r m a -
tions. On the other hand we have gained the invariance under BRST transformations,
that act as follows on the c o o r d i n a t e s of the string:
~X = ICX' ~b = - 21x'
+ l[cb' + 2c'b]
(24)
6c = Icc' where
1
is a c o n s t a n t G r a s s m a n n parameter.
The v a r i a t i o n of L a g r a n g i a n
(15) under the t r a n s f o r m a t i o n s
(24)
is a total d e r i v a t i v e 6L = ~[IcL]
(25)
i m p l y i n g the invariance of the c o r r e s p o n d i n g action. It is easy to see that the product of two t r a n s f o r m a t i o n s
(24) is
i d e n t i c a l l y vanishing. The g e n e r a t o r of the t r a n s f o r m a t i o n s Q = } dz:c(z)
[TX(z)
By using the c o n t r a c t i o n s
(24) is the BRST charge:
+ 1 Tg(z)] :
(21) and
(26)
(22) it can be shown after some
c a l c u l a t i o n that: Q2
1 $ c' ''( = 2-4 (D-26) ] d~ ~) c(~
T h e r e f o r e the q u a n t u m BRST charge is n i l p o t e n t only if
(27)
D = 26
This
implies that our q u a n t i z a t i o n p r o c e d u r e is c o n s i s t e n t only for the critical d i m e n s i o n
D = 26
In this case the BRST charge commutes with the V i r a s o r o generators:
for any
n .
In c o n c l u s i o n if
D = 26
the gauge fixed action
(5) is i n v a r i a n t
under two i n d e p e n d e n t and very important transformations: c o n f o r m a l transformations.
BRST and
It is useful harmonic
to expand
oscillators.
x
(z)
, b(z)
and
c(z)
in terms of the
They are given by: co
X (Z)= q c(z)
=
coz
ip iogz + i n:[I ~nl =
both sides of
constants
g~ 2 (z-~)
(31) and setting
we get the contraction
In order to derive some more discussion We can introduce
(31
the contraction
to zero the two integra-
(21). (22) from the mode expansion
is needed. the ghost number
current (32
j(z) = :C(Z) b(z): Using the contraction
j (z) j (~) =
Tg(z)
(22) it is easy to show the following
i 2 (z-~)
j(~) = ~ / ~ j(~) z-~
OPE's: (33
+
j(~) (z-~) 2
3 (z-~) 3
Because of the extra term j(~) is not quite a conformal A = 1 . In terms of the mode expansion defined by
(34)
field with
Jn = [ n+l n z
j (z)
(33)
They
and
(34)
L
' Jm
Jm
' Jn]
can
be
(35)
i mp ly :
- m
3n+m
- ~ n(n+l)
(36)
~n+m;0
(37)
= n 6n+m;0
checked
directly
using
the
expansion
in h a r m o n i c
oscilla-
tors: Jm = [ k
: Cm_ k b k
Lg = ~ n m
(n+m)
: b
:
= c
c n b_n
:
n
(38)
and
the
(39)
more
commutator
number
b_]nl
b
(39)
and
~ b +in I
-n
if
n _< 1
if
n > 2
the
one
J0
complicated
c
-n
n
gets
that
J-i
.+ = - 31
lq>
(36)
for
is n o t
(42)
= qlq>
implies +
universes
1 and
cosmology,
the
as a b r a n c h i n g IRW2>
2 can
evolve
+
of the wave
of pure function.
a superposition
to the
....
3
development
representing
a
I
+
Figure
expression
represent
density
to mixed For
states
example,
can
a pure
of R o b e r t s o n - W a l k e r
matrix
IRWI> oo as
~
is
when t -~oo .
The
scale
factor
the same as that of the stiff matter
a expands
(P=~
dominated
because of the massless scalar fields ( ~ , 04~, ~T and ~T ) . , ~
const, but
--~ and --~ 9 - oo
zero, hence this asymptotic
(~ ~
for QS = QT = 0 ). 4 solution is not F x K.
For the case (II), t ~
when
~ ~ ~o-O.
and ~T -~
The internal space shrinks to
a ~ t 2/3~ asymptotically,
is the Friedmann universe with matter fluid of P = ( / - i ) ~
.
which
Since ~.~r ~SI~T-> some
constants, the internal radius approaches some constant. All solutions approach 4 the F x K space-time for t -> ~o. Therefore, the Friedmann universe with a static internal space is a unique attractor in our dynamical system. From the above solutions, kinetic terms of the 'axions' for
~
and
~T
)
we find one important role of QS and QT ( the
(17)
Eq.(3.10)
if QS~ 0 and OTTO,
,
f e,/
shows that there are maxTmum values
i.e.
and
Those kinetic terms provide potential barriers which prevent away to arbitrary large values. stay near the preferential
minimum
~
and Jr from going
This result is important,because (~
the universe must
the Planek scale) when the gluinos condense,
to reach the present state for natural initial conditions. The effective 4-dim action (2.14)
contains the proper Einstein action,
it seems that this model guarantees the'Einstein gravity.
That is, however,
then not true
because the original coupling of a massless scalar to the 4-dim gravity was removed -6 from the action by the Weyl rescaling ( the factor b ), but appeared in the 4-dim world interval
(2.11).
We need a potential for
~
scalar in our model, so that it fixes the value of ~_
, which is the massless JBD and guarantees the 4-dim
Einstein gravity.
IV.
THE EINSTEIN GRAVITY AS AN ATTRACTOR
When the universe expands and the temperature (~/i~),
drops below some critical value
the gluinos of the largest gauge group (e.g. E~) condense.
We find the
59
gluino-condensation potential (14) .
As discussed by Rohm and Witten (16) , the VEVs
of H
may be also induced through an instanton solution at the gluino condensation MNP and its value e is quantized as
C = C)~ - ~ C o
(4.1)
.)
where n is an integer and c
o
is determined by the geometry of the internal space.
If we take into account the Chern-Simon term, however, we find an instanton solution, through which the quantized value c can change to the other value ( c ). Bubbles n n+l may be formed through the quantum tunnelling. Since c changes through the quantum tunnelling,
the potential V also changes with time.
tunnelling ( ~
The time scale of quantum
tQT ) is not yet known because the explicit instanton solution
not been obtained.
has
In this section, we shall discuss the evolution of the universe
mainly for the case that tQT>>
I/H ( CASE (I))
tQT 0. I
^
With number around in t h e third ticles
regard
to t h e t h i r d
of massive p
= 0
particles
which
zeroth
order
term reflects , but
t e r m of
of the mutual the
also
q > 0 . Therefore,
underlying
process, the
and
interaction.
factor
factor
these created as c o m p a r e d I/2 has,
I/E O r e m a i n s
that
with
is the
interval
o u t of t h e b a c k g r o u n d
The
appearance of the
particles
in t h i s
16p=o12
the m o m e n t u m
created
fact that not only one half
the r e g i o n
the
we recall
per unit volume
are gravitationally
in a d d i t i o n
16p=012 w h i i e
(4.5)
the
unchanged.
incoming
are decaying
second
case,
of this
term
parin
and its
to b e r e p l a c e d
by
75
The
three
go back
resonant
m e a n value
proportional
the t e m p o r a l
mean v a l u e
IFL
> =
bounded
familv
il
the r e s o n a n t in ~tot,
terms,
provided
contributions
of the outcome
expansion i.e.
(4.6)
also the
we have to work happening
To do so, we make is true
out
in
use of
for all
we find that also
the terms
we repalce
of
of taking
of the p r o c e s s e s
relation
laws,
in Q t o t
But instead
of t a n g e n t - s p a c e s .
~ .L~i.~z" +~--|.Because0 ) . this
monoton!c
to T
processes.
of the two m i n k o w s k i - t y p e
the continuous
4
theories
However,
SUSY YM theory
via d i m e n s i o n a l
to consider
for c o n s i d e r i n g and string
D=I0
in D=4.
this diffi-
[14] w h i c h
reduction.
the SUSY twistor
is a p os s i b l e
pro-
In such a
formalism
in D>4.
relation between
twis-
[41-44].
D>4
The D=4
twistors
can be d e f i n e d
in several
equivalent
ways,
for
example i)
as the f u n d a m e n t a l
SU(2,2)=S0(4,2) ii)
representation
of D=4 conformal
as the p a r a m e t r i z a t i o n
xified Minkowski iii)
of the
four-fold
covering
group. of the totally
null
2-planes
in comple-
space C 4
as a bundle
over
S 4 describing
all possible
complex
structu-
res on S 4 We
shall discuss
i) T w i s t o r s We define
briefly
as conformal
twistors
the e x t e n s i o n
as fundamental
In such a way one can introduce on the choice described
linear
spinors
twistors
of D) they can be real,
by the following
of these
definitions
to D>4.
s~inors.
complex
vector
of SO(D,2). for any D, and
(depending
or quaternionic.
spaces
for 4~D~I0
They are
(see e.g.
[45])
D
4
5
6
7
8
9
T
C4
H4
H4
H8
C 16
R 32
Table
I. D - d i m e n s i o n a l spinors
It should bed by a pair l exifie d
be added
R 32
as the f u n d a m e n t a l
that for any D the c o n f o r m a l
of Lorentz
rotation
twistors
10
groups
spinors. as well
This
conformal
spinors
decomposition
are descri-
is valid
as for the real one,
with
for comp-
arbitrary
signature. In such a f r a m e w o r k
the d i m e n s i o n s
D=6 and D=10
are s e l e c t e d
becau-
149
se they correspond
to quaternionic
complex descriptions following
table
extensions
of D=4 spinors and D=4 twistors.
(see e.g.
spin covering
Let us write
D=6
the
D=I0
SL(2,C)
SL(2;H)
DL(2;0)
C2
H2
R16~02
(4;H)
U (4;0)
H4
R32=04
spinors
spin covering
SU(2,2)=
of Conf.group
=U
U
(4;C) C4
Conf.fundamental
of the
[46])
D=4
of Lorentz group Weyl
and octonionic
spinors Table 2. The relation of complex numbers with D=4,quaternions with D=6'and octonions with D=I0
where
U
(F=C,H or O)
(4;F)
:
qAHABqB
i.e. U a describes ii) Twistors Following D=2k twistors planes dence
= inv
antiunitary
H + = -H group.
In particular
one can chose H=
0
as pure spinors.
[1,42-44]
one can adopt the view that in even dimensions
are pure conformal
spinors,
in C 2k. We obtain the following
describing
totally null k-
generalization
of the correspon-
(A) for even D>4:
point
in T(2k)
where T(2k) Pure
denotes
~
totally null k-planes
the space of twistors
spinors are obtained by imposing
ints on 'brdinary"projective dimension n=2k-1
fundamental
in C 2k
(A')
for D=2k. r=2 k
k(k+1) linear constra2 spinors, with complex
SO(D+2;C)
i.e. they are described by quadric Qq, with complex di-
mension q= k(k+1______~) We obtain the following complex manifolds 2 bing "ordinary" projective spinors and twistors: "ordinary"
projective
conf.
spinors
descri-
twistors
D=4
CP(3)
CP(3)
D=6
CP(7)
Q6
150
"ordinary"
projective
conf.
spinors
twistors
D=8
CP(15)
QI0
D=I0
CP(31)
Q15
Table
3. From o r d i n a r y
We s e e
that for D=4 o r d i n a r y
for D=6 one needs r=16.
to pure
to impose
In p a r t i c u l a r
for D=6
can be identified,
for D=8 r=5,
[42] that the p u r i t y condi-
as the c o n s i s t e n c y extended
Twistor
twistors
it can be shown
tion follows
iii)
and pure
spinors.
one constraint,
uation
(2.1)
conformal
condition
and for D=10
for the Penrose
incident
eq-
from D = 4 to D=6.
space as bundle
over
S 2k d e s c r i b i n g
all p o s s i b l e
comp-
lex structures. In the d e s c r i p t i o n space one can e x t e n d the e x t e n s i o n
of self-dual
the gauge
fields
connections
space as fiber bundle
cally by CP(1)=
SU(2) Because U(J] "
extended
(compactified)
from S 4 to CP(3)
is pure gauge. Such a c o n s t r u c t i o n
tion of twistor
that CP(1)=
on
SO(4) U(2)
' i.e.
over
leads
S 4 with
SO(4)=SU(2)xSU(2)
locally
Euclidean provided
to the introduc-
fibers
described
one can w r i t e
P T ~ S 4 X u( SO(4) 2 ) " This
relation
to any even k and we obtain locally for D=2k (see e.g. [47-49])
Counting
real
(4.1)
dimensions
with the d i m e n s i o n s
iii)
also
can be
PT ~-- S 2k x SO(2k) U(k)
as pure
lo-
spinors).
2k+k(2k-1)-k2=k(k+1),
obtained
From
from our second
the i d e n t i f i c a t i o n
we get the a g r e e m e n t definition
(i.e.
of the d e f i n i t i o n s
twistors ii)
and
one gets
D=4
D=6
D=8
D:I0
D P (3)
Q6
QI 0
QI 5
+CP(1)
+CP(3)
+ Q6
+ QI0
S4
S6
s8
s I0
Table
4. Twistor
The twistor of selfdual Finally
bundles gauge we
bundles
written
fields
shall
in D=2k
above
in D=2k,
consider
(k=2,3,4,5).
should be u s e f u l k=2,3,4,5.
for the d e s c r i p t i o n
151
iv)
Supersymmetrization
Only
for D=6
classical group
for D>4
the
spin c o v e r i n g s
SO(D,2)
Lie group,
and S0(6,2) = U
(4;H)
can be
Ua(4;H)
where
of t w i s t o r s
supersymmetrized
÷
as follows
(D>4) [50].
is d e s c r i b e d
by a
The D=6 c o n f o r m a l spin
[51]
U U(4;H)
the b o s o n i c
(4.2
sector
of the
SUSY e x t e n s i o n
of D=6
conformal
group
is
U
(4;H) × U(N;H)
The D=6 H4
conformal
H4 =
where We
see t h e r e f o r e [53]
c o n formal
purity
can be
mal a l g e b r a the
÷ H 4;N =
that
spinors
described
i) There
exists
variables
to make
in N=2,3
point
jectories".
Lie
the
a close
matrices
superalgebras
following
relation
SUSY
harmonic
as
"ordi-
of D=I0
[46]
(see
of
confor-
leads b e y o n d
[54]).
strings
superspace
by actions
selecting
In p a r t i c u l a r
twistorial and D=4
string
one can
[58],
exploiting in
minimal
scalar
Q2N_3 =
some notions
lines,
world
of twis-
and their
sheets
as
dynamics
"string
in any d i m e n s i o n
by a pair of null
of null
and the p a r a m e t r i c
space 15)
[49,57].
show that
Eisenhart
and harmo-
bosonic
lie on the quadric
is d e s c r i b e d
[60],
methods
recalls the a m b i t w i s t o r
along null
parametrization
for D=3 and M o n t c h e u i l
twistor
The additional
superspace
propagate
comments:
between
[55,56].
uivi=0) , w h i c h
of h a r m o n i c
of a b o s o n i c
[59]
supersymmetrization
of view has been p r e s e n t e d
Massless
is d e s c r i b e d
D=3
Z2-graded
the t w i s t o r s
It is not clear how the c o n d i t i o n The
by 4×4 a n t i h e r m i t e a n
to e x t e n d e d
The d i s c u s s i o n
The
c o o r d i n a t e s 14
of c o n v e n t i o n a l
supersymmetrize
for D=6.
((ui,v i) 6 CP N-I x CP N-I,
43].
(4.4
quaternionic
framework
of
Remarks.
nic a p p r o a c h
gation
in the
of a s s o c i a t i v e
like
ii)
by the SUSY e x t e n s i o n
(q1...q4;@1 .... @N )
one can only
We w o u l d
torial
are d e s c r i b e d
supersymmetrized.
framework
5. Final
(4.3
0° r er are G r a s s m a n n - v a l u e d i + Oi
=
superalgebras nary"
(8) × Sp(2n)
superspinors
(ql...q4)
@i
= SO
tra-
the propa-
curves
[58,
curves
has been g i v e n
for
formulae
of W e i e r s t r a s s
[59]
[61]
for D=4 has been;derived.
152
The e x p l i c i t e
parametrization
of null
red by H u g h s t o n
and Shaw
the d e s c r i p t i o n
of s u p e r s t r i n g s
has been given iii)
The
placement space
in
of QFT
zed only
nature
nature
complex
of s u p e r t w i s t o r s
nonchiral
to
superspace
ap p r o a c h
space by QFT
Such a p r o g r a m m
spinor
space
in twistor was also
approach between
of space-time
coordinates,
(see
is the re-
or a m b i t w i s t o r
investigated
[64,65].
is an analogy
of e l e m e n t a r y
to physics
in the
It should be al-
which
can be locali-
(2.4))
objects
with u n o b s e r v a b l e
consti-
(quark or p r e o n models).
It is t e m p t i n g tuents
in D=4
in terms of two twistors
- composite tuents
63]).
that there
- composite
and the a p p l i c a t i o n
of twistor
in M i n k o w s k i [62,
of R z e w u s k i ' s
so o b s e r v e d
for D=6 has been c o n s i d e -
[41].
"strong version"
(see e.g.
framework
[42],
curves
to describe
as due to the
stor space. [66-68],
Some
where
they were
the c o n f i n e m e n t
fact that they are
investigations
the notion
r e l a t e d with
in some
r e l a t e d with
of q u a r k - t w i s t o r
strings
of u n o b s e r v a b l e sense
locali~d
in twi-
such an idea were made
variables
on conformal
consti-
was proposed,
supergroup
in and
manifold.
Acknowledgments The author w o u l d le discussions, tional
Centre
completed.
and Prof.
talk was p r e s e n t e d
This paper ki on his nates
to thank
dr. L. H u g h s t o n
A.
for the h o s p i t a l i t y
Salam
for T h e o r e t i c a l
We w o u l d
perunification
like
like also
Physics,
where
to m e n t i o n
these
I would
like
70 -th birthday,
should be more
valuab-
at the Interna-
lecture
notes were
that the first version
at the F i r s t Torino Meeting
(23.IX - 27.IX
for several
on U n i f i c a t i o n
of t h ~
and Su-
1985).
to dedicate
to my teacher,
who teught me first
fundamental
that the
than the ones
prof.
J. R z e w u ~
spinor
described
coordi-
by space-time
fourvectors.
FOOTNOTES I. These g e n e r a l i z a t i o n s vitons);
see e.g.
were
defining
und
fields
states
(photons,
gra-
[4,5].
2. One can show that the coupling and.gravity
"googly"
does
not impose
of m a s s l e s s
particles
any r e s t r i c t i o n s
to external
YM
on these b a c k g r o -
fields.
3. For N=4 D=4 tegr~bility
SUSY YM theory
along
it is not k no w n how to derive from the in-
SUSY-extended
null
lines
the
internal
sector
selfdu-
153
ality c o n s t r a i n t for the field strenght superfield. 4.It should be m e n t i o n e d however, of SUSY in real E u c l i d e a n
that in these papers
the structure
space has not been taken into account.
For
the d i s c u s s i o n of s u p e r - s e l f - d u a l i t y with more e x p l i c i t e d i s c u s s i o n of E u c l i d e a n SUSY see
[22].
5. For local string s u p e r a l g e b r a
see
[27]
, for i n t e g r a b i l i t y see
[28].
6. This formula is due to Penrose, but some authors did put forward earlier ideas that space-time coordinate can be e x p r e s s e d as composite in terms of spinor c o m p o n e n t s - see e.g. 7. For N=4 one gets as internal
[29]
(see also
symmetry groups SU(4)
[30]).
[34].
8. We call superspace
SCM~N)" chiral because the v a r i a b l e s @~ and 0~" can u 1 1 be o b t a i n e d from 4 - c o m p o n e n t complex Dirac spinor by chiral projecticns ~(I±Y5)
(for Y5 diagonal).
9. From
(2.15b)
one gets for the. last. term of 6z &~ the e q u a t i o n i 6 z ~+
+@8 ~i:0 w h i c h is solved by 6z~8=i@@~8~ if we put ~i=@iz8. 10. The formula flag m a n i f o l d 11. In formula
(2.18)
(see e.g.
can be e x p l a i n e d g e o m e t r i c a l l y as s u p e r s y m m e t r i c [9]).
(2.19) one can recognize the known r e l a t i o n between the
real, chiral and antichiral 12. From
(3.3)
superspace coordinates
follows that ~ ~
is p u r e l y imaginary.
13. The S U S Y - e x t e n d e d null lines defined in [6] in fact do not lie on the s u p e r - l i g h t - c o n e
(3.13), because the part d e s c r i b i n g the transla-
tions along G r a s s m a n n directions
is missing.
not invalidate however the c o n c l u s i o n s 14. The q u a t e r n i o n i c morphisms
supergroups
This s i m p l i f i c a t i o n does
in [6].
as q u a t e r n i o n i c n o r m - p r e s e r v i n g endo-
in superspace were c o n s i d e r e d r e c e n t l y in [52].
15. In ref.
[57] even the name "isotwistor
superspace was p r o p o s e d as
more a p p r o p r i a t e than "harmonic superspace".
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S U P E R S Y M M E T R I E S OF THE D Y O N +
Eric D'Hoker Department of Physics Princeton University Princeton, New Jersey 08544 U.S.A. Luc Vinet Laboratoire de Physique Nucleaire Universite de Montreal C.P. 6128 Succ. "A" Montr(~al,Quebec H3C 3J7 Canada
Contents
Introduction A. Spectrum supersymmetries of particles in a Coulomb potential I. 2. 3. 4. 5.
The The The The The
4-dimensional system 3-dimensional system quantum numbers OSp(2,1 ) representations spectrum of HI)
B. Hidden symmetries of a spinning particle in a dyon field I. 2. 3. 4.
Symmetries of H (generalization of the Runge-Lenz vector) Supersymmetries of H Structure relations Spectrum analysis a la Bargmann
Acknowledgements References
* Seminar delivered by Luc Vinet in January 1986 at the Laboratoire de Physique Theorique et des Hautes Energies, Universite Pierre et Marie Curie (Paris VI).
157
INTRODUCTION Over the last two years or so, we have investigated the rble of superalgebras as dynamical algebras in Quantum Mechanics[l]. The first problem we analyzed[2,3 ] was that of a Non-Relativistic spin-I/2 particle in the field of a Dirac magnetic monopole which was shown to possess an OSp(1,1) dynamical superalgebra. We also observed [4] that this system can be generalized to accomodate a I/r2-potential and further noted the presence of an N = 2 superconformal symmetry in such instances. These interesting observations allowed us to obtain the spectrum and wave functions of the above systems from group theory alone. A famous problem with dynamical symmetries is certainly that of a spinless charged particle in a Coulomb potential. It possesses an 0(4) invariance algebra which explains the "accidental" degeneracy of the spectrum and all its states fall into a single irreducible representation of 0(4,2). A natural question that one can ask then, is the following: Can we find supersymmetries in the presence of a I/r-potential? We came up with the following answer. A. Consider the Hamiltonian D 2
-
-
(i)
(~-q)2- q~ + ~F.2
~ri~ i
42r 2
i= 1,2,$
r3
where ADi is the vector potential for a magnetic monopole of unit strength,
' - ( ° ' ° I 0o
.
0 ) :,0
(2)
2 and ~. is a free parameter. ItD describes the quantum dynamics of two spin 0 particles and one spin I/2 particle with electric charge - 11e in the field of dyons with electric charge e and magnetic charges respectively (q¥ I / 2 ) / e and q/e. We have found that HI) admits an OSp(2,1) spectrum supersymmetry which we used to obtain its spectrum and eigenfunctions[5]. B. In the special case i~ = 2q, the two lower components of HI) read
HI = Ho
112- q Bi°i
Bi - ri
rs
(3)
with
=
_
~+2r
2
(4)
158
It happens that the spectrum of H1 possesses high degeneracies. These are understood by viewing HI as the supersymmetric partner of H01]2 which is known to have the same spectrum structure as the Coulomb problem (with q=0). The constants of motion responsible for the accidental degeneracy of HI were obtained and embedded in an 0(4)~U(212) invariance superalgebra of the combined H0~2~)H 1 system [6], Their knowledge allowed for an analysis ~z la Bargmann of the spectrum of H1 [7] It is these results that I would like to expand upon in the course of this talk.
A. SPECTRUMSUPERSYMMETRIESOF PARTICLESIN A COULOMB POTENTIAL
In order to derive the spectrum supersymmetries of HD we shall use dimensional reduction to establish its connection with a 4-dimensional oscillator-like Hamiltonian. The supersymmetries of our 3-dimensional problem will then be inferred from those of this 4-dimensional system. It will be convenient to coordinatize ~4 with 2 complex variables za, a = 1,2 and their complex conjugate z a. We shall denote the corresponding vector fields by Oa = 0/O~ , ~a = O/Oza • Let r i, i = 1,2,3 be the standard Cartesian coordinates on ~3. Dimensional reduction will be effected via the Hopf map "
{2\ ri=,
{o} ---,
,IIi(Z)
=
3 \{o}
~'ao'ibzb
i- t,~,z
(s)
where a i stand for the usual Pauli matrices. This projection defines ~4 k{0} as a U( 1 )bundle over ~3\{0}. (Summation over repeated indices will be understood throughout.)
I. The 4-dimensional svstem
Consider the supercharges
(5) Izl"
159
with ~. a free parameter and the rls verifying
{~,.~b}=
0
(7)
=
We shall use the following realization of this Clifford algebra •
,(o
t(
,oo,)
rlz=-'~
~t="~ )_os
o o)+to 2)
-o 1-to 2
(8)
0
The anticommutators involving Q and Q~ are given by
[ Q , 0 } = {Q+,0+} = o
(9)
and H
= ½{~ .0 )} -
0a~'a +
(t0) T-~-z(~.-c)
-
z~,
izl4
where C = X +5:_
--
X = Zaa a - Za8 a
X-
' and
(
03 0 0 0
)
o)
(I)8)
(11b)
0 o_i 2
Note that X is the generator of the U( I )-action on the fibers of ~4 \{0} ~ ~3 \{0}. Now it is not too difficult to see that we can adjoin to Q, Q* and H, two more odd generators (S, St) and 3 more even generators (D, K, Y) to form an OSp(2,1 ) realization. Indeed one can check that
S =
Zaqa
S*=
Zaq)a
(superconformal)
(12a)
o =½(z°o,+:.:~+21
(dilations)
(!2b)
K = ~'aZ a
(conformal)
(I2e)
160
together with Q, Q) and H satisfy the structure relations that characterize the superalgebra OSp(2,1). These are
{Q,Q)} = 2H
{ S , S t } =, 2K
{Q,S )} = - 2 D - 2 i Y
{ Q+, s ] - - 2D + 2iY
[H,S] = -|Q
[H,S t ] ---iQt
(13c)
[K,0] =
[K ,0 t] = is +
(!~)
is
(1~)
(13b)
[D.Q]=-~'~ [D.S1= ~S [D.Q'1=-~O' [D.S+1=~S+ ['~.Q1=½~ [.~.s]= ½s [,,,,Q+].. -~ Q+[y.s+]--~s + [H.D] = iH
[H,K] = 2LD
[D.K] = iK
( 1u)
(1~j)
with all the other { ] equal to zero, We remark that all the above charges are invariant under the generated by
ji - - ' (~-z
i (~b - -ZaOabi ~ b ) + a oab
~i
i = 1,2,3
SU(2)-action
(14)
We also note that C = 2× + ~ commutes with ji and with all the 0So(2.1) eenerators. This observation will play a crucial r61e. In summary, the full symmetry algebra of the 4-dimensional problem that we have just defined is OSp(2,1) (~ SU(2) (~ U(I)
(is)
2. The 3-dimensional system Let us take the following superalgebra element :
(16) mB
",.
~,,a'a. (-
+ Iz) 2 +
b
- 2~
Izl 4
161
and introduce the eigenvalue equation :
R~,
=
(-2E) ~ •
(~7)
In order to project this equation from [~4 \{0} to ~3 \{0}, we shall require that the 4-dimensional wave function • be equivariant under the U(1)-action generated by X. More precisely, we shall take • in the U( 1) representation with weights ¢-~ ai,g ( q -
= di,g
q,
(
,
I
q
Equivariance under this representation is expressed by the condition X~= (q - : / 2 ) ~ or equivalently C ~ = 2q~ (tg) Let us point out here that the symmetries of the projected system shall be those of the 4-dimensional system which preserve this constraint. Since C is central, it means that the basis elements of our OSp(2, I ) ¢ SU(2) realization all generate symmetries of the 3-dimensional problem. To carry out the projection it is convenient to introduce the Euler coordinates O
'r = ¼ ( ) + ~ s ) Z
(31b>
+(q-x)
Finally, from the representation theory of 0(2,1), we know the eigenvalues of R to be given by
= (~.~. + .)
°.0.,.2
~32>
164
if those of CO are written in the form (33)
Here. A I , & , X . [ I ( j + I ) - I o ( J o + I ) + ( q . ~ ' ) 2 ]½__ 1 ( I - X ) ~ ' + 2
1
(34)
]n summary, we have the following eigenvalue equations to characterize the states of our system ' j21j,m,~,X,n>
= j(j+l)lj,m,~,~,n>
(3se)
}3 I j , m , ~,X, n> = In I j , m , ~,X, n>
(3Sb>
,~ l j , m , & , X , n >
(3Sc>
ySl
= & Ij,m,~,,X,n>
j,m, ~ , ~ , n> -
R lj,m,&,X,n>
X I j,m, &,)~, n>
= (A
(35d)
+n)lj,m,&,X,n> ,X-
(3se>
4. The 0SD(2.1) reoresentations The action of the remaining 0Sp(2.1) generators on the I j. m. &. ~. n > state vectors has been obtained recently [6]. This is most easily achieved by going to a Cartan-type basis for 0Sp(2.1 ). Introduce the following ladder operators B+"' ½ [ K - H + 2iD l L
F, = -~rLs + 0]
(36e)
F? = ¢F~)+
(36~>
The 0Sp(2. I ) structure relations then become
[R,6,_] = ± B,
{F"'~F"'"}- o
[B+.B ] = -2R
OF:_,F~}- B.
¢3~.>
C~+'_.F:}" ~ +--"
_L,R L,R L L | R [)~,~,. 1- ±½F_~ [~',F.]=-½F_. [Y,F~]- ~F+ L,R
lB.,. F,. ]-o
L,R
[B+ F"'"] =-T-F,.
~'"'
165
After a little work, one find that B± and F±(L,R)act as follows on our basis states: 8+lj,m;~.,X,n>
L
- [
(~ ;°)(~ ;o+,) ,&,
,&,
-
(3~)
^
F+ [ j , m ; e , - 1 , n> = (38b)
).m;-,,...± -~>
a aa[(,±a> ~.~,~ ½()-a)+.] R
F± I j,m;
^
n>
o~,-1,
(300)
-
,,~ ½^ a_£[(i±&)Aj,R,x + I g( I - ~,)+ n] i,m; 11, , n± -~> L
F±Ii,m;
t,l,n>
=
-
1 :t 1 + n
+a [~.~,.~ ~-~ I + +
R
F ± i j , m ; 1, 1 , n ) L F±lj,m;-!,1,n>
s + n
]~l j,m;
!,-1
I i,m,-I :I .
'½ )
n+~+
(3~1)
n+~-~> _
-
0
(3~)
-
0
(3o0
R
F_+ i j,m;-1,1, n> ffi
a+[ ()~))A),~,,x+ ~I + ~1 + 0 ,]~' Ii,m' -a_ [~'")5.~.~
I ± 1+n
~
,
' ½~>
1,=1 , 1').+-~+
(300)
i j,rn;-1,-1,n±~-~>
where
(39)
= o~
2A i,&,X
By going to a coordinate realization, solving for the ground state and applying repeatedly yhe ladder operators, the wave functions can then be obtained simply (see reference [6]).
166 5, The spectrum of HD
The spectrum of HD can now be straight forwardly gotten. We have arranged our equations for E to be the eigenvalue of HD. Now from eqs, (17) and (35e) we have R == ( - 2 E i ~ =
(Aj,&,~ll)
(40)
It trivially follows that
En4'~J~
•,
=I
2 (Aj,&,~ n) 2
(41)
In the special case ~, - 2q the two lower components (~C= - I ) of HD become H, -
½~2_
_I
r
+ q2
~'2
_ eB I'Oi
(42)
with V " -- (p =eR) =
~×~,
.~, = g R D =
(43)
g[-'. ro
I_a tr(XI a ).
Proposition 2.4. establishes a link between the formalism which is used in /I/, /2/ and our present approach. Originally the r-matrices were considered as functions with the values in J ~
~
. In our
approach we associate with such functions linear operators. The ellip-
184 tic r-matrix (2.15) was indeed (for n = 2) the first example of a classical r-matrix 6ver studied /I/. 2.4
Let us now indicate how the present formalism may be used to
produce integrable Lax equations. (a)
We proceed in several steps.
The pairing
Y.,-
y > -
(2.,7)
tr,~ D
"~
is non-degenerate and allows to identify ~ )
with its dual. Note also
that (~#3~) ~ _~ ~(~). Another model fer the dual to ~ by ~ -
- ~.~
~
~
[
~
is provided
The two models are related by a map
which assigns to a rational function on ~ satisfying (2.12) the set of its principal parts at ~ 6 ~ . space
~b
The Poisson submanifolds in the
are easy to describe. In particular, we have
Proposition 2.4.
Functions with simple poles at ~-6 ~
submanifold of ~ ( ~ ) a ¢
(~+)~.
The symplectic leaves lying in it
coincide with the eoadjoint orbits of (b)
Let I ( ~ )
form a Poisson
be the algebra of
~ r
I ~
-- S ~
Ad G-invariants on ~
The algebra of Casimir functions of ~
(~I ~ ) = ~(~l~)
is generated by the func-
tionals of the form
%
where
~ r(O..3 ) ,
o~v.-e e [ a¢ -4 , gCJ~, L=C~..,v-~@~6~
By restricting these functionals to the orbits described above we get Hamiltonians in involution giving rise to Lax equations of the form
(2.i9)
d._..~L = [ L,P'I ]
~
[, ~ £ &(£)
/ t'l=P ° (d~,9[[])
They are usually referred to as Lax equations with the spectral parameter on an elliptic curve. By applying Theorems 1.1, 1.2, we may systematically construct such equations and their solutions (cf./15/). 2.5.
The examples considered so far give rise to finite dimensional
systems admitting Lax representations L~
d[/~
= [ [l ~ ]
#
where
are matrices PoSsibly depending on spectral parameter. In many
cases it is natural to assume that Li~ variable
also depend on a spatial
x . Lax equation then takes the form
~
i ~
~ [ Ll M ~
.
There is a natural way to include such equations into the present iormalism.We explain it in brief since it will be of importance in the
185 study of dressing transformations (see § 6. below). Let ~
be a Lie algebra with an invariant scalar product. It will be
convenient to assume that ~ i s
a matrix algebra. We denote by G the
corresponding matrix Lie group. For the time being the reader may assume t h a t ~
is finite dimensional. However, in realistic applications
is always a loop algebra (see below). ~=
C°O(~/Z
~ ~). Suppose R ~
~
C ~C/~/~/
~/
End ~ s a t i s f i e s the Yang-Baxter
identity (1.8) . We extend it t o ~ ~
Put ~
by setting (RX)(x) = R(X(x)). Let
be the corresponding Lie algebra with the Lie bracket (1.4).
There is a 2-cocycle on ~ d e f i n e d
by
Y).
(2.20) Let Put
oen,
,.e
ex,eo
(2.21) ~ C ~ , y )
*oo
ooo o e
¢/
~ L ~(X,y)
~ 4_ ~ C x , R y )
Proposition 2.5. (i) Formula (2.21) defines a 2-cocycle on ~ ( i i ) L e t ~ # be the corresponding central extension o / / ~ Then (~/~l ~ ) is a double Lie algebra. It is particularly nice when the operator R is skew with respect to the inner product o n ~ . sion
~
In that case
c0~ = 0 , so the central exten-
splits. Hence the orbits of %
they are clearly
and
%
coincidei
"continuous products" of orbits o f ~ .
Since in the sequel we shall be dealing almost entirely with this case, it is worth giving a formal definition. Definition 2.1 algebra if
A double Lie algebr: (~, ~
(i) the operator
Baxter identity (1.8) duct on ~
and R
R ~
End ~
) is called a Baxter Lie
satisfies the modified Yang-
; (ii) ~here is a (fixed) invariant inner pro-
is skew with respect to it.
Let us now describe the Casimir functions on ~ Proposition 2.6.
Let us i d e n t i f y ~
.
with its dual by means of the
inner product
so that
--
@ ~. The coadjoint action of
by
(2.22) ~
X.(L
~)
=
(
I X , L]
on
,~X,
is given
o),
186
It integrates to the action of G
(2.23)
A~ ~
Notice that
given by
cI
=
-~
,
(2.23) coincides with the gauge transformations which are
connected with the linear differential equation
(2.24) Let
L~L
be the fundamental solution to (2.24) normalized by the condi-
tion
(2.25)
~ L (O)
=
~
•
(the identity matrix)
By definition, the monodromy matrix
T(L) = ~L(1).
Theorem 2.1. (Floquet).
Two points (L,e), (L',e') (e ~ 0)
same coadjoint orbit in
~#
if and only if
T(L), T(L') are conjugate in Corollary . where ~ Note.
e' = e and the matrices
G'.
The Casimir functions on
~@)is
are of the form L - - ~ ( T ~ L ) )
a central function.
it is clear now that the codimension of orbits in
equal to rank
~ L = ~ o There are also precisely
~
the algebra of Casimir functions on each hyperplane
~
lie on the
~
is
generators of e~= const ~
0 in
. Hence to get sufficiently many integrals of motion provided by
theorem 1.1. we must assume
~ = cw~ . This is the ease when ~
is a
loop algebra. Theorem 2.1.
shows in particular that our geometric approach incorpo-
rates the conventional inverse spectral transform methods which are based on the study of the auxiliary linear problem (2.24) . An extremely important point is the study of Poisson properties of the monodromy map which we now state. For ~ ~
~
~G ( )let
nition ~y l ~ ,
~l ~t
~ ~
be its left and right gradients. By defiand
I Theorem 2.2. tionals [_~ ~
x)=IgJ
'
Let ~ 4 , ~ ~ C "~ ~ G ~ • The Poisson bracket of the func(T(LI)
is given by
C~×,, x;)-,, C~c~,),x-), )- which coincides with (3.12)
. We leave it to the reader to prove the
last assertion which is done similarly. As a corollary of Proposition 4.3. we get Theorem 4.1.
(i)
action.
(ii) Let us identify the quotient space with ~
Natural action ~
~/~Ais
a Poisson group ~.
Then this action is given by the formula
(4.4)
("t~,~): ~ l'--~ Z'~ ~ ( ~-~ ~-t~ ~
In particular, the subgroup
(4.5)
~:
~
~-*
~
~
~gC~is
= ~ (~-tX-4~ ~) -
acting via
( ~-I ~ - 4
k_ ~ ) ~
This action is a Poisson group action and its orbits coincide with the symplectic leaves in ~ ( e q u i p p e d with{~Sklyanin bracket). We shall call (4.5) the dressing action, it may be regarded as an analogue of the %he coadjoint action° Proof.
Since natural projections
form a dual pair, we are in a position to apply a general theorem from /16/. It asserts that if ( ~I W') is a dual pair of Poisson mappings, then the symplectic leaves are obtained by blowing up points in the double fibering (~s~ ~) , i.e. they are the connected components of r~'CrF -4)
(~.The
projection map ~
~-~
G~\~
•
@
is given by
whence This makes the last assertion obvious. (All the rest is perfectly evident). Note. The result we have quoted is a slightly refined version of a theorem due to V.Drinfel'd. In a dual fashion we may give a description of symplectic leaves in ~
O/@
. Note first of all that
@
serves as another model for
199 ¢T% J__
the quotient spaces 6~/~
, (D'\ c~'. Canonical projections are then given
by
~
(4.6)
[X.,~)
#
Corollary I.
~
#I
-~
9C~ "4
Symplectic leaves in ~
are mapped onto conjugaey clas-
ses in
~
under the canonical mapping
Proof.
Both groups are different models of the same quotient space.
Corollary 2.
Casimir functions on ~
ture described in Proposition 4.4 on
~
m : ~
_, ~
with respect to the Poisson stru~
are precisely the central functions
•
For completeness we give an explicit formula for this Poisson structure Proposition 4.7. (4.7) ~ ~ ~ l ~ u ~
The quotient Poisson structure on 6 i s given by ~ < ~(X)~
y'>t V, ; - y'=
where ~ = V~ , X'= V~+ ,< y =
We leave the proof ~o the reader (cf. the proof of Proposition 4.3). As another application of the reduction technique we give a proof of Proposition 5.11. Proposition 4.8.
Canonical projections
J# C~, - ~ )
i
:~/~
form a dual pair. The proof is the same as in Proposition 4.3 (Note the sign difference in the Poisson bracket on ~
!)
Corollary ~quip ~ / ~
~
G~ \ B with t~e product Poisson structure. Canonical embedding "~,~Z~,'R~ L ~ I ~ × 6-R\~is a Poisson mapping.
It is easy to check that the quotient Poisson structures on ~ / ~ #
~R\~
are again given by (3.21) - (3.22), (2.28) - (3.12), respectively.Since $~. ~aC ~ is an open subset this finishes the proof of Proposition
(3.~I).
200
§~.
LAX EQUATIONS ON POISSON LIE GROUPS: A GEOMETRIC THEORY
We start with the simplest theorem on the subject which will then be generalized te include Lax equations for lattic~ systems. Throughout this §
we assume that
~is
a Poisson Lie group and that its tangent
Lie bialgebra is a Baxter Lie algebra. Recall from the end of §4 that there are two different Poisson structures on ~ (3.12-),
(4.7).
This suggests ~hat
which are given by
we may use them to construct inte-
grable systems in almost the same way as in Theorems 1.1, 1.2.
As we
shall see now, this is indeed the case. Denote by
I(~)
Theorem 5.1.
the space of Casimir functions for the bracket
(i)
(4.7)
Casimir functions of the Poisson bracket (4.7) are
in involution with respect to the Sklyanin bracket (3.12-). (ii)
Let ~ E I ( ~ )
. The equation of motion defined by ~
with respect
to the Sklyanin bracket has the Lax form
& (iii)
Let
z x+(t)
be the solutions to the factorization problem (1.11)
with the left hand side given by The integral curve of the equation (5.1) starting at L @ ~
is given by
The proof is parallel to the proof of Theorem 1.2. Observe first of all that left and right gradients of a function ~
I(@)
coincide. This
makes (i), (ii) directly obvious from the definition of Sklyanin bracket. Proposition 5. I. projection , ~ ~ tonian
~
(5.4) (
I(Gr), h ~
Recall that
by !
)
W
~ is included into a dual pair (4.6) . Projections
of the integral curve i n ' o n t o
the quotient spaces ~ / £ ~
reduce to points since the reduced~Hamiltonians Since
~
be the standard
• The integral curves of the Hamil-
=~,~
on ~ ~l~)are given
,
Proof.
: (x,y)#.~ xy -I
Let ~ , ~..~
is both right- and l e f t - ~ - i n v a r i a n t
,
are Casimir functions. we have
201
Obviously, V~y = (x' ,x') G d where X' = V~ (xy
) is ti~e-indepen-
Now (5.5) follows immediately.
dent.
Consider the action
~
× ~
~
Notice that the subgroup (~,e)
defined by
~ ~ is a cross section of (5.4) on an
open cell in ~9 . Hence we get a canonical projection
whose fibers coincide with ~ g -orbits in Proposition 5.2.
(i)
The action (5.5) is admissible.
(ii) The
quo-
tient Poisscn space is canonically isomorphic to ~(-R, g). We shall prove a more general statement below (Theorem 5.4). To finish the proof of Theorem 5.1 observe that f o r ~ E i ( G ) ~ = ~ , ~ hence (5.4) defines a quotient flow on
@(-~i~ith
Hamiltonian
Projecting the flow (5.4) down to C_~ gives (5.3). We shall indicate a generalization of Theorem 5.2
which is suited for
the study of lattice systems. Recall from Proposition 3.4
that we may
use more general Poisson brackets given by (3.10), with the left and right R-matrices not necessarily coinciding. This observation is used to twist the Poisson bracket on Let ~ b e
.
an automorphiem of a Baxter Lie algebra (~, R) i.e. an ortho-
gonal operator
"~
automorphism of ~ conjugation
(5.7) Let
~
Aut ~
which commutes with R. It gives rise to an
which we denote by
~rx ~--~
$:
~t'I(gm)
~
k~
g~-~ g
. Define the twisted
by
~L'~
be the space of smooth functions on ~ i n v a r i a n t
with
respect to twisted conjugations. Theorem 5.3.
(i)
Functions
~
~ ~I(G)
pect to the Sklyanin bracket on G by a Hamiltonian ~ ~ ( ( ~ )
(5.8) with
B =
~= ~ R
LA- 6L
have
.
are in involution with res-
( i i ) Equations of motion defined
the following form
,
L(~,
(V~(L)~A=z(Z).(iii) Let
x~ ( t )
, x_ ( t ) be the
202
solutions to the factorization problem (1.11) with the left hand side given by
The integral curve of equation (5.8) defined by ~ ~ ~ ( ~ )
(~.1o) ~he p r o o f
L(~) is
=
~:~ (~).
.L
x± (~)
are given by
based on t h e use o f a t w i s t e d
Poisson structure
on
Extend 9 t o
(~.~)
~c(×,y)=
(X, ~ Y )
and put
We also put
~C'~.~ =
~
()~)
=
t ( ~; ~)(~ ) 2"
Equip ~ w i t h the Poisson bracket (3.10) with Proposition 5.3.
R = Z R d , R' = R d .
(i) The natural action of ~ o n
translations is a Poisson action.
~(~,~]
by left
( i i ) The natural action of
on
~'R~,R~) by rig.t translationsis a right Poisson action This is a corollary of Proposition 3.6 s i n c e
~G
C
~(g4
; -Ra3,
are Poisson subgroups. Proposition 5.4.
Canonical projections
~: :Bc~R~, ~ ) -" ~/~ ~,
$'~ ~B(~R~'~ )
~ ~
\D
are dual to each other. Both quotient spaces are naturally modelled on ~
. Projections ~, ~l
are given by
Proposition 5.5.
Symplectic leaves with respect to the quotient
Poisson structure on ~ a r e
orbits of twisted conjugations (5.7).
Proof. it suffices to compute .~ ( ~ _ 4 ( Clearly,
~,-t(~)
~..
{(..~-4 ~ : ~ )
= [~-~,~-~ Corollary.
))
/ ~ ~ G ~ ( ~Oj-'s [~))'-,,. , ~ G ~ ~
Casimir T'unctions of the quotient Poisson structure on
are invariants of twisted conjugations. A generalization of formula (4.7) for the quotient Poisson structure on G
is given by
(5,14) a ~ v , * ~ , d . = -{R(×),Y>- < Rex,), y'> ~ - < ~,, -c.y> Now everything is ready for the proof of Theorem 5.5.
203 Proposition 5.6. Hamiltonian
Let ~ ~ I ~ G )
h T on ~ ( ~ l ~ 3 a r e
,
hT = fo ~ . Integral curves of the
given by (5.4).
We leave the proof to the reader since it is completel~
parallel to
that of Proposition 5.1. Consider the action
k
(5.15)
,~ ~ ,
Theorem 5.4.
G R xb~
~,
(i)
~+
To check
~
,
+ ~
The action (5.15) is admissible.
Poisson bracket on ~ / ~ Proof.
~ g i v e n by
(i)
( i i ) The quotient
coincides with the Sklyanin bracket.
we use Proposition 4.1.
Observe first of all that by combining left and right translations we get a Poisson group action:
C.R~, .R~ x SO~Rd,~ ) x ~ h ~ , ~ )
-->
SO(-~, ~ ) :
We have changed the sign of the ~oisson bracket on the second copy of ~so
as to consider left actions (More generally, if there are two
commuting Poisson group actions ~ x ~ - ~ ;
~ J b ~
] their combi-
nation gives rise to a Poisson group action of ~ x ~ ( w h i c h is equipped with the product structure). Now, ~ g is embedded into ~ X ~ Since the tangent
via
Lie bialgebra of
~
is (d @ d, ~ _ ~ A @ ~ , ~ )
our claim follows, by virtue of Proposition 4.1,
from the following
lemma. Lemma I. ~ C
Proof of the l e n a . nihilates
~
is a Lie subalgebra in ~ -~t~l.
~ • &
An element (X I ' X2 ' ¥I , Y2 ) ~ d @ d
(~,~,
Since there are natural Lie algebra embeddings '
R ~
~1~
an-
if and only if
Equivalently,
'
~
d ~
~ , ,~L3,~
~
' ~R
, ~_~'=~
~ dR d '
it suffices to check that
implies
R_([I.,~,]R
) .
~.~-C~;,~]~)
=o
The last assertion follows immediately from the Yang-Baxter identity.
204
We now come to the proof of the second assertion of Theorem 5.4. Observe that the subgroup (G, e)~-~ is again a cross section of the action (5.15) on an open cell in ~ .
The canonical projection
?: ~__~ C_T is now given by
X' = V~ , Y = V~ of
HT
, Y' = V ~
. It is easy to compute the gradients
. Their restrictions to the surface ( ~, e ) C ~ are given by
v,,~ --
( x, x;.
-~-×_,)
,
v ~ , : (x', x~. - ~ x _ ) .
Similar formulae hold for the gradients of
~r.'~'~,~
=
( x', x'-y"
H~
. Now
+--r y_}
After substituting these expressions into the definition of ~ W, ~ I(~,U~ we get after some remarkable cancellations
s
-- t
4)--
f r,
Note; Unfortunately, T do not know how to extend to the present case the qualitative argument which w~ have used in the proof of Theorem I .2.
This argument is now replaced by a direct computation.
Let us now apply Theorem 5.3
~o the difference Lax equations. Let
(~, R) be a Baxter Lie algebra, ~ t h e Put ~ - "
~
~l~r~N.We shall regard elements of ~
ping~//V~ into ~
(5.16)
corresponding Poisson Lie group.
. Equip ~
( X, x/ >
_---
as functions map~
with the natural inner product
~"
< /k(" , "x/~, >
h
and extend t~ ~ En~ ~
(~)~, -- ~(X~) . ~his makes (~,R) a Baxter Lie algebra.t1Equip ~ w i t h the product Poisson struc-
ture. Clearly, G
is ( ~ ~ )
is a Poisson Lie group and its tangent Lie bialgebra
We shall denote elements of G by S = (el...... S ~ )
Define the mappings
Functions ~
(5~)
~
t o m by setting
~j
T:
G--~
~
by
satisfy the linear difference system
= ~
~,
, %=
while T is the monodromy matrix associated with (5.18)o Obvieusly, one has Proposition 5.7.
The monodromy map T : G - ~
is a Poisson mapping.
205 This property of the Sklyanin bracket has served as a motivation for the whole theory. The quantum version of this statement goes back to R.Baxter. Let ~
Aut ~
(5.19)
~,
be the cyclic permutation
(
×~ . . . .
, x,)
~
( X~,
Clearly, the twisted conjugations
L ~
x~,
g L
×~ . . .
-~'-1 g
X~.~ )
coincide with the
gauge transformations for (5.18) induced by left translations ~
~
in its solution space. The operator (5.19) preserves
the inner product (5.16) and commutes with R .
Hence
applies to the present situation. The space
Theorem 5.3
is described by the
following simple theorem. Theorem 5.5. ("Floque~") gauge orbit in 6
~
.
(ii)
The algebra
, L --~ ~ (~- [ L ) )
As a c o r o l l a r y of Theorem 5.3 Theorem 5.6.
Two elements L, L' g ~ l i e
(i)
Functions
,
h~
,Ig I ( ~ )
tion~of motion with the Hamiltonian
-
(iii)
~
Let (gm) ~ (t)
~_,
mI(~)
T ~
I
is generated by the
(~
•
we ~et
respect to the Sklyanin bracket on ~
~
on the same
if and only if their monodromy matrices ~(L), T'(L')
are conjugate in ~
functions
(i)
-
. h~
are in involution with
( i i ) The Hamiltonian equais given by
~
L~,
be the solutions to the factorization problem
(I .11 ) with the left hand side given by
cLoll) The integral curve::of (5.20) with the origin at is given by
(5.22) Note.
=
(
% . )+_
A completely different approach to the study of difference Lax
equations was described by B.Kupershmidt / ~ / .
These two approaches may
be linked together by a discrete version of the Drinfel'd-Sokolov theory / lq/. However, a detailed analyses of this link goes beyond the scope of the present paper.
206
~6. DRESSING TRANSFOrmATIONS.
In the present paragraph we return to the study of Lax equations on the line described in §2.5.
Our notation will be close to that intro-
duced there, the only difference being that we now drop out the periodicity condition. Thus let (~, ~ R )
be a Baxter Lie algebra. Let ~; (=~
be the corresponding dual Poisson Lie groups.
I. Let~= ~ ( (6.1}
~ , ~ ) / ~=C~,~,~.We define an inner producto,~ by
x, Y>
Clearly,~,~)is
=
*
~ p~ ,,,,, ~ Equip ~
~
by setting
~_ ~ ~D
with the standard Poisson structure. Then (6.17)
is a Poisson group action. Proof.
Let us rewrite (6.12) using left-invariant frame
where
~,.[~]
=
~.
~i~)
-~ . g ~ o
~ ( ~ ;
-
We have
Since V
(~J(x)g) = Adg-1(
~(~J(x)))_
our assertion immediately
follows from the definition of Poisson group actions. By projecting down the action (6.17) to the quotient space G ~ I W ~ we get the following transformation
Theorem 6.2.
V~©
Formula (6.18) defines a right Poisson group action
...~ V .
This is an immediate corollary of Lemma 5.
211
Indeed the actions diagram,
(6.17), (6.~8) are included into a commutative
W~L9
~W/ V
We point< out the following special cases (i)
The action o f ~ C ~ i s
serves (ii)
~
a Poisson group action. This action pre-
and its restriction to
The action of ~ ~ r C ~ i s
given simply by ~
~o coincides with (6.5)
a Poisson group action. Clearly it is
. The quotient space is isomorphic to The natural projection is given by 7V:
~.~.
(x, x) = ~
(iii)~ The action of subgroup ( e , ~ ) ~
~
on ~ is given by (6.6). Un-
like the preced~n~ cases, this is not a Poisson group action. Indeed, (0, ~ ) & =
(~,
0) is no~ even a Lie subalgebra in dRd
Of course, formula (3.16) provides a precise information on the "nonconservation"
of Poisson brackets even in this case. Now, however,the
right hand side is not defined in intrinsic terms, it depends rather on the Poisson structure on the larger group
~__O(~leJor
on its
tangent Lie bialgebra. 6.4.
Dressing transformations
and dynamics.
For concreteness we shall assume througho.~t this n£ that = sl (2, ~ [ ~ ,
~-J] ~ . The standard decomposition
gives rise to an r-matrix on .~
(6.20)
%
-
which is skew with respect to the inner product
(6.2~)
< w, ",/> =- ¢~,X=o
In physical literature
4=,- ×(~) Y(~,).
(6.20) is often referred to as the (classical)
Yang r-matrix. Let
El
~j
G_
be the corresponding Lie groups (i.e. the loop group
of sl(2) and its subgroups consisting of absolutely convergent Laurent series). The factorization problem associated with (6.20) is the standard matrix Riemann problem. Let ~ c sl(2) be the standard Cartan
212
subalgebra, ~ C G
x~= ~ (6.22) ~(~,~)=
its centralizer in
, t=(~,
.... , ~,,...)
~ ± = H ~ G.~
@j
/
~.x= ~
. Let
~ x~.
V o c ~ ~x = ~c~) ® ( e ~x, ~ . x )
Let ~ ~o be a wave function obtained from ~ by a dressing transformation (6.18). Since ~÷ C ~ centralizes ~o , we may assume without any loss of generality that
(4,
j
c G_
the element g_ being defined uniquely up to a right factor which belongs to H_ . Dressing transformation defined by g_ transforms the "free wave function" ~ ( x , t ) into ~ ~I~ ~
the element
adg~ IX
being defined uniquely. By projecting (6.24) back
to W o ~ V/$(rwe get tk~
Formula (6.25) d e f i n e s an a c t i o n of H+ on the ~ - o r b i t P r o p o s i t i o n 6.3,
Vector fields
of ~o
•
on Ve which correspond to t h i s a c t i o n
are Hamiltonian. By contrast, the apparently more simple action VX ~ - *
V
given by
(6.24) i~ not Hamiltonian. This is easily checked using formula (3.16) to control the non-conservation of Poisson brackets under the transformations (6.24). This observation, however slightly puzzling, does not contradict of course to Proposition 6.3. Indeed, the point is that embedding H+ C_~ ~ given by with some fixed g_ ~ ~_is not a group homomorphism.
213
REFERENCES
I.
Sklyanin E.E. equation.
2.
On complete integrability
Preprint
LOMI. E-3-79, Leningrad:
Belavin A.A., Drinfel'd V.G. Yang-Baxter
of the Landau-Lifshi~z
equation.
LOMI
On the solutions
, 1980.
of the classical
Funct. Anal. and its Appl~
, 16(1982),
159-180. .
Semenov-Tian-Chansky Anal. and its ~ppl.
4.
M.A.
What is the classical r-matrix.
17(1983)
Kostant B. Quantlzation and representation of the Research Symp. on Representations
.
Notes Series,
Drinfel'd V.G.
structures
bialgebras
theory.
- in Proc.
of ~le grou~s,
1977, London Math. Soc.Lect. Hamiltonian
Funct.
, 259-272.
Oxf.,
1979, v.34.
on Lie groups, Lie
and the geometrical meaning of Yang-Baxter
equations.
Sov. Math. Doklady 27 (1983), 68. 6.
Zakharov V.E.
, Shabat A.B.
Integration of nonlinear equations
by the inverse scattering method. 13 (1979) .
8.
, 166-174.
Date E., Jimbo M., Kashiwara M., Miwa T. for soliton equations.
Proc.Japan.
3806-3816;
Physica
(1982), 343-365
18 (1982),
1077-1119.
Segal G., Wilson G. Publ. Math. I.H.E.S.
.
II. Funct. Anal. and its Ap~l.
Wilson G.
Habillage
4D
Transformation
groups
Acad. Sci. 57 A (1981), • Publ.RIMS Kyoto Univ.
Loop groups and the equations
of KDV type,
61 (1985), 4-64. et fonctions ~T , C.R.Acad.Sci.
Paris,
299 (1984), 587-590. 10.
Semenov-Tian-Chansky group actions.
11.
M.A.
Dressing transformations
Publ. RIMS F~voto Univ. 21 (1986)
Reyman A.G., Semenov-Tian-Chansky systems, Math.,
M.A.
Reduction
affine Lie algebras and Lax equations.
54 (1979), 81-100;
63(1981)
, 423-432.
and Poisson
, N6. of Hamiltonian
I, II . Invent.
214
12.
Adler M., Moerbeke P.
Complete integrable systems, Euclidean
Lie algebras and curves. Adv.Math., 38 (1980), 267-317. 13.
Reyman A.G., Semenov-Tian-Chansky M.A.
A new integrable case of
She motion of the 4-dimensional rigid body. Comm. Math.Phys. (lo~j
14.
Cherednik I.V.
Definition of R-functions for generalized
affine Lie algebras. 15.
Funct. Anal. Appl.
Keyman A.G., Semenov-Tian-Chansky M.A.
17 (1983), 243-244. Lie algebras and Lax
equations with the spectral parameter on an elliptic curve, Zapiski Nauchn. Semin. LOMI (in Russian), v.155 (1986). 16.
Weinstein A.,
Local structure of Poisson manifolds, J.Diff.Geom.,
18 (1983), 523-558. 17.
Karasjov M.V., To appear in Sov. Math. izvestija (1986).
18.
Kupershmidt B.A.,
Discrete Lax equations and differential-
difference calculus, 19.
Asterisque 123 (1985).
Drinfel'd V.G., Sokolov V.V., Lie algebras.
Equations of KdV
type and simple
Sov. Math. Doklady, 23 (1981), 457-462.
ON MONTE CARLO SIMULATIONS OF RANDOM LOOPS AND SURFACES
M.Karowski Institut f u r T h e o r i e der E l e m e n t a r t e i l c h e n Freie Universit~t Berlin A r n i m a l l e e 14 D-IO00 B e r l i n 33
1 Introduction
I would l i k e
to report
ration
with
"Freie
Universit~t
concepts
[I-5]
R.Schrader,
Random l o o p s
regions
theories
Symanzik's [7]
F.Rys,
Berlin".
in different
Quantum f i e l d [6].
on some r e s u l t s
W.Helfrich,
of
polymer
can be u n d e r s t o o d
follows.
The " p a r t i t i o n "
description for
the
function
in
and s u r f a c e s
collabo-
at t h e
are u s e f u l
physics.
can be f o r m u l a t e d
theories
obtained
and H . J . T h u n
i n terms
of random w a l k s
of e u c l i d i a n
simple
quantum f i e l d
case o f f r e e
bosons
as
of t h e t h e o r y (l
reads
in regularized
form on a l a t t i c e
Ld (2
where by
~)
is
a Ld-dimensional
vector
and t h e m a t r i x
~
is
given
216 The i n t e g r a t i o n s
in e q . ( 2 )
= l/,Z
can be performed
.t
-
[ - t,-
=
exe
Since the m a t r i x
7_ T
~
each term ~ t r ~
(I- m] tr
"connects"
in e q . ( 4 )
(4)
T~
n e a r e s t neighbours on the l a t t i c e ,
r e p r e s e n t s a sum over a l l
l e n g t h ~ c o n s i s t i n g of simple bonds. Thus the p a r t i t i o n can be w r i t t e n
Z
=
in terms of a s t a t i s t i c a l
loops of function
system
>-- ~
~×p
(5)
~o~.
where the sum extends over a l l
( p o s s i b l y o v e r l a p p i n g ) one-loop con-
figurations.
The c o n f i g u r a t i o n a l
energy - T , 9 - ~ . ~
to the t o t a l
l e n g t h of the loop.
Expanding the e x p o n e n t i o n a l
(5) we get a r e p r e s e n t a t i o n analogous to the w e l l r a t u r e expansion f o r c l a s s i c a l
Z
= 7__ ~×p
in eq.
known high tempe-
spin systems
~s~
I~ Z
I s~
is proportional
I
CO~. where a c o n f i g u r a t i o n
contributing
s e v e r a l disconnected l o o p s .
to the sum now may c o n s i s t
The e n t r o p y f a c t o r s
model. For the I s i n g model One o b t a i n s a f t e r
of
cz depend on the
resummation a sum over
n o n o v e r l a p p i n g loops
_ #~
-#
=T-e
a
:o.i~.
The 0(N) n o n l i n e a r
=tK~
)
~ -model in the l i m i t
"
(7)
N~0 d e s c r i b e s s e l f - a v o i -
ding random walks [ 8 , 9 ] . The polymer f o r m u l a t i o n of fermions leads (due to P a u l i ' s
princip-
217
le)
a l s o to s e l f - a v o i d i n g
The " p a r t i t i o n
loops.
function"
for
T h i s can be seen as f o l l o w s
free
fermions
[I].
is
F
(8)
11"
x
where sgn = s g n ( ~ ) ,
sgn(l+l=)x~(~ )
and
z
I
~=~ (Note t h a t
for
Kogut-Susskind
dependent number.)
(9)
~_m~t " fermions
Any p e r m u t a t i o n
~
~ is
can be taken a product
m u t a t i o n s I ~ .... J~s
which can be r e p r e s e n t e d
intersecting
The f e r m i o n
loops.
partition
as
function
as an x-
of c y c l i c
s
per-
oriented
non-
can be w r i t t e n
as
= where t h e
(I0)
sum e x t e n d s over a l l
avoiding
loops of t o t a l
tistical
system by
configurations
length ~.
Introducing
of o r i e n t e d
self-
an a u x i l i a r y
sta-
polymer : we can w r i t e
--
:
s.
N->O
where t h e
of t h i s
expres-
(14) over a l l
The e n t r o p y
[9,11]
sum i s
(g-~)
to
~,
sum e x t e n d s
area
expansion
to eq.(6)
two d i m e n s i o n a l
factors
closed
surfaces
c s depend on t h e model.
of
In t h e
one g e t s
now r e s t r i c t e d
to
self-,avoiding
surfaces
and ~( i s
the Euler characteristic. The s t r i n g
quantization
problem
over random s u r f a c e s [ 1 2 ] presented to
by s e l f - a v o i d i n g
self-avoiding
viour
of random s u r f a c e s
Moreover, faces
to
there
should
fermionic
l e a d to
in their
are p o s s s i b l e
strings
a better
state
physics play
phase t r a n s i t i o n s
surfaces in
of e n t r o p y expected to
p o l y m e r s ~ 13]
field
theories
tion
eqs.(ll,12). (loop
involving
gas model)
fermions
in
liquid
surfaces
for
loops
i n [ 2] to
on t h e c r i t i c a l
model
the
on a b a l a n c e
is
which is
be
sheet
a natural useful
to
and i n t e r f a c e s .
simulation of
be u s e f u l relies
two d i m e n s i o n a l
on t h e b a s i s
model
crystals)
sulphur [15 ] . Self-
model [ 1 8 ]
was proposed
the e x c l u d e d volume r e p u l s i o n
of
random s u r f a c e
a Monte C a r l o
A statistical
in the context
L 17J . They can a l s o
of f l e x i b l e
solid-on-solid
we d e v e l o p e d
in
might L I6]
interfaces
properties
of t h e
of
di-
role
whose s t a b i l i t y
t h e r o u g h e n i n g of c r y s t a l
In r e f . [ l ]
sions
dimensions
. The s e l f - a v o i d i n g
generalization describe
three
and e n e r g y of t h e i r describe
(e.g.
polymerisation
of microemulsions
of
and s u r -
i n two and t h r e e
defect-line avoiding
beha-
understanding
an i m p o r t a n t
. They have been s t u d i e d
understanding
may be r e l a t e d
of random w a l k s
polymer p h y s i c s [ 1 3 J mediated
are r e -
continuum limit.
random c h a i n s
L 14J and t h e e q u i l i b r i u m
as summation
particles
of t h e c r i t i c a l
applications
and s o l i d
Self-avoiding
as f e r m i o n i c
An i n v e s t i g a t i o n
theories
statistical
mensions.
walks,
surfaces.
gauge and s t r i n g
has been f o r m u l a t e d
. Similar
method f o r of the
quantum
polymer formula-
in d=2,3,
and 4 dimen-
s t u d y the i n f l u e n c e equilibrium
of
properties
219 of s t a t i s t i c a l
line
systems.
The c r i t i c a l
e v a l u a t e d by means of the " c r i t i c a l loop gas model
i n two d i m e n s i o n s .
diagrams of s e l f - a v o i d i n g mensions w i t h exponents
surface
R,~,~,
In r e f . [ 3 ]
tension
and ~ were
and c u r v a t u r e
intersecting
d i m e n s i o n s were e v a l u a t e d i n r e f . [ 5 ] .
for
the
we e x p l o r e d the phase
random s u r f a c e models i n t h r e e
and ~ f o r
Monte C a r l o s i m u l a t i o n s
exponents ~ , ~ , ~ ,
window" method i n r e f . [ 4 ]
energies.
surface
For o t h e r
and f o u r
di-
The c r i t i c a l
gas models i n t h r e e investigations
of random walks and s u r f a c e s
and
see r e f e r e n c e s
in
El-5J
2 Models The models to be c o n s i d e r e d lattices with
Ld w i t h
periodic
are d e f i n e d
on s q u a r e ,
boundary c o n d i t i o n s
ILdl ~ I 0 4. The p a r t i t i o n
functions
cubic,
hyper c u b i c
in d=2,3,4-dimensions
are d e f i n e d
by
c6(~ where the s e t s of c o n f i g u r a t i o n s E(c)
depend on the s p e c i f i c
gurations
of l i n e s
~i
the l a t t i c e four
links
tained
= ~closed
intersecting
c,~ i
at a common v e r t e x
for
loops
point.
loops
(link)
(17)
(surfaces)}.
(18)
in
of l i n k s c
each v e r t e x Thus t h e l i n e s
(link).
at a v e r t e x ,
two t y p e s of c o n f i -
(surfaces) }
a collection
For C6~sa
(plaquettes).
energies
surfaces):
loops
each v e r t e x
(plaquettes).
are a l l o w e d to touch nected t h i s
comprises
and the c o n f i g u r a t i o n a l We d i s t i n g u i s h
self-avoiding
such t h a t
i n two l i n k s
intersect
model.
(two d i m e n s i o n a l
~sa = { c l o s e d A configuration
~
(link)
The e n e r g i e s may i n c l u d e
con-
may not
distinct
be c o n s i d e r e d
in
in two or
in c is
(surfaces)
But two l o c a l l y
they will
(plaquettes)
is contained
surfaces
as d i s c o n -
t h r e e terms
and
(2o)
220
f o r surfaces. The f i r s t , loop length #
the tension term is p r o p o r t i o n a l to the t o t a l
(surface area
s).
The i n t e r s e c t i o n energy is p r o p o r t i -
onal to the number of i n t e r s e c t i o n points
i
(links ~).
The t h i r d con-
t r i b u t i o n s are curvature energies. They can also be understood as chemical p o t e n t i a l terms of the t o p o l o g i c a l q u a n t i t i e s : number of loops n
and Euler c h a r a c t e r i s t i c X,
r e s p e c t i v e l y . We are i n t e r e s t e d in the
nature of phase t r a n s i t i o n s ( f i r s t and c r i t i c a l
exponents of these models.
3 Monte C a r l o
For s i m p l i c i t y A configuration ratively
of
Method
I shall
describe Starting
change i n
one has t o make s u r e figurations
an o l d
square.
by o c c u p i e d ones and v i c e that
one g e n e r a t e s ci.
In t h e
in
fig.l.
no c r o s s i n g s samoles o f
heat
bath
terms of
random l o o p s .
can be g e n e r a t e d on a c o m p u t e r i t e -
from
a unit
such changes a r e d e p i c t e d
simulation
t h e method i n
(c.f.eqs.(17,18))
as f o l l o w s .
one by a l o c a l empty l i n k s
or second o r d e r ) , phase diagrams
configuration
This
versa.
means t h e The f o u r
one g e t s possible
For t h e
self-avoiding
appear.
By a Monte C a r l o
equilibrium
updating
a new
replacement
ensembles of
of
types
case
(17)
con-
p r o c e a u r e we s e q u e n t i -
"'-- ii
"I-i Figure l .
;;--If
Local changes of loop configurations within a
plaquette a l l y sweep a l l
d ( d - l ) / 2 Ld
plaquettes
new c o n f i g u r a t i o n with p r o b a b i l i t y
of the l a t t i c e and accept the
221
P = Wnew/(Wol d + Wne w) where t h e
w
(21)
are t h e B o l t z m a n n f a c t o r s exp(-E/kT).
T h i s means we t a k e (equally ciple
the
old
be a t t a i n e d
condition"
is
a probability is
stable
initial after
t h e new c o n f i g u r a t i o n
distributed
we r e t a i n
in the unit
one. after
sufficiently Obviously
distribution
under t h i s
A
obtained
is
calculated
after
about f i v e
is
than
reach
up" p e r i o d .
the
P, o t h e r w i s e can i n
prin-
"ergodic
set of c o n f i g u r a t i o n s
to
Moreover,
with
the B o l t z m a n n f a c t o r starting
from
N ~I03
sweeps t h r o u g h
(22)
an a r b i t r a r y
such an e q u i l i b r i u m The thermal
as t h e mean o v e r complete
less
many i t e r a t i o n s a large
we e x p e c t t o "warming
a pseudo-random number
allowed configuration
proportional
procedure.
configuration,
if
interval)
Since every
satisfied.
an a p p r o p r i a t e
riable
(22)
set
average of
a va-
configurations the
each
lattice
(23) The c o m p u t a t i o n s
are u s u a l l y
done i n
"thermal
. . . . T m a x , T m a x - a T , . . . , T m i n where we s t a r t
at
cycles"
Tmi n, Tmin+aT,
low t e m p e r a t u r e
from t h e
empty l a t t i c e .
4 Some R e s u l t s
A) S e l f - a v o i d i n g In r e f . [ 2 ]
l o o p gas
we c o n s i d e r e d
self-avoiding
l o o p gas systems
in d=2,3,
and
4 dimensions (24) C ~ ~$~
We "measured" the fluctuations
the
average
length
_~
(proportional
(proportional
to
to the energy) the
specific
and
heat)
in
222 thermal sists
cycles.
For low t e m p e r a t u r e s
of a few small
with temperature gurations
loops.
and < ~ > _
Z
others
loops, on ~
heat decreases a g a i n .
lattice
maximum.
l a r g e ones.
and -#";~ k=o I#-#~,.~I -r (28
X i s the s u s c e p t i b i l i t y
given by
o~ #-~ (29 and
h
a magnetic f i e l d
introduced
by the replacement ##-~ # ~ - h ~
eq.(24). We were able to f i n d which i s l i m i t e d , rature side,
a t e m p e r a t u r e regime w i t h i n
close to T c r i t ,
(where d i v e r g i n g thermodynamic q u a n t i t i e s and the end of the c r i t i c a l
rection-to-scaling "critical
window"
slope of the l i n e a r logarithmical
plot
portion
round o f f )
In t h i s
i n c r e a s i n g system
and ~ where c a l c u l a t e d from the
of the corresponding q u a n t i t y
near T c r i t .
on one
(where c o r -
on the o t h e r s i d e .
(whose e x t e n s i o n i n c r e a s e s w i t h exponents ~ , ~ ,
region,
rounding tempe-
regime away from T c r i t
terms become i m p o r t a n t )
s i z e ) the c r i t i c a l
the c r i t i c a l
by the f i n i t e - s i z e
Furthermore,
the exponent
in a doubly ~
was de-
termined from the i s o t h e r m at the c r i t i c a l t e m p e r a t u r e . We found I s i n g - l i k e values f o r a l l exponents c o n s i d e r e d .
224 B) S e l f - a v o i d i n g In r e f . [ 3 ]
surfaces
we i n v e s t i g a t e d
The E u l e r c h a r a c t e r i s t i c 7
= 2
self-avoiding
is defined
C m.o~
-
surface
gas systems
by
~k..Z)
(31)
where ncomp (nhand) i s the number of connected components (hand] e s ) of the s u r f a c e • F i g . 3 shows the average energy <s> and the
I
1.5
I
I
........ <S)/104 cs~/103 1.0 ....... "°•°'.l.o...l°"
1.0
• i,,O,o:~T
°
°'l "%
.5
% ".o~.
.5
%"•"'•..••..,••.
.0
.0
• ..i"
............ -.5
|..
°t:..°.................. ;::~:|ljl|jlIe,..o,•.°..
•
.... ,'':'3<X~/104 i
--•D
.1
g '
"5
.9
i
L
.5
.7
a)
obtained
10 3 and (b)
Euler characteristic lations
for
the s e l f - a v o i d i n g
surface
The average s u r f a c e <s> and E u l e r c h a r a c t e r i s t i c
on (a)
for
configuration
.9
B
b)
F i g u r e 3. Monte C a r l o r e s u l t s gas model.
•
3
10 4 l a t t i c e s .
obtained
the c a s e ~ = O . consists
were
in
"~-cycles"
by Monte C a r l o simu-
In the l o w - t e m p e r a t u r e
predominantly
of small
phase a t y p i c a l
s e p a r a t e d compo-
225 nents.
In t h e h i g h - t e m p e r a t u r e
typical
configuration
many handles
like
phase
consists
a sponge.
Note t h a t point.
= 0.353
dimensions cating
(in
a first
we found
order P tran
with
(in
an energy jump ~ < s>L -4
transitions in t h r e e
are i n
and f o u r
~
phase t r a n s i t i o n
= 0.68
object
(Fig.3a)
with
the data
at
3-dimensions).
(32)
v a n i s h e s at the c r i t i c a l to s c a l e i n v a r i a n c e .
hysterisis
D
connected
dimensions
m i g h t be r e l a t e d
(Fig.3b)
n e g a t i v e and a
phase t r a n s i t i o n
the E u l e r c h a r a c t e r i s t i c
This f l a t n e s s
is
of a s i n g l e
In t h r e e
show e v i d e n c e of a s e c o n d - o r d e r ~crit
~
In f o u r
loops i n the # - c y c l e s
indi-
at
4-dimensions) = 0.45.
agreement w i t h
(33)
The observed d i f f e r e n t
t h o s e of l a t t i c e
t y p e s of
gauge t h e o r i e s
dimensions.
i
i
i
low t e m p e r a t u r e
low temperature 0,5
phase
.0oo,,,,
""i'ii'ii" ",,,
0
/
I \"V
0.5
a)
Figure
./..1 phase/" or°poet w1
/
1 I
I
I
I
0
1
P
t.1
2
b)
4. Phase diagrams f o r
d i m e n s i o n s showing t h r e e order transition lines.
-
model
phases,
(30)
first
in (
(a) t h r e e )and
and
second-
b) f o u r (. . . .
)
226 For nonvanishing chemical p o t e n t i a l
for
the Euler c h a r a c t e r i s t i c
/~>0 we found the phase diagrams d e p i c t e d in f i g . 4 .
For l a r g e ~
new phase appears separated from the others by f i r s t - o r d e r tions.
This " d r o p l e t
sisting
(/~=0)
phase" i s r e l a t e d to a new ground s t a t e con-
of simple cubes, each touching
In r e f . [ 5 ~
the c r i t i c a l
e i g h t o t h e r s at i t s
behaviour of the s e l f - a v o i d i n g
corners.
surface gas
in t h r e e dimensions was i n v e s t i g a t e d by the c r i t i c a l
method. Analogously to the loop gas case,
&,#, ~,
a
transi-
window
I found I s i n g exponents
and &.
C) I n t e r s e c t i n g
surfaces
An i n t e r s e c t i n g
s u r f a c e gas model in t h r e e dimensions
c ~E; p r e v i o u s l y discussed in [ 1 8 ]
was i n v e s t i g a t e d in r e f .
it
approaches the s e l f - a v o i d i n g
i s the I s i n g model and i t
(for/w=O)
in the l i m i t
# ~
order t r a n s i t i o n cal
lines
and t r i c r i t i c a l
It
and mean f i e l d
shows f i r s t -
points.
window" method I obtained I s i n g - l i k e lines
For #~ =0 model
By means of the " c r i -
behaviour along the c r i t i -
behaviour at the t r i c r i t i c a l
p o i n t s which i s
D) H a u s d o r f f dimension A model of a s i n g l e s e l f - a v o i d i n g
random s u r f a c e in t h r e e d i -
mensions w i t h the f i x e d t o p o l o g y of the sphere was considered in ref.[5]
e
At the c r i t i c a l point ~ c r i t = 0.53 gyration diverges like
(35)
[19]
is
and second
expected in t h r e e dimensions.
: Z
(30)
. The phase diagram d e p i c t e d in f i g . 5
symmetric w i t h r e s p e c t to ~ - 7 - # - 2 # £ . tical
[5].
the average radius of
(36)
227
|
I l
\ l
(1)
~
(2) \
o
\ \ \
(3)
o
Figure
05
5.
Phase d i a g r a m o f t h e
showing a d i s o r d e r e d magnetic ( .....
phase
(3)
) transition
points
intersecting
, a ferromagnetic
separated lines.
by f i r s t -
At t h e i r
(
juncture
surface (2)
gas model
and a n a n t i f e r r o ) and s e c o n d - o r d e r
are t r i c r i t i c a l
(a).
The c r i t i c a l of t h e
(I
6
exponent V
is
related
to
the
"Hausdorff
dimension"
surface
~N : 1/,# defined
at ~ = ~ c r i t
~, where s(R)
= ]-~ is
by
/~.~P.
the part
of the
w i t h r a d i u s R, such t h a t Carlo result is
the
~k = 2 . 3 0 i n good agreement w i t h in [20~.
(37)
(38)
surface surface
contained passes i s
_+ o.o~
a Flory-type
formula
in
a sphere
centre.
The Monte
(39) dw = 2 1/3 d e r i v e d
228 References I.
M.Karowski, (1985)5
R.Schrader,
and H.J.Thun,
2.
M.Karowski, H.J.Thun, Gen.16(1983)4073
3.
M.Karowski
and H.J.Thun,
4.
M.Karowski
and F.Rys,
5.
M.Karowski,
J.Phys.A:
6.
J . F r ~ h l i c h , in ' P r o g r e s s in Gauge F i e l d T h e o r y ' , ed.G. t ' H o o f t et al (NATO Advanced Study I n s t i t u t e S e r i e s B Ro 115) (Plenum, New York 1984)
7.
K.Symanzik, in 'Local Quantum T h e o r y ' , P r o c . l n t . S c h o o l of Physics ' E n r i c o F e r m i ' , Course XLV, e d . R . J o s t (Academic, New York 1969), p.152
8.
P.de Gennes, Phys. Lett.38AC1972)339
9.
A. Maritan
I0.
K.Wilson,
II.
B.Durhuus, (1983)185
~.Helfrich,
Commun.Math.Phys.97
and F.Rys,
J.Phys.A:Math.
Phys.Rev.Lett.54(1985)2556
J.Phys.A:
Math.Geno19(1986)2599
Math.Gen.(in
and C.Omero,
press)
Phys.Lett.lO9B(1982~51
Phys.Rev. DlO(1974)2445 J.Fr~hlich,
12. A . P o l y a k o v ,
and T.Jonsson,
Nucl.Phys.B225
Phys.Lett.lO3B(1981)207
13. P . J . F l o r y , ' P r i n c i p l e s of Polymer C h e m i s t r y ' C o r n e l l U n i v e r s i t y Press,1969) 14. F.Rys and W . H e l f r i c h , 15. J . C . W h e e l e r , (1980)1748
(Ithaca,
S.J.Kennedy,
and P . P f e u t y ,
Phys.Rev. L e t t . 4 5
Phys. L e t t . l O 2 A [ 1 9 8 4 ) 4 2 0
17. P.de Gennes and C.Taupin,
J.Phys.Chem.86[1982)2294
18. J.D.Weeks, ' O r d e r i n g in S t r o n g l y F l u c t u a t i o n Systems' ed T . R i s t e (Plenum, New York 1979)
20. A . M a r i t a n
and J . G r e e n s i t e , and A . S t e l l a ,
N.Y.:
J.Phys.A15(1982)599
16. T.Hofs~ss and H . K l e i n e r t ,
19. T . S t i r l i n g
FS9
Condensed M a t t e r
Phys. L e t t . 1 2 1 B ( 1 9 8 3 ) 3 4 5
Phys.Rev. L e t t . 5 3 ( 1 9 8 4 ) 1 2 3
FIELD THEORETIC
METHODS
WITH
IN CRITICAL
PHENOMENA
BOUNDARIES
AJvL Nemirovsky
The James Franck Institute
The University of Chicago, Chicago, IL 60637
ABSTRACT
Recent work on field theoretic methods in critical phenomena with boundaries by the author and collaborators is described. The presence of interfaces and boundaries in critical systems produce a much richer set of phenomena than that of infinite sized systems. New universality classes are present and interesting crossover behavior occurs when there is a relative variation of additional length scales associated with either the size of the system or the boundary conditions (BC) satisfied by the order parameter on the limiting surfaces. A recently proprosed crossover renormalization group approach is very well suited to study these rich crossovers. Since functional integrals provide an indefinite integral representation of field theories, Feynman rules in configuration space are independent of geometry and BC. Renormalization of field theories with boundaries is discussed and various geometries and BC are considered. Application of field theoretic techniques are described for studying conformational properties of long polymer chains in dilute solution near interfaces or in confined domains. Also, related problems in quantum field theories with boundaries are presented.
The work I present here was performed in collaboration with ICF. Freed. Also, Z-G. Wang and J.F. Douglas have contributed to some of the work described below.
1, INTRODUCTION Experiments and computer simulations can only probe finite systems with limiting surfaces. On the other hand, theoretical studies of phase transitions (PT) usually consider infinitely extended systems. Although surface effects can, in general, be neglected in large systems, these effects become very relevant near a second order PT point as the correlation length grows unbounded? Critical singularities at a second order PT only occur in the
230
thermodynamic limit as they are rounded off in finite systems. On the other hand, systems which are of infinite extension in two or more dimensions and which are unbounded in the remaining directions, show interesting dimensional crossovers as the transition is approached. 2 Then, it is important to extend theoretical approaches to understand finite systems with limiting surfaces. Phenomenological finite-size scaling methods are widely used to extrapolate computer data to the thermodynamic limit, 2 but there are many aspects of finite size scaling which remain to be described by fundamental theories such as renormalization group (RG) methods. Such a fundamental theory becomes more important as interest extends to the study of particular finite systems with interacting boundaries. This is because universality classes of finite systems are more restricted than those of unbounded systems. The finite systems are characterized not only by the dimensions of the embedding space and of the order parameter but also by the geometry of the system and the boundary conditions (BC) for the order parameter on the limiting surfaces. 2 Here I discuss the application of field theoretic RG techniques to study critical phenomena in the presence of boundaries. The systems may be finite (or semi-infinite) along one (or several) of their dimensions, but they are of infinite extent in the remaining directions. Examples include systems which are finite in all directions, such as a (hyper) cube of size L, and systems which are of infinte size in d' = d - 1 dimensions but are either of finite thickness L along the remaining direction (e.g. a d-dimensional layered geometry) or of semi-infinite extension, etc. The presence of geometrical restrictions on the domain of systems also requires the introduction of BC (periodic, anti-periodic, free surfaces) for the order parameter on the surfaces. Critical systems with boundaries or interfaces display a very rich set of phenomena because the (totally or partially) finite and semi-infinite cases contain several competing lengths and hence have interesting crossover behaviors as these length scales vary with reslx~t to each other. These additional lengths are either associated with the finite size of the system in one or more of their dimensions or to the boundary conditions on the order parameter ~.1.2 Consider, for example, a semi-infinite critical system with a scalar order parameter which satisfies either the Neumann or the Dirichlet BC at the surface. These two cases belong to different universality classes called the special and ordinary transitions, respectively. 1 A surface interaction parameter c is usually introduced as (1/¢)(~/~n~n
= c where (3¢~/~n) stands for the normal derivative o f ~ at the limiting surface 3i2.1 Then, as c
ranges from zero to infinity the system crosses over from the special to the ordinary transition. These transitions are characterized, among other things, by different surface critical exponents. 1 On the other hand, systems that are bounded in one direction but of infinite extent in the remaining ones show a very interesting dimensional crossover as follows: In the critical domain, but away from the critical point, the behavior is dominated by the non-trivial 3d bulk fixed point, while as the transition is approached the 2d fixed point controls the physics. 2
231
Section 2 shows that Feynman rules of field theories in configuration space are independent of geometry and boundary conditions, so they are identical to the well-known rules for unbounded systems. Geometrical constraints and boundary conditions are implemented through the explicit form of the zeroth order two point .corrclati0n function. Semi-infinite critical behavior is briefly discussed in Section 3 where we inla'bduce a model of two coupled semi-infinite critical systems which possess a very rich physics. Section 4 considers the renorrealization of field theories with boundaries and discuss a crossover renormalization group approach that is very well suited to describe interesting multiple crossovers present in these field theories with boundaries. Section 5 deals with other interesting geometries. We begin by briefly discussing curved surfaces and edges, and then pass on to layered geometries with various boundary conditions (such as periodic, anti-periodic, Dirichlet and Neumann), and to cubic and cylindrical geometries. An important conclusion is that the usual eexpansion technique can be utilized to study any geometry and boundary condition as long as the smallest finite system size is not much smaller than the bulk correlation length of the system. Field theoretic methods can also be utilized to study the statistics of long polymer chains in solution near (liquid-liquid, liquid-solid) interfaces or in confined domains (such as a polymer chain in a cylindrical pore). This is the theme of Section 6. Finally, in Section 7 we present some analogies between the statistical mechanical problems of the preceding sections and related problems in quantum field theories.
2. Indefinite Integral Representation of Field Theories Functional integrals provide an indefinite integral representation of the differential equations of a field theory. However, this representation does not contain a complete specification of the boundary conditions. Hence, the same functional integral representation of a field theory applies for various boundary conditions.3 Consider, for example, an O ( N ) N-vector scalar ¢4 field theory in d = 4 - ~ dimensions in a region of the space ~ with a (d-l) dimensional boundary 3f~. The partition function Z[J] is a functional of the external source J given by Z [ J ] = ~D [¢]exp [ - F { ¢ } - ~ddxJ ( x ) ~ x ) ] ,
(2.1)
where F is the free energy functional, D [¢] represents the sum over all configurations of the order parameter ¢(x), x is a d-dimensional position vector inside the region fL to ~ T - To, with Tc the (mean field) bulk critical temperature, and uo are the bare reduced temperature and coupling constant, respectively. It is possible to formally integrate Eq. (2.1) over ¢(x) to obtain
232
.°.pI
t
,,,
where N is a normalization constant such that Z[J = 0] = 1 and G (°) is the bare propagator (two-point correlation function) which is the solution to the usual Klein-Gordan wave equation ( - V 2 + to)G (°) (x, x') = 8(a)(x - x ' ) ,
(2.3)
Eq. (2.3) is satisfied in the region f~ and it must be supplemented with appropriate boundary conditions at ~f~. Equivalently, G(°)(x, x') in (2.2) is only properly defined when boundary conditions are specified. The integral representation (2.2) of the ~p4 field theory is indefinite and applies to arbitrary boundary conditions which are implemented through the properly chosen propagator G (°). Coordinate space Feynman rules follow from (2.2), so they are independent of the explicit form of G(°)(x, x'). position space diagrammatic
rules remam
unchanged from
Hence, the above discussion implies that those o f
an infinite volume
theory, but
that the appropriate zeroth-order propagator G (°) (x, x') must be utilized. Chapter 14 of Ref. 4 contains expressions for the zeroth order two-point Green's function (in the context of the heat conduction problem) for a wealth of geometries and boundary conditions. Translationally invariant systems have G(°)(fx, x'l) but, in general, the presence of interacting surfaces breaks this symmetry making G(°)(x, x') * G(°)(Ix - x'l). The n-point Green function also depends on all n coordinates rather than on n-I coordinate differences as in full space. Diagrammatic expansions for unbounded systems are more conveniently performed in momentum space5 where the translational invariance of the theory is reflected in momentum conservation conditions. The "most" convenient choice for finite systems depends on geometry and BC as discussed in Ref. 3. In the following sections we discuss various geometries and BC.
3. Critical Behavior at Surfaces Semi-infinite critical systems have been studied by several workers using a variety of methods as described by Binder in his comprehensive review on the subject) Renormalizadon group techniques have proven to be one of the most powerful theoretical techniques to study critical phenomena at surfaces. An excellent review by Diehl describes recent advances in this area. 6 Thus, the topics presented below sketch out very recent results which, in general, are not covered in either review. The interested reader may find the details in the reviews of Refs. 1-and 6 and inthe original papers.
3.1. Semi-Infinite Geometry. Two Coupled Semi-Infinite Systems. We begin with the usual Ginzburg-Landau free energy functional of (2.1a) in a semi-infinite geometry. Thus, the region f2 is the positive half-space z>0 bounded by the (d-1)-dimensional flat surface 0 ~ at z = 0.
233
The position vector x of (2.1) is decomposed into its Cartesian components p and z with p a (d-1)-dimensional position vector parallel to the surface ~f2 at z = O. Mean field theory predicts the appearance of four phase transitions depending on the values of the reduced temperature t o and the surface interaction parameter c o (introduced through the boundary conditions satisfied by G (°) at 0f~ as discussed in Sec. 1). 1 These phases are depicted in Fig. la. For co~--~0the system orders at the bulk critical temperature to = 0. When c o is large, or more precisely when Co.~t] a , the transition is called ordinary, while for small values of co, such that co,~tio a , it is known as the special transition. For c00 and -tll2O(z)O(z ") + G~°~O(-z)O(-z ")
(3.2)
,- G~O(z)O(-z') + a ~ 0 ( - z ) 0 ( z ' ) ,
The functions G~,~ O, G~,~ ~, G ~ and G ~ are presented in Ref. 7 and some limiting cases are of interest. When go = 0, then G ~ ) = G~°)= 0, while G ~ ) and G~°) become identical m the two-point correlation functions of semi-infinite systems with surface interaction parameters CA.oand ca.0, respectively. 1 When ~o ~ 0, the two semi-infinite regions are coupled. The ~ 0 ~
limit produces the single surface interaction model of Bray and
Moore 9 with c ~ M) = c,A.o -~ CB.O. As can be seen, the model of two coupled semi-infinite systems describes a very rich physical situation. Even the exactly solvable Gaussian theory with uA.o= us,0 = 0 is of interest and is far from trivial.
4, Renormalizatlon of Field Theories with Boundaries As stated in Section 2, Feynman rules of field theories in configuration space are independent of geometry and boundary conditions. These constraints are implemented through the explicit forms of the two-point functions so, for the problem of two coupled semi-infinite systems, the usual flee propagator is replaced by the Fourier inverse of (3.2), and standard Feynman ruless are utilized to evaluate diagr,uns. Nevertheless, the breaking of translational invarlance introduces novel features in these problems such as the presence of one-particle reducible primitively divergent diagrams as shown in Fig. 3 and as discussed in length by Ref. 6. Ren0rmalization of field theories in presence of interacting boundaries has been studied by Symanzik, 1° by Diehl and Dietrich, H and more recently by Diehl 6 and by us.3 In addition to bulk renormalization constants Zo, Zf and Z,, which remain unchanged by the presence of surfaces (as do the I] function and the fixed points {u* }), it is necessary to introduce two additional renormalization functions Zc and Z1 required to renormalize surface interaction parameters and the fields on the surface. In the two coupled semi-infinite systems problem Zc becomes a 2 x 2 non-symmetric real" matrix.7
237
(o) FIGURE
ooo
(b)
c)
3
A new feature associated with the breaking of translational invariance m the existence of one-particle reducible primitively divergent diagrams. For example, the bare two-point functions G, G~ and G~I with 0. I and 2 points on the surface of (a), (b) and (e), respectively, have different singularities, thus requiring different renormatization constants as discussed in Section 4.
With a semi-infinite geometry it is useful to work in a mixed momentum-configuration space representation. For example, we define the bare n-point connected Green's function G } ") (Pi, zl, Co, to, uo) [with Co the column vector (Co, ~0) for the problem of coupled semi-infinite systems] as
Ga('~)(p,,z,, Co,to.
Uo) =
I (2~)"-' " •' exp(-ipvpw)l](an)d-~Sa-1(XPO i=1
(4.1)
Gs(~)(Pi, zi, Co, to, uo)
where Gt}')(pl, zi, ¢o, to, uo) is the bare n-point function in configuration space, and the 8 function reflects the fact that momentum is conserved in the direction parallel to the surfaces. The renormalized Green's function G/¢~)(pi, zi, e, t , u, 1¢,) is then given by
Gfl°(pi, zi, e, t, u, ~) = Z~""/2 [Z1('~)]-l/2G/~")(pi, zi, Zc c,Z, t, S E I ~ Z w u ) ,
(4.2)
where R is a parameter having dimensions of (temperature)1~2 used to define a dimensionless coupling constant u and Sa is lhe area of a sphere of unit radius. Minimal subtraction dimensional regutarization is the most
widely used technique to renormalize these field theories. 6 Minimal subtraction has ZI ") of the form Z~") = [Zi(u)] '~, if zl = 0, i = 1, 2 ..... m, and zi ~ O, i = m ~- 1 ..... n
(4.3)
Although this renormalizauon procedure is very convement to study the physics near a given fixed point [bP] such as the special FP, the ordinary FP, the bulk FP, etc., 6 minimal subwacdon techniques are not well suited to describe the rich crossovers of field theories with boundaries with two or more competing fixed points.12 We have recently proposed a crossover RG approach that is very convenient for studying critical phenomena with several competing lengthsJ 2 Amit and Goldschmidt ~3 utilize mathematically similar techniques to fully describe the bicritical crossover. In our coupled systems problem the surface normalization constants
238
are taken to depend on the extra lengths through the dimensionless combination ~cz, c/w., and 3/~c. This dependence emerges naturally by imposing appropriate normalization conditions on the two- and four-point Green's functions of the theory. In contrast, minimal subtraction dimensional regularization has the normalization constants independent of these lengths but only dependent on u and e. Due to the explicit 1c-dependence of the renormalization constants in addition to the usual implicit ~cdependence through the dimensionless coupling constant u, the renormalization group equations become more involved, but they now describes the full crossover between all fixed points. Consider, for example, a semi° infinite critical system near the special transition. We have evaluated to one-loop approximation ,2 the full zdependence of the surface susceptibility Xlx(z) describing the response of spins in a plane at a distance z away from the surface at z = 0 due to a magnetic field applied on the same plane. At the non-trivial fixed point u ° , the crossover RG equation implies the following scaling form for the renormalized surface susceptibility 7~,n
Xa,n(z, t, u* ,It) = ~-t + 2v(1- n)t*(1 - n)g (x, y ) F ( x ) ,
(4.4)
where v and 11 are the usual bulk exponents, y = Kz and x = ~z(t/K2)v. Ref. 12 presents the functions g(x, y)
and F(x) to O(e), and here we only give some interesting limiting cases. We always consider the asymptotic limit t < ~ ,
but the magnitude of ~
remains at our disposal.
Thus, y =w.z is always larger than
x = (re.)(t/~)" and three regimes exist. The x--coo limit gives 12
g (x, y) --->exp[ex-U2exp(-x)] ,
(4.5a)
F(x) --* 1 + 0 [exp(-x)] ,
(4.5b)
where higher order corrections, varying as x -l exp (-x), have been dropped. Eqs. (4.4) and (4.5) imply that bulk behavior is approached exponentially fast for 0_l, we obtain 12 g ( x , y ) = C ( y ) x -[(N + 2~(N + 8)le
F(x) = 1 + O(x, xlnx) ,
(4.6a)
(4.6b)
where C ( y ) is a finite function of y. The form predicted by (4.4) and (4.6) is in accord with scaling assumptions and previous calculations using minimal subtraction. 6 We stress that this near surface behavior is a
direct donsequence of the full crossover renormalizadon group approach. ,There is no need to utilize operator product expansion techniques, t2 Instead, these techniques are only required in the usual minimal subtraction
239
approach because the standard RG equation does not contain information about the near surface behavior. 6 Finally, as y - t 0 we find 12
g (x, y) = C ( y ) (x/y) -tOy + 2~(N + 8)]~
(4.7a)
F (x ) = 1 + O(x, xlnx)
(4.To)
d'(y) = I - [(N + 2)/(N + 8)]e y ln(y/2).
(4.7c)
Hence, as r.z--~0, 7~.~(z) reduces to 7~.11 = 7~m( z = 0) as expected physically and in contrast to the results of the minimal subtraction renormalization approach. It is interesting to note that 7~.xl(z) is a continuous function of z for 0 ~ I, where ~ is the correlation length, e-expansion techniques can be utilized to describe corrections to bulk quantities due to the finite extent of the system. Similar general results were derived by us for the effects of interacting boundaries where L is a parameter associated with surface interactions (as briefly summarized in Fig. lb). The theory is illustrated for the N-vector model in a layered geometry with periodic, anti-periodic, Dirichlet and Neumann BC where the correlation functions and susceptibilities are evaluated to O (e). Away from the critical point and when ( L / ~ ) ~ ,
we find that first order contributions to scaling functions due to finite size are
exponentially small, proportional to exp(-L/~), for periodic and anti-periodic BC, while these corrections behave as (~L) for free surfaces. This is in accordance with previous numerical calculations and results obtained from various models. 2 As the scaling variable (L/k) approaches unity, we show that first order in e corrections to scaling amplitudes become comparable with zeroth order terms. This marks the beginning of a dimensional crossover where
241
expansion methods break down. The finite size scaling literature 2 usually states that dimensional crossover occurs when the bulk correlation length becomes comparable to the typical system size L. While this is demonstrated by us to be true for a layered geometry with periodic or Neumann BC, it does not hold for example, for a layered geometry with anti-periodic or Dirichlet BC for which the e-expansions are well behaved even at the bulk critical temperature T, .3.1s
Close to the transition a region of dimensionally reduced physics emerges. Layered systems near the shifted critical temperature and semi-infinite geometries near the surface transition have d" = d - l .
Of course,
different geometries, such as an infinite cylinder, a cube, etc., give different ae. We discussts two mechanism for producing dimensional reduction (the emergence of d'-dimensionat physics out of an underlying ddimensional system): a geometrical one (e.g., a layered geometry very close to the shifted critical point), and an interaction drive one (e.g., a semi-infinite system close to the surface transition). An L dependent d" dimensional effective free energy functional for the lowest mode of the order parameter (massless mode) is evaluated by integrating out the higher (heavy) modes. Our approach presents some conceptual difficulties that still remain to be understood to fully describe the dimensional crossover. Can the crossover renormalization group approach be applied to this problem? We are presently investigating this interesting possibility.
5.3. Cubes, Cylinders and Other Geometries. Dynamical Critical Phenomena and First Order Transitions Our recent work 3 and that of Ref. 15 show that the usual c-expansion techniques can be applied to study any geometry and boundary conditions as long as the bulk correlation length of the system is not much larger than the smallest dimension of the system. As the system approaches arbitrarily close to the critical (or pseudocritical) point, the e-expansion break down. Related techniques to our effective free energy functional method have been proposed by Brtzin and Zinn-Justin,19 and by Rudnick et al.20 to investigate the deep critical region for cubic and cylindrical geometries with periodic BC. Their approaches do not present the technical difficulties of ours as discussed above, since these authors only consider systems with no true critical points. Brtzin and Zinn-Jusdn have also proposed a 2 + e expansion to study finite size effects in critical phenomena below Tc.19 Since then, several authorsm have extended the methods of Refs. 19 and 20 to study finite size effects on dynamics and in first order transitions always for systems with no true critical points and with periodic boundary conditions.
6. The Statistics of Polymers in Various Geometries. The study of conformational properties of long, flexible polymer chains near penetrable (liquid-liquid) or impenetrable (liquid-solid) interfaces or in various confined geometries (e.g., polymer chains in cylindrical or
242
spherical pores) has a variety of important practical applications. These applications include cohesion, stabilization of colloidal particles, chromotography reinforcement and floccalation. Also, we note that finite-size effects are present in computer simulations of polymer systems. Simulations generally employ periodic boundary conditions to remove the surface interactions, but the finite size of the computer still affects the computed thermodynamic properties. Therefore, systematic extrapolation of the simulation data is required in order to describe properties of the infinite system. It is, therefore, of theoretical interest to understand how the thermodynamic limit is approached as the size of the system is increased. The statistics of long flexible polymer chains with excluded volume in dilute solutions is well known to belong to the same universal class as that of the O(N) ¢4 field theory with N = 0. 99 This holds not only for unbounded systems but also for systems with interacting interfaces and those in confined geome~ies. Thus, most of the results for critical systems discussed in the previous sections can be transcribed to corresponding polymer problems. We have used powerful field theoretic techniques to study the conformational properties of polymers near interacting impenelrablez3 and penetrable z4 interfaces and polymer chains in confined geometriesz~ such as polymers between two parallel plates with various polymer-surface interactions on the limiting surfaces or polymers near the outside surface of a repulsive sphere. Some of the rich array of situations that can now be treated using renormalization group methods are illustrated in Fig. 4.
(a)
(b)
(d)
(c)
(e)
These figures illustrate some interesting systems involving a single polymer chain with excluded volume and interacting boundaries in several geometries. These geometries can now be studied by employing the RG methods discussed in Section 6 [as long as the radius of gyration of the polymer chain is not longer than the smallest dimension of the system]:
(a)
A polymer attached to a sphere with an interacting surface.
243
(b)
A polymer in the shell formed by two concentric cylinders.
(c)
A polymer near a sphere formed by two different solvents, e.g. oil and water. The quality of these two solvents is, in general, different. Furthermore, the interfacial region can be such that one side of the interface attracts the polymer whereas the .other side repells it.
(d)
A polymer in an edge where the power law exponent for some property(ies) can depend on the edge angle.
(e)
A polymer in a cone.
7. Quantum Field Theories with Boundaries
It is well known that there are many analogies between statistical mechanics and quantum field theories (Qk-'T) for unbounded systems.5 For example, the Green's functions of the QFI"s are the analogues of the correlation functions in statistical mechanics, and Z[J] of (2.1) can be viewed as the generating functional of Euclidean self-interacting scalar QFT. Successive derivatives of Z[J] respect to the external source J produce all the Green's functions of the theory. These analogies, of course, also hold when boundaries are present.
The Casimir effect, the attraction of two neutral and parallel plates in a vacuum environment, predicted and experimentally confirmed several years ago, is the earliest example of boundary effects in QFT. 26 An interesting example of the scalar "Casimir effect" in statistical mechanics, as discussed by Diehl,6 is provided by fluctuation-induced force between two plates with a binary fluid mixture at its consolute point held in between. Systems that are of infinite extent in two or more of their dimensions and finite in the remaining directions such as a layered geometry in d=3 dimensions, display 3d physics away from the shifted critical temperature (but inside the critical domain) but ae=2-dimensional physics in the deep critical region.2 Dimensional reduction, the emergence of a quasi ar dimensional physics out of an underlying d dimensional system, is one of the main ingredients of the Kaluza-Klein theories. 27
In fact, Kaluza-Klein masses are the analog of
experimentally observed 2s shifts in critical temperatures of finite size systems from those of the bulk.
We have used the analogy between finite size problems in a periodic layered geometry and similar problems in finite temperature field theories to demonstrate how e-expansion techniques can be employed to study finite systems away from the shifted critical point as described in Sec. 5.2. At finite temperatures 1]-1 (where = (kT) -1, k is Boltzmann's constant and T is the absolute temperature) the causal boundary conditions of field
theories in real time are replaced by periodic boundary conditions with period !3 in Euclidean time. 29 Thus, a
244
finite temperature field theory is identical to one contained between two-parallel (hyper) plates with periodic boundary, conditions. The (hyper) planes are perpendicular to the Euclidean time direction, and the periodicity is
8. ACKNOWLEDGEMENT I am grateful to H.J, de Vega for his kind hospitality at paris VI and to K. Binder. H,W. Diehl and E. Eisenricgler for useful discussions. This research is supported, in part, by NSF grant DMR 83-18560.
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See, for example, P.-G de Gennes, Scaling Concepts in Polymer Physics (Comell University, Ithaca, 1979) and references therein.
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H.B.G. Casimir, Proc, [{on. Ned. Akad. Wetenschap., BS1, 793 (1948). Experimental evidence is discussed by M J . Sparnaay; Physica, 24, 751 (1958).
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For a description of Kaluza-Klein theories, see, for example, E. Witten, Nucl. Phys. B186, 412 (1981); A. Salam and J. Strathdee, Ann. of Phys. 141, 316 (1982).
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