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o ) 8 o ( « » - ) . n-=o ^ V ! 0 d(fn,p; p d(fn> i L 2 0 2(&) for every 2 i-> S(ui,tp) = (n+1)
oo
Xo(n°)8 (n )) To
obtain
operator
this
(ill) of
= °.
can
in
block-operator
be
any
decomposable
r ( 3 0 , defined
b
1
( b (b)"
0
B
0
0
a n d ( b ) = \(b)
represented
on
b y the in
b)
J C
=
1
are described
the
"ket"-vectors
T ( J C ) as
+
X
that
2
T h . 1,
u
remark
B : %. - » 3 C , b ) * = (b*
* = [ \f ( n " , n ° , n+)] e
[KT](n-)=
should
o f a triangular
C © X® C, w h e r e in
we
= 1© b © b ® ©
n
= (
0
result
K = © b ® n=o
exponentiation
= I - L ( «p (n),^(n)) n=o n !
+
0
I? ( n
! / n
u
nJJ)K
I n
o
n
- \
+
v(n:,n
0
+n§,n;+n?
+ n;).
(2.1)
Here
K ( n ) i s the function o f four variables
values
<x_( o n
o> - » 3^( »
+
n
n
^ ( n ) _ { <£»n j K ( ° I ng n
n
acting o n k ® K( ° I ng n
({ k f
n
+
®
. . .® k {1
k f
V n
n
n
given
operator-
o n the s y m m e t r i c a l tensor
powers
operators
e
i
n
e
ac< ) n
n
+
n
°)
= (b) * { b)®"° n
(2.2)
n
f or a k e %_ a n d n = n
n
* ) k®( ° n? j
0
+ n g as
® Bk® °° n
m e a n s the s y m m e t r i c a l t e n s o r p r o d u c t
° ]
/ n!
t
°),
with
+
| = ( b ) * { b ) ® " ° ® B ® °° ® ( b ® ° } n? / n
n
by
+ n
n = ( n^) ^ * _
of
k® " \
n } = S Iy w i t h
®
(bk®
n ?
n ! n ! £ +
0
)
k
V
l
®
v = 0 , + , g i v e n b y the s u m over a l l partitions
llyl = iiy f o r Iy = { i = 1, . . . , n I V ; = v )) .
44 Theorem
2. oo
Let K = ©
n=o
K'"> be tkdecomposable
( 2 . 1 ) , given in the
operator
pseudo-Fock space r ( J C ) over K,=C © "K® C by a linear span of the kernels (2.2). Then the kernel operator K J , defined by the pseudo-isometry oo
[ J y ] ( n% n ° , n
+
) = 8 (n") y ( n ° ) ,
[ J * V ] (n) = X ^ V < > - °> m=o m ! m
0
of the pre-Hilbert space to a continuous operator
< - )
n
T(30 into r ( J C ) , can be uniquely
2
3
extended
oo
[ \ (K)
¥
X („") X
] (m) =
n+n+=m
on the completion
n =
~h-
( " " ' ' +> n
+ ") .
n
(2-4)
!
n
0
K
ll\jrll(a) =
J of F(30 with respect to the poly norm
II a * \f II, a e 3 , where g/ven by fhe kernel
b = t ( b ® ) . Moreover,
\:K
the map
t (K),
oo
K(n-,n,n )
= X
+
n
defines a ^-representation a ^-semigroup associated
B
"
n
,
;
n", n, n
= 0 , 1, 2 , . . .
+
of the decomposable
in the pseudo-Fock S%7) of poly-Hilbert
CB
for
9
b -* b ® ,
representation
p ( b ) = e*- , into the
with an infinitely divisi ble state
operator ^-algebra
(2.5)
^-algebra (b)
space
J, such that
t (I) = I , t ( K * K ) = t ( K ) * i ( K ) ,
where
(n) = K ( n * ) * ,
n *= [
n +
°
(K*K)(n)
|i
=
n
m; <
1
\ for ni I
n i
\ n„° y
r =n -m +s
+
+
r
• K («•"««.
(• means a pointwise multiplication
n = f
n i
n
;
InS K *
! !
M
°
+
M
" -
+
U ° + m °,
s
+
»
|, and n{|
n°- m° / +
)
spanning (2.7)).
It i s s u f f i c i e n t to s h o w that t h e o p e r a t o r s
X X ( b)®"+ ® n+n+=m n"=o
B ® ® (b® ~ } n
n
+
(2.6)
Proof.
[ b v ] (m) = e « W
.
R
V
( -+ n
m)
45 corresponding unital tors
to
the
ft-semigroup b ® on
k e
(2.2),
kernels
fi
= ( b
8
give
Ibe
8
[bk®](m)
=
e*< >
£
b
(n) £
m
jmj { b)®( - > ® ( B k ) ® n
m
^
{
b
®°
)
\j/(n) = k ® , n
®
+
the
opera-
B
k
®
m
J =
k®
](m)
•
n
= e
m
of
o f the d e c o m p o s a b l e
^
n"=o
+
X(b) + (bk
a ft - r e p r e s e n t a t i o n [
T ( K ) . L e t us p r o v e it o n the e x p o n e n t i a l v e c t o r s
n+n =m
e
8
X ( b )
(B k + b))® .
+
m
n=o
A p p l y i n g this f o r m u l a [a~ft~c k® ](m)
due b
for
=
b = a ft c , w e o b t a i n
e < * x
a
c
>
+
< * a
(a~ft"c k + a ft c ) ) ®
k
e
X ( a ) * + X(c) + (a*c) + (a^Ck + (ck tf^
* Q ^
e
X ( a ) * + (a*(Ck + e» ( *
+ a*))®
A
to
k + c))
( C
t h e p r o p e r t i e s (1.5)
a n d (1.6)
b ) , (b. H e n c e o n the s p a n o f t (a®*c®)
that i s
= t
((a*c)®)
t ( b ® ) = J * b® J
operators
b®
on
p r e - F o c k space sequences
n
n
c = a ft b e
e
p = e*.
The
is a
e W
of
x
(b)
#)-)®m
+
a
+
<
=
_
=
ck
[a*c
and
X(b)
T(JC)
the
ft-semigroup
$,
II (a) = J * ( a * b ) ® J v b -> b
associated
product formula operators
b®
(2.6)
T
c a n be
n
i
definition
s
II = II y
n
n
II (c)
the
sequences defines a closed infinitely
the k e r n e l obtained by
of
divisible
a linear
state
span
n i
{ a ft c ) ® ° ® ( A * C ) ® ° ® (a ft c ® " ° ) = n
n
( ( a * ) + ( a * c ) + ( c ) ) * { [ A * c) + a * ) ] ® " " ® ( A * C ) ® ° n
n
of
the l i n e a r e x t e n t i o n
of ( a ft c )
=
i n t o the c o n t i n u o u s o p e r a t o r s
with for
II \f II (a)
i s a l s o f u n d a m e n t a l due to
of a l l fundamental
f
of
a l s o o n the f u n d a m e n t a l
n
n
2*. H e n c e the m a p
of
i n the ^ - a l g e b r a
{\|f }-» {t ( b ® ) y j .
{t ( b ® ) \)/ ) n
maps
= t (a®)* t ( c ® ) , of
b = i (b®)
25 as the o p e r a t o r
the
JQ w e h a v e
ft-representation
II (a) = II J * b®J v
of
decomposable
m
c
=
m
w i t h r e s p e c t to a l l the s e m i n o r m s
r(2Q
of p o l y - H i l b e r t space ft-representation
* )
= a~ft~c = a*c
i s c o r r e c t , b e c a u s e the s e q u e n c e II i ( b ® ) v
^
{k® I k e
L e t us d e f i n e
r(9Q.
{\p }
+
the p s e u d o - F o c k s p a c e
II J * a * ® J v II , a e
for
c
® [(c + ( a * C ] ® ° } n
46 y
=
_ n ^ L _
( a*)® r+°
*)r
( a
*
c )
(
where
+
a
•
+
means
£
® (a*
C®
- ^ 7
° " ® (c®
m
}
s
,
+
a
+
( ®(
c
+
n«"-mo") } .
C
a pointwise multiplication
o f the elementary kernels
a n d c ® , d e f i n e d as
£ ® |mo- + m - , liU° + m ° ,
ri
+
{ a*)®
Remark
] . /
®
r ?
1
0
(b)
0
1
0
0
0
1
(b
0
1
0
0
1
satisfying
with
,
® (a*
C® ° M
®
(c® °} . (2.7) s
b
D
=
+
e
( c ) _ (a)+(c)
(cA? _
e
1
0
0
1
b)
0
0
1
0
properties
e o
b
= e
block-operators
-
1
0
0
0
B
0
0
0
1
.
atct =
with
with
b t
B =
A
( b ) = (a) + ( c ) ;
C;a°c° =b°
with
denote e
e
" (c)«
(2.7)
« b ) = b* = < > ,
the operators
(aA?
the elementary
b ) = a) + c ) , a ° c 8 = b °
(b = ( a + (c , and
(a)
m +
+
NS
the exponential
ao"Co- = b o
e
0
m
(a^c)
si \ _ t^) nV+m °,/
® / si, \ no + m i ,
2.
=
e
c
A * c ) ® ° ® (A * C)®
tef us introduce
e
£
S
A*®( n i + m ° ) g , ( i!r®( nii+ m ) } . { ) ® ( m £ + m - ) g,
+
b°.
)
m+
C®(ni'+m„") g
a*®
c
® A * c ) ® ° ® (A * O ®
. { i!r)®(n °-m °) g a
( a
i
i
A
+ b
having
(b^®), e
) = b : ,
B
N
= b ° ,
[ (a + (c ] A°
j^e/i
=
b
l
(
on 7 the corresponding
A ^ a ) A ^ c ) _ A£ [a) + c)] e
eC"A°
C
>
AN C
representation
N
=
2
g
)
properties
(AC)
N
,
b e 2'->b = i ( b
8
)
47 of arbitrary
2 » , associated
it-semigroup
p=e
state
can be represented
x
b = e
X ( b
It f o l l o w s
> e °> B A
from
b
N
(2.9)
the decomposition 1
(b
0
B
b)
0
0
1
0
(b)
1
0
0
0
1
0
0
1
b)
0
0
1
0
0
1
-
b
Example
(b) "
i
.
=
1
0
0
1
(b
0
0
B
0
0
1
0
0
0
1
0
0
1
-
b. bo bo b ° +
the triangular block-operator
£ ,
of the
b A
1
Let
divisible
product
e< °
b =
of
an infinitely ordered
(2.8):
operators
the m a p
with
as the normal
b a n d the multiplicative property
of
b = i (b®).
1. £
be a complex
E u c l i d e a n space w i t h a n i n v o l u t i o n
= % , ( 2 otj %{f = E a*£? , V § i e
the c o m p l e x
space o f pairs
£ , a; e C
^ - » %* e
a n d « = Cx £
b = (P,r|),peC,r|e £
be
w i t h the b i n a r y
ft-operation ( a , S ) f t ( Y , q ) = ( Y + S # c + a * , q + S ) = (P, defined
b y the scalar product
semigroup
= (5 I q) = £ * c , i "
w i t h the neutral element
b * = (P*, T| ) #
a n d the associative (b. t l ) - ( Y > S ) = (P+
The
product
•
the s i m p l e c t i c f o r m selfadjoint funciton on
p ({$, t i ) = eP
- T h e n ® i s a ft-
under the i n v o l u t i o n
operation
;
i f the i n v o l u t i o n
otherwise
# is isometric
i t represents
:
the canonical
TJ - q - q - T| = i s ( q , T i ) , c o r r e s p o n d i n g t o
s (c,,r|) = 21m ( n I q)
n = n *, | = q
the C C R algebra,
operators
£
relations ( C C R )
e = (0,0) = 0
E
T l ' S + Y . Tl + S ) , i l - q = ( T l * l q )
is commutative,
II i;* II = II i; II , V !; e commutation
n)
#
J
§
.
o n the real space
T h e positive-definite
R e £ of
and normalized
i s t h e g e n e r a t i n g f u n c t i o n a l o f a G a u s s i a n state generated
i n the G N S representation
b y the
48 b = e representing
< ^e <
I 0 > e
+
A
b = (p ,TI)
v a c u u m state operators,
A +
, A + ( ri) = A ( r i ) * #
i n the F o c k s p a c e f
J , where
A, A
are
+
the
w i t h respect
annihilation and
The projection
< j
function
A.(P , n ) = P
(a* + \ # ? + 7 ) S , =
X
;
<
ft # q)K , = II X
5
K*
Y
a/ye C
(Je C
5,?£l
5.C€ £
I f f
Z K „= 0 ,
p (b) = < 0 I b I 0 > The
ft-semigroup
and
X (0,0)
devisible generating
indefinite-form representation
of
space
triangular block
with
a£ = 0 ,
nil
M
* 0,
2
=
0,
hence
function. , n ) = (e. , 7t(P , n )
A.(P
E. = C © £
is g i v e n
© C
0
by
matrices
u < v , a : = 1 = at
ag = 1 ,
,
a? = E / ,
a ; = p*.
0
Example
2.
Let
S
be an operator
*-algebra
c l o s e d u n d e r the p o l y n o r m
vectors
, n)*,
= X (P
o f 2J i n the p s e u d o - E u c l i d e a n
a =^,
k e
3C,
br|(a) = n ( a b ) a*
= P*
i s an i n f i n i t e l y
canonical
7t|(p , n ) = [ a£ ]
Ki
n#)
X (P*,
p
the
creation
s a t i s f i e s the c o n d i t i o n s o f ( i i ) :
a.ye C
e.)
the
H 1-5)1.
+
if
to
s a t i s f y i n g the C C R [ A ( r,*), A (q) ] = (
X
over E
defines
and
for a l l
on
2
8 £ ^(JQ
i n a p r e - H i l b e r t space
II k II (b) = II b * k II , b e
«
o f the
a —>r|(a) be a c o n t i n u o u s l i n e a r m a p a,b e
S . The Hermitian operation
the s t r u c t u r e o f
the u n i t a l
"kef-
H -»
1
with
a ft c = c + a*c +
ft-semigroup
with
e = 0 , b * = b * , a • b = b + ab + a , V a,b e <S. D u e to not
b • c = (1 + b)(l
necessary
associative and
to
a Poissonian
in
the
the
with
X(b)
of
b
®,
1 , where one
state
p(b)
+
symmetric
e
x
p
on
{
Fock
x ( b )
the
N
by
, defined
(]n(i
space
II
2
a continuous
This
generating algebra
is
n*(b)
= T)
• is
definite positive
functional generated
operators
+ b)) + X(b)}
J
by
i s the
current algebra. the
that the p r o d u c t
is A b e l i a n . The positive
H
X ( b * b ) = II n ( b )
G N S representation = e A (ti(b))
= e
1 i s the i d e n t i t y o p e r a t o r i n 5£,
can easily find,
a n d it i s c o m m u t a t i v e i f
normalized function
linear f o r m
in
+ c) -
belonging
over
e *
%_
with
respect
to
the
(b*)*
vacuum
49 state
I0 >
creation
£
, where
f
and
N(b)
annihilation
commutation
i s the n u m b e r
operators
operator,
A (k°),
satisfying
A . ( k ° ) , k ° , k*, e
+
with
%_
the
relations [ N ( b ) , A ( k ° ) ] = A (bk°) , [ A . ( k ) , N(b) ] = A ( k b ) , +
+
0
0
[ N ( b * ) , N ( b ) ] = N ( [ b*,b ]) , [ A . ( k ) , A ( k ° ) ] = k k ° I . +
0
The
linear
respect to
*-function
X.(b)
the o p e r a t i o n
K a X.(a* + a*c + c) K =
Z
K
= 0 . Hence
p(b)
K
c
a,c e
b
with
= <
n(b) II > 0 2
b
be B
0 Ib I 0 >
is
an
infinitely
devisible
B
be
generating
function.
The
canonical
indefinite-form
= (e. , J t ( b ) e . ) o f "S i n the p s e u d o - E u c l i d e a n
\(b)
positive-definite
K * X(a*c) K = II £
£
c
Z
condionally
ft:
a,c e
if
is
0
is g i v e n b y the with
a£ = 0 ,
ft-semigroup u. < v ,
of
space
representation
£ . = C ffi E
triangular block-matrices
a : = 1 = at
,
ag = b * ,
©
0
C
7t|(b) = [ aJJ ]
a$ = r|(b*),
a
=
0
ri(b)*,
a ; = X.(b)*.
References 1.
R.L.Hudson
stochastic 2.
and
evolutions.
V.P.Belavkin,
ftltering.
K.R.Parthasarathy, Comm.
Math.
Quantum
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stochastic
Matematico
Universita di R o m a
II, R o m a ,
3.
A
V.P.Belavkin,
calculus
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Probability Lecture
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Volterra,
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Ito's
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Fuctorizable representations
K.R.Parthasarathy
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S c i . 5 (1970), pp
Continuous
nonadapted
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Central
Positive Limit
Mathematics,
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Quantum Theory, 10.
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infinitely Academic
W.vonWaldenfels.
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Positive
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450-466.
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M.Schiirmann. A
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V.P.Belavkin.
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Quantum Probability and Related Topics Vol. VII (pp. 51-65) © 1 9 9 2 World Scientific Publishing Company MINUSCULE
WEIGHTS
AND RANDOM
WALKS
51
ON LATTICES
Philippe Biane L a b o r a t o i r e de probabilites T o u r 56 3 ° 6tage, Universite P a r i s 6 4, place Jussieu, 75252 P a r i s cedex 05
Introduction: This
paper
is a continuation
studied
quantum
section
we
will
random walk type
of three
walks
recall
the
basic
where
we
In the f i r s t
associates
G is a compact
that
[51,
a
quantum
and ip a positive
group
in the case o f the group
SU(n) and
Markov chains on and that they are
probabilistic
conditionning. classes
of
there
which
[41, group.
a r e two n a t u r a l c l a s s i c a l
by
function
[3],
a compact
lattices which c a n be deduced f r o m this quantum random walk, related
trace
of
construction
on G . In [41 we proved
normalized
preceding papers
on the dual
(G,
to any couple
function
the
random
The aim of
groups
constructions, sections
and positive
2 to 4
type
such
as
functions.
We
notion f o r this problem is that o f a minuscule weight In
[5]
we
introduced
them
in
order
completely
extend
this
to
study
the
the notions
of
i n the case
description
simply connected compact
to
of the
G be a compact
left
translation
following, A[G)
boundary element
the group case
of
SU(n). a
this
will
and
analysis
see
that
Doob's to
other
the
basic
(cf Bourbaki [71). of
quantum
random
and exponential In the last
general
walks,
and described
section,
semi-simple,
we
will
connected,
L i e group.
1) C o n s t r u c t i o n o f a q u a n t u m Let
Martin
primitive
killing
is to extend
Markov c h a i n on the d u a l
of a compact
group and d(G) be its von Neumann algebra,
operators
X ,
geG,
on
L (G). 2
(cf
Dixmier
group:
generated by [10]).
(In
the
we make the assumption that L (G) is separable).
has a s t r u c t u r e o f H o p f - v o n Neumann algebra with respect
to the coproduct
A, the counit c and the antipode i given by continuous linear extension of A(X ) = X 8X , e(X ) = S (g) and i(X ) = X - i . g
(For Let
g
g
the t h e o r y o f Hopf algebras, v
and i^r be states
g
c
g
g
see Abe [ID.
on i 4 ( G ) , and W = >0(G)8...8^(G)8...
the i n f i n i t e
tensor
52 product being taken w i t h respect to the product state u> = il>8ve>...<s>v... F o r neN, define j : d(G)-¥W by n
j (A ) = A ®...A s i d a i d a . . . n e E t w i t h n+1 A f a c t o r s . g
One has: j
where i
n
is the embedding of ^ ( G )
= i o(id°A)"°A n
into the f i r s t n+1 f a c t o r s o f W.
8 n + 1
L e t Q be the completely positive map defined by: MGUd(G).
Q = (id®v)°A : then:
Q(A ) = 1>(A ) A g g s and g-w(A ) is a positive type function on G . g One has, f o r any a , . . . , a ed(G): 1 n J
(1.1)
w(j (a )j (a )...j (a )) = 0 ( a Q ( a . . . Q ( a 0
This
shows that
Markov ai(G).
0 1 1
n
under
the state
Q(a ))...)) n-1
1 2
n
n
w, the morphisms
chain with t r a n s i t i o n operator
Q and i n i t i a l
(j^,^*
form
a quantum
law ip on the a l g e b r a
(see A c c a r d i , F r i g e r i o , L e w i s [2]).
L e t us r e c a l l some results o f [31, [41, [14]. Let T be an abelian subgroup of G , and 7 be the subalgebra of d(G) generated by (A , geT}. ? is isomorphic to 4(T), which is isomorphic to 1°°(T) where T g is the c h a r a c t e r group of T . T h e r e s t r i c t i o n s o f the maps a
commutative
QC7)c3",
so that
process
(i.e. the j' (a) k
where
keIN
(j ) k
and a e 7
k g
^*
to ?
form
commute),
and
by f o r m u l a (1.1) we c a n interpret them as a c l a s s i c a l Markov
chain on T , w i t h t r a n s i t i o n operator Q | ^ and i n i t i a l l a w tp, ^. Let
i? be the center
the
set
of
classes
is proved that
o f ai{G). T h i s algebra of
irreducible
the r e s t r i c t i o n s
is isomorphic
representations
of the maps
of
t o 1 ( D where
G.
In
[3],
(j ) • to B f o r m k k€lN
a
[4],
T is
[14] it
commutative
process. When
v
is
a
central
state,
(i.e. v(ab)=v(ba))
Q(6)cB
allows us to i n t e r p r e t (j ) • as a c l a s s i c a l k kelN t r a n s i t i o n operator Q . „ and i n i t i a l law I/I . „ .
so
Markov
that chain
formula on
T
(1.1) with
G
I w I In [4] we proved that when G=SU(n) the group of nxn u n i t a r y m a t r i c e s of determinant 1, T is the diagonal subgroup (or any m a x i m a l torus), and i>(A ) is 1 the positive type function - x t r a c e , these two Markov chains c a n be r e l a t e d by 8
some
probabilistic
generalise sections.
this
operations
relation
to
(killing, other
and
Doob's
L i e groups
and
h - t r a n s f orm). states
in
the
We next
will 3
53 2)
Some notations
and facts
from
L i e group
and L i e algebra
theory:
Before s t a t i n g the results on quantum random walks, we need some notions which can
be
found
representations
in
any
textbook
(see f o r example
L e t G be a compact,
connected,
on
compact
Lie
group
and
Lie
algebras
[61, [71, [8), [9], [13]). simply connected
semi-simple
L i e group,
with
Lie algebra 9 . Let T c G be a m a x i m a l t o r u s , and P be the weight lattice associated to T . W is the Weyl group o f G. We choose Let
{0^,
a Weyl chamber i n o, i=l,..l),
(w,
i=l,...l)
and C
be
the
weights of G w i t h respect to C . {<x ,
its closure.
positive
roots
and the
fundamental
i=l,..l} is the dual basis of
V
Let p be the h a l f sum of positive roots,
then 1
p = 1=1 Let P
= PnC, P
= P n C , then
1 and
so that:
(2.1)
P
= p + P
++
+
F o r any x e P , let e(x) be the corresponding c h a r a c t e r of T . J-r
e(x)(8)
X 6
de
is a
minimal
projector
7,
of
and corresponds
to
1
(the
(x)
indicator f u n c t i o n of x) in the isomorphism between J and 1 (T). From
Cartan-Weyl
irreducible an
element
irreducible and l*(r}.
theory,
representation of
P .
By (2.1),
representation For xeP
we
know
of
we w i l l
the
highest
weight
we denote by r
and d
on G ,
X
corresponding
irreducible
equivalence
an element
x
class
weight of
of
an
which is
P ^ w i t h the
x - p , and so we identify
1° (I >
>
)
the character and the dimension
X
representation,
x
is
a
central
function
and i f
= d f F T g T X dg
n x
then
the
by its highest
identify
++
of
that
of G is determined
n
is a minimal c e n t r a l
x J
projection
X
isomorphism between
c
I
^
g
of 6",
which correspond
to 1
in the '
iS" and 1 (P
).
x
'
54 The r e s t r i c t i o n of x
to T is given by Weyl's f o r m u l a : X
L
X
(2.2)
= — )
x
det(w)
e(w(x))
det(w)
e(w(p))
f urthermore:
L We shall use the notation
det(w) e(w(p)) = n(e(a ) - e ( - a )). Y
det(w) e(w(p)) = ?,
wWV We now a r r i v e to the definition of a minuscule weight: An
element
u of P
is a minuscule weight
if
it s a t i s f i e s
one of the f o l l o w i n g
equivalent conditions: i)
the set of weights
weights w(u), weW,
of
1, weW,
ii) f o r any i=l
the representation
with
dominant
weight
u a r e the
and each weight has m u l t i p l i c i t y one. a^(w(u)) e {-1,0,1}.
(see Bourbaki [7], p 127 proposition 6 , properties i) and iii) ) The important property of minuscule weights
is the f o l l o w i n g ,
which is a
consequence of i i ) : (2.3) i f x e P
then x+w(u) e P
++
f o r any
weW.
+
The list of a l l minuscule weights of simple L i e algebras is given in [71 pl29.
3) Comparaison of two Markov chains: In this p a r a g r a p h , G is a compact
connected,
simply connected
semi-simple Lie
group, and we f i x u a minuscule weight of G. L e t K be the normalized c h a r a c t e r of the representation w i t h highest
weight u.
By
to
property
i)
of
minuscule
weights, in
\W\
Y
u,„ weW
the
restriction
of
K
T
the
restrictions
is
e(w(u))
L e t v be the weight given by v(A ) = K(g). g
We
first
of the (j )
determine
explicitly
the
transition
operators
of
• to ST and £ .
Theorem 3.1: i) T h e r e s t r i c t i o n of (j ) • to S defines k k€[N increments being
.,),.
V
| 1 V |
wW
a random walk on P, the law of its
S W
l
»
ii) If p(x,y) a r e the t r a n s i t i o n probabilities of this Markov c h a i n , p(x,y) = iii) F u r t h e r m o r e , f o r any weW,
[ e(x)(9) e(y)(6) ic(e) de. p(x,y) = p(w(x)),w(y))
then
55 proof: the i s o m o r p h i s m between In the
y
and l^IP),
A ey
corresponds to the
f l
function
x-»e(x)(8) on P . QU )
= K(6)A
e
= -p-L- £
e
e(w( ))(8)A M
Qn^pjIet.JteJKy) =
so
e
£^e(w(u)-y)(e)
Since the A ^ generate
7,
e(y)(6)
|J" e(y)O')
K(6) =
t h i s proves the f i r s t p a r t . K ( 9 ' ) e(x)(6')
orthogonality r e l a t i o n s of the functions
X
V(P) V = (
Ep (J
=
P
e ( y ) l e , )
that:
( x
f ^ < >( '! 'j J
- y ' ^x)
K(e,)
e
=
x
i
T
e
de'j
e(x)(9)
by
the
e(x)(.). 1 5 1
de
K ( e )
x
e
d
e(x)(8) A
e
de
f l
which proves i i ) . The
last
relation follows
process is i n v a r i a n t by
Theorem The
from
the
fact
that
the
law
of
the
increments
of
the
W.
3.2:
restriction
of
the
(j ) • k keIN
to
B
defines
a
Markov
chain
on
P
++
with
transition probabilities: d q(x,y) = - p x
r
x (g) X (g) J
G
x
*(g)
dg.
y
proof: QOI ) = y
Y L xeP
1< .y> n ^ x x
J
= d f x~Tg) yj~ y
K(g) A dg = g
d I xeP
'
++
J
x
G
*
{
g
,
)
^y' '' 8
K
(
g
>
)
orthogonality r e l a t i o n s between
Let of
p°(x,y) p(x,y)
be
to
the
P
.
p°
transition defines
a
d g >
]
d
'
xf
**
(
g
)
G
the
\
d
g
b
y
the c h a r a c t e r s .
probabilities
on
P
+ +
obtained
submarkovian kernel on
P
+ +
by
which
restriction corresponds
to the r a n d o m walk defined by p k i l l e d at i t s f i r s t e x i t time of P ..
Lemma
F o r any
3.1:
x,y€F
tt
p°(x.y) =
^ d e t ( w ) p(x,w(y)) = -^j
£^
V ^ d e t ( v w ) p(v(x),w(y))
proof: Let W
x,y
eP
+ +
a c t s simply t r a n s i t i v e l y on the Weyl chambers,
so that w(y)*P^
+
for
w*id.
56 By property (2.3) and theorem the
f
sum
det(w)
3.1) i) p(x.z) = 0 i f x e P ^
p(x,w(y))
a l l the terms
are 0
and z £ P ,
so t h a t i n
+
except
w=id,
a n d the
vkw f i r s t equality holds. The second equality follows f r o m i i i ) o f theorem 3.1. L e m m a 3.2: = [ x (g) X (g) K(g) dg. J x y
p°(x,y)
G
proof: Applying lemma 3.1, then theorem 1 i i ) , we have: p°(x,y)
= y i j
\W\
L
r
£^det(vw)
d e t ( v w )
J
d e t l v )
= TiJ7T f X (9) x~Te) I W I J ^ . x y =
J Q
I
<
e ( v ( x ) ) ( e )
e ( v ( x ) ) ( e )
K
p(v(x),w(y)) =
(e)
e
w
) (
l\
w
e ( w (
detM
y
) ) ( e )
)
K < e )d e
l?(e)| de by Weyl's f o r m u l a (2.2) 2
y (g) 5 (g) K(g) dg because x y
x . X . and K a r e c e n t r a l functions x y
so that
one can use Weyl's integral f o r m u l a (cf [8], [9] ).
From
this
we deduce
the following
result
which
i s the m a i n
result
of
this
paragraph:
Corollary 3.1: i) x - x l i s a p°-harmonic x . d ii)
q(x,y)
= ^ - p°(x,y),
function on P ++ so that
q is obtained
from
p° by r e l a t i v i s i n g
with
X
the harmonic function d. iii) Every positive p°-harmonic
function is p r o p o r t i o n a l t o x-»d .
proof: ii) comes f r o m
lemma 3.2, a n d the f a c t
that d is p " - h a r m o n i c follows
from
the f a c t that Q | g is a Markov operator, hence q a Markov kernel. iii)
is
positive given
consequence
q-harmonic
by a
following appears (see
a
verifies
property in some
of
functions
the results
hypothesis
of minuscule tensor
[51 proposition 12).
of
a r e constant.
power
h ^ of weights: of
[5],
where
we
One has to v e r i f y
[51, but this every
follows
prove that
that
easily
with
highest
v
f r o m the
irreducible representation
the representation
the
the state
of G
weight
u
57
fj
E x t e n s i o n to
In
[3]
dual
and of
some non centered
[16],
SU(2).
an
example
In
this
of
a
random
walks:
non-centered
paragraph,
we
random
study
walk
analogous
was
given
results
in
on
the
the more
general f r a m e w o r k of the preceding sections. L e t us f i r s t r e c a l l some results f r o m
[51.
L e t G be a compact group, and 4(G)
its von Neumann algebra.
4(G)
4(G)
has
classes
a
central
decomposition
= ®_M , x€r x
where
of i r r e d u c i b l e representations of G and f o r each
x
is
x
M «M x
*
Let
imbed 4(G)
us
into
the
product
= n M , xer
4(G)
'-algebra
the
d
set
of
(C). x
equipped with
the
4(G)
we
x
product
4(G)
topology.
is dense
in 4(G)
and f o r
any subalgebra
£
of
4(G).
c a l l B its closure in
The coproduct A can be extended in a unique way to a map A: <4(G)-» 4(G) where
we
use
a
which
contains
completed
tensor
4(G)
product
4(G)
a
«
n
" x,y€I
x
r
4(G)®4(G)
=
j] M a j | M xer eT x
as
a dense
a
4(G)
M eM
y
subspace
for
the
following
lemma,
product
y
y
topology. Let
T
be
an abelian closed 3
identify
and
respectively.
i?
( Ja?
subgroup
of
with
the
algebras
and
B e E completions
G,
of
with the all
of
complex
S a ^ and
functions Sag
on
in 4(G)
we T
will
and
a 4(G)
r
will
be the algebras of a l l complex funtions on T x T and TxT ).
Lemma
4.1:
i) L e t U be a complex f u n c t i o n on T , then the
J
U = £ U(x) converges
in 4(G)
and
the
uniquely in t h i s
way
ii)
complex
Let
V
be
a
sum
is
function
on
x(e) X e
7.
in
series
Every
then )
r,
d
e
element
V(x)
II
of
7
can
converges
be
in B
written
and
x6T every element o f £ is of this f o r m , proof: We f i r s t show t h a t the series l l
y
_
in
T
But II y
of
the
x(6)
\
The
de
n
=
0
if
x(e)
\
g
de converges
f o r every y
'
x
does not
appear
in the
decomposition
e
T
representation
dimensional, terms.
J
£ U(x) J x£T
y ^
and the
so
that
the
series
sum
is
in 7
since
irreducible representations
contains partial
only sums
a f i n i t e number are
in 7.
The
of
G are
of
non
function
finite
zero U can
be
58 recovered f r o m U i n the following way: U(x)
is the component
| x(8) X U de on the algebra J 9
of
a
corresponding to
T
the t r i v i a l representation of G . i f V. is an element
Conversely,
v e r i f y that K = £ U(x) J ii) follows
x(8) \
1
of
Q
we define
U by the f o r m u l a
above and
d6; this proves i ) .
by the same kind of arguments as i).
We now r e c a l l a basic d e f i n i t i o n f r o m [51.
Definition An
4.1:
element
a
of
4(G)
is
called
primitive
if
A(a)
=
aaa.
It
is
called
an
exponential i f it is p r i m i t i v e and positive.
Lemma
4.2:
If U is a f u n c t i o n on T w i t h values in C such that U(xy) = U(x)U(y), U
=
£ U(x) x€T
elements
x(9)
J
.
A
de
0
is
a
primitive
element;
all
then the
^
i n 7 a r e of this f o r m .
ti is an exponential i f and only i f U takes i t s values i n IR . proof:
A( £ u(x) J x(e) A dej = £ u(x) J x(e) A ®A de E
x€T
E
xST
T
E
T
By F o u r i e r analysis, f o r every continuous f u n c t i o n * oi
f *(e,e) de =
V
! f *(e,e') y(e) y(e') de de'
so that:
A(lO =
£ u(x) x€T
J J " x(e) y(e) y(e') A 8 A , d e de'
£
e
j? utzy) ^j" iTeT A
f z€T
y£T
= | £ U(x) J
e
y€T 0
dej
y T e ! A , de'j 7
0
^
x(e)
A
E
dej8^ £ U(x) J
x(6) A
FL
dej
since U(yz) = U(y)U(j
X6T
T h i s proves
the f i r s t
assertion,
the second comes f r o m
lemma
The last point is easy to check and we leave it to the r e a d e r .
We now r e c a l l the main result of [5].
4.1 i).
primitive
59 If
v
the
is a state solutions
on 4 ( G ) , and Q is the associated
in 4(G)* o f
the equation
completely
Q(a) =
positive a
a a (where
eR
map, then
is a
+
fixed
number) c a n be described as f o l l o w s : | f dm(f) where S is the set o f exponentials
every s o l u t i o n is o f the f o r m a =
in 4 ( G ) , a n d m is a positive Borel measure on &, uniquely determined by a, and supported a=l,
by the set o f
f
such
that
vlf)
= a.
but t h e p r o o f extends t r i v i a l l y to the case
Definition
([51 deals
J G
An immediate consequence
8
8
o f this d e f i n i t i o n i s :
L e t v be a c e n t r a l s t a t e ,
then f o r any ae4(G) (or 4(G) ), i>(a) = v ( E [ a ] ) g
L e t
~7^T-
v\
v
(since
choose
is
an
4(G)«4(G)...«4(G)®...
central,
initial
with
v(0a) =
central
respect
i>(0
state
to the state
1/z
be the state
a#' ),
so
2
ill
and
u.
=
ill ,»v
v,
that W
call
9
0
Lemma
the case
= [ A aA - l dg e £ .
E
state).
with
4.2:
F o r any a e 4 ( G ) , l e t E „ [ a l
v.(a)
only
above).
0
the
is a algebra
,e...e>v,®...
4>
4.3:
i) L e t ce6\ a n d m
ii) If a = (a )
_
e4(G),
x x€l
(c) commutes
m
then E „ [ a ] = 19
with every j (a), a e 4 ( G ) . n
(tr(a )
d"
X
„
1
X
id,. )
„
.
M x€l x
iii) E g [ 0 ] is an i n v e r t i b l e element o f 4 ( G ) .
proof: i) It is enough t o prove i t f o r a=A . g
We have: j (A ) = n
The
first
term
morphism ksm, ii)
j
g
(A
m
g
i n this
and c
m
l8
ln
m)
g
product
and A
commutes
commute.
with
j
T h e second
m
(c) because
term
j
is an algebra
m
commutes
with
any J ( ' . a
k
ae4(G).
F o r any x e T , J
linearly
A^a^A^-i
have
that,
by S c h u r ' s lemma,
i
A = W
d g is an element
on a, so i t is t r ^ ( A a ^ ) i d ^
must
E„[A b A - l ) = G g x g
tr(id
M
x
iii)
)(id® aA ' " i8ide...).
x
M
F
1
= d"
X
E„[b ] e
x
of the center
of
M
depending
f o r some A e M . F o r any g e G , b eM , one X
so
that
A AA - l = g
g
A.
x
is
irreducible,
so
A is a multiple o f i d ^ . Since E g l i d ^ ) = i d ^ one has X
1
id
X
X
M
X
F o r any x e T T h e component
o f 0 on M
is positive invertible so that i t s
60 t r a c e is >0 and (tr(0 )
d"
x
Theorem Under
w
M
)
„ is invertible.
x€l
4.1:
the state
the r e s t r i c t i o n
0
w i t h i n i t i a l law 0^ ^
on r,
id
1
x
of the (j ) k
k€lri
• to B is a Markov
whose t r a n s i t i o n operator
process
is given by:
proof: Let c ,c , . . . , c eB. o 1 n u . t j (c >...j (c )> = u t j (c )...j t c ) 0 ® * W r W r 0 0 0 n n 0 0 n n = w(j (C )...j (C )j (0) 0 ( 0 ) " ' f(0)~ ) 0 0 n n n = u ( j (c (c 0) i/>(0f W0)~ ) 0 0 n n = w(j (A )j (c (c 0) 0(0)"' (A -l)) n
n
)
=
n
1
n
g
O
O
n
n
n
n g
f o r any g e G , since v is c e n t r a l =
f « ( j (c J 11
n
G
W0)"")
(c A 0A - 0 0(0)"' W 0 ) " " dg n g
g
= 0(c Q(c . . . Q ( c Q(c E „ [ 0 ] M 0 ) ~ " ) . . . ) 1 2 n-1 n 6 by f o r m u l a 0
10
Remark:
(1.1) n-1 0
0 In this
n
theorem
it is of
crucial
importance
that
0
(the i n i t i a l
state)
is a c e n t r a l state. T h i s result can be compared to the f o l l o w i n g : if
is a Brownian motion
invariant
by
rotation
w i t h a constant
then
IXJ
is
drift
a
in R
d
homogeneous
whose Initial Markov
law
is
process
(see
P i t m a n - Y o r [15]).
We
can now describe
have just
the M a r t i n
boundary
of the Markov chain on I~ that
we
constructed:
L e t G act on S by conjugation
e (h) = A -lhA and let S/G g
g
be the o r b i t space
g
Remark: T h e r e is a n a t u r a l in S/G
determines
Proposition The
Q. , 0IE lr
identification
between S/G
and E^IS],
so that
any element
a function on T\
4.1:
harmonic
functions
a r e of the f o r m
J
g
/
(E„[0]) ' f B G
dm(f)
where
61 is a measure on S/G
uniquely determined, and supported by the elements
such
that i>(f) = i>(0) (v is well defined on S/G since v is central), proof: The
equation
the
to: Q ( E „ [ 0 )
Q , ( a ) = a is equivalent
9
solutions
are
of
the
form
a
=
a) = W0) 6
-
(E„[0])~
h
E „ [ 0 ] a, so that 6
dl(h)
positive measure supported by the set of h such that i>(h) = i>(0). If a solution a is i n § then a = ( E - I p l f f h d(e (I))(h) where 1
»
image
measure
I by e . By uniqueness, e unique measure £ on S/G such that a =
h
d£(h)
5
where
s
e (I) =
I,
so that
being
a
e (I) is the
e
there
exists
a
s
the image measure on S/G of I.
is
S/G
3
Let
of
Jg
1
us now come
back
to the situation
of
§ 2 : G is a compact,
connected,
simply connected, semi-simple L i e group, T a maximal torus, and a a minuscule weight. Let 0 be an exponential
in 7
and U the corresponding function on P by
lemma 4.1.
Theorem
4.2:
i) T h e r e s t r i c t i o n of (j ) * to 7 defines k kelN U(w(u) ) 5 , ,
a random walk on P, the law of its
L
increments being
Y
U(w(u) )
wkw ii) If p , ( x , y ) a r e the t r a n s i t i o n probabilities of this Markov chain, then -l p , ( x , y ) = p(x,y) U ( y - x ) i/(0) .
0
proof: As i n the proof of theorem 3.1, we compute i> (A > = v(0) 0
Q
u(0AJ = i>(0)
_1
|l0
(p)
(e(.)(e))(y)
=
U(w(u)) e(w(u))(8) so that:
wkw
9
.
V
-1
6
W0)"
1
£ ^
U(w(u))
e(w(u)-y)(9)
This proves i) ii) follows f r o m i) and theorem 3.1 i). As i n 53, we introduce the submarkovian kernel p° on P restriction
of
p..
0
It
gives
the
transition
which is equal to the
probabilities
of
the
random
walk killed a t the boundary of P^ . Let
us c a l l
q.
4.1,
and V the function on P
0
the t r a n s i t i o n + +
probabilities
of
the Markov
chain
of
corresponding to Egl0] by lemma 4.1 i i ) .
We have the f o l l o w i n g extension of the results of § 3 :
theorem
62 Theorem
4.3:
i) q ( x , y ) = v( )-' ,
V
m
^
9
0
x
, ,
y
)
q(x,y)
V(y) y U(x)
,
d
v T x l 3 - 0177 V
=
'
x
y
)
X
In p a r t i c u l a r , the Markov chain on 6" is obtained f r o m the random walk on 7 by k i l l i n g at the exit , Vd function JJ-.
time
of P then taking the h - p r o c e s s **
w i t h the
p°-harmonic 9
proof: i) is a d i r e c t consequence ii) follows
We
can
of theorem 4.1.
f r o m c o r o l l a r y 3.1, theorem 4.2 i i ) , and p a r t i ) .
deduce
from
these
results
the
Martin
boundary
of
the
killed
random walk: Corollary
4.1:
The e x t r e m a l positive p ^ - h a r m o n i c functions a r e : E [ h ] ^ where h is an exponential such that vlh) = v(0). g
proof: this follows
We
f r o m proposition 4.1, and theorem 4.3 i i ) .
now give
a
concrete
example,
with
the group
SU(3) and T
its
diagonal
subgroup. As
i n Bourbaki [6] planche
lattice cubic u
j
i n C generated root
is
a
of
by u
1. T h e roots
minuscule
weight
dimensional representation of The
weights
X , p.
276,
we c a n represent
= ^ (2+j)
the l a t t i c e
and °> ~ \ (l+2j) where 2 2 - j ; j ; -1; j ; - j >.
a r e {1; which
is
2
the
dominant
weight
j
of
=
P as the e
2 1 7 r / 3
the
;
s
basic
a
3
SU(3).
2 a r e u , j u , j u . In the coordinates u,v w i t h respect to i i j the dimension function is -p uv(u+v). 6 exponential associated to a f u n c t i o n U on P , given in (u,v) x z where x, z e R . T h e law of the associated r a n d o m walk is qS + rS 2 where l l x y z , ,-i p = ; q = — - — r = and y = (xz) x+y+z x+y+z x+y+z
w(u), weW
1
the basis o> , u 1 2 We choose an coordinates by given by pS + l
u
v
+
r
This
random walk
kernel
J
k i l l e d at the exit
p°, and by c o r o l l a r y 4.1,
in (u,v) coordinates by:
of the Weyl
the extremal
chamber
p°-harmonic
u,viO
has t r a n s i t i o n
functions
a r e given
63
*(u,v) = p r " ( a - b ) " ( b - c ) " ' ( c - a ) " '
a
v
b
u
1 where
a,b,c
eR
and
satisfy
c"
u
1
abc
=
1
1,
a+b+c
=
x+y+z.
When
some
of
a,b,c
coincide, it is understood that a limit is to be taken. The
function
completely
4
depends
symmetrically
on
the
parameters
a,b,c
so
that
it
is
determined by the positive number ab+bc+ca, which can vary in some
i n t e r v a l . T h i s i n t e r v a l is the M a r t i n boundary of the killed random walk.
5)
Exponentials
and
p r i m i t i v e elements
for
Lie
groups:
In this p a r a g r a p h we f i n d a l l primitive elements when G is a connected, connected folklore we
do
semi-simple of
Lie
not
group
have
a
and
below.
First
4(G)
as
the
Although this result
algebraic
reference
give the proof identify
L i e group.
for
dual
group
the
of
precise
a
theory
(see
Dixmier
result
that
we
bialgebra
of
simply
seems to be p a r t of [111,
need,
polynomial
the
3.8.16),
so
that
functions
we
on
an a l g e b r a i c group: Let
y
be
the
dimensional
algebra
and we give coefficients
it the
is
J
is a
1
£(7> ,C)~V x
£(T,C).
with
x
n
M ®M
"
x
r
x , y£l now
on
a
G
which
union
of
are
finite
coefficients
of
dimensional
Ts>P is isomorphic to the
is a bialgebra f o r
take
the
finite
subspaces,
algebra
TJPs'P
coproduct 3:
x.
f where V is the space of xer x x dual is then a direct product
sum
Its
H i l b e r t subspace of J
f (g) G
The dual of
4(G)-> We
direct
representation
space
is
of
given
Pigs').
an a l g e b r a i c
the
T
inductive topology.
identification, T
With this
functions
of f i n i t e dimensional representations of the group G x G .
by: 3(P)(g,g') = T
of
representations.
y
for
x
X
dg
g
e
L (G), M ,
we
x
have
we
an
identify
£(?> ,C),
an element
i d e n t i f i c a t i o n of
4(G)
of
each f
of
with
the m u l t i p l i c a t i o n map on T is then the coproduct A :
* JE(7'®7 ,C)«.4(G) 8
4(G).
,
G
and if
Z
coefficients
a
compact,
semi-simple
connected,
simply
connected
Lie
group K. Let
k
be
its
Lie
algebra,
and
let
K
c
be
the
complex
simply
connected
group w i t h L i e a l g e b r a keik. Kj, is a complex algebraic group. By
Weyl's
"unitary
representation Kj,,
so
the
algebraic
on T,
that
of T
trick"
K can be
can
be
group K
(cf
(131)
extended
identified
we
uniquely to
with
Let xe4(G)
know
be
and A(x)(P®0) = x(P)x(Q) = x(PQ),
the
that
any
finite
an algebraic
algebra
of
representation
polynomial
a primitive element,
dimensional
x
functions
is a
linear
of on map
so that x is an a l g e b r a morphism f r o m
64 'P to
c. By elementary algebraic geometry, there exist a unique element g of
KCsuch that x(P) = P(g) for every P (x is evaluation at g), and conversely, every element of KC determines such an algebra morphism, so that the set of primitive
elements
can
be
identified
with
KC'
.
Every
element
of
k
is
a
derivation on 'P, which composed with evaluation at zero gives an element of A(G)"'!e('P,C) such that .II(a) = a®id + id®a, and a
be written exp(a+ib), and only
if
it
is
= -a. Every element of KC can
and exp(a+ib)' = exp(-a+ib),
of
the form
exp(ib) .
Hence,
so that it is positive if the
set
of
exponentials
is
identified with the homogeneous space KC/K '" IS, by polar decomposition.
Theorem 5.1: The set of primitive elements of A(K) is in one to one correspondence with K ' C the correspondance being as above; in this correspondance, exponentials correspond to elements of the form exp(ib) where bek.
Corollary 5 .1: Using Cart an coordinates on KC '" KAK, we see that ISIK '" AlW (the action of K on IS being by conjugation).
I would like to thank Y.Benoist and P.Polo for some useful discussions.
References: [II E.Abe: Hopf algebras, Cambridge tracts in mathematics, vol 74, Cambridge Univ .press, 1980.
[2]
L.Accardi,
A.Frigerio,
J .T.Lewis:
Quantum
stochastic
processes,
Pub!.
R.I.M.S. Kyoto Univ . 18 p 97-133 (1982).
[3] Ph. Biane: Marches de Bernoulli quantiques, Seminaire de Probabilites XXIV, Lecture notes in Mathematics, Springer 1990.
[4] Ph.
Biane:
Quantum random walks on the dual of SU(n) , Probability
theory and related fields, 89, p 117-129, Springer 1991.
[5]
Ph.
Biane:
Equation
de choquet
Deny sur
Ie
dual
d'un
groupe
compact, pre print, Universte Paris 6, 1991. [6],
[7],
[8]
N.
Bourbaki:
algebres de Lie, chapitres 4
Elements
de
mathematiques,
Groupes
et
a 6; chapitres 7,8; chapitre 9 , Hermann.
[9] T.Brocker, T.tom Dieck: Representations of compact Lie groups , Graduate texts in Mathematics n098, Springer.
[10]
J.Dixmier:
Les
C' -algebres
et
leurs
representations.
1964.
[11] J .Dixmier: Algebres enveloppantes, Gauthier-Villars, 1974.
Gauthier-Villars,
65 (12]
A.W.Knapp:
Mathematics [13]
Representation
series,
Naimark,
n'36,
Stern:
theory
of
semi
simple
Lie
Princeton
groups,
1986.
Theorie
des
representations
des
groupes,
Editions
MIR, Moscou. [14]
K.R.
Parthasarathy:
de Probabilites [15]
J.W.
[16]
W.
Probabilites
Lecture
Pitman,
M.
of
Durham
Proceedings Springer,
XXIV,
A
Yor:
generalized
notes Bessel
Biane's
in mathematics,
Springer,
processes
indefinitely
and
conference,
Lecture
notes
The
process
of
Seminaire
process,
in
1991. divisible
Mathematics
laws, n°851,
1981. von
Waldenfels: XXIV,
Lecture
notes
Markov
in mathematics,
total
Springer,
spin, 1991.
Seminaire
de
Q u a n t u m P r o b a b i l i t y a n d R e l a t e d Topics V o l . V I I (pp. 67-77) © 1992 W o r l d Scientific P u b l i s h i n g C o m p a n y A N
E X A M P L E
Abstract.
O F A G E N E R A L I Z E D
67
B R O W N I A N
Marek
Bozejko
Roland
Speicher
M O T I O N
II
We continue our investigations of the generalized Brownian motion con-
sidered in the first part of this paper. of the operator Pi
Namely, we show that a non-trivial kernel
can only occur for z a root of unity and we characterize the
Gaussian distribution of our random
1.
variables.
INTRODUCTION
In [ B S p ] we p r e s e n t e d a r e p r e s e n t a t i o n o f the r e l a t i o n s (for a fixed fi € ( — 1,1)) c(f)c (g) +
-
M
c ( p ) c ( / ) = < f,g +
> 1
(f,g
€ £ (2R)), 2
w h e r e c ( p ) i s t h e a d j o i n t o f c(g)
(in a full Fock space equipped w i t h a twisted
scalar procduct)
is linear.
+
a n d g >—» c (g) +
W e such received a n i n t e r p o l a t i o n
b e t w e e n t h e b o s o n i c , free a n d f e r m i o n i c r e l a t i o n s . I n d e p e n d e n t l y f r o m o u r w o r k , G r e e n b e r g [ G r e , F i v ] p r o p o s e d the s a m e r e l a t i o n s as a first ( n o n - r e l a t i v i s t i c ) field t h e o r y t h a t a l l o w s s m a l l v i o l a t i o n s of t h e e x c l u s i o n p r i n c i p l e (i.e., o f F e r m i s t a t i s t i c s ) o r o f B o s e s t a t i s t i c s a n d he a r r i v e d at s i m i l a r r e s u l t s as i n [ B S p ] . T h e r e l a t i o n for fixed f = g w i t h ||/|| = 1, n a m e l y c c p e r a r e d i n t h e c o n t e x t of t h e t w i s t e d SU (2) a //-oscillator.
+
— / i c c = 1, h a s also a p +
(e.g. [ B i e , M a c ] ) a n d i t i s t h e r e c a l l e d
q
I n t h i s r e s p e c t we a r e d e a l i n g w i t h a n ensemble of / / - o s c i a l l a t o r s
which are //-independent.
N o t e t h a t t h i s f o r m of i n d e p e n d e n c e is m o r e
adequate
for t h e / / - o s c i l l a t o r t h a n t h e t e n s o r p r o d u c t . H e r e we w a n t t o c o n t i n u e o u r i n v e s t i g a t i o n s . the ' t w i s t e d ' s y m m e t r i z a t i o n operator
I n p a r t i c u l a r , we w i l l e x a m i n e
o f [BSp] m o r e c a r e f u l l y a n d show t h a t
a n o n - t r i v i a l k e r n e l c a n o n l y o c c u r at r o o t s o f u n i t y . c h a r a c t e r i z e t h e s p e c t r a l m e a s u r e o f c(f)
2.
S O M E INVESTIGATIONS
F u r t h e r m o r e , we w a n t to
+ c (/). +
O NT H EO P E R A T O R
P
Z
T h e m a i n o b j e c t i n o u r i n v e s t i g a t i o n s i n [BSp] w a s t h e o p e r a t o r P^ = 0 P ^ d e f i n e d o n t h e f u l l F o c k s p a c e T o f L (IR). 2
n
'
It w a s - for each n - g i v e n b y a
r e p r e s e n t a t i o n o f s o m e e l e m e n t o f t h e g r o u p a l g e b r a C 5 „ o f the s y m m e t r i c g r o u p S „ . S i n c e t h i s r e p r e s e n t a t i o n i s a f a i t h f u l o n e , we c a n r e s t r i c t o u r a t t e n t i o n t o t h i s g r o u p a l g e b r a itself a n d w i l l from now on denote by P
Z
= © pj
the element o f
t h e g r o u p a l g e b r a . W e h a v e a l s o r e p l a c e d \i b y z i n o r d e r to i n d i c a t e t h a t we d o
68 n o t r e s t r i c t to \i € ( - 1 , 1 ) a n y m o r e but c o n s i d e r g e n e r a l z € C . e x a m i n e t h e c o m p o n e n t s pf™' for a r b i t r a r y , b u t f i x e d F i r s t we r e c a l l t h e d e f i n i t i o n of P^ \ S
n
now
D e n o t i n g the n u m b e r of i n v e r s i o n s
n
T6
We will
n. of
b y z ( x ) , i.e. * M
:
= # { ( » . ; ' ) e { 1 , • • •, n}
| i < j , «•(») > 7 r ( j ) } ,
2
we p u t p(»)
:
=
^
2
»"(»)»
CS„.
e
O u r m a i n p r o b l e m w i l l c o n s i s t i n f i n d i n g the z S C , for w h i c h Pi™' is n o t i n v e r t i b l e . W e w i l l s h o w t h a t t h i s c a n o n l y o c c u r at r o o t s o f u n i t y . F o r p r o v i n g t h i s we n e e d some facts about the s y m m e t r i c g r o u p S W e s h o u l d n o t e t h a t for 7r, C 6 5
and its length function I(TT).
n
we d e n o t e b y ircr t h e p e r m u t a t i o n g i v e n by
n
(7rcr)(i) = cr(7r(i)), i . e . we t h i n k of a p e r m u t a t i o n as a r e a r r a n g i n g of l a b e l s , n o t of numbers.
T h i s s h o u l d be k e p t i n m i n d i f one w o n d e r s a b o u t s o m e
discrepancies
i n t h e o r d e r b e t w e e n o u r f o r m u l a s a n d the c o r r e s p o n d i n g f o r m u l a s i n t h e b o o k of C a r t e r [Car]. F i r s t , we f i x s o m e n o t a t i o n s . W e w i l l use a f i x e d n o t a t i o n for s o m e e l e m e n t s S „ , n a m e l y we d e n o t e t h e t r a n s p o s i t i o n (j <-> j+1) e l e m e n t of S
n
of
b y itj (j = 1 , . . . , n — 1), t h e u n i t
b y e a n d the m i r r o r i n g p e r m u t a t i o n Tr Tr 7r . . . 7 r „ _ 7 r _ 2 . . . i r j T i b y 1
7Tn, i . e . 7To(fc) = n — (k — 1). It is easy to see t h a t T w i t h t h e greatest l e n g t h , n a m e l y i(iro)
=
«
2
1
1
n
is t h e u n i q u e e l e m e n t o f
0
S
n
•'.
n
T h e s i t u a t i o n t h a t A i s a s u b s e t of B is d e n o t e d b y A C B , w h e r e a s t h e s y m b o l * C ' i s r e s e r v e d for p r o p e r
subsets.
N o w p u t S := { 1 , 2 , . . . , n -
1} a n d d e n o t e b y Wj
5 „ w h i c h i s g e n e r a t e d b y 7T; for i £ J. subgroups.
For example,
we have Wq> =
i s o m o r p h i c to a p r o d u c t of p e r m u t a t i o n Wj
£ 5
f c l
x • • •x 5
for J C S t h e s u b g r o u p
S u c h s u b g r o u p s of S „ a r e c a l l e d {e}
and W
=
s
of
parabolic
N o t e t h a t Wj
S. n
is
groups: w i t h fc! + • • • + k
f c r
= |J|.
r
B y | J | we d e n o t e t h e n u m b e r of e l e m e n t s o f t h e set J . W e p u t
Pi (J)
•=
n)
I n p a r t i c u l a r , we h a v e P , P
2
( n )
(0) =
L E M M A
1.
( n )
(5) =
Yl
P i
z
n
i
(
)
T
)
%
, P i
G
n
C
)
W
j
£
C
( l , . . . , -
5
"-
1) =
(k
e. a) F o r a i l
J
b) W e Jiave E J C S ( - 1 )
C S there exists R (J) n) z
| J |
^ " ( J ) = z*' }
(,roL
=
€ CS
n
such
that
<
„)
a
n
d
69 c) For aJJJ,J'cSwe d) We
have:
W
TQ^WJ-
\ %f g Wj,
{ T T J X J,
=
r
irj. g
Wj,}.
have
JJ ~ ' 1
p ( " ) . p|») _ p ( " ) . p ( » )
=
z +1
i=l
p(») . p( ) z -1
_ p(") . p(n) _ T T -1 z 11
n
REMARKS:
^
. p(») Z
~
1
+ 1
+
p(n) -1 •
1
z
1) T h e first t w o s t a t e m e n t s a r e m u c h m o r e g e n e r a l l y v a l i d , n a m e l y n o t
o n l y f o r 5 „ b u t f o r a l l f i n i t e C o x e t e r g r o u p s . T h e s t a t e m e n t s a) a n d (at l e a s t t h e a n a l o g u e i n t h e t r i v i a l r e p r e s e n t a t i o n of) b) a r e w e l l k n o w n f a c t s i n t h i s c o n t e x t a n d c a n , f o r e x a m p l e , b e f o u n d i n [ C a r ] . T h u s we w i l l o n l y g i v e a s h o r t s k e t c h of their proof. 2) P a r t a) is a g e n e r a l i z a t i o n of t h e d e c o m p o s i t i o n p{n)
p(»-l) .
=
(
e
+
Z
V
_
n
i
+
z
2
N o t e a l s o t h e t r i v i a l case R \s)
=
n z
e x p r e s s i o n R \j) z
T n
_ i l
r n
_
2
+
. . .
.
+
F o r J not of such a s i m p l e f o r m
e.
i s n o t s u c h easy. F o r e x a m p l e , for n = 4 a n d J = {2}
n
= (1 + ZTT ) • (1 + Z*\ + zr 2
2
+ z V i i r j + z 7Tiir + ZTT 2
2
3
3
the
we h a v e
+
2
+ z 7r 7r27r + z -KiW TZ2 + z 7r3 7r 7ri + 3
1
3
3
3
3
4
2
4
+ z TT -jr Tr ir 1
2
3
5
\
+ z Tr 7r37r 7r + z
2
1
2
•K -K V 'K 'K\).
1
X
2
3
2
It s h o u l d b e n o t e d t h a t for p a r t b) a l l s u b s e t s J C S a r e n e e d e d . 3) P a r t c) is n o t v a l i d for a l l C o x e t e r g r o u p s b u t i t d e p e n d s o n t h e s p e c i a l 'geom e t r y ' of S .
It h a s a n i c e i n t e r p r e t a t i o n i f one ' v i s u a l i z e s ' S
n
PROOF:
F o r J C S w i t h J = {k(l), Dj i.e.
Dj
geometrically.
n
a) c o m p a r e [ C a r , t h e o r e m 2.5.8] ...,
k(r)}
: = {TT € S„ | ir(k(i))
(fc(i) g S : = { 1 , . . . , n + 1)
< r(k(i)
1}) we define
for a l l i = 1 , . . . , r } ,
i s t h e set of a l l p e r m u t a t i o n s w h i c h d o n o t i n v e r t t h e o r d e r of a l l k(i)
k(i) + 1. Dj
n
n = 3 a n d J = {2} for f i x e d
J
<
i s a s u b s e t , a l t h o u g h n o t g e n e r a l l y a s u b g r o u p of 5 . F o r e x a m p l e , for we h a v e Dj
e v e r y e l e m e n t it £ S
= {e, X j , i r 7 r } . O n e s h o w s n o w by i n d u c t i o n t h a t 1
n
2
h a s a u n i q u e d e c o m p o s i t i o n of t h e f o r m
w h e r e TTJ £ W r , - i v £ D j . F u r t h e r m o r e one has t h a t i(ir) the definition
gives the r e q u i r e d d e c o m p o s i t i o n P z
( n )
=
P
(
Z
N
)
( J )
•
R \j). z
n
b) c o m p a r e [ C a r , t h e o r e m 9.4.5] W e have JCS
/CS
*€#j
= i(itj)
TT =
+ i(irj).
TTJTTJ,
Then
70 S i n c e TT € Dj
is e q u i v a l e n t t o 7t(k
J = {kj(l),
+ 1) for a l i i = 1 , . . . , O ,
< 7r(k
where
we get
kj(rj)},
JCS
*es„ *(kj(i))<#(M0+i)
•= 1
=
£
*GS„
(
TJ
E
(-i)
JCS
| J |
)^*^-
» ( M 0 ) < * ( M 0 + i ) •=1,...,TJ
N o w define I * := { i | 7r(i) < Tr(i + 1)}. as YIJCI- (~l)' 'i 7r =
7r
0
we get
S =
c) A s s u m e 7r = 0
ro
with f j
WJWJJ
€ Wj,
of a
TJ- g W J - a n d J,J'
j =
with 1 <
k
TJ
we c o u l d pass to 7r =
XJ/TTJ
0
C S. T h e <
assumption
n such that i <
a n d take into account that k
1
irj(i)
7 r ( i ) = (7rj7Tj.)(i) = irj'(irj(i)) 0
B u t t h i s is a c o n t r a d i c t i o n to 7To(l) = = P[
n)
• a = p[
n)
=
x
a s s u m e kj < kj>. T h e n we h a v e for i < kj:
all a g S ,
kj
W i t h o u t a n y r e s t r i c t i o n we c a n a s s u m e kj < kjr.
< kj.
d) W e h a v e
S i n c e i * = 0 o n l y for
z'(' ^7ro.
J C 5 i m p l i e s the e x i s t e n c e implies
T h e n t h e s u m i n t h e b r a c k e t s is t h e s a m e
w h i c h is one i f J * = 0 a n d z e r o o t h e r w i s e .
J
< kj < kji, < kj' <
kj
Otherwise So let us
k^-i.
which yields
n.
n. a n d
n)
=
•a = ( - l f ^ P ^
for
which implies
n
p(n) . ( n ) p
=
p
(
n )
. („) p
=
J2
(
2
>V))
. W P
=
JJ 1 = 11
z
i + 1
. p(")
e£S„
and
P
(").
P
w
=
?
pt»)
w .
= (
•p ^ = n
j 2 (-tf'i)
1
1 ~- ( - *~) • ' • 1
(
z
)
,
+
1
PQ.
"
"€S„
0 PROPOSITION
z<jtT P R O O F :
1. M
T i e element
P
z
:= {z g C | z (
a) z £ 7 > >
k
=> p[
n)
( n )
h + 1 )
g C S „ i s i n v e r t i b J e H and only = 1
for
if
s o m e lb g { 1 , . . . , n -
is i n v e r t i b l e
W e p r o v e the a s s e r t i o n b y i n d u c t i o n o n n. F o r n = 1 ( P i ^ = e, ( P i ' ) 1
always) and n = 2 ( P
z
( 2 )
1}}.
= e + Z5T , ( P i ^ ) " t
1
1
= e exists
- 1
= j ^ j f e - z w i ) exists if z
2
/
1) t h e
a s s e r t i o n is t r u e . N o w a s s u m e i t i s t r u e for m < n — 1 a n d c o n s i d e r n . L e t z g
C
71 with z £ T W .
T h i s implies z 0 T< )
for a l l m < n.
m
W j i s i s o m o r p h i c t o a p r o d u c t S/
For J C 5 := { l , . . . , n -
m a p p e d o n t o t h e p r o d u c t of t h e c o r r e s p o n d i n g
. S i n c e fc; < n, a l l P ^ * ' , a n d 1
h e n c e a l s o t h e i r p r o d u c t P i ^ ( J ) , a r e i n v e r t i b l e . T h u s we h a v e by l e m m a
la)
n
n)
•P
By
1
2
is
n
tr
R{ (J) = P^W
1},
U n d e r t h i s i s o m o r p h i s m Pi \j)
x • • • x S/ .
Sl
for all J C 5 .
( n )
p a r t b) of l e m m a 1, t h i s i m p l i e s 1
|J|
(
)
1
(£(- ) ^ " (-0~ K
(n)
i(To)
= z *o -
(-l) 4 >(S) | s |
n
JCS =
w h i c h y i e l d s ( n o t e TTOTO. = (z^-Vo +
z
^o)
7 r o
_(_l)|S|
e j
e)
(-l)l le)(£(-l)l lp(")(J)- )Pi") s
J
1
JCS
= ( Z ' M T , + (-l)l l )(z (' ->Vo s
( 2i('») _
=
S i n c e 2 i ( i r ) = n(n
the e x p r e s s i o n z ^ * )
1 is different
from
is i n v e r t i b l e , w h e r e IJ-^Z^-'TTO +
U -
z
-
0
n
(P (»))"i =
e)
| S |
e
— 1) a n d z g T^ \
0
(-l)
r
i) .
z
zero, w h i c h implies that P *
,
e
(-l) - e)(^(-l)l lpi' >(J)- ). n
1
J
l
1
JCS
b) P , Let
P
( n )
2
is i n v e r t i b l e => z £ T ( '
( n )
n
be i n v e r t i b l e . Hence, by P .
( n )
= Pz
( n )
( J ) • i t i ( J ) > aU P i n )
n )
( J ) for J
C
5 : = { 1 , . . . , n — 1} a r e i n v e r t i b l e , t o o . T h u s , as i n p a r t a ) , we get (£ _i)mpM(j)-i)p(»)
=
(
z
*(^)
T o
_
(-i)i i . s
e
JCS
N o w assume z € Z * * ) , i.e.
there exists a k € { l , . . . , n -
1} w i t h z ^ *
- 1
) =
W i t h o u t r e s t r i c t i o n w e c a n a s s u m e k = n — 1, i . e . z ^ " " " ) = 1. T h i s i m p l i e s 1
(-i)i i )(£(-i)
(***>*b +
5
e
| / |
1
Pi" (^)" )^ ' )
1
(
l )
=
JCS
a n d b y t h e i n v e r t i b i l i t y o f P^™* (
^ o )
T
o
+
(
_
1 )
|5|
e )
(^ _ (
JCS
1 )
Ulp(n)
( J )
-l)
=
0.
o
1.
72 W e w i l l s h o w t h a t t h i s is n o t p o s s i b l e . W e h a v e o f c o u r s e t h a t P ^ j y and
1
e
CWj
thus £ ( - l ) l l p ^ ( J ) J
where W
:= U ; s ¥ j . c
= : w € CW
1
Note that W
:= { £
|a . 6
aw w
C},
i s o n l y a s m a l l p a r t of S , for e x a m p l e , for n
n = 3 we have W
= {e,^,^} C S
I n d e e d , we h a v e ir such that TTJ/ =
=
3
0
•W n W
=
{e,-K ,TT ,^1^2,^2^1,^1^2^l}1
0, i . e .
2
t h e r e e x i s t n o e l e m e n t T T J - i n s o m e WJI
7Tn7T/ for 7Tj i n s o m e W j w i t h J , J ' C S.
T h i s is only another
f o r m u l a t i o n of l e m m a l c ) . F r o m t h i s f o l l o w s t h a t o u r a b o v e e q u a l i t y (T ±e)w 0
= 0
c a n o n l y h o l d for w = 0, w h i c h w o u l d y i e l d z ( °^7r — ( — l ) ' ' e = 0. T h u s we a r r i v e I
7r
5
0
at the desired c o n t r a d i c t i o n a n d c a n conclude t h a t z ^
T* K n
0 R E M A R K S :
1) T h e p r o p o s i t i o n i m p l i e s t h a t Pj?^
Since P Q " ^ = i d and Pji
h a s n o k e r n e l for /x £
(—1,1).
d e p e n d s c o n t i n u o u s l y o n jx, we get a n o t h e r p r o o f for the
s t r i c t p o s i t i v i t y o f P^™^ i n t h e i n t e r v a l \i G ( — 1 , 1 ) . 2) B y l e m m a I d ) , s y m m e t r i z e d a n d a n t i - s y m m e t r i z e d v e c t o r s a r e a l w a y s eigenvect o r s o f P i " ^ a n d a r e c a n o n i c c a n d i d a t e s for t h e k e r n e l o f P i " ^ i n t h e case z € B u t t h e k e r n e l i n t h i s case is m u c h b i g g e r a n d w e h a v e n o t y e t b e e n a b l e t o d e r i v e its exact f o r m .
3.
D E T E R M I N A T I O N OF
T H E G A U S S I A N
DISTRIBUTION
N o w we c o m e b a c k t o fj, £ (—1,1) a n d w a n t t o f i n d t h e s p e c t r a l m e a s u r e of c(/)
+ c (/) +
w i t h respect
to the v a c u u m expectation
state, i.e.
the G a u s s i a n
d i s t r i b u t i o n b e l o n g i n g t o o u r B r o w n i a n m o t i o n . W e w i l l r e s t r i c t t o t h e case ||/|| = 1, w h i c h c o r r e s p o n d s t o v a r i a n c e = l . F o r t h i s , t h e w h o l e o f o u r r e l a t i o n s d o e s n o t p l a y a n y r o l e , we a r e o n l y d e a l i n g w i t h one o p e r a t o r c : = c(f)
and its adjoint c cc
+
— fic c
+
fulfilling the relation
= 1.
+
O p e r a t o r s f u l f i l l i n g t h i s r e l a t i o n ( w h e r e u s u a l l y q is u s e d i n s t e a d of u) h a v e a l s o o c c u r e d i n c o n n e c t i o n w i t h t h e t w i s t e d SU (2) q
under the n a m e of 'g-analogue
of
h a r m o n i c o s c i l l a t o r ' (see for e x a m p l e [ M a c , B i e , C K L , V a S ] ) a n d s o m e of t h e f o l l o w i n g c o n s i d e r a t i o n s h a v e a p p e a r e d t h e r e i n a m o r e o r less e x p l i c i t f o r m . In o r d e r to f i n d the s p e c t r a l m e a s u r e of c + c expectation
+
w i t h respect
to the
s t a t e we s h a l l m a k e a c o n n e c t i o n o f t h i s o p e r a t o r w i t h
p o l y n o m i a l s a n d t h e n l o o k u p t h e c o r r e s p o n d i n g m e a s u r e i n [Asl]
vacuum
orthogonal
73 L e t us denote by p
= /
:= (c ) U +
n
n
0
t h e n - p a r t i c l e s t a t e , t h e n we h a v e
n
1 — u — — P n - U 1 - fi n
(c + c )p +
+(1
=
n
Pn+1
+ n + ••• + / i
where we agree u p o n p _ i = 0 . of o u r s t a t e s , n a m e l y (n >
n _ 1
n
1
=p„+i +
I n o r d e r t o c o m e c l o s e r to [ A s l ] we m a k e a c h a n g e
0) *
1
i=0
In t e r m s o f t h e q
)p _
**
t h e a b o v e r e l a t i o n reads as
n
(c + c + ) q „ = (1 -
u
n + 1
)q
+
n + 1
N o t e t h a t t h e s p e c t r a l m e a s u r e of c + c
—^—q -i, u n
1 -
w i t h respect
+
to the s t a t e q n
for n ^ m a n d t h e of t h e f o r m
<
fixing
qo,(c
q_! =
+ c)q +
k
0.
>
0
=
0
d e t e r m i n e d b y t h i s r e l a t i o n t o g e t h e r w i t h t h e fact t h a t q
and q
m
Q is
are o r t h o g o n a l
I n d e e d , this a l l o w s to c a l c u l a t e a l l m o m e n t s
a n d since c + c
+
is a b o u n d e d
operator,
this
d e t e r m i n e s t h e m e a s u r e u n i q u e l y . B u t the a b o v e r e c u r s i o n f o r m u l a is e x a c t l y t h e s e t t i n g o f t h e t h e o r y of o r t h o g o n a l p o l y n o m i a l s . W e o n l y h a v e t o t h i n k of the as o r t h o g o n a l p o l y n o m i a l s Q (x)
the c o n s t a n t f u n c t i o n 1 a n d of c + c
+
c o r r e s p o n d i n g H i l b e r t s p a c e L (IR,
v).
2
c + c . +
n
as t h e m u l t i p l i c a t i o n o p e r a t o r w i t h x i n t h e T h e n v w o u l d b e t h e s p e c t r a l m e a s u r e of
T h u s we h a v e t o d e a l w i t h the r e c u r s i o n f o r m u l a
xQ (x) n
These
q
w i t h respect t o s o m e m e a s u r e v o n IR, of fl as
n
= (1 - n )Q (x)
orthogonal
n+1
+ - i - Q 1— P
n+1
polynomials
((?_!
- i ( x )
n
are well k n o w n .
They
= 0, Q
1
=
1).
are c a l l e d c o n t i n u o u s
5-
H e r m i t e p o l y n o m i a l s , w h i c h c a n be c o n s i d e r e d o n one side as a s p e c i a l case of t h e A l - S a l a m - C h i h a r a p o l y n o m i a l s ( w h i c h are t h e m s e l v e s q-Wilson
ultraspherical polynomials. s p e c t r a l measure of c + c is g i v e n b y dv(x)
A*)
a s p e c i a l c a s e of
p o l y n o m i a l s ) , o n t h e o t h e r side as a s p e c i a l case of the c o n t i n u o u s +
/
V
w i t h density
^ - ) ( * )
•
±
^ r
L
s
i
n
*
n ^
1
-
where - cos0. F o r fi = 0 t h i s r e d u c e s
q-
T h e corresponding measure v - w h i c h is the wanted
- c a n be r e a d off f r o m f o r m u l a 3.30 or 3.31 i n [Asl] a n d
= u'(x)dx
= X ( - 3 / V T O
the
to
^(x)
=
X(-2.2)(^)^V l-(»/2) , /
2
-
e-
2
V I
2
} ,
74 w h i c h i s - as we a l r e a d y k n o w ( [ V o i , S p e ] ) - t h e d i s t r i b u t i o n o f / + I*, n a m e l y W i g n e r ' s s e m i c i r c l e d i s t r i b u t i o n . (It s h o u l d be r e m a r k e d t h a t t h e c o r r e s p o n d i n g f o r m u l a i n [Spe] i s n o t q u i t e c o r r e c t . ) F o r fi - » 1, t h e d e n s i t y v'(x)
tends to l / v 2 i r e x p ( - i / 2 ) , i.e. i n this l i m i t we /
2
a l s o get t h e r i g h t d i s t r i b u t i o n , n a m e l y t h e G a u s s i a n o n e . F o r n —• - 1 , t h e m e a s u r e v t e n d s t o l / 2 ( 5 _ b + b, +
+ £x), i.e. to the d i s t r i b u t i o n of
t
where 6 is the fermionic annihilation operator.
I n t h e a p p e n d i x we h a v e p l o t t e d t h e f u n c t i o n v'(x)
f o r different v a l u e s o f q = ji
i n o r d e r t o see t h e i n t e r p o l a t i o n b e t w e e n t h e f e r m i o n i c , t h e free a n d t h e b o s o n i c case.
4.
A N O T H E R
R E P R E S E N T A T I O N O F O U R RELATIONS
W o r k i n g i n the Fock space of a one-dimensional H i l b e r t space (i.e. considering o n l y o n e / / - o s c i l l a t o r c c — / / c c = 1) o n e w o u l d define t h e c r e a t i o n a n d a n n i h i l a t i o n +
+
operators by the following ladder
U _ _ J i S i n c e t h i s a l l o w s n o c a n o n i c e x t e n s i o n t o t h e case o f m o r e / / - o s c i l l a t o r s , w e c h a n g e d this p i c t u r e to the equivalent one
1 l~]
1 + // + / i
2
1 I T1+ M 1 i _ J l Instead of this we could also have used the l a d d e r
1 + / / + //
2
Q
1 + pt i_J
i 1
U _ J i H e r e we want to consider the question h o w the canonic extension of this l a d d e r ( d e s c r i b e d b y o p e r a t o r s d(f))
i s r e l a t e d to t h e o n e u s e d t i l l n o w ( d e s c r i b e d b y c ( / ) ) .
W e s h a U see t h a t t h e c o n n e c t i o n i s v e r y t r i v i a l , n a m e l y g i v e n b y a d j u n g a t i o n i n the full F o c k space. L e t u s r e c a l l t h e s e t t i n g o f o u r u s u a l p i c t u r e : W e h a v e i n t h e f u l l F o c k s p a c e !F o p e r a t o r s c ( / ) , c + ( / ) , s u c h t h a t / H-> C + ( / ) i s l i n e a r a n d / K * c(f) T h e y fulfill the relations c(f)c (g)-pc ( )c(f)=
+
9
is anti-linear.
75 a n d c(f)
a n d c (f)
c a n be made to adjoints of one a n o t h e i by choosing a new
+
s c a l a r p r o d u c t < 77, { > : = < 7>,P„£ > , w h e r e
is a strictly positive operator for
M
u £ ( - 1 , 1 ) . T h e a d j o i n t n e s s o f c(f)
a n d c + ( / ) is equivalent to c ( / ) * P
M
= P c+(/). < 1
T h e s y m b o l * denotes here a d j u n g a t i o n i n the full Fock space w i t h respect to t h e u s u a l s c a l a r p r o d u c t < , > . F u r t h e r m o r e c + ( / ) = l*(f)
w a s d e f i n e d as t h e left
creation operator. N o w w e c a n e a s i l y d e s c r i b e t h e o p e r a t o r s d(f)
belonging to the third ladder
above. W e define := c
d(f)
+
( / r = 1(f)
«,+(/) := c(/)*. M o r e e x p l i c i t e l y , d(f)
1(f) i s t h e left a n n i h i l a t i o n o p e r a t o r a n d d (f)
=
is a
+
' t w i s t e d ' left c r e a t i o n o p e r a t o r g i v e n b y d (f)h +
® • • • ® h
l
= f ® hi ® • • • ® h
n
+ phi ® f ® h ® • • • ® h + • • • +
n
+ / i "
_
2
n i ® • • • ®
1
® /
® / i
T h e n e x t l e m m a s h o w s t h a t t h e o p e r a t o r s d(f),d (f)
n
n
+ M / i i ® • • •® h n
n
® /.
b e h a v e e x a c t l y as t h e o p e -
+
rators c ( / ) , c + ( / ) . 2 . a ) The operators
L E M M A
c(f),c+(g),
namely
(f,g
d(f),d (g)
The operators a scalar
c) If we denote a bounded
< , >
measures
PROOF:
11
with
M
to adjoints
respect
to < , >
by
as c(f)
on T^.
M
and has the same norm are the
to the vacuum
by
1
1
of T with
of one another
< 77, £ >**:=< 77, P " ^ > .
the help of P'' :
with respect
+
for
can be made
+
a n d of d(f) + d (f)
+
b)
and d (f)
the completion
d) The spectral c (/)
+
d(f)
on T
operators
=< f,g > 1.
- pd (g)d(f)
product
operator
as the
2
+
b)
the same relations
L (IR))
d(f)d (g)
choosing
MM
+
e
expectation
, then d(f)
is
state of c(f)
+
same.
a) T h i s follows b y a d j u n g a t i o n ( w i t h respect t o < , > ) of the equation c(f),c (g). +
W e have
dlfyP'
1
c(/)*P„
=
P c+(/),
i.e.
M
d (/)P 4
M
=
which implies
P^d(f)*,
= P 7 d + ( / ) . T h i s i s e q u i v a l e n t t o t h e a s s e r t i o n . N o t e t h a t P' /
1
1
for p. e ( - 1 , 1 ) c) C o n s i d e r 77 € T a n d p u t £ : = P~ Tj.
T h e n we h a v e
l
< 77,77 > " = < n,P^n
> = < P^VtP^n
>=<
and < d (f)n,d (f) +
+
n
>" =< d+WP^P^d+iftP^
>
=
l
=<
ll
c (m,p^(f)i> +
=
+
>„.
exists
76 T h i s implies the assertion. d ) W e o n l y have t o n o t i c e t h a t t h e s p e c t r a l m e a s u r e i s u n i q u e l y d e t e r m i n e d b y d(f)Q
= 0 a n d the relation d(f)d (f)-pd (f)d(f) +
=<
+
1.
f,f>
References A s k e y R . , I s m a i l M . : Recurrence
relations,
continued
fractions
and
ortho-
M e m o i r s of the A M S , v o l u m e 49, n u m b e r 300, 1984
gonal polynomials.
B i e d e n h a r n L . C . : T h e q u a n t u m g r o u p SU (2) q
a n d a q-analogue of the boson
o p e r a t o r s . J o u r n . o f P h y s i c s A 22 ( 1 9 8 9 ) , L 8 7 3 Bozejko M . , Speicher R . : A n E x a m p l e of a G e n e r a l i z e d B r o w n i a n M o t i o n . C o m m u n . M a t h . P h y s . 1 3 7 , 519-531 ( 1 9 9 1 ) C a r t e r R . W . : Simple
groups
Chaichian M . , Kulish
of Lie type. J o h n W i l e y & S o n s , 1972
P . , L u k i e r s k i J . : 5 - d e f o r m e d J a c o b i i d e n t i t y , q-
oscillators a n d g-deformed infinite-dimensional algebras.
Phys.
Letters B
237 ( 1 9 9 0 ) , 4 0 1 - 4 0 6 F i v e l D.I.: Interpolation between Fermi a n d Bose Statistics U s i n g Generalized Commutators. Greenberg
P h y s . R e v . L e t t . 65, 3361-3364 (1990)
O . W . : (J-mutators a n d violations of statistics.
U n i v e r s i t y of
M a r y l a n d P r e p r i n t 9 1 - 0 3 4 , 1990 Macfarlane A . J . :
O n g-analogues of the q u a n t u m h a r m o n i c oscillator a n d
t h e q u a n t u m g r o u p SU(2) . q
J o u r n . of P h y s i c s A 22 ( 1 9 8 9 ) , 4581
Speicher R . : A N e w E x a m p l e of 'Independence' a n d ' W h i t e N o i s e ' .
Probab.
T h . R e l . F i e l d s 84 ( 1 9 9 0 ) , 141-159 Vaksman L . L . , Soibel'man g r o u p SU(2).
S.Ya.:
A l g e b r a of functions
on the quantum
Translation from Funktsional'nyi A n a l i z i E g o Prilozheniya
V o l . 2 2 , N o . 3 ( 1 9 8 8 ) , 1-14 Voiculescn
D.:
I n Operator Theory,
Symmetries
Algebras
of some reduced
and their
Busteni, Romania,
Connection
free p r o d u c t with
Topology
C*-algebras. and
Ergodic
1983, L e c t u r e Notes i n M a t h e m a t i c s 1132,
S p r i n g e r V e r l a g , H e i d e l b e r g 1985
Instytut Matematyczny,
Uniwersytet
Wroclawski, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Po-
land Institut fur Angewandte Mathematik, Universitat Heidelberg, Im Neuenheimer Feld 294, W-6900 Heidelberg, Federal Republic of Germany
77
Q u a n t u m P r o b a b i l i t y a n d R e l a t e d Topics V o l . V I I (pp. 79-91) ©
1992
79
W o r l d Scientific P u b l i s h i n g C o m p a n y
M i n i m a l
s o l u t i o n s
i n classical a n d
q u a n t u m
stochastics
A . M . Chebotarev Moscow USSR
Institute
Applied
1.
for Electronic
109028 Moscow,
Engineering
B.Vusovski
Mathematics
,
S/12 ,
Department
T h e i d e a o f m i n i m a l s o l u t i o n a p p e a r e d o r i g i n a l l y i n t h e t h e o r y of g e n e r a l i z e d
P o i s s o n processes [1], w h e r e a t r a n s i t i o n p r o b a b i l i t y is d e s c r i b e d b y a K o l m o g o r o v F e l l e r e q u a t i o n . I n t h e s i m p l e s t case t h i s e q u a t i o n l o o k s as follows j P{x, t
t |r, 0) = j
* ) | ( P ( x + q, t\T, 0) - P(x,t\F,
p(dq)\C(q,
0)),
2
(1.1
w h e r e P\t=o = lr(«r)i -fr is t h e i n d i c a t o r f u n c t i o n of a set T 6 S , (X, E ) is t h e m e a s u r a b l e p h a s e s p a c e of a process.
P a r a m e t e r s of t h i s process are a n i n t e n s i t y
k of j u m p s a n d a d i s t r i b u t i o n m o f j u m p s f r o m a p o i n t x £ X t o a set B £ E : k(x)
= J n{dq)\C{q,x)\ ,
m(B\x)
2
= J n(dq)\C(q,
X
x)\ /k(x). 2
B
T h e M a r k o v j u m p p r o c e s s is c a l l e d regular
i f i t p r o d u c e s a finite n u m b e r of j u m p s
o n a n y f i n i t e i n t e r v a l w i t h p r o b a b i l i t y one. T h e K o l m o g o r o v - F e l l e r e q u a t i o n c a n be t r a n s f o r m e d to a n i n t e g r a l f o r m w h i c h is e q u i v a l e n t t o (1.1) o n t h e set S+ of p o s i t i v e b o u n d e d s o l u t i o n s w h i c h are m e a surable w i t h respect to x a n d t P(x,t\T,0)
= +
exp(-k(x)t)I (x)+ T
f / cir e x p ( - / f c ( z ) ( t - r ) ) f c ( x )
0
exp(-k(x)t)Ir{x),
m{dq\x)P(x,T\T,0)
(1.2)
x
o If o n e t a k e s as P (x,t\T,Q)
r
t h e s o l u t i o n of t h e d i s s i p a t i v e p a r t of e q u a t i o n (1.1)
t h e n t h e sequence
P „ ( x , t | r , 0 ) = Po(x,<|r,0) + ^
dTe W- h(x) k
T
J^midq^P^ix
+
q,T\T,0)
80 w i l l be p o s i t i v e , b o u n d e d a n d m o n o t o n e :
0 < P „ < P „ + i < 1. H e n c e t h e r e exist
t h e least u p p e r b o u n d P ( x , t | r , 0 ) = l . u . b . P ( x , t | r , 0 ) b e i n g a s o l u t i o n of n
f r o m S + . If t h e r e e x i s t a n o t h e r s o l u t i o n P o f e q u a t i o n (1.2) f r o m S+, T h e r e f o r e P is c a l l e d the minimal conservative
(1.2)
then P < P .
A s o l u t i o n o f (1.1) o r (1.2) is c a l l e d
solution.
i f i t preserves t h e t o t a l p r o b a b i l i t y : P(x,t\X,0)
= 1. O n e c a n e a s i l y
p r o v e t h a t t h e c o n s e r v a t i v e m i n i m a l s o l u t i o n is u n i q u e i n t h e class o f c o n s e r v a t i v e solutions f r o m S
+
a n d it is also a c r - a d d i t i v e p r o b a b i l i t y m e a s u r e o n E . N e c e s s a r y
a n d sufficient c o n d i t i o n s for c o n s e r v a t i v i t y o f t h e m i n i m a l s o l u t i o n w e r e o b t a i n e d i n [2],[3]. W e m e n t i o n here t h e c o n s e r v a t i v i t y c o n d i t i o n i n o p e r a t o r f o r m :
l i m Q " ( l ) ( x ) = 0,
»-»oo
Q(f)(x)
E x a m p l e s and counterexamples
=
1
1 +
1/K(X)
/ m(dq\x)f(x J
+
q).
X
[3].
(a) F o r h o m o g e n e o u s C a u c h y d i s t r i b u t i o n of t h e j u m p s
m(B) = »
-
/ JB
Hil j
+
qh-'^q
t h e i n t e n s i t y of t h e j u m p s fc(x) = ( l + | x | ) ] n ( 2 + | « | ) generates t h e r e g u l a r M a r k o v j u m p process. R e g u l a r i t y is v i o l a t e d i f Jfc(x) = (1 + \x\) l n
1 + a
( 2 + \x\), a > 0.
(b) F o r h o m o g e n e o u s n o r m a l d i s t r i b u t i o n o f t h e j u m p s t h e i n t e n s i t y fc(z)
= ( l + |x| )ln(2 + |x|) 2
generates t h e r e g u l a r M a r k o v j u m p process. R e g u l a r i t y is v i o l a t e d i f k(x)
= (1 + | x | ) l n 2
1 + a
( 2 + |*|), a > 0.
F o r a p p l i c a t i o n s of r e g u l a r j u m p processes t o t h e P a u l i a n d D i r a c e q u a t i o n s
see
[9]-
2 . L e t H = L,2{X,do-)
b e a H i l b e r t s p a c e of s q u a r e i n t e g r a b l e f u n c t i o n s o n
w i t h respect t o a a - f i n i t e m e a s u r e a a n d let B(H)
X
be the v o n N e u m a n n algebra
81 o f a l l b o u n d e d l i n e a r o p e r a t o r s i n H.
T o each t r a n s i t i o n p r o b a b i l i t y corresponds
a b o u n d e d l i n e a r m u l t i p l i c a t i o n o p e r a t o r P (Ir)
€ B(H)
t
Pt(IrMx)
= P(x,t\r,0)
:
\\P (Ir)\\ <
I.
t
T h i s o p e r a t o r - v a l u e d f a m i l y satisfies t h e L i n d b l a d e q u a t i o n (see (^j ~CjP t
w h e r e AoB
= 0,
t
= 1/2(AB+BA),
m a p p i n g o n B(H).
CL(B)
= 9(B)-9(1)
°B
[4]) (2.1)
+ i[H,B],
a n d $(•) is a c o m p l e t e l y p o s i t i v e
[H, B] = HB-BH
I n t h e p a r t i c u l a r case of t h e K o l m o g o r o v - F e l l e r e q u a t i o n we
have
9(B) W
= J
p(dq)C(q,
= ^
-yD;BD C(q,
+ «),
D,1>(x)
9(1)
•),
t
= i,(x-q),
k(-),
=
H = 0,
P\
=
i=0
b u t i n t h e g e n e r a l case we c a n t a k e a n a r b i t r a r y i n i t i a l v a l u e f r o m T h e i n t e g r a l f o r m o f e q u a t i o n (2.1) similar to
= W 'BW
t
t
= exp(-Gt),
t
=
t
r
B(T-l). B
€ B(H)
is
(1.2)
P (B)
where W
w i t h i n i t i a l v a l u e P\ =o
I (-),
+
t
/' W *_ 9(P (B))W ^ Jo t
T
T
t
H e r e W*BW
G = l/29(I)+iH.
(2.2)
dr,
T
= P
t
t
( 0 )
( P ) is t h e s o l u t i o n
of d i s s i p a t i v e p a r t o f (2.1) a n d e q u a t i o n (2.2) a p p e a r s as a D u h a m e l e q u a t i o n . T h e sequence P
t
( n )
(P) = P
(
( 0 )
(P) +
/ Jo
W*_ 9(Pl - \B))W ^ dr n 1
T
t
T
has t h e s a m e p r o p e r t i e s as i n a b e l i a n case b o t h for b o u n d e d a n d u n b o u n d e d pletely positive mappings 0 < P
t
( n )
com-
9(-):
(B) < P
(
(5) <
( n + 1 )
H e n c e t h e r e e x i s t t h e s o l u t i o n P (B)
=
t
• /
l.u.b.P,
( n )
VP S
B(H)+.
( P ) of (2.1)
a n d (2.2)
which
is m i n i m a l i n t h e class o f b o u n d e d c o m p l e t e l y p o s i t i v e a n d s t r o n g l y c o n t i n u o u s solutions of (2.1). I n a s i m i l a r w a y one c a n c o n s t r u c t a c o m p l e t e l y p o s i t i v e e x t e n s i o n of t r a n s i t i o n p r o b a b i l i t y for a g e n e r a l M a r k o v p r o c e s s i n R L = i ( A ( x ) V , V ) + (&(*), V ) + 2
/ JR"\{O}
n
w i t h infinitesimal operator
p(dq)\C(q,
x)\
2
(D*
- l - ( q , V)) .
82 T h e L i n d b l a d e x t e n s i o n o f L a c t s as m a p p i n g £ ( • ) w i t h u n b o u n d e d o p e r a t o r - v a l u e d coefficients d e n s e l y d e f i n e d o n B(H):
*(J3) =
H = o
*„(J9)=
-£Hj, o
/
/
i(dj)C(flf o*(i?;--rW,-/)C( ,.), >
g
•>0<|g|
*x(B) =
/
A*(d9)C(ff, -TDlBDaCiq,
* ( f l ) = - ( V , AV'BAV'V),
H
a
Jf„ =
/ Kdq)C*(q, Jo<|j|
1
•),
=
i ( ( 6 , V ) + ( V , 6)),
•) f s i n 1 / i ( V , ?) - l / i ( V , ? ) ) C ( g , •) V /
I n o r d e r t o t r e a t e q u a t i o n s ( 2 . 1 ) , (2.2) w i t h u n b o u n d e d c o e f f i c i e n t s , w e h a v e t o i n t r o d u c e some notations a n d definitions. The
l o w e r i n d e x * w i l l d e n o t e a b i l i n e a r f o r m . T h e b r a c k e t s [...]
will be.used
i n t h r e e different w a y s . It w i l l d e n o t e t h e l i s t o f t h e a r g u m e n t s o f a b i l i n e a r f o r m . It w i l l a l s o d e n o t e c o m m u t a t o r o f o p e r a t o r s a n d a s e l f - a d j o i n t o p e r a t o r w h i c h corresponds t o a closed densely defined bilinear f o r m w h i c h is b o u n d e d f r o m below
[5]. W e d e n o t e b y CPn(Ti.)
t h e set o f a l l n o r m a l c o m p l e t e l y p o s i t i v e l i n e a r b o u n d e d
m a p p i n g s P : B(H) —+ B(H) a n d b y C P n * ( 7 - t ) t h e c o r r e s p o n d i n g set o f t h e m a p p i n g s P„ : BCH) —t B (H)
o f t h e v o n N e u m a n n a l g e b r a B(7i) i n t o t h e set o f b i l i n e a r
t
forms P,{B)[M] For
= (4>,P(B)rP),
P(-) €
CPn(H).
a g i v e n t o p o l o g i c a l v e c t o r space T w h i c h i s dense i n H w e d e n o t e b y
CPn,(H,T)
t h e c o m p l e t i o n o f CPn„(7i)
i n locally convex topology of u n i f o r m
c o n v e r g e n c e o n t h e c o m p a c t sets w h i c h i s d e s c r i b e d b y s e m i n o r m s
w h e r e A a n d B a r e a b s o l u t e l y c o n v e x a n d c o m p a c t sets i n B(T-l)
and i n T . In
w h a t f o l l o w s t h e coefficient $(•) o f t h e L i n d b l a d m a p p i n g £ ( • ) w i l l c o r r e s p o n d t o an element f r o m Let
CPn (H,T). t
u s s u p p o s e t h a t t h e r e e x i s t a p o s i t i v e s e l f a d j o i n t o p e r a t o r 4> > I a n d a
closed densely defined s y m m e t r i c operator H w h i c h characterize t h e i n f i n i t e s i m a l
83 operator — G of a strongly continuous one parameter contractive semigroup W
:
t
contractions W
= exp(-Gt),
t
W> 6 D(H)
Gifi = iH4> +
n £>($),
2 ess d o m G C D(H)
(2.3)
n Z?(*)
w h e r e ess d o m G i s a n y e s s e n t i a l d o m a i n o f t h e c l o s e d o p e r a t o r G . M o r e o f t h i s , let C J J p r ) n D f » C D(G),
D(G ) 2
O u r last a s s u m p t i o n requires t h a t W
t
DjG )
is b o u n d e d
(2.4)
= H.
2
a n d continuous i n T .
This
a s s u m p t i o n i s f u l f i l l e d i f T is a B a n a c h s p a c e s u c h t h a t
l M l r =
+ W i l l i
>
U n d e r t h i s a s s u m p t i o n $ , 6 CPn,(H,T) of c o m p l e t e l y p o s i t i v e b o u n d e d
W>GZ>(G).
(2.5)
if $ = $ ( / ) = [$,(/)] a n dif the family
mappings
*«(•) = e x p { - e $ ( / ) } $ ( - ) e x p { - e $ ( / ) }
G GPn(7Y)
is n o r m a l f o r a l l e G ( 0 , 1 ] . L e t t h e a s s u m p t i o n s ( 2 . 3 ) , ( 2 . 4 ) , (2.5) h o l d a n d $ , G CPn.
T h e n there
exist t h e s e q u e n c e o f n o r m a l c o m p l e t e l y p o s i t i v e a n d s t r o n g l y c o n t i n u o u s o n
R
+
mappings such that
P P
w h e r e
t
PI"\B)
(
t
( 0 )
( P ) = W'BWt,
( n )
=
P
t
( n )
(B) = [P
( B ) . [ ^ ] = (*»**) + / Jo
( n )
(P),] ,
^*.(P.
( B
-
1 )
(B))[0,_„^_.],
W y^; , vb G /-?(G) (see [5]).
W
(
t
t
T h e family of operators
i s m o n o t o n e a n d b o u n d e d , h e n c e t h e least u p p e r b o u n d f o r
P$ \B), N
B G B(H) defines t h e m i n i m a l s o l u t i o n o f i n t e g r a l L i n d b l a d e q u a t i o n
P (B) t
= W *BW t
t
+
f
dTW *_ $,(p (B))w t
T
T
t T
Jo H e r e t h e b r a c k e t s [...]
denote the correspondence
between
a set o f p o s i t i v e
d e n s e l y d e f i n e d a n d u n i f o r m l y b o u n d e d b i l i n e a r f o r m s o n D(G) a n d a set o f p o s i t i v e bounded
operators.
T h e m i n i m a l s o l u t i o n P i s c a l l e d conservative t
i f P < ( / ) = / V t > 0.
84 T h e o r e m 2 . 1 . [6] Let the assumptions Then (1)
the following
The minimal namical
(2)
statements
solution
are
(2.3) -(2.5) hold and $ » ( • ) 6 CPn»(H,
of the integral
n
Lindblad
is a c o n s e r v a t i v e dy-
equation
semigroup; —• 0 as n —• oo w h e r e Q(X)
Q (I)
T).
equivalent:
=
[Q.(X)]
/•oo Q.(X)[4>,ip]=
c-*dt*,(X)\w 4>,Wt1>];
/
t
Jo (3)
Equation
Q(X)
(4)
Equation
C*(X)
= X has no solutions
T h e o r e m 2 . 2 . L e t the assumptions dynamical (1)
semigroup
The infinitesimal with
in
B(H)+;
= X * o n ess dom G has no solutions of the
be conservative. mapping
theorem
in
B(7f)+.
2.1 hold
and
the
minimal
P<(-)
coincides
Then
C of the minimal
dynamical
semigroup
C: ]im±(P (B)-B)-C(B))
= 0
1
in the strong •R. C B(H)
and in c-weak
topology
for any B from
the dense resolvent
set
: p OO
/ Jo
domC = 11TZ={B:B=
(2)
If C : dom C D A —• A for a given
P,(•) : A -
e
_ <
P ( A ) dt, A £
B(H)};
4
von Neumann
A C B(H),
subalgebra
then
A.
O u r last t h e o r e m i n t h i s s e c t i o n gives c o n d i t i o n s sufficient f o r c o n s e r v a t i v i t y of t h e m i n i m a l d y n a m i c a l s e m i g r o u p i f H is s e l f a d j o i n t o p e r a t o r . T h e o r e m 2 . 3 . Let the assumptions i{$ip,
Vr,£D(G ),Ve 2
where $
e
2.1 hold and
•£}>)} > - c ^ a r V )
Hip) — (Hip, 9tp)}
t{(*i7,*ij) -
of the theorem
(*($ ) e
1 / 2
> —c(ip, $ip)
r ,*(* ) ?
e
1 / 2
r )} > /
let
V
Vip G ess dom
G*
G
-c{n,$n)
6(0,1],
= $ ( / ) ( / 4- e $ ( 7 ) ) , - 2
t h e n the minimal
dynamical
semigroup
is
conser-
vative. If t h e m i n i m a l d y n a m i c a l s e m i g r o u p is c o n s e r v a t i v e , we w i l l c a l l regular
its
i n f i n i t e s i m a l m a p p i n g £(•)• S o m e e x a m p l e s of r e g u l a r m a p p i n g s w e r e d e s c r i b e d i n H
85 3.
L e t H b e a H i l b e r t s p a c e a n d T = J (L2(R+))
L2(R+)-
For h =
we w i l l d e n o t e b y B(h)
Ti ® T
b o u n d e d o p e r a t o r s i n h . L e t a > 0 a n d let S Li{R+)
b e a B o s o n F o c k space over
r
of simple functions / : R
W e will denote by E
a
von N e u m a n n algebra of a l l be a linear subspace i n
= S {R+) a
—» C, |/| < a t a k i n g a finite n u m b e r of v a l u e s .
+
a dense l i n e a r v e c t o r s u b s p a c e i n J- g e n e r a t e d b y finite sets
A
of e x p o n e n t i a l v e c t o r s {yj(f),f
£
S }. a
F o r a n y d e n s e s u b s p a c e s D C H a n d E C 'f L(D,
E) w i l l d e n o t e a n a l g e b r a i c
t e n s o r p r o d u c t D ® E C h , i.e. a dense l i n e a r s u b s p a c e i n h g e n e r a t e d b y a l l finite sets o f v e c t o r s h ® xp, h £ D,ip
6
E.
I n t h i s s e c t i o n w e w i l l c o n s i d e r t h e f o l l o w i n g l o c a l l y c o n v e x t o p o l o g i e s o n 13(h) i n d u c e d b y t h e s y s t e m s of s e m i n o r m s (see [7]): t o p o l o g y ; (c) (D, £ ) - s t r o n g p
M
(a) n o r m a l t o p o l o g y ;
(b) s t r o n g
topology induced by seminorms
(u)
= \\u(h® VOIIh
h£D,yb£E;
(d) i ? - l o c a l l y b o u n d e d t o p o l o g y i n d u c e d b y s e m i n o r m s
p^(u)
=
sup — \\h\\
,
C o m p l e t i o n of B(h)
B (H,E)
C
e
E.
w i t h r e s p e c t t o 7 5 - l o c a l l y b o u n d e d (or (D, 75)-strong) t o p o l -
ogy w i l l be d e n o t e d by S loc
v
\\mm
n
l o c
( 7 i , £ ) (or B'(D,E)).
It is easy t o see t h a t /5(h)
C
B (D,E). 3
T w o - p a r a m e t e r s i n t e r v a l - p a r a m e t r i z e d f a m i l y u[r, t) € B"(D,E ),
0 < r < t <
S
oo, is c a l l e d interval-adapted
if
u [ r , t ) h ® % * ) ( / ) e « ® . F ( X
U[T, t)h ® vb{f)
2
M ) )
V / I G D ,
v / e s
(3.1)
= (u[T, t)h ® V>[r,<)(/)) ® V>[0,r)(/) ® % « , ) ( / ) •
where W / )
=
W(-)l[r,,))e^2[T,t)).
T w o - p a r a m e t e r s i n t e r v a l - a d a p t e d f a m i l y is c a l l e d vacuum-family S[r, t)A ® *(/)
= (^wfr
if
® ^ [ r , o ( 0 ) ® V-[o,r)(/) ® V>[t,oo)(/),
(3-2)
w h e r e W ^ [ T , t ) is a t w o - p a r a m e t e r s f a m i l y of l i n e a r b o u n d e d o p e r a t o r s i n Ti w h i c h depends o n r e s t r i c t i o n /(-)l[r,t)S u p p o s e t h a t u[r,t)
is b o u n d e d a n d i n t e r v a l - a d a p t e d f a m i l y , S[T, t) is b o u n d e d
v a c u u m f a m i l y a n d n[r,t)
= L A[r,t) 0
+ L A [r t) i
9
i
6 B"(D,E ) a
a r e o p e r a t o r s o f c r e a t i o n , a n n i h i l a t i o n a n d c o n s e r v a t i o n ([7]).
w h e r e A*,
A,
A
86 Let U[T, t)g, g eh
L e m m a 3.1.
in h with respect
and LiS[r,
i)q, q e L(D,
to r and t (0 < T < t < oo) | | ( u M ) - I)g\\ -
0,
h
and
m [ r , t) = S[T, t)+
be norm
a
t
h
stochastic
integral
U[T, O ( £ o A ( d £ ) + L A*(d€))S[Z,
f
continuous
- I)q\\ - » 0
\\Li{S[T t)
as t — T J. 0. T h e n t h e r e exists interval-adapted
E)
let
t)
x
=
J[r,t)
n lim
S[r,t)+
V«[T f )/ife,{ l
n—*oo '
w h i c h h a s t h e following (1)
the
limit
L(D,E )
(3.1)
i)5Ky+i,*)eB'(A««.)
m [ r , t ) i s interval-adapted
in B (D,E ); s
a
; u[r,t)q,
q
e
in h a n d ||(m[r,<) — /)9||h — * 0 as t — r j 0;
continuous
(p,m[r,t)g)
j +
properties:
exists
is norm
a
J
'
1
=
•'['•.*)
i j
for any ff = £ f f i ,
g =
h ®vj(fi)eL(H,E ),
i
i
a
i
q = £
9j>
9> = « j ® W j )
6
> E ); a
i (3) \\m[T,t)q\\
2
df|HT,0£(Io^(0
= ||S[r,r)g|| 4- / 2
+ ii)5K,t)ffill .
(3.3)
J
I n t h i s s e c t i o n we c o n s i d e r a c o n s t r u c t i o n o f v a c u u m f a m i l i e s S a n d 5 * a s s o c i a t e d w i t h t h e d i s s i p a t i v e p a r t s N a n d N* o f a d j o i n t q u a n t u m s t o c h a s t i c e q u a t i o n s d u[r,t)
= -u[r,t)(N(dt)
t
d u*(t, T
s—
T) = u*(t,
](N*(dr)
1
'
M(dt)), - M*(dr)),
T
l i m U\T, t) = I, t-rlO
-
s—
l i m u*(t,
(
T] =
I
<-r!0
where N(dt) N*{dt)
= Gdt + A(dt)
+ (R*L)*A(dt),
= G*dt + A(dt)
- L*A(dt),
M(dt) M*(dt)
= RA(dt) = R\k(di)
+ -
LA*(dt), LA*(dt)).
3
4 )
87 Thus N(dt)
- M(dt)
= N{dt) - M(dt)
Gdt + ( 7 - R)A(dt) N{dt)
+ N*(dt)
N*(dt)
=
+ (R*L)*A(dt)
-
L*A*(dt),
- N*(dt)N(dt)
=
M'(dt)M(dt),
+ N(dt) - N{dt)N*(dt)
=
M(dt)M*{dt).
I n w h a t f o l l o w s w e w i l l s u p p o s e t h a t TJ i s a s e l f a d j o i n t o p e r a t o r i n Ti, R 6 B(Ti) a u n i t a r y o p e r a t o r a n d L 6 C(Ti)
a closed densely defined operator such that
> I. H e n c e R*L i s c l o s e d , ( ( 7 ? * L ) * ) * = R*L, L*L i s p o s i t i v e a n d s e l f a d j o i n t
L*L
(see [5]). F o r a n y c € C , |c| < a l e t u s c o n s i d e r o n e p a r a m e t e r b o u n d e d s t r o n g l y c o n t i n u o u s s e m i g r o u p s Wf
G
c
H
c
= iH
c
a n d W£ i n Ti w i t h i n f i n i t e s i m a l o p e r a t o r s — G
+ i$
c
,
G
c
= iH
c
+ i *
= H+±:{(R*Lc*y-R'Lc'},
$
c
= * (7),
$ (X)
$
c
= $ (7),
*
c
C
c
, H
= (L + RcyX(L
c
c
c
and— G
c
- I ( L * c - L c * ) ,
= -H
c
+
( X ) = (R*(L - Ic))*XR*{L
3
6
Rc), - 7c).
Hence G
c
1 — , . v , . 1 , .2 = iTJ+ ^ I ' l + ( J T L c ' ) * + -|c| , 2
*
G
_ ™ r 1 r * r _ r * - _i_ l u = - i f f + -L*L - Vc +-\c
. c
2
W e w i l l s u p p o s e t h a t a s s u m p t i o n s (2.3)-(2.5) a r e f u l f i l l e d f o r o p e r a t o r s G a n d c
G
c
a n d that their domains do not depend o n c D(G )
= D(G)
C
C D(L),
7J(G ) = D ( G ) C 7J(L), C
D(G)
C 7J(iZ*L)*;
D(G)
C 7J(L*)
for c 6 C , |c| < a w h e r e a i s s u f f i c i e n t l y s m a l l p o s i t i v e c o n s t a n t . Now
w e c a n define e x p l i c i t l y t h e v a c u u m f a m i l i e s S[T, t) a n d S[r,t)
described
earlier b y (3.2). L e t
W^[r,t)
= W
f o r a n y /(•) € E The
r
a
2
r
C
2
t
- r))
s u c h t h a t / ( T ) = /(*>), r S [*,-—x,*>).
next f o r m u l a describes the m a i n properties of the v a c u u m famihes.
88 Lemma
Let the assumptions
v a c u u m families
S and 3* are contractive (1)
e-
(2)
d S[r,
(2.3)-(2.5),
l i m 3[T,t)=.I\
(3.7)
such
and for G .
c
Then
c
that
l i m §*(i,r]
s-
hold for G
=
I,
l|5|| = l , ||5*|| = 1; t) = -S[T,
t
on
t) = N(dr)S[r,
T
t)
L(D(G),E )\ a
r] = -N*(dt)S(t,
d S*(t, t
on L(D(G), (3)
d S[r,
t)N(dt);
1-];
d S*{t,
T]N*(dr)
r] = S*{t,
T
E ); a
de\\S[t,t)g\\
= \\M(dt)S[t,t)g\\
2
d \\§*(f,r]g[\
L(D(G),E ), a
= \\M*(dOS'(t,T]g\\ Vg
2
S
Vg e
2
N o t e t h a t p r o p e r t y (2) f o l l o w s f r o m (3.6)
(3.9)
€ L(D(G),E ).
2
a
a n d (3.8)
a n d p r o p e r t y (3) e a s i l y
follows f r o m e q u a t i o n (3.5). Under
C o r o l l a r y 3.3. sequence stochastic
of
uniformly
Duhamel
the
assumptions
bounded
imposed
strongly
in Lemma
continuous
3.2
there
exist
interval-adapted
the
quantum
integrals
u [r,t)
= S[r,t)+
n
u ^[T,OM(dC)S[^t),
[
n
(3.10) £T(i, r ] = S*(t,
f
r}+
u ^(t, n
r],
(\M*{dt)5*ti,
J\r.t)
w h e r e M and M*
in (3.4), itn = 5 , fij = S* and u ,
were introduced
n
b o u n d e d e x t e n s i o n s of uniformly
bounded
and densely
fi*
denned
operators
= u [r,t)X
®
are linear (3.10).
T h e w e a k e q u a t i o n s for t w o - p a r a m e t e r s f a m i l i e s P (t,r](X) n
= u* (t,r]X n
® Iu [r,t), n
P [r,t)(X) n
n
Iu* {t,r) n
f o l l o w s f r o m L e m m a 3.1. (g,P (t,T](X)g)
= (g[T,t),X®Ig[T,t))
n
[
(M(df)
(g, P [r, n
[ J[r,t)
F F
t)(X)g)
[T,o,p»-i(f,r](jfMde)sK t)) >
= {g(t, r],X®
Ig(t,
(M*(dOmr},P -il^t)(X)M*(dO~g((,r}), n
+ (
)
-
3
n
)
2
)
r])+ (
3
'
1
89 where g€L(D(G),E ),
g[r,t)
a
=
S[r,t)g; g eL(D(G),E ),~g(t,r}
=
a
F r o m ( 3 . 9 ) , ( 3 . 1 1 ) a n d (3.12) i t f o l l o w s t h a t P ,
P „ are contractive a n d monotone
n
(g,P (X)g)
< \\X\\• \\g\\l (g,P (X)g)
n
< \\X\\ • \\g\\l
n
0 < P _ ! ( X ) < P ( X ) < \\X\\-I, n
0 < P „ _ ! ( X ) < P „ ( X ) < ||X|| • / ,
n
for a l l X 6 B(H)
S*(t,r]~g.
u n i f o r m l y w i t h r e s p e c t t o r a n d t . H e n c e t h e r e e x i s t t h e least
+
upper bounds P(t,r](X) = l.u.b.P (f,r](X), B
P [ r , t)(X)
= l . u . b . P [ r , t)(X),
V X e
n
T h e o r e m 3 . 3 . L e t t h e a s s u m p t i o n s (2.3)-(2.5) G
c
and G . c
which l.u.b.P .
(4)
for
operators
Langeven
equations,
and (3.12) for P = l.u.b.P ,
P
n
=
properties to r and t
r ] ( - ) , P [ r , <)(•) a r e n o r m a l a n d completely
positive;
P(t,
P ( t , T](-) and P [ r , t ) ( - ) a r e
P and
P are minimal and strongly
the minimal
and C (-) c
infinitesimal
solutions
conservative
solution
P [ r , s)(P[s,
of (3.11),
(3.12)
t)(X)) = P[r, t)(X); in the
bounded
positive,
solutions; if and only if the mappings
for all c £ C , |c| < a , w h e r e £ (2.1)
of all
is u n i q u e in the class of bounded,
P and P are conservative
mappings
class
solutions;
and conservative
solutions
are regular
r](X);
continuous
measurable
the minimal
semigroups:
r ] ( X ) ) = P(t,
positive
strongly (5)
be fulfilled
of the weak integral (3.11)
have the following
on B(H)+;
P(t, s](P(s, (3)
equations
and (3.7)
P ( t , r ] ( X ) , P [ T , t ) ( X ) a r e b o u n d e d , s t r o n g l y c o n t i n u o u s with respect and positive
(2)
with limiting
These solutions
n
(1)
Then there exist the solutions
coincides
B(H)+.
with coefficients
c
and C
c
are the
C (-) c
Lindblad
(3.6).
N o t e t h a t i f t h e s t o n g l i m i t s exist u[r,f)=s-lmu„[r,(), n
u*(t,r]
= s - lim u*(t,r] n
t h e n t h e y s a t i s f y t h e l i m i t i n g i n t e g r a l e q u a t i o n s (3.10) a n d u*(t,r]u[r,t)
= P(t,r](I),
u[T,t)u*(t,T]
=
P[r,t)(I),
(3.13)
90 where P a n d P are the m i n i m a l solutions for integral L a n g e v e n e q u a t i o n s . H e n c e u a n d u a r e t h e m i n i m a l s o l u t i o n s of t h e l i m i t i n g e q u a t i o n s ( 3 . 1 0 ) i n t h e c l a s s of bounded strongly continuous two-parameters interval-adapted solutions u*(t, T]U[T, t) < v*(i,
T]V[T, t),
U[T, t)Q*(t, r] < v[r, t)v*(t,
r]
w h e r e v a n d v* are a n y o t h e r b o u n d e d s t r o n g l y c o n t i n u o u s i n t e r v a l - a d a p t e d s o l u t i o n s . It is easy t o see t h a t P a n d P a r e c o n s e r v a t i v e i f a n d o n l y i f u a n d u* are isometries. T h e next t h e o r e m shows t h a t the c o n s e r v a t i v i t y of P a n d P implies the strongly convergance (3.13). T h e o r e m 3.4. are regular strongly
for all c £ C , |c| < a.
continuous
u* satisfy Strongly
L e t the assumptions
of the theorem Then
interval-adapted
equations
(3.4)
continuous
there exist strong
two-parameters
on L(D(G),E )
and
a
interval-adapted
the class of bounded
strongly
3.3 be fulfilled
solution
continuous
limits
famiUes
and C (-), (3.13)
c
are
which
of isometries.
on
C (-)
c
u and
L(D(G),E' )correspondingly. 3
of each equation
interval-adapted
(3.4)
is unique
in
solutions.
I n o r d e r t o p r o v e t h a t u = u we w i l l s u p p o s e t h a t \\LWf\\ < k t f o r s o m e p £ ( 0 , 1 ] , k > 0. and W
c t
1 + f i
,
\\LW \\ < k t t
c
(3.14)
1 + p
S u c h a s s u m p t i o n s h o l d for a n a l i t i c a l s e m i g r o u p s
if I I ^ G c + o-/)- !! < 1
fcH-H-P
a n d \\L(G +erl)- ]]
Wf
< k\o-\- <> i n a sector
1
C
l+
o f a n a l i t i c i t y of t h e r e s o l v e n t o p e r a t o r s . U n d e r a s s u m p t i o n s (3.14) t h e i n t e g r a l s
j
... j f
\g,S[r,T )M(dn)...M(dT )S[r ,t)h),
j
...J
(g,S\t,T \M\dT )...M*(dr )S*(T T])~h),
1
n
n
are a b s o l u t e l y i n t e g r a b l e f o r a l l g £ L(H,
E ), a
n
n
U
L
h £ L(D(G),
E ),
h £ L(D(G),
a
E ). a
F r o m h e r e we o b t a i n t h e f o l l o w i n g r e s u l t . Lemma integral
3.5.
L e t (3.14)
equations
and
the
assumptions
(3.10) are equivalent
U » [ T , t) = S[r, t)+
of Theorem
to following
3.3 hold.
equations
S[T, o m k k - I K , t ) ,
! J[r,t)
u* (t, r] = S*(t, r] + n
I
S\t,
SM'idOQ^tf,
T].
Then
the
91 If t h e i n f i n i t e s i m a l m a p p i n g s C (-)
a n d C (-)
c
are r e g u l a r , t h e n u
c
n
a n d fi* c o n -
verge s t r o n g l y a n d f r o m (3.15) f o l l o w s t h e s a m e d i f f e r e n t i a l e q u a t i o n s for
u*(t,r],
u * ( t , T ] w i t h r e s p e c t t o r a n d for u [ r , t), u[r, t) w i t h r e s p e c t t o t. F r o m t h e u n i q u e ness of s t r o n g l y c o n t i n u o u s i n t e f v a l - a d a p t e d b o u n d e d s o l u t i o n s of e q u a t i o n s
(3.4)
f o l l o w s e q u a t i o n u* = u* w h i c h gives u n i t a r i t y of u [ r , t ) . T h u s w e c o m e t o t h e final r e s u l t . T h e o r e m 3.6. G
c
and G
L e t t h e a s s u m p t i o n s (2.3)-(2.5)
and let the mappings
c
T h e n t h e r e exist u n i q u e strongly equations
C (-), c
C (-)
continuous
and (3.7),
(3.14)
be fulfilled
for
for all c £ C , |c| <
be regular
c
interval-adapted
unitary
solution
a. of
(3.4).
S o m e e x a m p l e s a n d c o u n t e r e x a m p l e s were c o n s i d e r e d i n [8] w i t h F . F a g n o l a a n d A . Frigerio.
References. [1] F e l l e r , W . , An Introduction
to Probability
Theory
v.I
and its Applications.
J o h n W i l e y & S o n s , 1968 [2] G i k h m a n , I.I. a n d S k o r o k h o d , A . V . , An Introduction chastic
Processes
to the
Theory
of
Sto-
S c i e n c o M o s c o w , 1977
[3] C h e b o t a r e v , A . M . , C o n d i t i o n s Sufficient for R e g u l a r i t y o f a M a r k o v J u m p P r o c e s s e s Probability
Theory
3 3 1, 1988 1 5 - 2 9 ( R u s s i a n )
and Its Applications
[4] L i n d b l a d , G . , O n t h e G e n e r a t o r s of Q u a n t u m D y n a m i c a l S e m i g r o u p s mun.
Math.
Phys.
Com-
4 8 2, 1976, 1 1 9 - 1 3 0
[5] K a t o , T , Perturbation
Theory for
Linear
Operators
Springer-Verlag Berlin,
H e i d e l b e r g , N e w Y o r k 1966 [6] C h e b o t a r e v , A . M . , C o n d i t i o n s N e c e s s a r y a n d Sufficient f o r C o n s e r v a t i v i t y of D y n a m i c a l Semigroups.
T h e P r o b l e m s of t h e M o d e r n M a t h e m a t i c s . T h e l a t e s t
A c h i e v e m e n t s V I N I T I 3 6 1989, 1 9 2 - 2 1 1 ( R u s s i a n ) [7] H u d s o n , R . L . a n d P a r t h a s a r a t h y , K . R . , Q u a n t u m I t o F o r m u l a a n d S t o c h a s t i c E v o l u t i o n Commun.
Math.
Phys.
9 3 1984, 3 0 1 - 3 2 3
[8] C h e b o t a r e v , A . M . , F a g n o l a , F . , F r i g e r i o , A . , T o w a r d s a S t o c h a s t i c S t o n e ' s T h e o r e m P r e p r i n t 315, D i p a r t i m e n t o di M a t e m a t i c a U n i v e r s i t a degli S t u d i di Trento.
1990
[9] K o n s t a n t i n o v A . A . , M a s l o v V . P . , C h e b o t a r e v A . M . , E x p e c t a t i o n f o r m u l a e for s o l u t i o n s o f C a u c h y q u a n t u m m e c h a n i c s . 3-23 (Russian)
Uspekhi
Mathem.
Nauk
4 5 6 1990,
Q u a n t u m P r o b a b i l i t y a n d Related Topics V o l . V I I (pp. 93-107) © 1 9 9 2 W o r l d Scientific P u b l i s h i n g C o m p a n y
93
M a r k o v i a n i t y f o r states o n v o n N e u m a n n algebras by Carlo Cecchini Dipartimento d i Matematica e Informatica d e l l ' U n i v e r s i t a d i U d i n e — V i a Zanon,
6
—
1. Introduction
33100
U d i n e — Italy
T h e theory o f v o n N e u m a n n a l g e b r a s c o r r e s p o n d e n c e s (cfr.
f o r i n s t a n c e , [ l l ] ) , h a s a l s o , as seen i n [ 6 ] (cfr. also [ 3 ] ) , a n o n c o m m u t a t i v e p r o b a b i l i t y s i g n i f i c a n c e . F o r a n y c o r r e s p o n d e n c e f o r v o n N e u m a n n algebras it i s possible
to c o n s i d e r
a canonically defined
m a p p i n g , the s t o c h a s t i c
coupling,
w h o s e p r e d u a l m a p p i n g p l a y s the r o l e o f the c l a s s i c a l t r a n s i t i o n p r o b a b i l i t y . W e s h a l l u s e i t , together w i t h the n o t i o n o f g e n e r a l i z e d c o n d i t i o n a l e x p e c t a t i o n f o r v o n N e u m a n n a l g e b r a (cfr. [ 4 ] , [ 7 ] , [ 8 ] , [ 9 ] ) to d i s c u s s the n o t i o n o f m a r k o v i a n ity f o r n o r m a l states o n v o n N e u m a n n a l g e b r a s w i t h respect to a g i v e n l o c a l i z a tion. T h i s w i l l be a triple o f trivially intersecting v o n N e u m a n n algebras
gener-
a t i n g the g i v e n v o n N e u m a n n a l g e b r a . S e c t i o n 2 w i l l b e d e v o t e d to p r e l i m i n a r ies a n d n o t a t i o n s , together w i t h a f e w a n c i l l a r y results; s e c t i o n 3 w i l l
contain
o u r m a i n results, i n c l u d i n g the g e n e r a l i z a t i o n s o f the m a i n c l a s s i c a l t h e o r e m s .
2. P r e l i m i n a r i e s a n d notations
In the f o l l o w i n g M
will
be a v o n
N e u m a n n a l g e b r a , S(M ) ( S (M )) the set o f its n o r m a l ( f a i t h f u l ) states. W e s h a l l f
a s s u m e M to act o n a H i l b e r t space f o r M a s s o c i a t e d w i t h co G S
f
w i t h a c y c l i c and separating vector
Q
(M)
W e s h a l l d e n o t e b y J Q (or s i m p l y J w h e n n o a m b i g u i t y arises) the i s o m e t r i c a l i n v o l u t i o n l e a v i n g Q f i x e d a n d b y VQ U
M
the s e l f d u a l p o s i t i v e c o n e f o r M i n
c o n t a i n i n g Q . I f i|i G S ( M ) , i p s a m f o r s o m e a > 0 , w e d e n o t e b y [TJI/ co]
the e l e m e n t o f M o b t a i n e d b y e v a l u a t i n g i n the p o i n t i / 2 the a n a l y t i c a l e x t e n s i o n o f the C o n n e s c o c y c l e (Di|>: D c o ) (cfr. t
[10]).
c l o s u r e o f the set { [ijV co] Q : Tp E S(M),
So [ ip/ co]QE
and
V^
is the
i|i s t x t o f o r a > 0 } . T h e n o t a t i o n co
f o r co G S(M), a G M , w i l l denote the f u n c t i o n a l co (•) = c o ( a a
+
a
• a). W h e n u s i n g
94 t h i s n o t a t i o n w e s h a l l f u r t h e r a s s u m e u n l e s s e x p l i c i t e l y stated that co(|a| }) =
L
so that c o E S ( M ). a
Let now N
be a v o n N e u m a n n s u b a l g e b r a o f M , co^= u)/
N
Q a vector i n H
co £ S (M),
and
i m p l e m e n t i n g co. W e s h a l l o f t e n i d e n t i f y w i t h { a Q : a E N}
M
G . N . S . H i l b e r t space
the
for N and Q .
W e shall denote b y E
Q
'
M
N
the o r t h o g o n a l p r o j e c t i o n f r o m H
to H Q
M
N
, and
b y e ^ ' ^ the g e n e r a l i z e d c o n d i t i o n a l e x p e c t a t i o n f o r co f r o m M to N (cfr. [ 4 ] , [ 7 ] , [ 8 ] , [ 9 ] ) . I f p i s the s u p p o r t o f co, t h e n , f o r a E M , e ^ ( a ) i n pN
i s the u n i q u e e l e m e n t
p s a t i s f y i n g the e q u a l i t y :
w i t h Q a v e c t o r i n H ^ s u c h that co(a ) = < Q , p a p Q . > . W e r e c a l l that w h e n e v e r there i s a co p r e s e r v i n g n o r m o n e p r o j e c t i o n
from
M
to N ( i n p a r t i c u l a r i n the c o m m u t a t i v e c a s e ) it c o i n c i d e s w i t h E ^ ' ^ . W e s h a l l denote b y
the v o n N e u m a n n s u b a l g e b r a o f N o f the p o i n t s w h i c h r e m a i n
f i x e d u n d e r e^' M
to
. T h e r e is a n o r m one p r o j e c t i o n ( c o n d i t i o n a l e x p e c t a t i o n )
r
p r e s e r v i n g co^. O n {p
apQ, a E
} the o p e r a t o r s J ^ Q
from
and T Q
A™ 7
c o i n c i d e , a n d they c o i n c i d e a l s o w i t h the i s o m e t r i c a l i n v o l u t i o n We
shall
associated p%
M
to
: S(N)
call
canonical
the c o n e
VQ
state
extension
(see [ 7 ] ) PQ
M
J fa. p
from
M
to N
( Q s e p a r a t i n g f o r N) the m a p p i n g
S ( M ) d e f i n e d as
[pQ' (^iv)](a)=
(
<^ ,^N>
M
N
a E M )
w i t h Wjy E V Q t h e v e c t o r r e p r e s e n t a t i v e o f
ty
N
i n VQ.
A s this d e f i n i -
t i o n d o e s not d e p e n d o n the c h o i c e o f the c o n e , w e s h a l l u s u a l l y s h o r t e n our
n o t a t i o n to O Q '
M
if
Q is
a
vector
i m p l e m e n t i n g co i n U
M
A
N
D
95
F r o m n o w o n w e s h a l l c o n s i d e r a set o f i n d i c e s A , to e a c h o f w h i c h w e associate a v o n N e u m a n n a l g e b r a Aj B C A w e denote b y A
(j G A ) a c t i n g o n a H i l b e r t s p a c e H
the v o n N e u m a n n
B
algebra
generated
A
. For
b y [J
Aj ;
/SB i f B = {j} o u r n o t a t i o n w i l l be s i m p l i f i e d to Ay W e s h a l l a s s u m e that i f B , C C A , B f i C = 0 , then A
flA
B
W h e n dealing with
i s the t r i v i a l a l g e b r a o f the m u l t i p l e s o f the identity.
Q
the v o n N e u m a n n algebras A
B
( B C A ) , w e shall simplify
o u r notations b y u s i n g s i m p l y B as a n i n d e x instead o f A w i l l b e s h o r t e n e d to E ^ ' ) . 8
A
and A- are m u t u a l l y c o m m u t i n g (i.e. f o r a^ G A
i
2.1 L e m m a .
Let
co G
b
B,
c
co (c) = co(fe
+
j ;
(e.g. e ^ B ^ C ( B 3 C )
aj G A j a a- = a^ a ^ i
C C A ,
c
P r o o f . W e have, for c G ( A ) So
S(A^,
= s u p p (u> ) . T h e n p™c s p "0= s u p p c o
p™c
B
W e s h a l l a l s o a s s u m e n o w that f o r i j E A , i / j ,
0
fc
B f l C = 0,
b £ A
B
,
c
c b ) «: || b || co ( c ) ;
+
2
co (c) = 0 i m p l i e s co (c) = 0 fc
I n this s i t u a t i o n (cfr. [6]), f o r C , D C B C A , C D D = 0 , a
canonical
state
extension
from
A
to
c
C U D = B and A
B
,
the
p
c , B
mapping
cp - * p ' ( c p ) | A f o r c p G S ( A ) a d m i t s a n e x t e n s i o n to a l i n e a r p o s i t i v e c o n c
c
B
c
D
c
C
t i n u o u s m a p p i n g f r o m (A ) c
to (A )
t
D
t
. Its d u a l m a p p i n g i s a l i n e a r , p o s i t i v e ,
a n t i c o m p l e t e l y p o s i t i v e ( i . e . i t s c o m p o s i t i o n w i t h the a d j u n c t i o n i s a n t i l i n e a r completely from co
c
co
B
G
Any
A
positive),weak to A .
D
c
operator
It w i l l
be
continuous
denoted
Xj"'
D
unity preserving
S ( A ) a n d s o m e c a n o n i c a l state e x t e n s i o n p f
c
contraction
i f f o r s o m e co s u c h c , B
w e have p
that
(co ) =
c , B
c
linear, positive, anticompletely positive, w e a k operator continuous unity
p r e s e r v i n g c o n t r a c t i o n from A c o u p l e ( A , A ). Q
D
co(c d ) = < J
Q
D
to A
I n p a r t i c u l a r Xj*'
c
c
w i l l b e c a l l e d a s t o c h a s t i c c o u p l i n g f o r the is c h a r a c t e r i z e d b y the e q u a l i t y :
A.^ (d) Q , c Q > c c C
c
with c G A , d G A , c
D
Q
c
a vector i n H
the i s o m e t r i c a l i n v o l u t i o n l e a v i n g it i n v a r i a n t .
A
i m p l e m e n t i n g the state c o a n d c
J
Q
c
96 Conversely, if K ' D
i s a s t o c h a s t i c c o u p l i n g f o r the c o u p l e (A , A )
C
D
ping cd — < J ^ X ' f d ) Q D
£
n o r m a l f a i t h f u l states c o tensor product o f A
C
a )
,c£2
f 0 c
on A
c
and A
>(cGA ,deA ) c
C
the m a p -
c a n be extended for a l l
D
to a state co o n t h e p r o j e c t i v e C * - a l g e b r a
C
(cfr. [12] I V ) ; the v o n N e u m a n n a l g e b r a A
D
g e n e r a t e d b y its G . N . S . r e p r e s e n t a t i o n w i t h respect t o co h a s a n o r m a l state c o w h i c h i s the e x t e n s i o n o f
co a n d h?' = X / c o
i f w e identify A
and A
C
B
B
with
D
t h e i r f a i t h f u l r e p r e s e n t a t i o n s w h o s e u n i o n generates A . B
W e note that i n the a b e l i a n case w e h a v e
^
= E / |A
C
3
0
D
F o r f u r t h e r p r o p e r t i e s a n d results o n s t o c h a s t i c c o u p l i n g s see [ 6 ] . 2.2 P r o p o s i t i o n (cfr. [6]). L e t A • B D C , D , C D D = 0 , C U D = B , C D G , B D G U D
= F . Then for co E S ( A ) with co = co |A B
B
c
B
Q
E
E S ( A ) w e have f
c
>?' = e ' * G
X'
c G
CO
D C
CO
CO
P r o o f . W e have, for d E A , g E A D
< ( J
= < J if
O
ti*
G
K° - (d)Q,g£2> 1
on
{g Q : g E A } G
H
Q
G
,
this
separating for A
A
c
<7
^
( d ) Q , g Q >
G
i m p l e m e n t i n g the state c o of
i s dense i n / Y implies
A . ° ' ( d ) a, g Q > =
= <E°'%
the c l o s u r e
G
:
= <x>(dg)=
c
Q is a vector i n H
involution
in
(d)Q,gQ>
G
Q
G
{g Q , and J
2^ (d)£2 G
, and both
=
: g E A A
( £
( f ^
G
G
}
G
i s the i s o m e t r i c a l
leaving Q
• }^ ) (d)
invariant. A s
Q , J ^ ' ( d ) Q are
C
G
G
G
(d)Q for all d E A
' £ f )
>^' (d) a n d ( e £ ' « G
G
and J
Q
X°' )(d)
D
are i n A
c
; since Q is G
, w e have
proved our claim.
2.3 P r o p o s i t i o n . co
B
Let A D B D
E S ( A ) , co = co | A B
(£; )(|c| )= D
2
c
B
c
E
|[(|co ) /co ]| c
D
D
C, D,
C U D = B,
S f ( A ) . T h e n for c E c
2
A
c
C n D =
w i t h co (| c |
2
)
=
0 1
97 P r o o f . N o t e first that f o r d E ( A ) D
w e have
+
\ c |
( c o ^ (d) = co (
)
s||c|| co(d); s o [ ( c o ^ / c o j i s w e l l d e f i n e d . N o w , f o r a l l d E A , i f Q i s a v e c t o r 2
D
i n R~ i m p l e m e n t i n g c o : < J p X A
(| c 1 )Q, d Q >= 2
w
=co ( | c 1 d)=(co ) ( d ) = c o ( [ ( c o ) / c o ] d [(co y c o j 2
B
c
D
D
c
[ K ^ H R d j O =
<
D
=
c
[(co ) /co ]Q>= c
I K ^ c )D^ D 1 1 ^
^ D
D
+
a
C0
n
D
c
D
'
a
s
p r o p . 2.1 this i m p l i e s o u r c l a i m .
m
3. Markovianity for triples of von Neumann algebras.
U n l e s s otherwise
stated i n this s e c t i o n A = { 1,2,3}; w e s h a l l a l s o o f t e n o m i t the i n d e x A f o r the sake o f conciseness.
3.1 P r o p o s i t i o n . L e t A b e a b e l i a n a n d co E S ( A ) . T h e n e £ ^ ' ^ ( a ) E 2
all a £ A 3
iff e^
3
2,3
*' does not depend o n a 2
Proof. L e t f £ ' ' ( a ) { 1
one
projection
^{2,3}-
W
e
h
2 }
from 3
V
e
^
A
2 3
for a l l a
2
j to A
^ { 2 , 3 } ( 2,3}) a
=co(<£ ' ( | a, | a { 1
E A
3
2 }
2
{ 2 > 3 }
f o r a l l 3{2 3} ^ ^ { 2 3} C o n v e r s e l y , let E *
'
2 3 1
2
)) =/(0\
* S
a n c
{
e t o
u
= E* ' *' 2
3
r
2
^
E
1
e£
3
{ 1
a n d it i s e n o u g h
2
= a )
'
|A
2 }
{
2 3
to s h o w
2
W
3
( i 2^ a
a
{ 1
'
3
2 }
a
= Kf) 2,3}( o, E
{
= co(
a
i
= W
< 2
'
(^'
3 }
'
2
{ 1
'
2 }
3
a
a
( 2 3))= a
a
a Ej ^ (a )) 2
2
2
3
which implies ^ ^ ( a 3 ) = e W ( a
3
a
) E A
2
.
is a
norm
2a
{ 1
{ 2 3 )
i pli
i r s t
m
for a l l a
( i 2 3))
}
( I 1 I {2,3})) =
c a t
'
2 }
(a
{ 2
,
3 }
))
i° n
ECA^cOj. T h e n for a
x
t
w e have:
( 3))
for
2
it p r e s e r v e s
i E A , a E A
A
Aj
E A ; then
3
3
( i 2 3)=Kf){2 3}( 2 3)=
= M
a
a
a
Kf) 2,3 ( 2
}
a
>
a
E
{ 2 , 3 }
'
2
a
( 3> ) = a
E(Aj )
+
a
2
98 T h e first c o n d i t i o n i n p r o p . 3 . 1
is a standard definition o f m a r k o v i a n i t y a n d
therefore i f c o G S ( A ) satisfies i t w e c a l l i t m a r k o v i a n w i t h r e s p e c t t o t h e l o c a l i z a t i o n (A., A A^)-
T h e second condition i s m o r e suitable f o r generalization to the
2
non-commutative situation.
3.2 T h e o r e m .
L e t to G S ( A ) w i t h C 0 j £ S ^ A j ) c O j
c o m m u t e w i t h A^ y 2
3
2
3
j G S (A { 3})> f
2
A ^ b e the v o n N e u m a n n s u b a l g e b r a o f t h e f i x e d
2i
under E ^ ' ^
2
and
the c o ^ j p r e s e r v i n g n o r m o n e p r o j e c t i o n f r o m A ^ 2 3
A ^ . T h e n the f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t
3
)
K
H
b)
{
2
^
3
K
c) o X a a 1
d ) X1'™
e) * 4
2
{ 2 3 }
K
( {2,3}))
^
= ^
a
3
}
'
2
1
1
0
r
a
l
l
a
i
e
A
l '
a
{ 2 , 3 }
3
2
{ 2 3 }
))
fora^A^
a
1
a l){2,3}
:
D
M
{2,3})t £ A
f
CA
2
{
2 3
}
which implies b). b ) => c ) F o r a G ( A j ) , t
a i
a
{
2 3 }
{
2
3
}
G A
{
i m p l e m e n t i n g co
a) =>b) B y a) f o r a l l a G A j
co(
{2,3}
f o r ^ G A ,
, { 2 , 3 }
m
A
2
P r o o f . I n the f o l l o w i n g let Q b e a v e c t o r i n
(° (
E
f o r a , G A,
2
)=to(a Ej ' >' (a
= X -
f
+
) = co | ( a a
{ 2 3 }
a
{
2
3
) = co
}
{ 1 2 }
and o u r c l a i m follows b y linearity
G A (
3 l
{
2
3
}
:
ej ^2
2
(a
{
2 3 }
))
2
3
}
A
i
points 2 3
j to
99 c) = > d) L e t p j
j
2 3
be the s u p p o r t o f <*>{ 3} 2
a n c
* ^
a
y ''
c
c
c
a
n
separating
u
v e c t o r f o r A i n H s u c h that co(a^ j ) = 23
= < Q,
P
{
2 3
a
}
{
23
p
}
F o r each a G
{2,3}
X
co ( a a t
=<J
i {
< 2 , 3 >
£2 >
}
+
(
3
l)
j £
2 3
a i
)Q,J
2
3
E < ' >' J 2
Q
3
A J
2
2
w e get J c ^ ' *
J
3
1
linearity for a G
A
x
(a
2
N o w , b y the d e n s i t y o f { a j rating for
{
3
a
}
(
) ) =
e) => a) F o r a E L A :
co(
a
3 l
= < A' J
2
{ 2 3 }
) = <J _
^'
{ 2
{ 2
'
*
Q
e (cfr. l e m m a 2 . 1 ) :
= C O
£ J ( 1 P{2,3> <2,3} P{2,3> )= A
3
)) =
{ 2 3 }
Q . >=<J
E AJ
2 3
{2)3}
\J>
2
( )Q, h
J } in
a
{
2
3
}
Q >.
a n d as £2 i s s e p a -
* (aA = X ^ ' (aA first f o r a 2
G
x
A *
and then b y
a
^1,2
)
(
=
GAy
1
1,12,3}
x
Z{ 3} ^ ^ { 2 3} *
x
o
2
X ^
3 }
( i)
3 }
a v
y
1,{2,3}
( X
"
e
j Q-.a^^
2 3
2 , 3
2
{ 23 }
d ) => e) B y p r o p . 2.2 w e h a v e f o r a {2.3},2
w
A
2
(
^ { 2 3}
'P{23} <2,3} P{2,3}
Q
) = co ( a , E J ' * '
23 }
2
3
(Aj) , a^
1
<J
R
3
> ( a , ) fi, a
• « £ ( (2,3 )
Q
A
}
"
R
3
}
(
)
u
r
a
*
r e a
fi >=<
> = co (
f l j
^ y established notations: ^
E/ ( a
(
{ 2 3 }
a
A
Q
^
fi
>=
)).
I f A is a b e l i a n the e q u i v a l e n t c o n d i t i o n s o f t h e o r e m 3.2 i n c l u d e the s e c o n d c o n d i t i o n i n p r o p . 3.1 c h a r a c t e r i z i n g the m a r k o v i a n i t y o f co ( c o n d . b ) , w i t h r e s p e c t to t h e l o c a l i z a t i o n (A., A , A ) . I n t h i s s p i r i t w e g i v e the f o l l o w i n g . 2
3
3.3 D e f i n i t i o n . A state co G S ( A ) w i l l b e c a l l e d m a r k o v i a n w i t h respect to the l o c a l i z a t i o n (A
v
A , A j ) i f it satisfies the e q u i v a l e n t c o n d i t i o n s o f t h . 3 . 2 . 2
W e n o t e that c o n d i t i o n c ) i n t h . 3.2 i s s t r o n g e r t h a n the c o n d i t i o n u s e d i n [5] to d e f i n e M a r k o v states f o r m a t r i x a l g e b r a s ; there it i s a s s u m e d that
w
( i a
A
{2,3}) =
w
( i a
e { 2
'
3 >
'
2
( {2,3})) A
100 with E ^ ' ^ ' 2
3
a general completely positive linear contraction (not necessarily
2
the C0p j - g e n e r a l i z e d c o n d i t i o n a l e x p e c t a t i o n ) f r o m A ^ 3
In
the c l a s s i c a l case (A
abelian)
m a r k o v i a n f o r the l o c a l i z a t i o n (A localization
(A
A,
y
a s t a n d a r d fact
t
2
3
general v o n N e u m a n n algebras the equivalence g e n e r a l i z a t i o n o f t h i s fact, as a b e l i a n s i t u a t i o n A ^ ' * ' * (aA 1
2
i s that i f a state co i s
A ) , t h e n i t i s a l s o m a r k o v i a n f o r the
A,
v
j to A j .
( " t i m e r e v e r s a l " ) . I n the c a s e o f m a r k o v i a n i t y f o r
A)
2
23
C
=
3
E ^
2
'
3
o f c o n d i t i o n s b ) a n d e) i s a
a n d , as a l r e a d y r e m a r k e d , i n the
A^
(aA. It i s p o s s i b l e , h o w e v e r , t o g e n e r a l -
*
i z e to n o n c o m m u t a t i v e v o n N e u m a n n a l g e b r a s t h e m a r k o v i a n i t y i n a d i f f e r e n t direction w h i c h is invariant under "time reversal"
3.4 T h e o r e m . L e t A
v
A, A 2
be mutually c o m m u t i n g . T h e f o l l o w i n g
3
condi-
tions are equivalent.
3
) [(»«, J * /
K
J
13
J =
tK
h 1
3
s u p p o r t o f (co ) a
/C0 2
]
Pu
3
) I K ^ / ^ l
c
) [C ,,
= [(
C O
£
, )2 i
/ C 0
2J
tK
commutes with
00
x
1
3
l
V
2 ]
W
f
0
r
3 1 1
A
T
i
a
6
3
3
G
3
A
a
2
P 3
A
[(co ) / c o ] a n d X,^' 2
3
K P
a
i
s
t
h
e
£
j
3 -
4
does not depend o n
2
„
3
a G A ).
V
3
3
3
G A
3
[(co )2/co ] a
2
c o m m u t e s w i t h [(CO ) / c o ] a n d A ' q
1
depend o n a j G
e
6
3
EA
d) f o r a
1
2
b
(a
3
0j
1
a
for
A
) [ K ^ V ^ J
2
3
2
2
does not
3
v
= [K ) / 1
0 tK^VK^] =
2
( 0
2] [ K
3
)
2
/
W
2
]
f
0
r
f o r 3 1 1
i
a
a
i
G
E
A
A
V
3
3
G
v 3 a
A
e
3"
A
3 '*P„ i s the
s u p p o r t o f (co ) . fl
2
P r o o f : W e s h a l l a s s u m e i n the f o l l o w i n g c o to b e f a i t h f u l w i t h 2
vector
corresponding
Q . T h e extension to the general situation is straightforward. W e shall
o f t e n use p r o p . 2 . 3 .
101 a) <=> b ) ( a n d s y m m e t r i c a l l y , f ) <=> e) ) . W e h a v e , b y the c h a i n r u l e
O n the other h a n d l(Pay
^ P * I
3
M = [M /o> ]
M
2
[(co ) / co ],
2
1
flj
fl
2
2
1
3
and our claim follows b ) => c ) ( a n d e) =>f)).
=tK ) i
/ "2] t K )
2
3
/ co ] Q, 2
2
w h i c h i m p l i e s the c o m m u t a t i v i t y o f [(to jj/coj fl
w
( I i I\1 31 ) = ^ K ^ V a
= < K
W
a
2
a ) M [K ) 3
= <2 C
2
2]
/C0
(I i I ) [ ^ a
J
2
a
)
2
andco(|a | a |a | )=co 2
1
= <
J
2
*i
2
( K f X K
2
"3
^ a 3
) / ^
a
1
n
Vf°
n
2t & a ^ M ®
) M
[K )2 3
2[K ) ^ ] 3
2
/C0
2]
V^J- ^'
a
>=
Q
>
=
Q
2
1
2
2
2 [ K ) / 2] >; W
2
Q
3
1 2
(0
°3
d ) . It i s e n o u g h to s h o w that c ) i m p l i e s the i n d e p e n d e n c e f r o m a o f :
,3,2 °i
W e have: < J X ' 3
2
2
(| a 1 ) [(co^yco,] Q , a [ ( c o ) / c o ] Q > = 3
2
2
fli
2
2
=< J [K ) /co ][(co ) /co ]Q,a J [(co ) /co ][(a) ) /co ]Q> = =< J
2
o
=X'
2
CO
2
s
(|a | a ) =
a
2
a
a
3
therefore X '
c ) <=>
2
3
^ -
' 2K « >
Q
a
0
w
3
2
2
a
2
2
2
[K ) /co ][(to ) /co ]Q,a J i
2
2
a3
2
2
2
2
fl3
2
2
2
fli
2
2
[ ( ( o ) / ( o ] [ co ) /co ]Q>= ai
2
2
fl3
2
2
102 =< h ^
=< J
^
2
2
(I
2
(I
a
a
iI'
X
i I)
2
K vp
^
'
a
2 [ K
3
)
2
2
2
/
W
2
]
>=
Q
>=
3
1 a 1 a, |
2
V ^ M
a
°3
= o> (| a ,
P
3
) = <J
A ' 3
2
(| a , | ) [(co
2
2
) /co ]Q, a 2
fl
2
2
[(to
This implies
X ' co "l 3
r
= X ' co
2
3
fl
) /co ]Q >. 2
2
1
1
2
c ) => b ) ( a n d d ) =>e) We have: V(co )] | = ^
I Kco
2
2
1 3
(I a 1 ) = ^
2
3
T h e r e f o r e [(o)^
a a
m
(|aj )=
2
| [(co U t o J | .
2
2
1
3
^) K a ) ] 2
2
x
^
2
= u
a
*°
^2
2 ( 3) [( a m
r s
o
m
partial isometry
e
u ( a ) G A ; s o [(co^ycoJQ = J [ (co^ ) /co ]Q= 2
"
J
2 [K
= J
=
j
2
2
so
3
u
L F L 3
) /K ) ] [K ) / 2] 2
2 ( ) K
U
3
2
2
3
2 ( s)
3
t D
as
2
=[(to
fl
1
l
3.5
[(»
J
3
2
2
they
and
q
are both
2
a
Remark
) /(co 2
13
= 2 2 ( 3) [ ( » « ^ J K ^ V ^
f 2
/ C 0
J
2]
U
a
= 2 2 ( 3> ^ K ^ V ^ l
Q
J
U
a
[(CO ) /co ][(co ) /co ]Q
) /co ][(co ) /co ] 2
t 0
2
« V 2l 2 [K )2
[(CO ) /co ]Q
A ;
3
2
5
=
2
K ^ V ^ z ]
0
=
y ^ y ^ ^ y ^ " ;
a
qa
2
2
2
2
if
2
2
2
a
invariant
2
2
under
correspond J
2
we
t o the s a m e state o n
get
[(co ) /co ]£2 fl a
2
2
=
Q and hence our c l a i m .
A
) ] = ( e< ' 1
2
3 } 2
is
abelian
it
is
easy
to
check
that
(|a | ) )2 . S o b y 3 . 1 the e q u i v a l e n t c o n d i t i o n s i n t h . 2
°1
3.4 c o i n c i d e w i t h the u s u a l m a r k o v i a n i t y . T h i s a n d t h e o b v i o u s
"time reversal"
s y m m e t r y o f o u r c o n d i t i o n s i n t h . 3.4 j u s t i f y the f o l l o w i n g
3.6 D e f i n i t i o n co G S ( A ) w i l l spect
to t h e l o c a l i z a t i o n (A
t h . 3.4.
v
be called
symmetric
markovian
with re-
A , A ) i f i t s a t i s f i e s the e q u i v a l e n t c o n d i t i o n s o f 2
3
103 I n [5] the n o t i o n o f M a r k o v states i s w e a k e n e d to M a r k o v c h a i n s ; i n t h e same spirit w e shall w e a k e n our notion o f m a r k o v i a n i t y . 3.7
Theorem.
f
( A
p
A
, A
2
a n d w i t h representing vector Q i n H
(xi ES (A ) l
L e t co G S ( A ) , w i t h
1
A
3
) mutually
commuting,
. T h e n the t w o f o l l o w i n g
c o n d i t i o n s are e q u i v a l e n t . a) there i s a v o n N e u m a n n a l g e b r a A™ j j generated b y t w o m u t u a l l y c o m 2
m u t i n g a n d t r i v i a l l y i n t e r s e c t i n g f a i t h f u l representations Jtj (A|) ( i = l , 2 ) a n d a ^{1,2}
it
2
G
(
S
°{1,2})
A <
S
U
C
n
t
h
3
t
W
2 "
W
h
e
r
e
}
^{1,2
S
IK (AJ 2
=
W
2 *
and
= ^{1,2}
" O l V s } )
b) C
<
2
,
Proof
3
}
(
3
i ) e
A
2
V a
1
G A
= ^
l
>
^ J ^
= < J
2
}
1 , { 2
K ( a
'
1
( l) ' {2,3}
3 }
3
) n
2
Q
3
( e ^ ( a
{
2
Q >
>
3
2
{2,3> " 5
1
3
, a^ j
1
J l
a
E
2 3
3
r 3
,
1 3
1 ^
A
l - {2,3} 3
e A
{2,3}-
A J
2
3
J
(JI^AJ),
w e have:
3
) ) ) =
}
2
{ 2 3 }
( ^{1,2} ( l( i))) ' {2,3}
w h i c h i m p l i e s kj-™
0
° < 1 {2,3} ) =
=
( A ^ ( ^ ( a ^ Q . e f >' ( a
1
f
1
El A
1
2
X
a
b e the s t o c h a s t i c c o u p l i n g f o r the c o u p l e
2
n ^ A j ) ) a n d cp"*^ ^ . T h e n , f o r a
J
( {2,3}»
2
a
a =>b L e t A^*
* {2,3} < o
ffi'
(«l( j>2(
Q
a
(a,) = ( i t "
1
) Q >=
Q
•A^
>
2
'
{ 1 2 }
• i t ^ ) GA
2
b => a . S e t «PMCV ) = <J ^ 2
2
| 2 3 |
(a )0,
then
2
to a state o n the C * - a l g e b r a p r o j e c t i v e
2
2
a Q>;
1
1 2
j b e the v o n N e u m a n n a l g e b r a g e n e r a t e d b y its
G . N . S . r e p r e s e n t a t i o n f o r cpJJ j a n d Jij ( A j ) ( i = l , 2 ) the c o r r e s p o n d i n g 2
presentations o f A
v
tensor
T h e n c l e a r l y t p =co 2
2
b y construction and
subre-
104
W
( i {2,3>) = a
=
a
<
K
h
U { 2
<J
-'
}
{2,3} K ' ( i)". a
3.8 D e f i n i t i o n co £ the l o c a l i z a t i o n
( l) 3
h{2 3}
£
J ' 2
3
1
'
2
Q
' {2,3} a
( 2,3 ) a
{
}
Q
Q
>
'
>= ^ . 2 }
i
( i a
2 4 2
( { 2 , 3 ))• a
}
S ( A ) w i l l be c a l l e d w e a k l y m a r k o v i a n w i t h r e s p e c t to A
(Aj, A,
A )
2
3
i f it satisfies the t w o e q u i v a l e n t c o n d i t i o n s o f t h .
3.7. The
above definition is motivated
3.9 R e m a r k A s c o n d i t i o n
b y the f o l l o w i n g
e) i n t h . 3.2
is s t r i c t l y s t r o n g e r t h a n c o n d i t i o n
b)
i n t h . 3.7, m a r k o v i a n i t y i s s t r i c t l y s t r o n g e r t h a n w e a k m a r k o v i a n i t y . 3.10
Corollary If A
i s a b e l i a n , m a r k o v i a n i t y a n d w e a k m a r k o v i a n i t y are
equivalent. P r o o f I n the a b e l i a n case, as a l r e a d y r e m a r k e d , A ^ = E^I ' 1
3.11
2 , 3
' - ! ' ! ( a ^ ; so 2
condition
3
b)
i n t h . 3.7
2
' ^ (a,)
=
g i v e s o u r statement.
E x a m p l e s a) C o n d i t i o n s e) i n t h . 3.2 a n d b ) i n t h . 3.7 s h o w h o w
to
c o n s t r u c t e x p l i c i t e x a m p l e s o f m a r k o v i a n a n d w e a k l y m a r k o v i a n states. N a m e l y , it i s e n o u g h
to take a n y t w o v o n N e u m a n n a l g e b r a s A
2
intersection acting on Aj
2 3
a H i l l b e r t space, consider
j t h e y generate a n d a state c o ^ j i n S ( A ^ j ) 2 3
coupling t h . 3.7,
A : Aj
-»
defines
a
A ^ state
3
j on
3
the v o n
and A
3
with
Neumann
trivial algebra
T h e n one takes a stochastic
s a t i s f y i n g c o n d i t i o n e) i n t h . 3.2 o r c o n d i t i o n b i n the p r o j e c t i v e
tensor
product C * - a l g e b r a of A j
a n d ^{2 3} T h > as p o i n t e d out i n s e c t i o n 2, the e x t e n s i o n o f t h i s state to the e n
v o n N e u m a n n a l g e b r a g e n e r a t e d b y its G . N . S . r e p r e s e n t a t i o n w i t h r e s p e c t to it has the r e q u i r e d m a r k o v i a n p r o p e r t i e s . b ) A n e x p l i c i t e x a m p l e o f a m a r k o v i a n a n d s y m m e t r i c m a r k o v i a n state i s the l i f t i n g o f a c o m m u t a t i v e M a r k o v state f r o m a c o m m u t a t i v e s u b a l g e b r a to the v o n N e u m a n n a l g e b r a v i a a p r o d u c t n o r m o n e p r o j e c t i o n . M o r e p r e c i s e l y , let A j , A,
A
B
respectively.
2
3
3
be m u t u a l l y c o m m u t i n g w i t h a b e l i a n v o n N e u m a n n s u b a l g e b r a s B
v
B, 2
105 3 L e t B b e the ( a b e l i a n ) v o n N e u m a n n
subalgebra o f A generated
by N
B
W e a s s u m e there i s a n o r m o n e p r o j e c t i o n
V
l o c a l i z a t i o n , i.e. s u c h that a. e ^
e (Aj) = B
e f r o m A to B r e s p e c t i n g the
a n d that
{
e ( a a a ) = e (aA t(a^ e ( a ) f o r 1
2
3
3
(i=l,2,3).
L e t c o b e a state o n B s u c h that e ^ f u ^ b j ) G B m a r k o v i a n i t y ) . T h e n the state co =co
B
f o r each b
2
B
£ Z?
3
(classical
3
• e is markovian and symmetric markovian
a n d m a r k o v i a n f o r the l o c a l i z a t i o n ( A , , A , A ) . 2
3
T o c h e c k m a r k o v i a n i t y , w e n o t e that B both in A |
2 3
j and in A ; so B
2
2
W e s h a l l s h o w n o w that X markovianity o f co
in th. 3.2. T h e m a p p i n g a
( l {2,3}> = < { 2 , 3 } ( ^ a
J
-
< 2
l
E V S > >
<
J
{
(b ) =
= ^ O ^
1
'
2
^
= co(E^<2,3}
= co ( a
'
B
t
a
{ 2
3 }
2
*
E
a
a
t
* e
A^ '* 1
2 , 3
* (aA E B
E
1
a
{ 2
) e (a
Q
,3
{ 2
3}
a
}
therefore a
{2,3}
°
Q > f
r
3
1
G
A
3
2
2
l '
a
{2,3}
& 4
{2,3}.
=b*Q, a n d f o r
2
) :
Q
1
2
2
*
=CO(E ( ) E 3 L
2
E
E
) ( l)
(a
3
{ 2
3 }
))
) (a ) aa 1
E
+
,
3 }
a
{ 2
{ 2
Q>=
( {2,3 ) = a
= co
}
(E
(a
t
3 }
))
=
) .
T h e m a r k o v i a n i t y o f co w i t h respect t o the l o c a l i z a t i o n (A by
coupling
to c h e c k that
J ^ j b Q= J b Q
) = ^ i ^
)))
1
t
Q
2
f o r a £ A p the
:
• e ( a ) i s clearly a stochastic
) Ol) >
£ B
(a )
a n d t h e r e f o r e c o n d i t i o n e)
2
) ( i ) > {2,3} >= ^ j , ' ' " * *
E
) ( i)
(E ( a
2
3
(aA
{ 2 , 3 >
6 ^ ( b
1
2,3}(*T ™
*
3 )
We have, since for each b
b
'
W e only need
3
3
1
S
f o r the c o u p l e ( A , , A ^ j ) . W
c o
- * E^' <2,3)
l
i s e x p e c t e d w i t h respect to c o ^ j
C A^.
i m p l i e s then
B
2
y
A , AA 2
symmetry. I n o r d e r t o p r o v e the s y m m e t r i c m a r k o v i a n i t y f o r co n o t e f i r s t that
follows
106
for all a . G A j (i=l,2,3). Indeed, w e have = co (| a , |
u> ^(* ) a
2
2
= co ( e ( | a | ) a e ( 2
1
|a | a )=co(e(|a | 2
3
2
1
| a | ) ) = a^ffi
2
3
2
2
(a^.
^
2
3
|a | a ))
2
O n the other h a n d : co(e(|a | )a
2 E
= co (E( |
E ( a ^ E (| a 1 ) ) = t o
2
1
As
[ K
E|
V
A s co
f
B
f
2 )
2
3
E
3
E
1
2
2
E ( | f l i f t
2
|a | ))) = 3
i^
2
(E ( a , ) ) .
i s a n o r m o n e p r o j e c t i o n this i m p l i e s that
a
A
|
H
(|a | ))=co( ( |a | a E(
l
2
]
=
K ^ i a j ) ! s(\aj)i 2
)
/
2
=
K^s)
^
W •
is m a r k o v i a n o n B abelian our c l a i m follows b y 3.5.
A s i m p l e e x a m p l e o f this s i t u a t i o n o c c u r s i f M i s the q x q m a t r i x a l g e b r a , N the s u b a l g e b r a o f d i a g o n a l m a t r i c e s i n M i n a g i v e n b a s i s , A
1
= M®I®I,
trace
A
2
= I®M®I,
preserving norm
A
3
= I®I®M,
A
B = N®N®N
= M®M®M, a n d E i s the
one projection f r o m A to B .
A k n o w l e d g m e n t . T h i s p a p e r w a s b e g u n d u r i n g a o n e y e a r v i s i t at the Institut f u r A n g e w a n d t e M a t h e m a t i k o f the H e i d e l b e r g U n i v e r s i t y t o w h o m , as w e l l as to t h e S o n d e r f o r s c h u n g s b e r e i c h
1 2 3 t h e a u t h o r w o u l d l i k e t o t h a n k f o r the s u p -
p o r t . It i s a p l e a s u r e t o t h a n k a l s o P r o f . W . v o n W a l d e n f e l s a n d P r o f . M . L e i n e r t for their most k i n d hospitality.
Bibliography 1.
L . A c c a r d i . N o n commutative
M a r k o v chains. International S c h o o l
of
Mathematical Physics, Camerino (1974) 268- 295 2.
L . A c c a r d i . Topics i n Quantum Probability, Physics Reports 169-192
77 (1981)
107 3.
L . A c c a r d i . C e c c h i n i ' s transition expectations
and M a r k o v chains. Q u a n -
tum Probability and Applications IV. Springer Verlag Lecture Notes
1396
( 1 9 8 9 ) 1-6. 4.
L . A c c a r d i - C . C e c c h i n i . C o n d i t i o n a l expectations
in von Neumann
alge-
bras and a theorem o f Takesaki. J.Funct. A n a l . 45 (1982) 2 4 5 - 2 7 3 5.
L A c c a r d i , A . Frigerio. M a r k o v i a n cocycles, Proc. R o y a l Irish A c a d .
83
A(1983) 251-263 6.
C . C e c c h i n i . Stochastic couplings Probability
for v o n N e u m a n n algebras.
and Applications IV. Springer
Verlag Lecture
Quantum
Notes
1396
(1989) 128-142 7.
C . C e c c h i n i . A n abstract c h a r a c t e r i z a t i o n o f c o - c o n d i t i o n a l
expectations.
M a t h . Scand. 66 (1990) 155-160 8.
C . C e c c h i n i - D . P e t z State e x t e n s i o n s a n d R a d o m - N i k o d y m t h e o r e m conditional
expectations
on
von
Neumann
algebras,
Pac.
J.
for
Math.
138(1989) 9-23 9.
C.
Cecchini-
D.
Petz.
Classes
of
conditional
expectations
over
von
N e u m a n n algebras. J . F u n c t . A n . 9 2 (1990) 8-29 10.
A . C o n n e s . S u r le t h e o r e m e de R a d o n - N i k o d y m p o u r le p o i d s
normaux
fideles semifinis. B u l l S c i M a t h S e c H I 97(1973) 2 5 3 - 2 5 8 11.
A . C o n n e s , V . Jones. Property T for v o n N e u m a n n algebras. B u l l . L o n d o n M a t h . S o c . 17 ( 1 9 8 5 ) 5 7
12.
M . Takesaki. Theory o f Operator Algebras I. Springer Verlag 1979
Quantum Probability and Related Topics Vol. VII (pp. 109-123) © 1992 World Scientific Publishing Company Q U A N T U M O F
F L O W S
F R E E D O M
A
N
W I T H
I N F I N I T E
D T H E E f t
P.K. Das
and
Kalyan B. Sinha Indian Statistical Institute
September 1990
D E G R E E S
P E R T U R B A T I O N S
Indian Statistical Institute Calcutta - 700 035
New Delhi 110016
109
110
1
Introduction
The general theory of quantum stochastic flow (or quantum diffusion) has recently been studied by Evans [1] for finite degrees of noise freedom and by Mohari and Sinha [2] for countably infinite degrees of freedom. In both cases the structure maps are bounded and in the second instance, they satisfy moreover a strong summability conditions. On the other hand, Evans and Hudson [3] considers stochastic perturbations of quantum stochastic differential equation (q.s.d.e.) of Hudson-Parthasarathy (H.P.) type [4] with time dependent coefficient operators. Here the time dependence of the coefficients are given exactly by quantumflowsof the type studied by Evans in [1]. This iB the stochastic equivalent of the following problem: let U and V satisfy in a Hilbert space the differential equations;
dV
=
iHoVdt
and
dU
=
iV* HVU
dt
with
U{0) = V(0) = / and H and Ho bounded selfadjoint. Then it is easy to see that VU satisfies: d(VU) = i(H + H)VU dt, i.e. VU gives the perturbed 0
motion due to the perturbation of H by HQ. Here we study the same question as in [3] i.e. the perturbation of a q.s.d.e. of the H.P. type but in the background of the theory of Mohari-Sinha [2] with countably infinite degrees of freedom of noise.
2
Notations and Preliminaries
All the Hilbert spaces that appear here are assumed to be complex and separable with scalar product < •, • > linear in second variable. For any Hilbert space Ti we write T{71) and B[f{) respectively for the boson Fock space over H and the C"-algebra of all bounded linear operat ors in H. Let W arid A." be twofixedHilbert spaces and let % = X (iR )\@ /C. and 2
0
W = «o©r(7*).
+
(2.1)
111
For Rn
y / € "H we denote by e(/) the exponential or coherent vector in r(7i)
associated with / and by £ the linear manifold of all vectors of the form u©e(/) with ti € He and / € W. Also we adopt the convention of writing ue(/) in place of u © e(/). Note that £ is dense in H. Choose and fix an orthonormal basis {cy/yij of K and set E'
k
=
|ej. > < ej|,j, k > 1. The basic quantum stochastic processes of the theoryare: A(M, =
|0I)
©£?) if , « , ! / > !
® *j'UM*^)
^ - '" ° ifi/>l,u = 0
if i = v = 0,
a
(
X
m
e
1
(2 ">)
=
r
where Af^ is the multiplication operator by the function <j>. The quantum Ho multiplication rule is then summarized as : =
d\}dA*
where
f*
run over
et,ii..n.
v > 0
= 0 if a = «/ = 0 =
Convention :
eyA.%:
(2.3)
6 " otherwise.
Here we have adopted the convention that the Greek indices
0,1,2,...
while the Roman indices run over
1,2,...
We shall also use
the summation convention that repeated Greek indices imply summation from 0 to co while repeated Roman indices mean summation from 1 to oo. An
adapted process
is a family
L =
{•£>£(*)}<><) of operators defined on £,
having each adjoint also defined on £, and such t hat for each t > 0 corresponding to the decomposition :
U = (Tic © r ( i [ 0 , 2
t] © £)) © T(L*[t,
oo) S) £ ) , Z(<)
admits the factorization L{t) = L © P. where L is an operator in the first t
t
factor while P is the identity operator in the second factor Hilbert space. Such a process is said to be square integrable if it satisfies furthermore for each v
112
and
t >
0: /' £
\\lt{,)ue{f)\\*d» {.),
(2.4)
f
where u,(t) = jftl + ||/(*)||)
A B in [4], one may form the (quantum) stochastic integral of a square integrable adapted process L :
In the sequel we shall restrict the K -valued functions / to belong to a dense subspace
M =
{/ € L (2!? ) © K\P{») 3
+
=<
e„f(s)
> = c
0 except for a finite
number N(f) of indices j and for all t > 0}. We shall write /,(*) — fi(g) for j > 1 and /<>(«) = /°f j) = 1 for all s > 0. We shall also denote by E(M) the linear manifold generated by ue(f),
€ M and u € Wo-
f
Then one has the following properties of the stochastic integral XU){u,v€H ,f, eM): 0
9
(i) A (t) exists as a strong integral on £{A4) and defines an adapted process, < ue(f), X(t)ve(g)
(ii)
(iii) If X'{t)
L' S
>=
t
< »e{f),
L*(s)ve{g)
> ds.
(2.6)
is another adapted square integrable process and
{L'f[s)}
=
f* f A*)g"(s) Jo
torn
<X'(i)ue(f),X(t)ve{g)>=
d*f, (s)g"(s)
f
t
•
Jo {< X'{*)ue(f
),Lt(s)ve(g)
< L*{ )ue(f).Ji{*)ve(a) a
(iv) ||A'(0«(/)|| < 2 exp (^(0) 3
> + < !*(•)«(/).X{i)vc(g)
> +
>}.
(2.7)
£ »>«
/*IWW«(/)l| A'/(«). a
•'
Cl
For the proofs of these properties, the reader is referred to [4] and [6].
( -«) 2
113
3
Unitary stochastic evolutions and quantum stochastic flows
Here we collect some of the results of [2] that will be needed in the next two sections. Lemma 3.1 : Suppose {A*} and {Bj,} be two families of bounded operators in a Hilbert space such that £ -AJ.-U and £ #JSjt converge strongly. Then £AJ5/t converges strongly. Suppose Lfc,5* be bounded operators in 7io and H be bounded selfadjoint such that : £ \\Lku\\ < C\\u\\ 2
3
for some
C >
0 and all u € «o-
S f 5* = SjSi* = 4l,
(3.1)
where the second sum is assumed to be strongly convergent. Next we set: St - 6*1
if
I„
if p>
1,^ =
- L ; S J
if ^ >
Uft
iff Then it is easy to verify the
\L\L
k
fi, v > 1
0
(3.2)
=o
if ^ = i/ = 0.
uniiaritg condition:
L% + V£ + L?Lt
= Lt + V;
+ LIU?
= 0,
(3.3)
where
EPC«lt
a
(3.4)
Conversely, one can show that if the family {£[,'} of bounded operators in satisfy (3.3) and (3.4), then they admit a representation (3.2) subject to (3.1).
114
Theorem 3.2 : Let {IJ} of bounded operators satisfy (3.3) and (3.4) (or equivalent!}- (3.1) and (3.2)). Then the q.s.d.e. dU{t) = L£JA"(i)U{t),
(3.5)
U{0)=J.
has a unique unitary adapted process as a solution. Let A be a unital *-suba!gebra of B[H ). 0
A family of linear bounded maps
AJ : A —' A is called structure map if AJffl
(i)
=o (3.6)
(«) (iii)
Aj(zy) = *K(V) + W*\9 + W*)Kiv),
w h e r e x, y <E . 4 . a n d (iv) i f there exists for each v > 0 c o n s t a n t s a a n d a f a m i l y {D^}, i
l€ r
v
> 0 , a c o u n t a b l e i n d e x set J „
o f b o u n d e d o p e r a t o r s i n Ho s u c h t h a t f o r x € A, u € Tto
£
PfrHP
<
£
II^MI ™ t h 2
(3-7)
£
We define a quantum
IIWP
<
a ||«|| . 2 M
stochasticflowon . A
2
as a family {jt}t>o of contractive
*-homomorphism from A into 5(H) satisfying for each x € A : (0 io(*) = *>
(ii) jt[I) = I and if(ar) is an adapted process, (iii) there exist structure maps AJ obeying (3.6) such that j (z) satisfy the f
q.s.d.e. = ix(Aj(*))dA;;(i).
(3.8)
Theorem 3.3 : Suppose that structure maps AJ satisfy (3.6) and (3.7). Then there exists a quantum stochastic flow j on A satisfying (3.8). Furthermore,
115
the map (t,*) -+ j (x) is jointly continuous in the strong topology of Ti with t
respect, to the strong topology of .4, and j satisfies the estimate for 0 < t < T t
£
\\*D»,lKX---D»*\r}Mf)\\*-
(3-9)
The proofs of Lemma 3.1 and of theorems 3.2 and theorem 3.3 can be found in PI
4
Perturbation of quantum flows
Given a quantumflowjt on .4 constructed as in theorem 3.3 with structure maps A£ satisfying (3.7) and assuming that the operators Z£ € .4 for all ft, v > 0, it follows that the time dependent coefficients jt(ZJJ) also satisfy a relation similar to (3.3). Then we may try to solve the q.s.d.e. similar to (3.5) with J<(If) replacing ££, viz. dU(t)
= j {L» )U(t)dA^ t
v
C'(0)
= /.
(4.1)
We have the main result in the next theorem. Theorem 4.1 : Let j be a quantum stochasticflowon A as in theorem 3.3. and let £{,' € .4 V ft, v satisfying (3.3) and (3.4). Then there exists a unqiue unitary adapted solution to (4.1). We break up the proof into a few lemmas. Lemma 4.2 : Assume the hypotheses of theorem 4.1. Then there exists a unique isometric solution of (4.1).
116
Proof : A B usual we set up the iterative scheme. U (t)
= I, U (i)
0
n
W e n o t e that by h o m o m o r p h i s m
= I + J* j,{L!)U -A<)d\;.
(4.2)
n
p r o p e r t y the s t r o n g c o n t i n u i t y of
and
(3.3)
for $ € H
£ Eiw.wii
= £
2
=
i/=0
< 5Z(PT+2ira)WT h u s by
(4.2), (2.8)
and
(4.3)
(4.3)
we h a v e t h a t
IIPMO-^OM/MI' <
£
C(f)
/'||i,(ir)[C^i(*)-^- (<)]«e(/)!| ^/(<) a
=
»<»<"•(/) <
( \\[V^ {M)-U^.,[s)]ue{f)\\^f(a)
<M/)
t
1
Jo
C7 (/)"|l«e(/)|| v \tTI*\
<
2
l
s h o w i n g t h a t U (t)ue(f) n
(4.4)
f
converges u n i f o r m l y for i €
[0, T ] , «
S i n c e a t e a c h s t e p o f t h e i t e r a t i o n t h e i n t e g r a l i n (4.2) p r o c e s s , t h e l i m i t ( d e n o t e d U(t)) c l e a r t h a t U(t) t h e n (7(f) -
satisfies
U'(t)
=
t h e o n e l e a d i n g t o (4.4)
(4.1)
o n £[A4).
tij.(Lt)W(»)
-
If U a n d W
U'(a)]dA*
this proves the uniqueness. a
=
€
A4.
adapted I t is also
are t w o s o l u t i o n of
(4.1),
a n d a n i d e n t i c a l c a l c u l a t i o n as
gives an estimate :
^p^H«e(/)|i
a
||f7«)«e.(/)||
defines an
e x i s t s a s a d a p t e d p r o c e s s o n £(JH).
\\[U - t / " 3 « e ( / ) l | < CAf)" and
€ Ho, /
Finally,
Ueeef/»|- follows f r o m
phism and strong continuity property of
the isometry
(4.1), (2.7), (3.3) j. t
2
of V
on
a n d the
£{.M)
i.e.
homomor-
117
Remark The perturbed equation (4.1) differs from the corresponding one in the work of Evans and Hudson [3] in that U appears on the right in (4.1) instead of on the left. This on the other hand is in conformity with the material in [2]. We continue to denote by U(t) the isometric extension (adapted) to whole of Ti and note that by (2.6) the bounded adapted process U'(t) satisfy the q.s.d.e. dU* = WjtiL?)dA*.
(4.5)
F(t)
(4.6)
We set
and observe that
F
2
= —F,F*
= U(t)U{ty-T, = F
and \\F||
Next we use the quantum
= 1.
Ito's formula (2.7) and (3.3) along with equations (4.1) and (4.5) to derive the q.s.d.e. satisfied by F(t): dF = {ji(L*)F
+ Fj (L?) t
+ 3,(11 )Fj ( t
IHW
(4-7)
with F(Q) = 0. To prove theorem 4.1 we need to show that (4.7) admits F(t) = 0 as the solution and for this we need the next lemma. Lemma 4.3 : Assume the hypotheses of theorem 4.1 andfixx 6 .4. Then the sum Zj* X
(x)
k u
converges strongly in TU- Set 0J(x) =
L^'x +
A£(*) + Zj" Aj(*).
Then for each f, there exist aj, > 0 , a countable set 7„ containing 2„ and bounded operators DC in Ti such that for u € 7io
»«=o
„€X„
where
El|Z^a|p
118 Proof: (i) If we set for fixed fi, v, A = L* and B = Aj(z), then thefirstpart k
k
follows from Lemma 3.1, hypotheses (3.7) and (3.3). (ii) Thus 0*(x) is well defined as a map from A into itself since V~ belongs to the algebra A and since the algebra A is strongly closed. Now
£ll*?(*)«ll <
3(£||i^«|! + f:i|A^)«j|
a
2
3
+ £ w > \ n
(4.8)
where we have written M» = £X**Aj(a!), By (3.3) it follows that ^||i^"as«|| = fc=i " -2 Re < xu,Ll,xu > < 2\\Ll\\ \\xuW . This and (3.7) yields that the contribution of thefirsttwo sums in (4.8) admit estimates of the type required. 2
2
Set. for n > m > and M£(n,m)
n
=
Q,M»(n.ni)
rk*\k
£ Z-**Aj(i) so that Af£ = k=m+l
is strongly Cauchy as m—
*=1
oo.
Mf(oc.O)
Next, we note that, by (3.3)
M=m+1
>*=1
TI
= -2 £ tfe < A/5(n m)u,A^(sr)«> T
/=m+l
< 2( £
||^(« m)«|| £ 3
)
/=m+l
HA^x^lD / , 1
2
i = m4 1
which leads t o £||M„>,rn)«||
3
and £||Af,"(n,0)tt|p p=i
< 4 £
||A*(«)u|P,
< 4£||A*(VH| < 4|kl| «2lMI . 3
3
I
ib=i
3
(4.9)
Thus {HM^fn.OJuH},, as an t - sequence is Cauchy in n and hence converges 7
in £ . On the other hand M£(n,0)u 2
-* M£u
by part (i). Therefore by (4.9)
£ l |M>|| < ||,W?t.|| + 4£||Af.( -)«|| . 3
f=0
a
a
h=l
3
(4.10)
119 The required result now easily follows f r o m (4.8) a n d (4.10).
Proof of theorem 4.1 : i(Fjtix))
•
B y the Ito's f o r m u l a (2.7) and (4.7) we get
= \jt(L',:)Fj (x)
+ FM9^))
t
+ MLi)FU9t{*))]^
>
which b y (2.8) leads to \\F(t)j (x)ue(f)f
<
t
3C(f)
f
£
du (s){\\j,[L:)Fj (x)ue(f)f f
t
+ IU.U^.(<(*))«(/>lr'}.
+mMi*))"eU)\f
(4.11)
Now l e m m a 3.1 a n d a calculation exactly similar to the part (ii) of the proof of l e m m a 4.3 leading to (4.10) shows that
<
<
i:\\j.(Ll)Fj (et(x))ac(f)\\ t
sup E i u ^ H i
3
Co
m
H*/.i«frJW#MP+«
ȣl
IMI=lfc=l <
E
+ 4 E
2
M
x
E
y
^
i
m
II^WM/MI
Kil
\\Fj.mx))ueif)\\ .
E
3
(4.12)
2
A p p l y i n g (4.3) a n d (4.12) to (4.11) we get
f
dv (s){\\F{,)j,(x)«e{f)\\ f
•M) Jo
+
2
E \\Fi*)3.W(*))Mf)\n „>«
(4-13)
Iterating (4.13) n times, we have that
Jo
Jo
Jo
(?) E I W ) i M < ^ ( * ) M / ) | | + (") E I I W * , ^ •«£ (*)M./)II 3
+•• •+
E
H W i « » •
1
• •^ (*))«e(/)|| i. ,
a
,
S
(4.i4)
120
Now by the bounds of theorem 3.3 and lemma 4.3. Ellit(.W)«e(./)|P -
h
v^T
to
l | e ( / , n
fl
/»,!• « « l K , , 0 < ! . < ? ' ( / ) 0 « < » 1
I
< W ) £^ ft
_ E fi&'i •<»/£
v
°8l.J.J
/
_.»2i/
,.2,Jtii|_l|2
II..113
< c f / K E ^=f- ( E «?)< E
A-
V * !
,,=o
3
p=0
< CM/)IWI ll«ll lE «?) a
a
An identical calculation gives ns that
E
IIM*E
- WM./)>II
3
SW)u.j|*|MPlE«a . -
Substituting the estimate (4.15) in (4.14) and observing that
(.4.15) = 1, we
have \)F(i)j,(x)«e(/)|| < C.(/) J
\\ \f || |p[l + £ <*?]*, x
u
and since n is arbitrary this leads to F(i)j (x)ue(f) t
= 0
V t > 0, x € . 4 , « € « , / € A i .
Setting x = I, we get the required result .
•
121
5
Applications
Let {-£(,'} and {AIJ} be two families of operators in the strong-closed algebra .4 satisfying the unitarity relations (3.3) and (3.4) and we define (L * M)l = I£ + M£ + I£U$.
since by (3.3) and (3.4).
^vIMI*
for « 6
'Ho,
£||2,;%l| = £P*«1| 3
2
(5.1)
<^
Ell^MP <
M ?
it follows by Lemma 3.1 that the sum in (5.1) converges
strongly and hence (L + Af ){J € A
€ A satisfying
In fact the set of
(3.3) and (3.4) forms a noncommutative group under • operation defined in (5.1) with 0 as identity and inverse {lT )$ v
=
H we use the representation
of (3.3) as in (3.2), then one has a more transparent interpretation of the * operation. Let ££ be represented by the pair {-£*,SJ|j, k > 1} and Af* by the pair {Mk,Tl\j,k NS
s
>
1} in the scheme (3.2). Then a simple calculation shows that is represented by the pair
( L * Af)5
S"J2^Jj, > l}. We consider A
= (B^Ii
-4 —
{N
=
k
(v(A),
L
f c
+ S$Mj,Ui
=
(ST){ =
equipped with the topology
induced by the convergence of £||oju|| < C||u|p V u g Ho, a = {a S .4} € -4. 2
3
j>i
Since .4 is strongly closed, A becomes a vector of space with an ,4-valued scalar product < a, b
£<*y&; , the sum converging strongly by Lemma 3.1. Then
S = {S } and L = {£*} act on A by (Sa)j = S a and by translation : 3
k
k
}
o —» {ajt + L * } respectively. In this picture the pair (L,S) forms naturally a group with group operation being the semidireet product viz. (L, S) • (M, T) = By what we have said above, it is then clear that these two
(£ + SM,ST).
groups are isomorphic. In the context of countable state Markov chains (see [2]), we set M = multiplication by m (jf) — x
\/t(y.xy)
if x
e and 0 otherwise V y,
x € A",
the state
space and T = identity and L = 0 and S = {S } with S the left translation x
x
122
in A*. Then it follows that (0,S) • Then
(S ,S) m
(M,I)
= (SA/,5) where
SM
= {S m } x
x
.
is the pair that represents the classical Markov chain. In other
words, the composition of a pure Brownian motion with a second quantization is equivalent to a classical Markov chain. Similarly the pairs ({SjW,}, {5*}) and ({-Sj ^}, {S ^ }) corresponding to two classical Markov chains compose 1
1
1
to give the pair ({5 m + k ), I) which correspond to a non-classical Brownian x
x
T
motion. Finally, as in [3] we want to relate the * - operation in (6.1) with the perturbation of diffusion constructed in the Section 4. Let V be the unitary solution to the q.s.d.e. dV
set jt[se)
= V"[x
= L*Vik%
V{0) = I,
(5.2)
© t)V, and consider the q.s.d.e : dU = UWiWd^,
U(0) = / .
(5.3)
We know that by theorem 4.1, (5.3) has an unique unitary solution and it is easy to see that VU satisfy d(VU)
= (L * M)t
VVih' . p
(5.4)
This is a natural quantum stochastic generalization of the perturbation problem mentioned in the introduction.
123
References 1.
M.P. Evans, Existence of quantum diffusion, fields,
PTOD.
Theory and related
81, 473-483 (1989).
2. A. Mohan and K.B. Sinha, Quantum stochasticflowswith infinite degrees of freedom and countable state Markov processes, Sankhya, 1990. 3. M.P. Evans and R.L. Hudson, Perturbation of quantum diffusions, University of Nottingham (U.K.), preprint
(1989).
4. R.L. Hudson and K.R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys.
93, 301-323 (1984).
5. K.R. Parthasarathy. K.B. Sinha, Representation of a class of quantum martingales II, Quantum Probability III, ed. L. Accardi and VV. von Waldenfels, Springer LNM
1303. 231-250 (1988).
Quantum Probability and Related Topics Vol. VII (pp. 125-137) © 1992 World Scientific Publishing Company KOLMOGOROV
125
F L O W S , D Y N A M I C A L E N T R O P I E S and M E C H A N I C S GERARD
G . EMCH
Department of Mathematics, University of Florida, Gainesville, F L 32611, U S A January 4, 1991 Abstract. The classical and quantum K-flows theories are reviewed, with special emphasis on the statistical mechanics of non-equilibrium phenomena; some recent results are listed, and four open problems are isolated.
0. I n t r o d u c t i o n . In the mid-sixties, the announcement of the results of the Russian mathematicians [38, 39, 2, 3, 40, 41] on the ergodicity of classical hard balls systems immediately impressed the statistical mechanics community, see for instance [45]: a mathematical theory in which Boltzmann's ideas on the physical foundations of statistical mechanics could be cast had been found, namely the theory of Kolmogorov systems. Since we believe that at the microscopic level, i.e., at the fundamental level of statistical mechanics, Nature follows the laws of quantum mechanics, the above contention required that one should also have a quantum gneralization of these classical Kolmogorov structures. Section 1 of this lecture sketches an overview of the historical development of these theories, classical and quantum. In Section 2, the general (i.e. quantum) definitions are presented first, followed by a discussion of the sense in which these definitions include the classical Kolmogorov flows as particular cases. Specific examples are described in Section 3. The physical background for some of these examples is discussed in Section 4. Some of the ergodic and hereditary properties of Kolmogorov flows are listed in Section 5 and 6. Finally, alternate definitions of dynamical entropies are considered in Section 7. 1. H i s t o r i c a l O v e r v i e w . To draw any such overview is a dangerous tasks, since a sketch requires many omissions: drastic choices have to be made that reflect then the particular biases one warps around one's own involvement in the development of a particular subject. As early as 1898, Hadamard [20] isolated the property of sensitive dependence on initial conditions that is exhibited by the geodesic flow on the Poincare half-plane H (the twodimensional simply connected Riemann manifold of constant negative curvature). It was not until the middle of this Century that the importance of this discovery came to be recognized. A group-theoretical exploitation of the action of SL(2,R) on H led Gelfand and Fomin [17] to the proof of the ergodicity of this flow, when resticted to the unit tangent bundle of a fundamental Poincare domain, i.e. to T\(H /T) with T co-compact. Soon thereafter, Anosov and Sinai [3, 37, 38] showed that this ergodic behavior was in fact the bottom rung of a ladder, the higher rungs of which move successively from the probalistic 2
2
2
Research supported by N S F Grants DMS-8801749 and DMS-8802672
126 notions of mixing, Lebesgue spectrum, and Kolmogorov flow, to the topological structure of Anosov foliations that manifests itself most readily in differential geometry when one studies the geodesic flow on a Riemann manifold with negative curvature bounded away from zero. This hierarchy is nowadays part and parcel of most mathematical courses on classical ergodic theory [5, 35, 42, 46, 12]. From a physical point of view moreover, these mathematical strutures allow to prove that classical systems of identical balls with elastic collisions (hard spheres) satisfy the Boltzmann ergodic hypothesis (metric transitivity); Arnold's book on mechanics [4] presents in Appendix 1 a graphic sketch of how the system formed by two hard disks moving freely on a square billard with purely elastic walls can be understood as a limit of a system with a very localized region of very high negative curvature: this region is responsible for the exponential sensitivity to initial conditions that entails the randomness in the evolution necessary for the ergodic hypothesis to hold. For the purpose of quantum statistical mechanics, it was then necessary to extend the probalilistic concepts, involved in the theory of the classical Kolmogorov flows, to the study of dynamical groups acting on non-commutative algebras. This program was started in the mid-seventies [14], developped in the early eighties [36], and carried out to the fullest extent by the Vienna school [29, 30, 31, 32, 33, 10]. The randomization that occurs in the course of the evolution of the classical Kolmogorov systems is measured by an invariant, called the dynamical entropy [24]; as we shall discuss in Section 7, the extension of this concept to the the quantum framework is less than straightforward [14, 11, 10]. 2. M a t h e m a t i c a l Definitions o f K - F l o w s . As pointed out in [31, 32] it is didactically better to split the original definition [14] into two categories: the purely algebraic axioms A, and the (quantum) probabilistic axioms f l . A X I O M S A.
; A C*-algebraic
unit I ; the action automorphisms
a : (t, N)
K-Flow
is a triplet {A/"; ct; A}
where Af is a C*-algebra
£ R x Af i-» ctt[N] £ A/" defines a group {a
of Af ; and A is a sub-C*-algebra
t
with
\ t e R]
of
of Af such that with At = ct \A}, we t
have:
(a)
t > 0 implies
(b)
At D A
\lA =M t
t (c)
l\A
=
t
CI
where V i n (6) means that the LHS is the smallest
C*-subalgebra
of Af that contains
all
A. t
AXIOMS f2. . A W*-algebraic
K-Flow is a quartet {A/ ; a; A;u>] where Af is a von
algebra; A is a sub-von Neumann-algebra
-
of Af; {A/ ; a; A} is a C -algebraic
however V i n the LHS of (b) is now the W-
-
rather that the C*-algebraic
Neumann
K-Flow,
where
l.u.b.; and ui is a
127 normal state of Af (i.e., w is a positive eiement is the predualAf. such that (d)
ui is faithful
(e)
u o a, = ui V t £ R
(/)
of Af with ;I >=
1)
AVtER
where {
As for the connection between the above definitions and the classical concepts, recall (see, e.g., [5, 35, 42, 46, 12]) that a classical K-flow is defined as a quartet {(X, E ) ; <j>; C; p) where (X, E ; p.) is a (non-atomic) probability space; {<j>t | t £ R] is a group of measure preserving transformations of (X, E ; u); and £ is a c-sub-algebra of £ such that: (a)
for positive t, (j>t[C] refines £
(&)
{MQ
()
{"MC] 11 € ii} has trivial past-tail
c
\ t£R}
generates E
To recover our formulation, let Af = C°°(X, E) (with ordinary ess-sup norm), the predual of which is the Banach space £ ( X , E ; p.). We realize C°°(X, E) as a von Neumann algebra by considering its elements as multiplication operators on the Hilbert space X
U = C (X,Y,;p); let further {a \ t £ R) be defined by (a,[N\)(x) = N(
t
t
the sets belonging to the cr-algebra £ ; and finally, let
It is then straightforward to verify that the quartet {Af; a; A; w) just constructed satisfies axioms A and 0 , i.e., is a W * - algebraic K-flow (where
characteriaitic feature of the
The
-algebraic K-flows obtained, in the manner just described,
from classical Kolmogorov flows is that Af (or equivalently A) is abelian. 3. E x a m p l e s o f K - F l o w s . The diffusion equation, both for its historical interest and for matters of principle, is an archetype for the situations encountered in classical non-equilibrium statistical mechanics; since the work of Einstein, we know that to such a macrospcopic evolution is attached a microscopic evolution, known as Brownian motion. For a detailed mathematical study of the classical flow of Brownian motion, see e.g., [21]; we only briefly recall here its principal ingredients. Let X
=
S* be the dual of the Schwartz space S of real valued, rapidely decreas-
ing functions on R; £(/l,/a,-.•>/»;#)
E be the
= {x € X \ (< x;h
>,<
x;f
2
>,-••
< x;f„
>) £ B},
where n runs
128 over the positive integers, {fi,f , • • • , / n ) runs over the collection of n-tuples of elements /,- G S, and B runs over the Borel subsets of i i " . To complete the description of the probability space on which the flow of Brownian motion is acting, choose 0 G i i with 0 > 1; let || . || denote the £ - n o r m on S; C : f G S i-» exp{-9||/|| /4} G i i ; and p. be the probability measure defined on (X, E) by: 2
2
2
C(f)
= I dp(x) exp ( - i < x- f Jx
>).
The action
t
t
t
0
2
2
This representation is characterized uniquely (up to unitary equivalence) by the existence of a vector ft, cyclic in Ti with respect to W(T) , and such that (W(f)ft, ft) = C(f) V / G T . Let then Af be the von Neumann algebra generated by the operators {W(f) \ f G T} . Let {a | t G ii} be the group of automorphisms of Af defined by continuity from <*t[W{f)] = W{U f) V / G T , with ( [ / , / ) ( « ) = exp{-ist}f(s) Vt,seR}. This give the group action. t
t
For the definition of the refining algebra A, choose a G i i fa G Ti defined by f
-
f
—
+
and consider the vector
1*
and the subspace T„ obtained as the closure of Span {U fa I * < 0} . A is defined as the von Neumann algebra generated by the operators {W(f) \ f G T ] . Finally the normal state w is defined by u : N G Af >-> (TVft, ft) G i i . It is then straightforward to verify that the quartet {Af; a; A; ui} is a W-algebraic Kflow; that Af is a factor of Type III\ with A as described above; and that the modular group of the normal faithful state ui is obtained as the extension to Af of cr \W(f)\ = W(\ f). We will refer to this K-flow as the quantum flow of Brownian motion (for a discussion of some of its physical attributes, see section 4 below). The examples of the classical and quantum flows of Brownian motion involve only (classical or quantum) probabilistic notions; it is however true that many of the examples of classical K-flows one finds in the mathematical literature, derive their K-properties from t
0
t
H
129 an underlying Anosov structure. This is in particular the case for the geodesic flow on surfaces of negative curvature (see section 2). The so-called Arnold cat is an inspiring caricature of this feature. Consider indeed the iterations {T \ n £ Z) of a linear map T of R onto itself defined by the matrix 2
n
detT = 1 ensures that T is a Lebesgue measure-preserving bijection; tr T > 2 further implies that T has two distinct eigenvalues Ai = A and A2 = A with A > 1 so that the eigenvectors of T give respectively a dilation direction and a contraction direction; furthermore, the fact that the matrix of T has integer entries implies that A is irrational, and that the eigenvectors of T have irrational slope; finally, this "integer entry" condition implies that one can restrict the evolution of the torus T = R jZ . The resulting discrete time dynamical system on this torus is the Arnold cat. It is almost [5] straightforwward to find a partition that equips this with the structure of a classical Kolmogorov system. For further reference, note that the abelian von Neumann algebra of the corresponding W*-algebraic K-system is £ ° ° ( X ) and is generated by the functions of the form W{n) : } , with ne Z . - 1
2
2
2
2
2
Recently, an interesting quantum extension of this classical system has been obtained [8]. Consider indeed the symplectic form a : (n,m) E Z x Z h-» n\mi — n 2 i £ Z, and the characteristic functional C(n) — (S o ; and choose a fixed, non-zero, rational number 9. One can then construct, by a proceedure analogous to that followed in the construction of the quantum flow of Brownian motion, a complex Hilbert space H, and a map W : n £ Z i - » W(n) E U(H) such that for all n and m in Z : W(n)W{m) = W(n+m) exp{i 27T 9 a(n, m)} . This representation is characterized uniquely (up to unitary equivalence) by the existence of a vector fl, cyclic in H with respect to W(Z ), such that (W(n)Cl,U) = C(n) V n e Z . Af is then the von Neumann algebra generated by the operators {W{n) | n e Z } ; and {at \ k £ Z} is defined by a [W(n)] = W{T [n]) V n £ Z and V k £ Z. Here again this quantum dymanical system can be equipped with the structure of a W*-algebraic K-system (the rationality of 6 appears to play here an essential role). 2
m
2
ni
2
2
2
2
2
k
k
2
This model brings to the fore a question raised already several times in this lecture: P R O B L E M 1. To generalize
to the quantum realm the notion
of Anosov
Row.
It may be useful to emphasize once more that, although some filiation relations are known between classical Kolmogorov flows and Anosov flows, these dynamical structures partake to very different mathematical categories: Kolmogorov flows appeal to probabilistic notions, whereas Anosov flows involve topology and/or differential geometry. Classical probabilistic notions do generalize naturally when one allows for non-commutative generalizations of the usual function spaces of the classical theories. For differential dynamical systems, the most effective tool so far has been the methods of geometric quantization; however, their full power does not seem yet to have been brought to bear decisively on the above problem.
130 4. P h y s i c a l B a c k g r o u n d f o r K - F l o w s . T h e q u a n t u m flow of B r o w n i a n m o t i o n discussed i n section 3 f u r t h e r generalizes its classical counterpart i n that it is associated to a q u a n t u m diffusion e q u a t i o n ; see [14, 15]. T h i s equation is most concisely described i n its integrated f o r m , i.e. as the semigroup of completely positive maps {71 = £ o a 0 1 \ t > 0 } acting i n the v o n N e u m a n n algebra Af t
generated by {W(z
\ z € C ) ; here t is the canonical i n j e c t i o n of Ma i n t o M,
f) a
a
and £
is the c o n d i t i o n a l expectation f r o m M onto Ma w i t h respect to the state u>. W e o b t a i n i n this manner: f ))
lt[W(z
a
= W(e-"
z fa) e x p { - 0 |z| (1 - e 2
2 a 4
)/4} .
T o see how this semigroup corresponds indeed to a q u a n t u m diffusion process consider, for each direction u = (x,y)
E C w i t h |u| = 1 i n the classical phase space of the q u a n t u m
h a r m o n i c oscillator described by Ma , the a b e l i a n v o n N e u m a n n s u b - a l g e b r a M generated by {W(buf )
\ b £ R};
a
u
of
M
a
note that each of these sub-algebras is stable under
{71 I t > 0 } . T h e action {(71). | t > 0 } o n the p r e d u a l (M ) u
t
of M
u
is t h e n verified to
be the integral s o l u t i o n of the differential equation for the diffusion i n a h a r m o n i c well, namely: {d
t
- D [d u
+ vid)d
2 (
+ v ; ( 0 ] } V>„«,t) = 0
K
w i t h „( . , t) e C} (R, d£) defined by
= J
J Jt
d£e-'
For completeness we m e n t i o n t h a t , i n terms of the defining constant a a n d the state o> of the K - f l o w , the diffusion constants are given by £>„ = a/(K fi) 2
tentials by V ( £ ) = Kl ?/2
with K
u
defined by < u\W(buf ) a
andH
= (P
2
+
>=
2
a n d the effective h a r m o n i c po-
= 2 t a n h ( / 3 f c / 2 ) / { / 3 ( / f c i + i f c - y ) } , where /? a n d k are 2
tr pp exp{-ib(xP
1
2
+ yQ)} , pp = exp(-fiH)/ti
exp(-/?#),
k Q )/2. 2
2
W e have thus o b t a i n e d a dissipative t h e r m o d y n a m i c a l process (the q u a n t u m diffusion process) as a "contracted d e s c r i p t i o n " [45] of a conservative d y n a m i c a l process (the q u a n t u m flow of B r o w n i a n m o t i o n ) . T o make proper contact w i t h s t a t i s t i c a l mechanics, one is f u r t h e r r e q u i r e d to give a mechanistic i n t e r p r e t a t i o n of the m o d e l i n terms of a H a m i l t o n i a n t h a t describes a recognizable energy assignment. T h i s has been done [13]: the diffusion is due to a n H a m i l t o n i a n interaction between the single h a r m o n i c oscillator considered above, a n d a n i n f i n i t e chain of h a r m o n i c oscillators t h a t serves as a t h e r m a l b a t h . T h e i n t e r p r e t a t i o n is however not s t r a i g h t f o r w a r d , as it involves c o n t r o l l i n g two l i m i t i n g procedures: l i m i t , and a l o n g - t i m e / w e a k - c o u p l i n g l i m i t .
the t h e r m o d y n a m i c a l
W h i l e the former is a t e c h n i c a l device that
allows to get r i d of irrelevant recurrences, the latter seems to p l a y a less u n i v e r s a l l y recognized role: it allows to separate the time-scale proper to the t h e r m o d y n a m i c a l process f r o m the time-scale of more transient phenomena. It is this l i m i t t h a t is responsible for the essential differences between the d y n a m i c s described b y the a c t i o n a responsible for the a p p r o a c h to e q u i l i b r i u m (the action f r o m w h i c h one b u i l d s the " c o n t r a c t e d d e s c r i p t i o n "
131 1 = £ o a o t), and the modular action a associated to the state u> (that sees only the weak-coupling part of the limit, and ignores its long-time aspect). This limit is in general very difficult to control, although 35 years have elapsed since it was pioneered by van Hove [22], who had approached it in terms of the selection of the most divergent diagrams in a pertubation expansion. A n exactly solvable model for the contribution of random impurities to the electric conductivity in a crystaline material has been studied [27] and it illustrates, now in a mathematically controlled manner, how the van Hove "diagonal singularity" is indeed responsible for an approach to equilibrium that is governed by a Markovian master equation. An interesting variant of the scaling limit scheme is presented in [1]. Together with the ergodic properties of K-flows to be discussed in the next section, the above remarks suggest the following: P R O B L E M 2. To augment the repertory of quantum hamiltonian systems which do exhibit, through controlable limiting procedures, an approach to equilibrium that is the contraction of a K-flow. A variant of this problem can be stated as follows: P R O B L E M 2'. To understand the van Hove scaling limit in terms of a reformuJation his "diagonal singularity" criterion that would allow to predict the occurence of the K-flow property. Another, more modest, variant of Problem 2 would be to address the "inverse problem"(for its classical background see, e.g. [44]), namely: P R O B L E M 2". Given an arbitrary dissipative semigroup of completely positive maps, find a non-commutative analog of the classical Kolmogorov-Daniell reconstruction theorem, and decide whether the conservative dynamical system obtained in this manner is necessarily a K-flow. Encouraging answers to that problem have been found for the particular case of quasifree evolutions [16, 6]. 5. E r g o d i c P r o p e r t i e s o f K - F l o w s . We consider here W-algebraic K-flows, since they present the most complete generalization of the probabilistic structure of classical K-flows. Without loss of generality, we can assume that Af is a von Neumann algebra of bounded linear operators acting on a Hilbert space Ti , that there exists in Ti a vector Cl, cyclic in Ti , and separating, for Af (and thus equivalently for the commutant Af' of Af) such that < ui;N >= (NQ,Q) VAT e Af. It then follows that there exists a group {U | t e R} of unitary operators, acting on Ti, and such that for all t € R we have U,Q = Q, and a,[N] = U,NU-, V N 6 Af. With A, = a,[A], let E be the closed subspace in Ti spanned by A Q • It then follows immediately that {(U ,E ) | (t,s) e R } generates a system of imprimitivity based on R. Since, moreover it follows from a quantum martingale theorem (see [14]), that t
a
t
3
2
s
f\A, 3
= CI
implies
f]E, 3
=
Cn,
132 we have that on the orthogonal complement of CO, : {(U,,E.) | (t,s) £ R } decomposes into a direct sum of irreducible representations of the canonical commutation relation on R. Consequently, we have for every W-algebraic K-flow: 2
LEMMA.
Let H be the generator of {U
t
\ t 6 ii}; then
(1) the discrete spectrum of H is {0} with multiplicity 1 (2) * 6 H and H * = 0 imply * = cti for some ceC (3) H has homogeneous Lebesgue spectrum on . This generalizes exactly the classical result. It is then only a matter to translate into the quantum language the ergodic consequences this lemma is known to have in the classical theory of K-flows; we list some of them below: THEOREM. (1) u; is extremal a*-invariant ; (2) N G Af and a,[N] = N V t £ R imply N = < w; N > i ; (3) u> - l i m b i c * , ce [N] = < w;N > I; (4) for every normal state il> on Af and every N £ Af : t
lim
(5) for every N N ,N U
2
< wjJVittffiVil-Ns > -
(—»±oo
(6) for every N N ,N ,N u
2
3
—
<w;N>;
& Af :
3
lim
I—*±oo
< w; iVi-A^ >
2
> = 0;
e Af :
t
lim
< w; ^3 [Ni , a J A y i JV4 >=
0.
t—*±oo
Note that (1) is the quantum equivalent of metric intransitivity: the state u> cannot be written as a convex combination of two invariant states; (2) asserts that there are no non-trivial constant of the motion; (3) and (4) are dual statements about the approach to equilibrium; (5) is a mixing property. The last of the above properties, namely (6), is a weak form of asymptotic abelianness, a condition that can be softly paraphrased by saying that, as any two observables (denoted here iVi and N ) move away in time from one another, they tend to commute; this last property is evidently trivially satisfied in the classical case. It should be noted that its strongest form, namely: 2
lim
||[JVi,a [/V ]]||=0 t
a
can usually not be satisfied in quantum systems for which the time evolution is coming straightforwardly (i.e., only through the thermodynamical limit) from a Hamiltonian that can be interpreted as a reasonable energy assignment; in view, however, of the discussion preceding Problem 2 above, one should be open to the possibility that some K-flows (in
133 particular those obtained in situations where a van Hove type of limit is involved) might nevertheless exhibit some form of norm-asymptotic abelianness; this is indeed, the case (see T h m . II.2 in [14]). A n intermediate form of this property, the strong asymptotic abelianness 3
-
lim
[Nt , a , [Ay] = 0
may be expected to hold in even broader circumstances (e.g. the time-evolution of the even part of the Fermi field algebra, i.e., the observable algebra) and this condition has been called upon to interpret K M S states as dynamically stable states [19]. This discussion of property (6) above suggests a further elaboration of problem 2, namely: P R O B L E M 3. To understand the meaning of the condition ianness in the context of the theory of K-flows.
of strong asymptotic
abel-
A remarkable result in this direction has been obtained by Benatti and Narnhofer [7]: they show that for an entropic K-system [31] (see section 7 below) where Af is a type III hyperfactor, and w is a tracial state on Af, invariant under {a, | t € R}, this action is strongly asymptotically abelian. Conversely, a W*-algebraic K-flow that is strongly asymptotic abelian is also an entropic K-system [29]. However, algebraic K-systems are known that are not strongly asymptotic abelian, e.g., the quantum version [8] of the Arnold cat reviewed in section 3 above; here again, it would be interesting to examine such situations in the spirit of Problem (2'). We should remark that stronger ergodic properties of algebraic K-flows can be obtained, some of them under conditions that are somewhat weaker that those involved in the theorem stated above. Two of these properties are : uniform ^-clustering or K-clustering [36]; and a quantum generalization of topological mixing [26, 23, 9, 33]. The first of these properties states that for every fixed A^i £ Af and e > 0, there exists T such that | < u;Nia-t[N ] 2
> -
< u>; JV
X
><
ui; N
2
> | < e < u;N*Nx
>i
z
>i
for all N £ A and all t > T. The second of these ergodic properties is that for any fixed non-zero N^,N £ Af there exists T such that 2
2
Ni a [N ] ^ O V t with |t|>T. t
2
6. A n H e r e d i t a r y P r o p e r t y o f K - F l o w s . Given a W*-algebraic K-flow, we denote by Af" the centralizer of Af with respect to u>, i.e., the von Neumann algebra Af" = {N £ Af |
NVteR}
of the fixed points of Af for the modular group of automorphisms {o
t
associated to ui; equivalently, Af" can be defined as Af" = {N € Af |< w; [AT, M] > = 0 V M £
Af};
\ t £ R} canonically
134 in particular, note that u> is a tracial state on Af". Physically, from the point of view of the quantum measuring process, Af" can be characterized as follows. Let £ be a. partition of the identity I by mutually orthogonal projectors Pk E Af; according to von Neumann [34], in the course of a measurement corresponding to this partition, a state i{> on Af is changed into a new state k where A =
J
and 4> :N h
e A f *
<
i ;
*
*
>
£
The following conditions are then equivalent on C ! (a) C H = w (b) C C . A f " . Hence the centralizer of A/" is the von Neumann subalgebra Af" C Af generated by the partitions of the identity that do not perturb the state u . Note further that, since a o a, = u o at V s, t £ R, we have that A/"" is stable under the action {at \ t £ R} • Let then a" denote the restriction of at to Af" , and A" be An Af" ; we use the same symbol to denote the state ui on Af and its restriction to Af" . We are now in position to state the principal hereditary property of W-algebraic K-flows, namely that the quartet {Af"\a";A";u>} is again a W^*-algebraic K-flows. This reduction is evidently effective only when u> is not a trace, and thus in particular when Af (or equivalently A) is not abelian. We say that the W-algebraic K-flow {Af ; a; A;u>} is regular if every maximal abelian von Neumann subalgebra in Af" is already maximal abelain in Af. A n example of a nonabelian regular K-flow is provided by the quantum flow of Brownian motion described in section 3. It is however not true that this property is universal, and one knows [14] that the K-flow property does not, by itself, even prevent in general the singular circumstance where the centralizer of the algebra Af is trivial. For non-singular K-flows, we will argue in the next section that the dynamical entropy should be defined as a property of the restriction of the evolution to the centralizer. t
s
7. D y n a m i c a l E n t r o p i e s . Let {Af, a,ui} be a W-algebraic dynamical system, i.e., a von Neumann algebra Af, a group {a | t £ iJ]of automorphisms of Af, and a normal state ui on Af with w o a = w V r £ R. t
t
A dynamical entropy for this dynamical system is a non-negative (possibly infinite) r jmber H"(a)= sup H"{M,a) where: (a) T is a family of "admissible" von Neumann subalgebras AA C Af (b) For AA £ T : H"(AA,a) is a dynamical entropy of AA with respect to a and ui. The indefinite character of the above prescription reflects the fact that a completely satisfactory answer to the following problem does not seem to have been achieved yet, although one has some definitely more specific ideas on what is likely to work.
135 P R O B L E M 4. To obtain a necessary and sufficient characterization of the family T, and of the entropy H"(AA,ct) for M E T, in such a manner that the resulting entropy H"(a) be mathematically adapted to the notion of K-Sow and be physically accessible to empirical testing. Some mathematical clues are obviously to be obtained from the classical theory where one has succeeded indeed to define a dynamical entropy that is strictly positive on all classical K-flows. In that theory, the elements AA of T are the abelian von Neumann algebras generated by finite (or countable) measurable partitions C, of the probability space of the flow. One has there: H"(M,a)
= lim H"(C \ a' ^] 1
V • • • V c<- [(}) n
n—*oo
= ^ ^ l
1
1
" ^
V a " [C] V • • • V a - [ C ] ) 1
where £ V C' is the coarsest partition refining f and C , H"(0 is the usual entropy of a partition, and H"(C | C') is the conditional entropy. The first remark of which one has to be aware when one tries to generalize this classical dynamical entropy to algebraic dynamical systems, is that the above two limits are conceptually different, although they nicely turn out to be equal. There do not seem to be any reason to expect this to generalize to the non-abelian case. A choice therefore has to be made. The second remark is that in the non-abelian case, measurements do in general change the state of the system, and increase its entropy; if one wants to have a dynamical entropy that expresses only the randomization that results from the evolution, one must restrict oneself to partitions that do not perturb the state. These have been charaterized in section 6 above as partitions in the centralizer Af" of Af. Note in particular, that since a o
s
3
t
n
n—*oo
where H"(AA) is the ordinary entropy of AA in the state ui. This seems (see in particular the first remark made after Problem 3 in section 5 above) to be a most promizing step
136 towards the understanding of the chaotic behavior to be expected f r o m conservative d y n a m i c a l systems associated w i t h the " c o n t r a c t e d " dissipative evolutions c h a r a c t e r i s t i c of n o n - e q u i l i b r i u m statistical mechanics. A s of this w r i t i n g , the second part of P r o b l e m 4 seems s t i l l t o be w i d e l y o p e n . REFERENCES 1. L. Accardi, Scaling Limits of Quantum Hamiltonian Systems, in "Present Proceedings." 2. D . V . Anosov, Ergodic Properties of Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature, Dokl. Akad. Nauk SSSR 151 (1963), 1250-1252; Soviet Math. Dokl. 4 (1963), 1153-1156. 3. D.V.Anosov, Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature, Trudy Mat. Inst. Steklova 90 (1967); Proc. Steklov Inst. Math. (AMS) (1969), 1-235. 4. V . I. Arnold, "Mathematical Methods of Classical Mechanics," (K. Vogtmann and A . Weinstein, transl.) Springer, New York, 1980. 5. V . I. Arnold and A . Avez, "Ergodic Problems of Classical Mechanics," W . A . Benjamin, New York, 1968. 6. F . Benatti, Quantum Dynamical Entropy and Entropic Dimension, P h . D . Thesis (1988). 7. F . Benatti and H . Narnhofer, Strong Asymptotic Abelianness for Entropic K-sysiems, preprint U W T h P h 1989-34. 8. F. Benatti, H . Narnhofer and G . L . Sewell, A Noncommutative Version of the Arnold Cat Map, Lett. Math. Phys. 21 (1991), 157-172. 9. O. Bratteli, G . A . Elliott and D. W . Robinson, Strong Topological Transitivity and C*-Dynamical Systems, J . Math. Soc. Japan 37 (1985), 115-133. 10. A . Connes, H. Narnhofer and W . Thirring, Dynamical Entropy of C*-algebras and von Neumann Algebras, Commun. Math. Phys. 112 (1987), 691-719. 11. A . Connes and E . Stormer, Entropy for Automorphisms of Hi von Neumann Algebras, Acta Math. 134 (1975), 289-306. 12. P. Cornfeld, S. V . Fomin and Ya. G . Sinai, "Ergodic Theory," Springer, New York, 1982. 13. E . B. Davies, Diffusion for weakly coupled quantum oscillators, Commun. math. Phys. 27 (1972), 309-325. 14. G . G . Emch, Generalized K-Flows, Commun. math. Phys. 49 (1976), 191-215. 15. G . G . Emch, Non-Equilibrium Statistical Mechanics, Acta Physica Austriaca S u p p l . X V (1976), 79-131. 16. G . G . Emch, S. Albeverio and J.-P. Eckmann, Quasi-free Generalized K-Flows, Rep. Math. Phys. 13 (1978), 73-85. 17. I. M . Gelfand and S. V . Fomin, Geodesic Flow on Manifold of Constant Negative Curvature, Usp. Math. Nauk 47 (1952), 118-137; Transl. Amer. Math. Soc. (2) 1 (1955), 49-65. 18. R . Haag, N . Hugenholtz and M . Winnink, On the Equilibrium States in Quantum Statistical Mechanics, Commun. math. Phys. 5 (1967), 215-236. 19. R . Haag, D. Kastler and E . Trych-Pohlmeyer, Stability and Equilibrium States, Commun. math Phys. 38 (1974), 173-193. 20. J . Hadamard, Les Surfaces a Courbures Opposees et leurs Lignes Geodesiques, Jour. Math. Pures Appl. 4 (1898), 27-74. 21. T . Hida, "Stationary Stochastic Processes," Princeton Univeristy Press, Princeton N J , 1970. 22. L. van Hove, Quantum-Mechanical Perturbations giving Rise to a Statistical Transport Equation, Physica 21 (1955), 517-540; The Approach to Equilibrium in Quantum Statistics, A Perturbation Treaiement to General Order, Physica 23 (1957), 441-480. 23. A . Kishimoto and D. W . Robinson, Dissipations, Derivations, Dynamical Systems and Asymptotic Abelianness, J . Oper. Theory 13 (1985), 237-253. 24. A . N . Kolmogorov, A New Metric Invariant of Transitive Sysaiems and Automorphisms of Lebesgue Spaces, Dokl. Akad. Nauk 119 (1958), 861-864; On the Entropy per Time Unit as a Metric Invariant of Automorphisms, Dokl. Akad. Nauk 124 (1959), 754-755.
137 25. R. Kubo, Statistical Mechanical Theory of Irreversible Processes, Journ. Phys. Soc. Japan 12 (1957), 570-586. 26. R. Longo and C . Peligrad, Noncommutative Topological Dynamics and Compact Actions on C*-Algebras, J . Funct. Anal. 58 (1984), 157-174. 27. Ph. Martin and G . G . Emch, A Rigourous Model Sustaining van Hove's Phenomenon, Helv. Phys. Acta 48 (1975), 59-78. 28. P. C . Martin and J . Schwinger, Theory of Many-Particle Systems I, Phys. Rev. 115 (1959), 1342-1373. 29. H . Narnhofer, Dynamical Entropy, Quantum K-Systems and Clustering, in "Quantum Probability (Heidelberg, 1988)," (L. Accardi and W . von Waldenfels eds.) Springer-Verlag, Berlin, 1990, pp. 286-295. 30. H . Narnhofer and W . Thirring, Mixing Properties of Quantum Systems, J . Stat. Phys. 57 (1989), 811-825. 31. H . Narnhofer and W . Thirring, Quantum K-Systems, Commun. math. Phys. 125 (1989), 565-577. 32. H . Narnhofer and W . Thirring, Algebraic K-Systems, Lett. Math.Phys. 20 (1990), 231-250. 33. H . Narnhofer, W . Thirring and H . Wiklicky, Transitivity and Ergodicity of Quantum Systems, Jour. Stat. Phys. 52 (1988), 1097-1112. 34. J . von Neumann, "Mathematical Foundations of Quantum Mechanics," (R.T. Beyer transl.) Princeton University Press, Princeton N J , 1955. 35. W . Parry, "Entropy and Generators in Ergodic Theory," W . A . Benjamin, New York, 1969. 36. W. Schroeder, A Hierarchy of Mixing Properties for Non-Commutative K-Systems, in "Quantum Probability and Applications to the Quantum Theory of Irreversible Processes," (L. Accardi, A . Frigerio and V . Gorini, eds.), Springer, New York, 1984, pp. 340-351. 37. Ya. G . Sinai, Dynamical Systems with Countably Lebesgue Spectra, Izvestia Math. Nauk 25 (1961), 899-924; Transl. Amer.Math. Soc. (2) 39 (1961), 83-110. 38. Ya. G . Sinai, Geodesic Flows on Compact Surfaces of Negative Curvature, Dokl. Akad. Nauk SSSR 136 (1961), 549-552; Soviet Math. Dokl 2 (1961), 106-109. 39. Ya. G . Sinai, On the Foundations of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics, Dokl. Akad. Nauk SSSR 153 (1963), 1261-1264; Soviet Math. Dokl. 4 (1963), 1818-1822. 40. Ya. G . Sinai, Ergodicity of Boltzmann's Gas Model, in "Statistical Mechanics, Foundations and Applications," (Proceedings of the IUPAP Meeting, Copenhagen, 1966; T . A . Bak, ed.) W . A . Benjamin, New York, 1967, pp. 559-573. 41. Ya. G . Sinai, Dynamical Systems with Elastic Reflections, Usp. Math. Nauk V . X X V N o 2 (1970), 137-189; Russian Mathematical Surveys 25 (1970), 137-189. 42. Ya.G. Sinai, "Introduction to Ergodic Theory," (V. Scheffer, transl.), Princeton University Press, Princeteon N J , 1976. 43. M . Takesaki, "Tomita's Theory of Modular Hilbert Algebras and its Applications," Lecture Notes in Mathematics No 128, Springer, New York, 1970. 44. H . G . Tucker, " A Graduate Course in Probability," Academic Press, New York, 1967. 45. G . E . Uhlenbeck, Concluding Remarks, in "Statistical Mechanics, Foundations and Applications," (Proceedings of the IUPAP Meeting, Copenhagen, 1966; T . A . Bak, ed.) W . A . Benjamin, New York, 1967, pp. 574-582. 46. P. Walters, " A n Introduction to Ergodic Theory," Springer, New York, 1982.
Walker Hall, The University of Florida, Gainesville, F L 32611, U S A
Quantum Probability and Related Topics Vol. VII (pp. © 1992 World Scientific Publishing Company
139-148) 139
Unitarity of solutions to quantum stochastic differential equations and conservativity of the associated semigroups Franco Fagnola Dipartimento di Matematica, Universita di Trento I - 38050 P O V O ( T N )
Abstract.
ITALY
W e s h o w h o w u n i t a r i t y of t h e s o l u t i o n of a q u a n t u m s t o c h a s t i c differ-
e n t i a l e q u a t i o n c a n b e d e d u c e d f r o m t h e c o n s e r v a t i v i t y of t w o a s s o c i a t e d q u a n t u m d y n a m i c a l semigroups.
1. Introduction Q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n d r i v e n b y t h e B o s o n F o c k space basic s e m i m a r t i n g a l e s [13], w i t h coefficients g i v e n b y c o n s t a n t o p e r a t o r s i n t h e i n i t i a l s p a c e , n a t u r a l l y arise i n t h e s t u d y of l i m i t s of q u a n t u m e v o l u t i o n s [2]. A u n i t a r y s o l u t i o n of s u c h e q u a t i o n s p r o v i d e s a d i l a t i o n of a q u a n t u m d y n a m i c a l s e m i g r o u p v i a t h e F e y n m a n - K a c p e r t u r b a t i o n t e c h n i q u e [1]. T h e case i n w h i c h t h e coefficients are u n b o u n d e d o p e r a t o r s is t h e m o s t i n t e r e s t i n g . S o m e r e s u l t s c a n b e f o u n d i n [4], [5], [6], [7], [11]. U n d e r c e r t a i n t e c h n i c a l c o n d i t i o n a s o l u t i o n c a n be e a s i l y c o n s t r u c t e d as i n [7], [11] b y a n e q u i c o n t i n u i t y m e t h o d s u g g e s t e d b y A . F r i g e r i o . A s s h o w n i n [5], t h e r e s u l t s of [4], [7]
together
w i t h a d u a l i t y p r i n c i p l e d u e to J o u r n e [9] (see also [8] i n t h e free noise case, a n d [12]) a l l o w i n g t o r e d u c e t h e p r o o f of c o i s o m e t r y t o t h a t of i s o m e t r y f o r a d u a l p r o c e s s l e a d t o p r o v e u n i t a r i t y u s i n g t h e c o n s e r v a t i v i t y of t h e a s s o c i a t e d q u a n t u m dynamical
semigroups.
I n t h i s n o t e we s h o w h o w t h i s c a n be d o n e i n a v e r y s i m p l e w a y u s i n g t h e c h a o s decomposition
of t h e B o s o n F o c k space a n d a n i n d u c t i o n a r g u m e n t o n W i e n e r
chaoses. T h e c o n s e r v a t i v i t y of a q u a n t u m d y n a m i c a l s e m i g r o u p i n g e n e r a l is a d i f f i c u l t problem.
H o w e v e r s i m p l e a n d e a s i l y v e r i f i a b l e sufficient c o n d i t i o n s h a v e b e e n
g i v e n b y C h e b o t a r e v i n [4] t o g e t h e r w i t h a c h a r a c t e r i z a t i o n t h e o r e m w h i c h is a q u a n t u m s t o c h a s t i c a n a l o g u e of F e l l e r ' s c o n d i t i o n f o r t h e s o l u t i o n d i f f e r e n t i a l e q u a t i o n s t o be p r o b a b i l i t y p r e s e r v i n g .
Kolmogorov's
I n s e c t i o n 3 we w i l l give a
s i m p l e p r o o f of C h e b o t a r e v ' s c h a r a c t e r i z a t i o n r e s u l t . T h i s note was w r i t t e n while the a u t h o r was v i s i t i n g the M o s c o w for E l e c t r o n i c Engeneering.
Institute
T h e a u t h o r w o u l d l i k e to express h i s g r a t i t u d e to
140 A . M . Chebotarev a n d A . S . Holevo for s t i m u l a t i n g discussions o n this subject. A f t e r t h i s n o t e was w r i t t e n we k n e w s i m i l a r t e c h n i q u e s h a v e b e e n u s e d a l s o b y M o h a r i a n d P a r t h a s a r a t h y i n [10],
[12].
2. U n i t a r y s o l u t i o n s L e t Ti be t h e B o s o n F o c k space over L
2
t i a l v e c t o r s ip(f)
L e t A, A ,
with / € L
a n d n u m b e r processes.
(R+).
2
( R + ) a n d let S d e n o t e t h e set of e x p o n e n A be t h e a n n i h i l a t i o n , c r e a t i o n
+
L e t h b e a c o m p l e x s e p a r a b l e H i l b e r t space ( t h e i n i t i a l
space) a n d let I? b e a dense l i n e a r m a n i f o l d i n h. W e f o l l o w t h e m e t h o d s a n d use t h e n o t a t i o n of q u a n t u m s t o c h a s t i c c a l c u l u s o n Ti as d e s c r i b e d i n [13].
A l l the
o p e r a t o r s d e f i n e d o n a t e n s o r p r o d u c t f a c t o r of h ® Ti w i l l be i d e n t i f i e d w i t h t h e i r c a n o n i c a l e x t e n s i o n t o t h e w h o l e space. M o r e o v e r we w i l l o f t e n o m i t t h e t e n s o r product ®. Let T
b e t h e s e c o n d q u a n t i z a t i o n of t h e r i g h t shift o n
(
/(*-<) ; 0
L (R+) 2
if s > t
:
if s < t
a n d let TZt b e t h e s e c o n d q u a n t i z a t i o n of t h e o p e r a t o r p o n L ( R + ) d e n n e d , for 2
t
all s G R
+
, by , , > f fit — s) ( * ' ) « = { } > )
if s < t i f , ; * .
u
F o r a l l b o u n d e d o p e r a t o r A o n h®H,
T AT* t
is a n o p e r a t o r o n h®F
(L (t, 2
+oo))
its c a n o n i c a l e x t e n s i o n t o h ® Ti w i l l be d e n o t e d b y r A r * (
D e f i n i t i o n 2 . 1 . A contraction cocycle
if, for all s,t
> 0, we X(t
valued
adapted
stochastic
process
(X(t))
t>0
is a
have + s) =
L e t X be a c o c y c l e a n d let X(t)
X{t)T X{s)T*. t
d e n o t e t h e o p e r a t o r TZ X*(t)TZf t
T h e following
r e s u l t h a s b e e n p r o v e d i n [9]. P r o p o s i t i o n 2.2.
( J o u r n e ' J The process
T h e process (X(t)) >o t
Let L i , L 1 L 3 , L 2
4
(X(t)) > t
0
is adapted
and is a
cocycle.
w i l l be c a l l e d t h e dual cocycle a s s o c i a t e d w i t h ( X ( t ) ) > . t
0
b e o p e r a t o r s o n h w i t h d o m a i n c o n t a i n i n g V s u c h t h a t the
d o m a i n of t h e a d j o i n t o p e r a t o r s a l s o c o n t a i n s T> a n d t h e f o l l o w i n g r e l a t i o n s h o l d f o r a l l v, u £ T> L)
= 0
(2.1)
+ (L!v,u)
+ (L\v,L\u)
= 0
(2.2)
(v,L u)
+ (L v,u)
+ (Liv,L u)
= 0
(2.3)
(v, L u)
+ (L v,
= 0
(2.4)
(v, L )
+ (L ,
(v,L )
lU
lU
2
4
lV
3
4
u) + (L , lV
lU
2
u) + (L v, 2
L u) 2
141 Consider the q u a n t u m stochastic differential equation f dU(t)
= U(t)
\
= I
U(0)
( X j d A ( t ) + L d A + ( t ) + L dA(t)
+
3
2
L dt) 4
( 2
where I denotes the i d e n t i t y operator.
W h e n the operators L i , L , L , L 2
3
4
5 )
are
also b o u n d e d t h e n (2.5) has a u n i q u e u n i t a r y s o l u t i o n (see [13]). M o r e o v e r i t is p o s s i b l e t o a s s o c i a t e w i t h {U(t))
the semigroup on
t>0
X S i n c e t h e o p e r a t o r s (Lj)* ,
E
(U(t)XU'(t)).
Q]
[L'j)*._
=1
B(h)
h a v e a dense c o m m o n d o m a i n , i t is p o s s i b l e
1
to s h o w as i n [3] t h a t t h e d u a l c o c y c l e U a s s o c i a t e d w i t h U satisfies t h e q u a n t u m stochastic differential equation j dU(t)
= U(t)
I
= I
U(0)
(L*dA(t)
+ L* dA(t) 2
+ LldA+(t)
+ L*d<)
(2.6)
T h e s e m i g r o u p o n B (h) a s s o c i a t e d w i t h t h e d u a l c o c y c l e U T (X)
= E
t
(2.7)
(p(t)XU-(t))
0]
has i n f i n i t e s i m a l g e n e r a t o r C s a t i s f y i n g , for a l l u, v £ V (v, C(X)u)
= (v, XL u) t
+ (L v,
T h e case i n w h i c h t h e o p e r a t o r s (Lj)*
Xu)
4
=1
+ [L v, 2
(2.8)
XL u) 2
are u n b o u n d e d has b e e n s t u d i e d i n
s e v e r a l p a p e r s (see, f o r e x a m p l e [4], [5], [6], [7], [10], [11], [12]). A c o n t r a c t i v e s o l u t i o n t o ( 2 . 5 ) , (2.6) c a n be c o n s t r u c t e d b y t h e e q u i c o n t i n u i t y m e t h o d e x p o s e d i n [7].
I n t h i s n o t e we w i l l g i v e a s i m p l e m e t h o d to d e d u c e i s o m e t r y (resp.
c o i s o m e t r y ) of t h e s o l u t i o n of a q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n w i t h u n b o u n d e d coefficients f r o m t h e c o n s e r v a t i v i t y of a c o r r e s p o n d i n g s e m i g r o u p . W e w i l l c o n s i d e r o n l y t h e case of one noise process. T h e f o l l o w i n g p r o p o s i t i o n c a n be p r o v e d as L e m m a (3.2) i n [7]. Proposition
(
£'."'] 3
i) for
)
n>0
2.3.
Suppose that for a l l j
o f bounded
operators
on h such
=
1 , . . . , 4 t h e r e exists
that:
allu£V s- l i m L^u
=
LiU,
a
sequence
142 ii) for all n G N the operators there exists
a contraction
to equation
(2.5).
( i y ) '
satisfy
0
valued
L e t U be a s o l u t i o n t o (2.5).
adapted
(2.1), (2.2),
process
(2.3),
(U(t)) >o
(2.4).
Then
is a
which
t
T h e s t r o n g l y c o n t i n u o u s f a m i l y of
solution
contractions
( T ( f ) ) > o o n h g i v e n b y t h e F o c k v a c u u m c o n d i t i o n a l e x p e c t a t i o n of t h e o p e r a t o r s t
(i.e. T (I)
U*(t)U(t)
g i v e n b y (2.7)) satisfies t h e e q u a t i o n
t
(v, T ( t ) u ) = (v,u)+
[ [{L v, Jo t
T(s)u)
+ {v, T(s)L u)
+ (L v,
4
f o r a l l u, v G T>. C l e a r l y , b e c a u s e of c o n d i t i o n ( 2 . 4 ) , T(t) (2.9).
T(s)L u)}
2
(2.9)
ds
2
= I f o r a l l t > 0 satisfies
T h e f o l l o w i n g P r o p o s i t i o n shows t h a t t h e u n i q u e n e s s of t h e s o l u t i o n of
(2.9) i m p l i e s t h a t U is i s o m e t r i c . P r o p o s i t i o n 2.4. itive
contractions
Suppose solving
c o n t r a c t i o n U solving Proof.
the only family (2.9)
(2.5)
is given
is
( T ( t ) ) t > o of strongly
by T(t)
continuous
= I for all t > 0.
pos-
T h e n every
isometric.
L e t us f i r s t r e m a r k t h a t , as a c o n s e q u e n c e of (2.4) a n d t h e a s s u m p t i o n
of t h e t h e o r e m , t h e e q u a t i o n (u,T(t»=
/ [(LiV,T(s)u) Jo
h a s t h e u n i q u e s o l u t i o n T(t) L e t f,g
G L
+
4
(L v,T(s)L u)]ds 2
= 0 f o r a l l t > 0 a n d a l l u, v 6
2
V.
W e w i l l p r o v e b y i n d u c t i o n o n N = m + n t h a t , for a l l
(R+).
2
+ (v,T(s)L u)
v, u G V a n d a l l n, m G N , we h a v e (U(t)vg° ,
U(t)uf®«)
m
= (v, u) (g® , m
/•»)
(2.10)
W e w i l l e v a l u a t e t h e r i g h t h a n d side u s i n g q u a n t u m I t o ' s f o r m u l a w r i t t e n u s i n g n - p a r t i c l e v e c t o r s i n s t e a d of e x p o n e n t i a l v e c t o r s . L e t us f i r s t c o n s i d e r N = 0. W e have then (U{t)v,U(t)u) +
/ Jo
=
[{U(s)L v,
F o r a l l t G R+
(v,u) U(s)u)
t
+ (U(s)v,
U(s)Liu)
+ (U(s)L v, 2
U{s)L u)] 2
ds
let T ( t ) b e t h e c o n t r a c t i o n i n B (h) d e f i n e d b y 0
(v,T (t)u) 0
T h e operator family ( 7 o ( t ) )
t > 0
= (U(t)v,
U(t)u).
is a s o l u t i o n t o t h e e q u a t i o n (2.9) t h e r e f o r e
c o i n c i d e s w i t h t h e i d e n t i t y o p e r a t o r for a l l t € R . +
T(t)
T h i s proves (2.10) w h e n
143 = 0. S u p p o s e n o w t h a t (2.10) h o l d s f o r a l l v,U £ 2?, f,g
N
g L (R+) 2
and
N' < N a n d let m , n b e t w o integers s u c h t h a t m + n — JV + 1. W e h a v e t h e n b y the q u a n t u m Ito's f o r m u l a a n d the i n d u c t i o n assumption
= (v,u) (g® ,f® )
(U(t)vg® ,U(t)uf®") m
m
r7(s)L
+ (U(s)vg® , m
+ J*
n
({U(s)l vg® ,U(s)uf®»)
/ ® " ) + (U(s)L vg® , 2
4 U
2
( ( L v , u) + (v, L u) + (L , L u) ) (g® ~ ,
+
( ( i j o , u) + (v, L u) + (Lav, L u) ) (
/
+ J
2
lV
m
2
3
) ds
U(s)L uf®»)
m
+ J
3
m
4
x
9
/«»)
l
0 m
ds
, Z ® " - ) f(s) ds 1
( ( L i t > , U) + (v, L u) + ( L i v , L u ) ) (g - ,f - )g(s)f(3) x
9m
x
1
9n
ds
1
T h e last t h r e e i n t e g r a l s v a n i s h b e c a u s e o f c o n d i t i o n s ( 2 . 1 ) , (2.3).
Denoting by
T j v + i ( t ) t h e c o n t r a c t i o n i n B (h) d e f i n e d b y (u,Tjv
+ 1
(U(t)vg® ,U(t)uf® ).
(r» =
m
for a l l u,v € TJ, w e find a g a i n t h a t ( T ( t ) )
( > 0
n
is a solution o f equation
therefore T j v + i ( t ) c o i n c i d e s w i t h t h e i d e n t i t y o p e r a t o r for a l l t € R + . proves (2.10) a n d c o m p l e t e s t h e p r o o f .
(2.9) This
•
C l e a r l y c o i s o m e t r i c i t y is e q u i v a l e n t t o t h e i s o m e t r i c i t y o f t h e d u a l c o c y c l e . T h e r e l a t i o n s h i p b e t w e e n u n i q u e n e s s o f t h e s o l u t i o n o f (2.9) a n d t h e c o n s e r v a t i v i t y of the q u a n t u m d y n a m i c a l s e m i g r o u p T is g i v e n i n t h e f o l l o w i n g s e c t i o n b y C h e b otarev's T h e o r e m .
3. C h e b o t a r e v ' s c h a r a c t e r i z a t i o n o f c o n s e r v a t i v i t y T h i s s e c t i o n i s i n s p i r e d b y s e c t i o n 3 o f [4]. W e w i l l s u p p o s e t h a t A
- T h e operator L
4
extends t o the irmnitesimal generator G of a strongly
c o n t i n u o u s s t r o n g l y c o n t i n u o u s c o n t r a c t i o n s e m i g r o u p P = (P(t))t>o
on h for
w h i c h V i s a core. T h e c o n d i t i o n ( A ) has t h e f o l l o w i n g consequences: - f r o m (2.4) i t f o l l o w s t h a t L has a c l o s e d e x t e n s i o n L t o t h e d o m a i n D(G) o f 2
G, - f o r a l l u € D(G) t h e m a p t H-» LP(t)u
is c o n t i n u o u s ,
- f o r a l l u, v € D(G) we have (v, Gu) + (Gv, u) + (Lv, Lu) = 0
(3.1)
144 M o r e o v e r we w i l l use t h e f o l l o w i n g t e c h n i c a l c o n d i t i o n B
- F o r a l l t > 0 a n d a l l u £ V, C
P(t)(V)
2
P r o p o s i t i o n 3 . 1 . Let us suppose be a strongly conditions
continuous
are
(v,T(t)u)
t h a t conditions
family
of positive
(A),
(B) hold a n d let
contractions
on h.
(T(t)) > (
0
following
have
= (v,u)+
ii) for all v,u 6 T> we
f [(Gv,T(s)u)-r{v,T(s)Gu) Jo
+ {Lv,T(s)Lu)]ds,
(3.2)
have
(v, T ( t ) u ) = (P(t)v,
P(t)u)
+
f (LP(t Jo
- s)v, T(s)LP(t
- s)u)
ds.
(3.3)
T h e d o m a i n V is a core f o r G a n d (3.1) h o l d s , t h e n e a c h T s a t i s f y i n g
(3.2) for e a c h v, u £ V satisfies (3.2) f o r e a c h v, u 6 D(G). u,v,
The
equivalent:
i) for all v, u £ T> we
Proof.
b e l o n g s t o t h e d o m a i n of G (i.e.
P(t)Gu
D(G )).
M o r e o v e r , f o r a l l such
the function t M
(v,T(t)u)
is d i f f e r e n t i a t e . T h e r e f o r e , f o r a l l s £ [ 0 , t ) , a n d a l l u , v £ T> we h a v e — (P(t
- s)v, T(s)P(t
-s)u^
= (LP(t
- s)v, T(s)LP(t
-
s)u)
H e n c e we o b t a i n (3.3) i n t e g r a t i n g o n [Q,i]. C o n v e r s e l y , i f T is a s o l u t i o n t o (3.3), b e c a u s e of a s s u m p t i o n ( A ) , (3.3) h o l d s f o r a l l u,v
6 D(G).
U s i n g the condition
( B ) , we c a n d i f f e r e n t i a t e t h e t w o sides of (3.3) a n d o b t a i n (v,T(t)u) + J
= (Lv,T(t)Lu)
+ { (P(t)Gv,P(t)u)
({LP(t-s)Gv,T(s)LP{t-s)u)
= (Lv, T(t)Lu)
(P(t)v,P(t)Gu)
+
+ (Gv, T(t)u)
+ (v,
I n t e g r a t i n g o n [0,<] we o b t a i n (3.2). D e f i n i t i o n 3 . 2 . Suppose
+
s
T(t)Gu) •
that the conditions
called c o n s e r v a t i v e if the only solution
{LP{t-s)v,T{s)LP(t-s)Gu))d y
(A),
(B) hold.
of the equation
(3.2)
The semigroup is given
for all t > 0. L e t C b e t h e b i l i n e a r f o r m o n B(h) (v,C(X)u) f o r a l l u, v 6
V.
= (v,XGu)
defined by + (Gv,Xu)
+
(Lv,XLu)
by T(t)
T is = I
145 Theorem following
3.3.
i) the semigroup ii)
( " C h e b o t a r e v ) Suppose
conditions
the equation
are
T is C(X)
that the conditions
(A),
(B)
hold.
The
equivalent: conservative,
= XX has no solution
in B
(h).
W e w i l l d i v i d e t h e p r o o f i n s e v e r a l steps P r o p o s i t i o n 3.4. T(
m , n
) of (3.3)
solution
Suppose
that condition
Then there exists a property:
solution for
all
and all t > 0 we h a v e
T of (3.3)
T (t)
< T(t)
(min)
Proof.
(A) holds.
the m i n i m a l s o l u t i o n w i t h the following
called
(3.4)
< I.
L e t u s define a sequence T „ b y T„(t) =
(u,T
n +
w h e r e u,v
p(typ(t)
i(t» = (P(t>,P(f» + 6 D.
f* / (LP(t Jo
- s)u,T (s)LP(t
- s)u)
n
(3-5)
ds
W e w i l l p r o v e b y i n d u c t i o n t h a t , f o r a l l n £ N , T „ is a s t r o n g l y
c o n t i n u o u s f a m i l y of p o s i t i v e c o n t r a c t i o n s o n h. M o r e o v e r t h e sequence ( T „ ( t ) ) „ > o is i n c r e a s i n g f o r a l l t > 0. also f o r a n i n t e g e r n .
T h i s f a c t c l e a r l y h o l d s for n = 0.
The
right-hand
side i n t e g r a l i n (3.5)
f a c t , b e c a u s e of c o n d i t i o n ( A ) a n d t h e s t r o n g c o n t i n u i t y of T , n
a continuous f u n c t i o n . Moreover, using the inequality 0 < T
n
Suppose it holds m a k e s sense.
In
t h e a r g u m e n t is < I a n d (3.1)
we
can m a j o r i z e it b y
/ Jo
ds = - 2 3 f e / {P(s)v,P(s)Gu) ^ Jo
\\LP(t - s)uf
ds ( 3
6 )
T h i s i m p l i e s b y p o l a r i z a t i o n that the
right-hand
continuous contraction T
M o r e o v e r , since T „ ( t ) — T „ _ i ( t ) > 0 f o r
n +
i ( < ) o n h.
side of (3.5) defines a s t r o n g l y
e v e r y t > 0, we h a v e (u, ( T
B + 1
( t ) - T „ ( f ) ) u) =
f (LP(t Jo
- s)u, (T (s) n
- T „ _ ! ( s ) ) LP(t
- s)u)
> 0
F o r e a c h t > 0 t h e sequence T „ ( f ) converges s t r o n g l y t o a c o n t r a c t i o n T ' ™ " ^ ) and the family ( T
( m i n )
(t))<>
0
satisfies (3.3). If T is a n o t h e r s o l u t i o n t o (3.3) t h e n
we c a n show by i n d u c t i o n that T„(t) <
T(t)
146 for a l l n g N a n d a l l t > 0. H e n c e w e o b t a i n (3.4).
•
F o r a l l A > 0 let u s c o n s i d e r t h e o p e r a t o r Q\ o n B(h) d e f i n e d b y ••oo (v,Q (X)u)= / exp(-Xs){LP(s)v,XLP{s)u) ds Jo x
U s i n g (3.1) a n d i n t e g r a t i n g b y p a r t s w e c a n e s t i m a t e t h i s b y \\X\\ t i m e s oo yoo J 2
/
e x p ( - A s ) \\LP(s)u\\
H e n c e Q\(X)
ds = - j
2
b e l o n g s t o B(h).
e x p ( - A s ) - \\P(s)u\\ ds < \\u\\ 2
o
2
M o r e o v e r t h e m a p Q\ : B(h) i-+ B(h) i s n o r m a l
a n d monotone. P r o p o s i t i o n 3 . 5 . Suppose
that condition
(A) holds.
Then
the following
condi-
tions are equivalent: i) T^ (t) = I for all t >Q, ii) s - l i m n - . o o Q1{I) = 0 f o r aJJ A > 0. n)
Proof.
L e t T „ b e t h e sequence d e f i n e d b y (3.5). F o r e v e r y i n t e g e r n > 0 a n d
e v e r y t > 0 let R (t)
=
n
A s o l u t i o n t o (3.3) i s g i v e n b y T(t)
I-T (t) n
= I f o r a l l t > 0. T h e r e f o r e {R (t)) > n
n
is
0
a sequence o f p o s i t i v e c o n t r a c t i o n s a n d , u s i n g (3.6) t o c o m p u t e Ro, we h a v e the recursion relations (u, Jlo(i)u) = I N I
- ||P(
2
2
= / ' \\LP(t - s)u\\ ds 2
J
(u,R (t)u)
=
n+1
for e a c h u € T>. L e t R
n
/ Jo
(LP(t
°
(3-7)
- s)u,R (s)LP(t
- s)u) ds
n
denote the L a p l a c e t r a n s f o r m of R
n
e x p ( - A f ) ( u , Rn(t)u) for a l l A > 0, u g V.
given by dt
T a k i n g t h e L a p l a c e t r a n s f o r m o f b o t h sides o f (3.7) a n d
a p p l y i n g F u b i n i ' s t h e o r e m t o t h e r i g h t - h a n d side w e c a n e a s i l y o b t a i n (u,R (\)u}
=
n+1
(u,Q (R (X))u\ x
f o r a l l u g V. T h e r e f o r e , since Ro(X) = Q\(I), R„W for a l l i n t e g e r n a n d a l l A > 0.
n
for all A > 0 we have (3.8)
= Ql (I) +1
N o w the conclusion follows f r o m t h e clearly
equivalent conditions: - t h e i n c r e a s i n g sequence ( T ( t ) ) „ > o n
converges s t r o n g l y t o I f o r a l l t > 0,
- t h e d e c r e a s i n g sequence (R (t))n>o
converges s t r o n g l y t o 0 f o r a l l t > 0,
- t h e d e c r e a s i n g sequence ( i ? ( A ) ) „ > o
converges s t r o n g l y t o 0 f o r a l l A > 0,
n
n
- t h e d e c r e a s i n g sequence Q"(I)
converges s t r o n g l y t o 0 f o r a l l A > 0.
•
147
Proposition conditions
3.6.
i) s - l i m , , - . ^ Ql(I) ii)
the equation
Proof.
Suppose
that
the
condition
(A)
holds.
Then
the
following
solution
for all A > 0.
are equivalent: = 0 f o r a l l A > 0,
X = Q\(X)
has no bounded
positive
F o r e a c h A > 0 a n d e a c h i n t e g e r n > 0, t h e f o r m u l a (3.8) y i e l d s
If t h e c o n d i t i o n i ) d o e s n o t h o l d t h e d e c r e a s i n g sequence ( Q " ( / ) ) > o converges n
t o a n o n z e r o p o s i t i v e o p e r a t o r X w h i c h is a s o l u t i o n of t h e e q u a t i o n X = b e c a u s e of t h e n o r m a l i t y of Q\.
Conversely, if there exists a positive
n o n z e r o o p e r a t o r X s a t i s f y i n g t h e e q u a t i o n X = Qx(X),
Q\(X) bounded
d i v i d i n g X by its n o r m
we c a n s u p p o s e t h a t X i s less o r e q u a l t h a n t h e i d e n t i t y / . W e h a v e t h e n X = Qx(X)
<
Q (I). X
H e n c e w e c a n p r o v e b y i n d u c t i o n t h a t t h e o p e r a t o r s Q"(I)
are b o u n d e d
b e l o w b y X f o r a l l n > 0. T h e r e f o r e t h e c o n d i t i o n i ) does n o t h o l d .
Proof,
(of T h e o r e m 3.3).
from
•
If T is n o t c o n s e r v a t i v e , t h e n t h e r e e x i s t s a p o s i t i v e
s o l u t i o n of t h e e q u a t i o n (3.2) w i t h T ( t ) < / for s o m e t > 0. T a k i n g t h e L a p l a c e t r a n s f o r m of I—T
a n d a p p l y i n g F u b i n i ' s t h e o r e m we c a n s h o w t h a t t h e c o n t r a c t i o n
X defined by (v,Xu)=f
Jo
exp(-Xt)(v,T(t)u)
is a n o n z e r o p o s i t i v e s o l u t i o n of t h e e q u a t i o n XX
dt Conversely, if there
= C(X).
e x i s t s a n o n z e r o p o s i t i v e s o l u t i o n of t h i s e q u a t i o n , t h e n , for a l l r > 0 a n d a l l u, v € T>, we h a v e (LP(t)v,
XLP{t)u)
= X (P(t)v,
XP(t)u)
- (P(t)Gv,
XP(t)u)
- (P(t)v,
XP(t)Gu)
T a k i n g t h e L a p l a c e t r a n s f o r m of b o t h sides w e o b t a i n (v,Qx(X)u)
= - J
-exp(-Xt)(P(t)v,XP(t)u)
dt =
A p p l y i n g t h e a b o v e P r o p o s i t i o n s we c a n c o m p l e t e t h e p r o o f .
(v,Xu).
•
References [1] A c c a r d i , L . : O n t h e q u a n t u m F e y n m a n - K a c f o r m u l a . R e n d i c o n t i d e l S e m i n a r i o M a t e m a t i c o e F i s i c o d i M i l a n o . V o l . X L V I I I (1978).
148 [2] A c c a r d i , L . , F r i g e r i o , A . , a n d L u , Y . G . : T h e w e a k c o u p l i n g l i m i t as a q u a n t u m f u n c t i o n a l c e n t r a l l i m i t . Commu.
Math.
Phys.,
[3] A c c a r d i , L . , J o u r n e ' , J . L . L i n d s a y , J . M . : cocycles.
1 3 1 , 537-570 (1990).
O n multidimensional Markovian
I n : A c c a r d i , L . , W a l d e n f e l s , W . , v o n (eds.)
a n d a p p l i c a t i o n s I V . P r o c e e d i n g s R o m e 1987 ( L e c t . p p . 59-67) B e r l i n Heidelberg N e w Y o r k , Springer, [4] C h e b o t a r e v ,
A . M . : The
theorem.
of the conservative
1396,
1989. dynamical
semigroups
and
P r e p r i n t M I E M n . l . M a r c h 1990.
its applications. [5] C h e b o t a r e v ,
theory
Quantum Probability
Notes M a t h . V o l .
A . M . , Fagnola, F . , Frigerio A . :
Towards
a stochastic
Stone's
Trento, I t a l i a , P r e p r i n t (1990).
[6] F a g n o l a , F . : O n q u a n t u m s t o c h a s t i c coefficients. P r o b a b .
Th.
Rel.
differential equations
with
unbounded
8 6 , 501-516 (1990).
Fields,
[7] F a g n o l a , F . : P u r e b i r t h a n d p u r e d e a t h processes as q u a n t u m flows i n Fock space. Sankhya,
5 3 , Series A
(1991).
[8] F a g n o l a , F . , M a n c i n o , M . : Free n o i s e d i l a t i o n of s e m i g r o u p s of c o u n t a b l e state M a r k o v processes. T r e n t o , I t a l i a , P r e p r i n t ( 1 9 9 1 ) . [9] J o u r n e , J . - L . : S t r u c t u r e des c o c y c l e s m a r k o v i e n s s u r l ' e s p a c e de F o c k . Th.
Rel.
Fields
Probab.
7 5 , 291-316 (1987).
[10] M o h a r i , A . : Q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s w i t h u n b o u n d e d
coeffi-
cients a n d d i l a t i o n of F e l l e r ' s m i n i m a l s o l u t i o n . P r e p r i n t , I . S . I . , D e l h i C e n t r e . (1991). [11]
M o h a r i , A . , Parthasarathy, K . R . :
O n a class of g e n e r a l i z e d
Evans-Hudson
flows r e l a t e d t o c l a s s i c a l M a r k o v processes. P r e p r i n t , I . S . I . , D e l h i C e n t r e Notes (1990). [12]
M o h a r i , A . , P a r t h a s a r a t h y , K . R . : A q u a n t u m p r o b a b i l i s t i c a n a l o g u e of F e l l e r ' s c o n d i t i o n f o r t h e e x i s t e n c e of u n i t a r y M a r k o v i a n c o c y c l e s i n F o c k s p a c e . P r e p r i n t , I.S.I., D e l h i Centre. (1991).
[13]
P a r t h a s a r a t h y , K . R . : A n Introduction D e l h i 1990.
to Quantum
Stochastic
Calculus.
New
Quantum Probability and Related Topics Vol. VII (pp. 149-163) © 1992 World Scientific Publishing Company F R E E
NOISE DILATION OF S E M I G R O U P S
C O U N T A B L E
STATE M A R K O V
149 OF
PROCESSES
F. Fagnola and M . Mancino Universita di Trento, Dipartimento di M a t e m a t i c a
I - 38050 Povo (TN)
Abstract. flows
Italia
W e s h o w h o w free noises c a n be u s e d t o c o n s t r u c t i n n e r q u a n t u m
g i v i n g u n i t a r y d i l a t i o n s of e v e r y u n i f o r m l y c o n t i n u o u s q u a n t u m d y n a m i c a l
s e m i g r o u p s a n d o f s e m i g r o u p s of c o u n t a b l e s t a t e t i m e c o n t i n u o u s c l a s s i c a l M a r k o v processes.
1.
Introduction
O n e of the most
i m p o r t a n t a p p l i c a t i o n s of q u a n t u m s t o c h a s t i c c a l c u l u s is t h e
d i l a t i o n of q u a n t u m d y n a m i c a l s e m i g r o u p s .
Various q u a n t u m stochastic calculi
b a s e d o n different q u a n t u m noises h a v e b e e n d e v e l o p e d (see [2] f o r a u n i f y i n g a p proach).
E a c h of t h e m has b e e n u s e d t o p r o d u c e d i l a t i o n s of a class of u n i f o r m l y
continuous q u a n t u m d y n a m i c a l semigroups.
T h e s e are r e a l i z e d v i a t h e q u a n t u m
F e y n m a n - K a c p e r t u r b a t i o n t e c h n i q u e d e v e l o p e d b y A c c a r d i i n [1] b y
conjugation
w i t h a u n i t a r y c o c y c l e w h i c h is t h e s o l u t i o n o f a q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n . T h e d i l a t i o n o b t a i n e d is a n o n c o m m u t a t i v e s t a t i o n a r y M a r k o v p r o c e s s i n t h e sense of K u m m e r e r [8]. T h e B o s o n F o c k s p a c e q u a n t u m s t o c h a s t i c c a l c u l u s c a n be u s e d t o c o n s t r u c t d i l a t i o n s of e v e r y q u a n t u m d y n a m i c a l s e m i g r o u p
with
b o u n d e d g e n e r a t o r as d o n e b y H u d s o n a n d P a r t h a s a r a t h y i n [6]. I n t h i s case t h e M a r k o v p r o c e s s g i v i n g t h e d i l a t i o n is a n E v a n s - H u d s o n flow. W h e n t h e g e n e r a t o r of t h e q u a n t u m d y n a m i c a l s e m i g r o u p is n o t b o u n d e d i t is n o t k n o w n i n g e n e r a l w h e t h e r a B o s o n F o c k space q u a n t u m stochastic calculus d i l a t i o n c a n be given. T h e answer is i n the affirmative for semigroups satisfying suitable conservativity c o n d i t i o n s as s h o w n b y C h e b o t a r e v i n [3]. T h e s t u d y of t h e s p e c i a l case of s e m i g r o u p s a s s o c i a t e d w i t h c o u n t a b l e s t a t e M a r k o v p r o c e s s e s is c a r r i e d o n b y M o h a r i a n d P a r t h a s a r a t h y i n [10],[11]. I n t h i s case t h e E v a n s - H u d s o n flow, r e s t r i c t e d t o a s u i t a b l e a b e l i a n * - a l g e b r a , is also c o m m u t a t i v e ; t h i s a l l o w s t o r e a l i z e t h e c o r r e s p o n d i n g c l a s s i c a l M a r k o v p r o c e s s e s as q u a n t u m flows i n t h e B o s o n F o c k s p a c e . I n t h i s n o t e w e s h o w h o w free noises i n t r o d u c e d b y S p e i c h e r i n [14] c a n b e u s e d
150 t o c o n s t r u c t u n i t a r y d i l a t i o n s o f t w o classes o f q u a n t u m d y n a m i c a l s e m i g r o u p s . F i r s t w e e x t e n d t h e free n o i s e q u a n t u m s t o c h a s t i c c a l c u l u s t o t h e case o f i n f i n i t e free c r e a t i o n a n d a n n i h i l a t i o n fields; t h i s a l l o w s u s t o o b t a i n a free n o i s e d i l a t i o n of e v e r y u n i f o r m l y c o n t i n u o u s q u a n t u m d y n a m i c a l s e m i g r o u p . T h e n w e p r o v e a n existence, uniqueness a n d u n i t a r i t y theorem for a q u a n t u m stochastic differential e q u a t i o n w i t h u n b o u n d e d coefficents d r i v e n b y i n f i n i t e c r e a t i o n a n d a n n i h i l a t i o n fields.
Its s o l u t i o n i s t h e u n i t a r y c o c y c l e w h i c h gives a d i l a t i o n o f t h e s e m i g r o u p of
every c o u n t a b l e state t i m e continuous M a r k o v process. It is t o b e n o t e d t h a t the sufficient c o n d i t i o n f o r u n i t a r i t y o f t h e s o l u t i o n t o t h e q u a n t u m s t o c h a s t i c different i a l e q u a t i o n w e c o n s i d e r , is t h e c l a s s i c a l c o n s e r v a t i v i t y c o n d i t i o n (see a s s u m p t i o n ( A ) i n section 4). T h e proof of existence of our q u a n t u m stochastic differential e q u a t i o n is done b y the Frigerio equicontinuity m e t h o d .
Isometricity is proven reducing the problem
t o t h e u n i q u e n e s s a n d c o n s e r v a t i v i t y o f t h e s o l u t i o n t o t h e c o r r e s p o n d i n g classic Kolmogorov backward equation. W e introduce here a new m e t h o d for p r o v i n g c o i s o m e t r i c i t y r e d u c i n g the p r o b l e m t o t h e i s o m e t r i c i t y o f t h e t i m e r e v e r s e d p r o c e s s w h i c h e s s e n t i a l l y satisfies, i n o u r case, t h e s a m e e q u a t i o n . T h i s t e c h n i q u e c a n b e u s e d a l s o f o r e q u a t i o n s d r i v e n b y the B o s o n Fock space basic martingales.
2. N o t a t i o n a n d p r e l i m i n a r i e s L e t ho, k b e c o m p l e x s e p a r a b l e H i l b e r t s p a c e s , let T b e t h e f u l l F o c k s p a c e over h = L ( R ) ® k a n d let H b e t h e t e n s o r p r o d u c t ho ® T. F i x o r t h o n o r m a l basis 2
+
{/j}jez* i nk and { e , }
i g z
i n h a n d w r i t e , f o r a n y u £ h,t E R + , j 0
£ Z*
«>(t)=(/ u(0>. i(
L e t Ai b e t h e set o f s t e p f u n c t i o n s u i n t h e H i l b e r t s p a c e h s u c h t h a t u (t) = 0 1
e v e n t u a l l y , s u p \u (s)\ < 1 f o r a l l j £ Z * a n d ||u|| < 1. F o r a l l e l e m e n t u o f M let s
N(u)
3
= m a x { | i | : u ' ( - ) is a n o n z e r o f u n c t i o n i n L ( R ) } . 2
+
L e t u s d e n o t e b y T> t h e set o f e l e m e n t s o f H w h i c h c a n b e w r i t t e n i n t h e f o r m x ®
where x is a n element of the linear m a n i f o l d generated b y { e ; } ,
e z
and (
is t h e fc-fold t e n s o r p r o d u c t o f k e l e m e n t s o f Ai f o r s o m e p o s i t i v e i n t e g e r k. T h e v e c t o r x ® £ w i l l b e d e n o t e d a l s o b y x£. F o r a l l £ = u i ® . . . ® ujt a n d h £ Z l e t n u +i h
h
b e t h e e l e m e n t « i ® . . . ® u -\ ® h
® . . . ® Uk i f h € {1, • - . , k}, t h e v a c u u m v e c t o r ft i f k = 1, a n d 0 o t h e r w i s e
151
We will often identify an operator defined on a (tensor product or free product) factor of ?t with its canonical extension to 1t. For every 9 E L2(~, C) consider the free annihilation and creation operators {lj,l; jj e-Z·} defined, for allj E Z·, by: I j (g)U1 ® ... ® Uk = (gj, uih2(R+)U2 ® ... ® Uk
1;(g)U1 ® ... ® uk
= (g/j) ® 1.11 ® ... ® Uk.
Let Atj, (resp. A[I) with t E [0,+00] be the *-subalgebra of 8(:F) generated by {lj(g)}jEZ- where 9 is an element of L2(O,t) (resp. L 2(t,+00)). Consider then the class £tj(V,?t) (resp. £:j(V,?t)) of operators with domain containing V, which are weak (resp. strong) limits on V of finite sums of operators of the form L ® A, where L is a bounded operator on ho and A is an element of Atj . We write £(V,?t) (resp. £"(V,?t)) when t = +00. The classes £[t(V,?t) and £it(V,?t) are defined in a similar way.
Definition 2.1. A process X is a family {X(t)}t~O of operators in £(V, ?t). The process X is called adapted if X(t) E £'j(V,?t) for all t E R+. For all z E ?t with chaos decomposition with respect to Fock space z = 2:~o Zk, let [z ® e] denote the vector in ?t with chaos decomposition 2:~o Zk ® Notice that we have lI[z ® IIzlllleli. Moreover the map z 1-+ [z ® e] is continuous with respect to the weak topology on ?t. With every A E 8(h o ) ® Aooj it is possible to associate two operators A+, A_ of 8(h o) ® Aooj ([5],[9]) characterized by:
e.
m::;
A+xe
= [(Ax!!) ® e]
(A_Y77, xe)
= (Y77, [(A· x!!) ® en.
(2.1)
Remark that, for all t E R+ and all X E £tj(V,?t) (resp. £:j(V, ?t)), the first formula (2.1) uniquely defines an operator X+ E £tj(V,?t) (resp. £:j(V, ?t)). We need the following extensions of the definitions of left and right stochastic integrals with respect to Ij , I;,j E Z·. Consider the class I2 of processes P strongly measurable on V, such that pet) E £tj(V, ?t) and
1t
IIp(s)xeIl2ds < +00
for all t E R+. Moreover consider the class I_ of processes P with the following property: there exists a sequence {p(n)}n>o in 8(h o)®Aooj such that for all t E R+ p(n)(t) is a member of 8(h o) ® Atj, the ~perator p(n)·(t) converges strongly on ho and for all t E R+ and all vector xe E V it holds: t
lim
m,n~CX)
1 0
2
II(p~n)(s) - p~m)(s))xell ds = O.
152
For all FE I_ one can define the process F_ as a strong limit on 1) of processes {F~n)} n>O ' Clearly F_ does not depend on the particular sequence chosen, because (2.1) implies the strong convergence of {F(n)·(t)xn}n~o for all t E R+ and for all
xn E 1). Observe that ([5] Prop. 3.1) the integrability from the right with respect to Ij, I; is ensured for elements in the class I2 thanks to (2.1) . Each element of I2 can be integrated from the left with respect to I; . Moreover, for all F E I_, one can form the left stochastic integrals with respect to Ij. The following lemmas can be proved by straightforward computations. Lemma 2.2. Let G,F be processes in I 2 , then for all x' e', x~ E 1), for all positive real numbers s, t with s < t and for all j E Z·, we have:
(i)
(G(s)x'e',F(s)lj(s,t)x~) = J.I u{(r)dr(G(s)x'C' , F(s)X771)
Lemma 2.3. Let G be a process in I2 and F be a process in I_, then for all x~, x'e' E 1), for all positive real numbers s, t and j E Z· we have: k-l
(G(s)x'C',/j(s,t)F_(s)x~) = L
1
J. u~+l(r)dr(G(s)x'C',F~o-\S)X77Q+l)
0=0
s
where F~o-) denotes the process 2:.13>0 IIpF_IIp+o-, and IIp is the orthogonal projection onto the {3-th chaos in:F. Now from lemmas (2 .2), (2.3) and a slight extension of Theorem 6.1 ([5]) we obtain the right and left Ito formulae. Suppose that for all J' E Z* , FJ· , GJ· , H J'· J{-}, Go,Ko, are processes in I2 nI_. Define, for all n EN,
x(n)(t)
=
t Go(s) ds + L {t dlj(s)Fj(s) + t dlj(S)Gj(S)} 10 10
10
y(n)(t)
=
11
O
Ko(s) ds
o
Then, for all x'e',x~ E
+
L {1 O
1),
1
dlj(s)Hj(s)
+
0
we have the left Ito formula:
t dZj(S)Kj(S)}.
10
153
E
+
E
(Y \s)x'(\(F t\s)xn )ui ( )ds
/
(n
]
a+1
+1 S
0<|>|
0<|>|
J
0<|>|
+
E
J
E
K
J
E
f -
S X
(n)
j
+
«^)-W4«,Jr W*f>«24-iW* w
1 / 0
V /Vi(*)*'€'.Gi(»)*0 o<| |<„-
(K (s)x'C,X^\s)xl)ds+ 0
7 0
N o w l e t F , - , G j , Hj,Kj(j
d s
-
(-) 2
2
/o
J
Xl"\t)
ds
a+1
o
/
0<|j|
+
o
f\ j( ) '('d^ )-\^a l)u (s)
E
0<|j|
+
o
G Z * ) , G , # 0 b e processes i n J . Define for all n G N : 0
= f G (s)ds Jo 0
+
Y 0 < | j |
2
<
{ f IJO
n
r<»>(t) = / ' /ro(s)ds + E {/' y
L
0
/ o
F,is)dl,is)+
Jo
^ ( - w * ) + / -
fGMdlXs)} J
*oo)<«;(*)} •
7 0
J
T h e n , f o r a l l x ' £ ' , x £ G I ? , we h a v e t h e r i g h t I t o f o r m u l a :
(Y<»>(i>r,*
Jo +
( B )
(*K)
=
J O
E
(/V
( n )
(*yr,F>(a)*»h)«iw
0<|>l
+
/'{F(")(s)x'nl ,_ (Gy) (s)x04_*^) t
i l
+
*
JO
+
[\(K ) (s)x'(',X( \s)xT, - ,)ui_ ,(s) Jo n
j +
k k
+ j\H (s)x'n X^\s)xc:}uY{r) j
u
ds
k
ds + j f { ^ ( ^ ' f i , G j ( s ) x n ) ( e , ( )
F i n a l l y we have the following results:
154 P r o p o s i t i o n 2 . 4 . Let G,{Fj,j and suppose
that there
€ Z*},{Gj,j a positive
exists
£
Then,
for each X
G B(h ),
0
t G R + , the following
X(t)
= X
exists
+ £
0
2
in h
X
€ J
2
(11-
satisfying,
for all
Let L
d
be bounded constant
denote
0
+
c, it
i( ) i) +
,
s
G
operators
G
ods|
in h , 0
such that H is self-
holds:
l i W s ^ M i '
the
operator:
kezThen
there
exists
U(t)
a unique
= 1+ f J
M o r e o v e r t h e adjoint
V{t)
= l
0
2
a process
( ^ / W
I
E
0
operators
equation:
and for some positive
f o r a l l a; i n h .
2
y
T h e o r e m 2 . 5 . Let H, {Lj}jez' adjoint
be bounded
c such that, for all x 6 ho
||G *||
there
0
€ Z*}
constant
unitary
U(s)
+
J £
/[g
U such
(-L'dl^s)
that:
+ Ljdl'^s))
[jez'
o
process
process
V
+ L*ods. 1
J
(2.4)
satisfies
W
W
~ m
a
)
L
i )
+L ds\ 0
V(s).
A s a n a p p l i c a t i o n o f T h e o r e m 2.5 we o b t a i n (as i n [6]) a d i l a t i o n o f e v e r y u n i formly continuous quantum dynamical semigroup.
155
3 . Cocycle properties of solutions to quantum stochastic differential equations Let 6 b e t h e r i g h t shift o n t
L (R+) ® 2
ifc d e f i n e d b y
.0
if x < t
and let T , b e t h e operator o n T defined b y TQ t
= ft ,
T ui
® . . . ® ujt = (Btut)
t
® . . . ® (fliujt).
F o r a l l s 6 R+ a n d a l l b o u n d e d o p e r a t o r X i n £(2>, W ) , t h e o p e r a t o r T,XT* 2
Definition
maps
o o ) ® fc) i n t o i t s e l f . T h e c a n o n i c a l e x t e n s i o n t o Ti w i l l b e d e n o t e d b y
ho ® F(L (s,
3 . 1 . A bounded
process
on Ti is a cocycle
{U(t)}
t>0
if, for all s,t G
R-I-, w e h a v e U(t +s)
(3.1)
= U(s)T U(t)V . 3
s
In this section we w i l l s t u d y t h e cocycle properties of solutions t o q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s s a t i s f y i n g t h e a s s u m p t i o n s o f T h e o r e m 2.5. n o w o n w e w i l l denote b y U a u n i t a r y process satisfying
From
(2.4).
L e t u s first s t a t e t h e f o l l o w i n g p r o p o s i t i o n .
Proposition
3 . 2 . T h e u n i t a r y p r o c e s s U is a
cocycle.
C l e a r l y , e x c e p t i n s o m e q u i t e t r i v i a l cases, t h e a d j o i n t U* t o t h e s o l u t i o n o f will n o t be a cocycle.
(2.4)
H o w e v e r , a c t i n g o n U* b y a u n i t a r y t i m e r e v e r s a l , w e c a n
associate w i t h i t a cocycle.
F o r a l l t 6 R+ l e t 1Z b e t h e u n i t a r y o p e r a t o r o n T t
defined by
TZtUx ® ... ® u where p is the operator o n t
{ p t f
k
t
L (R ) ® k 2
>
{ s
... ®
= (p ui)®
(p u )
defined, for a l l s
+
t
k
€ R+,
by
i f s X .
>-\f(s)
C l e a r l y , f o r a l l t E R+, w e h a v e TV = TZ . t
t
L e t U{t) d e n o t e t h e u n i t a r y o p e r a t o r TZ U*(t)TZ . t
t
T h e following result holds.
156 P r o p o s i t i o n 3 . 3 . The process Proof.
{U(t)}
and is a
is adapted
t>0
cocycle.
A d a p t e d n e s s f o l l o w s f r o m a d a p t e d n e s s o f U* a n d t h e f a c t t h a t , f o r a l l
t £ R + , t h e u n i t a r y o p e r a t o r 72., leaves V i n v a r i a n t a n d i s a n e l e m e n t o f £ ] t
(V,H).
L e t u s p r o v e n o w t h e c o c y c l e p r o p e r t y . F o r a l l s, t € R + , u s i n g t h e c o c y c l e p r o p e r t y of U w e c a n w r i t e t h e i d e n t i t i e s
7i u; n t+a
+3
=
t+a
iz ,r u:r* u "ii , t+
=
t
t
t
t+
(n tW*rjTi )(7z u;Ti ) t+a
t+a
t+s
t+3
T h e c o c y c l e p r o p e r t y f o l l o w s t h e n f r o m n e x t L e m m a 3.4. •
L e m m a 3 . 4 . For all s,t € R
and all X e C (V,H),Y t]
+
= T TZ X1Z T*
TZ X1Zt+ t+a
a
= 7e
n Yiz a
a
a
< + s
t
t
r^Fr*7e
€ C (V,H) we h a v e [t
(3.2)
a
t + s
.
(3.3)
T h e p r o o f c a n b e g i v e n b y d i r e c t c o m p u t a t i o n o f m a t r i x e l e m e n t s o f b o t h sides of ( 3 . 2 ) , ( 3 . 3 ) u s i n g t h e p r o p e r t i e s o f free p r o d u c t s . I t i s q u i t e s i m i l a r t o t h e p r o o f of L e m m a 3.6 a n d w e o m i t i t . T h e u n i t a r y c o c y c l e U i s t h e d u a l c o c y c l e ( i n t h e sense o f J o u r n e [7]) a s s o c i a t e d w i t h U. W e c a n w r i t e t h e q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s a t i s f i e d b y U. P r o p o s i t i o n 3 . 5 . The adapted quantum
to the
differential equation
stochastic
X(t)
U is the u n i q u e u n i t a r y solution
process
= l + j* X(s)
(-Ljdl'is)
| g
+ L]dlj{s))
+ LI ds J .
processes
such that F(t),
(3.4)
L e t us first prove the following L e m m a 3 . 6 . Let F,G be two bounded £ ( ] ( P , H) for any t € R point
for functions
+
adapted
. For all
a n d all t 6 R
u,u', there exists
+
which
is a
G(t) S continuity
h(t) > 0 s u c h t h a t f o r all s,h E ( 0 , h(t)) we
have (F(i)z'e,r Tl G(ayR. Tixt) t
Proof.
h
(3.5)
= (F(i)x'i',T G(a)J^x£).
k
t
F o r s i m p l i c i t y w e s h a l l p r o v e ( 3 . 5 ) o n l y i n t h e case w h e n
of t h e first c h a o s .
S a y £ ' = u , £ = u w i t h u,v € M.
n u m b e r s u c h t h a t v, u a r e c o n s t a n t o n (t, t + h(t)).
S i n c e F, G a r e i n C (V, S
c a n r e s t r i c t o u r s e l v e s t o t h e case w h e n t h e o p e r a t o r s F(t),G(s) of the f o r m F(t)
= A ®A 0
t
,
G(s) =
£ are vectors
L e t h(t) b e a p o s i t i v e r e a l
B ®B, 0
Ti.) w e
w i t h s < h(t) a r e
157 w i t h A ,B 0
e B(h ),A
0
0
e S ( . F ( £ ( 0 , r ) ® k)),B.
€ Z ? ( . F ( L ( 0 , s ) ® Jfc)). W e
2
t
have
2
then (Ao A x'v,
TtTZhB^hT'tBoxu)
t
= (A x'
(A A x'v,r B r;xu) 0
S i n c e T TZhB TZhT"t t
t
t
€ B(F(L (t,
(A x'v,T 1Z B,TZ r;u) t
t
h
0
0
a
1
t t
n
t
t
b y t h e f u n d a m e n t a l p r o p e r t i e s of free
+
+ (A v T Tl B 1Z T* u ) t
th
t
h
a
h
t
(A v ,T 1l B,n T* u ) t
+
[t
[t
t
h
h
t
t]
(A v T Tl B 1ZKT* u ) t
[u
t
h
s
t
[t
(A v ,u )(n,r Tz B n T Q)
=
t
t]
t
t]
h
h
s
m t
+
{A v ,Q)(a,T TZ B,TZ r;u )
+
(A n,n)(v ,r 7i B,Tz v u )
t
t
t]
t
k
t
[t
a n d t h i s , b e c a u s e o f t h e a s s u m p t i o n o n u,v,
t
t]
t
s
t
[t
h
h
t
n
[t
c a n be w r i t t e n i n t h e f o r m
= ( A ^ . ^ x ^ s . a ) + {A v ,ci)(n,r B v u }
+
[t
(A n,n)(v r B v* u ). t
F i n a l l y a s i m i l a r c o m p u t a t i o n s h o w s t h a t (A v,T B r*u) t
s a m e way.
t
we have t h e n = (Atv^TtTlkB.Tl^Ut])
h
t
(A x',B x}(A v,T B,r;u}
+ 0 0 ) ® k)),
2
3
0
=
a
p r o d u c t s , f o r a l l h < h(t)
,B x)(A v,T 1l B ,Tl T u)
0
t
[u
t
t
t
[t
can be w r i t t e n i n the
s
•
L e m m a 3.7.
For ail
functions
we
u,u'
G T> and all t G R
+
which
is a continuity
point
for
have
i Proof.
( x V - ^ + W ^ ^ ^ - ^ W
+ (x'S',U(t)L* x(;)} 0
.
(3.6)
B y t h e c o c y c l e p r o p e r t y , f o r a n y t G R + w h i c h is a c o n t i n u i t y p o i n t for
a n d f o r h > 0 s m a l l e n o u g h , we h a v e
u,u'
(x'C,
(U*(t
+ h) - U(t))x£)
= <£*(*>'£', r ^ f y ^ / i ) - i ) f t
f
c
r ; * o = (u*(t)x't',rt(u*(h)
-
i)r* x() t
b e c a u s e o f L e m m a 3.6. D e n o t e Y*(h)
= T (U*(h) t
-
l ) r ? a n d n o t i c e t h a t Y\h)
can be w r i t t e n i n the
form
Y
f r
I I
t+h
L)Y\s
~ t)dlj(s)
- LjY\
s
- t)dl*j(s)
+ L* Y\s 0
- t)
ds
158 T h e r e f o r e , b y L e m m a 8.1 ([5]), w e o b t a i n (x't\(U(t
+
h)-U(t))xt)
t+h {x'(.\U{t)L]Y\s
/
j
-t)x )u{(s)
ds
ni
t+h
.
/ t+h {x't^UWL'Ytis-VxQ
/ N o w (3.6)
ds
f o l l o w s b e c a u s e t i s a c o n t i n u i t y p o i n t f o r u,u'
a n d Y*
is s t r o n g l y
c o n t i n u o u s o n T>. •
Proof
(of P r o p .
Z = U — X.
3.5).
Let X
b e t h e u n i q u e u n i t a r y s o l u t i o n t o (3.4)
is a c o n t i n u i t y p o i n t f o r
u,u' j {x'(',Z(t)xO
= 0
t
w h i c h i m p l i e s t h a t Z(t) o n V.
a n d let
U s i n g t h e f o r m u l a e (8.1) ([5]) a n d L e m m a 3.7 w e h a v e , f o r a l l t w h i c h
= 0 f o r a l l t g R + , b e c a u s e o f t h e s t r o n g c o n t i n u i t y of Z
•
4 . Dilation of semigroups of countable state Markov processes through unitary evolutions in the full Fock space C o n s i d e r a f a m i l y (qij)ijez
of r e a l n u m b e r s s u c h t h a t qij > 0 f o r a l l i ^ j Y
Hj = 0
f o r
a 1 1
•
(4.1) (4.2)
L e t qi d e n o t e —qu, c l e a r l y we h a v e gt > 0. It i s k n o w n ([13]) t h a t o n e c a n assoc i a t e w i t h (o s a t i s f y i n g t h e K o l m o g o r o v b a c k w a r d e q u a t i o n s PiA*) = E « t f « W kez
Ptf(0) = ' « -
M o r e o v e r s o l u t i o n t o (4.3) is u n i q u e a n d satisfies t h e c o n d i t i o n
jez
(4.3)
159 i f a n d o n l y i f t h e f o l l o w i n g c o n d i t i o n ( T h e o r e m 7 [13]) h o l d s : ( A ) f o r s o m e A > 0 t h e set o f e q u a t i o n s
(A + 9 , ) x ; =
Y
(-)
1H i x
4 5
has n o b o u n d e d n o n negative solution other t h a n Xj = 0 f o r a l l i e Z . I n [10] M o h a r i s h o w s t h a t u n d e r t h i s a s s u m p t i o n t h e u n i q u e M a r k o v c h a i n s a t i s f y i n g (4.3) c a n b e r e a l i z e d as a c o m m u t a t i v e i d e n t i t y p r e s e r v i n g q u a n t u m flow i n a B o s o n F o c k space. I n this section under the a s s u m p t i o n ( A ) w e w i l l construct a n i d e n t i t y p r e s e r v i n g q u a n t u m flow o n Ti g i v i n g a d i l a t i o n o f t h e s e m i g r o u p by
jez.
generated
T h e q u a n t u m flow i s r e a l i z e d b y c o n j u g a t i o n s w i t h r e s p e c t t o a u n i -
t a r y operator valued a d a p t e d process satisfying a q u a n t u m stochastic differential e q u a t i o n d r i v e n b y free noises. L e t /in = / ( Z ) b e t h e i n i t i a l s p a c e w i t h c a n o n i c a l o r t h o n o r m a l b a s i s
(th)hez-
2
L e t S b e t h e u n i t a r y right s h i f t o n /in d e f i n e d b y 5 e j , = e +i n
be t h e o p e r a t o r d e f i n e d b y D(N) = { x € h : J2k k \xkf
< + ° ° } and Nk =
2
F o r a l l j S Z * l e t {^j(k)}
for all h 6 Z and N ke .
e
k
b e a sequence of c o m p l e x n u m b e r s such t h a t
kez
l\>(*0|
=9kk+j-
2
W e w i l l study the q u a n t u m stochastic differential equation
dU(t) U(0)
= U(t) E
i
e
z
.
- A (iV)5-^/ (<) - |\\ iN)\ dt)
(S>\j(N)dl*(t)
J
>
3
2
(
4
g
)
= 1
F i r s t we prove t h e existence of a solution b y t h e Frigerio equicontinuity m e t h o d as o u t l i n e d i n [4]. F o r a l l n e N l e t
M(k)
= \ >W 10 X
x
J
a n d let
tf|*l<_n otherwise.
b e the unique u n i t a r y solution t o the q u a n t u m stochastic differential
equations w i t h bounded
f dCT(-)(*) = UM(t)
£
coefficents
(5'A( (JV)dZ-(t) n )
- Tf\N)S->dl it) }
- ^
(N)\
2
dt)
{ E/<">(0) = 1 (4.7) A s t r a i g h t f o r w a r d c o m p u t a t i o n u s i n g the Ito f o r m u l a yields the f o l l o w i n g
160 L e m m a 4 . 1 . For all n 6 N , e £
G V and all s, t G R + with
h
J (U^(t)
- t/< >(») e t f n
<(t-
h
T h e a b o v e l e m m a s h o w s t h a t , f o r a l l x'£', x£eT>
s < t we
have
(4.8)
s)q . h
f u n c t i o n s {x'£',
?7
( n )
( - ) x £ ) are
equicontinuous. T h e n b y the A s c o l i - A r z e l a theorem, using t h e separability of the H i l b e r t spaces i n v o l v e d a n d a d i a g o n a l i s a t i o n p r o c e d u r e , w e c a n f i n d a s u b s e q u e n c e w h i c h we w i l l s t i l l d e n o t e b y { U ^ }
n
>
a n d a c o n t r a c t i o n v a l u e d process U such
0
that, for a l l t G R + w -
l i m U (t)
=
(n)
V(t).
n—•oo
Moreover one c a n show that a) U i s s t r o n g l y c o n t i n u o u s o n T>, b ) t h e r i g h t - h a n d s i d e series i n (4.6) i s s t r o n g l y c o n v e r g e n t o n X>, c) U is a s o l u t i o n t o (4.6). W e c a n n o w prove t h a t , u n d e r t h e a s s u m p t i o n ( A ) , U is u n i t a r y . W e w i l l
first
p r o v e t h a t U is i s o m e t r i c . T h e o r e m 4 . 2 . If condition isometric Proof. all e £ h
U(t)e £. n
solution
( A ) holds
then U is isometric.
it is the only
to ( 4 . 6 ) .
W e w i l l show b y i n d u c t i o n on / t h a t , for a l l t G R + a l l h G N a n d e V w i t h N(£)
< I t h e sequence {C^' '(t)e/i£}„>Q converges strongly to n
C l e a r l y we c a n s u p p o s e t h a t £ i s a u n i t v e c t o r . L e t us f i r s t p r o v e s t r o n g
c o n v e r g e n c e w h e n / = 0. F o r a l l k £ N l e t eh®
Moreover
be the orthogonal projection onto
T and put =
Pkk(i)
\\P U(t)e f. k
h
B y t h e I t o f o r m u l a we have t h e n
Phk{t)
= 6hk + Y
qhjPjk(s)
}
J
ds.
(4.9)
o
Because of a s s u m p t i o n ( A ) , K o l m o g o r o v b a c k w a r d equation has a u n i q u e solution satisfying the condition 5 > * * w it
=
1
f o r a l l t G R + . T h e r e f o r e we o b t a i n
n ^ K i i
2
= E i i It
p
^ W
e
* i i
2
=
1
-
161 T h i s i m p l i e s , b y a well k n o w n p r o p e r t y o f weak convergence, c o n v e r g e s s t r o n g l y t o U(t)e .
>o
n
t h a t t h e sequence
Suppose now that, for a l l f £ R
n
+
,
h' £ N a n d e .T) £ V w i t h ^(77) < / - 1, w e h a v e t h e n H
Urn U
s-
( n )
(t)e
U{t)e ,T).
=
h i r l
H
n—•oo F o r e v e r y t £ R + a n d e v e r y e £ £ V w i t h N(£) = I a n d ft £ N w e h a v e h
= l i m (V< >(t)e £,
(U(t)e £,U(i)e ,r,) k
n
h
= 0
U^(t)e , )
k
h V
n—>oo b e c a u s e [/("' i s i s o m e t r i c a n d v e c t o r s e/,£, ef,<»7 a r e o r t h o g o n a l .
Using again the
Ito f o r m u l a we o b t a i n \\U(t)e £\\ h
2
= 1- q
f\\U(s)e (\\ d
h
= l + q t-q h
w h i c h i m p l i e s ||?7(t)eft£||
=
2
s t r o n g l y t o U(t)e £.
h
Jo
h
f
Jo
2
+ q
S
f
h
Jo
\\U(s)e \\ ds 2
h
\\U(s)e £\\ ds 2
h
1. T h e n t h e s e q u e n c e { £ ^ " H * ) f c £ } n > o
converges
e
S i n c e J 7 ' ' i s i s o m e t r i c a n d f o r a l l t £ R + t h e sequence n
n
c o n v e r g e s t o U(t) s t r o n g l y o n T>, h e n c e U i s i s o m e t r i c .
Uniqueness fol-
lows f r o m a s i m i l a r argument u s i n g t h e uniqueness of solutions t o K o l m o g o r o v backward equations under the assumption ( A ) . • W e w i l l n o w p r o v e c o i s o m e t r i c i t y o f U. W e r e d u c e t h e p r o b l e m t o t h e i s o m e t r i c i t y o f t h e s o l u t i o n o f t h e right q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s a t i s f i e d b y t h e c o c y c l e U d e f i n e d i n s e c t i o n 3. P r o p o s i t i o n 4 . 3 . If condition
( A ) holds
then
the unique
isometric
(4.6) i s
coisometric.
Proof.
B e c a u s e o f P r o p o s i t i o n 3.5, f o r a l l n £ N t h e p r o c e s s U <7
( n )
(t) =
(n)
solution
to
defined b y
n u *{t)n t
(n)
t
satisfies t h e q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n U \t) (n
= 1+
£
f
V X*) {n
Q
(\ ;\N)S-idlj(s) (
- Si\\ \N)dr (s) n
3
-
i|A^ (iV)| ds) n )
2
T h e a b o v e e q u a t i o n i s t h e s a m e o f (4.7) e x c e p t f o r t h e s i g n o f A / s . T h e r e f o r e f o r a l l t € R + , U (t) M
c o i verges s t r o n g l y t o V(t). T h i s s h o w s t h a t U(t) a n d U*(t)
are u n i t a r y operators.
•
162 References [1] A c c a r d i , L . : On the quantum
Feynman-Kac
Rendiconti del Seminario
formula.
M a t e m a t i c o e Fisico di M i l a n o . V o l . X L V I I I (1978). [2] A c c a r d i , L . , F a g n o l a , F . , Q u a e g e b e n r , chastic
J . : A r e p r e s e n t a t i o n free quantum
sto-
C e n t r o M a t e m a t i c o V . V o l t e r r a , P r e p r i n t 18, R o m e J a n u a r y
calculus.
1990. T o a p p e a r i n J . F u n c t . A n a l . [3] C h e b o t a r e v , A . M . : T h e theory
[4] F a g n o l a , F . : P u r e birth
and pure
S a n k h y a , 5 3 , Series A
space.
of the
conservative
dynamical
semigroups
and
as q u a n t u m Sows in
Fock
P r e p r i n t M I E M n . l . M a r c h 1990.
its applications.
death
processes
(1991).
[5] F a g n o l a , F . : O n q u a n t u m stochastic
integration
with
respect
to
"free"
noises.
C e n t r o M a t e m a t i c o V . V o l t e r r a P r e p r i n t N . 3 7 M a y 1990. T o a p p e a r i n Q u a n t u m Probability and Applications V I . [6] H u d s o n , R . L . , P a r t h a s a r a t h y , K . R . : Stochastic completely
positive
of uniformly
continuous
A c t a A p p l . M a t h . 2, 3 5 3 - 3 9 8 ( 1 9 8 4 ) .
semigroups.
[7] J o u r n e , J . - L . : Structure
dilations
des cocycles
s u r l'espace
markoviens
Probab.
de Fock.
T h . R e l . F i e l d s 7 5 , 291-316 (1987). [8] K u m m e r e r , B . : S u r v e y on a theory cesses.
of non-commutative
stationary
I n : A c c a r d i , L . , v o n W a l d e n f e l s , W . (eds.)
Markov
pro-
Q u a n t u m Probability and
A p p l i c a t i o n s I I I . P r o c e e d i n g s , O b e r w o l f a c h 1987. ( L e c t . N o t e s M a t h . , v o l . 1303, p p . 1 5 4 - 1 8 2 ) . B e r l i n , H e i d e l b e r g , N e w Y o r k : S p r i n g e r 1988. [9] K u m m e r e r , B . , S p e i c h e r , R . : Stochastic S F B - p r e p r i n t 602, Heidelberg [10]
M o h a r i , A . : Q u a n t u m stochastic cients and dilation
of Feller's
Integration
on the
differential
minimal
equations
solution
to classical
Markov
with
processes.
unbounded
space.
I n : A z e m a , J . , M e y e r , P . A . , Y o r , M . (eds.)
bilites X X I V 1988/89.
(Lect.
0^.
coeffi-
Evans-Hudson
flows
P r e p r i n t , I . S . I . , D e l h i C e n t r e N o t e s (1990).
[12] P a r t h a s a r a t h y , K . R . , S i n h a , K . B . : M a r k o v c h a i n s as Evans-Hudson Fock
Algebra
P r e p r i n t , I . S . 1.(1991) .
[11] M o h a r i , A . , P a r t h a s a r a t h y , K . R . : On a class of generalized related
Cuntz
1990.
Notes M a t h . , vol.
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in
S e m i n a i r e de P r o b a -
1426, p p .
362-369).
Berlin,
H e i d e l b e r g , N e w Y o r k : S p r i n g e r 1990. [13] R e u t e r , G . E . H . : Denumerable semigroups [14]
Markov
on I. A c t a M a t h . 9 7 , 1-46.
S p e i c h e r , R . : A New
Example
processes
contraction
(1957).
of "Independence"
T h . R e l . Fields 8 4 , 141-159 (1990).
and the associated
and
"White
Noise".
Probab.
163
Note added in proof The main result presented here (see Theorem 2.4) has been extended by the second author to arbitrary semigroups and quantum stochastic differential equations with infinite "free" noises processes of the form dU(t) where pg(t) =
t,p (t) 0
=
= V(t)Lidp)(t)
/;(*),p?(<) =
l*j(t) and p)(t)
(E) for i,j
€ Z* are the "free"
analogue of Poisson processes. It has been shown, assuming that the coefficients L'j can be approximated by a sequence of bounded operators (Zrj)( ) converging strongly on a dense submanifold of the n
initial space such that the solution of the quantum stochastic differential equation dU {t) n
=
U (t)(L!)^dp)(t) n
is a contractive cocycle, and that LQ is the infinitesimal generator of a strongly continuous contraction semigroup, that: i) there exists a unique contractive cocycle U satisfying (E), ii) the cocycle U is an isometry (resp.a coisometry) if and only if the quantum dynamical semigroup associated with U (resp.with U) is identity preserving.
Quantum Probability and Related Topics Vol. VII (pp. 165-173) © 1992 World Scientific Publishing Company
165
A Q u a n t u m Characterization of Gaussianness
G e r h a r d C . Hegerfeldt Institut fur Theoretische P h y s i k Universitat Gottingen, Germany
Abstract Gaussianness extends the notion of normal random variable. A quantum analog of a theorem of K a c and Loeve is proved and a sharpened quantum version of a theorem of Cramer is given. Simple examples from quantum mechanics and quantum fields are presented.
1. I n t r o d u c t i o n a n d S t a t e m e n t o f t h e P r o b l e m In classical p r o b a b i l i t y theory a r a n d o m variable is b y definition n o r m a l i f the corresponding p r o b a b i l i t y measure has a G a u s s i a n density or, equivalently, i f its characteristic function is G a u s s i a n . W e w i l l c a l l these properties " G a u s s i a n n e s s " . T h e r e are various classical results r e l a t i n g t o Gaussianness of w h i c h we m e n t i o n some. (i) C r a m e r ' s T h e o r e m .
It states t h a t i f Xi a n d X
2
n o r m a l ( " G a u s s i a n " ) t h e n b o t h X\ a n d X
are independent a n d X\ +
X
2
are n o r m a l (Gaussian). T h i s result has
2
been carried over t o t h e q u a n t u m case by m e [1] a n d b y E m c h a n d myself [2]. (ii) M a r c i n k i e w i c z ' T h e o r e m .
It states t h a t i f t h e characteristic f u n c t i o n of X is the
e x p o n e n t i a l of a p o l y n o m i a l t h e n X is G a u s s i a n . A n analogous q u a n t u m result has been o b t a i n e d b y B a u m a n n a n d myself [3]. (iii) C o n n e c t i o n w i t h C e n t r a l L i m i t T h e o r e m .
A l s o i n this case there are a number of
q u a n t u m results, cf. e.g. [1], [4], [5], [6], [7], [8]. (iv) T h e o r e m s b y K a c a n d Loeve.
L e t X = (Xi,X )
have independent
2
components.
A s s u m e t h a t i t s r o t a t i o n b y a n angle ip also has independent components. K a c [9] proved that i f this is true for a l l
0,1,2,suffices.
T h e basic i d e a , of t h e proof of this last result is a m a z i n g l y simple. the characteristic functions factorize i n two coordinate systems.
B y assumption
T a k i n g logarithms one
obtains a n equality of t h e f o r m /i(z) + / where x' =
y.x + vy , y' =
2
W = Si(x') +
—ux + fty , fi =
d/dy t o b o t h sides gives 0 =
{&(*•)
g (y') 2
cos tp , v =
sin tp. A p p l y i n g d/dx a n d
166 for all x',y'. Thus, if J.l1I:f 0, one has g~ =
canst =
g~ .
This implies the statement. It should be noted that positive definiteness of the characteristic functions has not been used, and the proof goes through for factorizing C 2 -functions. In fact, the theorem holds for LP-functions, as shown recently by Carlen [12]. In this paper we carryover Loeve's result to a noncommutative setting. We first turn to the question of how to formulate an analogous quantum problem. To this end we note that classical moments E(Xj" Xl) or, more generally,
define a linear functional on the abelian (polynomial) algebra generated by abstract elements Xl and X 2 . Independence is mirrored by
We now simply make the algebra nonabelian and introduce for the general quantum case the following notations.
M
-
a countable set (our basic set)
P(M) - the polynomial algebra over C in noncommuting indeterminates from M PI (M) - all polyn.omials of degree 1 P(M)"- all linear functionals on P(M) Note that C can be regarded as a subset of P(M) and that PI (M) is the finite linear span of M. For T E P(M)" and ai E M, i 1, ... ,n
=
is defined as a formal power series, and similarly
provided T(l) :f O. For a normalized functional, i.e. T(I) P(M)" is defined through
1, the cumulant TC E
(1) For an abelian algebra this agrees with the classical definition of cumulants through moments. The definition is adapted to the "bosonic" case. For a "fermionic" definition see [13].
Definition . TEP(M)*,withT(I) = 1, is called Gaussian if, for al, ...,a n EM, TC(al ... an) = 0 for n > 2 If T(I) :f 0 it is called Gaussian if T fT(I) is Gaussian. This agrees with the definition in case of moments and random variables . All Gaussian functionals are known and can be easily constructed; cf. e.g. [1] .
167 Definition. Let M = M UMb be a disjoint n o n t r i v i a l decomposition of M, and let T £ P(M)*. T h e n M a n d Afj are ( m u t u a l l y ) independent w i t h respect t o T (or T - i n d e p e n d e n t ) i f for any m o n o m i a l p ( w i t h factors f r o m M) a
a
T(p)
=
T( ) Pa
(2)
T(p„)
where p„ is o b t a i n e d from p by o m i t t i n g a l l factors f r o m Mb, a n d s i m i l a r l y for p . b
E . g . , for p = a , 6, « t h e above c o n d i t i o n reads T ( a i &i a ) = T(ai
2
2
If M is some other basis of P\(M) a n d M = M U M some disjoint n o n t r i v i a l decomposition one m a y define independence of M a n d Mb i n a s i m i l a r way by replacing M by M i n the definition. a
b
a
N o w we consider b o t h M = M 0 A f a n d M = M„ 0 M . L e t M = {a , a ,...} , A f = {61,62,...} a n d s i m i l a r l y for Af„ a n d Aft. T h e element of M a n d A?(, are finite linear combinations of the f o r m a
t
k
a
a
2
t
a
a, = J2 « < + Eft* Q
*i
=
(3)
a
a
< + Z)
6
*
() 4
Definition. T h e decompositions ( M „ , A f t ) a n d ( A f , A f t ) are m u t u a l l y interlocking the right-inverses of the matrices (an),($kj) exist.
if
0
Interlocking w i l l replace the c o n d i t i o n \iv ^ 0 from above, i n s u r i n g t h a t the elements are t h o r o u g h l y m i x e d . If A f is finite t h e n i n t e r l o c k i n g implies t h a t the matrices are n x n and that M has 2n elements. To o b t a i n Gaussianness b y the methods of this paper we need p o s i t i v i t y w h i c h enters i n the following f o r m . L e t an i n v o l u t i o n a 1—• a* from M t o M be given a n d let it be extended i n an obvious way t o a n i n v o l u t i o n on P(M). In this case P(M) becomes a *-algebra. E l e m e n t s i n P(M) of the f o r m £ p*p,- , p; £ P(M), are called positive. Definition,
m 6 P(M)*
is positive i f it is nonnegative o n positive elements of
P(M).
N o w we c a n f o r m u l a t e our m a i n result. Theorem
1.
L e t A f be a countable set w i t h i n v o l u t i o n a n d P(M)
generates. L e t M be another basis of i ' i ( A f ) , a n d let M be disjoint decompositions (M ,
Mt)
a
=
the *-algebra it
Af„ U Mb a n d M
Remarks.
0
M
a
0 Mb
a
are i n t e r l o c k i n g . L e t m be a positive linear f u n c t i o n a l o n P(M).
and Mb as w e l l as M
=
w h i c h are i n v o l u t i o n - i n v a r i a n t a n d for w h i c h (M ,Mb) If b o t h
and M
a
a n d Mb are independent w i t h respect t o m , t h e n m is G a u s s i a n .
1. W i t h o u t loss of generality one can assume m ( l )
=
1 so t h a t m is a state.
A state c a n be characterized by the associated G e l f a n d - N a i m a r k - S e g a l representation of P(M).
F o r a G a u s s i a n state this representation is w e l l - k n o w n . It consists of the direct
s u m of a n a b e l i a n representation a n d a Fock representation of the canonical c o m m u t a t i o n relations [3]. 2. T h e p o s i t i v i t y c o n d i t i o n can be replaced b y the m u c h weaker c o n d i t i o n m(l) 0. T h i s requires a different m e t h o d of proof a n d w i l l be discussed elsewhere. N e i t h e r is the existence of a n i n v o l u t i o n needed.
168 3. A s i m i l a r result can be proved for the F e r m i case using a p p r o p r i a t e definitions of Fermi-Gaussianness a n d Fermi-independence. T h i s also w i l l be discussed elsewhere. 4. T h e c o u n t a b i l i t y of the basic set M can be replaced by c o n t i n u i t y c o n d i t i o n s on the functionals.
2. A n I n t e r m e d i a t e R e s u l t W e first prove i n an intermediate step a result w h i c h does not need the p o s i t i v i t y of the linear f u n c t i o n a l nor that P(M)
is a *-algebra.
P r o p o s i t i o n 1. L e t M be a countable set a n d P(M) Let M be another basis of P\{M),
a n d let M
decompositions for w h i c h (M ,M\,) a
T(l)
^
0. If b o t h M
and M
a
b
t h e n , for any c i , c , . . . G P\(M) 2
=
=
M
a
UM
b
be disjoint
are i n t e r l o c k i n g . L e t T G P(M)"
a
a
M\, a n d M
a
a n d (M ,M\,)
as well as M
the p o l y n o m i a l a l g e b r a i t generates.
M ii
and M
with
are independent w i t h respect to T
b
a n d any n,
T(e*™...
e "<") = x
exp P(x ...,x ) u
(5)
n
where P is a p o l y n o m i a l w h i c h is at most q u a d r a t i c i n each i , . T h i s is a l m o s t , - b u t not q u i t e , Gaussianness for w h i c h P w o u l d have t o be of at most second degree i n a l l variables simultaneously. T h e methods used here cannot exclude terms like x\...x . F o r the proof we need some lemmas. n
L e m m a 1. If M a n d M are T - i n d e p e n d e n t t h e n , w i t h i n an argument of T, elements of M„ m a y be c o m m u t e d w i t h those of Mi,. a
b
P r o o f . L e t pi a n d p
2
T(pi
be m o n o m i a l s w i t h factors from M.
F a c t o r i z i n g T(pi
a b p ) and 2
b a p ) according to the definition of independence one obtains the same expressions. 2
L e m m a 2. L e t M f r o m M. W i t h M„ power series
a n d M be T - i n d e p e n d e n t . L e t p\ a n d p be m o n o m i a l s w i t h factors = {5i,...} a n d M = {fti,...} one t h e n has i n the sense of formal
a
T(piexp
b
2
b
Xi&i
+
J2
=
Vityfa)
x, a,} exp
Tfaexp
»i
(6)
P r o o f . W e w r i t e the l.h.s. of (6) e x p l i c i t l y as a f o r m a l power series, o b t a i n i n g a s u m of terms w i t h T a p p l i e d t o m o n o m i a l s . B y L e m m a 1, elements f r o m M w i t h those f r o m M ,
a
m a y be c o m m u t e d
a n d regrouping the power series gives the statement.
b
L e m m a 3 . W i t h the assumptions a n d notations of L e m m a 2 one h a s , if T ( l ) T
fa
exp =
where p
la
r(p
x, 5,-} l 0
exp
exp {J2
y,- 6 j } p ) 2
b
1, (7)
2
« i } p „ ) T(p\
=
exp {J2
etc. are defined i n analogy to p„ a n d p i n (2). b
P r o o f . T h e statement follows from the definition of independence.
Vi ~ j}p~2b) b
169 L e m m a 4. If M
and M
a
T ^
=
are T - i n d e p e n d e n t a n d T ( l )
b
exp
1
x|">a. +
= r (n.
exp { £
=
1 then
yj^Jj
£
r (n
xj") g , } )
(8)
K
e
x
s^})
p
P r o o f . T h i s follows by i n d u c t i o n f r o m L e m m a 2 a n d 3 . L e m m a 5. W i t h the assumptions of P r o p o s i t i o n 1 a n d w i t h T ( l ) is the e x p o n e n t i a l of a power series i n { a f } . (bilinear) i n x<">
:=
=
1 , the r.h.s. of (8)
w h i c h , for fixed K , is at most q u a d r a t i c
....). T h e same is true w i t h 5; a n d bj replaced b y a, a n d bj i n
(X[ , K)
(8). Proof.
W e m a y assume T ( l )
and ( 4 ) . W e define
:=
=
1 . O n the l.h.s. of 8 we insert for a, a n d bj f r o m (3)
(£<">,...), i|W :=
...) a n d
£<*> := £ «« & + E * 0 vi"
:=
£
A . - «{«> +
(9)
4
£
(10)
where a l l the sums are finite. T h e l.h.s. of (8) can then be w r i t t e n as
T (llU
exp { £ £ < * > «* + E
•
^ J )
(11)
B y L e m m a 4 , applied to A / a n d M , this equals a
b
r (n. { E C?^})
r (n. e x {EV^**}) •
(12)
P
T h i s therefore equals the r.h.s. of (8). W e now pick a fixed K a n d w r i t e X ; for x ' " ' a n d so o n . W e m a y t a k e the l o g a r i t h m s of (8) a n d ( 1 2 ) a n d o b t a i n an i d e n t i t y of the f o r m 0
AW
+ Mr)
=
«i(€) +
(13)
where the other indeterminates have been suppressed i n the n o t a t i o n .
P e r f o r m i n g the
p a r t i a l derivatives w i t h respect to x,- a n d yj one gets zero for the l.h.s. a n d thus by (9) and
(10)
off
Ofi
^
dn ' k
Oi\k
B y t h e i n t e r l o c k i n g property, the right-inverses ( a ^ ) a n d (a,^ ) exist a n d hence 1
»(€) = - E /»« «tf
A-i
1
*
(is)
170 Since M a n d M are b o t h bases, the t r a n s i t i o n f r o m the i n d e t e r m i n a t e s ( x , y ) t o ( £ , 17) corresponds to a coordinate change, and thus £ a n d TJ c a n be regarded as independent indeterminates. Therefore b o t h sides of (15) must equal a constant, yrt say, so t h a t ei{€) die
a£
w
=
•
(16)
t
A s a consequence, 5i(£) is at most b i l i n e a r i n £• F o r 52(f) the argument is analogous. F r o m (9), (10) a n d (13) the statement of the l e m m a then follows. P r o o f o f P r o p o s i t i o n 1. W e m a y w r i t e c,- as a finite linear c o m b i n a t i o n c, =
£ e
pn a
+
c
£ k
p
b .
ki
k
A p p l i c a t i o n of L e m m a s 4 a n d 5 y i e l d the statement of the p r o p o s i t i o n .
3. A S h a r p e n e d Q u a n t u m V e r s i o n o f t h e M a r c i n k i e w i c z T h e o r e m a n d P r o o f of M a i n R e s u l t In [3] i t was proved t h a t if the n - t h order cumulants of a state m a l l v a n i s h for n > some Af (i.e.
if m (a ...a ) c
1
=
n
0 for n > N)
N,
then m is G a u s s i a n . T h i s is a q u a n t u m
version of the M a r c i n k i e w i c z T h e o r e m . It can be sharpened i n the following way. T h e o r e m 2 . L e t A f be a set w i t h i n v o l u t i o n a n d P(M) the * - a l g e b r a i t generates. L e t m be a state on P(M). A s s u m e t h a t for a n y a i , . . . , a „ £ M there is a p o l y n o m i a l Pa ...a {z ,...,z ) such that n
l
i
n
m{e™...
=
e*"°»)
exp P
0 1
... „( 0
Z l
,*.)
(17)
or, equivalently, such t h a t ...«*•)
m°{a\\..a»> for sufficiently large AT,-, N Gaussian. Proof.
t
0
(18)
> iV™(tt ..»o ) a n d a l l kj > 0, i l
=
B
W e first show t h a t (17) implies (18).
differentiating kj times w i t h respect t o Zj, j z =
=
l,...,n.
T h e n m is
T a k i n g the l o g a r i t h m of b o t h sides and i, a n d A^ times w i t h respect t o 2,- at
^
0 one o b t a i n s , u p to f a c t o r i a l factors, the l e f t - h a n d side of (18) (cf. [3] or [1]). Since
we are differentiating a p o l y n o m i a l this w i l l vanish for large A ^ . Conversely, if (18) holds a n d if the l o g a r i t h m of the l.h.s. of (17) were not a p o l y n o m i a l , t h e n differentiation at z
=
0 w o u l d lead to a sequence of nonzero terms as on the l.h.s. of (18) w i t h the value
of (J2 kj +
Ni) s t r i c t l y increasing. B u t this is a c o n t r a d i c t i o n t o (18).
W e now prove Gaussianness. B y l i n e a r i t y of m , the v a l i d i t y of (18) for a,- 6 M implies c
the same for linear combinations, i.e. a,- G P\(M).
T h e f o r m a l power series o n t h e l.h.s.
of (17) converges absolutely since the r.h.s. does. F r o m C o r o l l a r y 2.8 of [1] i t follows for a; G P\{M)
t h a t i n the associated G N S representation, x = n
n
e
z,
Mat)
ir , w i t h c y c l i c vector <j>, m
(19)
171 exists as a n operator o n D
=
T
i n the sense of strong convergence.
TT(P(M))4>
one c a n proceed as i n the proof of L e m m a 4.2 of [3]. - a j , one has
Px(M),a\
(4>, e' *
(e* *Mt%
M
Schwarz's i n e q u a l i t y i m p l i e s Re P (z)
3
does not exceed 2.
ai
e
ai
For n
=
"f" '/ ^). 1
2
z) a n d t h i s i n t u r n s implies t h a t the
< P (Re
ai
degree of P ajgflj, a
Now
For an h e r m i t i a n element a, 6
2 a n d 3 i n (19) a n d h e r m i t i a n elements
one shows i n a s i m i l a r way t h a t the p o l y n o m i a l s are at most q u a d r a t i c . T h e rest
of the proof is as t h a t i n [3]. F r o m C o r o l l a r y 4.1 of [3] one has for the G N S representation associated w i t h a G a u s sian state m , for a i a n d a e
This implies that namely m ( a i o ) .
JT(<JI)
2
G
Pi(M),
*(°l)
e
"t°a)|£
—
e
^fj)
">(aia )/2 f(ai)+*(
e
a n d i r ( a ) satisfy C C R ' s w i t h a possibly degenerate b i l i n e a r f o r m , 2
2
W e now prove T h e o r e m 1, our m a i n result. W e c a n assume, w i t h o u t loss of generality, m(l)
=
1 since, by p o s i t i v i t y a n d Schwarz's inequality, m ( l )
=
0 would imply m
=
0.
T h e n , P r o p o s i t i o n 1, w i t h T replaced by m , shows t h a t c o n d i t i o n (17) of T h e o r e m 2 is satisfied. H e n c e m is G a u s s i a n . Remark.
F r o m Gaussianness it follows t h r o u g h (20), t h a t 7r
m
gives a representation
of the C C R ' s p l u s a n abelian p a r t . T h i s is seen by decomposing P\(M) such t h a t the b i l i n e a r positive-semidefinite f o r m (a, 6)
:=
=
MQ • © A f j ' 1
m(a*6) vanishes on
and
is positive-definite o n A f j ' . T h e representation of the C C R ' s i s , b y Gaussianness, a Fock 1
representation.
4. S i m p l e E x a m p l e s Lie algebras. G i v e n a r e a l L i e algebra L we c a n pick a basis a n d take t h i s as the basic set A f . T h e n the t o m p l e x i f i e d L is P\(M).
In P(M)
we c a l l l j , the i d e a l generated by
the L i e - a l g e b r a r e l a t i o n s . P ( A f ) / I t is the u n i v e r s a l enveloping algebra of L.
A linear
f u n c t i o n a l o n the u n i v e r s a l enveloping algebra of L c a n be e x t e n d e d t o a linear f u n c t i o n a l o n P(M)
v a n i s h i n g o n Ii,
a n d conversely a linear f u n c t i o n a l o n P{M)
on Ix, induces a linear f u n c t i o n a l o n the u n i v e r s a l enveloping algebra. a * - a l g e b r a i f one defines a*
:=
—a for a G L.
w h i c h vanishes P(M)
becomes
A c y c l i c representation 7r of L by
s k e w h e r m i t i a n operators i n a H i l b e r t space, w i t h c y c l i c vector <j>, defines a state
on
P ( A f ) through m^ai
.... a„)
:=
W e now assume t w o decompositions
(|,
(21)
7r(ai) ... ir(a )<j>) . n
of L i n t o two disjoint subs paces V , Vt a n d V , V i a
a
respectively, L
=
V + V , a
b
L
=
V. + V i
(22)
w h i c h are i n t e r l o c k i n g i n a n obvious extension of the definition i n Section 1. If one has f a c t o r i z a t i o n s analogous to (2), (<j>, • • • x(aA • • • n(bj) --4>) =
(••• ^r(a,) • • • <j>) (<j>, • • x(bj) •••<
172 w i t h a, € V , bj e ' V i , a n d s i m i l a r l y for a ; G V , bj G V , t h e n a
a
is G a u s s i a n a n d it is
b
given b y (20). B y t h e r e m a r k at t h e e n d of Section 3, L has t h e n t o b e h o m o m o r p h i c to a C C R algebra plus a n a b e l i a n algebra. Quantum
W e consider t w o particles w i t h p o s i t i o n a n d m o m e n t u m oper-
mechanics.
ators Qi a n d Pi, i satisfy C C R ' s .
=
1, 2. F o r s i m p l i c i t y we take t h e i r mass t o b e 1. T h e Qi a n d Pi
W e assume existence of t h e corresponding W e y l o p e r a t o r s a n d a cyclic
vector. W e i n t r o d u c e Qc QR
:=
\(Qi + Q )
Pc •= P1 + P2
:=
Q1-Q2
PR :=\(Pi-P2)
2
(23) (24)
•
These are t h e operators for t h e center-of-mass a n d for t h e r e l a t i v e m o t i o n , a n d they also satisfy C C R ' s . W e assume p h y s i c a l l y t h a t t h e particles are i n d e p e n d e n t l y prepared w h i c h implies f a c t o r i z a t i o n a n d independence of {Qi,Pi}
a n d {Q ,P } 2
i n t h e sense of
2
Section 1. O n e m a y now restrict t h e representation t o t h e W e y l operators o f the centerof-mass s y s t e m a n d t o t h e subspace generated by t h e i r a c t i o n o n t h e c y c l i c vector. If this restricted representation is extremal t h e n f a c t o r i z a t i o n also for t h e center-of-mass and t h e relative m o t i o n system follows f r o m t h e c o m m u t a t i v i t y of t h e corresponding v o n N e u m a n n algebras.
Hence independence of
a n d {Q ,P } 2
2
together w i t h
e x t r e m a l i t y of {Q , P } i m p l y Gaussianness a n d F o c k representation. W e r e m a r k t h a t for c
c
this s i m p l e example one m a y also w o r k w i t h t h e t h e o r e m of Loeve d i r e c t l y , w i t h o u t using T h e o r e m 1. Quantum
A g a i n we consider a s i m p l e e x a m p l e .
fields.
A s s u m e we have t w o fields
d i s t r i b u t i o n s so t h a t t h e operators of interest a r e t h e smeared fields
a n d j i n a c o m p l e x test f u n c t i o n space V .
A s s u m e t h a t t h e fields ipi a n d ip are 2
independent, i n t h e obvious sense of f a c t o r i z a t i o n as above. W e consider " r o t a t e d " fields [H,v
If
±
0)
and
ifi
2
Mf)
=
v- v i ( / ) + v
92(9)
=
-v
2
tpx(g) + fi
are also independent t h e n a l l fields are G a u s s i a n .
T h e proof of t h i s is f a i r l y obvious. T h e role of Pi(M) is t a k e n b y V © V a n d P(M)
2
adapted to
P i c k i n g a_basis S
i n V, M
¥>(/>?)* =
a
(V, V),
•= V i ( / ) +
fU,9) a n d define i n v o l u t i o n o n P(M)
M
=:
is t h e tensor algebra generated b y V ffi V. W e define
=
a
a n d M correspond t o { ( / i / i , vfi)} b
(
corresponds t o ( 5 , 0 ) a n d M
a n d {{-vfi,
b
to (0,5).
Then
/i/,)}„ respectively. I n case V does
not coincide w i t h t h e finite l i n e a r span of 5 one obtains i n t h i s w a y G a u s s i a n for any subspace of V w h i c h is t h e finite linear span of a countable set of elements. T h i s implies Gaussianness for a l l of V.
173
References [1] G . C . Hegerfeldt, N o n c o m m u t a t i v e Analogs of P r o b a b i l i s t i c Notions a n d Results. J . F u n c t . A n a l y s i s 64 (1985), 436-456 [2] G . G . E m c h a n d G . C . Hegerfeldt, N e w C l a s s i c a l Properties of Q u a n t u m
Coherent
States. J . M a t h . P h y s . 27 (1986), 2731-2737 [3] K . B a u m a n n a n d G . C . Hegerfeldt, A N o n c o m m u t a t i v e M a r c i n k i e w i c z T h e o r e m . P u b l . Res. Inst. M a t h . S c i . ( K y o t o ) 21 (1985), 191-204 [4] R . L H u d s o n a n d C D . C u s h e n , A Q u a n t u m M e c h a n i c a l C e n t r a l L i m i t T h e o r e m , J . A p p l . P r o b . 8, 454 -469 (1971) [5] N . G i r i a n d W . v o n Waldenfels, A n A l g e b r a i c V e r s i o n of the C e n t r a l L i m i t T h e o r e m , Z . WahrscheinUchkeitstheorie verw. Gebiete 42 (1978), 129-134 [6] W . v o n Waldenfels, A n A l g e b r a i c L i m i t T h e o r e m i n the A n t i - C o m m u t i n g Case. Z. WahrscheinUchkeitstheorie v e r w . Gebiete 42 (1978) 134-140 [7] M . S c h i i r m a n n u n d W . v o n Waldenfels, A C e n t r a l L i m i t T h e o r e m o n the Free L i e G r o u p , i n : L . A c c a r d i a n d W . v o n Waldenfels ( E d s . ) , Applications
Quantum
Probability
and
III, Proceedings, Oberwolfach 1987. L e c t u r e Notes i n M a t h e m a t i c s 1303,
Springer, B e r l i n 1988 [8] G . C . Hegerfeldt,
P r i m e F i e l d Decompositions
a n d Infinitely D i v i s i b l e States on
B o r c h e r s ' Tensor A l g e b r a . C o m m u n . M a t h . P h y s . 45 (1975), 137-152 [9] M . K a c , O n a C h a r a c t e r i z a t i o n of the N o r m a l D i s t r i b u t i o n , A m . J . M a t h . 61 (1939), 726 -728 [10] J . M . L i n d s a y , Ortho-Independent
States of the C C R . A l g e b r a , P u b l . Res. Inst.
M a t h . S c i . ( K y o t o ) , 20 (1984), 585 - 593 [11] M . Loeve i n : M . L e v y , Processus Stochastiques et M o u v e m e n t B r o w n i e n . P a r i s 1948, p. 337-338 [12] E . A . C a r l e n , S u p e r a d d i t i v i t y of F i s h e r ' s I n f o r m a t i o n a n d L o g a r i t h m i c Sobolev I n equalities, J . F u n c t . A n a l y s i s , i n press [13] R . F . Streater a n d D . M a t h o n , Infinitely D i v i s i b l e Representations of C l i f f o r d A l g e bras, Z. WahrscheinUchkeitstheorie verw. Gebiete 20 (1971), 308 [14] D . V o i c u l e s c u , S y m m e t r i e s of some reduced free p r o d u c t C*-algebras. I n : H . A r a k i et a l . ( E d s . ) , O p e r a t o r Algebras a n d t h e i r C o n n e c t i o n w i t h T o p o l o g y a n d E r g o d i c T h e o r y . Proceedings B u s t e n i , R o m a n i a , 1983. L e c t u r e Notes i n M a t h e m a t i c s 1132. S p r i n g e r , B e r l i n 1985.
Quantum Probability and Related Topics Vol. VII (pp. 175-202) © 1992 World Scientific Publishing Company TIME-ORDERED EXPONENTIALS IN Q U A N T U M
175 STOCHASTIC
CALCULUS* A.S.Holevo Steklov M a t h e m a t i c a l Institute V a v i l o v a 42, Moscow, U S S R
§1.
Introduction.
1. In q u a n t u m stochastic c a l c u l u s an important role is p l a y e d b y the solutions U = U(t) o f l i n e a r q u a n t u m stochastic differential equations (q.s.d.e.)
d U = (L(,dA + L ] d A + L ^ d A * + L j d t ) U ; t > 0 ;
where
L
= L (t)
a
U(0) = I ,
are o p e r a t o r - v a l u e d functions i n i n i t i a l H i l b e r t space J t ; d A , d A ,
a
0
+
dA
(1.1)
2
are basic stochastic differentials i n the F o c k space r = T ( L ( R ) ) +
[1].
I n this paper w e define and study time-ordered exponentials U(t) = e^p
} (M dA + M,dA + M dA 0
2
+
+ M dx),
(1.2)
3
o where
M
a
= M ( / r ) are functions w i t h values i n L ( J t ) , w h i c h f o r m an inportant class a
0
of solutions o f q.s.d.e. (1.1). T h e
representation
o f solutions o f (1.1) i n the f o r m (1.2) is
p h y s i c a l l y r e l e v a n t a n d i m p l i e s a n u m b e r o f interesting f o r m u l a s . T h e u n i t a r y t i m e ordered exponentials w i t h M
0
= 0 were introduced i n a heuristic w a y i n [2]. I n [3] these
were s h o w n " m e a n - s q u a r e " convergent i n a n o n - F o c k case. W e develop a general method w h i c h p e r m i t s , b y a r e d u c t i o n to the case conditions o n K
0
® T
e
M (t) a
, where
(1.1) w h e r e
L
a
r
c
dim K
0
= 1, to e s t a b l i s h under suitable
the strong c o n v e r g e n c e of the t i m e - o r d e r e d exponentials (1.2) o n is the span o f exponential vectors i n T , to the s o l u t i o n o f q.s.d.e.
are related to
M
a
This work has been supported by the Deutsche
b y the f o r m u l a s (1.7) b e l o w . F o r s i m p l i c i t y we Forschungsgemeinschaft.
176 restrict to the case o f o n e - d i m e n s i o n a l
q u a n t u m n o i s e . T h e g e n e r a l i z a t i o n to the
m u l t i d i m e n s i o n a l case presents n o s p e c i a l analytic d i f f i c u l t i e s , a l t h o u g h it is e v e n more interesting f r o m an a l g e b r a i c p o i n t o f v i e w due to the t e n s o r i a l nature o f the process A(t)
(cf.
[4]).
I n c l a s s i c a l stochastic c a l c u l u s the c o r r e s p o n d i n g n o t i o n is that o f m u l t i p l i c a t i v e stochastic i n t e g r a l , w h i c h is u s u a l l y defined as a s o l u t i o n o f the c o r r e s p o n d i n g linear s.d.e. [5] , [6]. B y u s i n g the standard e m b e d d i n g o f the W i e n e r (or P o i s s o n ) process into the F o c k space [1] it is possible to substantiate the direct d e f i n i t i o n o f the m u l t i p l i c a t i v e stochastic i n t e g r a l against W i e n e r (resp. P o i s s o n ) integrand as a c o r r e s p o n d i n g t i m e ordered e x p o n e n t i a l . 2. A f u n c t i o n M ( t ) , t e [0,t]
with values i n L ( H ) is c a l l e d s i m p l e i f there exists a 0
division 0 = to < 1, < ... < t
o f the interval [0,T]
= T
0
M (t), t e
[0,T]; ( a = 0,1,2,3) w i t h v a l u e s i n L ( H ) 0
is strongly
is a l i m i t o f sequence o f s i m p l e functions i n the n o r m
II M ( - ) 11^ =
a
= ess sup II M ( t ) II, t 6 [0,T], function
M,(t) , M (t)
(and
thus they
functions
M (t) 3
i n the
T
o
a
is a strongly admissible q u a d r u p l e , then the q.s.d.e. (1.1) has
unique adapted solution defined o n H ® T M (t)
approximating
a
[0,T]. W e put
&Aj = A ( t p - A ( t ) , H
e
(see T h e o r e m 2).
is a strongly a d m i s s i b l e q u a d r u p l e . L e t us f i x a quadruple of
a
M (t)
+
simple
2
o
+
are l i m i t s o f
{II M ( t ) II dt , j II M ( t ) II dt ). B y m o d i f y i n g methods o f [1], [8]
one can p r o v e that i f L ( t )
division of
are square integrable functions and
2
T
corresponding norms
s i m p l e functions
for t E [tj_,, tj). A q u a d r u p l e o f Lebesgue
H
admissible i f M ( f )
F r o m n o w on
(1.3)
such that M ( t ) = M ( t )
measurable functions
i s an i n t e g r a b l e
N
M ( T ) , and let (1.3) a
Atj = t - t j _ , , A A j = A ( t j ) - A ( t ) , }
H
be the
corresponding
A A j = A(tj) -
A(t _ ), j
1
AMj = M o d ^ A A j + M ^ t j ^ A A j + M (tj_,)AAj + M ^ t j ^ A t j . 2
W e set f o r m a l l y V j = exp A M j = X
(AMj)",
(1.4)
177
n
= v ,.,v,.
V j
(i.5)
N
IN
T h e o r e m 1. T h e operator
f l V j where V j are understood as the series expansions <
N
(1.4) is defined as a s t r o n g l y convergent series on of q u a d r u p l e s maxAtj^o, 1 <j
M ( t ) approximating
M (t)
a
n
H
® Te . I f there is a sequence
0
s u c h that f o r c o r r e s p o n d i n g
a
divisions
then N
sup
II [ n
V o e Ho II
v
0
Vj-U(T)] y
U(f)
0
= a(M ), Lj = M b(M ), L
0
®
I! —> 0 ;
V
e
V
T ,
(1.6)
e
II < 1
u n i f o r m l y in T < T , where
L
0
'
0
1
0
2
is the s o l u t i o n o f the q.s.d.e. (1.1) w i t h
= b(M )M 0
,L = M
2
3
3
+ M c(M )M , 1
0
(1.7)
2
where a(z)=e*-l , b ( z ) = ^ - ,
C
Z
(z) = ^ = t i 2
(1.8)
z
are entire a n a l y t i c functions o f Z In this sense the s o l u t i o n o f the q.s.d.e. (1.1) w i t h L
g i v e n b y (1.7) is described b y
a
the time-ordered e x p o n e n t i a l (1.2). R e v e r s i n g relations (1.7) gives
M
0
= a'(L ) , M , = L , b ' ( L ) , M 0
where
0
a'(z) = l n ( l + z) , b'(z) =
= b'(L )L
2
0
l n ( 1
+
z )
2
, M
, c'(z)
3
= L
=
3
l n ( 1
+ L,c'(L )L , 0
+
Z
are w e l l d e f i n e d e.g. for
z
) ~ z
z
(1.9)
2
.The relations (1.9)
2
IIL II < 1. 0
In p h y s i c a l a p p l i c a t i o n s o f quantum stochastic c a l c u l u s one is interested i n the case of u n i t a r y U ( t ) . C o r o l l a r y 1.
Let
M
0
= hp , M
3
= - i H , 0
where
(p =
0
= H (t) 0
are
178 hermitean, M , = iT(t)* , M
= iT(t) , then the unitary operator v a l u e d process
2
U ( t ) = exp i | ((pdA + T * d A + T d A - H d T ) is the solution o f the e q u a t i o n +
0
0
d U = [ ( W - I)dA - L * W d A + L d A
+
- (iH + \ L*L)dt ] U ,
with W = e'
L = ib(i
0
+ T*d(
where d(z) = (z - s i n z ) / z . 2
T h e f o l l o w i n g c o r o l l a r y describes s i t u a t i o n i n w h i c h t i m e - o r d e r e d and ordinary exponentials c o i n c i d e . C o r o l l a r y 2. L e t M ( t ) , Mp(s) be c o m m u t i n g for a l l a , B; t, s. T h e n a
U(t) = exp j ( M d A + M , d A + M d A 0
2
+
+ M dr), s
o where the exponential is defined as the series e x p a n s i o n strongly convergent o n H , is the s o l u t i o n o f the q.s.d.e. (1.1) w i t h L
0
® T
c
g i v e n b y (1.7).
a
3. C o n s i d e r the ordinary differential equation i n L ( J t ) 0
dX = K X L * + M X + X N * ; t>0; dt where
X(0) = X ,
(1.10)
K , L , M , N are n o r m - c o n t i n u o u s functions w i t h values i n L ( H ) . O n e can give a 0
n u m b e r o f stochastic representations for the s o l u t i o n o f equation (1.10) i n terms of timeordered exponentials. D e n o t i n g b y
the v a c u u m c o n d i t i o n a l e x p e c t a t i o n f r o m
L(K
0
[ exp } ( K d A + M d t ) ] X [ e~xp f ( L d A + M d t ) ] * . o o t > 0 , be the standard W i e n e r process. B y the d u a l i t y t r a n s f o r m [1], [7]
X(t) = E W(t) ,
L (W) 2
0
0
c
Let
E
to L ( K ) and u s i n g the theorem we have
® r)
0
is i s o m e t r i c a l l y embedded onto T so that W ( t )
maps into A ( t ) + A ( t ) +
and the
mean-square c o n v e r g e n c e corresponds to the strong c o n v e r g e n c e o n the v a c u u m vector \|/(0). W e then deduce f r o m the theorem that the m u l t i p l i c a t i v e stochastic integral U(t) = exp} [ L d W + ( N - y L ) d x ] 2
o exists as the mean-square l i m i t o f the time-ordered products
<— N
IT V j , w h e r e
179
Vj
= e x p [ L ( T j . , ) A W j + ( N ( T j _ , ) - 2 L ( T j _ , ) ) A t j ) i n the sense that 2
r
sup VoeHo II Vo
11 2
•>
N
E l l [ [1
V j - U ( t ) ] \j/ " 0
- » ° .
O-Ai)
J
1
as m a x Atj - > 0 and satisfies the s.d.e. dU = ( L d W + Ndt)U. W e then get the f o l l o w i n g stochastic representation X ( t ) = E [ exp j ( K d W + ( M - \ K ) d t ) ] X 2
o [ exp f ( L d W + ( N - \ L ) d x ) ] * . 2
0
In the same w a y , let 2
n(t)
be the P o i s s o n process o f the i n t e n s i t y
L >
max t e [0,T]
2
m a x { H L ( t ) l l , IIK(t)ll ). B y u s i n g the e m b e d d i n g o f the P o i s s o n p r o c e s s into the F o c k space [1], [7] we see that U ( t ) = exp} [ ln(I + X T
1 / 2
L)dn + (N - X
, / 2
L)dT ]
o is d e f i n e d as the m e a n - s q u a r e l i m i t i n the sense (1.11) o f t i m e - o r d e r e d products and satisfies the s.d.e. dU = [ >/
1 / 2
L d n + ( N - >. L)dt ] U . 1/2
W e then get the stochastic representation
X(t) = E (exp} [In (1 + r
1/2
K)dn + ( M - A. K)dr]) 1/2
o X {txp {[In (1 + X~ L)dU m
+ ( N - X L)dt]) m
*.
o The representations are e a s i l y g e n e r a l i z e d to the case where the right-hand side o f (1.10) contains several terms o f the s i m i l a r f o r m (cf.
[4]).
180 §2. E x p o n e n t i a l s of
basic
processes.
1. W e consider the F o c k space r = T ( L ( R ) ) 2
= [ f , f , ( x ) , ... , ffa,
V
where f
... , T ) , ... ] ,
0
e C ,
0
f (x,,... , x ) n
Tj , . . . , x € R n
n
is a s y m m e t r i c square integrable f u n c t i o n o f the variables
n
such that
+
II y II = < v I v > = L 2
^
"=°
f ... f I f „ ( x , , . . . , x ) I d i , ... d x < n
'
0
x„), are zero, is denoted
V
f
n
\|/ for w h i c h a l l components, e x c e p t for f ( x , , . . . , n
. F o r any h e L ( R ) 2
n
]„ ( x „ ... , x ) = j h ( x n
.
0
T h e n-particle subspace, consisting o f
[ A(h)
consisting o f elements
+
n + l
) f
n + 1
the a n n i h i l a t i o n - c r e a t i o n operators
+
( x „ ... , x „ V J ) d x n
n + 1
,
(2.1)
0 n
[ A (h) +
(where
y ]„ ( 1 , ,
... , T „ ) =
I
h ( T j ) f _,
( x „ ... , T j , ... , T ) ,
n
(2.2)
n
X j means that the v a r i a b l e
X : is omitted) and f o r
h e
L°° ( R ) +
the gauge
operator n
[ A(h)
V
]„ ( x . . . , x ) = [ I l f
n
h(xp ] f
are defined on the invariant d o m a i n
( x „ ..., x )
(2.3)
n
consisting o f \|/ satisfying
f - f l f n ^ . - . - g I d t , ... d x „ < ~
IS
2
0
0
for a l l c E R A (t) = A ( l +
n
+
[ 0
. T h e c o r r e s p o n d i n g basic processes are d e f i n e d as
+
l ]
),
A(t) = A ( l
[ 0 l]
) , where
l
[
0
l
]
A(t) = A ( l
is the i n d i c a t o r o f the i n t e r v a l
[ 0 t
]) >
[0,T]
(see [1], [2], [7] for more detail). 2
For f € L ( R ) +
V
(f)
the exponential vector \y(f) e T
m
= [1, f(x), ... , f ( x , ) - . . . - f ( x ) , n
...]
is defined as (2.4)
181 D e n o t i n g b y (f,g) = J f ( x ) g(x) dx the inner product i n L ( R ) 2
+
w e have
o < V (g) I V ( 0 > The f u n c t i o n
= exp (g,f) .
(2.5)
z —> \|f (f + z g ) f r o m C to F is entire analytic. In particular the m a p f
-> V|/ (f) f r o m L ( R ) to T is strongly differentiable i n any d i r e c t i o n ; denoting b y V +
h
the derivative i n the d i r e c t i o n h we have A (h) y ( f )
A(h)v(f)
V
Denote b y r h
... V
h
n
e
(2.6)
V
V
+
V
= (h,f) ( f ) ,
A (h) (f) = V
=
h V
(f) ,
(2.7)
(f).
h f V
(2.8)
the span o f exponential vectors and T ' the span o f vectors o f the form e
f o r a l l n ; h , , . . . , h „ ; f e L ( R ) . F r o m (2.6) - (2.8) it f o l l o w s that F '
v(f)
2
+
e
is a n invariant d o m a i n o f A ( h ) , A ( h ) , A ( h ) . +
2. W e are g o i n g to define a n d study the exponentials A ( m ) ] i n r . T h e k e y fact is that r +
2
exp [ A ( m ) + A ( m , ) + 0
is invariant under these exponentials.
c
L e m m a 1. L e t m € L ° ° ( R ) ; i n , , m , e L ( R ) , then the exponential e x p [ A ( m ) 0
+
2
2
+
0
+ A ( n . j ) + A ( m ) ] is defined as a p o w e r series strongly convergent o n r +
2
e
and
exp [ A ( m ) + A ( m , ) + A ( m ) ] \|/(f) = +
0
=
2
V|/(f + a ( m ) f + b ( m ) m ) • e x p ( m , , b ( m ) f + c ( m ) m ) . 0
0
2
0
0
2
(2.9)
Proof. T h e relation P ( z ) \if(f) = \)/(f + a ( z m ) f + b ( z m ) z m ) • 0
0
2
• exp z ( m , , b ( z m ) f + c(zm ) z m ) ; z € C , 0
defines operators
0
(2.10)
2
P ( z ) w i t h the invariant d o m a i n r
e
, satisfying
182 P(z, + z )
= P(z,)P(z ).
2
(2.11)
2
B y l i n e a r i t y and strong c o n t i n u i t y
P(z)
is extended to
T' e
. U n s i n g (2.6) - (2.8)
one
chesks that d£<0) y = [ A ( m ) + A(m,) + A ( m ) ] y dz +
0
(2.12)
2
for y = y ( f ) , and then again u s i n g (2.6) - (2.8) this r e l a t i o n is e x t e n d e d to
T '. e
F r o m (2.11), (2.12) d S .
0
A s seen f r o m (2.10) the f u n c t i o n
n
2
z —> P ( z ) y ( f )
from
C to F is entire a n a l y t i c and
therefore P ( l ) y(f)
= X
-h
(0) y ( f )
S
= X
-JT [ ( m ) + A(m,) + A ( m ) ] A
+
0
2
y(f)
n
where the series is strongly convergent. T h i s proves (2.9). F r o m this l e m m a and (2.5) w e get useful formulas
< y ( g ) I exp [ A ( m ) + A ( m , ) + A ( m ) ] y ( f ) > +
0
2
=
= < y ( g ) I y ( f ) > • exp [(g, a ( m ) f) + (g, b ( m ) m ) + 0
+
0
2
( m , , b ( m ) f) + ( m , , c ( m ) m ) ] ; 0
0
(2.13)
2
II exp [ A ( m ) + A ( m , ) + A ( m ) ] y ( f ) II = +
0
= exp [ I e ° f + b ( m ) m m
where
0
2
2
2
I + 2ke(m!, b(m ) f + c(m ) m )] , 2
0
l-l denotes the n o r m i n L ( R ) +
0
(2.14)
2
as distinct f r o m the n o r m ll-ll i n r .
T h e f o l l o w i n g result is the m a i n t o o l for the r e d u c t i o n o f the case o f arbitrary i n i t i a l space H°
to the case d i m H
0
= 1 under consideration.
L e m m a 3. L e t X ( h ) = A ( h ) , X ^ h ) = A ( h ) , X ( h ) = A ( h ) , X ( h ) = j°h(T)dt. 0
2
+
3
o T h e n for f, g e
L (R ) 2
+
183 I < \|f(g) I X ( h , ) - . . . - X ( h ) i i
where
i p
p
V
( f ) > I < < \|/(lgl) X ( l h l ) - . . , - X ( l h t ) v ( l f l ) > i i
i = 0, 1, 2, 3 f o r s = 1
1
i p
p
,
(2.15)
p.
s
P r o o f . B y the d e f i i n i t i o n (2.4) o f the e x p o n e n t i a l vector
I < y ( g ) I Jt^ ( h , ) - . . . - X , ( h ) y ( f ) > I < £
l < g® I X - ( h , ) - . . . * , (h ) f n
p
where
g ® " i s the v e c t o r i n r
(2.8) e a c h o f the operators and the i n n e r p r o d u c t
P
w i t h the component
n
X (h ) i s
transforms r
s
g(T,)-...-g(x ). A c c o r d i n g to (2.6) n
into either o f T _i , r m
m
< g® 1 X (h,)-...-X (h ) f ® > n
> I , (2.16)
m
p
n,m=0
P
i ]
i
m
p
m
or r
m + 1
_
is a n integral expression,
linearly dependent o n g ® , h , , . . . , h , f ® . A p p l y i n g repeatedly to this e x p r e s s i o n the n
m
p
inequality I J F I < I J F I w e get I < g®" I X ( h , ) - . . . - X ( h ) f ® > I < < Igl®" I X ^ C l h i O - . - . - X i (Ihpl) l f l ® > . i i
i p
m
p
m
Substituting
this into (2.16) and s u m m i n g gives (2.15). 3. W e n o w pass to operators i n the space
H =H
0
® T , where H
i n i t i a l H i l b e r t space. T h e n o r m a n d the i n n e r p r o d u c t i n H
0
is arbitrary
w i l l b e denoted a g a i n 11-11
and <•!•> , a n d the situation o f n . l corresponds to d i m J t = 1 . L e m m a 3 i m p l i e s 0
P r o p o s i t i o n 1. L e t rtij
-
be nonnegative numbers such that the series
t
A =X X m , : X : (lhV°l)-...-X= (lh„ l) converges strongly o n T. a n d let M j . n i =0,l,2,3 " '" (s=l n) n)
1
s
be operators i n H
A =
n
1
n
s u c h that II
Q
X X n
1
- II < i . . . i m
T
1
n
. then the series
M , , X ( h , ) - . . , X i (h (
i =0,1,2,3
1-n
s
(s=l,...,n)
n )
1
n
n)
)
(2.17)
n
converges s t r o n g l y o n H ® r a n d 0
c
II A (\|/ ® V|/(f)) II < II A v ( l f l ) 11-11 V 0
M o r e o v e r f o r t w o s u c h operators
A ,B
0
II •
(
2
-
1
8
>
184 I < B (\y ® V|/(g)) I A (\\i ® v|/(f)) > I < < B \|/(lgl) I A \)/(lfl) > • l l y H , Q
0
0
and the r i g h t - h a n d sides i n (2.18), (2.19) are m o n o t o n e l y coefficients
m=
j
fact that A ( h ) , A ( h )
II A(\|/ ® \)/(f)) II 0
are f o r m a l y adjoint, and A ( h )
2
as i n n e r p r o d u c t a n d use the
is f o r m a l l y s e l f - a d j o i n t o n r '. e
I n what f o l l o w s we u s u a l l y start f r o m an operator expansion
i n c r e a s i n g f u n c t i o n s o f the
. T h e relation (2.19) d i r e c t l y f o l l o w s f r o m l e m m a 3 and elementary
estimates. In p r o o f o f (2.18) we w r i t e +
(2.19)
2
A
( 2 . 1 7 ) , a n d c o n s t r u c t an e s t i m a t i n g o p e r a t o r
convergent expansion in
T. Clearly
A
in
Jt
A
g i v e n b y formal
given by
a strongly
is b y no means u n i q u e , a l t h o u g h this is not
reflected by the notation. C o n s i d e r e.g. the f o r m a l p o w e r series A
m e x p [ M A ( t ) + M,A(t) + M A ( t ) + M3t] 0
=
X
2
+
s
[ M A ( t ) + M , A ( t ) + M A ( t ) + M t ]", 0
ns=0
+
2
3
'
where by L e m m a 1 the p o w e r series s t r o n g l y c o n v e r g e s o n p o w e r series for
A
strongly converges on K
® r
0
e
r
. B y P r o p o s i t i o n 1 the
e
and
II exp [ M A ( t ) + M , A ( t ) + M A ( t ) + M t ] Vj/ ® \)/(f) II < 0
+
2
3
2
0
< II e x p [ m A ( t ) + m , A ( t ) + m A ( t ) + m t ] \)/(lfl) ll -ll y 0
+
2
2
3
0
II . 2
(2.20)
F r o m (2.14) the right side is
I
9
II exp [ m A ( t ) + m , A ( t ) + m A + ( t ) + m t ] \|/(lfl) II = exp f F , ( r ) d T , 0
2
3
o
(2.21)
where F i C O = [ (1 + a ( m ) ) lf(T)l + b ( m ) m 0
+ 2 [ b ( m ) m , lf(r)l + m ()
0
3
2
]
2
+
+ m,m c(m ) ] 2
(2.22)
0
is a l o c a l l y integrable f u n c t i o n depending on f and m
a
.
185 §3. M o d i f i e d s t o c h a s t i c
i n t e g r a l a n d solutions of l i n e a r q.s.d.e.
1. E x i s t e n c e and u n i q u e n e s s o f the s o l u t i o n o f q.s.d.e. (1.1) u n d e r s o m e g e n e r a l c o n d i t i o n s w a s e s t a b l i s h e d i n [1]. H e r e w e g i v e a m o d i f i c a t i o n o f this c o n s t r u c t i o n , i n c l u d i n g the d e f i n i t i o n o f stochastic i n t e g r a l , so as to get operators defined o n the w h o l e Jt
0
® T
e
and not a subspace o f it as i n [1]. T h i s m o d i f i c a t i o n seems natural w h e n d e a l i n g
w i t h t i m e - o r d e r e d e x p o n e n t i a l , although it applies to a n a r r o w e r c l a s s o f integrands as compared to [1]. A q u a d r u p l e o f o p e r a t o r - v a l u e d functions on H
® r
0
M (t) V
o
a
defined
(for s i m p l i c i t y o f n o t a t i o n s w e o m i t the s e q u e n c e i n d e x ) s u c h that f o r a l l
0
e
M ( t ) , t e [0,T] ( a = 0,1,2,3)
is c a l l e d a d m i s s i b l e i f there exists a sequence o f s i m p l e adapted functions
c
tt
0
, # e
T
c
ess sup II [ M ( t ) - M ( t ) ] V|/ ® v|/(f) II -> 0 0
0
1 2
0
u
T
V
o
(3.1)
2
V
_
f II [ M ( t ) - M ( t ) ] y 3
3
0
® \|/(f) II dt -> 0 .
o T h e quadruple M ( t ) is c a l l e d strongly a d m i s s i b l e i f a
T
ess sup II M ( t ) - Mo(t) II - » 0 , [ II M , ( t ) - M , ( t ) II dt -> 0 , 0
2
2
2
0
[ l l M ( t ) - M ( t ) II dt —> 0 . 3
(3.2)
3
o F o r s i m p l e adapted functions
M ( t ) the stochastic integral a
I(t) = j [ M ( t ) d A ( T ) + M , ( T ) dA(r-) + M ^ T ) d A ( r ) + M ( x ) d x ] = +
0
3
o = { [ dA(T) M ( T ) + dA(r) M , ( x ) + dA (x) M ( T ) + dr M ( T ) ] 0
+
2
3
o is d e f i n e d as the c o r r e s p o n d i n g s u m [1], [8]. A s s h o w n o n p p . 2 9 2 - 2 9 4 o f [8] the following inequalities hold
186 ti sup II | M ( T ) d A ( T ) vj/ ® \|/(f) II ti < t o ()
0
<
(Ifl + V Ifl + 1 ) W 2
sup II J M , ( x ) d A ( T ) ti < t o
f l f ( x ) l II M ( T ) y 2
0
® V(0 M dx
,
2
0
® v|/(f) II < Ifl .,J\\\ M i ( x ) \ | f ® V ( f ) H dx o
1
sup II f M ( x ) d A ( x ) \|f (8) y ( f ) II < (ifl + V Ifl + l ) •/>/} II M ( x ) y tl < t b 1
+
2
2
0
,
2
0
2
0
® \|/(f) l l d x 2
.
It f o l l o w s that sup II I ( t , ) V Q ® V ( 0 II < C (Ifl) • { f l f ( x ) l M ( x ) ti
2
0
V
® (0
o
H dx + 2
V
+ } [ II M , ( x ) VKo ® y ( f ) II + II M ( x ) v|/ ® \|/(0 II ] dx + o 2
+ where
2
(3.3)
2
0
[jllM (x)v|/ ®y(f)lldx] } , o 3
2
0
C(lfl) is a constant depending o n Ifl .
Let
M ( t ) be an a d m i s s i b l e q u a d r u p l e , M ( t ) a sequence o f a p p r o x i m a t i n g s i m p l e a
a
adapted q u a d r u p l e s . F r o m (3.1), (3.3) it f o l l o w s that the s e q u e n c e o f
corresponding
stochastic integrals I(t) c o n v e r g e s s t r o n g l y o n H
I(t) w h i c h is
0
® T
e
to a n operator
c a l l e d the ( m o d i f i e d ) stochastic integral o f the a d m i s s i b l e q u a d r u p l e
M ( t ) . A s i n [1] it a
can be s h o w n that
< y
® V ( g ) I KO % ® i|/(f) >
0
= |< v o
0
® V(g) [ o( ) 1
M
T
f
( > + T
+ M ^ x ) f(x) + M ( x ) g(x) + M ( x ) ] \|f ® \|/(f) > dx 2
for a n y y
0
2.
e H
0
3
(3.4)
0
and f , g e L ( R ) .
Inequality
2
+
(3.3) o b v i o u s l y
extends
to s t o c h a s t i c
integrals
of
admissible
q u a d r u p l e s . F r o m this i n e q u a l i t y it f o l l o w s that the stochastic i n t e g r a l I(t) is a continuous
187 function o f
t:
sup II [ I(t + At) - I(t) ] 0
L e m m a 4. L e t M ' ( t )
0
II-> 0
as e ^ O
(3.5)
be a strongly a d m i s s i b l e quadruple,
a
w i t h values i n L ( H )
\|/ ® v(f)
an adapted f u n c t i o n
w h i c h is continuous i n the sense (3.5). T h e n
M '(t)
is an
admissible quadruple. Proof. Let M ' ( t ) a
be a p p r o x i m a t i n g s i m p l e adapted quadruple for
M '(t) a
and let
(1.3) be the c o r r e s p o n d i n g d i v i s i o n of [0,T]. B y using (3.2), (3.5) one shows that simple adapted f u n c t i o n s w h i c h are e q u a l to quadruple
M '(f)
In p a r t i c u l a r , i f I(t)
N4 *(tj_ )
1
on [ t j _ t j )
1
p
approximate
the
i n the sense (3.1). M '(t) a
is strongly a d m i s s i b l e q u a d r u p l e , then
M ' ( t ) I ( t ) , where a
is a m o d i f i e d s t o c h a s t i c i n t e g r a l , is an a d m i s s i b l e q u a d r u p l e and therefore
the
stochastic integral f [ M ' ( T ) I(x) d A ( T ) o 0
+ M , ' ( T ) I ( i ) d A ( x ) + M ' ( i ) I(x) d A ( x ) + M ' ( x ) I ( i ) d i ] +
2
3
is defined. F r o m (3.3) we get the estimate
II [ J ( M ' ( x ) I(x) d A ( x ) + M , " ( T ) I ( T ) d A ( i ) + M ' ( x ) I(x) d A ( x ) + 0
+
2
o + M ' ( x ) I ( T ) dx) ] vjr ® v|/(f) II < C(lfl) • {u(x) II I ( i ) v|/ ® v(f) 3
2
0
0
II dx , 2
(3.6)
o where u(x) = ess sup II M ' ( t , ) II • l f ( x ) l + II M , ' ( x ) II + II M ' ( x ) II + 2
0
2
2
2
H
+ } II M ' ( x ) II d i • II M ' ( i ) II o 3
3
is an integrable f u n c t i o n . 3. L e t us turn to the l i n e a r q.s.d.e. (1.1). W e put
(3.7)
188 dL = L dA + L,dA + L d A 0
+
2
+ L d t and write (1.1) in the integral f o r m 3
U(t) = I + j d L ( x ) - U ( x ) . o
(3.8)
T h e o r e m 2. L e t L a ( t ) b y a strongly a d m i s s i b l e q u a d r u p l e , then there e x i s t unique adapted continuous f u n c t i o n on H o ® r
c
satisfying (3.8). It is g i v e n b y the series
U(t)=X I (t)
(3.9)
n
n=0
strongly convergent on H
0
® r
c
, where
I (t) = j d L ( x ) I _ , ( x ) ; n = 1,2,... ; I (t) = I . o n
n
(3.10)
0
P r o o f . F r o m the r e m a r k after L e m m a 4 it f o l l o w s that f o r s t r o n g l y a d m i s s i b l e quadruple and f r o m II I (t) n
V
o
LaW
stochastic integrals I„(t)
t u e
are w e l l defined b y (3.10) o n H
0
® T
e
(3.6) ® \(/(f) II < C(lfl) • j u(x) II I _,(x) \|/ ® V|/(0 N dx . o 2
n
2
0
B y iterating this inequality we obtain
II I„(t)
V
o
® y ( f ) II < C ( l f l ) " . [ J ^ ( t ) dx n! o 2
II v)/ ® y ( f ) H •
(3.11)
2
0
It f o l l o w s that the series (3.9) strongly converges on
H
0
® r
a n d defines a function
e
U ( t ) w h i c h is continuous i n the sense (3.5). F r o m L e m m a 4 it f o l l o w s that the stochastic i n t e g r a l i n the r i g h t - h a n d side o f (3.8) is w e l l - d e f i n e d . F r o m (3.11) it f o l l o w s that the stochastic integration and the s u m m a t i o n over n can be interchanged. T h e n (3.8) follows f r o m (3.10), (3.9). U n i q u e n e s s o f the s o l u t i o n U(t)
on H
0
® T f o l l o w s f r o m standard c
argument [1 J. In case d i m K
( )
= 1 the s o l u t i o n U ( t ) is g i v e n e x p l i c i t l y as an e x p o n e n t i a l of basic
operators. L e m m a 5. L e t l ( t ) be an a d m i s s i b l e quadruple o f functions w i t h v a l u e s i n C . The a
s o l u t i o n of q.s.d.e.
189 dU = (l dA + l,dA + l d A 0
2
defined on r
+ l dt) U ; t > 0 , U(0) = I ,
+
3
, satisfies
c
U(t) V|/(f) = y ( f + l
l f + l
m
0
[ 0 l ]
l ) • exp | [ 1,(T) f ( t ) + l ( x ) ] d t . 2
(3.12)
3
0
If l ( t )
are obtained f r o m another a d m i s s i b l e quadruple
l
0
a
0
= a(m ),
1,
= m, b ( m ) ,
2
2
1 = m
0
3
3
m (t) a
b y f o r m u l a s (1.7) i.e.
+ m,m c(m ), 2
(3.13)
0
then the s o l u t i o n is g i v e n by U(t) = exp [ A ( l
| 0 l J
m ) + A(l 0
+ A (1
m,)
| 0 l ]
+
i == exp | ( m d A + m , d A + m d A o 0
2
+
[ 0
l]
i m ) + Jm (r) d r ] = o 2
3
+ m dx).
(3.14)
3
P r o o f . T h e relation (3.12) defines adapted process
U(t) , the stochastic integral o f
w h i c h is w e l l defined. W e have to show that t U(t) = I + j ( l d A + l , d A + l d A o 0
2
+
+ l d-r)U(T) ; 3
t>0.
A c c o r d i n g to f o r m u l a (3.4) it is enough to check that for any f, g e
< ( g ) I U(t) y ( f ) > = < V(g) I V
+)
o
< V ( g ) I [ 'o CO
L (R ) 2
+
¥(0 > +
fCO + l ] ( t ) f ( t ) + l a W
+
1
U
W
V(0 > d t ,
and this f o l l o w s f r o m (3.12), (2.5) b y direct c o m p u t a t i o n . T o p r o v e the second statement it is enough to compare (3.12) and (2.9). 4. T h e f o l l o w i n g l e m m a is an a n a l o g o f P r o p o s i t i o n 1 for m u l t i p l e stochastic integrals. Lemma 6 + L dA 2
+
L
e
t
L
a( )
t>
l
+ L d t and I (t) 3
e
3
strongly a d m i s s i b l e quadruple, d L = L d A + L , d A + 0
be defined by (3.10) i.e.
n
I (t)=)dL(T ))dL(T _,)...|dL(T,). n
n
0
n
0
0
190 L e t l ( t ) be an arbitrary admissible quadruple of functions w i t h values i n C
such that
a
II L ( t ) II < l ( t ) B
a
; t€
[0,T] .
Put d l = l d A + l , d A + l d A 0
2
+
(3.15)
+ l d t and 3
Wt) = )dl(T ) f d l d , , . , ) . . . } dl(t,).
T h e n for \|/ e H
0
0
I < I (t) v n
(3.16)
2
n
0
0
0
, and f,g e
L (R ) 2
+
® V ( 0 11,„(0 V o ® V ( g ) > I S < W O ( l f l ) I Ln(t) t|f(lgl) > II V V
0
0
II
(3.17)
2
and the right-hand sice is a monotonely increasing function o f the quantities l„(t). Proof. Let L ( t ) a
be an a p p r o x i m a t i n g s i m p l e adapted q u a d r u p l e f o r
(1.3) be the c o r r e s p o n d i n g d i v i s i o n o f I (t) n
L ( t ) and let a
[0,TJ. F r o m the p r o o f o f L e m m a 4 it f o l l o w s that
can be approximated strongly on H ® T 0
c
b y the s u m
N
I„(t) = X
[ L ( t ) A A j + L . d j . , ) A A j + L (tj_,) A A j 0
H
2
+ L ( t j _ , ) At, ]
+
.
3
In the same w a y I _,(t) can be approximated b y the s i m i l a r s u m I _](t) . It f o l l o w s that n
I (t) n
n
can be approximated b y N
f (t)
= E
n
[L (tj_ )AA +...]I _ (tj_ ). ( )
1
J
n
1
1
F o r s i m p l i c i t y o f notations w e e x p l a i n it only for the first term i n the squared brackets : II L ( t ) A A j ( I ^ t j . , ) - L ^ t j . , ) ) 0
H
V
o
®
V
(0 H <
< II L ( t j _ , ) II • II A A j V(f[j_,) H • H ( I . , ( t j . , ) - I _ , ( t ) ) 0
where
fjj_, is the restriction of
n
f onto
[tj_,,T]
n
H
and f u j
II . is the r e s t r i c t i o n o f
f onto
[0,tj_,). C o n t i n u i n g i n this w a y we can a p p r o x i m a t e
I (t) n
b y a s u m o f the f o r m (2.17).
191 A p p l y i n g P r o p o s i t i o n 1 and passing from sums to integrals we get (3.17). Let
M ( t ) , t e [0,T] be a strongly a d m i s s i b l e quadruple, then it is easy to show a
that L ( t ) defined by (1.7) is again a strongly a d m i s s i b l e quadruple. P u t 0
m ( t ) = It M ( t ) II .
(3.18)
II L „ ( t ) II < l ( t ) ,
(3.19)
a
a
Then B
where
l
a
are obained f r o m
m
a
b y relations (3.13).
L e m m a 7. L e t U ( t ) be the solution o f q.s.d.e. (1.1), where 2 from M ( t ) b y relations (1.7). T h e n f o r \\i e K , f e L ( R ) a
a
0
L (t)
(3.20)
2
0
are obtained
+
II U ( t ) } y ® v|/(f) II < e x p } F , ( T ) dx , o
a
o
where F , ( x ) is g i v e n by (2.22) w i t h m
a
= m (x) a
defined b y (3.18).
P r o o f . A p p l y i n g T h e o r e m 2, L e m m a 6 and P r o p o s i t i o n 1 w e get
II U ( t ) v
where
0
® v|/(f) II < II U ( t ) \|/(lfl) II • II v|/ II , 0
U(t) = X
i (t) n
and I (t) n
are defined b y (3.16) w i t h
l (t) a
n=0
inequalities (3.19). B y (3.14) 0(t) = e x p [ A ( l
[ 0 l |
m ) + A(l 0
U s i n g (2.21) w e get (3.20).
[ 0 ( )
m,) + A ( 1 +
( 0 l]
m ) +} m ( x ) d x ] . o 2
3
a p p e a r i n g i n the
192 § 4 . E s t i m a t e s for
time-ordered
products.
1. W e n o w pass d i r e c t l y to the p r o o f o f T h e o r e m 1. L e t
M (t) , t e a
[0,T] , be a
strongly a d m i s s i b l e quadruple of functions with values i n L ( J t ) . F i x an a p p r o x i m a t i n g 0
quadruple o f s i m p l e functions M ( t ) and let a
0 = t„ < t, < ... < t
N
be the corresponding d i v i s i o n of
= T
(4.1)
[0,T|. Denote
m „ ( t ) = II M „ ( t ) II , m ( t ) = II M„(t) II , 0
„(t) = max { m „ ( t ) , m ( t ) } .
(4.2)
a
A s (3.2) h o l d s , we have T
ess sup I m„(t) - m,,(t) I —* 0 , l i m , ,(t) - m , (t) I dt -> 0 , 0 < l
2
T
{I m3(t) - m ( t ) I dt -> 0
(4.3)
3
o and the same w i t h m ( t ) a
replaced by
m (l)
.
a
Let V j be defined as in (1.4), i.e. V j = exp [ M
0
(tj.,) A A j + M , (tj_,) A A j + M (t^) 2
AA*
+ M ( t ) Atj ] . 3
H
N
P r o p o s i t i o n 2. T h e operator
H ®r 0
c
n
Vj
is defined as a s t r o n g l y c o n v e r g e n t series on
and (4.4)
where
F,(t)
is g i v e n by (2.22) w i t h m „
replaced b y
m (t). a
193 N
Proof. We have y(f) = ® \|/(fj) , where fj is the restiction of f onto L tj_j, tj). Denoting K j , = H o ® T ( L [O.tjl ) we identify H ^ with H 2
}]
® I <= H .
Then Vj can be considered as an operator acting in Jtjj . We have < <— ii n v ( f ) i i j V
< n
0
nv
j V
(fpii . H
where II A \|f(fj)
=
sup II A(v„ ® v(fj» II , Voc Kj.il ll F HS 1 l
0
for an operator A acting in H j , . According to (2.20), (2.21), where the role of H
0
is
played by H j _ , n
l
II Vj (fj) H V
H
-
l j
< exp^j
F,(t)dt, ij-i
whence (4.4) follows. *— N
We now describe the strategy of proving (1.6). We have U(T) = If U j , where Uj = Uj(tj) and Uj(t)
is the solution of the q.s.d.e. (1.1) on
[t _ ,tj] .satisfying j
1
Uj(tj_,) = I . To prove (1.6) we thus have to evaluate <
<—
iiCn V j - n U j ) <-
¥ o
® (f)n V
=
2
*-
<-
= < (n ^ - n Uj) v ® v(f) 1 1
<-
_n ^ (v - u ) n Uj v ® y(f) > (
0
k
k
0
(4.5)
We are going to show that «-
<-
<¬
i < n V : v|f ® v(f) i n V : (v - u ) n u y ® ytf) > i < pi j=k+l j=1 k
0
J
k
0
(
J
J
(11 L (t) - L (t _,) II f(t) I f(t) I + II L,(t) - L , ^ , ) III f(t) I + 0
0
k
~ _ k ~ + H L ( t ) - L ( t _ ) l l f ( t ) + I I L ( t ) - L ( t _ , ) II ) dt + ( f F (t)dt) ] . tk-i l
2
2
k
1
3
3
k
2
2
(4.6)
194 Here L ( t ) and L „ ( t ) are obtained f r o m a
M ( t ) and M ( t ) b y f o r m u l a s (1.7), f(t) is R
a
a nonnegative square integrable function, C is some constant and F ( t ) is a nonnegative 2
integrable f u n c t i o n . T h e functions
f
and F
2
depend o n
f
and o n q u a n t i t i e s
defined b y (4.2) and thus o n the d i v i s i o n (4.1); h o w e v e r as (3.2) h o l d s , f L
2
[0,T]
and F
in L
2
1
m (t) a
has a l i m i t i n
[0,T] . N
Q u i t e s i m i l a r estimate holds for the q u a n t i t y w i t h
fl Vj
i n the left h a n d side
N
replaced b y n
U j . Therefore (4.5) w i l l be estimated b y the quantity
C [ f { II L ( t ) - L ( t ) II f(t) I f(t) I + II L , ( t ) - L , ( t ) II I f(t) I + II L ( t ) - L ( t ) II f(t) + o 0
0
2
~
Ik
+ II L ( t ) - L ( t ) II ) dt + irrax { 3
3
„
T „
F ( t ) dt • ( F ( t ) dt ] 2
2
,
o
lk-1
w h i c h tends to zero as (3.2) holds and
2
m a x At; - » 0 . T h u s to p r o v e T h e o r e m 1 it is 1 < j < N
J
sufficient to establish (4.6). T h e p r i n c i p a l fact w h i c h f o l l o w s from the c o n d i t i o n that M ( t ) take their values in a
L(Jt )
is that
0
j-th
factor
i n the t i m e - o r d e r e d
products
acts
e f f e c t i v e l y in
2
K
0
® T ( L [ t j , , , tj]) . W e have the f o l l o w i n g m o d i f i c a t i o n o f P r o p o s i t i o n 1 : P r o p o s i t i o n 3. L e t mf*
Aj = 1 1 n
where h
hJa
( n
s
J )
where Then
(I h < f
are zero outside M|»
J n
i =0,l,2,3 (s=I n)
Xj
j n
(I
I) ,
(4.7)
e L(H ) ( )
[tj_,,tj] , converge strongly o n r
(h 7 l)-....X (
'
s
i n
I) • ... • X
1
s
Aj = I S "
Xj
mf i =0.1.2,3 (s=l n)
be nonnegative numbers such that the series
i n
)
( h M ) ,
j
c
. Let (4.8)
n
are such that II M®
i n
II < m®
i n
and similarly for Bj , Bj
195
i< n
Av® I
v(oi
0
n
v|/ ® \ | / ( o > i < n
< Aj\)/(ifji)iBj\|/(ifji)>II\t/ II
0
0
P r o o f . N o t e that by P r o p o s i t i o n 1 the series d e f i n i n g
A j converges o n
(4.9)
2
H
0
® r
.
c
2
M o r e o v e r , the operator
Aj
acts effectively on
K
® T(L
0
(tj_j,tj)). T h e p r o o f o f
(4.9)
is a direct c h e c k u s i n g (2.19). W e stress again that for an operator
Aj
g i v e n b y (4.8) the e s t i m a t i n g operator
Aj
can be c h o s e n i n m a n y w a y s . F o r the quantity o f interest we then have «-
i < n
<
<-
Vj
n
j
V
V
o
<¬
® \|/(f> i . n , ^ +
(ifi)iv
j
¥
(v
- u )
k
k
(ifi)>.{
W
n
u
i
V ( )
®
V
(f)
< v ^ ^ ) ' ( v ^ u < V
J
k
V
(|f |) I V k
k
> i <
)
v(iM)>
^
}
(
4
1
0
)
YOfkD >
k
where we have taken V j = U j = exp { A ( 1 ,
, uo) + A (1,
, p.,) + A ( l +
(
, u^) +}
u ( t ) dt } , 3
IN M a k i n g this c h o i c e we use the d e f i n i t i o n (1.4) of and the fact that
£
Uj
u ( t ) > m a x (II M ( t ) II , II M ( t ) II) for t 6 [tj_,,tj) . F r o m L e m m a 1 a
a
n
V j yOfjl) = v|/(lfl) exp ^
where
V j , the argument o f L e m m a 7 for
is the restriction to
,
(4.11)
[ t j ^ t j ) o f the f u n c t i o n
f(t) =
[1 + l ( t ) ] l f ( t ) l + ? , ( t )
(4.12)
a
[ l,(t) I f(t) I +
(4.13)
0
and J
f
=
l (t) | d t . 3
ij-i Here functions
I (t) a
l ( t ) = a(rn (t)) , 0
0
are obtained from
m (l) a
1, ( t ) = m , ( 0 b ( m ( t ) ) , 2
2
0
by the relations (3.13) i.e. ? (t) 3
= m (t) 3
+ m,(t)
m (t) 2
c(m (t)) . 0
196 Note that, as promised, f(t) is a nonnegative square integrable function, converging in L (0,T) , as (3.2) holds, to the function [1 + l (t)J I f(t) I + l (t) . 2
0
2
From (2.14) »
l
J
~
I < Vj v(lfjl) I Vj V(lfjl) > I = exp j
F,(t)dt .
Thus the product in the right-hand side of (4.10) is estimated by exp j F](t)dt . A s the o approximation conditions (3.2) hold, F,(t) tends to F,(t) in L ( 0 , T ) 2
exp|F,(t)dt
and the quantity
remains bounded by a constant depending on m ( t ) and f(t). a
o It remains to evaluate
< V
k V
( l f l ) I ( V ^ U k ) VCffcJ) > / < V vC(f l) I V v(lf l) > = k
k
k
k
= < ( l f l ) I ( V ^ U ) (|f l) > exp {-} V
k
k
v
k
F,(t) dt}
k
(4.14)
Ik—1
and in particular to explain how to choose ( V - U ) . The last factor is less than one but k
k
it is convenient to retain it since it absorbs similar factors arizing from estimates of the first factor. We put V
k
- U
k
= A
B +C
k +
k
k
,
where A
k
= V
k
- I - [
L ( t _ , ) A A + L , ( t _ , ) A A + L (t _,)AAi^ + L ( t _ , ) A t ] , 0
k
k
k
k
2
k
3
k
k
* B
k
= |
{ |L i(tk-i) (
_
M O 1 dA(t) + | L,Ct _,) - L,(t) ] dA(t) + k
i -i k
+
C
k
[ L ( t _ , ) - L (t) | dA (t) + [ L ( t _ , ) - Lj(t) ] dt } , 2
= - U
k
k
+
2
+ I+ f
3
dL(t) = - J f
'k-l
=2 ik-1
n
k
C1L(T„) f
d U V i ) - f
Ik—I
and let Vk~^U
= Ak + k + C • B
k
where A , B , C k
k
k
k
are to be defined and evaluated below.
lk-1
d
L
^i)
197 2. T h e case o f
B
is the most straightforward. A c c o r d i n g to L e m m a 6 we can take
k
Ik
B
k
= J
{II L ,(t) - L ( t _ , ) II d A ( t ) + II L , ( t ) - L , ( t _ , ) II d A ( t ) + (
0
k
k
+ II 1^(0 - L ( t _ , ) II d A ( t ) + II L ( t ) - L ( t _ , ) II dt } . 2
+
k
3
3
k
T h e n , u s i n g (3.4),
< y (ft) I B V (lf D > = < V (f ) I V (lf D > • k
•f
k
k
k
{ II L ( t ) - L ( t _ , ) II f(t) lf(t)l + II L , ( t ) - L , ^ ) 0
0
k
II lf(t)l +
lk-1
+ II L j ( t ) - L ( t _ , ) II f(t) + II Lj{t) - L j ( t _ , ) II } d t . 2
k
k
T h e factor < y (f ) I v (lf l) > k
= exp j
k
{[1 + ? (t)l lf(t)l + l (t) lf(t)l} dt 2
0
2
lk-1
is absorbed b y the factor
exp { - f
F,(t) dt}
=
i -i k
= exp {-{
{( [1 + Lj(t)] lf(t)l + l (t) ) + 21,(t) lf(t)l + 21 (t)}dt } 2
2
3
(4.15)
lk-1
i n (4.14) and thus the term w i t h B 3. S i n c e C
k
= £ f
II L ( t ) II < l ( t ) a
a
d?(T„)f"
gives the first integral term i n the estimate (4.6).
k
then according to L e m m a 6 we can take
d?(v,)-f
2
did,)
,
n=2 , lk-1
where
'k-1
lk-1
dl = l d A + l , d A + l d A + l d t . Then
Ck = V
0
k
2
- I - f lk-1
and b y (4.11), (3.4)
dl(t)
+
3
198 < y (f ) I C \|/ (lf l) > = < V)/ (f ) I v (f ) > exp o k
k
k
k
k
-
k
ft ~ - < V (f ) I V (!f l) > - < V (f ) I V (ICkO > | 2(0 ik-i k
k
F
k
d t
-
where F (t)
= l ( t ) f(t) lf(t)l + l,(t) lf(t)l + l (t) f(t) + i ( t ) ,
2
0
2
(4.16)
3
F r o m (4.12), (4.13)
ft < y (f ) I C y (lf l) > exp a k
k
k
= < v (f ) I \\i (lf l) > • exp |
k
k
k
F ( t ) dt , 2
lk-1 hence < Y (f ) I C y (lf l) > = < v (f ) I y (lf l) > • c( { F (t) d t ) • ( f F (t) d t ) lk-1 lk-1 k
k
k
k
< \|/ (f ) I C y (lf l) >
T h e factor The function
k
k
c(z) =
k
z
z tends i n L (0,T) tk ^ c( J F ( t ) dt ) lk-1 2
.
2
is again absorbed b y the factor (4.15) i n (4.14).
k
e — 1— z
2
is l o c a l l y b o u n d e d . A s (3.2) h o l d , the f u n c t i o n
** F (t) 2
2
to an integrable f u n c i t o n (depending
on
f(t) and m ( t )
),
remains b o u n d e d . T h u s the c o n t r i b u t i o n o f the term w i t h
C
a
therefore k
has the
f o r m as the second term in the right-hand side of (4.6). 4. T h e most d i f f i c u l t case is that o f L
a
= L (t a
k
l
)
A
and o m i t the subscript k of
W e also denote
AX
0
k
. F o r b r e v i t y w e denote
AA ,... k
= dA , A X , = dA , A X , = d A
oo
°°
X
0
n
-h-Mj ; _j n! L
we can write
3
= M
L, = M ,
+
, AX
3
3
+ M,
Mo"' ;
n=l
X
n=l
~ 7 Mo" M , , 2
n
L
2
=
= M (t _!), a
k
= At . T a k i n g into account
°°
X
a
.
that L =
M
X
n=l
jr "•
M
2
;
199
A = k
"
X -4? X
n=l
M . -... Mi A X -... A X : + Y
ip=0.1.2.3 (I>=l,...,n)
n !
'l
;
'n
'l
'»
n
+ M , M o " ' ( A A A A " " - A A ) + Mo"' M , ( A A 1
+ M , Ml'
- L { Mn ( A A - A A ) + N
tS
n!
AA
N L
1
- AA ) +
+
+
M ( A A A A " " A A - At) } ,
2
2
2
where X
+
means the summation over all possible sets ( i , , . . . , i„) , where L runs
ip=0,1,2.3 (P=l n)
P
the values 0,1,2,3 expect for the terms MS"
1
M
2
AA"~'AA
(for
+
n = 1,2,
... ) ,
Mn A A " , M , Mo * A A A A " " ' , M, M"," M 2
2
AA AA " N
2
A A
+
(for n = 2, ... )
and M A(t) (for n s= 1). 3
The computation relations [ AA,AA
] = AA
+
+
, f AA.AA
I= -AA
(see e.g. [4]) can be rewritten in the form AAAA
= AA
+
(I + A A ) " , A A A A = (I + A A ) A A
+
,
whence AA"AA
+
= AA
+
(I + A A ) "
A A A A " = (1 + A A ) " A A .
,
Using these relations (4.17) can be given the form
A = k
k
X n=l
n !
, X M: - . . . - M : AX: - . . . - A X : i =o.i,2,3 'i 'n i (P=l n)
+ M , Mo"' [(I + A A ) " " ' - I] A A + Mn"' M + M , MU" M 2
2
• [AA
+
a
with
AA
2
+
[(I + A A ) " " - I] + 1
(2 + A A ) " " A A + ((I + A A ) " " ' - I) At ] }
m = m (t a
k
I
k
(4.18)
.
2
According to Proposition 1 we can take for A replacing M
- L {M^ ( A A " - A A ) + n!
+ y ^
p
the expression obtained from (4.18) by
). Introducing the quantities l
a
related to
m
a
formulas (3.13) and using the definition (1.8) of functions a(z) , b(z) , c(z) , we get
by
200
A
= {exp ( m A A + m,Aa + m A + m At) - I - 0 A A + i,Aa + I A + l At)} +
K
0
+
2
3
0
+
2
3
+ {a (m AA) - a (m ) A A } + 0 1 , {b ( m (I - AA)) - b ( m ) } A A + 0
()
0
+ rn AA {b(m (I-4A))-b(m )} +
2
0
+ m, m { A A 2
+
+
0
C
0
(4.19)
( m (2 + AA)) AA + [c(m„ (I - AA)) - c(m ) ] At} . 0
0
We now explain that the contribution into < y ( f ) I A y ( l f l ) > k
of each term in the
k
right-hand side of (4.18) can be evaluated by a quantity C • [ j
F (t) dt J . 2
•j-i Using (2.13) and (3.4) we find that the contribution into < \)/(f ) I A \|/(lf l) > of k
K
k
the first term in (4.19) is equal to < ( f ) I \|f(lf l) > [exp f V
k
F (t) d t - l - f
k
V
3
lk-1
= < v|/(f ) I ( l f l ) > = c (j k
F (t) dt ] =
3
'k-1
F (t) d t ) • [ j
k
F (t) dt ]
3
lk-1
3
2
,
lk-1
where F (t) = i (t) f(t) if(t)i + i,(t) if(t)i + i (t) f(t) + i (t) 3
0
2
3
and l (t) are simple functions related to m (t) by formulas (3.13). A s in n.3 this can a
n
Ik
be evaluated as C • [ j
„
,
F (t) dt ] " . 3
lk-1
Since l (t) < l (t) , we can replace F (t) by F (t) . a
a
3
2
From (2.13) we get <\t/(g)lexp(mAA)x|/(f)> = <\|/(g)h/(f)>
exp[(e -l)f m
gO) f(t) dt] ,
lk-1
Then for the second term in (4.19) we have < \|/(f ) I [ a ( m AA) - a (m ) AA ] V|f(lfl) = k
0
()
.
ik
= < y ( f ) I \)/(lf l) > (exp 1 f k
k
0
„
.
k
k
lk-1
k
0
lk-1
= < V(f ) I y(lf Q > c ( f
i
f(t) lf(t)l dt - 1 - 1 {
lk-1
F (t) dt) •if 4
lk-1
F (t) dt ] , 4
2
^
f(t) lf(t)l dt } =
(4.20)
201 where
F ( t ) = 1 f(t) lf(t)l 4
0
, so that again
F (t) < F (t) 4
and the c o n t r i b u t i o n of the
2
second term has the raquired f o r m . In c o m p u t i n g the r e m a i n i n g three terms we use the identity AA (f)
[f
=
V
f(t) dt ]
\|/(0 ,
lk-1
w h i c h f o l l o w s f r o m (2.6). T h e n the factors ik
factors
m, J l
f(t) dt or
m
2
ik
.
j
f(t) dt
k-l
m,AA
or
m AA 2
result i n n u m e r i c a l
+
when c o m p u t i n g the corresponding
terms
i -l k
in < y ( f ) I A v(lf l) >, k
k
k
and w e are left w i t h the expressions o f the f o r m < v(f )
I [ b (m
k
0
(I + A A ) ) - b ( S ) ] ( l f l ) > ()
and the same w i t h b(z)
V
,
k
(4.21)
replaced by c(z). These can be calculated e x p l i c i t l y by using the
relations i
b (mA) = m
m
f e
i K A
m
K) e
c ( m A ) = —7 [ (m n
dK ,
0
K A
dx
and (4.20). F o r e x a m p l e (4.21) is equal to < (f ) V
k
I \)/(lf l) > • b(ffi ) • c (1 k
0
f(t) lf(t)l dt ) • [ T j
f
0
0
lk-1
f(t) lf(t)l dt ] .
tk-1
Therefore the contribution o f the third term is estimated as tk
C •}
.
Ik
1,0)
lf(t)l dt • j
«c-l
.
l,(t) f(t) lf(t)l dt
,
lk-1 Ik
„
w h i c h is a g a i n less than C • t j
2
F ( t ) dt ] 2
>
a n [
j
s o
o
xh
n
u s
t
h
e
i n e q u a l i t y (4.6) is
'k-1
established and this completes the p r o o f of the T h e o r e m 1. In the same w a y one can establish the convergence to products
fl W: j
where
'
= exp f
' ij-i
f M dA + M,dA + M dA n
2
U(t)
o f the
time-ordered
+ M d t ] . F r o m this the 3
C o r o l l a r y 2 f o l l o w s . T h e scheme o f p r o o f o f other statements is g i v e n i n [4] , [7] ; it can be made rigorous b y u s i n g the results and methods o f this paper. It is natural to ask for
202
the extension of the Theorem 1 to the caase where M ( t ) are adapted processes with a
nontrivial dependence on the past ; however the present method of proof seems not to extend to this case.
Acknowledgements. The author is grateful to Professor W. von Waldenfels for the invitation to the Institute fur Angewandte Mathematik, which in particular made this publication possible. He also acknowleges useful discussions with Dr. M . Schiirmann, P. Glockner and R. Speicher.
References
[1]
Hudson R . L . , Parthasaratky K . R . Quantum Ito formula and stochastic evolutions. -
[2]
Hudson R.L., Parthasaratky K . R Construction of quantum diffusions. - Lect. Notes
[3]
von Waldenfels W . Ito solution of the linear quantum stochastic differential
Commun. Math. Phys., 1984, v.93, 301-323. in Math., 1984, v.1055, 173-198. equation describing light emission and absorption. - Lect. Notes in Math., 1984, v. 11055, 3 8 4 ^ 0 4 . [4]
Holevo A . S . Stochastic representation of quantum dynamical semigroups (in Rus sian). - To appear in Proc. of Steklov Math. Inst., 1989.
[5]
Daletsky Y u . L . Multiplicative operators of diffusion processes and differental equations in vector bundles (in Russian). - U M N , 1975, v.30, 209-210.
[6]
Skorokhod A . V . Operator stochastic differential equations and stochastic semigroups (in Russian). - U M N , 1982, v.37, 157-183.
[7] [8]
Holevo A . S . Quantum stochastic calculus (in Russian). - Preprint 1988. Meyer P.A. Elements de probabilites quantique. - Lect. Notes in Math., 1986, v.1204, 186-312.
Quantum Probability and Related Topics Vol. VII (pp. 203-209) © 1 9 9 2 World Scientific Publishing Company
203
STRUCTURE RELATIONS FOR FERMIONIC FLOWS R L Hudson and P A Shepperson Mathematics Department University of Nottingham Nottingham NG7 2RD UK 1. Introduction A notion of quantum stochastic flow has been introduced [1] which generalises the classical notion of stochastic flow on a manifold in two ways. FirsUy the manifold or state space is quantized, by replacing the algebra of complex-valued smooth functions thereon by a noncommutative *-algebra, which may be obtained by deforming the multiplication of the former algebra in the important special case of a quantum group. Secondly, the classical Ito stochastic calculus of Brownian motion is replaced by a noncommutative quantum stochastic calculus. In all work hitherto this calculus has been either one- or mulU-dimensional Boson calculus in Fock space. In this paper we investigate the use of Fermionic stochastic calculus in this context. At first sight it may seem surprising that there is a separate Fermionic theory to be explored, at least in the case of one-dimensional stochastic calculus, since the Boson and Fermion theories are known [2] to be equivalent in so far as the Fermionic creation and annihilation processes B* and B can be realised in Boson Fock space through the stochastic differential prescription dB* = RdA*.
dB = RdA,
(1.1)
where R is the parity process and A and A the Bosonic creation and annihilation processes. However, as noted in [2], the presence of the factor R, arising when Fermionic stochastic differential equations are converted to Bosonic using (1.1), creates new types of unitary processes which in turn generate new 'inner' Fermionic flows by conjugation. A further variety of such inner flows is found by Z grading the initial algebra and coupling it to the Fock space by the Z -graded or ami-commuting tensor product, so that the driving coefficients of the stochastic differential equations for unitary processes, which are operators belonging to the initial algebra, no longer commute with differentials. The characterisation of such unitary processes is explored in §2; it turns out to be very little different to that of the case when the Z -grading is trivial of [2]. However the corresponding notion of flow got by conjugating is more complex. To obtain a closed system of stochastic differential equations, or equivalently the analogue of the characterisation of flows through structure relations of the Boson case, we need to consider a 4-tuple of maps corresponding not only to conjugations by the unitary process U but also by its image under the grading automorphism together with the two crossconjugations where the grading automorphism is applied only to one of U and U~ . i
2
2
2
2. Unitary processes driven by elements of a Z -graded initial algebra 2
We consider the usual Hilbert space framework for one-dimensional quantum stochastic calculus, namely X is the (Boson) Fock space over h = L (U+), with exponential vectors yrf.f), / e h, X is the initial Hilbert space and X = X ®X. We assume X is Z -graded, that is, there is a distinguished self-adjoint unitary operator R e B(X ), the eigen subspaces X of which corresponding to eigenvalues + 1 define the even and odd sectors of X . We then define the parity (or reflection) process R = (R(t): t £ 0) by the actions 2
B
0
0
0
2
Q
0±
0
R(t)u®y(f)
R is self-adjoint unitary valued.
=
«o"®lK-^[o.,]/ ^«.-)A +
ueX .feh, 0
204 We introduce the usual gauge, creation and annihilation processes of (Boson) quantum stochastic calculus, defined by A(/)«®tt
A (/)«®
for ueh , feh. 0
^utSyfa'*^)
to
£
=o
«®£^/+£^ .,])
V
0
Fermion creation and annihilation processes fl* and B are defined by the stochastic
integrals B (0 = f
R(s) dA\s),
f
B(t) = [' R(s) dA's).
(2.1)
It may then be verified, by straightforward extension of the case H C, R = 1 [2] that, for / £ * , t S 0, 0
0
Jo
Jo
where <> is a realisation of the Fock representation of the canonical anti-commutation relations over L ( K ) with vacuum u<0). In other words each b*(f) is a bounded operator, the map / >-> b*(f) is linear, the canonical anti-commutation relations t
2
+
{b(f),b\g)) = < / , « > l j hold, where b(f) = ( i *(/))*. we have
Kf)v(0) = 0, and K is generated by the vectors [b\f„)...b\f ) (V). l
n = 0.1
V
/,
/.eA).
Now let there be given an initial *-algebra A c B(K ) invariant under the grading automorphism p :T—> R TR . We are interested in constructing unitary-valued processes satisfying stochastic differential equations of the form 0
0
0
0
dU = U(.L dA-rdB^L +L dB+L dt), 1
2
3
l/„=l,
4
(2.2)
where the Lj are elements of A identified with their ampliations L , ® 1 to H. (The notation UdB^L^ is adopted for convenience; it could be replaced by Up (L )dB to restore the differential to its conventional location on the right hand side, or indeed by dB^p,(U)p (L) where p, is the product grading Ti-* R't)UR(tY'). 0
i
2
0
In what follows it will sometimes be convenient to drop the subscripts 0 and / from the grading automorphisms Po.Pi (contexts will resolve ambiguities). Assuming the existence of £/, the adjoint process l / will satisfy +
dU
1
= (L\*dA + dB*L,$-rL dB+Lidt)U . 2
>
t
(2.3)
Necessary conditions for the unitarity of the solution of (2.2) are now obtained from the differential form (1.1) of (2.1) and the quantum Ito formula applied to 0 = d(l) = d(UU*) =
d(U U), f
by equating to zero the coefficients of dA, dA*. dB and dt. (Independence of these differentials follows from (1.1) and the corresponding Boson result which has been established rigorously by
205 Vincent-Smith [5] and Lindsay [4]. The result is 0 = f/(L,+Lf +Z.,Lf )U = Lt +L,+LfZ. , 1
0 = l?(L +LJ+£ Lf)(/ 2
1
1
= LJ+ij+LfLj,
t
0 = f/CLj+LJ+LjLf )U = LI+Lj+LfL,, r
0 = £/(L +tJ+L LJ)t/ 4
3
t
= LJ+^+LJLj,
where - denotes the action of the grading automorphism. These relations are satisfied if and only if the 4-tuple is of the form (W- l.L. -L*W,
Ui-IL*L),
where W.L.He A with W unitary and H self-adjoint. Then (2.2) becomes dU = U((W-\)dA
+ dB*L-L*WdB
+ 'iH-} L*L)dt),
V
t
Q
= l.
(2.3)
We note also the corresponding stochastic differential equations for U , 0 and 0*, f
dU* = ((W -l)U*M-dB*W*LU*+L*U*dB-(,iH+iL*L)U dl), r
V2 = 1,
f
dU = U((W-l)dA-dB*L+L*WdB
+ (iH-±L*L)dt),
= ((# M)<M+
t
,
(2.4)
U = 1,
(2.5)
0
62 s i .
(2.6)
By straightforward extension of the proofs of Theorem S.l and 6.2 of [2] (which are the case R = 1) we have the following. 0
Theorem 2.1. Let W.L.H be bounded operators on K with W unitary and H self-adjoint. Then (2.3) has a unique solution, which is unitary-valued. 0
3. Fermionic flows We may attempt to construct an 'inner' flow j from the solution U to the equation (2.3) by setting, for each / e R+, j,(X)
=
U.X8W,- . 1
However the differential of j,, computed from the quantum Ito formula and (2.4), is found to be dj<X) =
j(WxW- -x)dA l
+ dB*U(LX-(WXW-YL)U-
+
1
+j(.i[H,X] -
U(XL*-L*'WXW- r)U- dB ,
1
l(L*LX-2L*(WXW )-L+XL*L)dl. l
We have chosen to ensure 'democracy' between left andrightby writing <JB on the left and dB on the right (because dA and dt are even it is immaterial whether they are written on the left or the right). However no other such convention will make it possible to write the coefficients of dB* and dB in the form j,(j3*(X)), j,{p'X)) respectively. To obtain a closed system of stochastic differential equations it is necessary to introduce the four families of processes T
y,i(X)=p ({/)v®lp*({/-i), i
It may be verified from (2.4) dm
U = 0,1.
= jiaiw)dA+dByi+ (pioc))+j' ^ i
K = p\W)XpKw- )-X,
A -» A are defined by (3.3)
l
Bl(X) = (-p?(L)X-{ 'W)X'-p HW~ )n-p L), i
0.2)
k+i
where + is addition mod 2, and the maps Xi.B^.B^',:
P
(3.1)
(2.6) that these satisfy
k
,
k
(3.4)
206
B*'(X) = Bi'X*)*,
(3.5)
T 'X) = i ( > ' ( f f ) ; r - V p * ( / / ) ) - £ ( p ' ( L » ^
(3.6)
k
Clearly the j' are linear injective maps satisfying the relations k
llfrV
=
m*
= J?(.x).
Jlfd) = 1
(3-7) (3-8)
if ' = *•
ii(xy) = ;/(*);'*(>)
(3.9)
(not summed)
(3.10)
for arbitrary X, y e A and /, A, / = 0,1. This motivates the following. Definition. Let A be a unital '-subalgebra of fl(#o) grading automorphism p : X i-> X . A f«rmion ,/Zoiv over 4 is a family / of linear injective maps (j^ : A —> fl(#). i , * = 0,1, < e R ) such that, for each x e A and i,k = 0,1, jl(x) = O'iOO i « R , ! is an adapted process satisfying the relations (3.7), (3.8), (3.9), (3.10), and there exist maps X' , B , Bp, T ', i,k = 0,1 from A to itself such that, for arbitrary x e A , t S 0, i , k = 0,1, w i t l 1
+
k
k
h
/£,(*) = * ® 1 + [' UilXiix^dA+a^yt'ifim+j'.
{fil\x))dB+j \4'x))dt}. k
(3.11)
If there exist W,L,H e A with V/unitary and / / self-adjoint such that A^, At. Af'> tl are given by (3.3), (3.4), (3.5) and (3.6), or equivalently the jl(X) are given by (3.1) where U is the solution of (2.3), then we say j is inner.
4. Structure relations It may be verified that the maps XI, Pi, Pi', x' inherit linearity from the j of (3.7) (3.9), satisfy k
k
and, in consequence
(Here the sign is + when K = X or r and - when K = /? or /?*), together with Xi'x)* = X?(x*), Pl\x)* = Pt'x*),
Pi(x)* = pt\x*)
(IE/I),
rl(x)* = T?>x*)
A i d ) =Pl(l) = PiHV = 4(1) = 0
(*eA), if i = *.
(4.2) (4.3) (4.4)
while, by differentiating (3.10) using the quantum Ito formula and using the injectivity of the maps j , we have for arbitrary i.l.k = 0,1, x.y e A, k
A/W) = Ai(x)y+xi *(y)+Ai(i)A,*(y), /
A W ) = Pfc)y+spfo)+p(A!«)A*(y),
(4.6)
A W ) = A WJ+^A *(>)+A Wp(A *(y)), f
t,
t
ti
/
T ; W ) = <(x)y+xzf{y)+pl'(x)Pt{y). Our main result is
(4.5)
(4.7) (4.8)
207 Theorem 4.1. Let XI, p , p ', x be linear maps from A to A satisfying the structure relations (4.1) (4.8). Then there exists a 6-tuple (o,B,z,w,l,h) comprising k
k
k
(1)
an endomorphism a of the unital '-algebra A,
(2)
a <J-derivation B of A, that is a linear map B : A —» A such that, for arbitrary x,y e A, P'xy) = B(x)y+rKx)~fl(.y)
(3)
a primitive x for the 2-cocycle Hp defined by ty(*.y) = -B\x)B(y), where B*(x) = f)(x*)*, that is a linear map x: A -» A such that, x(x,y) = xr(y) + r(x)y+B (x)B(y), i
(4)
a unitary element w of A such that w* = H\
(5)
an element / of A satisfying / = w*l,
(6)
(4.9)
an odd self-adjoint element h of A,
such that, for each x e A, Xi(x) = w o\x)w' -x, i
Bg(x) = B(.x),
B?(X) = -w/sc*r - 1
T °W = 0
Tl
(4.10)
k
B \x) = w*B(x) + t,\
B^'x) = B(x*r.
BZ\X) = -pa*r'w*
p}°(x) = p(x*)*w+i*,
p}\x) =
x(x).
.
0
B,\x) = -par.
- [*,
-pa*)-',
x \x) = x(x) + (ih-il*t)x+lp(x), 0
V) = x(x)-x(ih+y*i)+pHx)i,
T , ' W = xor.
Conversely, given such a 6-tuple (a,p,x,w,l,h) the maps Xi,p' ,pl',x (4.12) and (4.13) satisfy the structure relations (4.1) (4.8). k
k
(4.12) (4.13)
defined by (4.10), (4.11),
Proof. Making the substitution X \x) = ai(x)-x, k
we find, from (4.5) and (4.1)
(4.14)
(4.4) that, for all x, y e A, i, k, I = 0,1, Orto) =
o (x)~ = tr' (JE), k
+1
*+i
0-^(1) = 1 if i = k. From these relations it follows that a =: o$ is an endomorphism of the unital *-algebra A, that w =: o"o(l) unitary, that w = w*, and that i s
ofa) = oia)oS(x)a^l)
= wW)*"*.
Hence X is given by (4.10). Setting i = k = 0 in (4.10) and applying p using (4.1) gives X}(x) = oix)~-x. On the other hand setting i = k = 1 and replacing x by i gives X}(x) =
208 wcKx)w~ -t. l
(4.9) follows by comparing these expressions.
Making the substitution (4.14) in (4.6)
we obtain ft'Wy+p(iv otJ)H'-*)j8 (>)-
B/(xy) =
,
(
t
( -!5) 4
Putting i = / = k = 0 in (4.15) we see that B =: B° is a 6-derivation of A. In particular p"o(l) = 0. By (4.1) B}(x) = -B(X)-. Putting x = y = 1 and i = / = 0, k = 1 in (4.15) and using (4.4), Q = 0 °m+#*jPi(t). 1
Let / = p o ( l ) .
B y (4.1), B°(\) =
(4.15)
and (4.15) gives - / " + * * / = 0 which gives (4.9) on applying p.
Putting i = / = 0, k = 1, Jt = 1 in (4.15) gives B(y) =
-wl+wBi(y),
that is, for arbitrary x e A B \x) = w*B(x) + l. 0
By (4.1) B \x)= a
-wB(x,--l
which completes the proof of (4.11). (4.12) now follows immediately from (4.3). Putting i = k = I = 0 in (4.8) we see that T = : T Q is a primitive for (which is a cocycle because B is a ^-derivation [3]). In particular To(l) = 0. Putting i = / = 0, k = 1 and x = y = 1 in (4.8) gives 0 = T (1) + T ( 1 ) + /*/. 1
0
0
,
We define h by setting
T '(l) = 0
m-iPf.
Then -ih-\Vl.
T,°(l) =
It follows from (4.3) that h = h* and from (4.1) that h is odd. Finally, putting i = k = 0, / = 1 and y = 1 in (4.8) gives r,°W = i(x)-x(ih
+
ll*I)-B\x)wl
and putting i = 1, * = / = 0, x = 1 and y = x gives z(.x)+(.ih-}l*l)x-wIB(x).
T \X) = 0
That xl(x) = z(£)' follows at once from (4.1). The converse may be proval by direct verification. 5.
•
Remarks
In the case of the inner flow corresponding to the solution of (2.3) the 6-tuple (a,B,T,w,l,h) Theorem 4.1 is given by <j(x) = WxW'
1
B(x) =
LX-
T(X) = i[H,x]-
i(L*Lx-L*a(x)L+xL*L)
w = WW* I= h =
-L-WW*L H-H-ii(L*WW*L-L*WW*L).
of
209 The process j (\) = UU* satisfies the equation l 0
d/o(D = Q[<sv-l)M
+
aB*l-l*waB+(ih-\l*l)at)U-\
As in the Boson case, outer flows are obtained when A possesses outer endomorphisms, for example if A = B(3C ) with H infinite dimensional. 0
0
References [1]
R L Hudson, Algebraic theory of quantum diffusions, pp 113-124 in Stochastic Mechanics and Applications, proceedings, Swansea (1986), ed A Trumda, Springer L N M 1325 (1988).
[2]
R L Hudson and K R Parthasarathy, Unification of Fermion and Boson stochastic calculus, Commun Math Phys 104, 457-470 (1986).
[3]
R L Hudson, Quantum diffusions and cohomology of algebras, pp 479-485 in Proceedings of 1st World Congress of Bernoulli Society, V o l 1, Tashkent 1986 ed Y Prohorov et at., V N U Press 1988.
[4]
J M Lindsay, Independence for quantum stochastic integrators, pp 325-332 in Quantum Probability and Related Topics VI, proceedings, Trento (1989), ed L Accardi et al., World Scientific 1991.
[5]
G Vincent Smith, Unitary quantum stochastic evolutions, Proc London Math Soc, 63, 401-425 (1991).
Quantum Probability and Related Topics Vol. VII (pp. 211-219) © 1 9 9 2 World Scientific Publishing Company FERMIONIC
211
HYPERCONTRACTIVITY
J M L I N D S A Y and P - A M E Y E R
B y r e f i n i n g the basic estimate i n [ L ] , best possible V
—> L
q
hypercontractive
stants for F e r m i second quantisation are obtained w h e n p and q',
con-
the conjugate o f
q,
are e v e n integers. T h e algebraic aspect o f the d u a l i t y transforms, between F o c k space and W i e n e r space (respectively Clifford
is further c l a r i f i e d . W e also o b t a i n an
space),
interesting property o f the O r n s t e i n - U h l e n b e c k s e m i g r o u p .
T h e paper is intended to
be s e l f - c o n t a i n e d , but m u c h o f the b a c k g r o u n d and relevant references m a y be f o u n d i n [ L ] . A n e x t e n s i v e b i b l i o g r a p h y for h y p e r c o n t r a c t i v i t y has recently b e e n c o m p i l e d ( [ D G S ] ) a n d there is a n i c e e x p o s i t o r y account o f gaussian
hypercontractivity—the
case o r i g i n a l l y treated b y E . N e l s o n — i n [ F a r ] . §1. P r e l i m i n a r i e s L e t 7" be the s y m m e t r i c measure space o f / = [0, T] or R+ w i t h its B o r e l cr-algebra and L e b e s g u e measure ( [ G u i ] ) .
T h u s F consists o f subsets o f / h a v i n g finite c a r d i n a l -
ity, for w h i c h w e s h a l l use G r e e k letters a,fi,co,..., written j ...da. Lemma:
and integration over F w i l l be
T h e f o l l o w i n g elementary i d e n t i t y i s f u n d a m e n t a l :
L e t g e L ' ( r x F ) , then // g(a,/3) dad/3 = J
I
g(a,a)dT
a a x
where a denotes the c o m p l e m e n t o f a i n r : a = F o r zeC,
z
N
tion operator X'
w i l l denote b o t h the function o n # := L ( T ) . 2
For c >
T h u s the f a m i l y {$ :
= \J$ . C
c
t\a.
x i-> z , and the c o r r e s p o n d i n g m u l t i p l i c a # T
0 let 9
C
:= {/:
N
c
& ~ c
are
dual
(exponential total i n 3 Let
C
under
vectors)
the
p a i r i n g (f,g)
: r >-> l l
N
2
X:=C\9
and
C
c > 0} i s decreasing as c increases and 9
U n d e r the n o r m g i v e n b y || • || : / i-> \\c f\\, 9 l
c feL (.F)}.
[ e t
C
X
=
9.
is a H i l b e r t space, m o r e o v e r &
j /(r)£(r)dr.
¥>(').
are
The
product
c
functions
l i n e a r l y independent
and
f o r e a c h c > 0. L e t <S be their linear span. +
be a c o m p l e x H i l b e r t space o n w h i c h a c o n j u g a t i o n (conjugate l i n e a r , isometric
i n v o l u t i o n ) K is defined.
C a n o n i c a l l y associated w i t h the p a i r (<&,/ST) are
and
212 (i) the G a u s s i a n space i n d e x e d b y the real H i l b e r t space 8. := [v e >fj: Kv = v] and (ii) the c o m p l e x ( w e a k l y c l o s e d ) C l i f f o r d algebra o v e r
denoted
%($, )
M
K
d
8
( i , A T ) r e s p e c t i v e l y ( [ S i l ] , [ G r 1]).
(&,K),
O n the other h a n d , the F o c k
functors c a n o n i c a l l y associate to the p a i r (•&,/?) ( i ) ' the s y m m e t r i c F o c k space o v e r
denoted &($,K)
respectively ([Coo]).
T h e s e associations d e t e r m i n e natural
isomor-
phisms
where L ( ( g ^ , 2
i J f
) ) i s I . E . S e g a l ' s n o n - c o m m u t a t i v e L - s p a c e w i t h respect to the nor2
m a l i s e d trace o n the C l i f f o r d a l g e b r a ( [ S e l ] ) .
A contraction C between
conjugated
H i l b e r t spaces ($,-,£,-) i = 1,2, w h i c h i n t e r t w i n e s K\ a n d K2, determines contractions
0 (C): ±
L C S ( $ , , £ , ) ) - * Ll^tf>2,Kz))
w h i c h respect the ' r e a l ' structure, t h r o u g h Bose
2
(respectively F e r m i ) second q u a n t i s a t i o n .
M o r e o v e r 0 (C)
restricts f o r p € ( 2 , ° ° ] ,
±
and extends f o r p e [ 1,2), to a c o n t r a c t i o n b e t w e e n the c o r r e s p o n d i n g / / - s p a c e s (see e.g. [ L ] ) . When
= L\l) and K is p o i n t w i s e c o m p l e x c o n j u g a t i o n b o t h
be identified with 9 = L (T) as f o l l o w s .
^
&($,K)
F o r / e 9 let / * = / ( 0 ) a n d let f*
2
0
:l
n
m
a
v
-» C
be the (anti-) s y m m e t r i c e x t e n s i o n o f (si
when
s ) i-» f{[si,...,s }) n
then the correspondence
n
< . . . < s„,
( / „ ) < - » / < - > ( / " ) i s a n i s o m o r p h i c one. +
O n the other hand
we m a y take the standard W i e n e r p r o b a b i l i t y space f o r @j$ j n i a n d the C l i f f o r d algebra
of
operators
a/p + a^-.fe
acting
on
L ( / ) } , for S ^ J C ) . R
t i o n a n d c r e a t i o n operators.
+
+
lp
ip
r
e
a
" d
9 —> L :=
which
2
is
generated
by
[c,p :=
are the u s u a l ( F e r m i ) F o c k annihila-
a
W e s h a l l w r i t e If
A convenient description o f D : D :n r^e
H
&($,K) = ^ e
±
f o r the c o r r e s p o n d i n g
//-spaces.
i s i n terms o f stochastic e x p o n e n t i a l s : (p W-\j
(psL (I).
O n the other h a n d D ~ is the inverse o f the m a p d e t e r m i n e d b y
2
(1.1)
213
ft"' %,
c
^i— 9n 0
c
c
S
n
where # 0 i s the F o c k v a c u u m vector: f r t f ^ I 0 An
appealing
for
notation
e
N
-
0-2)
eL
^ ^ otherwise. T
the i s o m o r p h i s m s
Z3
is
1
i n terms
of
multiple
Wiener/Clifford integrals: D :f^j^f( )dW , +
T
D~:fr->j
T
e m b o d y i n g the property o f chaos completeness
f{t)dC . x
o f the W i e n e r a n d C l i f f o r d processes
([BSW]). §2. F o c k space estimates W e want to estimate i n t e g r a l - s u m c o n v o l u t i o n products. function p :
Fxrxr
—> C w h i c h is b o u n d e d
F o r z e C and a measurable
i n m o d u l u s b y 1, a product
f*g
is
defined f o r suitable c o m p l e x v a l u e d functions / , g o n F b y
f*g:r>->
£
p(a,a,a)f(auco)g((ova)z 'dco.
j
act
2#a
p
Precise c o n d i t i o n s o n a p a i r (f,g) for f*g below.
to be defined are g i v e n i n P r o p o s i t i o n 2.1
W h e n z = 1, a n d p i s i d e n t i c a l l y equal to o n e , this i s the Wiener product
introduced i n [ M a a ] .
W h e n z = 1 and p(a,ca,P)
=
(_I)»(«U«,*M»
#{(a,6) e a x / J : a > b], the Clifford product i s obtained ( [ M ] ) .
the s y m m e t r i c a n d a n t i - s y m m e t r i c (exterior) products r e s p e c t i v e l y . analysed i n [ L P 1]. W h e n z i s p o s i t i v e , z
2
W
H
E
R
E
„
(
A
3) _
<
P u t t i n g z = 0 gives F u r t h e r cases are
m a y be thought o f as a v a r i a n c e — w e shall
therefore w r i t e a f o r | z |. Proposition
2.1: If / e ^
belongs to X' = \J 9 . C
a
and j e S j
where ab > a
2
then f*g
i s defined a n d
Moreover
c>0
|| */**fl * I V c W C
for e a c h c S 0 a n d 5 > 0.
+ c*)"/! lVc
2
+
J
- V
+
c /«H
(2.1)
2
I n p a r t i c u l a r * is w e l l - d e f i n e d o n e q u i v a l e n c e
classes
(modulo n u l l functions). Proof:
A p p l y i n g the C a u c h y - S c h w a r z i n e q u a l i t y i n L (r)
a n d i n C , a n d the r
2
L e m m a (three times) together w i t h the b i n o m i a l identity £ act
x
*<*y#a _ ( + yf x
T
g
i
v e
s
214
\\c f*8\\ N
«/dTc
2 # T
[
/da(cr^)
S
# a
\f(auco)\(a/^f \g(mua)\ a
i
[ OCT
s / d r c H "
I . act
^-
£ /dr[c (l+.r )]* 1
2
=
//dad
f a
= / dT[c (s 2 +
{/(to(A) "* |/(auffl)| /dv(o »- ) i
I
T
[c (l+5- )] V 5) ^ 1
2
#
2
5( T £
2
#
+
# a
2
S
2
2
,
# v
12 |«(vua)| }*'
)'V"l/(aw)| /dv(ffVy |s(vua)| 2
v
2
2
^ (01
+
2
|/(auffl)| //d/3dv[c (l+i- )] V i" ^ 2
1
2
2
C ) >|| ||Vc
2
a
d T ' t c ^ l + j - ^ + o- *- ]* '!^')!
2
2 A
2
t o
/to(
+ l) + o s]*VW\ j
2
= ||Vc
< t e
2
+
C
2
1
1
#
2
1
2
)^|| . 2
• C o r o l l a r y 2.2: L e t / e 9 (i)
For
M, y, w
a n d j e S j where
a
2
> o" , 2
^ provided u (ii)
>
+o
N
f * g \ \
^ w
1
* l l V ^ / f l
(2.2)
I N ^ f f i
.
For c e R, *
flc /**| W
p r o v i d e d (a - c )(b 2
Proof:
2
c)
2
P u t t i n g c = ^w-a \\^ f*g\\ sN
f r o m w h i c h (2.2) f o l l o w s .
2
2
> (c + rj ) . 2
2
2
a n d s = u/w-l i n (2.1) g i v e s « l l V ^
/ l l
w
P u t t i n g s = (a -c ) 2
N
l^w^-u- )- -^!! 1
1
I(c + o ) i n (2.1) g i v e s
2
l k " / * g | | « \\a f\\ | | V w h i c h i m p l i e s (2.3).
(2.3)
2
c
2
2
+(c +o- ) /( 2
2
2
a
2
-c ) 2
w
g
||
215 (2.1), (2.2) a n d (2.3) are a c t u a l l y e q u i v a l e n t , h o w e v e r i t is the f o r m (2.2) that i s ripe for generalisations to products o f m o r e than t w o f u n c t i o n s . Theorem uf
2.3:
+ . . . +v~
l
L e t fi e 9^ ._^i v
for
i = l
where
n
U\,...,v
> a
n
and
2
< a~ , then h := / i * ( . . . * ( / „ _ i */„)..-) i s defined a n d b e l o n g s to X'.
l
2
In fact i f w » a
2
then
* n
\\J^ fi\\
(-)
N
2
4
i=i as l o n g as w Proof:
_
1
> u f +
...+v~ . l
Iteration o f (2.2) y i e l d s the result.
• (2.4) i s a c t u a l l y v a l i d f o r a n y b r a c k e t i n g i n the p r o d u c t ; a s s o c i a t i v i t y o f * m a y be characterised b y a f u n c t i o n a l i d e n t i t y for p ( [ L P 1 & 2]). C o r o l l a r y 2 . 4 : I f fe\J[9
: a > C N / H - 1 ) then the n - f o l d p r o d u c t / * ( . . .
a
*(/*/)...)
lies i n X', a n d satisfies \\c f* . . . */|| « ( | | V « c N
2
+
(«-l)a
z A
>|) .
(2.5)
B
F o r the W i e n e r p r o d u c t (p = l , z = 1) the p r o d u c t functions p r o v i d e m a x i m i s e r s f o r (2.4). P r o p o s i t i o n 2 . 5 : L e t p = l , z = 1 a n d let w,vi,...,v w~ . l
T h e n e q u a l i t y i s r e a l i s e d i n (2.4) i f f
t
= n
Vj
2 1 satisfy t > f + . . . + u „ !
n
_ 1
=
where q>i = v~
some p e L ( / ) . R
Proof:
U s i n g the r e l a t i o n s
Tt^Tt^eK,^),
IM
2
=
E
W
|
2
(-) 2 6
this i s e a s i l y v e r i f i e d .
•
216 §3. D u a l i t y a n d h y p e r c o n t r a c t i v i t y W e first w i s h to tighten the algebraic correspondence under the d u a l i t y transforms D between F o c k space and W i e n e r space (respectively Clifford
([LM]).
space)
±
Viewing
as a subspace o f 9, it m a y be seen b y i n s p e c t i o n that
L (I) 2
c f=9*-f
f
9
L e t S~ be the l i n e a r span o f {S ) 0
seen that @~ is dense i n each
o
1}
r
c
u [fi
(3.1)
* _
*
It i s easily
2
n
9. C
T h r o u g h o u t this section z = 1 and p{a,to,p)
= ( iy(<* ~"».<»u/J) ±
,
)
w
i
m
c
o
r
r
e
s
p
o
n
d
i g n
products denoted b y * . ±
T h e o r e m 3 . 1 : L e t v\,...,v
> 1.
n
(i)
I f w :=
> 1 then the maps
...+V' )' 1
1
(fi,...J )e9^^x...x9^^^
f * ...* f e9^zi
n
1
±
±
n
are c o n t i n u o u s . (ii)
I f fisSF^pi
for i = ! , . . . , «
where ti£" + . . . + » „ "
S ^ , then /
*+ . . . * + / „ e 9
t
and (3.2)
D (f * ...* f )=D (f )...D (f ) ±
Proof: (ii)
l
±
±
± n
±
l
n
(i) i s c o n t a i n e d i n T h e o r e m 2 . 3 .
T h e v e r i f i c a t i o n o f (3.2) f o r / j e ( S i s s t r a i g h t f o r w a r d u s i n g (2.6) a n d (1.1). T o +
+
p r o v e the general statement w e e x p l o i t the density o f S
+
each i let (ff ^)
be a sequence i n 8
m
(D (ff )) +
D (/ +
m)
1
( m )
is )...Z) (/ +
almost n
( m )
everywhere
convergent.
) converges to D (f )...D (f ) +
in 9
C
f o r every c > 0. F o r
w h i c h converges to ft i n 9^z\
+
+
l
also c o n v e r g e s to D ( / i *+ . . . * + / „ ) i n L \ . +
n
Then
D {f[ +
m)
for which
* ... */ +
+
(m) n
) =
almost e v e r y w h e r e a n d , b y (i), it
T h e result f o l l o w s .
( i i ) _ T h i s m a y be p r o v e d b y a f u n c t i o n a l a n a l y t i c m o d i f i c a t i o n o f the a b o v e p r o o f . B y (3.1), (1.2) and the a s s o c i a t i v i t y o f * _ , (3.2) holds f o r / , e S~. N o w i t i s sufficient to c o n s i d e r the case w h e n n = 2, since iteration w i l l then g i v e the result f o r any n. Thus let fe9^zx,ge9^zi
where i r ' + i ; -
1
=S |. I f (/<">),(g< >) m
are sequences i n S'
217
w h i c h approximate / and g i n
^
r e s p e c t i v e l y t h e n , f o r each m ,
9^z\
- > 0 as n ->
so that D " ( / cation £r(/
( m )
( m )
)D-(«
operator
) = D-(f *_g )
( n )
(m)
on
Lp-^fi^
L ,
is
closed,
so
*_g).
(m)
Z)
(g)
B u t the left m u l t i p l i ¬
lies
i n its
domain
N o w let m —> oo and repeat the above
n)
r e p l a c i n g Lp-tfi^
-> D~(f
w
) L > ~ ( g ) = D-(f( *_g).
»,
w i t h the right m u l t i p l i c a t i o n operator Rp'd)
t
o
and
argument
o b t a i n the result.
• T h e f o l l o w i n g p r o p e r t y o f the O r n s t e i n - U h l e n b e c k s e m i g r o u p is another consequence of our b a s i c estimate (2.4). P r o p o s i t i o n 3.2: T h e n for FeL , p
L e t (P,)
b e the O m s t e i n - U h l e n b e c k s e m i g r o u p o n W i e n e r
where p~
GeL%
+
l
+ q~
\P
as l o n g as ( 1 + e " ' ) " Proof:
> (1+e~T
1
1
« \\PrF\\L\\\PsG\\ \
(3-3)
L
+ ( l + 0
-
1
-
S i n c e the O m s t e i n - U h l e n b e c k s e m i g r o u p o n F o c k space is ( e
v a l i d for F, G eD (X)
b y T h e o r e m s 2.3 and 3.1.
+
density o f D (X) +
in L
+
space.
< 1,
l
- t W 2
) , (3.3) is
T h e f u l l result f o l l o w s thanks to the
f o r p > 1 and H o l d e r ' s i n e q u a l i t y .
• : 2 $ p « oo} a n d { £ > * ( # , . ) : c > 1}
[L
p ±
provide
two increasing families of
norms
w h i c h we c o m p a r e n e x t . P r o p o s i t i o n 3.3:
L e t p > 2 and / e 9, then
at least i f p i s an e v e n integer, so that L+
+ e
D (^^rr) ±
c L
p
t
.
I n p a r t i c u l a r , L> * (^VT+c^
at least for c e ( 0 , 2 , 4 , . . . ) .
Proof:
L e t / e ^ / T + 2 a where U E N
then, b y C o r o l l a r y 2.4 and T h e o r e m 3.1,
1
c
218
... * ± / ) | | 2 ( l ) -
= \\D (f*±
+a
±
=
ll/*±-*±/ll
S
(\\f\\^2af
folQ
Product
+a)
w h i c h i m p l i e s (3.4).
• L - b o u n d s o n n - t h order chaos were also f o u n d i n [ B o n ] . p
O u r final result concerns
h y p e r c o n t r a c t i v i t y o f the second q u a n t i s a t i o n m a p s . T h e o r e m 3 . 4 : L e t C: &i —» #2
b e a c o n t r a c t i o n i n t e r t w i n i n g c o n j u g a t i o n s Ki and
K2 w h i c h satisfies \\Cf$(q-l)/(p-l) where
T h e n 0 (.C)
q,pe[l,~].
is contractive L ( « f ?
±
# l i J f l )
) -> L ^ O S ^ j ^ ) ) — a t
least w h e n p a n d o ' are e v e n integers. Proof:
A s e x p l a i n e d i n [ L ] i t i s sufficient to c o n s i d e r the case w h e r e (#i,ATi) =
(£2.^2) =
( L ( / ) , c . c ) . F u r t h e r m o r e , since
±
1
p
contractive for
each p ( w h e n C * 0 ) , i t suffices to c o n s i d e r the case w h e r e C i s a m u l t i p l e o f the i d e n t i t y a n d thus 0 (C)
i s o f the f o r m c .
±
N
I f p =s 2 a n d ( q )
2
< (p-l)
_ 1
then, by
(3.4),
« 11/11 T a k i n g adjoints and e x p l o i t i n g L -L q
=
\\DHJ1\\L\.
duality, w h i c h holds also for non-commutative
q
L - s p a c e s , gives ?
\\c^D (f)\\L\<WD^nui ±
w h e n e v e r C2 « {q' - 1 )
_
1
= ( < ? - ! ) . P u t t i n g these together g i v e s the r e s u l t .
219 Remarks:
1. T h e F e r m i o n i c case o f this theorem lends further support to the conjec-
ture o f L . G r o s s ( [ G r 2 ] ) . 2.
I n finite d i m e n s i o n s the e v e n integer case o f gaussian h y p e r c o n t r a c t i v i t y y i e l d e d to
quite crude e s t i m a t i o n ([Nel]) together w i t h a c o m b i n a t o r i a l estimate ( [ S i 2]). 3.
Since
||# (C)e ||* = exp{(p-l)||c>|| } for p e [ l , ° o ) , stochastic +
2
v
p r o v i d e m a x i m i s e r s for < P ( C ) as maps L% —> L . +
exponentials
T h e converse question o f i d e n t i f y -
p +
i n g a l l the m a x i m i s e r s i s harder. It is s h o w n i n [ B r L ] , and also i n [ C a L ] , that w h e n # is finite d i m e n s i o n a l a n d c
= (q-
Gaussian.
[ L i e ] for a recent c o m p r e h e n s i v e
2
We
recommend
l ) / ( p - 1 ) , the L
q
-» L
p
m a x i m i s e r s for c
N
are a l l
analysis o f this and
related questions.
References [BSW]
C . B a m e t t , R . F . Streater and I . F . W i l d e : T h e I t o - C l i f f o r d i n t e g r a l , / . Anal.
[Bon]
A . B o n a m i : E t u d e des coefficients de F o u r i e r des fonctions
de L (G),
H . J . B r a s c a m p and E . H . L i e b : B e s t constants i n Y o u n g ' s i n e q u a l i t y , its c o n verse, a n d its g e n e r a l i z a t i o n to more than three functions, Adv. (1976)
[CaL]
Ann.
2
22 (1970) 3 3 5 - 4 0 2 .
Inst. Fourier [BrL]
Funct.
48 (1982) 1 7 2 - 2 1 2 .
in Math.
20
151-173.
E . C a r l e n and M . L o s s : E x t r e m a l s o f functionals w i t h c o m p e t i n g /. Funct. Anal.
symmetries,
88 (1990) 4 3 7 - 4 5 6 .
[Coo]
J . M . Cook:
[DGS]
E . B . D a v i e s , L . G r o s s and B . S i m o n : H y p e r c o n t r a c t i v i t y : a b i b l i o g r a p h i c a l
Soc.
T h e mathematics
o f second quantisation, Trans.
Amer.
Math.
74 (1953) 9 - 3 9 .
r e v i e w , in " H o e g h - K r o h n m e m o r i a l conference p r o c e e d i n g s , 1 9 8 9 " . [Far]
W . G . F a r i s : P r o d u c t spaces and N e l s o n ' s i n e q u a l i t y , Helv.
[Grl]
L . Gross: Existence
Phys. Acta.
48
(1975) 7 2 1 - 7 3 0 .
Anal. [Gr2]
and uniqueness
states, / .
Funct.
10 (1972) 5 2 - 1 0 9 .
L . G r o s s : H y p e r c o n t r a c t i v i t y and l o g a r i t h m i c S o b o l e v inequalities for C l i f f o r d - D i r i c h l e t f o r m , Duke Math.
[Gui]
of physical ground
A . Guichardet: "Symmetric L N M 261
(1972).
the
J. 42 (1975) 3 8 3 - 3 9 6 .
H i l b e r t spaces and related t o p i c s " .
Springer
220 [Lie]
E . H . L i e b : G a u s s i a n kernels have o n l y g a u s s i a n m a x i m i s e r s ,
[L]
Inventiones
(to appear).
Math,
J . M . L i n d s a y : Gaussian hypercontractivity
revisited, / .
Funct.
Anal.
92
(1990), 3 1 3 - 3 2 4 . [LM]
J . M . L i n d s a y and H . M a a s s e n : D u a l i t y t r a n s f o r m as
*-algebraic
p h i s m , in " Q u a n t u m p r o b a b i l i t y and a p p l i c a t i o n s V : P r o c e e d i n g s , 1 9 8 8 " (ed. L . A c c a r d i , W . v o n W a l d e n f e l s ) .
isomor-
Heidelberg
S p r i n g e r L N M 1442
(1990)
247-250. [ L P 1]
J . M . L i n d s a y and K . R . Parthasarathy: C o h o m o l o g y o f p o w e r sets w i t h a p p l i cations i n q u a n t u m p r o b a b i l i t y , Commun.
[ L P 2]
Math.
Phys. 124 ( 1 9 8 9 ) 3 3 7 - 3 6 4 .
J . M . L i n d s a y and K . R . Parthasarathy: R i g i d i t y o f the P o i s s o n c o n v o l u t i o n , in " Q u a n t u m p r o b a b i l i t y and a p p l i c a t i o n s V : P r o c e e d i n g s , (ed. L . A c c a r d i , W . v o n W a l d e n f e l s ) .
[Maa]
Heidelberg
1988"
S p r i n g e r L N M 1442 (1990) 2 5 1 - 2 6 2 .
H . M a a s s e n : Q u a n t u m M a r k o v processes i n F o c k space d e s c r i b e d b y integral k e r n e l s , in " Q u a n t u m P r o b a b i l i t y a n d A p p l i c a t i o n s II: P r o c e e d i n g s , H e i d e l berg 1 9 8 4 " , (ed. L . A c c a r d i and W . v o n W a l d e n f e l s ) .
S p r i n g e r L N M 1136
(1985) 3 6 1 - 3 7 4 . [M]
P-A.
Meyer:
Elements
de
probabilites
quantiques
I,
in
"Seminaire
de
P r o b a b i l i t e s X X " . S p r i n g e r L N M 1204 (1986) 1 8 6 - 3 1 2 . [Nel]
E . N e l s o n : Q u a n t u m fields and M a r k o v fields, in " A M S s u m m e r institute on p a r t i a l d i f f e r e n t i a l equations, B e r k e l e y 1 9 7 1 " .
[Seg]
I.E. Segal: A non-commutative
extension o f abstract i n t e g r a t i o n , Ann.
[Si 1]
B . S i m o n : " T h e P(
57 (1953) 4 0 1 - 4 5 6 ; C o r r e c t i o n , Ann. Math.
Math.
58 (1953) 5 9 5 - 5 9 6 .
2
i n P h y s i c s , P r i n c e t o n , N J , 1974. [ S i 2]
B . S i m o n : A r e m a r k o n N e l s o n ' s best h y p e r c o n t r a c t i v e Amer. Math. Soc. 55 (1976), 3 7 6 - 3 7 8 .
M a t h e m a t i c s Department Nottingham University University Park Nottingham N G 7 2 R D U.K.
estimates,
Proc.
D e p a r t e m e n t de M a t h e m a t i q u e U n i v e r s i t y L o u i s Pasteur 7 rue R e n e
Descartes
6 7 0 8 4 Strasbourg FRANCE
Quantum Probability and Related Topics Vol. VII (pp. 221-249) © 1992 World Scientific Publishing Company
221
O n a class of generalised Evans-Hudson flows related to classical M a r k o v processes by A. Mohari and K.R. Parthasarathy Indian Statistical Institute, Delhi Centre 7.S.J.S. Sansanwal Marg, New Delhi 110 016
1
Introduction
Let (G, fi) be the measure space where G is a locally compact second countable topological group and /i is afixedleft invariant Haar measure on G. Consider a complex valued Borel measurable function ton G x G and the associated and not necessarily bounded operator (Qt
Qi
on the Banach space L°°(G)
defined by
I \I{*,V)\\MM*) Ja
=
I «(*,»WW-^))<»
(i i)
Ja
where dy = dp(y) and q (x,y) t
= \t(x,yx- )\ A(x- ), 1
3
(1.2)
i
A being the modular homomorphism on G satisfying u(Ex) — /J(£)A(Z) for all Borel sets E C G and elements x € G. If / satisfies the condition {H ) 0
es6.sup J \i{x, y)| = ess.sup j q {x, y)dy < oo J
te0
l60
t
then Qi is a bounded operator on £ (C7). Furthermore Qi is the generator of co
a uniformly continuous Markov semigroup with state space G. It was shown in [10] how the associated Markovflowcan be realised as an Evans-Hudson flow on the abelian von Neumann algebra
L°°(G).
This is the end result of a series
222
of notes [7-11]. In the present paper we continue this analysis by dropping the hypothesis (Ho) and investigating the more delicate problem under hypotheses of the following kind : c (K) = ess.sup{/ q (x, y)dy, x € K) < oo
(ffj)
t
0
(
for every Borel set K C G with compact closure; (H )
d (K) = ess.sup{ J q (x, y)dy, x € G} < oo
2
t
K
t
for every Borel set K C G with compact closure; (H )
lim _ f qt(x, y)dy = 0 for every compact set K C G,
3
I
00
K
oo indicating one point compactification ; (Hi)
l(x, y) is continuous in x for everyfixedy .
We emphasize that the present analysis is only a modest beginning. However, it seems to give rise to many interesting questions on the relationship between classical Markov processes and Evans-Hudsonflowswith not necessarily bounded structure maps in the context of Feller's boundary theory.
2
Preliminaries on quantum stochastic calculus
We briefly recall some of the basic results of quantum stochastic calculus [4,6,12] in the new notation developed in [10,11]. All the Hilbert spaces referred to in this paper are complex and separable with scalar product < •, • > which is linear in the second variable. If H is any Hilbert space B(K) is the algebra of all bounded operators on H. If Hi, i = 1, 2 , n are Hilbert spaces and «; € Hi for each i the product vector If
<S>"=\ i u
®"=i % ' often denoted u u ...u„.
^
8
1
is an operator in Hi with domain D the operator X\ in ® " t
domain {iijvluj £ ampliation
of
X\
Di,v£
®" Hi)
to ®"=i
Hi.
satisfying
m3
If
Xi
Xi(uiv)
is an operator in
= (XiUi)v Hi
=1
2
Hi with
is called the
with domain
Di
its
ampliation Xi to ®y=i Hj is defined by using the natural isomorphism between
223 the H i l b e r t spaces
= 1
a n d H; ® U
Hj
® • • • ® 7f;_, ®
x
® • • • ®
ft,,
when
1. M o s t o f t e n we d r o p t h e " a n d d e n o t e a n o p e r a t o r a n d i t s a m p l i a t i o n b y
t h e s a m e s y m b o l . I n p a r t i c u l a r i f Xi i s a n o p e r a t o r i n Hi for e a c h i = w e m a y d e n o t e XiX
• • • X„
3
b y XiX?...X
1 , 2 , n
i f there is no c o n f u s i o n .
n
L e t h i & be H i l b e r t spaces a n d let 0
® T(L (1R )
® k) ,
H
=
h
h
=
/lo ® (dfe-oo © ifc ©
^
h
=
n ® T ( L ( i R ) ® Jt)
ft
2
0
+
© fc ® A; © h
0
0
2
(2.1)
(ttoo)
;
0
(2.2) (2.3)
+
where F indicates boson Fock second q u a n t i z a t i o n , e , ±
Ge
± 0 0
are unit vectors
and
a r e c o p i e s o f t h e o n e d i m e n s i o n a l H i l b e r t s p a c e <S. I f (X, T, v) is a m e a s u r e
s p a c e a n d H i s a H i l b e r t s p a c e w e v i e w t h e H i l b e r t s p a c e L?(y)®H as a s p a c e o f W - v a l u e d f u n c t i o n s o n X respect to
frequently
which are n o r m square integrable w i t h
v.
Choose
a n d fix a n o r t h o n o r m a l b a s i s {e;, i € S}
b a s i c n o i s e p r o c e s s e s {A'-, i,j
6 S U {— oo, o o } }
i n it a n d i n t r o d u c e
the
of q u a n t u m stochastic calculus
as d e f i n e d i n [10] b y
=
A|
ALoo
=
A |
A r
=
A|
A!U<)
=
A| _
=
A ^ = 0 f o r a l l i, j <= 5 U { - c o ,
A"
w h e r e Aj, i,j
i,j€S,
Aj
0
0
e j > < e i
e
_
-
>
e i > < e
t
|, <
e
j
| =
„| =
o o > < e o o
Aj,
j €
S,
A j , i e 5 ,
|(t) =
0 oo}
€ S a r e t h e c o n s e r v a t i o n ( o r e x c h a n g e ) p r o c e s s e s a n d Ai, A], i € S
a r e r e s p e c t i v e l y t h e a n n i h i l a t i o n a n d c r e a t i o n p r o c e s s e s a l o n g e,.
224 Let D
„„(*) M
0
C h be a linear manifold. Introduce 0
=
/ ' ( l + IM<)l| )<*«. Jo
neL (R )®k,
=
[u\u G L ( i R ) ®
< u(i), i > =
2
a
2
(2.4)
+
e
+
identically int
0
f o r a l l b u t a finite n u m b e r o f i G S}, A4
0
=
{u|u i s a s t e p f u n c t i o n o n 2 R
+
with
(2.5) compact
support and taking values i n the linear manifold generated by{e„.GS}}.
(2.6)
L e t L = { L J , i , j G S U { — o o , o o } } b e (D ,M)
- a d a p t e d processes i n H w i t h
0
L'j = 0 w h e n e v e r i = o o o r j = — oo a n d
/ 'i:ii^(«)M«)ii ^.(»)< 2
o
for a l l / € f o ) u.
u
oe
€ .M,< > 0,e(u) being the exponential vector associated with
W e t h e n s a y t h a t L G E,(D ,M)
a n d L is s t o c h a s t i c a l l y i n t e g r a b l e . B y
0
t h e m e t h o d s o f q u a n t u m s t o c h a s t i c c a l c u l u s t h e r e e x i s t s a (D , 0
M)-adapted
process Af, satisfying
A ( 0 ) = 0 , d\
L
L
(See,
for example
= Y)M L
d
w
» h r e s p e c t t o (D , M) l
0
, [8] , S e c t i o n 2 7 i n [12]).
With
(2.7)
t h e help of ampliations
i n t r o d u c e the o p e r a t o r s L ( t ) i n H (See (2.1) - (2.3)) b y
W)
= E ^ W N •J
>< >\> e
(2.8)
LeE(D ,M). 0
D e n o t e b y F t h e u n i q u e u n i t a r y flip o p e r a t o r i n h s a t i s f y i n g
j F / e - o o = f e , Ffe x
for a l l /
x
= fe-^,
Ffu
=
fu
G h , u G k. L e t { £ } , ( £ } ) ' } b e a n a d j o i n t p a i r o f (D , A 4 ) - a d a p t e d Q
0
p r o c e s s e s i n Ti w i t h L'j — 0 w h e n e v e r t = o o o r j = — o o . F o l l o w i n g B e l a v e k i n
225
[1] define L = {(L )}} by h
k
(L%
=[{LiY
if
i,j€S,
if i = — o o , j 6 S, if j
= oo,«' €
if i = — o o ,
5,
j = oo,
otherwise . Suppose that L, L € E(D , M). Then {A , AL } is an adjoint pair of (D , A i ) b
0
L
6
0
- adapted processes in H. If L(t) is defined by (2.8) then L\t) = F(£L'(<)»|e, >< \)F ei
>J
where appropriate ampliations are used. We are now ready to state the first and second fundamental formulae of quantum stochastic calculus. Proposition 2.1 : Let
L, M € IL(D , M). 0
Suppose
X ( i ) = X(0) + A (i),Y(i) = Y(0) + A (t),< > 0 L
Af
are (D , A^)-adapted processes where A^,, AM are defined by (2.7). Then for all 0
f,g € D , it, v € M, t > 0 the following relations hold : 0
(i)
< fe(u),
X(t)ge(v)
>=<
/e(«), X(0)ge{v)
+ / „ ' < / ( e ^ + u ( « ) ) e ( « ) , L(s)g{v{s)
>
+ e«,)e{v)
> ds
226 (ii)
< X{i)fe(u),
Y(t)ge{v)
>=<
+ /„'{<
X{0)fe(u),
Y{0)ge(v)
+ « ( « ) ) < « ) , M(«) (t;(-) + f l
>
)e(v)
>
eoo
+ < FL(«)F/(e_
00
4- u(»))e(«), r(»)fl(«(«) + e )e(») >
-I- < FL(»)F/(e.
M
+ «(«))e(ii),
M
M(a) (t>(*) + fl
e^e^)
>}da
where L(i), M(«) are defined by (2.8). Proof : This is a straightforward translation of the two fundamental formulae of quantum stochastic calculus in terms of the operators L(t), M(t) in the enlarged Hilbert space Ti. defined by (2.3). (See, for example, Section 27, [12]). • Corollary 2.2 : Let ||[X(t) 2
{X(t)}
be as in Proposition 2.1. Then
XMlfeiu)]]
2
<
2 -<"> / "
+ e )e( )|| ri,
e
TO
Jti
2
U
for all 0 <
Proof : This is once again a basic identity which is derived from (ii) in Proposition 2.1 by putting L = M and X — Y. Once again we refer to Section 27 , [12]-
•
Remark : In Proposition 2.1 it is interesting to note the coordinate-free expressions on the right hand sides of (i) and (ii). For example, consider t h e algebra 1(A) =
{L\L
€
=
B^Lfe^
closed under the involution L
—» L
b
L'fe^
where
= 0 for all / in
b
0
fe^Lgej
0
l{h)
is
Define the operators
L = FL*F.
L) € B(h ) by the identity < f,L)g >=<
h }.
>,/,
0
227
for each i,j G S U {—oo, oo}. Then L = {Lj} can be interpreted as an element of Bj(h ,
With this understanding we obtain the following proposition as
Ai).
0
an immediate consequence of Proposition 2.1 and Corollary 2.2. Proposition 2.3 : Let L - adapted process to
AL
Then there exists a unique (h ,
G I{h).
0
in H satisfying A i ( 0 )
2
+
Jb)
with respect
^L'jdAl
for every orthonormal basis {e<,i € 5 } . Furthermore
(h ,A4) 0
< /e(«), A (t) e(v) >= / ' < f(u(s) L
for all t
= 0,
L (jR ) ®
+
5
> 0, f,g G
A , «,»£ 0
eo
a
® Jt. If Af
L {IR ) 2
e_ )e(u), L (v(«) +
+
G 1(A)
)e{v)
eoo
> d«
is another element then
< A (<)/ (u),A (t) e(v) > £
e
w
fl
= /„•{< A («)/(e_ x
0o
-I- u(«))e(u), Af(«) (»(«) + )e(v) > 5
u(«))e(u),
+ < FLFfie-^
+
+ < FLFfie^
+ u(*))e(u),
eoo
A (»)g(v(s)
+ e )e(v)
M
Mg(v(a)
for all t > 0 , / , g G A , u, v G L (R ) 2
0
>
x
+ e„)e(v)
>}ds
® Jt.
+
Proposition 2.4 : Let L G 1(A). The stochastic differentia] equation U(0) = l,dU = (dA )t/ with respect to(A„,A<) £
has a unique unitary operator valued (A , A4 (-adapted process as a solution if 0
and only if L + L + LL b
b
= L + L + LL = 0 . b
b
In such a case dU* = U*dA t L
with respect to (A , M) 0
.
The process U is independent of the basis {e,,i G 5 } and \\[U*(h) - D * ( « i H / « W | | for all 0 < ti < i
2
a
< 2 e - < " > | | ( « ) | | / " \\L»Hu(s) +
< oo, u G L ( 2 R + )
e
2
e )\\ d, 00
1
228 P r o o f : T h i s is m o s t l y a restatement o f t h e basic result o n u n i t a r y evolutions i n F o c k s p a c e f r o m [4]. S e e a l s o [8j a n d S e c t i o n 27 i n [12].
3
A quantum stochastic differential equation with unbounded coefficients
W e shall n o w exploit the results of Section 2 i n t h e special case w h e n h
0
L (G) 2
i n (2.1) - (2.3) where G is a locally c o m p a c t second c o u n t a b l e
= k—
topological
g r o u p w i t h a l e f t i n v a r i a n t H a a r m e a s u r e /J a n d L i s a n o t n e c e s s a r i l y
bounded
o p e r a t o r i n h of a s p e c i a l f o r m i n P r o p o s i t i o n 2.4. Define t h e u n i t a r y o p e r a t o r U i n L {G 2
x G ) by
(Uf)(x,v)
D e n o t e b y L\{G),
1
2
L ( G ) f o r L (G).
x G).
L (G
2
2
V
( G x G) .
2
(3.1)
x G ) the respective linear manifolds of functions with
L (G
c o m p a c t s u p p o r t i n L (G), 2
f G L
= f(y- z, )
2
F i x a n o r t h o n o r m a l b a s i s {e;, i G S } C
R e c a l l i n g f r o m ( 2 . 2 ) t h a t h = L ( G ) © L (G 2
2
e x p r e s s a n y e l e m e n t / G h as a 3 - c o m p o n e n t c o l u m n
x G) ©
L (G) 2
vector
\
/=
/o
, /
±
G i
2
( G ) , /
0
G L
2
( G x G )
(3-2)
/+
L e t i(x,y),x,y (Ht)
G G be a complex
valued Borel function satisfying condition
o f section 1 where q is g i v e n b y (1.2).
o p e r a t o r t:
t
L ( G ) - + L (G 2
2
Denote by the same letter / the
x G) g i v e n b y
f GL (G)
{if)i*,v)=l{x,y)f{x),
2
(3.3)
with m a x i m a l d o m a i n a n d /* its adjoint.
O n c e a g a i n f o l l o w i n g B e l a v e k i n [1]
i n t r o d u c e t h e o p e r a t o r s L = L(t)
= L(£)
and L
b
k
i n h v i a the matrices with
229
operator entries L = L(t) =
0
-f
0
U-l
0
b
h
(3.4)
UI
0
o L = L(i, U) =
-ft
0
-|n
ru-
o u*-i
-t
o o
o
(3.5) /
Let D = {f\f±
6 Ll(G), f
0
(3.6)
e L ( G x G)} C /» 2
We shall establish a few elementary lemmas in order to study the stochastic differential equation dV
=
VdA i. L
Lemma 3.1 : For any / € L\(G) \\lf\?
<
Q(sup /)||/|! , 2
P
\\rif\\ <
Q ( S U
P
P
/)||/||,
where supp denotes support and c/(-) iB as in {Hi). Proof :
The first inequality is immediate from
(3.3)
and the second follows
from the fact that Ct is multiplication by the function J
a
\t(-, y)\ dy. 3
•
Lemma 3.2 Let / € L\{G x G) and supp / C A , x K where K K? C G are 2
lf
compact. Then \\e'U-f\\
Proof :
2
KctiK^Ki)"'
By definitions and Schwarz's inequality II^VII
2
= <
J\J((x,y)f(yx,y)dy\ dx 2
f(j\t(*,v)\ dy)(j\f(v*,v)\ dy)di 2
2
230
By hypothesis, if f(yx,y)
/ 0 then
G A\,y G K and, in particular, x G
yx
2
tfj-'AV Thus by (1.2) and (tf,) ||r{/7H
2
{ess. np - J (x,y)dy}\\f\\
<
S
=
xeK
lKl
7
qi
^(^-^xXi/ii . 2
• The next lemma is basic to the whole paper. We call it the approximation lemma. Lemma 3.3 (Approximation lemma) : Let / satisfy (Hi). Choose and fix an increasing sequence {A*„} of compact neighbourhoods of the identity in G such that G = U K . n
n
Define t (x,y)
=
n
U(x,y)]lx£K , n
0 L
=
n
for every n. Then L„, L* G
1(h)
for every n. For anyfixed/ G rt ,
L(t ),
L = b
n
and L„ +
L(l )
n
L
b
n
b
n
b
n
b
n
+ LL n
D L f = Lf, L f
where D, L, L are defined by (3.2) -
Proof : The boundedness of L I
otherwise,
n
b
= L„ + L + L L„ b
n
= Lf b
b
n
n
~
0
for all sufficiently large
(3.6).
and L are immediate from (3.4), (3.5) with b
n
replaced by £ and Lemma 3.1, 3.2. The fact that n
L fe^ n
x
= L^fe^ =
0
follows from the property that thefirstcolumn and the last row in L are zero. n
The remaining parts are obtained by straightforward verification.
•
Using Proposition 2.4 and the approximation lemma construct the sequence {V ] of unitary processes satisfying n
V (0) n
= 1, dV = V d\ t , n
n
L
n
n = 1, 2,...
(3.7)
231
with respect to (h , M ) where M is defined by (2.6). Following an idea of 0
0
0
Frigerio as outlined by Fagnola in [2] we shall examine the asymptotic behaviour of {V } as n —• oo. n
Proposition 3.4 : The sequence {V„} defined by (3.7) admits a subsequence {V } Ti
satisfying the following :
(i) w.limfc-,00 V^,,(<) = V(t) exists for every t > 0 ; (ii) {V(i)} is a contraction valued
(L (G),
-adapted process
L (R+)®L (G))
2
2
2
for which sup | < V , [V (i)
lim
nt
*— oo
-
V{t)]fe(u)
> | = 0
(
for 0 < T < oo, V> € H, f € L\{G), u € Mo;
(iii) For each 0
< T < oo, f € L\(G),
u € Mo
there exists a positive constant
c = c(/, u, T) such that W(t ) 2
-
< c ( i - Ufl ,0
V{U)]fe(u)\\
< i, < i
2
2
2
< T .
Proof : Consider the sequence {p \ of continuous functions on 1R defined by n
(t)
Pn
=<
+
tp, V (t)fe(u)
>,n
n
= l,
2,...
where lp € H, f € £*(
<
HV'lll|[K.(<2)-V;(< )]/e(»i)||
<
||V|||K«)||{2e-(")r||^
1
f® (,)
\\ d») 2
u
Jti
\
t
1
By the last part of the approximation lemma we have for all sufficiently large n 0
232
for all 0 < i, < < < T .
(3.8)
2
Furthermore |p„(t)| < ||V>|| ||/e(u)|| for all t, n. Hence, by the Arzela-Ascoli theorem {/>„} is conditionally compact in the topology of uniform convergence on compacta. Using the separability of the spaces involved and a diagonalisation procedure extract a subsequence {V^,,} satisfying (i) and (ii). To prove (iii) observe that for any rj> <
€ U, f € L\(G),
[V{i ) - V(U)]fe(u)
>=
a
u €
lim < rp,
M
0
[V {t ) nk
3
-
V». (<,)]/>(«) >,
K—• OO
the convergence being uniform in 0 < ii < ia < T. Using (3.8), taking supremum over all unit vectors ^ and by the definition of A
•
Proposition 3.5 : For each i, j € 5 U {—oo, oo} there exists a unique operator {L )'j h
with domain
:
2
(L ))g b
3
2 C
f&L (G),g<=Ll(G)
(i) < /,
in L (G) such that the following holds for every
L (G)
>=<
(ii) E l K ^ j s i r
f
® ti, L g ® ey > for each i, ; G S U { - O O , oo} ; b
= U9
® eilf
b
for each
jeSU
{-oo,oo}.
i
Proof : Immediate .
•
Using Proposition 3.5 we conclude that the constant adapted processes {{L ))} b
constitute an element in /L(L (G), M ). 3
Now define the
0
{L\(G),
M )0
adapted process A^i satisfying A*(Q)
(3.9)
= M A * =
y with respect to (L (G), M ) 3
0
Proposition 3.6 : Let {V(t)} be as in Proposition 3.4 and let A i be as in £
(3.9). Then {V{t)(L ))} is in b
JL{L\{G),
V(Q) = l,dV =
and
M) 0
Y V{L ))d\\ J
b
233
with respect to
(L\{G), M )
(3.10)
•
0
Proof : Since V(t) is a contraction the stochastic integrability of {V(t)(L*)J} r
is immediate from Proposition 3.5. By (i) in Proposition 3.4, (3.7) and Proposition 2.1 we have for any f,g € £ (G),u, v € M , 2
lim*^ <
a
< fe{u), V(i)ge(v)
>=
=<
+ lim _ /„' < /(e.,*, + u(«))e(u),
fe(n),ge(v)
>
fc
fe(u),
V ,(t)ge(v) n
>
00
V (s)L g(v{s) b
ni
+ e ^ v )
nk
> ds.
Since, by the last part of the approximation lemma, L g(v(s)+e b
l
=
)
oa
L g(v(s)+ b
eoo) for all large k we get < /e(u), V{t)ge(v)
>
=
< fe{u),ge(v)
+ f
< f{e-oo
Jo
which implies
(3.1Uj .
Proposition
3.7
>
+ «(*))e(u),
V(s)L g(v(s) b
+
)e(v)
> ds
eoo
•
: The equation (3.10) has a unique solution
{V(<)}
where
each V(i) is a contraction. Proof : Suppose V is another contraction process satisfying (3.10). Define W(t) = V(i) - V'(t) so that
with respect to
(L\(G),
Ll(G),
0
< fe(u),
M ). 0
By thefirstfundamental formula for any
f,g
€
u,v€M ,
W(t)ge(v)
>= / ' < /(e_ +u(»))e(u), Vy'( )I (v( )-|-e )e(v) > 00
a
i S
a
O0
ds .
Jo
(3.11)
234
In particular , W(i)ge(0)
< ft(0),
>=
- ± / ' < /e(0), W(,)l*lge(Q)
(3.12)
> ds
I Jo
Recalling the sequences {K„}, {t } in the approximation lemma observe that n
(3.12) implies < We(0), W(t)I
ge(0)
>
Km
= ~ \ [ < /jf./e{0), where I
Kn
W( )I J'J ge(0) t
K
(3.13)
>
n
is the indicator of K . Define the bounded operators Af„(<) in L*(G) n
by < /, M (t)g n
>=<
I Je(0),
W(t)I ge(0)
K
> for all f,g € L\G).
Kn
Then (3.13) can be expressed as ^
Since
= - -M tJ ,M (0) 1
n
= 0.
n
n
is a bounded operator it follows that M (t) = 0. Since n is arbitrary n
we get < /e(0), W(t)ge(0) For any u € L (R ) 2
f
®
>=
0.
L (G) write 2
e(«) = §«<"> n=0
where
is the n-particle component of e(u). For anyfixedti, v 6 £ (JR ) ® a
+
L (G) define the bounded operators Af„(r,«; u, v; t) in -L (G) by 2
2
< / , M„(r,
u, v; t)g >=<
(l J)vS \ K
r
W(i){I g)v^ Kn
> .
We have just now shown that M „ ( 0 , 0;«, v; t) = 0. We shall now prove by induction that all the M 's vanish. Suppose n
Af (», /; u, v; i) = 0 for i < r,j < s and i = r,j < $. n
235
Since (L
+ L + LL )f b
=
h
for all / € D where
0
D
is defined by ( 3 . 6 ) we obtain
from (3.11), the analyticity of the map v —• e(v) and the induction hypothesis < f,
1
Af„(r, »\u, v;
r'
t ) g > = - -
M„(r, »\ u, v; T)C t„g
< f,
> dr
n
i Jo
which implies that M (r, a; u, v;t) = 0 . Thus induction and the arbitrariness n
of n imply that W(t) = 0 .
•
Proposition 3.8 : The operators
in the unique solution of ( 3 . 1 0 ) are
V(t)
isometric. Proof: By (ii) in Proposition 2.1, for f,g < V{t)fe(u),V{t)ge(v)
>=<
V » / ( e _ » + «f*))e(u),
+
f{< Jo
+
< V{,)FL Ff{e.
+
< V{,)FL Ff(e-„
i
€ Ll(G),
+
b
Replacing / , g by //(„/, IK„9
0
fe{u),ge{v)
V(.)L g[v(.)
+
h
+ u(«))e(u),
to
u, v € M ,
V{B) (V(B)
>
)e{v)
>
+ e )e(v)
>
Kao
x
9
u(.))e(u), V(»)L g(v(a)
+ e )e(v)
b
respectively,
we have
x
>}d*(2.U)
being as in the approximation
{K } n
lemma, define the operators N (t) in £ (G) by 2
n
< f,N (t)g
>=<
n
V(t)I fe(Q),V{t)I ,ge(0) Km
>
K
.
Then (3.14) yields after putting « = v = 0
^f- = -\(N„r e n
n
+ rj N ) n
n
+
r N e„ n
(3.15)
n
where N (0) = IK„ is the operator of multiplication by IK,- B y the definition n
of the bounded operator t (See ( 3 . 3 ) ) it follows that n
N„(i)
= IK„
of (3.15). Thus < V ( i ) / e ( 0 ) , V(t)ge(0)
> = < fe(0),ge(0)
>
.
is a solution
236
Now we proceed by induction as in the proof of Proposition 3.7. Define the bounded operators < /, N (r, n
in L (G) by
N (r,s;u,
v;i)
a;u,v;t)g
>=<
n
2
K(i)( W ) «
We have, just now, shown that N„{0,0;
( 0
>
V{t)(I g)v"
>
Kn
a, v\ t) =< u , v <0)
(0)
>
IK.-
.
N O W
suppose
that N {i,
=<
j;u,v;t)
n
for i <
«W
r,j
„0) >
< a
j
and
K n
i < r,j
= a .
We shall prove the same relation for i = r, j = «. Since (L
+ L
+ LL )f
b
b
= 0 for
all / € D where D is defined by (3.6) it follows from the inductive hypothesis, (3.14) , the analyticity of the map v —* e(v) and a fairly routine but somewhat tediouB
calculation that
N„
=
N„(r,
u, v; •) obeys the differential equation
> I .-
(3.15) with N (0) =< u >, (r
n
a;
Repeating the same arguments we
K
conclude that < V(<)/«
for all / , g € L (G),u, v 2
( r )
, V(i)gv^
>=<
fu \gv^ (T
>
.
and 1. In other words
€ M ,r,a 0
is an isometry
V(t)
for every i.
• We now summarise our conclusions in the form of a theorem. Theorem 3.9 : Let t(x,y),
x,y
€
G
be a complex valued Borel measurable
function satisfying (Hi). Define the operators L , L
b
in L (G) © L (G x G) © 2
2
L (G) by (3.4), (3.5). Then the quantum stochastic differential equation 2
V{0)
= l,dV=
VdA > L
=
J^VLfdAl
with respect to (L (G), A4 ) has a unique isometric solution V . 2
0
L
237
Proof : This is immediate from Proposition 3.4 - 3.8 . • Remarks : Suppose / satisfies operators
V (t) L
Then it is known from
(H ). 0
in Theorem 3.9 are unitary. Furthermore if ^ 6
[7,11]
that the
I°°(G)
and the
operator of multiplication by <j> is also denoted by
(3.16)
L
yields a commutative Evans-Hudson flow with initial algebra
L°°(G)
which
describes a classical Markovflowwith generator
(1.1).
It is an
Qt
defined by
open problem, in general, tofindnatural conditions on I to ensure the unitarity of Vc(i) for all t. Eventhough V (t) is only an isometry under (Hi) it is expected L
that the flow j\ defined by
(3.16)
=
0
f
o
is commutative, i.e. , r
a 1 1
tufa
€ L°°(G),ti,h
> 0.
It is also expected that the semigroup {Tt} on ^(G) defined by Tt(
JEjf-(
where E denotes the vacuum conditional expectation, is the minimal Markov 0
flow with generator
Qt
on the domain
L\(G).
When {£„} is as in the approximation lemma it is not clear from our analysis what happens to the sequence of the induced Evans-Hudsonflows{jt'}-
We
shall comment on this in the next section. When G = Z and
t(i,l) t(i,-l) l(i,j)
=
t>0
=
y/JTi,
=
0
£ > 0
otherwise
(3.17)
238
where {A;} and {^} are sequences of positive numbers Qt describes birth and death processes. For a detailed analysis of the special case of pure birth and pure death processes in the framework of Evans-Hudsonflowssee Fagnola [2j.
4
A n Evans-Hudson equation with unbounded structure maps
Let t(x, y), x, y € G be a complex valued Borel measurable function on G x G. We shall frequently view
L°°(G)
as the abelian von Neumann algebra of all
bounded multiplication operators in L (G).
An element
2
and the
<j> € L°°(G)
operator of multiplication by $ as well as its ampliations are denoted by the same symbol
approximation lemma of Section L (G) 2
®
© L\G)
=
0(4>), 6 (4>), 6 (
n
in
L
h
e (<j>) =
L
=
L
n
k(
n
Define the operators
3.
© (Se^) by 0(4,)
where L, L
n
b
n
L
n
h
n
L
n
n
n
are the operators defined in the approximation lemma and <j> €
L°°(G).
Lemma 4.1 : Let I satisfy support,
0(4>),Q (4>) n
(Hi) — (H ). 3
For any
and 8 (4>) are bounded and n
U m f J I M r f ) - H+)\\ + WLW
-
6(
with compact
239
Proof : Elementary algebra shows that
o
0 » O ) - H4)
n
n
U*4>U({„ -1)
=
0
/
' o 4>(i- - e* ) + ({* - r)u*4>u i* u*4>ut - ru'< n
n
n
n
+(te-i' i )4> n
o
u*
o
o
n
n
o
n
o
From definitions it follows that for all sufficiently large n 4>(t -r ) n
= o,
Qt = o.
n
Furthermore the following inequalities hold :
\\u'4>u(t -i)\\ = \\(r -r)u'
n
< \\rU'4>U(l -t)\\ n
lklU{ess.sup , / < ||^|U{ess.sup
x€n
\\r u*
qi(x,y)dy}
112
l€K
n
/
I M Uess.suP l 6 K ,
,
qi(x,y)dy}, /
<,,(*, j/)
where prime ' indicates set complement. The required result is immediate from
Now consider the sequence {V } of unitary processes satisfying (3.7). We N
shall analyse the behaviour of j} W = n)
v v:,4> n
e L°°(G).
(4.1)
as n —• oo . Denote by Lf(G) C L°°(G) the linear manifold of elements with compact support.
240
Lemma 4.2 : Let
2
+
2
there exists a positive constant c depending on
i
2
such that for all
u
< oo
™ \\bX\
n < ce-(">K(< ) -
!/„(
2
where i/ is given by (2.4) . u
Proof : From Proposition 2.1 and the consequent quantum Ito's formula dj\ (
n
(See [10]) with respect to (L (G),
L (Hl )
2
< 2e"Wtf
2
n
< 2e^>||/e(t )|| (su t
+
n
2
Pn
(4.2)
n
By Corollary 2.2
® L (G)). 2
+
\\V (s)6 (
eM)
eoo
)e(u)|| d 2
\\9 (
a
•
n
The required result is immediate from Lemma 4.1.
•
Lemma 4.3 : Let Co C L°°(G) be any countably generated linear manifold. Then there exists a family {jt,t > 0} of linear maps from C into B(ft) satis0
fying the following: (i) {j (
t
i>
e
® L (G))-adapted process for every
L (IR ) 2
2
+
Co;
(ii) j o W = )\\ < mu ; (iii) For every / e L (G), u £ 2
positive costant c = c(/, u, \\\jt, (
L {R ) 2
+
® L (G),T
>0,
2
o
there exists a
T) such that j., W]/e(«)|| < c K ( t ) - „„(<,)] ; 2
a
(iv) There exists a subsequence {j*"*)} of (4.1) such that for every
2
+
lim sup |
J1-.00 o < l < T
2
< V , \j t \) ~ (
nk
j.W]/e(u) > | = 0
241 and, i nparticular ,
w.linu^
= j,{
j\ (<j>) h)
P r o o f : Define
n=l,2,...
p (t)=,j \
n
B y L e m m a 4 . 2 , {p„} i s a u n i f o r m l y b o u n d e d a n d e q u i c o n t i n u o u s f a m i l y o f c o n t i n u o u s f u n c t i o n s o n Bi+. B y u s i n g t h e A r z e l a - A s c o l i t h e o r e m , t h e s e p a r a b i l i t y o f Co a n d a l l t h e H i l b e r t s p a c e s i n v o l v e d t h e r e q u i r e d r e s u l t s a r e p r o v e d e x a c t l y along t h e s a m e lines as P r o p o s i t i o n 3.4. D e n o t e b y C(G),C (G)
•
a n d Co(G) r e s p e c t i v e l y t h e s p a c e s o f a l l c o m p l e x
C
v a l u e d b o u n d e d c o n t i n u o u s f u n c t i o n s o n G, a l l e l e m e n t s o f C(G) w i t h
compact
s u p p o r t a n d a l l e l e m e n t s o f G(G) v a n i s h i n g a t c o . T h e n w e h a v e t h e f o l l o w i n g c o r o l l a r y t o L e m m a 4.3.
C o r o l l a r y 4 . 4 : T h e r e e x i s t s a f a m i l y { j , , t > 0} o f l i n e a r m a p s f r o m
C (G) 0
i n t o B(H) s a t i s f y i n g t h e f o l l o w i n g : (i) F o r e v e r y
IMU
a
n
d
i s a n a d a p t e d p r o c e s s for w h i c h
{jt(<j>)}
<
jo(4>) = 4>;
( i i ) F o r a n y / € L (G),u 2
<= L (R ) 2
> 0 a n d <j> e C (G) t h e r e
® L (G),T 2
+
0
e x i s t s a p o s i t i v e c o n s t a n t c = c ( / , u,
-i.,(^)]/e(«)ll
WtM)
2
< cK(< ) a
(iii) T h e r e exists a subsequence H, f
€ L (G), 2
u € L (M ) 2
+
® L (G),
of (4.1) such that for every ^ €
2
*.(*»)]
€ G (G) 0
and T > 0
l i m sup | < ip,ti (
242 Proof : First choose C to be a countably generated dense linear manifold in 0
C (G), C
apply Lemma 4.3 and extend in the obvious manner to C
= C (G).
0
0
•
Our aim now will be to show that {j } defined by Corollary 4.4 obeys an t
Evans-Hudson equation in a weak sense. To this end we introduce for each <j>
€
C (G) C
the bounded operator
0'j(
U {-00,00} in L (G) by the 2
relations < /, 9)(4)g >=<
f ® *, 0{
L (G)
(4.3)
2
j
where {ei,i € 5} is the orthonormal basis in L (G) chosen and fixed at the 2
beginning of Section 3. A routine computation shows that 9)(
=
JttW-famyhjMdy,
=
/ {x,y)[<j>(yx)-4>(x)\ej{y)dy,
=
J Ux,y)[<j>(yx) -
-
J
=
0
l*( tV)| Wl/< ) ~
Lemma 4.5 : Let £ satisfy
3
€
L°°(G)
B
i i€S,j
» = -00,
=-00,jeS,
d
= oc j =
00,
otherwise
(4.4)
— (#4). Then for any
(Hi)
G
C (G), C
9)(<j>) € C (G) for all i, j € S U { - 0 0 , 0 0 } .
(4.5)
0
Proof: From (4.4) it is clear that \9)(
and
.„,'€?.
t
a;
0)(
e C(G). For i,j <= S,
j h(j,)
Ci
(i,)|
\
Hi^Kx' ) 1
where A ' = supp
E S.
Let
i = —oo,j
Then by (4.4) and Schwarz's inequality we have for x £ supp 4> \ej^(
2
<
J\£(x,y)\ \
2
€ 5.
243 < ML j
Jeupp4>
ji(*,v)dy.
By (-^3) the right hand side tends to 0 as x —• 0 0 . A similar argument holds for #Lj(^), t
€ S.
When
i =
—00,
j = 00
and a; £ supp
IC°0)(*)I<(/
Jsupp^
<j>
This completes the proof.
•
Proposition 4.6 : Let {j,} be as in Corollary 4.4. Then for all f,g € L (G),
u, v S A4 ,
2
0
C
< fe(u),j {4>)ge(v)
>=<
t
+ £
fe{u),
>
e \)g(v(s)
/ ' < / ( - o o + « ( * ) ) « ( « ) . O ' . ( ^ W ) ® |e, X e
k
+ t ^ v ) > ds
(4.6) where all but afinitenumber of terms in the summation on the right hand side vanish. Proof : Fix /, g, u,v,4> as in the statement of the proposition and consider {jj"*^} as in Corollary 4.4. By thefirstfundamental formula < / e ( « ) , j , (
>=<
( nt)
+ £
fe(u),
X
/' < f(e. +u(s))e(u),(j^\(O y W)®\e, x
nt
>
}
e | ) ( t . ( » ) + e ) e ( i > ) > ds i
f l
0 0
(4.7) where the number of terms on the right hand side within the summation depends only on u, v € M o • By Lemma 4.1 and (4.1) we have
iim||i("'>(((?„,);(^))-jr'>(
lim ||(
-
= 0 for all •., .
Now the required result follows from (4.7) and (iii) in Corollary 4.4.
•
244 Remark
:
Equation
(4.6)
may be interpreted as a 'weak' or 'generalised'
Evans-Hudson equation : djtW
= 5>(<W))
with respect to (L (G), M ) for each 4> <E G (G). 2
P r o p o s i t i o n 4.7
0
C
Let t satisfy (if,)
Suppose
- Lff ). 4
V
L
is the unique isometric
process of Theorem 3.9 and it(
v {t)4>v {iy,4>£Co(G). L
L (G), u,v e M , i> € G (G) and t > 0
Then for all
2
0
< i^{u),it(
f
+ T
L
< f(e-oo
+
C
>=<
u(t)e(u),j. (0LW) L
fe{u),
® \a ><
>
e \)g(v(t) k
+ )e(v)
>
eoo
dt
(4.8)
where all but a finite number of terms in the summation on the right hand side vanish. Proof:
Define the family {rj"*} of linear maps from G (G) into B(H) by 0
r?\
where
V„
is given by
/' < /( -~ e
L
n
By the first fundamental formula
(3.7).
< M«),rt°Wfle(f)
+ £
V (t)
+ «(»))«(«).
> = < fe{u),
>
(r^WnYM)) ® |e. X g(v(t) +
eoo
)e(v) >
dt
where 0 is in Lemma 4.1. Using Lemma 4.1 and the weak convergence of n
rt"\<j>) to jf(4>) as n - +
oo
for every <j> € C (G) we obtain 0
(4.8).
•
245 C o r o l l a r y 4 . 8 L e t / s a t i s f y (TY,) - (H ). S u p p o s e t h a t t h e r e e x i s t s a u n i q u e t
f a m i l y o f l i n e a r m a p s {j ,i > 0 } f r o m C (G) i n t o B(U) s a t i s f y i n g t h e f o l l o w i n g : t
0
(1) , o W = ^ J U . W I I < I M U
(2) < fe{u),j {4>)ge(v)
>=< fe(u),
t
+ Z.^/o < / ( e _ c o + u ( , ) ) e ( u ) , ( j . ( < W ) ) ® |e. X for a l t f,g € L ( G ) , u , v € M ,4> 2
0
e \)g(v(»)
+ e ^ v )
K
> de
€ G ( G ) , < > 0 . T h e n t h e s e q u e n c e {j\ } o f n)
C
(4.1) s a t i s f i e s t h e f o l l o w i n g : (i) s . l i m , ^ . . ^ ^ ) (ii) F o r a l l 0 < t
Proof:
lt
= j (4>) = }{•(<(>) f o r a l l
0
< T < o o ,
2
G (G), 0
B y t h e u n i q u e n e s s h y p o t h e s i s jt(4>) — it(4>) f o r a l l
Fur-
t h e r m o r e , f o r f i x e d t,j? i s a * p r e s e r v i n g ( b u t n o t n e c e s s a r i l y u n i t a l ) h o m o morphism.
Each
is a * preserving homomorphism.
B y C o r o l l a r y 4.4,
jl"\<j>) c o n v e r g e s w e a k l y t o jt(4>) a s n —• o o a n d h e n c e ( i ) o b t a i n s . M i M> \jt?\
= 0 f o r e a c h n w e o b t a i n ( i i ) from ( i ) .
4 . 9 L e t I s a t i s f y (H ) — ( r 7 ) . x
4
Suppose
Since, b y •.
that there exist t w o
d i s t i n c t f a m i l i e s { i , , < > 0 } , i = 1 , 2 o f l i n e a r m a p s f r o m C (G) i n t o B(U) (,)
0
satisfying the following: ( i ) jfty
= <j>,
|#(#)||
<
I^IU a n d
{;ffy)}
is a n a d a p t e d process i n W ;
( i i ) F o r a n y / , g € L ( G ) , u,v € M ,4> € C ( G ) , < > 0 2
< fe{u),it\
0
0
>=< / e ( u ) ,
246
+ E / ' < /(«-<* + «WM«). Ctf'WV) ® h >< «*!)*(«(«) + ~ H ) > e
v
Ja
Then there exists a A > 0 and a nonzero totally finite complex valued measure fi\ on G such that / Qt(+)(m)dpx{m) Ja
where Proof:
= A /
= Q (
Define j' (
£ ( G ) ,
e
C (G)
all 0 €
C
(1.1).
= ^ (<£) - j» (<£). l)
By hypothesis there exist
3,
f,g
€
u,» € Mo,to > 0 such that
>*o
1
but
< /« . (r)
j '(^)fl (
>=0
u ( , )
for
r < m, « < n and r < m,» < n.
Define 7A(» =
/ " e- ' < Jo
; '(*)s«<"' > J t .
A
(4.9)
e
Then |7AWI<2|WU|I/||[| ||||^||||« I!.
(4.IO)
W
?
Integrating by parts on the right hand side of (4.8) and using (ii) we get 7*(*)
=
x
~ l " * ~ l
M
j
t
t m )
-i'A4>)gv
( n )
> dt
= A " y/ o e- * 1
=
A
_
,
}
A<
7 A ( W ) ) .
(4.11)
Since 7> is a continuous linear functional on G ( G ) it follows that there exists 0
a totallyfinitecomplex valued measure fix B u c h that jx(
4> € G ( G ) . 0
Now the required result follows from (4.10).
Ja
x
B
fo
247
Theorem 4.10 Let i(x,y),x,y
€ G be a complex valued Borel measurable
function satisfying (Hi) - (#4) and let Q be the operator denned by (1.1). For t
any A > 0 let B\ be the set of all totallyfinitecomplex valued measures ft on G satisfying
Suppose that
= 0 for every A > 0. Then there exists a unique family
> 0} of * preserving homomorphisms from C (G) into B(H) satisfying 0
the following (i) M
< II0IU and {j (
>
t
0 is an adapted process in
(ii) {jt} obeys the generalised Evans-Hudson equation dj (
H\
,* jt(0j(^))dA*(f)
with respect to (L (G), A4„) for every
0
k
are determined by (4.4). Furthermore there exists an isometric adapted process { V(t)} satisfying ;,(«£) = V(t)
The family
{jt(
for all
t > 0,
G (G). 0
0,4> € G0(G)} is commutative.
Proof: This is just a summary of Proposition 4.6, 4.7, 4.9 and corollary 4.8.
a Remark When G = Z and / is given by (3.17) conditions (Hi) — (H ) are t
fulfilled. Following Feller [3] and Karlin and McGregor [5] it is possible to choose {Ay} and {n ) in (3.17) such that the conditions of Theorem 4.10 are }
fulfilled. It is possible that j ( l ) < 1 for some t. The adapted family {i«(l)} of t
projections will determine a stop time at which the associated 'Markov Chain' escapes to the boundary. We hope to examine this problem in greater detail in our future investigations.
248 REF E R E N C E S
[1] Belavekin, V.P. A new form and '-algebraic structure of quantum stochastic integrals in Fock space, Rendiconti del Seminario Mathematico e Fisico di Milano, (1990) [2] Fagnola, F.: Pure birth and pure death processes as quantumflowsin Fock space, Universita di Trento preprint (1990), To appear in Sankhya, Ser. A. [3] Feller, W.: On boundaries and lateral conditions for the Kolmogorov differential equations, Ann.Math 65(1957) 527-570. [4] Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolutions, Commun. Math.Phys. 93 (1984) 301-323. [5] Karlin, S., McGregor, J.L.: The differential equations of birth and death processes and the Stieltjes moment problem, Trans. Amer. Math. Soc. 85 (1957) 489-546. [6] Meyer, P.A.: Elements de probabilites quantiques, Seminaire de Probabilites, Springer LNM 1204, 186-312(1986), 1247, 34-80(1987), 1321, 101-128 (1988), 1372, 175-185 (1989). [7] Meyer, P.A.: Chaines de Markov finie et representations chaotique,
Stras-
bourg preprint (1989). [8] Mohari, A., Sinha, K.B.: Quantum stochasticflowswith infinite degrees of freedom and countable state Markov processes, Sankhya, ser. A. 52, Part 1(1990) 43-57 [9] Parthasarathy, K.R., Sinha, K.B.: Markov chains as Evans-Hudson diffusions in Fock space, Seminaire Strasbourg 24(1990) Springer LNM 1426.
249 [10] Parthasarathy, K . R . : Realisat ion of a class of Markov processes through u n i t a r y evolutions in Fock space, I.S.I. D e l h i Centre preprint (1990). [11] Parthasarathy, K . R . : A n a d d i t i o n a l remark on unitary evolutions i n Fock space, I.S.I. D e l h i C e n t r e preprint (1990). [12] Parthasarathy, K . R . : A n Introduction to Q u a n t u m Stochastic C a l c u l u s , I.S.I., D e l h i C e n t r e Notes (19901
Quantum Probability and Related Topics Vol. VII (pp. 251-259) (c) 1992 World Scientific Publishing Company
251
Note on 5 -Entropy Naohumi Muralri D e p a r t m e n t o f I n f o r m a t i o n Sciences C o l l e g e o f Science a n d E n g i n e e r i n g Tokyo Denki University H a t o y a m a - M a c h i , H i k i - G u n , S a i t a m a 350-03, J a p a n
5 - E n t r o p y for states o f C * - d y n a m i c a l s y s t e m s i n t r o d u c e d
Abstract
by O h y a is a m e a s u r e o f c o m p l e x i t y o f a s t a t e , d e f i n e d f r o m t h e v i e w p o i n t that g i v e n a r e f e r e n c e set 5 o f states w h i c h is c o m p a c t c o n v e x , t h e a m o u n t of u n c e r t a i n t y o f a state
convex
sets 5 a r e C h o q u e t
simplexes.
We
also p r o v e t h e i m p o r t a n t r e l a t i o n s b e t w e e n 5 - e n t r o p y a n d a n o t h e r e n t r o p y introduced b y Connes andNarnhofer-Thirring. 1.
INTRODUCTION
Q u a n t u m m e c h a n i c a l e n t r o p y o f states o f q u a n t u m systems w a s first i n t r o d u c e d by v o n N e u m a n n , i n t h e case o f systems d e s c r i b e d by d e n s i t y o p e r a t o r s p on H i l b e r t spaces [vN]. V o n N e u m a n n entropy 5 ( p ) = — t r p l o g p expresses t h e a m o u n t of u n c e r t a i n t y c o n t a i n e d i n p , a n d h a s c o m p l e t e a n a l o g y t o the S h a n n o n e n t r o p y i n i n f o r m a t i o n theory, a n d i s e s s e n t i a l i n q u a n t u m s t a t i s t i c a l m e c a n i c s [ v N , B R ] . T o r i g o r o u s l y describe q u a n t u m s y t e m s o f i n f i n i t e degrees o f f r e e d o m , such as q u a n t u m l a t t i c e systems or q u a n t u m fields, t h e m e t h o d o f C * - a l g e b r a s i s very useful, because i n s u c h theory we m u s t d e a l v a r i o u s difFerent representations o f o p e r a t o r algebras [ H K , S 1 ] . For s e m i f i n i t e v o n N e u m a n n algebras, t h e e n t r o p y of a n o r m a l state tp was given as S(ip)
b y S e g a l [S2],where r i s a t r a c e a n d D is the d e n s i t y of tp w i t h
= —rDlogD
respect t o r. F o r C * - a l g e b r a s , O h y a has i n t r o d u c e d a n o t h e r e x t e n s i o n o f v o n N e u m a n n e n tropy, d e n o t e d b y S (
t o describe e n t r o p y t r a n s m i s s i o n between t w o C * - d y n a m i c a l
systems [ 0 2 ] , e s p e c i a l l y t o a p p l y i t i n q u a n t u m i n f o r m a t i o n t h e o r y [ 0 4 ] . O h y a ' s e n t r o p y S (
s y s t e m 5 , a c o m p a c t c o n v e x set o f states c o n t a i n i n g tp. W e c a l l i t 5 - e n t r o p y because we t a k e s e v e r a l 5 here. M a i n p r o p e r t i e s o f 5 - e n t r o p y was p r o v e d i n [02] w i t h o u t the c o n t i n u i t y . T h e c o n t i n u i t y o f 5 - e n t r o p y was p r o v e d i n [ 0 M ] o n l y for the cases t h a t 5 are C h o q u e t s i m p l e x e s , such as the set o f a l l K M S states, or the set o f a l l i n v a r i a n t states i n G - a b e l i a n s y s t e m s . I n the p a p e r [03] O h y a i n t r o d u c e d q u a n t u m m u t u a l i n f o r m a t i o n I (tp;A*) s
b a s e d o n q u a n t u m r e l a t i v e e n t r o p y S(i>,tp) i n t r o d u c e d by U m e g a k i
[U3] a n d g e n e r a l i z e d b y A r a k i [A2] a n d U h l m a n n [ U I ] . M u t u a l i n f o r m a t i o n is r e l a t e d to t h e 5 - e n t r o p y S (tp), s
a n d i s v e r y essential i n q u a n t u m c o m m u n i c a t i o n t h e o r y [ 0 1 ] ,
252 because it describes the t r a n s m i t t e d i n f o r m a t i o n f r o m a n i n p u t s t a t e tp t o a n o u t p u t state A'tp t h r o u t h a c h a n n e l A*. H e also n o t e d [03] t h a t i n some p h y s i c a l s y s t e m s his m u t u a l i n f o r m a t i o n is given b y the f o r m u l a /•*(*>; A * ) = J
S{A*u,A' )
w h e r e fi is some measure p s e u d o - s u p p o r t e d 5.
A n o t h e r q u a n t u m e n t r o p y H (N)
(1.1)
dii(u>),
v
o n t h e set exS
o f a l l e x t r e m e p o i n t s of
different f r o m 5 - e n t r o p y
V
was i n t r o d u c e d by
C o n n e s , N a r n h o f e r - T h i r r i n g i n [ C , N T ] , a n d t h e y develop t h e n o n - a b e l i a n K o l m o g o r o v S i n a i t h e o r y [ C N T ] . T h e i r d e f i n i t i o n of e n t r o p y uses the a b o v e f o r m u l a . So we want to c l a r i f y the r e l a t i o n s of the O h y a 5 - e n t r o p y e n t r o p y ( C N T e n t r o p y for s h o r t )
w i t h the C o n n e s - N a r n h o f e r - T h i r r i n g
H (N). V
O u r results consists of two p a r t s ( § 3 , 4 ) .
I n §2 we p r e p a r e s o m e n o t a t i o n s a n d
t e r m i n o l o g y . I n §3 we generalize the c o n t i n u i t y of 5 - e n t r o p y to t h e cases t h a t 5 are not necessarily C h o q u e t S i m p l e x e s , s u c h as the set of a l l i n v a r i a n t states i n n o n G a b e l i a n systems or m o r e general c o m p a c t c o n v e x sets. I n §4 we s t u d y the r e l a t i o n of the O h y a 5 - e n t r o p y w i t h the C N T entropy. 2. D E F I N I T I O N O F 5 - E N T R O P Y L e t (.4, G, a) be a C - d y n a m i c a l s y s t e m , t h a t i s , A b e a u n i t a l C * - a l g e b r a ( i n t e r p r e t e d as observables of a q u a n t u m s y s t e m ) , G a l o c a l l y c o m p a c t g r o u p ( i n t e r p r e t e d as a g r o u p of some s y m m e t r i e s of the s y s t e m ) , a a c o n t i n u o u s a u t o m o r p h i c a c t i o n of G on A.
T h i s t r i p l e t is used to describe general q u a n t u m d y n a m i c a l s y s t e m s i n c l u d i n g
systems of i n f i n i t e degrees of freedom [ B R ] . L e t 6 or 6(A) states over A,
b e t h e s t a t e space of a l l
5 be a c o m p a c t c o n v e x subset of © i n the weak* t o p o l o g y .
Ohya's
5 - e n t r o p y for a s t a t e tp i n 5 , defined b e l o w , expresses t h e u n c e r t a i n t y of a s t a t e tp m e a s u r e d by a referece s y s t e m 5 . W e choose several 5 to m e a s u r e v a r i o u s u n c e r t a i n t y of a same state tp, for e x a m p l e , the state space 6 , t h e space o f a - i n v a r i a n t states 1 ( a ) , the space of a - K M S states K(a) others.
d e s c r i b i n g t h e r m o d y n a m i c e q u l i b r i u m states, a n d
W h e n we consider a n o r m a l state tp on a W ^ - d y n a m i c a l s y s t e m (M,G,
where A t is a v o n N e u m a n n a l g e b r a a n d a is a w e a k l y c o n t i n u o u s a c t i o n of G o n
a), M,
we t a k e v a r i o u s n o r m closed c o n v e x sets 5 of n o r m a l s t a t e s . F o r a n y m a x i m a l m e a s u r e fi i n t h e C h o q u e t o r d e r i n g , p s e u d o - s u p p o r t e d o n 5 a n d r e p r e s e n t i n g a s t a t e tp:
let / J ( / i ) be t h e f u n c t i o n given by H(fi)
= supj{-
^
p(A)lo /i(A)} f l
w h e r e the s u p r e m u m i s t a k e n over a l l finite p a r t i t i o n A of 5 .
253 [02] (1) C-algebra
Definition2.1
v e r s i o n : T h e S-entropy
S (
of a state tp
w i t h respect t o a c o m p a c t c o n v e x set 5 c o n t a i n i n g tp is the i n f i m u m of H(p)
for a l l
possible pi's. (2) V o n N e u m a n n a l g e b r a v e r s i o n : T h e S-entropy
S (tp) s
of a n o r m a l state tp w i t h
respect t o a n o r m closed convex set 5 of n o r m a l states, c o n t a i n i n g ip is the i n f i m u m of H(fi.) for a l l p o s s i b l e / i ' s , where p. i s a t o m i c , s u p p o r t e d i n e x t r e m e p o i n t s of 5 , a n d representing tp. I n t h e a b o v e d e f i n i t i o n we t a k e t h e c o n v e n t i o n t h a t i n f
for a d e n s i t y o p e r a t o r p on a c o m p l e x H i l b e r t space:
T H E O R E M 2.2 [ O U ] : Let {pj-J sional projections,
= 1,2, • • • } be pure states identified
= 1,2, • • •} be positive
{Xj\j
5(£)A,-ft) <
A
i Equality
holds if {pj,
numbers
;
with one
dimen-
with sum 1, then,
l o g A,-.
i
A } are Schatten ;
decomposition
of p.
5 - e n t r o p y posesses several n a t u r a l p r o p e r t i e s as " q u a n t u m e n t r o p y " , a n d has been a p p l i e d to q u a n t u m c o m m u n i c a t i o n t h e o r y [04]. 3.
C O N T I U I T Y OF 5 - E N T R O P Y
O n e o f t h e m a i n p r o p e r t i e s of v a r i o u s q u a n t u m entropies is t h e i r lower s e m i c o n t i n u i t y w i t h respect to the convergence of q u a n t u m states. T h i s p r o p e r t y is essential i n the a p p l i c a t i o n o f q u a n t u m entropies to the s t u d y of models of q u a n t u m s t a t i s t i c a l m e c h a n i c s . C o n c e r n i n g t h e c o n t i n u i t y of 5 - e n t r o p y , the followings are k n o w n . L e t tpj be a net o f states over A c o n v e r g i n g i n the n o r m to a s t a t e tp. D e n o t e by 7 ( a ) set o f a l l a - i n v a r i a n t s t a t e s , a n d b y K(a)
the
t h e set o f a l l a - K M S states at the inverse
t e m p e r a t u r e /3 = — 1, d e s c r i b i n g e q u i l i b r i u m of the s y s t e m w h e n G = R t h e real l i n e . T H E O R E M 3.1 [ O M ] : Iftpj
are all a-KMS S W(tp) K
states,
then so is tp, and
J C ( a )
(^ ). J
3
THEOREM system,
3.2
[ O M ] : If tp, are all a - i n v a r i a n t states i n a G-abelian
C*-dynamical
then so is tp, and 5
/ ( o , )
(v)
a
3
I n t h e a b o v e t w o t h e o r e m s , t h e a p p e a r i n g c o m p a c t convex sets are C h o q u e t s i m plexes [ A I ] , some k i n d o f i n f i n i t e d i m e n s i o n a l s i m p l e x e s . I n a C h o q u e t s i m p l e x 5 , a n y
254 state tp i n S has the u n i q u e r e p r e s e n t a t i o n as a b a r y c e n t e r of a m a x i m a l m e a s u r e o n S. B u t t h i s is n o t t h e case for g e n e r a l q u a n t u m s y s t e m . I n q u a n t u m t h e o r i e s , t h e state spaces i n c o n s i d e r a t i o n do not a l w a y s f o r m C h o q u e t s i m p l e x e s .
So we m u s t e x t e n d
the a b o v e theorems t o the s i t u a t i o n t h a t S do n o t f o r m C h o q u e t s i m p l e x e s . t h i s p r o b l e m , we get t h e f o l l o w i n g s .
c o n v e r g i n g i n t h e n o r m t o a s t a t e tp, a n d A" A.
A s for
L e t tp, be a net of states over a C * - a l g e b r a A the e n v e l o p i n g v o n N e u m a n n a l g e b r a of
L e t us assume t h a t A is separable for s i m p l i c i t y .
T H E O R E M 3.3 [ M ] : S (
S (tpj).
G
6
T H E O R E M 3.4 [ M ] : If a is a c o n t i n u o u s a c t i o n of compact
S (tp) I(a)
T H E O R E M 3.5 [ M ] : If a is an a c t i o n of diciete
/ ( a )
group
(y> )J
group
;'
I{a)
G, and if A"
and if A"
is G-Rnite,
then
/(o)
T H E O R E M 3.6 [ M ] : If a is a c o n t i n u o u s a c t i o n of separable dual A*,
then
S (
separable
G,
is G-Rnite, S (
kminfS j
<
I{a)
group
G, and A has
the
then J(a)
(»>,•)•
W e o n l y give here the d e t a i l e d p r o o f of T h e o r e m 3.3.
T h e proof of others w i l l
a p p e a r i n [ M ] . A t first we p r e p a r e t h e f o l l o w i n g l e m m a 3.7 c o n c e r n i n g the e x t r e m e b o u n d a r i e s of t h e c o n v e x sets of n o r m a l states o n f a c t o r s . L e t A f be a v o n N e u m a n n a l g e b r a , 6„(M) LEMMA
3.7:
type III factor, PROOF.: tor M.
t h e convex set of a l l n o r m a l states over (i) If M is a type II factor, then ex<5 (M) N
M. = <j>. (ii) If M is a
then ex&„(M)
o--finite
=
(i) L e t r be the u n i q u e f a i t h f u l n o r m a l s e m i f i n i t e t r a c e o n the t y p e I I facE a c h n o r m a l state ifi € 6 „ ( A f ) is r e p r e s e n t e d by t h e u n i q u e r - i n t e g r a b l e
o p e r a t o r T = - j f 6 L (M,T)
the R a d o n - N i l c o d y m d e r i v a t i v e o f ifr, f r o m S e g a l ' s n o n -
X
c o m m u t a t i v e i n t e g r a t i o n theory. Since a n y t y p e I I f a c t o r posseses no m i n i m a l p r o j e c t i o n , we c a n d e v i d e T i n t o n o n - t r i v a l p o s i t i v e r - i n t e g r a b l e o p e r a t o r s T = T
X
+ Tt(Tr
# 0,Tj ^ 0,T
t
> 0,T
T^,T : 2
> 0).
2
L e t tj)\, ip2 be n o r m a l p o s i t i v e f o r m defined by ^ = T ( T - ) , ^ = r(r ). 1
1
r
T h e n we have V> = V"i + i>i- B y n o r m a k z i n g V ' i i V ' 2 > d e c o m p o s i t i o n o f ^ i n t h e convex set 6 (M). N
p o s a b l e , a n d hence e z © „ ( A f ) = <j>.
w
e
g
e t
a
n
o
n
trivial extremal
T h e r e f o r e each n o r m a l s t a t e is d e c o m -
255
(ii) For a normal state i> G 6 (M) on Af, i> e ex& (M) is equivalent to tr^M) = B(Ti^,), where (*y,7i^) is the G N S representation associated to tj) and B(7i^,) is the set of all bounded operators on H^,- Let us assume that ex6 (M) ^
n
n
n
*V(A4) = B(W^)
Since any
: M -> S ( W v , )
v
is a *-isomorphism or zero map. B y the way, 7ty is not a zero map because of Ti^, ^ 0. So x^, is a *-isomorphism and hence M is a type I factor. This contradicts with the fact that A t is a type I I I factor. Therefore ex& (M) =
3 . 3 : Let M
PROOF OF T H E O R E M
be the enveloping von Nuemann algebra of
A,
and ir be the canonical embeding of A into Ai. Since the Banach dual A * of A is canonically isomorphic to the predual Ai, of M, i.e. .4* = A 4 . : tp —* ^>(y? = ij> o 7r),
there is a natural isomorphism between two closed convex sets 6(A) and ©„(A 1). So there is a natural bijection between the extreme boundaries G(A) and @ (M). Let us decompose the enveloping von Neumann algebra M as follows. Let {-fjjigr, be the set of all minimal central projections Fi G Proj(Z), where Z is the center of M. Using minimal central projections, we get the discrete decomposition of the enveloping von Neumann algebra Ai /
n
M
= (QitL^Mt)
® (e
m
€
L
l
M
m
)
® (®n€L,M ) n
©
M,,
where L = L\ + L2 + I> , and A t j (resp. Ai ,M„) is the factors of type I (resp. type II, type in) induced from minimal central projections of type I (resp. type II, type in) and Ai is the von Neumann algebra induced from central projection I — YlktL ^hBy the way, since the central support z(tp) of any extremal state tp G e z © ( . 4 ) is a minimal central projection, the extremal state tp is identified as a extremal normal state i n ex& (Aik) on some factor Mh(k G L ) . Any factor representation of a separable unital (7*-algebra is quasi-equivalent to some factor representation on a separable Hilbert space. Since any type n or tr-finite type in factor possesses no extremal normal states, we get ex<5 (Aih) = 4> for k 6 £2 + LB- So any extremal state tp G e z © ( . 4 ) is identified as a extremal normal state Y> 6 e z © „ ( A t i ) on a type I factor Mi(l G £ 1 ) Denote by ©<j the set {tp G S ; 3 / i which is atomic, supported in e z © ( . 4 ) , and representing tp}, and by K the set {k G J\tpu £ © i } - We first consider the case that K is cofinal with J. Let Tl>,ip),(k 6 K) be the normal states on M canonically corresponding to states tp,tp . tp converge to tp in the norm topology of © ( . 4 ) , and hence ip), converge to ij> i n the norm topology of & (Mi), where Mi = ©igLjA^i. 3
m
T
n
n
k
h
n
256 S i n c e 6 „ ( A 4 / ) is i d e n t i f i e d as a n o r m - c l o s e d subset of © „ ( A 4 ) , t h e s t a t e ip is also i n &n{Mi),
a n d hence tp is i n 6 j .
o p e r a t o r s p ,p h
S i n c e i> ,i>
£ © n ( A - t j ) are i d e n t i f i e d w i t h d e n s i t y
h
on a H i l b e r t space W j = e i i , 7 i j , where W is a f a i t h f u l r e p r e s e n t a t i o n e
(
space of M.\ , a n d since pj converges t o p i n the t r a c e n o r m , we h a v e S (
< l i m i n f S(p ) h
= S(p)
= liminf k
k
S (
h
f r o m the lower s e m i c o n t i n u i t y of v o n N e u m a n n e n t r o p y i n t h e t r a c e n o r m [ O U ] . T h e r e fore we have S ( y > ) <
l i m i n f y S (ipj).
6
liminf; 5 ® ^ )
When K
6
is not c o f i n a l w i t h J,
we
have
= oo. So we get t h e desired r e s u l t . |
T h e o r e m s 3.4, 3.5, 3.6 c a n be p r o v e d u s i n g c o n d i t i o n a l e x p e c t a t i o n s [U2] a n d the K o v a c s - S z u c s ' s e r g o d i c t h e o r e m [ O U ] . W h e n we consider as 5 g e n e r a l c o m p a c t c o n v e x subsets o t h e r t h a n © or 1(a)
or K(a),
T H E O R E M 3.8 [ M ] : If 5 is metrizable
we get and exS
is weakly * closed,
S (
S (tpj)
s
for a n y net of states
then
s
in the weak* topology
to a state (p.
T h e d e t a i l e d p r o o f of t h e above t h e o r e m s w i l l a p p e a r i n [ M ] . 4. 5 - E N T R O P Y A N D C N T E N T R O P Y L e t A be a u n i t a l C * - a l g e b r a , ip a s t a t e o n A, a n d N £ A be a finite d i m e n s i o n a l C ' - s u b a l g e b r a . T h e n the C o n n e s - N a r n h o f e r - T h i r r i n g e n t r o p y H (N) V
H {N)= V
sup
is g i v e n b y
^ft.SfefJV.^JV),
(4.1)
where s u p r e m u m is t a k e n over a l l finite d e c o m p o s i t i o n o f a s t a t e
and the C N T entropy. and C N T entropy
of-
T h i s is p r o v e d b y u s i n g t h e t h e o r y of r e l a t i v e e n t r o p y a n d sufficiency
in [H0T1,H0T2].
I n the f o l l o w i n g d i s c u s s i o n s , we m a i n l y d e a l w i t h v o n N e u m a n n
algebras, i n s t e a d of C * - a l g e b r a s , for s i m p l i c i t y . A t first, t h e f o l l o w i n g h o l d s T H E O R E M 4 . 1 : F o r a n y s t a t e
=
C*-algebia
A,
H (A). V
T h i s result was e s t a b l i s h e d b y O h y a a n d P e t z , r e c e n t l y ( p r i v a t e
communication).
Its d e t a i l e d p r o o f w i l l be a p p e a r e d i n [ M O P ] . U s i n g t h i s t h e o r e m we get t h e f o l l o w i n g .
257 T H E O R E M 4.2:
be a C*-dynamical
Let (A,G,a)
S*'\ )
system
=
V
with G compact,
then
H (A ). a
v
Let
T H E O R E M 4.3:
For any normal
state ip on a von Neumann S ' 6
T H E O R E M 4.4:
S^'fo)
T H E O R E M 4.5:
For any a-KMS
=
l
M
\
V
)
=
algebra
Ai,
H,{M)
H (M"). V
normal
state
S ^W)
=
K
H {Z). V
We only give here the proof theorem 4.4. The proof of others will appear in [MOP]. P R O O F : Since the fixed point algebra Ai" is sufficient for the invariant states 7 (a) [H0T1], there exists a normal conditional expectation e : Ai —» Ai , preserving all <j> € 7 ( a ) (i.e., <j>(e(A)) = <j>(A),A £ Ai). Then the associated channel e*, the transpose of e, maps all normal states on Ai into the invariant states 7„(a), affinely, and this gives the isomorphism between closed convex sets S „ ( A 4 ° ) and I (a). So any finite decomposition in 6 (Ai ) is equivalent to one in I (a), and we have n
a
n
a
n
a
n
S^ \
= S "( "\
M
n
=
a
H r '(M ) ¥
= sup{Y HS(iiMM )\v\M a
J
e
e
»(^ )> a
i
= up{Y,r>jS(r \M ,y[M );
j
a
a
= Y,HVi{M ,p
a
a
i
i
€/»(«)}
i
= ^{Y y. S( \M MM )-MM i
j
=
a
=
a
;'
d
M
Vi
a
a
a
= Y.WilM'w
€ e (M)} n
i
BAM")
I
We are now studying the relations among various quantum entropies in order to understand their conceptual meanings and to clarify the correspondence of classical entropy and quantum entropy. ACKNOWLEDGEMENTS The author is greatly thankful to Prof. M . Ohya for a number of suggestions and encouragement for this work, and he also thanks to Prof. M . Ohya and Prof. D . Petz allowing him to report the results of our joint work [MOP]. The author would like to express his sincere thanks to the organizing comittee for the kind hospitality.
258
References [AI] Alfsen, E . , Compact Convex Sets and Boundary Integrals, Springer-Verlag, BerlinHeidelberg-New York, 1971. [A2] A r a k i , H . , Relative entropy for states of von Neumann algebras, P u b l . R I M S , Kyoto Univ., vol.ll(1976), pp.809-833. [BR] Bratteli, O. and Robinson, D . W . , Operator Algebras and Quantum Statistical Mechanics II, Springer-Verlag, New York, 1981. [C] Connes, A . , Entropie de Kolmogorov-Sinai et mechanique statistique quantique, C . R . Acad. Sci. Paris t 301 1,1 (1985). [CNT] Connes, A . , Narnhofer, H . and Thirring, W., Dynamical entropy of C*-algebras and von Neumann algebras, Commun. M a t h . Phys., vol.112 (1987), pp.691-719. [CS] Connes, A . , and Stormer, E . , Entropy of automorphisms of Hi- von Neumann algebras, A c t a M a t h . , vol.134 (1975), 289. [HK] Haag, R . and Kastler, D., A n algebraic approach to quantum field theory, J . M a t h . Phys., vol.5(1964), pp.848-861. [HOT1] Hiai, F . , Ohya, M . and Tsukada M . , Sufficiency, K M S condition and relative entropy in von Neumann algebras, Pacific J . M a t h . , vol.96 (1981), pp.99-109. [HOT2] Hiai, F . , Ohya, M . and Tsukada, M . , Sufficiency and relative entropy in '-algebras with applications in quantum systems, Pacific J . M a t h . , vol.107 (1983), pp.117¬ 140. [M] M u r a k i , N . , Remarks on continuity of entropy of states on C- dynamical sytems, to appear in J . M a t h . Phys. [MOP] M u r a k i , N . , Ohya, M . and Petz, D., Note on entropy of general quantum states, preprint. [NT] Narnhofer, H . and Thirring, W., Fizika, vol.17 (1985), 259. [01] Ohya, M . , On compound states and mutual information in quantum information theory, I E E E Transactions on Information Theory, vol.29(1983), pp.770-774. [02] Ohya, M . , Entropy transmission in C*-dynamical systems, J . M a t h . A n a l . Appl., vol.100 (1984), pp.222-235. [03] Ohya, M . , State change and entropies in quantum dynamical systems, Lecture Notes in Mathematics, vol.H36(1985), pp.397-408, Springer. [04] Ohya, M . , Some aspects of quantum information theory and its application to irreversible processes, Rep. M a t h . Phys., vol.27 (1989), pp.19-47. [05] Ohya, M . , Information dynamics and its applications to optical communication processes, to appear in Lecture Notes in Physics. [OM] Ohya, M . and Matsuoka, T., Continuity of entropy and mutual entropy in Cdynamical systems, J . M a t h . Phys., vol.27 (1986), pp.2076-2078. [OU] Ohya, M . and Umegaki, H . , Quantum Mechanical Entropies, Kyoritsu Shuppan, Tokyo, 1984. [SI] Segal, I. E . , Postulates for general quantum mechanics, A n n . M a t h . , vol.48(1947), pp.930-948. [S2] Segal, I. E . , A note on the concept of entropy, J . M a t h . M e d i a n . , vol.9 (1960), pp.623-629. [UI] Uhlmann, A . , Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in interpolation theory, Commun. M a t h . Phys., vol.54(1977), pp.21-32
259 [U2] U m e g a k i , H . , C o n d i t i o n a l e x p e c t a t i o n i n a n o p e r a t o r a l g e b r a , T o h o k u M a t h . J . , vol.6(1954), pp.177-181. [U3] U m e g a k i , H . , C o n d i t i o n a l e x p e c t a t i o n i n a n o p e r a t o r a l g e b r a V I , K o d a i M a t h . Sem. R e p . , vol.l4(1962), pp.59-85. [vN] v o n N e u m a n n , J . , M a t h e m a t i s c h e G i u n d l a g e n der Q u a n t e n m e c h a n i k , S p r i n g e r , B e r l i n , 1932.
Quantum Probability and Related Topics Vol. VII (pp. 261-274) © 1992 World Scientific Publishing Company
261
APPLICATIONS O F FINITELY CORRELATED STATES T O T H E GROUND STATE P R O B L E M OFQ U A N T U M SPIN CHAINSt*
Bruno Nachtergaele * Instituut voor Theoretische Fysika, Universiteit Leuven, B-3001 Leuven, Belgium
We review recent results on Finitely Correlated States on Quantum Chains. Particular attention is given to applications in Quantum Spin Systems. It is illustrated how the techniques of Finitely Correlated States can be used to obtain rigorous and exact information on the ground states of VBS models. Using this new method one can obtain detailed results on the ground state degeneracy, on ground state entropy, on correlation functions and on the nature of the spectrum of excitations in the ground state.
1. I n t r o d u c t i o n . Almost all of the material I want to present in this paper can be regarded in two different ways, in a perfectly parallel manner: either one can consider the results as a systematic analysis of a special class of Discrete Time Quantum Stochastic Processes, or as a study of the physical properties of the so-called Valence-Bond-Solid-models. The latter are Quantum Spin Systems for which the ground state problem can be solved exactly. Recently they called quite some attention in the physics literature in connection with a conjecture of Haldane [16]. The ground states of these models can be constructed explicitely and their properties can be studied in great detail due to a kind of Markovian structure of the correlations. In the context of Quantum Probability this structure was first introduced by Accardi in [3] under the name Quantum Markov Chains and further studied in [18]. In a paper of Affleck, Kennedy, Lieb and Tasaki [5] the simplest example of a VBS-model was analized in detail. That these states and their generalisations were examples of "Quantum
t Research financially supported by the Fondo Nacional de Desarrollo Cientffico y Tecnologico (Fondecyt, Chile), project Nr 90-1156 # Talk held at the Vth Workshop on Quantum Probability and Applications, december 30,1990-january 4, 1991, New Delhi * Onderzoeker I.I.K.W. Belgium
262 Markov States" was first realized in [10] by Fannes, Nachtergaele and Werner. The same authors then undertook a detailed study of the structure of these states and derived a series of remarkable properties of the VBS-models [11-14]. In this contribution I want to review some of the most typical results of these works. It was not possible to give all the proofs at full length. For the reader who is interested in proofs, sufficient indications are given.
2. F i n i t e l y C o r r e l a t e d States We are concerned with Stationary Discrete Time QSP with a. finite state space, i.e. we consider the following structure: • A i = (^2jA,A
C Md,l
G A, where Md denotes the C*-algebra of complex d x d
matrices and the tensorproduct is the minimal C*-tensor product; • LJ is a translation invariant state of A z; t for all i e 1 we consider the natural embedding ji : A —> A z which identifies the i
t h
factor in A zFor any A C 2 we define A\
=
\/i^ji(A).
completely specified when we know uj(j\(Ai)j2(A2) Ai,...,
A„
G A.
^he translation invariant state u is • • • j (A„)) n
for all re > 1 and for all
We are now interested in describing the correlations between
"past"
and "future" which are present in OJ. Let us introduce the following set of "conditional functionals" of w. for all Y € A^-oop) "Y
: .4[i,oo) -
€ : OJY(X) = u(Y ® X)
where Y
liOQ
) = Ai-
that the functionals u> form a linear subspace of the dual of -4[i ). Y
i00
It is obvious
One might expect
that correlations have a particularly simple structure in cases where this linear space of functionals is finite-dimensional. Therefore we define:
2.1 D e f i n i t i o n . The translation invariant state u> is Finitely Correlated if V = {u;y|K £ A(-oo,o]} is a finite-dimensional linear subspace of A^
>.
To give an idea of what kind of states are contained in the class of Finitely Correlated States (FCS) we now give some examples of both classical and quantum processes (states) that are finitely correlated.
263 a) Classical Processes: one then takes A to be the abelian algebra of diagonal matrices, i.e. A =
C({l,...,d}).
• Any stationary Markov process is finitely correlated. Indeed it is straigthforward to check that dimV < d. • Let (Xi)jg x be a finitely correlated process taking values in a finite set I (i.e. a stochastic process such that the functional denned by the associated probability measure, is finitely correlated). Take non-negative numbers F ,x y:X
finite set and such that for all x, ^
s
F
G I,y
G J, where J is another
= 1. Then the process ( F O i e Z defined by
yjZ
- »..*i'--^,*„Prob(A-
ProHY,., = » ! , . . . , 1 7 . =y„)=
F
i l
= x
u
..., X
=
in
x) n
Xl,...,X„
is again finitely correlated. In particular functions of Markov processes are finitely correlated. It also follows that e.g. a two-block factor of a finitely correlated process with finite state space, is again finitely correlated. • The set of finitely correlated measures is also closed for taking convex combinations and conversely, the ergodic components of a finitely correlated measure are finitely correlated. • A process is called one-dependent if for any i G 2 the "past" (-Xy)jj. In [1] the authors study a class of examples of one-dependent processes taking values in {0,1). In particular they study one-dependent processes with the property that the probability of finding 3 consecutative l's vanishes. This implies that the process is finitely correlated (it is not true that any one-dependent process is finitely correlated). It is actually not difficult to see that for a one-dependent process with vanishing probabiltity for finding m consecutative l's, one has dimV < m.
b) The most interesting examples of F C S for Quantum Spin Chains are obtained by a slight generalisation of a construction of Accardi and Frigerio [4]:
2.2 C o n s t r u c t i o n / D e f i n i t i o n . A translation invariant state u> of Ax Correlated
State (C*-FCS)
completely positive
map mapping A®B
into B, with B an auxiliary C*-algebra
1; p is a state of B. Further these objects have to i) 1E(1
A
® I ) = e
1
B
ii) p(E(I.4 ® B)) = p(B)
is a
C*-Finitely
if it can be given in terms of a triple (p, US, I), where E is a
for all B G B
satisfy:
with unit
264
(1)
with for all A £ A }E
A
:B -
B : E A ( B ) = E ( A m B)
To give an idea of in what sense C * - F C S are simpler than general F C S , we first studied the abelian ("classical") case:
2.3 T h e o r e m . A classical stationary
process with finite state space is a function of a
Markov process with a finite state space iff it is a C*-FCS
such that B can be taken to be
abelian.
The proof of this theorem and other results on finitely correlated measures can be found in [9]. What we learn from this theorem is that interesting phenomena have to be sought mainly in cases where B is non-abelian.
2.4 R e m a r k . Replacing,
in the definition of C*-FCS,
ordered vector space V, E by a completely
positive
B by a Bnite-dimensional
matrix-
map such that E i leaves a suitable
positive element e £ V invariant, and taking for p a positive linear functional
on V, one can
express any FCS by a formula of the type (1).
3. G r o u n d S t a t e s o f Q u a n t u m S p i n C h a i n s 3.1. T h e G r o u n d State P r o b l e m Our initial motivation for this study came from problems in Quantum Spin Systems. There are several ways in which Quantum Spin Hamiltonians arise [7]. Here we want to focus on Ground States for a particular class of Hamiltonians. A chain of spin s, with s £ |IN, has a single site algebra A = Mi,d translation invariant nearest neighbour
interaction
selfadjoint element h = h* £ Mi ® Mi.
Putting
= 2s + 1.
A
for the spin chain is determined by a = ji ® ji+i(h)
one defines the local
Hamiltonians for the chain as: for all m < n £ 2 n-1
(2) 1 =m
265 For any C * -algebra C denote by S(C) the set of all states of B. A state u> of A[ ^ 0
m
is then
called a local ground state of the spin chain with local Hamiltonians (2) if
M
wo(-H[ ,„])= m
«(%,,»])
A ground state of the infinite chain is then a state w of Az which is a weak limit of local ground states: for a choice of increasing intervals (A„) u{A)= for any strictly local A G Ax-
lim w „ ( A ) A
(3)
We remark that this is not the most general setup, for
we are not considering the most general type of boundary conditions (see e.g. [6]). We strongly believe however that for the so-called VBS-models we are interested in here, the above definition of ground state includes all ground states of the model satisfying arbitrary boundary conditions. To solve the ground state problem for a Quantum Spin Model, i.e. to determine the set of limiting states (3), one should at least be able to recognize some regularity in the structure of the local ground states. A complication that arises in a generic model is that for a finite volume ground state its restrictions to smaller volumes do not produce local ground states of these smaller volumes. To state it in a different way, the minimization of the energy cannot be done locally: the energy n-1 ^[m,n](#[m,„])
=
w
[ m , n ] ( £
attains its minimal value in this particular state but the same state does not minimize the terms in the sum separetly, not even approximately. Nevertheless exact ground states or at least the exact values of the ground state energies have been obtained for quite some models: the classical models, the Heisenberg ferromagnet, the one-dimensional Heisenberg antiferromagnet and its generalisations (see e.g. [17,7]). In the following we will show that quite a huge class of interesting models exists, where this complication does not occur: the so-called generalised VBS-models. In these models the energy can be minimized locally. This phenomenon, we believe, is closely connected with the fact that the corresponding ground states are finitely correlated. 3.2 Some basic examples: integer-spin V B S - m o d e l s [5,2,10] The interactions we want to define now are SU(2)-invariant and a lot of the calculations are based on the representation theory of SU(2) (see e.g. [8]) . The basic facts are the following: for each half-integer s there is an up to unitary equivalence unique irreducible
266 representation of dimension d = 2s + 1. The generators of this representation are denoted by S ,S x
y
and S', and they satisfy [ S ° , S ' ] = * $ > „ , ^ 7
It is convenient to put them together in a vector S . One then has S* • S* = (2s + 1)1 We also write S i for jj( S ) etc. The representation itself is denoted by D ,
i.e. for all
3
g £ SU(2), D,(g)
is a unitary element of Md- Reduction of tensorproducts of irreducible
representations of SU(2) is given by the Clebsch-Gordan series: for juh D
® D
h
h
S D_ lh
|+i © • • • © D
©
M
For any 0 < it < 2s we will denote by subspace in €
d
h
+
£ |N:
h
the orthogonal projection onto the spin=fc
® C .
Let s = l , 2 , 3 , . . . and define the following interactions for a spin-s chain:
h=
where the coupling constants
£
are arbitrary but strictly positive. These interactions can
be seen as modifications of the Heisenberg interaction (= S j • S ;+i); e.g. for s = 1 one has hi,i+i = J {^ 2
+ - S i • S ; i + i ( S i • S ,+i) } 2
+
For higher values of the spin there are one or more non-trivial continuous parameters which may vary between certain limits. In fact the methods we will describe below apply to a much larger class of models (see [12]) and SU(2)-invariance is in fact not an essential feature. A characteristic feature is that the local Hamiltonians are positive operators and that, for any of these models, one can find a C*-finitely correlated state u) such that w(hi,i+i) = 0. It is then obvious that the local restrictions of this state are local ground states and therefore, according to our definition, u; is an infinite-volume ground state of the model.
For the
integer-spin models defined above, such a state u> is constructed as follows: the auxiliary algebra for the FCS-construction is taken to be B = M,+i,
the state p =
7^-yTr and
the CPU-map E is defined in terms of the up to a phase unique intertwining isometry V :C
s + 1
-» C
2 s + 1
® C
+
1
that satisfies
D.(g) ® D, {g)V /2
= VD, {g) /2
for all g £ SU(2)
267 One then has the following fundamental theorem:
3.1 T h e o r e m . Let s = 1,2,3,
For any ground state U>Q of the model with
h, there is a constant c > 0 such that wo < c u , ui being the C*-£nitely
interaction
correlated
state
given above.
In general we will call a VBS-model a model with an interaction h such that a theorem similar to the previous one, can be derived:
3.2 D e f i n i t i o n . An interaction interaction
h > 0 for a Quantum
iff we can hnd a C*-hnitely
Spin Chain will be called a VBS-
correlated state w such that w(hi,i+i)
= 0 for aij
t £ 2 and such that for any finite volume A and any local ground state U>A of the model, there exists a constant c > 0 such that ui\ <
cu\A\.
It is now clear that solving the ground state problem for a VBS-model is equivalent to decomposing the C * - F C S ui into its pure components. This motivates for the following section.
4. D e c o m p o s i t i o n T h e o r y The crucial object in the decomposition theory of C * - F C S is the transition operator P = E t . This is not amazing because P obviously governs the asymptotic behaviour of the correlation functions of the state. In the following we assume the the algebra B is generated by {Eyi o • • • o E y i ( l ) | j 4 i , . . . ,A„ I
n
£
A}
This is a matter of working in a "minimal" representation of the C * - F C S (see [12]). Then the following statements can be proved: 1) A l l eigenvalues A of P satisfy |A| < 1, and obviously 1 itself is an eigenvalue of P because P ( l ) = I . 2) The eigenvectors corresponding to the eigenvalue 1 are all of the form and where {P } a
is a resolution of the identity.
cP, a
a
c
a
S C,
268 3) The ergodic decomposition of our limiting state uio is given by u
=
0
Or where the u
a
are translation invariant C*-finitely correlated states generated by a
completely positive map E bra P MkP a
a
a
which is the restriction of E to the invariant subalge-
of Mk- These states w cannot be further decomposed into translation a
invariant states. The eigenvalue 1 of P ( - ) = E ( l „ - ® • ) is non-degenerate. a
0
4) It is possible however that the uj can be decomposed further into periodic states. This a
depends on further (complex) eigenvalues of the operators P . One can show that the a
eigenvalues of modulus one of P „ (the peripheral spectrum of P ) is a cyclic group of 0
roots of unity: there is a positive integer n such that the eigenvalue of modulus 1 are a
given by {exp(iq/n )\q = 0 , 1 , 2 , . . . , n — 1} and they are n a
can then find n
a
a
a
simple eigenvalues. One
periodic states ui ,o, • • • , U a , i i „ - i of period n , which are translates a
a
of one another and such that -1
The results l)-4) hold for the general situation of C*-finitely correlated states.
In
particular the completely positive map E does not need to be of the form V * • V (as is the case in the examples given up to now). If it is of this particular form with one single isometry, we say that E is pure. Then we can prove the following: 5) The periodic states
iji
a>q
are pure, i.e. they cannot be further decomposed in whatever
states. So we have obtained the complete description of the decomposition of a VBS-state (like the ground states given in the previous section) into its pure components. In particular we see that translation invariance can be spontaneously broken and also a discrete local symmetry. Spontaneous breaking of a continuous local symmetry (like the SU(2)-invariance) is excluded in VBS-models, except in cases where the the phenomenon of residual entropy occurs ([13]). A n example of breaking of translation invariance is given by the MajumdarGhosh model (see e.g. [19]).
5. R e n y i - e n t r o p i e s a n d e n t r o p y e s t i m a t e s Any state a; of a quantum spin chain A z is completely described in terms of a set of
269 density matrices P{ ,... } 6 ®^Md, such that m
n
u{A)
= Tr p{ ,... )A m
all A £
for
n
A{ ...„) mj
The von Neumann entropy £ / „ . } of the state u> restricted to the volume m i
n
{ m , . . . n] C 2 is then defined by: = - T r P{ ,...„) In
S{m,...n)(u)
m
m
For q > 1 the local Renyi entropy
A related quantity was introduced by Renyi [20]. Ji|
p{ ,...„)
m
„j(w) of order q is defined by: ^ .... m
n )
M=T^lnTr(
P { m
,...„ )' }
In our case we are particularly interested in translation invariant (or periodic) states. In this case it is known that only the von Neumann entropy has sufficiently nice properties to guarantee the existence of its density s(u>) [6]:
For the finitely correlated states however, it is possible to obtain the existence of the Renyi entropy densities r'(u>): r'H
= Umsup^i |_„ t
i
n )
H.
n—oo
under conditions which are fulfilled for a generic finitely correlated state. In fact one can express the density in terms of the defining objects (IE, p). It is obvious that S{n,...„)W
=
Umit«
m n
}
H
For the densities we only have the following inequalities:
5.1 P r o p o s i t i o n . Let u be a translation invariant state of the chain algebra A z , then r (o>) < r"(w) < s(u>) 52
for 1 < 91 < g • 2
P r o o f : Fix n € IN and let P{_ ...„} be the density matrix corresponding to the restricni
tion
W{_
N I
... } N
of ui to A{_„,...„}.
Let {A,} be the set of eigenvalues of />{-„,...„} repeated
according to multiplicity. Obviously the A ; are non negative, add up to 1 and ^- ,...n)H = n
T ^ l n E
A
?
270 First we show that by Holder's inequality: R\- ,...n)(")
< R\- ,...n)H
n
K
f O T
n
91 < 92
( ) 4
Indeed, t
t ^ ^ ^ j
<
2
^ ( 9 i - i ) / ( ? 2 - i )
^ ( » - « i ) / ( « - ' )
/-y^
i _
i
^ ^
^}2^(?l-l)/(?2-l) 1
and so (4) follows by taking logarithms. As S i_„ ... (a>) = l i m ,
>
n}
} i l
#'_„,...„}(">), we obtain:
*?-„,...«}(">) < « ? - » . . . . > } ( " )
S
The Proposition now follows by dividing this inequality by 2n + 1 and taking the lim sup.
We now derive a "replica trick" which will enable us to compute the Renyi entropy in specific examples and thus, by applying Proposition 5.1, we will obtain estimates for the von Neumann entropy. Let {Ai,...A } q
be a set of trace class operators on an Hilbert space Ti and let {e; \
!; = 1,2,...} be an orthonormal basis for Ti. Observe that for q = 1,2,
Tr
H
:
Ai • • • A , = Y,( ' ' "'' i >) e
Al
A
e
i
=
( * i . A e )(e
£
e
1
il
i2
, Ae ) 2
i3
• • • {«,-, ,
Ae ) q
i%
fi,.,,*,
=
(e,, ® • • • e j , , Ai ® • • • A
£
= Tr g,... 7 i
w
(Aj ® ---A,)
s
e
i3
®•••e
if
® e ) 4l
S
where
S r #i ® • • •4 *-* q
® • • • 09 ® 91
4>\,- .
is the cyclic shift to the left on H ® •••Ti. We can apply this to a density matrix p corresponding to a state UJ to get: Tr p« = u®
(S)
(5)
271 In order to show how this relation can be used to compute the Renyi entropies of integral order for a finitely correlated state LJ generated by ( E , p), we introduce the following operators: for q = 2,3,4,... F<«> : ®*Mk -
®"M
g, ( g,, Mkj
r-
k
•• X i - ¥^\X)
= E < ® ( S ® X) ? )
with E
(® ) . ( « ^ 9
g
M j k
)
.
® • • • X , ® Kj ® • • • Y, H * ELXi ® H ) ® • • • E ( X , ® Y„
5.2 P r o p o s i t i o n . Let LJ be a finitely correlated state generated by ( E , p), then for q = 2,3,... *
in particular r'(u>) exists and
>-'H>-FrinR(F('»),
(6)
where for a linear operator A , R(A) denotes the spectral radius of A . P r o o f : Observe first that for A C 2 u;
A
® • • • U>A = w | ^ (,)
A
where w<»> is a finitely correlated state on a product of g copies of the chain: (Md ® • • • Md)x = ®Mj
. The state u/ > is a product state on A z
® • • • - 4 2 , generated by ( E * ' ' , ® ' / > ) .
s
According to (5) we have to compute w' '(rS{i ... y) where •?{].,...„} = ®"Si. ,
l
copy of the cyclic shift on the C ® • • • € d
d
n
at the i
t h
Here
is a
site of the product chain. So, applying
the defining formula (1) of a finitely correlated state, we calculate the expectation value of
*<,,...„}(")= " F T
^
( , )
(S
{ 1
,...n ) }
= - ^ T ln(®'p)((E)"(®
h(®V)((F
(
,
)
Wl))
As E ^ ' ' is a linear transformation on a finite dimensional space r«(w) = l i m s u p i / i j , n—.oo
exists and satisfies (6).
•
In order to prove that in equation (6) one has in fact equality, one would use a result similar to the classical Perron-Frobenius theorem. In our case however, it is not immediately
272 clear in general that the mappings F
( ? )
possess the necessary positivity. For the Renyi
entropy of order 2 it is not so difficult to obtain the following result: 5.3 T h e o r e m . Let LJ be a finitely correlated state generated by ( E , p) and suppose that F ' ' has only one eigenvalue of maximal modulus, 2
r (oj)
then:
= - In R ( E
2
( 2 )
).
E x a m p l e : Consider the finitely correlated state LJ on the spin 3/2 chain generated by the following E and a p that satisfy the compatibility conditions. Define with the up to a phase unique intertwining isometry V which satisfies: As/2 ® -°i/2
®
D
W
V =
VD
1/2
the unity preserving completely positive map E :Mi ® M
-> M
2
A ® 5 H
2
E ( A ® B) = V(A
® 1
2
® B) V
It is then straightforward to see that the state LJ generated by E is a ground state for the interaction Pl?
+i
= ^ ( 4 9 5 +972 S*,- • ti+i + 4 6 4 ( t , • s'.+i) + 64( S*. • "S i ) ) 2
j +
3
This ground state is ergodic but far from pure (note that E is not pure). It has a nonvanishing entropy density which we can estimate using Theorem 5.3. 5.4 P r o p o s i t i o n . 0.654 . . . = - ln
5
+
/ ^ = r {uj) 2
< S{LJ) < | l n 3 - | l n 2 = 0.685...
6. Characterization of pure F C S Finally we mention here two general results on pure C*-finitely correlated states. The complete proof of these theorems is quite lengthy and we cannot reproduce it here. Part of it appears in [12] and the remaining arguments will be the subject of a future publication [15]. 6.1 T h e o r e m . Let LJ be a C*-finitely then the following
are
equivalent:
correlated state generated by minimal triple (p, E , 1),
273 i)
is a pure state.
ii) ui is clustering and has vanishing entropy density. iii) The map E is pure (i.e. of the form E ( X ) = V'XV
with V an isometry) and such
that the transition operator P has trivial peripheral spectrum (i.e. the eigenvalue 1 is non-degenerate and there are no other eigenvalues of modulus 1). iv) There exists a finite range interaction h>0
such that w is the unique ground state of
the corresponding VBS-model.
The finite range interactions h which appear in iv) of the Theorem above, share a remarkable property: 6.2 T h e o r e m . The Quantum Spin Models mentioned in iv) of Theorem 6.1 are all "massive": they have a spectral gap in the ground state, i.e. there exists a constant 7 > 0 such that for all A £ Ax u>(A-[H,A]) > 7 M A ' A )
- KA)| ) 2
References 1) J . Aaronson, D. Gilat, M . Keane and V . De Valk: An algebraic construction of a class of one-dependent processes, Ann. of Prob. 17, 128-143 (1989) 2)
3)
D.P. Arovas, A . Auerbach, and F . D . M . Haldane: Extended Heisenberg models of antiferromagnetism: analogies to the fractional quantum Hall effect, Phys.Rev.Lett. 60,531-534 (1988) L . Accardi: Topics in quantum probability, Phys.Rep. 77, 169-192 (1981) Proc.R.Ir.Acad.
4)
L . Accardi, and A . Frigerio: Markovian Cocycles, (1983)
83A(2),
251
5)
I. Affleck, T . Kennedy, E . H . Lieb, and H . Tasaki: Valence bond ground states in isotropic quantum antiferromagnets, Commun.Math.Phys. 115, 477-528 (1988)
6) 0. Bratteli, and D.W. Robinson: Operator algebras and quantum statistical mechanics, 2 volumes, Springer Verlag, Berlin, Heidelberg, New York 1979 and 1981 7) W . J . Caspers: Spin Systems, World Scientific 1989 8) A . R . Edmonds: Angular Momentum in Quantum Mechanics, Press, New Jersey 1957
Princeton University
9) M . Fannes, B. Nachtergaele, and L . Siegers: "Functions of Markov Processes and A l gebraic Measures", Preprint K U L - T F - 8 9 / 1 0
274 10)
M . Fannes, B. Nachtergaele, and R . F . Werner: Exact Ground States of Quantum Spin Chains, EuToyh.ys.Lttt. 10, 633-637 (1989)
11)
M . Fannes, B. Nachtergaele, and R . F . Werner: Valence bond states on quantum spin chains as ground states with spectral gap, J.Phys.A:Math. Gen. 24, L185-L190 (1991)
12) M . Fannes, B. Nachtergaele, and R . F . Werner: "Finitely correlated states of quantum spin chains", submitted to Commun. Math. Phys. 13) M . Fannes, B. Nachtergaele, and R . F . Werner: "Entropy Estimates for Finitely Correlated States", KU.Leuven-Preprint K U L - T F - 9 1 - 9 14) M . Fannes, B. Nachtergaele, and R . F . Werner: "Ground States of VBS-Models on Cayley trees", KU.Leuven-Preprint KUL-TF-91-21 15) M . Fannes, B. Nachtergaele, and R . F . Werner,: "Characterization of pure C*-finitely correlated states", in preparation 16)
F . D . M . Haldane: Continuum dynamics of the 1-D Heisenberg antiferromagnet: identification with the 0(3) nonlinear sigma model, Phys.Lett. 9 3 A , 464-468 (1983)
17) Y u . A . Izyumov and Yu, N. Skryabin: Statistical Mechanics of Magnetically Ordered Systems, Plenum, New York 1988 18) B. Nachtergaele: "Working with Quantum markov States and their Classical Analogues", in L . Accardi and W . von Waldenfels (Eds.), Quantum Probability and Applications V , Springer Lecture Notes in Mathematics 1442, Berlin 1990 19) B. Nachtergaele: "Uniqueness and degeneracy of ground states for some quantum spin chains", to appear in the Proceedings of the Second Latinamerican Workshop on Nonlinear Phenomena, September 1990, Santiago de Chile 20) A . Renyi: " O n measures of entropy and information", Proc. 4 Berkeley Symp. Math. Statist. Prob. 1960 University of California Press Berkeley 1961 ,A
Quantum Probability and Related Topics Vol. VII (pp. 275-297) © 1992 World Scientific Publishing Company
275
Entropy in Quantum Probability I by Denes P e t z M a t h e m a t i c a l I n s t i t u t e o f t h e H u n g a r i a n A c a d e m y o f Sciences H - 1 3 6 4 B u d a p e s t , P F . 127, H u n g a r y
T h e notion of entropy was introduced b y C l a u s i u s i n order to
Introduction.
d i s c u s s t h e t h e r m a l b e h a v i o u r o f p h y s i c a l s y s t e m s . A f t e r the w o r k o f C l a u s i u s , B o l t z m a n n , G i b b s a n d others, the beginning of the m a t h e m a t i c a l treatment of e n t r o p y was d u e t o v o n N e u m a n n a n d S h a n n o n . T h e i r s t a r t i n g p o i n t s were v e r y different, v o n N e u m a n n was m o t i v a t e d b y q u a n t u m m e c h a n i c s a n d S h a n n o n f o u n d e d c o m m u n i c a t i o n theory. T h e e n t r o p y o f a state d e s c r i b i n g a p h y s i c a l s y s t e m is a q u a n t i t y e x p r e s s i n g the uncertainty o r randomness of the system. S h a n n o n regarded this uncert a i n t y a t t a c h e d t o a p h y s i c a l s y s t e m as t h e a m o u n t o f i n f o r m a t i o n c a r r i e d b y t h e s y s t e m , so t h a t t h e lack o f e n t r o p y o f a state c a n b e r e a d as the i n f o r m a t i o n c a r r i e d b y t h e state. H i s i d e a comes f r o m t h e f o l l o w i n g c o n s i d e r a t i o n : If a p h y s i c a l s y s t e m h a s a l a r g e u n c e r t a i n t y a n d o n e receives i n f o r m a t i o n o n t h e s y s t e m b y s o m e p r o c e d u r e , t h e n t h e so o b t a i n e d i n f o r m a t i o n is m o r e v a l u a b l e t h a n t h a t r e c e i v e d f r o m a s y s t e m h a v i n g less u n c e r t a i n t y . It i s u n d e r s t o o d i n p r o b a b i l i t y t h e o r y t h a t t h e n o t i o n o f ( S h a n n o n o r m e a sure t h e o r e t i c ) e n t r o p y has successful a p p l i c a t i o n s i n a v a r i e t y o f s u b j e c t s b e cause i t d e t e r m i n e s t h e a s y m p t o t i c b e h a v i o u r o f c e r t a i n p r o b a b i l i t i e s i n t h e course o f i n d e p e n d e n t t r i a l s . F i r s t we w i l l discuss t h i s p h e n o m e n o n for f i n i t e q u a n t u m systems. Entropy.
B y a finite q u a n t u m s y s t e m w e m e a n a n a l g e b r a o f m a t r i c e s w h i c h
is s t a b l e u n d e r t a k i n g a d j o i n t . ( I n o t h e r w o r d s , a finite q u a n t u m s y s t e m i s a finite
d i m e n s i o n a l C * - a l g e b r a . ) If A is s u c h a n a l g e b r a t h e n there i s a l i n e a r
f u n c t i o n a l T r w h i c h takes the v a l u e 1 a t e a c h m i n i m a l p r o j e c t i o n . It is " t r a c i a l " i n t h e sense t h a t Tra6 = Tr6a
( a , b £ A).
E v e r y f u n c t i o n a l u> o n A is d e t e r m i n e d b y a d e n s i t y o p e r a t o r D
w
€ A i n the
form ui(a) = T r D
a
w
(aeA).
T h e e n t r o p y S(ui) o f a f u n c t i o n a l u> is defined b y m e a n s o f i t s d e n s i t y o p e r a t o r as
S(UJ) = Trr](D ) u
(n(t)=-t
logt).
276
T h i s n o t i o n w a s i n t r o d u c e d b y v o n N e u m a n n i n 1927 a n d w e t e r m i t v o n N e u m a n n e n t r o p y o r s h o r t l y e n t r o p y (cf. [1]). I n f a c t , v o n N e u m a n n d e f i n e d the entropy i n such a way for the algebra of a l l b o u n d e d operators o n a given H i l b e r t space. I n t h i s case a n y t w o m i n i m a l p r o j e c t i o n s c o u l d b e t r a n s f o r m e d into each other b y means of a u n i t a r y operator w h i c h i s , of course, belongs to o u r o p e r a t o r a l g e b r a . T h e r e f o r e i t is n a t u r a l t h a t t h e m i n i m a l p r o j e c t i o n s m u s t h a v e e q u a l w e i g h t s a n d i t is a q u e s t i o n o f n o r m a l i z a t i o n t h a t w e choose t h i s weight t o b e one. H o w e v e r , i f t h e t h e o p e r a t o r a l g e b r a i s n o t a f a c t o r , f o r e x a m p l e , A = M ( C ) ffi M ( C ) t h e n there i s n o i n t e r n a l c o n n e c t i o n b e t w e e n m i n i m a l p r o j e c t i o n s o f t h e first s u m m a n d a n d t h o s e o f t h e s e c o n d one. N o w we just note this point w h i c h might become i m p o r t a n t . 2
3
L e t A b e a finite q u a n t u m s y s t e m w i t h a f a i t h f u l s t a t e ui. T h e b r a i c t e n s o r p r o d u c t A = A ® ... ® A i s a g a i n a finite q u a n t u m t h e p r o d u c t f u n c t i o n a l ui = u> ® ... ® UJ is a s t a t e o f A . U s i n g i d e n t i f i c a t i o n s t h e i n c l u s i o n (A„,ui ) C (A ,ui ) h o l d s for n < m n
n
n
n
m
m
n - f o l d algesystem and t h e obvious a n d we set
( A x > , W o o ) = U { ( - 4 „ , o ; ) : n 6 IN} . n
O n t h e * - a l g e b r a Aoo t h e r i g h t shift e n d o m o r p h i s m 7 i s d e f i n e d f o r a i ® a ® 2
• •. ® a „ G A
n
as
7(01 ® a ® . . . ® a „ ) = / ® a i ® a ® • • • ® n € 2
a
2
A +i n
a n d ujoa is i n v a r i a n t u n d e r 7. N o w p e r f o r m t h e G N S - c o n s t r u c t i o n w i t h t h e state oioo a n d a r r i v e a t t h e t r i p l e t ( i r , W , Q). W e i d e n t i f y A through its faithful representation w w i t h a subalgebra of the generated v o n N e u m a n n algebra M = *{A. )" C B(H). T h e n o r m a l state x
X
ui{a) = (n,aQ)
{a€M)
is a n e x t e n s i o n o f ojoo a n d the e n d o m o r p h i s m 7 e x t e n d s t o Ai s u c h t h a t t h e r e l a t i o n w 0 7 = u> is prese™ed. ( F o r t h e sake o f s i m p l e r n o t a t i o n we d o n o t use a new letter for the extension.) T h e f o l l o w i n g w e l l - k n o w n r e s u l t m a y b e c a l l e d t h e weak l a w o f l a r g e n u m b e r s (for i n d e p e n d e n t finite q u a n t u m s y s t e m s . ) I n t h e above d e s c r i b e d s i t u a t i o n t h e f o l l o w i n g s t a t e m e n t s h o l d .
L e m m a 1.
(i) If a G A t a n d 7(a) = a t h e n a G C I. (ii) F o r e v e r y a G Ai t h e sequence S (a) = n~ (a + 7(a) + . . . + 7 " ( a ) ) converges t o u>(a)I i n t h e s t r o n g o p e r a t o r t o p o l o g y . n
- 1
l
(iii) If a G Ai a n d J C IR is closed i n t e r v a l s u c h t h a t ui(a) £ J a n d p i s t h e s p e c t r a l p r o j e c t i o n o f S (a) c o r r e s p o n d i n g t o t h e i n t e r v a l J t h e n p —> 0 i n t h e s t r o n g o p e r a t o r topology. 3a
n
n
n
L e t u s fix a p o s i t i v e n u m b e r e < 1. F o r a w h i l e w e say t h a t a p r o j e c t i o n Qn € A
n
is r a t h e r sure i f u>n(Q ) > 1 ~ £• O n t h e o t h e r h a n d , t h e size o f n
277 Q , t h e cardinality of a m a x i m a l pairwise orthogonal family of projections n
contained i n Q ,
is g i v e n b y T r „ Q „ .
n
the algebraic trace functional o n A
n
( T h e subscript n i n T r „ indicates that is m e a n t here.) T h e t h e o r e m b e l o w says
t h a t t h e v o n N e u m a n n ' s e n t r o p y o f ui governs a s y m p t o t i c a l l y t h e size o f r a t h e r sure p r o j e c t i o n s : A r a t h e r s u r e p r o j e c t i o n i n A„ c o n t a i n s a t least exp(nS(u>)) pairwise orthogonal m i n i m a l projection. ([2]) U n d e r t h e a b o v e c o n d i t i o n s a n d w i t h t h e a b o v e n o t a t i o n
Theorem 2.
the limit relation l i m — inf {logTr„ Q
n
: Q
e A
n
n
is a p r o j e c t i o n , u > „ ( Q ) > 1 — e} = S(ui) n
n—>oo n
holds. Proof.
If D
denotes t h e d e n s i t y o f u> t h e n o n e c a n see e a s i l y t h a t
n
n
n-1 -\ogD
=
n
^
7
i
(-logZ) ) 1
i=0
w h e r e 7 s t a n d s f o r t h e r i g h t s h i f t . T h e sequence
(7*(— l o g D i ) )
behaves as
i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e s w i t h respect t o the s t a t e Woo. M o r e p r e c i s e l y , t h e p r e v i o u s l e m m a a p p l i e s for a = — l o g Di a n d says t h a t - log-D -+5(w)I n n
s t r o n g l y . L e t P(n,6)
be t h e s p e c t r a l p r o j e c t i o n o f t h e selfadjoint
— n l o g £ > „ c o r r e s p o n d i n g t o t h e i n t e r v a l (S(ui) _ 1
operator
— 6, S(u>) + 6). A c c o r d i n g
to (iii) of L e m m a 1 one has (1)
P(n,6)^I s t r o n g l y f o r e v e r y 6 > 0. I n p a r t i c u l a r , u(P(n,S))
= {P{n,8)n,Q)
-> 1
as n —» 00 a n d P(n, 6) is a r a t h e r sure p r o j e c t i o n i f n is large e n o u g h . It follows f r o m t h e d e f i n i t i o n o f P(n,6) D
n
that
P(n, S) e x p ( n S ( w ) - nS) < P(n, 6) < D
n
e x p ( n 5 ( w ) + nS)
(2)
w h i c h gives - l o g T r „ P(n,6)<S(u) n
+ 6.
S i n c e 5 > 0 w a s a r b i t r a r y we e s t a b l i s h l i m sup n—•oo
ft
inf{logTr
n
Q „ : Q „ } < S(u>).
(3)
278 To prove that S(w) is actually the limit we shall argue by contradiction. Assume that there exist a sequence n(l) < n(2) < .. . of integers, a number t > 0 and projections Q(n(k)) E An(k) (k = 1,2, .. . ) such that
(i) woo(Q(n(k)) ~ 1 - c, (ii) log Trn(k)Q(n(k)) :::; n(k)(S(w) - t). The bounded sequence (Q(n(k))h has a weak limit point in the von Neumann algebra M, say T E M . Instead of selecting a subsequence we suppose that Q(n(k)) -+ T weakly. It is straightforward to show that from (1) the weak limit
Q(n(k) )P (n(k) , 8)
-+
T
follows . Consequently, lik~~fwoo(Q(n(k))P(n(k),8)) ~ weT) ~ 1- c.
(4)
Using the first part of (2) we estimate
TrQ(n(k))
~ ~
TrQ(n(k))P(n(k), 8) Tr Dn(k)Q(n(k))P(n(k), 8) exp(n Sew) - n8)
= exp(n S(w) -
n8)w oo (Q(n( k ))P(n(k), 8))
and liminf (lk) log Trn(k)Q(n(k)) k-oo n
~S(w)-8+ k-oo lim (l )logwoo (Q(n(k))P(n(k),8)) . n k The limit term on the right hand side vanishes due to (4) and we arrive at a contradiction with (ii) if 0 < 8 < t . This proves the theorem . 0 Contrary to the commutative case the state space of a quantum system is not a Choquet simplex in the sense that states admit several extremal decompositions . For example, for A = M 2 ( (; ) the general form of a density matrix IS
D=~(1+.a 2
b-Ic
b+iC) 1-a
(5)
where a, b, C are real numbers and a 2 + b2 + c2 :::; 1. Thanks to the affine correspondence D <---> (a, b, c) we can visualize the state space as a ball (of radius 1) and surface points correspond to pure states. Let 'P be a state of a finite quantum system and 'P = I:i Ai.,pi be an extremal decomposition (that is, every .,pi is pure). Approaching from information theory one might think that the entropy of 'P is - I:i Ai log Ai. This , however, would not be satisfactory because the Ai 'S are not in general the probabilities of mutually exclusive events. In fact,
279
l o g A,-
(6)
a n d t h e e q u a l i t y h o l d s i f a n d o n l y i f t h e e x t r e m a l d e c o m p o s i t i o n 23-^1^1 is o r t h o g o n a l . T h i s w a s o b t a i n e d i n [3] a l o n g t i m e ago a n d here i t c o u l d be d e d u c e d b y m e a n s of t h e r e l a t i v e e n t r o p y as w e l l . T h e i n e q u a l i t y (6) is i n t e r p r e t e d as f o l l o w s . I n t h e sense o f i n f o r m a t i o n c o n t e n t , t h e m o s t e c o n o m i c a l e x t r e m a l dec o m p o s i t i o n is t h e o r t h o g o n a l o n e , w h i c h is i m p l e m e n t e d b y t h e d e n s i t y m a t r i x . Relative entropy.
T h e r e l a t i v e e n t r o p y is a n i n f o r m a t i o n m e a s u r e r e p r e s e n t -
i n g t h e u n c e r t a i n t y of a s t a t e w i t h respect t o a n o t h e r s t a t e . H e n c e i t i n d i c a t e s a k i n d of d i s t a n c e b e t w e e n t h e t w o states. I n i n f o r m a t i o n t h e o r y t h e r e l a t i v e e n t r o p y S(v, fi) of t w o finite p r o b a b i l i t y d i s t r i b u t i o n s v =
« j , , . K „ ) and
ft = ( A j , A , . . . , A ) o n a n n - p o i n t space is u s u a l l y defined b y 2
n
i f A ; = 0 i m p l i e s « j = 0,
(7)
otherwise . I n t h e q u a n t u m case ([23]) t h e e n t r o p y of w w i t h respect to tp is d e f i n e d b y 5(u;, ) = ¥ J
f (
1 V i ?
"( s -- e^) l o
I ?
l o
+00
i f s u p p D,,, >
suppU,
(8)
otherwise .
H e r e s u p p D ^ , d e n o t e s the s m a l l e s t p r o j e c t i o n p s u c h t h a t xp(p) =
ip(I)- I n
p a r t i c u l a r , S(w, tp) is a l w a y s finite i f t h e d e n s i t y of tp h a s s t r i c t l y p o s i t i v e e i g e n v a l u e s . ( S u c h a tp is c a l l e d f a i t h f u l . ) W h e n D
v
commutes w i t h D
and their
u
eigenvalue l i s t s a r e ( A i , A , . . . , A „ ) a n d ( « i , . . . , t„) 2
respectively, then 5 ( . ,
.)
r e d u c e s t o t h e c l a s s i c a l e x p r e s s i o n (7) d u e to K u l l b a c k a n d L e i b l e r . A l t h o u g h w e m o s t l y speak o f t h e r e l a t i v e e n t r o p y of states i t is c o n v e n i e n t t o a l l o w us a n d tp i n t h e d e f i n i t i o n of S(u>, tp) t o be a r b i t r a r y p o s i t i v e f u n c t i o n a l s . N o w we a p p r o a c h q u a n t u m r e l a t i v e e n t r o p y a x i o m a t i c a l l y . O u r c r u c i a l p o s t u l a t e for t h e r e l a t i v e e n t r o p y i n c l u d e s t h e n o t i o n of c o n d i t i o n a l e x p e c t a t i o n . L e t us r e c a l l t h a t i n t h e s e t t i n g of o p e r a t o r a l g e b r a s c o n d i t i o n a l e x p e c t a t i o n (or p r o j e c t i o n of n o r m one) is d e f i n e d as a p o s i t i v e u n i t a l i d e m p o t e n t l i n e a r mapping onto a subalgebra. N o w w e l i s t t h e p r o p e r t i e s of t h e r e l a t i v e e n t r o p y f u n c t i o n a l w h i c h w i l l be used i n an axiomatic characterization: (i) C o n d i t i o n a l e x p e c t a t i o n p r o p e r t y : A s s u m e t h a t A is a s u b a l g e b r a of B a n d t h e r e e x i s t s a p r o j e c t i o n of n o r m one EoiB tp. T h e n for e v e r y s t a t e LJ of B S(Lj,tp)
o n t o A s u c h t h a t tpoE =
= S(u>\A,
o E)
holds. ( i i ) I n v a r i a n c e p r o p e r t y : F o r e v e r y a u t o m o r p h i s m a of B we have S(u>, tp) = S(u> o a,tp o a). (iii) Direct s u m property: Assume that B =
B\ ffi B . 2
L e t tp (a ffi b) 12
=
\tpi{a) + (1 — A)
a 6 Bi,b (1 -
2
€ Bi a n d s o m e 0 < A < 1. T h e n 5(u>i2,
\)S(u> ,tp ). 2
2
+
280 ( i v ) N i l p o t e n c e p r o p e r t y : S(tp,tp) = 0. (v) M e a s u r a b i l i t y p r o p e r t y : T h e f u n c t i o n (to,tp) i-> S(uJ,tp) is m e a s u r a b l e o n t h e s t a t e space of t h e f i n i t e d i m e n s i o n a l C * - a l g e b r a B ( w h e n tp is a s s u m e d t o be f a i t h f u l ) . T h e p r o p e r t i e s ( i ) - ( v ) are w e l l - k n o w n p r o p e r t i e s of t h e r e l a t i v e e n t r o p y f u n c t i o n a l . A m o n g t h e m the c o n d i t i o n a l e x p e c t a t i o n p r o p e r t y is t h e m o s t c r u c i a l a n d i t w i l l be d i s c u s s e d b e l o w . ( I n f a c t , t h e c o n d i t i o n a l e x p e c t a t i o n p r o p e r t y includes the nilpotency.) T h e q u a n t i t y SBS(U^) =TrZ?
iogLDl'*D-WJ )
(9)
2
w
has a l l t h e p r o p e r t i e s ( i i ) - ( v ) a n d c o i n c i d e s w i t h t h e r e l a t i v e e n t r o p y f o r c o m m u t i n g d e n s i t i e s . ( V a r i a n t (9) of t h e r e l a t i v e e n t r o p y a p p e a r e d i n [4].) It was o b t a i n e d i n [5] t h a t (10)
SasQ*,*) > S(u,
> T r A ( l o g A + l o g B)
for p o s i t i v e d e f i n i t e m a t r i c e s A a n d B. ( F o r r e l a t e d t r a c e i n e q u a l i t i e s , see [28].) T h e r e l a t i v e e n t r o p y S(us,tp) is t h e i n f o r m a t i o n a l d i v e r g e n c e o f ui f r o m tp. In t h i s s p i r i t t h e c o n d i t i o n a l e x p e c t a t i o n p r o p e r t y h a s a r a t h e r n a t u r a l i n t e r p r e t a t i o n . T h e i n f o r m a t i o n a l d i v e r g e n c e S(u>,
One
c o m p o n e n t is t h e d i v e r g e n c e o f ui f r o m ip o n t h e s u b a l g e b r a A. T h e o t h e r c o m p o n e n t is c o m i n g f r o m t h e e x t e n s i o n p r o c e d u r e of a s t a t e o n t h e s u b a l g e b r a t o t h e w h o l e a l g e b r a B. R e l a t i v e t o a s t a t e tp ( o r r a t h e r t o t h e c o n d i t i o n a l exp e c t a t i o n E), t h e n a t u r a l e x t e n s i o n of UJ\A t o B is o b v i o u s l y UJ o E. H e n c e the s e c o n d c o m p o n e n t of S(u>,
u
- log
D ). v
A v a r i a n t of T h e o r e m 2 h o l d s for t h e r e l a t i v e e n t r o p y . L e t us use t h e s e t t i n g of T h e o r e m 2 b u t i n a d d i t i o n let tp be a f a i t h f u l s t a t e o n t h e finite q u a n t u m s y t e m A. W e w r i t e tp
n
and
a n d tpao for t h e c o r r e s p o n d i n g p r o d u c t states of
Aoo, r e s p e c t i v e l y . I n s t e a d of t h e t r a c e , tp
n
projections i n A
n
w h i l e ui
n
A
w i l l m e a s u r e t h e size of t h e
r e m a i n s b e i n g i n t e r p r e t e d as p r o b a b i l i t y . Set
281 R(n,e) = i n f { l o g i p „ ( Q „ ) : Q
n
g A
n
is a p r o j e c t i o n , u>„(Q„) > 1 -
e}
for p o s i t i v e e.
Theorem 4. lim
([5])
l i m — R(n, e) -
—S(us, tp)
T h e r e l a t i v e e n t r o p y m a y be defined for l i n e a r f u n c t i o n a l s of a n a r b i t r a r y C * - a l g e b r a . T h e g e n e r a l d e f i n i t i o n goes t h r o u g h v o n N e u m a n n algebras a n d n o r m a l states. L e t Ai be a v o n N e u m a n n a l g e b r a w i t h n o r m a l states tp a n d us. T h e r e l a t i v e e n t r o p y S(u>, tp) w i l l be defined b y m e a n s of t h e s p a t i a l d e r i v a t i v e o p e r a t o r . O f c o u r s e , for finite q u a n t u m systems the n e w d e f i n i t i o n reduces to the p r e v i o u s one b a s e d u p o n d e n s i t y m a t r i c e s . L e t Ai a vector { Ai' of Ai. selfadjoint
act o n a H i l b e r t space Ti so t h a t us = u>{ is a vector state g i v e n b y 6 Ti. T h e v e c t o r £ 6 Ti i n d u c e s a v e c t o r state us'^ o n t h e c o m m u t a n t T h e s p a t i a l d e r i v a t i v e A(tp/us'^) is at o u r d i s p o s a l . T h i s is a p o s i t i v e o p e r a t o r w i t h s u p p o r t [A4£] s u p p tp a n d
(i) llA^tp/ui'^) / ^|| = tp(apa*) for every a € Ai a n d for the s u p p o r t p r o j e c t i o n p of us, (ii) Ai£\ is a core for t h e r e s t r i c t i o n of Aitp/u)^) ! to the closure of Ai£. 1 2
2
1 2
( R e c a l l t h a t s u p p tp, t h e s u p p o r t p r o j e c t i o n of tp, belongs to Ai
and
[Mf;]
s t a n d s for the o r t h o g o n a l p r o j e c t i o n o n t o t h e closure of t h e l i n e a r m a n i f o l d M(\. T h e l a t t e r p r o j e c t i o n is a n element of Ai'.)
A r a k i defined the r e l a t i v e
e n t r o p y as follows ([7]). cf
\_
!>{u,
f +°°
i
[-(logAitp/us'^O
f
££
suppip ,
otherwise.
N o t e t h a t { 6 s u p p tp is equivalent t o s u p p us < /
A{tp/J :)= t
Jo
Ad£
{
-
1 1 >
s u p p tp. L e t
A
be t h e s p e c t r a l d e c o m p o s i t i o n . M o r e precisely, (11) is m e a n t as - ( l o g A( /u,' )t, V
(
t) = - f
logM(Ex,£,0 - j ~
log\d(E\{\, f ) .
(12)
Since log A < A a n d
rxd{Exu)=n^%M) fii = i/2
2
Jo we see t h a t t h e s e c o n d t e r m i n t h e r i g h t h a n d side of (12) is a l w a y s finite. So S(u),tp) is finite o r i n f i n i t e d e p e n d i n g o n t h e i n t e g r a l
l o g Xd{Ex(, £)• F r o m
282 o u r p r e s e n t a t i o n i t is n o t c l e a r ' w h e t h e r t h i s d e f i n i t i o n is i n d e p e n d e n t
o
t e
a u x i l i a r y v e c t o r £ ( b u t T h e o r e m 6 b e l o w shows t h a t i t is so). It is d e s i r a b l e to get r i d of t h e d o m a i n p r o b l e m i n (11)
caused b y t
l o g a r i t h m i c f u n c t i o n . T h e f o l l o w i n g e q u i v a l e n t d e f i n i t o n is e s s e n t i a l l y d u e
e to
U h l m a n n who embedded it into a quadratic interpolation machinery S( ,V>) = - J j m t - ^ | | 4 ( W 4 ) ' / 2 e i | 2 W
S i n c e for a g i v e n
- ll£ll ) •
<)
2
13
A > 1 ( 0 < A < 1 )
lim t - ( A - l ) = logA 1
(14)
<
(-•+0
i n c r e a s i n g l y ( d e c r e a s i n g l y ) , t h e m o n o t o n e convergence t h e o r e m of i n t e g r a l s e n sures t h a t (11) a n d (13) are e q u i v a l e n t . T h e s p a t i a l d e r i v a t i v e o p e r a t o r i n v o l v e d i n the d e f i n i t i o n of t h e r e l a t i v e e n t r o p y is a r e p s r e s e n t a t i o n d e p e n d e n t selfadjoint o p e r a t o r . It is u s e f u l to have at o u r d i s p o s a l a f o r m u l a e x p r e s s i n g t h e r e l a t i v e e n t r o p y o f t w o states i n o p e r a t o r a l g e b r a i c t e r m s i n a r e p r e s e n t a t i o n i n d e p e n d e n t way. W e r e c a l l t h a t t h e C o n n e s c o c y c l e is a o n e - p a r a m e t e r f a m i l y of c o n t r a c t i o n s . F o r n o r m a l states tp a n d u> of the v o n N e u m a n n algebra M
[Dip, Du]
= Aitp/rp')"
t
a c t i n g o n t h e H i l b e r t space H t h e C o n n e s c o c y c l e
A W ) - "
•
(* €
m)
w h e r e xj>' is a f a i t f u l n o r m a l f u n c t i o n a l o n t h e c o m m u t a n t Theorem
5 ([8])
Ai'.
If tp a n d u> are n o r m a l s t a t e o n a v o n N e u m a n n a l g e b r a
s u c h t h a t S(
o
1
<
1).
I n p r i n c i p l e T h e o r e m 5 gives p o s s i b i l i t y to c o m p u t e r e l a t i v e e n t r o p y .
On
i n j e c t i v e v o n N e u m a n n a l g e b r a s the finite d i m e n s i o n a l a p p r o x i m a t i o n a l l o w s a convenient
d e t e r m i n a t i o n of t h e r e l a t i v e entropy. L e t Ai
be a n i n j e c t i v e
von
N e u m a n n a l g e b r a w i t h n o r m a l states tp a n d u). T h e n S(u>,
w h e r e a r u n s over a l l c o m p l e t e l y
u n i t a l m a p p i n g s f r o m a finite d i m e n t i o n a l a l g e b r a i n t o Ai.
positive
(See [29] f o r
the
d e t a i l e d p r o o f . ) N e x t we discuss a n o t h e r f o r m u l a d u e to K o s a k i , w h i c h is t o o c o m p l i c a t e d for c o m p u t a t i o n a l p u r p o s e s b u t , o n t h e o t h e r h a n d , i t s h o w s a l l t h e m a i n p r o p e r t i e s of t h e r e l a t i v e e n t r o p y . T h e o r e m 6. ([9]) A s s u m e t h a t TV is a l i n e a r subspace of a v o n N e u m a n n a l g e b r a Ai a n d t h a t I is i n c l u d e d in N. If i V is dense i n t h e s t r o n g * - o p e r a t o r t o p o l o g y , t h e n for a n y tp,u> £ Ai* w e o b t a i n S(u;,
I
/ u,(y(t)*y(i)) Jl/n
+t~ ip(x(t)x(ty) i\ t d
1
w h e r e the first s u p is t a k e n over a l l n a t u r a l n u m b e r s n, t h e s e c o n d one o v e r a l l step f u n c t i o n s x : ( 1 / n , oo) —• N w i t h finite range a n d y(t)
= I —
x(t).
283 K o s a k i ' s f o r m u l a c o n t a i n s v e r y i m p o r t a n t , nevertheless n o t c h a r a c t e r i s t i c , p r o p e r t i e s of t h e r e l a t i v e entropy. W e say so because t h e f o r m u l a is b a s e d u p o n o p e r a t o r c o n v e x i t y o f t h e f u n c t i o n — l o g x a n d several o t h e r e n t r o p y l i k e q u a n t i t i e s s h a r e s i m i l a r p r o p e r t i e s . N o w we t u r n t o the c o n d i t i o n a l e x p e c t a t i o n p r o p e r t y (used a l r e a d y for a x i o m a t i c d e t e r m i n a t i o n of the r e l a t i v e e n t r o p y ) .
Theorem 7.
([10]) L e t A f b e a v o n N e u m a n n a l g e b r a a n d A f i a v o n N e u -
m a n n s u b a l g e b r a . A s s u m e t h a t t h e r e exists a f a i t h f u l n o r m a l c o n d i t i o n a l e x p e c t a t i o n E of A t o n t o A f i - If fi
a n d us are states of A f i a n d A t , r e s p e c t i v e l y
then S ( u , 9 i o £ ) = S(u>\Mi,ipi) + S(u>,u> o E).
T h e s i m p l e s t d e f i n i t i o n of t h e r e l a t i v e e n t r o p y has b e e n g i v e n b y s t a t i s t i c a l o p e r a t o r s , i n t h e m o r e g e n e r a l s i t u a t i o n s t a t i s t i c a l o p e r a t o r s are n o t a v a i l a b l e (due t o t h e l a c k of t r a c e f u n c t i o n a l ) a n d the s p a t i a l d e r i v a t i v e o p e r a t o r is the c r u c i a l o b j e c t i n t h e d e f i n i t i o n . F o r p o s i t i v e f u n c t i o n a l s of a n a r b i t r a r y C * a l g e b r a A t h e r e l a t i v e e n t r o p y m a y be d e n n e d t h r o u g h t h e e n v e l o p i n g v o n N e u m a n n a l g e b r a A**.
L e t u s r e c a l l t h a t the s e c o n d d u a l A"
of A is t h e
d o u b l e c o m m u t a n t o f A i n i t s u n i v e r s a l r e p r e s e n t a t i o n . S o every s t a t e xjs of A i n i t s u n i v e r s a l r e p r e s e n t a t i o n . So every state rjs of A a d m i t s a u n i q u e n o r m a l e x t e n s i o n tp t o A**. S(rpi, fa) =
W e set
S(fa,fa)
w h e r e t h e r i g h t h a n d side is defined for t h e n o r m a l states fa a n d fa. K o s a k i ' s f o r m u l a supplies us w i t h a n e q u i v a l e n t d e f i n i t i o n .
Jl/n
H e r e t h e d o u b l e s u p r e m u m is t a k e n as i t was d e s c r i b e d i n T h e o r e m 6. T h e G N S - c o n s t r u c t i o n m a y be u s e d t o h a v e a t h i r d e q u i v a l e n t . L e t (H,
stand
for t h e G N S - t r i p l e t for t h e u n i t a l C * - a l g e b r a A a n d t h e p o s i t i v e f u n c t i o n a l tp of A. L e t us be a p o s i t i v e f u n c t i o n a l o n A. W e w r i t e us for the n o r m a l state of n(A)"
such that
u>(7r(a)) = us(a)
(a 6
A)
i f a n o r m a l f u n c t i o n a l w i t h t h i s p r o p e r t y exists. ( W h e n us exists one says t h a t LJ is q u a s i c o n t a i n e d i n tp.) W e s h a l l check t h a t (16) If us e x i s t s t h e n S(us,tp) =
S(us,tp) follows o b v i o u s l y f r o m K o s a k i ' s f o r m u l a .
N o w a s s u m e t h a t us is n o t q u a s i c o n t a i n e d i n tp. O n e k n o w s t h a t i n t h i s case the central support z
u
of us is n o t s m a l l e r t h a n the c e n t r a l s u p p o r t z
v
of tp.
284 T h e r e f o r e t h e r e e x i s t s a p r o j e c t i o n p i n A** s u c h t h a t tp(p) = 0 a n d u)(j>) ^ T h i s y i e l d s S(u>,tp) = + 0 0 ) . P r o p e r t i e s of t h e r e l a t i v e e n t r o p y f u n c t i o n a l o f states of a C * - a l g e b r a are e a s i l y d e d u c e d f r o m t h e v o n N e u m a n n a l g e b r a case. W e s u m m a r i z e t h e m o s t i m p o r t a n t ones. Theorem
T h e r e l a t i v e e n t r o p y o f p o s i t i v e f u n c t i o n a l s of a C * - a l g e b r a
8.
possesses t h e f o l l o w i n g p r o p e r t i e s . (i) (u>,tp) t-» S(ui,tp) (ii) y ( i i i ) S(u,
is c o n v e x a n d w e a k l y l o w e r s e m i c o n t i n u o u s .
< 2S(u,
- uf
S(u>, tp ) i f ipi <
=
1.
tp .
2
2
(iv) F o r a u n i t a l c o m p l e t e l y p o s i t i v e m a p a : Ao —» A\ t h e r e l a t i o n
S(uioa,tpo
holds.
a) < S(to,tp) (v) F o r u = J^i^i
w
e
h
a
v
e
( 'f)
S
= 2D"=i
S(ui,to)
+ S"=i
u}
S(ui,ip).
T h e s e p r o p e r t i e s h a v e b e e n o b t a i n e d i n different p a p e r s at different levels of g e n e r a l i t y , (i) is r e l a t e d t o [10] a n d [12], ( i i ) i s e s s e n t i a l l y f r o m [13], ( i v ) is a r e s u l t f r o m [8] a n d (v) is f r o m [15]. It is n o t e w o r t h y t h a t one c a n i n f e r e a s i l y t h e s t r i c t c o n v e x i t y of t h e r e l a t i v e e n t r o p y i n t h e s e c o n d v a r i a b l e . L e t tp,u)\,u> be states a n d a s s u m e t h a t S(tp,u>i) 2
a n d S(tp,u> ) 2
are finite. T h e n t h e i d e n t i t y (v) gives + (1 - X)S(w ,tp)
XSfajtp)
- S(Xui
2
= A5(wi,Awi +(1
1
+ (1 - X)u> ,tp>) 2
- A ) w ) + ( 1 - A ) 5 ( w , A w ! + (1 - X)u> ). 2
2
2
If for 0 < A < 1 t h e left h a n d s i d e v a n i s h e s t h e n u>i = to
2
a c c o r d i n g to property
( i i ) . T h e s t r i c t c o n v e x i t y is c l e a r a l s o f r o m t h e f o l l o w i n g e s t i m a t e . Theorem 9.
L e t tp a n d u , be states o n a C * - a l g e b r a ( i = l , 2 , . . . , n ) . T h e n
n
n
0 > S(22 XiWi,ip) i=l
-
n XjSjuij,
i=l
tp) > ^ A,- l o g A , i=l
p r o v i d e d t h a t A ; > 0 a n d Y^i=i ^' ~ !• Proof. S i n c e t h e first i n e q u a l i t y is j u s t c o n v e x i t y , we c o n c e n t r a t e o n t h e s e c o n d one. B y i d e n t i t y (v) of T h e o r e m 8 we h a v e n i=l S i n c e S(u>j,^2"
=1
n
n
i=l
j=l
A , w ; ) < S(u>j, Xjuij)
n i=l
= — l o g A ; the s t a t e m e n t is p r o v e d .
•
T h e n e x t r e s u l t a p p e a r e d i n c o n n e c t i o n w i t h t h e d e f i n i t i o n of the d y n a m i c a l e n t r o p y i n [14] a n d c o u l d be c a l l e d m a r g i n a l i n e q u a l i t y . Theorem 10.
L e t (<£>;,*) be a finite f a m i l y of p o s i t i v e f u n c t i o n a l s o n a C o -
a l g e b r a . If tp = J2j,k
= T,k 9i,k a n d tp
2) k
= £\ tp
jik
then
285
j
J,*
T h e o r e m 11.
k
([11]) L e t K. be a c l o s e d convex set i n t h e s t a t e space of t h e C * -
a l g e b r a A a n d tp be a s t a t e of A. If F : K —> I R U { + o o } is a l o w e r s e m i c o n t i n u o u s convex f u n c t i o n a l a n d d = i n f |F(u>) + S(u>,
£}
is finite, t h e n t h e r e e x i s t s a u n i q u e ip € K s u c h t h a t F(ip) W e choose a sequence (u> ) C K s u c h t h a t l i m _ .
Proof.
n
n
+ S(>p, ip) = d.
0 0
{ F ( w ) + S(u>„, ip)} n
= d. F r o m t h e c o n v e x i t y of F a n d the i d e n t i t y (v) of T h e o r e m 8 we infer F ( w „ ) + S ( w „ , tp) + F ( w ) + 5 ( w , m
> 2 d + S lu/ ,
-
n
1+5
m
p)
lw
m
,
I .
Hence
i f n a n d m are b i g e n o u g h . T h e i n e q u a l i t y ( i i ) of T h e o r e m 8 guarantees t h a t (CJ„) is a C a u c h y sequence a n d i t converges t o a state ip 6 K. ( i n n o r m ) . F r o m lower semicontinuity F(t/>) + S(iP,ip)
< l i m i n f { F ( u ; ) + S(ui , } n V
n
< d
n
a n d ip is a m i n i m i z e r . S i n c e a n y m i n i m i z i n g sequence is convergent the m i n i m i z e r m u s t be u n i q u e . T h e o r e m 12.
•
L e t tpi ® ip2 a n d u>i be n o r m a l states of t h e tensor p r o d u c t 2
v o n N e u m a n n a l g e b r a Mi ® ip )
S(ui2,lfil
Proof.
2
® M
= S(wi,ipi)
2
a n d let Wj = ^ i 2 | A ( i (i = 1,2). T h e n /
+ 5 ( w i 2 , W l ®
(16)
2
T h e e q u a l i t y h o l d s t r i v i a l l y i f the s u p p o r t of ui\ is s t r i c t l y l a r g e r t h e n 2
t h a t of ip\ ® ip . H e n c e we m a y a s s u m e t h a t
O ® J H
ip [b)a 2
extends to a faithful n o r m a l conditional expectation E w h i c h
preserves t h e s t a t e i p i ® ip . H e n c e T h e o r e m 7 a p p l i e s a n d u>i 2
yields the statement. A c c o r d i n g to the previous theorem
2
0
E = u>i ® tp
2
•
286 S(u> ,wi
®f> )
u
+ 5(wi2,a>j ® U > )
= S(u ,tp )
2
2
2
(
2
1 7
)
1 8
)
a n d c o m b i n a t i o n o f (16) a n d (17) leads t o t h e i d e n t i t y 5 ( ^ 1 2 , ^ 1 ® fl)
= 5 ( w ! , V l ) + S(w ,
2
2
(
2
W e claim that S(U>12,¥>1 ® V 2 ) > 5 ( w i , V i ) + S(u! ,tp ) 2
( ) 19
2
w i t h the above n o t a t i o n . Moreover, i f the equality holds a n d the relative e n t r o p i e s are f i n i t e , t h e n u = Wi ® w . T h e p r o o f o f (19) is n o t h i n g b u t a reference 2
to the m o n o t o n o c i t y of the relative entropy. N o w w e t u r n t o t h e C * - a l g e b r a v e r s i o n o f (19). L e t tpi ® tp a n d u>i b e 2
states o f t h e ( p r o j e c t i v e ) t e n s o r p r o d u c t A\ ® A
2
a n d let u>; = u\ \Ai 2
2
o f t h e C * - a l g e b r a s A\ a n d A
2
(i = 1,2) as before. If u>i is n o t q u a s i c o n t a i n e d i n
2
2
t h e n (19) h o l d s t r i v i a l l y w i t h S ( w t , V i 0 V 2 ) = + ° ° - H W12 i s q u a s i c o n t a i n e d 2
i n tpi ® 1^2 t h e n we m a y pass t o t h e G N S - r e p r e s e n t a t i o n a n d m a y refer t o t h e v o n N e u m a n n a l g e b r a i c v e r s i o n o f (19). Theorem
13.
L e t tp b e a p o s i t i v e n o r m a l f u n c t i o n a l o f t h e v o n N e u m a n n
a l g e b r a Ai a n d let t G IR. T h e n t h e set rC(tp,t)
=
{u>€M* :S{u>,tp)
consists o f n o r m a l f u n c t i o n a l s a n d i s a c o n v e x c o m p a c t set w i t h respect t o t h e topology.
a(M,,M)
If S(u!,ip)
Proof.
is finite t h e n u> is q u a s i c o n t a i n e d i n tp a n d s o u> is n o r m a l .
T h e r e f o r e t h e o(Ai„,M) T o show c o m p a c t n e s s
a n d t h e o(Ai*,Ai)
topologies
i t suffices t o n o t e t h a t IC(ip,t)
coincide o n
is c l o s e d a n d
IC(tp,t).
bounded.
B o u n d e d n e s s follows f r o m w ( I ) ( l o g w ( Z ) - l o g ¥>(/)) < a n d IC(ip,t)
S(u, ) V
is closed a n d c o n v e x as a consequence o f lower s e m i c o n t i n u i t y a n d
convexity of the relative entropy.
P e r t u r b a t i o n o f states. H SA
sa
•
F o r a finite q u a n t u m s y s t e m A w i t h h a m i l t o n i a n
the free e n e r g y f u n c t i o n a l at inverse t e m p e r a t u r e 3 i s d e f i n e d as
F(u>)= (ff)-isH W
for a s t a t e u>. T h e c a n o n i c a l s t a t e tp m i n i m i z i n g t h e free e n e r g y f u n c t i o n a l possesses t h e d e n s i t y -0H
e
Tre-""'
287 T h e p e r t u r b e d s t a t e [tp \p is t h e c a n o n i c a l state for t h e H a m i l t o n i a n H + h. H e n c e [tp ]p represents t h e e q u i l i b r i u m state of t h e p e r t u r b e d p h y s i c a l s y s t e m i n w h i c h t h e energy o f e a c h state a h a s been i n c r e a s e d b y o(h). T h e d e n s i t y o f h
h
[tp% is e
-a(H+h)
T -/»(«+'>)' r e
I n t h e finite q u a n t u m case [tp ]p m i n i m i z e s t h e f u n c t i o n a l h
a n d f o l l o w i n g D o n a l d ([15]) we use t h i s p r o p e r t y as a d e f i n i t i o n i n t h e general case. D u e t o t h e s i m p l e t r a n s f o r m a t i o n
the v a l u e 0 = 1 w i l l b e fixed m o s t l y a n d we w r i t e [tp ] i n s t e a d of h
L e t M b e a v o n N e u m a n n a l g e b r a a c t i n g o n a H i l b e r t space H. T h e convex set o f n o r m a l states o f M. w i l l be d e n o t e d b y E,(A4).
W e are g o i n g t o d e a l
w i t h l o w e r - b o u n d e d selfadjoint o p e r a t o r s affiliated w i t h M. L e t u s r e c a l l t h a t a selfadjoint o p e r a t o r i s a f f i l i a t e d w i t h Ai i f a l l i t s s p e c t r a l p r o j e c t i o n s b e l o n g to Ai.
It w i l l be a l l o w e d t h a t + o o is a n eigenvalue of t h e o p e r a t o r . M o r e f o r m a l l y ,
a n e x t e n d e d - v a l u e d l o w e r - b o u n d e d (selfadjoint) o p e r a t o r (affiliated w i t h Ad) c a n be defined b y a s p e c t r a l d e c o m p o s i t i o n .
w h e r e c G IR, p is a c e r t a i n p r o j e c t i o n i n M a n d E\ is a s p e c t r a l measure w i t h values i n t h e v o n N e u m a n n a l g e b r a p^Mp 1
a n d g i v e n o n t h e i n t e r v a l [c, oo).
N o w we are r e a d y t o define [tp \ for a n o r m a l state tp of M h
g e n e r a l i z e d o p e r a t o r h G Ai . ext
d = mi{F{u>)
a n d for a
If
= 5 ( w , >p) + h(u>) : u €
£,(M)}
is finite t h e n d u e t o T h e o r e m 11 there is a u n i q u e state [tp ] of M s a t i s f y i n g h
N o t e t h a t i f h(tp) is finite t h e n [tp ] exists a n d t h a n k s to S([tp ],tp) h
k
< +oo the
state [tp ] is a l w a y s n o r m a l . M o r e o v e r h
supp[y]
A
< s u p p tp.
It follows also f r o m t h e d e f i n i t i o n t h a t (20)
288 w h e n e v e r q is a p r o j e c t i o n i n Ai w i t h q > s u p p i p . P r o p e r t y (20) h a s t h e consequence t h a t o n e c a n e a s i l y pass t o a n a l g e b r a o n w h i c h tp i s f a i t h f u l . O n e o n l y needs t o r e p l a c e h b y ( s u p p
H
w h e r e q s t a n d s for I-p. that
W e are g o i n g t o s h o w t h a t [tp °° ] k+
S(ip, tp) = - l o g T r qe-* *H
p
= i>- O n e c o m p u t e s
- xp(k) .
k
H e n c e i t suffices t o s h o w t h e i n e q u a l i t y S{u,tp) + <xi{k) + ocu,(p) > - l o g T r g e ^ ' * -
(21)
-
for e v e r y state ui o f A. T h i s t r i v i a l l y h o l d s i f ui(p) ^ 0, so w e m a y a s s u m e s u p p a; < q. W e h a v e S(u>,tp) + u(h) > - l o g T r e - - ' H
for e v e r y h G A . (22) b e c o m e s sa
(22)
1
F i n a l l y r e p l a c e h here b y —H + qHq + k + np (n G IN) a n d
S{u,Lp) + uj(k) > - l o g T r e - ^ ' - ^ - " " = - \ogTi(qe- -
qHg k
+
e~ p). n
T a k i n g t h e l i m i t n —> oo w e c o n c l u d e ( 2 1 ) . N o t i c e i n t h i s e x a m p l e t h a t s u p p [
p
k
W e define Cf){tp,h) = m{{±S(u,tp)
+ h{Lo) : to G
B*(M)}
g i v e n a s t a t e tp G S,(M) a n d a n extended-valued lower-bounded operator h a f f i l i a t e d w i t h t h e v o n N e u m a n n a l g e b r a Ai. T h e n cp(tp, h) < + o o is e q u i v a l e n t t o t h e c l a i m t h a t [tp ]p e x i s t s , a n d i n t h i s case h
c (^h) 0
= ^S([op W) + [tp }(h). h
k
W e w r i t e c(tp, h) w i t h t h e u n d e r s t a n d i n g t h a t 8 = 1. T h e r e l a t i o n [tp ] k+k
=
{['P ] ] h
k
is v e r y m u c h e x p e c t e d a n d t e r m e d c h a i n r u l e .
T h e o r e m 1 4 . ([15]) L e t k = k* G M a n d h b e a n e x t e n d e d - v a l u e d l o w e r b o u n d e d selfadjoint o p e r a t o r a f f i l i a t e d w i t h t h e v o n N e u m a n n a l g e b r a Ai. L e t
289 be a n o r m a l state of M
s u c h t h a t c(y>, h)
<
oo. T h e n [
A+k
]
=
c(
5 ( < 7 , [v^]) + o-(fc) + c f o ft) = c( >, ft + fc) + 5(
(23)
holds. L e t A be a C * - a l g e b r a w i t h a fixed s t a t e
defined o n t h e s t a t e space a n d t h e f u n c t i o n c r->
c(ip, k) defined o n t h e selfadjoint p a r t of A m a y be c l a r i f i e d b y m e a n s of some s t a n d a r d c o n v e x a n a l y s i s . L e t V a n d U t w o l i n e a r spaces p l a c e d i n t o d u a l i t y b y a pairing <
. , . > . T h e spaces V a n d U w i l l be e n d o w e d w i t h t h e t o p o l o g i e s
cr(V, U) a n d cr(U, V ) , respectively. L e t F be a f u n c t i o n of V i n t o IR U { + o o } . T h e n the f o r m u l a = s u p { < v,u
F*(u)
: v £ V}
> -F(v)
(24)
defines a f u n c t i o n F * of U i n t o I R U { + o o } w h i c h is c a l l e d the c o n j u g a t e f u n c t i o n of F. ( F * is c a l l e d the L e g e n d r e t r a n s f o r m of F as well.) It is o b v i o u s t h a t i n (24) we c a n confine ourselves t o those v s u c h t h a t F(y)
is finite. T h i s process
c a n be r e p e a t e d a n d leads to t h e s e c o n d c o n j u g a t e F * * , w h i c h is a f u n c t i o n of V into I R U { + o o } : F " ( u ) = sup{< » , » >
- F * ( u ) : u€U}
(25)
A b a s i c r e s u l t i n c o n v e x a n a l y s i s says t h a t i f F is a lower s e m i continuous convex function then F = F * * . Choose now U =
and V
A
3a
=
A* , h
the r e a l l i n e a r space of h e r m i t i a n
f u n c t i o n a l s of A. W i t h respect to t h e d u a l i t y < v, k >=
v(k)
(k e A
sa
and v £
A\)
t h e c o n j u g a t e f u n c t i o n of TPf^) — I / 3 5 ( u ; , V 3 ) \ +oo _1
i f ui is a state otherwise.
is n o t h i n g else b u t —c^(
Large deviations.
+ c (
: k £ A' }. a
(26)
I n m a t h e m a t i c a l p h y s i c s one often meets the p r o b l e m of
d e t e r m i n i n g the a s y m p t o t i c b e h a v i o u r of a sequence t r a c e (exp H ) n
w h e r e (H ) n
(n = l , 2 , . . . )
is a sequence of selfadjoint o p e r a t o r s a c t i n g o n some H i l b e r t space.
T h e p r o b l e m arises, for e x a m p l e , i n s t a t i s t i c a l m e c h a n i c s of q u a n t u m l a t t i c e
290 s y s t e m s . I n the one d i m e n s i o n a l , case t o each n g Z a c o p y B is a s s o c i a t e d . If J #2-
C Z
then
denotes C„ j
n
of a C * - a l g e b r a
a n d S w i l l be w r i t t e n for
B
e
n
T h e n - f o l d t e n s o r p r o d u c t # [ ! , „ ] represents n i d e n t i c a l s y s t e m s i n t e r a c t i n g
w i t h each o t h e r . If t h e l o c a l h a m i l t o n i a n s R~ g B [ i j c a n be d i a g o n a l i z e d t h e n n
lim
i n
^-logTre-"""
(27)
n—>oo np
m a y be t r e a t e d b y p o w e r f u l p r o b a b i l i s t i c m e t h o d s . R o u g h l y s p e a k i n g , t h e seq u e n c e (H )
is m o d e l l e d b y a sequence of r a n d o m v a r i a b l e s a n d t h e l a r g e
n
d e v i a t i o n t e c h n i q u e is u s e d . L e t (/in) be a sequence of m e a s u r e s o n a (sufficiently r e g u l a r ) t o p o l o g i c a l space X . T h e sequence is s a i d t o o b e y t h e l a r g e d e v i a t i o n p r i n c i p l e w i t h c o n stants ( V ) a n d rate f u n c t i o n £ : X n
-* I R
+
U {oo}
if the following conditions
are s a t i s f i e d : (i) T h e l e v e l sets { i g X : £ ( x ) < i] are c o m p a c t for e a c h t g IR . (ii) l i m s u p „ V
n
_ 1
l o g fi {F)
< — i n f {C(x)
: x g F}
w h e n e v e r F C X is a closed
l o g / j ( G ) > — i n f { £ ( x ) : x g G}
w h e n e v e r G C X is a n o p e n
n
set. (iii)
liminf
n
V
n
- 1
n
set. T h e s t r o n g r e l a t i o n of t h e l a r g e d e v i a t i o n p r i n c i p l e t o t h e t h e r m o d y n a m i c l i m i t (27) w i l l be t r a n s p a r e n t f r o m t h e n e x t r e s u l t , a c c o r d i n g t o V a r a d h a n ([22]). T h e l a r g e d e v i a t i o n p r i n c i p l e is e q u i v a l e n t t o t h e l i m i t r e l a t i o n Jm
y
= s u p { / ( x ) - C{x)
J e " dp.n{x) v
f(z)
:xeX}
(28)
for e v e r y c o n t i n u o u s b o u n d e d f u n c t i o n / : X —> IR. T h e m e a n field i n t e r a c t i o n is one of t h e s i m p l e s t e x a m p l e w h e r e t h e l a r g e d e v i a t i o n t h e o r y is n o t s u i t a b l e for t h e d e t e r m i n a t i o n of t h e l i m i t (27). L e t the l o c a l h a m i l t o n i a n be of t h e f o r m n = J 2
H ,n) [m
'
i = m
1 h
i
+
"
TT n — m +1
X i X
'
( ) ' 2 9
^
v
i j = m
w h e r e hi g B\ a n d x,- g B\ are copies of s o m e selfadjoint o p e r a t o r s ftj g
B\
a n d x j g 0 j , r e s p e c t i v e l y . T h e e x p r e s s i o n m e a n field reflects t h e s e c o n d t e r m of (29). If t h e s i n g l e s y s t e m is d e s c r i b e d by a finite d i m e n s i o n a l C * - a l g e b r a B\ t h e n t h e l o c a l free e n e r g y d e n s i t y at t h e inverse t e m p e r a t u r e 0 is g i v e n b y
F (B) n
= i l o g T r exp l-fi
L e t ip be t h e s t a t e of B
x
f^hi
- ^
£
1,1,1
w h i c h possesses t h e d e n s i t y
•
(30)
291 a n d let tpoo be t h e c o r r e s p o n d i n g p r o d u c t state of B. I n a n o t h e r n o t a t i o n we have
F»(P)
= -fa*»$'E*i*i>P)-
() 32
It is not a r e a l r e s t r i c t i o n i f we p u t 0 =
1 i n the sequel. I n o r d e r t o m a k e
contact w i t h p r o b a b i l i t y t h e o r y , let us assume for a w h i l e t h a t B is a b e l i a n , or at least [h,x]
= 0. T h e n ipco c o r r e s p o n d s t o a p r o d u c t m e a s u r e w h i c h is the
j o i n t d i s t r i b u t i o n of t h e sequence (x,)
of i d e n t i c a l l y d i s t r i b u t e d i n d e p e n d e n t
r a n d o m v a r i a b l e s . I n t h i s p r o b a b i l i s t i c t r a n s l a t i o n (30) b e c o m e s
t
1 1 " - log / e x p ( - n ( - V Xi) )du. a J n f~*
(33)
2
If fi is t h e d i s t r i b u t i o n of j Yl?=i < * sequence (/i„) o b e y s t h e large deviation principle w i t h the rate function £ given below. x
n
L(u)
= Jexpuxidfi(u)
and
n e n
t
n
e
£ ( x ) = s u p { u i — l o g L ( u ) : u £ IR}.
(34)
( T h e large d e v i a t i o n r e s u l t for t h e m e a n s of i d e n t i c a l l y d i s t r i b u t e d i n d e p e n d e n t r a n d o m v a r i a b l e s is a r e f o r m u l a t i o n of a c l a s s i c a l t h e o r e m p r o v e d b y C r a m e r . ) T y p i c a l l y £ is a c o n v e x f u n c t i o n w h i c h vanishes at t h e e x p e c t a t i o n v a l u e m of x\. O n e k n o w s f r o m t h e l a w of l a r g e n u m b e r s t h a t i f F C IR is a closed set such that m £ F then 1
"
Prob(-y"x,-e.F)^0 n 'r—' i=l
as n —> oo. C o n d i t i o n (ii) of the large d e v i a t i o n p r i n c i p l e tells us t h a t t h i s convergence is e x p o n e n t i a l l y fast a n d its speed is s h o w n b y the r a t e f u n c t i o n . F r o m the a b o v e d i s c u s s i o n we w a n t to c o n c l u d e t h a t t h e r m o d y n a m i c s of m e a n field i n t e r a c t i o n s is s t r o n g l y r e l a t e d to the large d e v i a t i o n p r i n c i p l e i n t h e case [h, x] = 0. A l l t h i s w i l l serve as a m o t i v a t i o n for us to d e v e l o p a n analogous t h e o r y i n t h e p u r e q u a n t u m case [h,x] ^ 0. T h e f u n c t i o n a l c f r o m (32) w i l l take t h e p l a c e of " l o g / e x p " here a n d the g u i d e l i n e w i l l be v e r s i o n (28) of the large d e v i a t i o n p r i n c i p l e . L e t 7 d e n o t e t h e r i g h t shift a u t o m o r p h i s m of the C * - a l g e b r a B. T h e l i m i t of the m e a n s n-1
*„(o)
=
-
lH«)
Y
(a 6 B",
n € IN)
t=i
is the subject of e r g o d i c t h e o r e m s . T h e C * - a l g e b r a i c ergodic t h e o r e m we are g o i n g to prove needs t h e f o l l o w i n g l e m m a . L e m m a 15.
L e t / be a p o l y n o m i a l a n d K € I R . T h e n for e a c h e > 0 t h e r e
exists 8 > 0 such that
+
292 \f(A) - f(B)\ < e w h e n e v e r A,B G B , 3a
Theorem 16.
\\A-B\\
< 8 a n d ||A||,
< K.
L e t g : [s,t] —• R b e a c o n t i n u o u s f u n c t i o n a n d b € B"
a
such
t h a t S p f e C [s,t]. T h e n l i m u(g(a (b)))
(35)
n
n—•oo
e x i s t s for every 7 - i n v a r i a n t s t a t e u> of B. Proof. It is a p l a i n consequence o f t h e W e i e r s t r a s s a p p r o x i m a t i o n t h e o r e m t h a t i t suffices t o prove t h e existence o f t h e l i m i t (35) w h e n g i s a p o l y n o m i a l . A f u r t h e r p o s s i b i l i t y of r e d u c t i o n is b a s e d o n L e m m a 15. S i n c e t h e l o c a l elements are n o r m dense i n t h e a l g e b r a A, w e m a y a s s u m e t h a t o is l o c a l , t h a t i s , there exists k G IN s u c h t h a t 6 G #[i,fc]F i r s t we c o n s i d e r t h e case k = 1. T h e n t h e S 6 C J U C H C 6 SJI (6) i s i n t h e a b e l i a n C * - a l g e b r a . . . ® C ® C ® . . . w h e r e C is t h e s u b a l g e b r a g e n e r a t e d b y a i n B\. A c c o r d i n g t o t h e r e p r e s e n t a t i o n t h e o r e m o f a b e l i a n C * - a l g e b r a s , C m a y be v i e w e d as a n a l g e b r a o f c o n t i n u o u s f u n c t i o n s o n t h e s p e c t r u m o f a a n d u>\... ® C ® C ® . . . c o r r e s p o n d s t o a n i n t e g r a t i o n w i t h respect t o a m e a s u r e fi. B y t h e i n d i v i d u a l ergodic t h e o r e m s (b) converges / / - a l m o s t e v e r y w h e r e a n d so does g(s (b)) for a n y c o n t i n u o u s f u n c t i o n g. T h e L e b e s g u e t h e o r e m t e l l s u s t h a t n
n
= iv(g(s (b))
Jg{s„(b))d[i
n
converges as w e l l us n —> 0 0 . T h i s c o m p l e t e s t h e p r o o f f o r b G A'". T h e g e n e r a l case 6 G
k]
be r e d u c e d t o t h e a b o v e d i s c u s s e d a b e l i a n
case b y c h a n g i n g t h e l o c a l i z a t i o n . F o r a n i n t e g e r £> k w e set e-k
i=o a n d w r i t e t h e a r b i t r a r y n i n t h e f o r m u • £ + r w h e r e u a n d r a r e integers a n d 0 < r < t. T h e n
HE
V ' « -
E
7
J <
( M I I < («*
+ r)t|6||-
(36)
T h i s gives t h a t
N
J=I
c a n be a r b i t r a r y s m a l l i f I is b i g e n o u g h ( w i t h respect t o k) a n d n is b i g t o o . A c c o r d i n g t o L e m m a 15
293
ll/('»W)-/(££V'(W<)ll 3=1
is s m a l l as w e l l . O n t h e o t h e r h a n d , for fixed £
"(/(^X> 'U/*)) J
j=o is a C a u c h y sequence. I n d e e d , t h e sequence i (bi/£) is b u i l t f r o m p a i r w i s e c o m m u t i n g o p e r a t o r s a n d t h e above a b e l i z a t i o n a r g u m e n t w o r k s . I n t h i s w a y we are a b l e t o c o n c l u d e t h a t u>(f(s (b))) is a C a u c h y sequence w h i c h w a s t o be proven. • ]t
n
Theorem 17. formula l ^ ^
x
([17]) F o r 6 £ S i a n d a c o n t i n u o u s f u n c t i o n / t h e v a r i a t i o n a l
, n f ( s
n
( b ) ) )
= m{{f(fab))
+ S(rP, ) Vl
:
6
holds. T h e o r e m 17 is a t y p i c a l e x a m p l e o f w h a t we s h a l l c a l l p e r t u r b a t i n a l l i m i t t h e o r e m . L e t A a n d C „ be C * - a l g e b r a s a n d ip a state of C„. A s s u m e t h a t p o s i t i v e u n i t a l m a p p i n g s a„ : A —* C „ ( n £ IN) a r e g i v e n . M o t i v a t e d b y T h e o r e m 17 (as w e l l as large d e v i a t i o n t h e o r y , i n p a r t i c u l a r V a r a d h a n ' s t h e o r e m ) t h e p e r t u r b a t i o n a l l i m i t principle is said to b o l d i f n
l i m -c(fa, n—»oo TI
= i n f { E & a ) + / ( « ) i us £ £{A)}
nf(A„(a)))
(37)
for every a = a* £ A, for every c o n t i n u o u s f u n c t i o n /
: I t - • Ht a n d for a
c e r t a i n w e a k * lower s e m i c o n t i n u o u s f u n c t i o n a l I : E(A)
—* 1 R U {oo}. T h e +
l a t t e r f u n c t i o n a l i " w i l l be c a l l e d r a t e f u n c t i o n a l . T o v i e w T h e o r e m 17 as a p e r t u r b a t i o n a l l i m i t t h e o r e m one chooses A == B i , C
n
tpoo a n d
= B,tp = n
a
n
=n- (id + 7 + ...+ " 1
7
_ 1
)|Bi
(neW).
T h e rate f u n c t i o n is the r e l a t i v e e n t r o p y i n t h i s case. T h e o b j e c t (B, Bj, 7,
294
5(w|%,n+m].r ooj%,n+m]) = ,
> S(a,|B[i ],^oo|B[,,„]) +
S{uj\B[
1|
],>fi \B[ i,n+ ])
n+lin+m
in
= S(w|B[ „],
l i m
00
n+
m
],(p |5[i, ]). 0 O
m
T h i s y i e l d s t h a t t h e n u m b e r s <„ = 5 ( w | B [ i , ] , Voo|B[i, ]) form a s u p e r a d d i t i v e n
sequence ( t h a t i s , t
> <„ + < )
n+m
S
M
(w,
V
o
o
)=
lim - 5 ( w , B
S i n c e S(u ,tp ) n
)=8up{-S(w
B I
^„):n6]N}.
is a w e a k * - l o w e r s e m i c o n t i n u o u s f u n c t i o n of u>, i t
n
i m m e d i a t e l y t h a t SM (W, foo) SM(U,?<X>)
V B
B
a n d so
m
(38) follows
is lower s e m i c o n t i n u o u s as w e l l . T h e q u a n t i t y
w i l l b e c a l l e d m e a n r e l a t i v e e n t r o p y (or r e l a t i v e e n t r o p y d e n s i t y ) .
T h e o r e m 18.
T h e m e a n r e l a t i v e e n t r o p y ui t-* S M ( W ,
s e m i c o n t i n u o u s afRne f u n c t i o n a l o n i 7 ( B ) . 7
Proof. 0>
It r e m a i n s t o show t h e affine p r o p e r t y . T h e o r e m 9 t e l l s u s t h a t S()Mi
> AlogA + ^logp.
+ fujl,ip )-\S(ui )-fiS(ul,
fVn
n
i f A + \i = 1. D e v i d i n g b y n a n d l e t t i n g n —> oo we o b t a i n t h a t S(Ao)
1
+ ^ w , ^ ) = A5(w ,¥> ) + ^5(w ,y> ). 2
1
2
0 0
0 O
A t t h i s p o i n t we c a n a n t i c i p a t e w h a t k i n d of a n a l o g u e of t h e v a r i a t i o n a l f o r m u l a (28) m u s t h o l d i n t h e q u a n t u m case. L e t 6 6 ^[""it] b e a l o c a l o p e r a t o r i n t h e q u a s i - l o c a l a l g e b r a B . B y t h e d e f i n i t i o n of t h e f u n c t i o n c we have c(
= c(^ o|S( , j,),n/(s (fe)))
< u(nf(s„(a)))
+ 5(a;|B[i „ t],( Soo|B[i, /t])
1
0
i
+
n +
B
f
n+
for a n y s t a t e w of B . D e v i d i n g t h i s i n e q u a l i t y b y n a n d l e t t i n g n - t o o w e a r r i v e at t h e f o l l o w i n g r e l a t i o n . Ibtt.sup
-c(v>«,, nf(s (b)))
<
n
n — oo n
K m c(
(39)
n—oo
w h i c h h o l d s for e v e r y t r a n s l a t i o n i n v a r i a n t s t a t e U J . It is o u r a i m t o s h o w t h a t t h e f o r m of (39) is a n a n a l o g u e of V a r a d h a n ' s t h e o r e m . W h i l e i n T h e o r e m 17 t h e p e r t u r b a t i o n a l l i m i t was s t a t e d for selfadjoint elements of B\ i n t h e n e x t t h e o r e m a l l q u a s i l o c a l elements are i n c l u d e d . C o r r e s p o n d i n g l y , t h e v a r i a t i o n a l e x p r e s s i o n is over a l a r g e r set. T h e o r e m 19.
F o r b 6 B ' a n d for a c o n t i n u o u s r e a l f u n c t i o n / t h e v a r i a t i o n a l A
formula lim
i c ^ o o , f(s (b))) n
= i n f { l i m w ( / ( s „ ( 6 ) ) + S (u>, M
V
o
o
):a,£
Z (B)\ Y
295 h o l d s w h e r e t h e n o t a t i o n S (B)
s t a n d s for t h e t r a n s l a t i o n i n v a r i a n t states of
y
B. T h e o r e m 19 s u p p l i e s us w i t h a n e w e x a m p l e of L e g e n d r e t r a n s f o r m a t i o n . C h o o s e U = B'
a
a n d let V be t h e r e a l l i n e a r space of h e r m i t i a n 7 - i n v a r i a n t
f u n c t i o n a l s o f B. W i t h respect t o the d u a l i t y < u,b >=
u(b)
t h e c o n j u g a t e f u n c t i o n of F(u)
— < [ +00
9°°l
i f 1/ is a 7 - i n v a r i a n t s t a t e otherwise.
is F*(b)
= -
lim -c(v> ,ns„(6)) 0 0
I m p r o v i n g t h e m e t h o d of [17] v a r i a t i o n a l f o r m u l a s were o b t a i n e d i n o r d e r t o e s t a b l i s h t h e G i b b s v a r i a t i o n a l p r i n c i p l e for different k i n d of m e a n f i e l d m o d e l s i n [19] a n d [24], see also t h e c o n t r i b u t i o n of R . F . W e r n e r i n t h i s v o l u m e . S i n c e t h e s e l e c t i o n of t h e r e s u l t s c o n c e r n i n g q u a n t u m e n t r o p y follows t h e interest o f t h e present a u t h o r , readers are suggested t o l o o k at o t h e r r e v i e w s of t h e s u b j e c t , for e x a m p l e [20] a n d [21]. A c k n o w l e d g m e n t . T h i s p a p e r was w r i t t e n d u r i n g the a u t h o r ' s s t a y at the University of Heidelberg. T h e
financial
s u p p o r t of S F B 123 a n d t h e v e r y k i n d
h o s p i t a l i t y o f P r o f e s s o r W . v o n W a l d e n f e l s are a c k n o w l e d g e d .
References [I] N e u m a n n , J . v o n ( 1 9 2 7 ) : " T h e r m o d y n a m i k q u a n t e n m e c h a n i s c h e r G e s a m t heiten", G o t t . N a c h r . p p . 273-291 [2] O h y a , M . , P e t z , D . ( 1 9 9 1 ) : " N o t e s o n q u a n t u m e n t r o p y " , p r e p r i n t [3] J a y n e s , E . T . (1956): " I n f o r m a t i o n t h e o r y a n d s t a t i s t i c a l m e c h a n i c s . I I " , Phys. Rev. 108,
pp.171-190
[4] B e l a v k i n , V . P . , S t a s z e w s k i , P . (1982): " C * - a l g e b r a i c g e n e r a l i z a t i o n of r e l a t i v e e n t r o p y a n d e n t r o p y " , A n n . I n s t . H e n r i P o i n c a r e , Sec. A 37,
pp.
51-58. [5] P e t z , D . ( 1 9 9 0 ) : " C h a r a c t e r i z a t i o n of t h e r e l a t i v e e n t r o p y of states of m a trix algebras", preprint [6] H i a i , F . , P e t z , D . (1991): " T h e p r o p e r f o r m u l a for r e l a t i v e e n t r o p y a n d i t s asymptotics i n quantum probability", preprint [7] A r a k i , H . ( 1 9 7 6 ) : " R e l a t i v e e n t r o p y for states of v o n N e u m a n n a l g e b r a s , P u b l . P I M S , K y o t o U n i v . 1 1 , pp. 809-833
296 [8] U h l m a n n , A . (1977): " R e l a t i v e e n t r o p y a n d t h e W i g n e r - Y a n a s e - D y s o n - L i e b c o n c a v i t y i n a n i n t e r p o l a t i o n t h e o r y " , C o m m u n . M a t h . P h y s . 54, p p . 2 1 - 3 2 [9] P e t z , D . (1986): " P r o p e r t i e s o f the r e l a t i v e e n t r o p y of states of a v o n N e u m a n n a l g e b r a " , A c t a M a t h . H u n g . 47, p p . 6 5 - 7 2 [10] K o s a k i , H . (1986): " R e l a t i v e e n t r o p y for states: a v a r i a t i o n a l e x p r e s s i o n " , J . O p e r a t o r T h e o r y 16, p p . 3 3 5 - 3 4 8 [11] P e t z , D . (1991): " O n c e r t a i n p r o p e r t i e s of the r e l a t i v e e n t r o p y of states of o p e r a t o r a l g e b r a s " , M a t h . Z e i t s c h . , 206, p p . 3 5 1 - 3 6 1 [12] D o n a l d , M . J . (1985): " O n t h e r e l a t i v e e n t r o p y " , C o m m u n . M a t h . P h y s . 105, p p . 1 3 - 3 4 [13] H i a i , F . , O h y a , M . T s u k a d a , M . (1981): " S u f f i c i e n c y , K M S c o n d i t i o n a n d r e l a t i v e e n t r o p y i n v o n N e u m a n n a l g e b r a s " , P a c i f i c J . M a t h . 96, p p . 9 9 - 1 0 9 [14] C o n n e s , A . (1985): " E n t r o p i e de K o l m o g o r o f f - S i n a i et m e c a n i q u e s t a t i s t i q u e q u a n t i q u e " , C . R . A c a d . S c i . PE i s Ser. I. M a t h . 301, p p .
1-6
[15] D o n a l d , M . J . (1990): " R e l a t i v e H a m i l t o n i a n w h i c h axe n o t b o u n d e d f r o m a b o v e " , J . F u n c t i o n a l A n a l . 91, p p . 1 4 3 - 1 7 3 [16] R a g g i o , G . A . , W e r n e r , R . F . (1990): " M i n i m i z i n g e n t r o p y i n a f a c e " , L e t t . M a t h . P h y s . 19, p p . 7-14 [17] P e t z , D . , R a g g i o , G . A . , V e r b e u r e , A (1989): " A s y m p t o t i c s of V a r a d h a n t pe a n d t h e G i b b s v a r i a t i o n a l p r i n c i p l e " , C o m m u n . M a t h . P h y s . 121, p p . 271-282 [18] P e t z , D . ( 1 9 9 0 ) : " F i r s t steps t o w a r d s a D o n s k e r a n d V a r a d h a n t h e o r y i n o p e r a t o r a l g e b r a s " , i n Quantum
probability and applications
V ed. by L .
A c c a r d i a n d W . v o n W a l d e n f e l s ( L e c t u r e N o t e s i n M a t h . 1442,
Springer),
pp. 311-319 [19] R a g g i o , G . A . , W e r n e r , R . F . (1989): " Q u a n t u m s t a t i s t i c a l m e c h a n i c s of gene r a l m e a n field s y s t e m s , H e l v e t i c a P h y s . A c t a 62, p p . 9 8 0 - 1 0 0 3 [20] W e h r l , A . (1978): " G e n e r a l p r o p e r t i e s of e n t r o p y " , R e v . M o d . P h y s i c s 50, pp. 221-260 [21] A r a k i , H . (1987): " R e c e n t progress o n e n t r o p y a n d r e l a t i v e e n t r o p y " , i n VIHth Int. Cong, on Math. Phys. ( W o r l d S c i . P u b l i s h i n g , S i n g a p o r e ) p p . 354-365 [22] E l l i s , R . S . (1985): Entropy,
large deviations,
and statistical
mechanics
(Springer) [23] U m e g a k i , H . (1962): " C o n d i t i o n a l e x p e c t a t i o n s i n a n o p e r a t o r a l g e b r a I V ( e n t r o p y a n d i n f o r m a t i o n ) " , K o d a i M a t h . S e m . R e p . 14, p p . 5 9 - 8 5 [24] R a g g i o , G . A . , W e r n e r , R . F . (1989): " T h e G i b b s v a r i a t i o n a l p r i n c i p l e for i n h o m o g e n e o u s m e a n field s y s t e m s " , t o a p p e a r [25] L i e b , E . H . , R u s k a i , M . B . (1973a):
" P r o o f of t h e s t r o n g s u b a d d i t i v i t y o f
q u a n t u m m e c h a n i c a l e n t r o p y " , J . M a t h , P h y s . 14, p p .
1938-1941
[26] L i n d b l a d , G . (1974): " E x p e c t a t i o n s a n d e n t r o p y i n e q u a l i t i e s for finite q u a n t u m s y s t e m s " , C o m m u n . M a t h . P h y s . 39, p p . 1 1 1 - 1 1 9
297 [27] L i n d b l a d , G . (1983): Nonequlibrium
entropy and irreversibility
(Reidel,
Dordrecht-Boston) [28] H i a i , F . , P e t z , D . (1991): " T h e G o l d e n - T h o m p s o n trace i n e q u a l i t y is c o m plemented", preprint [29] P e t z , D . (1991): " O n e n t r o p y f u n c t i o n a l s of states of o p e r a t o r a l g e b r a s " , preprint
Quantum Probability and Related Topics Vol. VII (pp. 299-315) © 1 9 9 2 World Scientific Publishing Company
299
FROM C L A S S I C A L GEOMETRY TO QUANTUM STOCHASTIC FLOWS: AN EXAMPLE.
Jean-Luc
Sauvageot
L a b o r a t o i r e de P r o b a b i l i t e s Tour 5 6 - 4
place
U n i v e r s i t e P.& M . C u r i e
Jussieu
F -
75005 P a r i s
Abstract: The foliation
transverse
heat
is
into
dilated
semigroup
on
the
C*-algebra
a quantum s t o c h a s t i c
of
a
Riemannian
process.
Introduction. The p u r p o s e constructed basic
idea
where
all
in is
of
this
paper
([S1],[S2])
on the
a s t a n d a r d one
the
is
heuristic
to
the
C*-algebra
([Ma]):
or
dilate
of
t o work
local
heat
transverse
semigroup
a Riemannian f o l i a t i o n .
i n an o r t h o n o r m a l frame
expressions
become
actual
The
bundle
and
global
equations. The diffusion K
main
part
operator
C -algebra,
can
of & ,
be
the
work
extended
to
becomes the
X
k
will
this
(k=l,..,q)
The
being
canonical
transport
along
the
crucial
fact
is
differential
D^=
operators
and
stochastic
("of
classical
show
that
transverse
orthonormal
evolution type",
rt
the
x
the
heat
frame
k
~X*
:
= LMW
(0.2)
dU
t
= U
in (0.2)
Hilbert
this
quite
admits space
an
J
^
d
thus
transverse
kernels
bundle
on
the
W where
B
classical
fq
which
k
+
^
by
of
in H associated
the
meet
driven
D
elementary
L (WxJt ) 2
bundle of
terminology
t
Now,
we
equations
i n the
vectors
skew-adjointness
dj
equation
k k
horizontal
transverse
(o.i)
the
to
the
it
r? = iy (x - x V ,
parallel
in
be
which generates
foliation. of
only
classical
(
D
geometrical solution: just
order
processes
Brownian
motions
[A2]):
.
context,
namely, the
first
2 ) d t
^ ( D ^ D d t l
is
the
stochastic
D. A p p l e b a u m
I V^k k
with
the
unitary
the
evolution
unitary
version
process
[i.e.
the
300 ,
flow
with
the
Radon-Nikodym
derivative
correction
..1/2
flu*!
^
°
f
^
classical flow with equation de
So
that
all
differential
what
=
t
we
V
*ii
(X
have
oQ
k t
to
)dB
k
do
equation with smooth
+ i(V, X o9 )dt . 2 4 t t 2
is
to
solve
coefficients,
a
classical
stochastic
in a manifold which is not
compact.
What we get in the end is -
a C*-algebra B extending the C*-algebra of the foliation fl = C (V,F) :
the orthonormal group 0(q) acts naturally on B, and fl appears as the fixed points algebra of B for this action; - a family
D ,..,D 1
-
of
self
adjoint
the diffusion
derivations on B;
2
q
operator
precisely the generator A
on B, the restriction of
which
to fl is
of the transverse heat semigroup;
- a quantum Evans-Hudson flow of classical type on B {j :
B<snL (w m ^
B ->
t
2
q
t
o
which solves the equation d
-
\
= Ek V k (
D
the restriction of this flow
process
dilating
the
) d B
k
+
5 VEk k (
D
2 ) d t
;
{j } to fl, which is the quantum
transverse
heat
semigroup
[but
is
no
stochastic longer
of
Evans-Hudson type].
I.UNITARY EVOLUTION ASSOCIATED WITH A CLASSICAL STOCHASTIC
FLOW
I.l.An evolution equation
A smooth manifold M is given, and a C°° measure u on M , in the Lebesgue If £ is a C°° vector field on M , one denotes by div^j the divergence
class.
of f relative to the measure u , that is the smooth function on M such that Jctfjdu = - Jdiv^Cf du , If
D
considered L (M,u), 2
is
V f C°° with compact support.
a
differential
operator
(when
convenient)
as
a
on
densely
M , then defined
D
will
closable
be
implicitly
operator
on
with domain E = C ° ° ( M ) [the vector space of smooth functions with
301 t compact s u p p o r t ] , and D w i l l denote the r e s t r i c t i o n t o 2) o f the a d j o i n t o p e r a t o r o f D. F o r instance, i f f i s a vector f i e l d , one has £ * = - f - d i v £ : 1 t * and f o r D = £ + - d i v £ , one has D = - D . 2 u On each
£
M a r e given is
i
q global
associated
C™ v e c t o r
the smooth
fields
£
i
q
;
for i = l,..,q,
* = - i ( £ + £ * ) = i-div £ , 1 2 1 1 2 u i
function
i
t
to
a n d the
t
f i r s t o r d e r d i f f e r e n t i a l o p e r a t o r D = £ + * = - ( € - € ) such that D =-D . i l 1 2 1 I 1 1 We s h a l l meet a u n i t a r y evolution equation o f t h i s type: (1.1.1)
dU = U [ J t
t
in
the H i l b e r t space
of
the q - d i m e n s i o n a l
measure.
q
= i
D»dB
+ i E^tD^Ddt]
i
L (M,n)®L (lf ), 2
2
q
where
B r o w n i a n motion,
W
with U = U q
is the space
q
equipped
with
of
trajectories
the canonical
Wiener
on M solution
of the
[1 denotes the identity o p e r a t o r on L ( W ) ] . 2
q
1.2.The p h i l o s o p h y o f t h e s o l u t i o n One
considers
the (classical)
stochastic
flow
e
t
s t o c h a s t i c d i f f e r e n t i a l equation
(1.2.1) de = V (£ ° 9 )dB + i ( V £ ° e )dt , t n t i V 1 t the R a d o n - N i k o d y m d e r i v a t i v e s d(u°9 ) 2
u
and t h e f a m i l y (6J As
noticed
implement
u
o f automorphisms of M x W :
e M x . u ) = (e"(x),u).
q
in
the f l o w
[AI], (9^};
the we
between the both equations,
family state
of
a
unitary
slightly
{U^}
more
solution
accurate
of
1.1.1
relationship
only i m p l i c i t e l y contained i n [ A l l :
1.2.2. Lemma The
unitary
U : f -> (p)
operators
f°8^
UZ
t
s t o c h a s t i c equation (1.1.1) above,
on
L (MxJC ) 2
q
satisfy
the
w i t h U =1.
Proof Let T space
t
be the (closable,
H = L ( M , u ) » L (W ); q
non bounded) the
(TJ
o p e r a t o r (p -> 0 ° 9
satisfy
a
t
on the H i l b e r t
stochastic
differential
equation: dT
= T [ T" £ ®dB
t The
functions
a r e equal t o T T
p
,
t *• ^1 being
+ •!•(£ £ ® D d t ] . 2
1
considered
2 *" l s
as
multiplication
operators
, a n d s a t i s f y the s t o c h a s t i c d i f f e r e n t i a l equation
on W,
302 dp" r
= dT . T * + T dT* + dT dT*
1
t
t t
t t
t t
«dB i(EC ®l)dt E£*®dB i(K* 8l)dt (£
= T
2
i +
= T l-2£*®dB t
+
2
i +
+ (2(2* -f (* ))9l)at]T 2
i
l
l
e^-DdtlT"
+
P
= p" [-2T](* »9 )®dB + ( V ( 2 * o e - £ (* ) ° 8 )®l)dt] , 1
the
three
first
2
lines
being
straightforward
computation,
a n d t h e l a s t one
U * =
coming f r o m
a n d thus
T 0 T * = p~\
t
= p [2^(*oe )adB + (£(2¥ °e T
t
2
i
+
t
Joe^aDdt]
and dfp
1 / 2
l
=
P
1 / 2
[
n * oo )®dB + - ( n * ° e
+ e t* )=e )®ndt] .
z
Compute now dU
=d(p-T, = p| (E(* /z
1
= d(p-).T <>
+ p
1 / 2
+ P
1 / 2
p - d T
+
e )8dB+i(E(* '>e +e (* 2
t
T [ V 6 ®dB
t
d ( p - ) d T
)oe ®i)dt]T +
i
+ -(J"
+
t
C® Z
t
Ddt] +
I I ( * »9 )®1]TJ? «l))dt
= p^ T [r*®dB+i(E(* +e (* »®l)dt] + /2
+
2
t
P ^ T J E € »dB
i
1
+ i ( £ C ® l)dt] + 2
i
+ P ^ T J F * , ? ®ndt = U J E(* e )®dB i+
=
U [V
q
i
1.3.Some
conclusions
The
stochastic
thus
unique)
exploding In manifold
I
evolution
solution
if
M
equation
i s compact
I
(D ®l)dtJ .
q
_
2
2 **i<±r
i
•
equation
a n d only
classical stochastic particular,
i
+ - r
DedB
t ^1=1 I
+ I l p ^ + ^ J * )+2* € )®lldt]
i
if
1.1.1
admits
t h e equation
choice
existence
between of
1.1.1 a d m i t s
(cf.the
[A3],[FS],[VS],etc.), quantum f l o w ,
testing
solutions
a
classical
the various
to a
o r merely
unitary defines
(and a non
f l o w on t h e m a n i f o l d M . unitary
point
of
solution
quantum checking
analitycity stochastic
manifolds,
conditions differential
t h e non e x p l o s i o n
as we shall do i n section III above.
whenever t h e
i n IEI] a n d
view
quantum one i n [AI]). A s long as we deal w i t h non compact a
a
1.2.1
which equation of
a
the
we g e t insure (cf.
classical
303 II.UNITARY E V O L U T I O N ASSOCIATED WITH A S U B B U N D L E O F T H E T A N G E N T BUNDLE II.l.Geodesical
stability f o r
subbundles
Throughout this section a
subbundle
?
of
the
are fixed
tangent
d i m e n s i o n a l , a n d the bundle y
L
a smooth
bundle
is
Riemannian m a n i f o l d M , and
TM. M
is
n
dimensional,
J
is
q
(p=n-q)-dimensional.
V i s the L e v i - C i v i t t a connexion on M , and u the Riemannian measure. T h e m a i n assumption is the f o l l o w i n g geodesical (II.1.1)
s t a b i l i t y of 7:
F o r any point x in M , and any vector £ i n y
, the geodesic
issued
at x i n the d i r e c t i o n € is supported by y . T h a t i s : the s o l u t i o n of the equation V ^ ^ c ' l t ^ O , c'(t) e f Notice
that
this
property
holds
w i t h c'(0)=f|,
satisfies
„, , ¥ t i 0 . clt)
for
the
normal
bundle
in
a
Riemannian
f o l i a t i o n (IMol,3.5).
II.1.2.Notations a n d t e r m i n o l o g y A smooth s e c t i o n o f J w i l l be c a l l e d a 1-vector
field.
y p
denotes
the orthogonal
f i e l d s onto y - v e c t o r
projection
from
T M onto
y
(and f r o m
vector
fields).
T o a p a i r £,T) o f y - v e c t o r
f i e l d s is associated the y - v e c t o r y y
field
T o a y - v e c t o r f i e l d TI a r e a s s o c i a t e d : - the
usual
divergence
div n = tracefTVsSj -> V ^ i i e T V ) ; y
- the y - d i v e r g e n c e One checks
d i v
y
1 )
=
t
r
a
c
e
C
9
diVg.fi) = f-div^T) + i)(f),
€
> P
v
£ ! T
e y
) •
f e C°°(M) .
II.1.3.Lemma I V T h e r e e x i s t s a global y - v e c t o r H
7
=
field 2
y
on M s a t i s f y i n g
^ ( d i v ^ - d i v ^
f o r any l o c a l f i e l d (c , . . , c ) of o r t h o n o r m a l bases of y . i J_ 2 V F o r any l o c a l f i e l d (TJ , . . , T ) ) of o r t h o n o r m a l bases of y , one has i p „y —p y „ 1
H
K
71 1 1
304
Proof: One has < - ^ p \ ^ ,
e>
= ^
k
V
^
,
V
= V
i
rv
p
>+r"
e
<e ,V
L^l=l V 7) k ^ J = l div(e ) - d i v _ e , . k J k
=
II.2.Diffusion
operator
and
stochastic
c >]
J
i
k
differential
,<e .57
-
"J=l
J
J
e >
Cj k
equation
associated
with
L e t (e , . . , c ) i q and consider: -
be
a
locally
the f i r s t o r d e r d i f f e r e n t i a l a
= e k k
defined
w i t h d> = - <E *k 2 J
-
of
orthonormal
bases
for
7,
operators
+
r
field
the d i f f u s i o n o p e r a t o r
, E > = - [div(E ) - d i v _ £ ] ; k 2 k 3k
T
A
= -£J
^ ^ k
'
k
II.2.1.Lemma
y IV
The
operator
2
A =
E
k
G
k
E
+
k
d
' y i ' k v
E
E
t
does
not
depend
on
the
p a r t i c u l a r choice of f i e l d of o r t h o n o r m a l bases f o r
j 2 ° / The o p e r a t o r A has the l o c a l f o r m A = - T E E + -divH IIH II = A 'Ic k k 2 4 It
does
bases
not 3,
for
differential Proof the
is
generator
deal
with
it
in section heuristical generator algebra a
depend and
quantum
a
thus
particular
appears
straightforward. of
a
in a
of
the
- -IIH
2
choice
as
+ -div=
a
of
II
.
4
a
field
globally
defined
of
orthonormal second
order
o p e r a t o r on M .
a
classical non
Of
course,
diffusion
commutative
A
is
process
context;
on
Markov
diffusion
Riemannian diffusion
semigroup
foliation").
process,
In
we
a
Dirichlet form, M.
in view
(non abelian) * - a l g e b r a A on ? as derivations, and A will
III a sense,
of
on
+ H
But of
should
to
purpose we
is
shall
dj
t
while the associated
= £ (J °9 >dB t
k
k
+
dilate
meet
a
heat
kernels
this
semigroup
quantum
i(j °A )dt t
7
u n i t a r y evolution s a t i s f i e s
to
meet
k
on
the evolution e q u a t i o n :
the into
stochastic
d i f f e r e n t i a l equation w h i c h is only l o c a l l y defined and has the l o c a l f o r m : (II.2.2)
thus
w h i c h the d act, in a actually appear as the
("transverse
order
our
[S2],
and
305 UI.2.3) where
dU the
Fock
A^
space
The
which
D
,
k
are
2
of
our
the
3^
can
apply
is
+ i(A ®l)dtl 7
provide
give
global
a
algebra
defined
[D* k
creation motions
the
globally
adjoint
and
a^dA
Brownian
to
widening
become
skew
r
the
work by
-
+
a n n i h i l a t i o n and and
q
equations,
formally
above
A*
a 8dA
L (R )sC , + f [A + A 1 . k k
aim
heuristical on
and
of
the p r o c e s s e s
= U rr
=
- D ], k
a
so
solution
B k
a
order
that
to
are
Fix
x
y
along a
i n M and £, in y
to
result
the
Boson
identified
with
those
two
algebra
differential
the
both
the
bigger
S,
operators
of
section
unitary
equation II.2.3, a n d the q u a n t u m s t o c h a s t i c d i f f u s i o n equation
II.3.Parallel transport of
in
meaning
into
A
first
processes
I
evolution
II.2.2.
geodesic.
. Let c
[t-»c(t)] be the
geodesic
issuing from
x
in the d i r e c t i o n
II.3.1.Lemma IV Tj(t) [t
+
For
e
y
T,
2V
admits
c(t)
C ( t )
TJ i n y
any
X
,
a
the
equation
unique
V
solution.
c > ( t )
T,(t)
=
This
solution
from
y
with
0
will
TI(0)
= TJ ,
denoted TJ
be
c
l
For
fixed
t,
the
TJ -» i)
map
c(t)
x
into
is
linear
and
isometric.
Proof: Work write we get
with
Tj(t) =
£
matrix
we get
local
V Mc
the s y s t e m dTj / d t k
The
a
k
k
of
= E «
oc(t)
k
is
k
field ,
so
(e , . . , e ) i q that
v^,
of
( t )
T)(t)
k
(t)V
bases
of
y,
and
,„,e + dii / d t . c , C tt) k k kj
equations
kj
(th),(t) J
with a
antisymmetric,
and
kj
(t)
so
=
its
primitive
(j,k=l,..,q). 0(t)
= j"\x(s)ds
;
an explicit solution Tjjt)
1
=
e
P l t )
T) ( t )
q where
orthonormal
the
isometry.
matrix
Hit)
e
is
; orthogonal:
TI
I
V
which
provides
linearity
and
306 II.4.Horizontal
vector f i e l d s on the o r t h o n o r m a l f r a m e
The o r t h o n o r m a l f r a m e (x,e),
where
x
is
a
1
in the euclidean space at
x
is
an
0(q)
bundle W
point
of
. W
transitive
M
and
bundle
r e l a t i v e to "3 is the
manifold of
pairs
= (c , . . , e ) is an o r t h o n o r m a l bases i q is a p r i n c i p a l bundle over M , and the f i b e r c
homogeneous
space
[t)(q)
is
the
orthogonal
group
i>
(only)
of o r d e r q l . We
fix
on
0(q)-invariant
W
the
v = j"
measure
p r o b a b i l i t y measure
i> dp.(x)
,
x
M
on W
where
x
is
the
.
II.4.1.Lemma IV §
[W^ 2V
Fix x
in M .
Any vector
a c —> § ( x , e ) 6 T j
£
.W
x
in ?
lifts
x
] along
c a n o n i c a l l y as
w h i c h is
a vector
field
(D(q)-invariant.
The map f; -•> f; is l i n e a r .
Proof: L e t (x,e)
t-»e
The c(t)
geodesic
a
be a point in W ^ .
notations being c(t) c(t) =(E ,..,c ) i q
those of
of
II.3.1,
this
bases
lemma of
3
provides , , c(t)
a
field
along
the
c.
Define f ( x , c ) =
(c(t),e
In
linearity,
order
local
field
neighbourhood UxO(q)
lemma
orthonormal
s
(y,u)
to
prove e(y)
=
U
of
c ( t )
)l
(c ( y ) , . . , c (y)) x,
.
t = 0
one
can
trivialize W
of
orthonomal
and
identify
-> (y,ue(y)) e W [where (uc) (y) s
by
extending
bases
W ^
of
to
= £ _ u e (y) J—1 kj J q
w i t h the orthogonal m a t r i x u(t) given by and, a f t e r
i d e n t i f i c a t i o n of
, k=l,..,q
£(x,u) = w i t h a = 0'(O)
t k W t ) )
u(t)
= u(0).e
P
w i t h UxO(q) : Oj.-u.a)
= (
3"
;
into in
Ux(D(q)
T h e p r o o f of lemma II.3.1 provides
\ =W
e
].
a by
307 II.4.2.Def inition For k=l,..,q,
a smooth
v e c t o r f i e l d X i s d e f i n e d on W by k X (x,c) k
For k=l,..,q,
a smooth
= c (x,e) . k
function *
i s d e f i n e d on W by
1 7 * (x,e) = - < H (x),e > k 2 k For
k=l,..,q,
a
first
order
3" [with H as d e f i n e d
differential
i n II.1.3].
operator
is d e f i n e d
on W
by D Smooth
functions
invariant functions
on
M
= X + * k k
k
will
be
systematically
considered
as 0 ( q ) -
on W , a n d conversely.
II.4.3.Lemma IV
If f i s a smooth
f u n c t i o n on M , one has X (f)(x,e) = e (f) . k k
and
L
x
2
It
2°/
(
f
)
= L
k
(
e
Z
lc
+
d i
V
1c
>
E
>( > =
E
f
k
J
A ? ( f
>
(
s
e
e
lemma
II.2.1)
k
F o r u i n 0 ( q ) , one has u(X ) = T > X k J Jk J
3V
F o r k=l,..,q,
one has
div X ( x , e ) = < H ( x ) , e > = 2* ( x , e ) . V k k k 7
4°/
Y X (* ) Mc k k
The function
on W i s (D(q)-invariant; = 2Y X (* ) + I I I I Mc k k
div
2
a n d one has
.
Proof: IV
L e t ( x , e ) belong
t o W , a n d c(t) be the geodesic
issuing
from
x in
the d i r e c t i o n e^. One h a s X ( f ) ( x , e ) = ^ f t c t t ) ) k dt *k »
= d t <
(
c
f ' P ^
a
d
= e (f) = <e, , p g r a d f>, a n d k k X 7
f
>|t=0 =
<<
(
t
)
.<.
(
t
)
P
Vad(f,>
|
t
=
0
3 =
p grad(f)>
.
k Extend
e
neighbourhood
as
a
field
of
orthonormal
bases
of
3
o f x. Then
<e , V k
E
p grad(f)> 7
= e (f)(x) 2
k
c ,p grad(f)> 7
k = e (f)(x) k 2
< V c , grad(f)> . e k k 7
defined
in
a
308
Compute -J^
• >i = div
c
? C j
(x).
k Hence t h e r e s u l t . 2V
L e t u be i n 0(q) and
lemma II. 4.1, W
function
on W ; compute,
using
and 2 V :
(uX ) 0 ( X , E ) = X (u*0)(x,u*e) = (u*e)~(u*0)(x,u e)
k
^ ~
»
#
= V u e (u 0)(x,u c )
J Jk j = V U E (<j>)(x,c) = [ u X (0)(x,c)
.
J Jk J 3V
From
2V
above,
t h e vector
field
V div X . X
is
k k
V
which implies t h a t , f o r f a smooth f u n c t i o n on M ,
(D(q)-invariant;
V div X . X (f)(x,e) = V. div X ( x . c ) . c ( f )
M t f k k
Mr
l> k
k
does not depend o n c . We get then, f o r f a n d g l o c a l l y d e f i n e d w i t h support,
a n d extending
c
as
a
field
of
orthonormal
bases
of
compact J
on a
neighbourhood o f x:
- J ML ^ v V f ) . g . d u
= - \u
- |
^(fl.^tgj.du
M
;
hence i;«uv (x. ).e (f) = A
e
t
+
E div k
= [V div e . c
Mt
4V
The 0(q)-invariance
.
£ k
F ^ . ^ t f )
+
- T d i v „ c . E ](f) . Mt 3 k k
k k
is obvious.
? e k
For f
a smooth
f u n c t i o n on M , one
computes | div / . f . d u M
= ->^| <E ,E ><E 7
w
= "
2
4 w
K
W
k >
grad
Thdv
=
t>.to 2
j M
[
W
V
+
divX .* ]f.du. k
k
-
II.4.4.CoroIlary IV D
f
k
The operators
= -D
k
2V
D
k
defined
above
are
formally
skew
adjoint:
, V k=l,..,q.
If f is a smooth
is d e f i n e d i n II. 2.
Proof: I V is lemma II.4.3, 3°
f u n c t i o n on M , one has Y, D (f) = A (f), where A
k k
309 For 2 V ,
compute k
= Mc I * ? k + Mr LdivX
x
k
k
=
+
2
" illH^II
?
(by 11.4.3,4")
2
4
and, f o r f a smooth f u n c t i o n on M : 5J divX .X (f)(x,e) k
k
= E <E ',e >
k
3
k
k
= E (f) =
£ (dive -div .c )e (f)
7
Fxf(f) T t
+ T d i v X . X (f) Mt k k
k
k
k
= A^(f)
+ F^div
-£k k k c
k
k
+ V (div e - d i v - . e )e (f) It k J k k
= Z&n -
g
c
V
e (f) k
•
( f )
The r e s u l t s comes f r o m lemma II.2.1.
_
II.4.5.Lemma With the (D > d e f i n e d as above, i f the evolution equation k
dU = U [T D ®dB + - r" (D ®l)dt) t t Mt=l k k 2 Mt=l k q
admits U
t
a
unitary
2
solution,
in
the
Hilbert
with U = 1 0
space
L (W)®L (W ), 2
Z
q
then
each
commutes w i t h the n a t u r a l a c t i o n of 0(q) onto L ( W ) ® L ( J f ) . 2
2
q
Proof: F i x u = = (u
) be a qxq orthogonal m a t r i x , and let J
-
u
be
the
k
2
unitary
operator
on
L (W,i>) a s s o c i a t e d w i t h the v - p r e s e r v i n g
a c t i o n o f u on W : (uf)(x,e) = f ( x , u * c ) [(u*c) =V u e l ; k J kj J - A(u) be the n a t u r a l i s o m e t r y on L (W ) s a t i s f y i n g q
X(u)B = V u B V k=l..q . k h Jk j T h e n the [U .ueX(u)] s a t i s f y the d i f f e r e n t i a l equation d[U .u®A(u)] = U . [ T D ®dB + -T (D ®l)dt].u®X(u) t t *"Tt k fc 2Mc k 2
= U .u®A(u).lT u u D®dB + i V ( D ® l ) d t ] t Mjk kl kj 1 J 1c 2
= U .u®A(u).[r D ®dB t
hence
U^.usAtu) =
usMuJ.U^
a unitary solution exists
by
(cf.1.3).
i[
unicity
k
of
k
the
+ -T ( D ® l ) d t ] ; Z
s o l u t i o n to
1.1.1,
as
soon
as
310 III.APPLICATION T O RIEMANNIAN FOLIATIONS
III.1.The t r a n s v e r s e v e c t o r f i e l d a s s o c i a t e d
Let
(V,F)
C^-Riemannian the tangent bundle,
a
([MoD.
7
tf
foliation,
Riemannian
manifold,
bundle o f
foliation:
IF is a p - d i m e n s i o n a l
V
induced E u c l i d e a n s t r u c t u r e is
will
q-dimensional, be
identified
where
with
is
integrable
bundle 7
V , and the quotient
on w h i c h the
invariant the
be
with a Riemannian foliation
n-dimensional
smooth
subbundle
= T V / I F is the
is
assumed
q = n-p
the
an
is
subbundle
to
the of
of
transverse
be
holonomy
codimension
of
orthogonal
to
TV
F. T h e c o t r a n s v e r s e bundle 7*c If
(x , . . , x ) is 1 n structure, the p coordinates, then
the
system
first
the fields
F.
The
last
L
local
coordinates
adapted
,..,d/dx ) f o r m a f i e l d of i p longitudinal riemannian
0/3x
The
square
root
(locally)
the
density
of
the
bases
the
foliated
Riemannian
of
the
vector
matrix
|G 1 11
J
is
to
(x , . . , x ) being the longitudinal 1 n (x , . . , x ) being the transverse one, p+l n
ones
= fg =<3/3x , 3 / 3 x >1 U lj 1 J l,J=l..p
determinant
1.
T * V is the dual bundle o f of
coordinates
q
vector
bundle G
and
a
is of
1 / 2
measure
its
on
the
l e a f m a n i f o l d , w i t h r e s p e c t to the l o n g i t u d i n a l measure dx . . d x . i p The d i f f e r e n t i a l f o r m s (dx ,..,dx ) form a f i e l d of bases « P bundle U . Moreover, the scalar products
N
only on the t r a n s v e r s e c o o r d i n a t e s
(x
p+1
,..,x
n
7 its
be the o r t h o g o n a l
-
.
For f
a
smooth
differential
global
df.
transverse
written,
function, What
has
differential
iy
been
d f
denote
shown
form
6
in any system o f c o o r d i n a t e s as 8
J,
p r o j e c t i o n f r o m T V onto let
= E"
the
depend
).
J Let p
$
of
+ K
p +
the
in [S2] (a
the
projection
is the
section
of
* or f r o m T V onto
existence 3
)
onto
*
3"
on V of
which
can
of a be
above:
d x+ id Log|G 7
7
| .
1 On
the
other
hand,
lemma
II. 1.3.
provides
a
globally
defined
t r a n s v e r s e v e c t o r f i e l d (a section of V) of w h i c h the l o c a l d e f i n i t i o n H = does
not
bases f o r
depend 1.
on
the
^ ( d i v ^ l - d i y ^ ^
particular
choice
of
a
field
(e ) k
of
orthonormal
311 III.1.1.Lemma The
transverse
associated
differential
t o J above
form
correspond
n
6 a n d the t r a n s v e r s e
one t o the other
vector
through
field E
the euclidean
? and J .
d u a l i t y between
Proof L e t (?) , . . , T ) ) be a local i p II. 1.3, one has t o show:
field
o f orthonormal
bases
f o r F . By lemma
<-I V TI , d x ) = <9 , d x > "1=1 T)I 1 p*k p*k P
for k = l , . . , q . First matrix G
n
T) = l
write
T * a d/dx u j
/3 = a * a i s the inverse
. a n d one has J * V TJ = ^1=1 T) l Moreover,
andlet a = (a ) ; ij i,j=i...,p
matrix
then, t h e
o f the longitudinal Riemannian
matrix
p d/dx . 1,J=I IJ d/dx j
one c a n w r i t e
(V.,„
I
dx ) = r
d/dx, j
d/dx
p*k
'
+k
£ £ +k = TH P g * . ^£=1 l j , p
p
lj
6
£
where the T a r e the C h r i s t o f f e l symbols, and the P a r e defined by: U U £ 2P = 3g „ / 3 x + 3g . / 5 x - 9g / 3 x . . lj il J j£ l 1J I One computes f i r s t 5
B
C V V ^
= - JtAi
6
^
a
V
s
j
x
=
" It-ii
S
g
"
p
t
k
/
a
x
j
•
and
ag/ax.g^" = -
F 0
Hf\t-t=i
\t
B
j
6
^j=i
agJ-^/ax = - £ < 2 „ . , j
6
^j=i
a/aXj
n
l / a X
«
|
i ,
d V d x >. j *k P
Then
W V
a
X
«
=
a L O g l G
and W hence
the
g
1
j
/
a
x
f
g
t
p
"
=
<
d
L
o
g
G
l
-
d
V k
>
;
result.
III.2.The t r a n s v e r s e
d i f f u s i o n o p e r a t o r o n t h e g r a p h 'S
We s u m m a r i z e t h e p r e s e n t a t i o n o f [ S 2 l . The
graph
§ o f the f o l i a t i o n
([Wi],[Co])
i s the m a n i f o l d
d r a w n on the leaves o f the f o l i a t i o n , defined up t o holonomy.
o f the paths
312 F o r y i n S, we w r i t e
y:x-*y
f o r a path y w i t h o r i g i n x a n d e x t r e m i t y
y. s and r a r e the " s o u r c e " a n d " r a n g e " maps f r o m S» onto V , w i t h s(y)=x a n d r(r)=y f o r y:x->y. T h e tangent is
equipped
bundle T V o f V is orthogonally decomposed
with
the Riemannian
structure
appears as orthogonally decomposed as
f o r which
a s T V = F © 3 ; S?
i t s tangent
TS = r * F ® s * F ® 3 ( § ) ,
where
bundle as an
E u c l i d e a n bundle, c a n be i d e n t i f i e d w i t h r 3 o r s 3, i n d i f f e r e n t l y . It 3
is a fundamental
(and thus
results
7(g))
o f section
canonical
vector
property
shares
of Riemannian foliations
the p r o p e r t y
II.1.1
that
the bundle
(cf.[Mo],prop.3.5).
Hence, the
II apply to the m a n i f o l d 'S and i t s bundle 3(S?): we get a g-(g) 3(S/) field H , a diffusion operator A , and a stochastic
evolution equation o f type 1.1.1 on a n o r t h o n o r m a l f r a m e bundle W over '§. On the other
hand, on the d i f f u s i o n
operator
A
b u i l t on the g r a p h of
the f o l i a t i o n ([S2]) c a n be w r i t t e n A
= - ( 3 ) 3
with
3
= d
+ - [ r 9+s 9] z
9 being the d i f f e r e n t i a l f o r m met i n lemma III. 1.1.
III.2.1.Proposition The d i f f u s i o n to section
operator
II above,
3(g) A
associated
and the d i f f u s i o n
w i t h the bundle 3"(S») a c c o r d i n g 3
operator
A
on §
deduces
that
associated
with a
Proof: Riemannian f o l i a t i o n i n [S2] (7.1), a r e equal. From
lemma
III.1.1, * * [r 9+s 9)
differential
form
correspond
one to
one
easily
on g
the the other
and the t r a n s v e r s e through
the vector
the E u c l i d e a n
duality
3*(
has the local f o r m
3 ( f ) = £ Ek(f)eIt + 2i< = 7
j so t h a t A
3
7(i?)
,ck>ek ,
has the local f o r m
A as i n II. 2 above.
J
= -(3 )V J
= - Y a a v. k k u
k = ek + kd>
with 3
y
transverse 37*?) field S between
313 III.3. E x t e n d i n g t h e f o l i a t i o n C
Following the
the
orthonormal
lines
of
-algebra
section
frame
bundle
denote
the
W
II,
we
have
associated
now
with
to
the
consider
to
consider
3{S)
subbundle
in
the
m a n i f o l d M = S». Let
C^tW) c
support on W , the
product
vector
space
of
smooth
and the » - a l g e b r a B ^ be the vector
and
involution
foliation (cf.ICol
similar
to
those
of
=
space C (W) the
f
_ cMy ,e)iMr . « ) i z [the f o r m u l a e x p l a i n s as f o l l o w s : J
let
and
y:x-»y
and
a l l the
in
measure
on
orthonormal bases in 3
y
r
r
an
orthonormal
of
writing y
integration
in
the
intermediary
the
leaf
,
, and thus in 3
algebra
bounded
fixed
*
operator
into JE(L (W))
,
2
ft = C * ( V , F )
of
of
the
y;
of
the
formula
being
with
respect
secondly,
e
in 3
,
V§)
n(0)
clearly
and its
made
to be
firstly, paths (at
the
least
Riemannian
considered
well
as
we
y^:x->z
an
as
an
orthonormal
to
L (W)
the
on B
CO
smooth
3 i/i -»
cftW) c ;
TI is
n o r m closure
foliation
be
0(q)
, of
algebra
which fl^
the
of
the
IS2],4.1.1).
, the map 2
on
can as
two
(i?)l.
appears
CO
fixed;
of
product z,
be
composition
on W induces an action of
in B
the
the
and thus
f o l i a t i o n (with the notations of For
equipped w i t h algebra
0 * ( r , e ) =
bases
as
point
containing
in 3
bases
point
smooth
compact
1
e
T h e a c t i o n of 0{q) fixed
r
ways
f^z->y, the
locally)
with
§5):
(c*«i/i)(r,e)
consider
functions
a
will
appears
f o r the a c t i o n o f the orthogonal group
e C°°(Wj c
*-algebraic be denoted
now
as
the
extends
representation by
fixed
B . The point
in a of
B^
C*-algebra
algebra
of
B
0(q).
III.3.1.Lemma The
differential
operators
(k=l..q) on W
associated
w i t h the
bundle
3(§)
by II.4.2 act on B as s e l f - a d j o i n t d e r i v a t i o n s : co D (0«i//) = D (0)«i/< + 0«D (i/i) D (0*) = D (0) V k k k k k
tp.i/i
e C°°(W), c
k=l..q.
Proof: is
obvious;
p r o o f o f p r o p o s i t i o n 4.3.2
Self-adjointness
in [S2J.
for
the
Leibnitz
formula,
repeat
the
314 III.4.Canonical
We have flow the
quantum flow
just
now t o put together
on the C * - a l g e b r a C*-algebra
associated
of
with the transverse
foliation
fl
kernels
the p r e v i o u s r e s u l t s t o get a q u a n t u m
IB, w h i c h is O ( q ) - c o v a r i a n t ,
the
heat
dilates
the
a n d whose r e s t r i c t i o n t o
transverse
heat
semigroup
c o n s t r u c t e d i n [SI] and [S2].
111.4.1. L e m m a With the D a s s o c i a t e d w i t h k
the s t o c h a s t i c e v o l u t i o n e q u a t i o n + V
d U = U IT? D ®dB
(D ®l)dt]
with U = 1
Z
a d m i t s a s o l u t i o n w h i c h is unique a n d u n i t a r y .
proof: By with the
1.3
the
above,
vectors
we
X^,
have
to
k=l..q,
p r o v i d e d i n IEI],
criteria
as s t u d i e d i n [S2], 6.4
check
does
that
not
VII, § 6 ,
the
stochastic
explode. and o f
Which
is
flow a
the s t r u c t u r e
associated
consequence of
of
the g r a p h S?
and 6.5.
111.4.2. P r o p o s i t i o n I V The stochastic flow
over B
j ; B 3 b H> j j b ) = U * ( b » l ) U satisfies
t e R
+
the q u a n t u m s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n
d j [Tt(0)l = Y t M
j [n(D e»]dB + i j MY D 0 ) ] d t c t k k 2 t M c k
V 0 e B . co
2
m 2°/
Its
restriction tA
heat s e m i g r o u p Concluding Notice
(e
to
fl
= C (V,F)
is
a
dilation
of
the
transverse
7
)
on fl.
remark: that
the quantum
stochastic
type ( c f . I E H l ) : j J B ) c B®ie(L (W)), 2
flow
(jj
on B is
of
Evans-Hudson
V t ; while i t s r e s t r i c t i o n t o fl is n o t .
315 REFERENCES
[All
D. Applebaum,
'Towards
a
quantum
theory
Riemannian manifolds', p r e p r i n t [A2]
D.Applebaum,
'On
a
class
of
of
stochastic
B r o w n i a n m o t i o n ' , t o appear i n t h e Trento [A3]
D.Applebaum,
'Unitary
stochastic calculus', [Co] A.Connes, in
[EH]
Pure
M.E.Evans
Lecture [E£]
evolutions
Notes
A.M.S.,
and
38
(1982),
R.L.Hudson,
in Maths.
theoremes (1974),
'Formules
in
quantum
lifts
d'annulation
and o p e r a t o r a l g e b r a s ' , Proc.
Symp.
quantum
diffusions',
p.68-89. equations
on
'Quantum f l o w s w i t h unbounded
manifolds,
de
pour
s t r u c t u r e maps
preprint.
l a moyenne, l e s formes
calcul
des
harmoniques',
Foliations,
perturbations
et
J.Funct.
17
Anal.
B i r k h a i i s e r , B o s t o n , 1988.
j . - L . S a u v a g e o t , ' S e m i - g o u p e de l a c h a l e u r d'un
quantum
p.274-291.
[Mo] P . M o l i n o , Riemannian [SI]
by
1982.
F . F a g n o l a and K . B . S i n h a ,
P.Malliavin,
driven
p. 521-628.
'Multidimensional
and f i n i t e d e g r e e s o f f r e e d o m ' , [Ma]
flows
horizontal
differential
Cambridge U n i v e r s i t y P r e s s , [FS]
on
proceedings
and
n"1303 (1988),
Stochastic
K.D.Elworthy,
diffusions
preprint.
'A s u r v e y o f f o l i a t i o n s
Maths.
classical
«
feuilletage
t r a n s v e r s e sur i a C -algebre
r i e m a n n i e n ' , C. R. Acad. Sc. Paris,
serie
I,
t.310
(1990),
p.531-536. [S2]
J . - L . S a u v a g e o t , same t i t l e ,
preprint.
[VS] G . F . V i n c e n t - S m i t h , ' U n i t a r y quantum s t o c h a s t i c e v o l u t i o n s ' , [Wi] H. E. W i n k e l n k e m p e r , (1983),
p.51-75.
'The
graph of a f o l i a t i o n ' ,
Ann.Glob.Geom.
preprint. Anal.,
1
Quantum Probability and Related Topics Vol. VII (pp. 317-329) Q 1992 World Scientific Publishing Company
317
T h e A z e m a M a r t i n g a l e s as C e n t r a l L i m i t s
Michael Schiirmann I n s t i t u t fiir A n g e w a n d t e M a t h e m a t i k U n i v e r s i t a t H e i d e l b e r g , I m N e u e n h e i m e r F e l d 294 6900 H e i d e l b e r g , G e r m a n y
ABSTRACT L e t (a„)„gpiJ be
a
sequence of ' g - i n d e p e n d e n t , i d e n t i c a l l y d i s t r i b u t e d ' elements of
a * - a l g e b r a . It follows f r o m o u r results that the m o m e n t s of the sums
for n —> oo converge t o t h e m o m e n t s of an A z e m a m a r t i n g a l e .
1. I n t r o d u c t i o n a n d m a i n r e s u l t . L e t q be a r e a l n u m b e r not e q u a l to 0 a n d let a > 0 be a 2 x 2 - m a t r i x w h i c h is assumed to be of d i a g o n a l f o r m unless q = ± 1 . L e t L (t) qi
be the s o l u t i o n of t h e q u a n t u m s t o c h a s t i c i n t e g r a l e q u a t i o n
(1.1)
T h e i n t e g r a l s a r e t o b e u n d e r s t o o d i n the sense of o f R . L . H u d s o n a n d K . R . P a r t h a s a r a t h y [7] o n B o s o n F o c k space T over C ® i ( R + ) . T h e s y m b o l s A*(t), A(t) 2
2
a n d A(t) denote the c r e a t i o n , p r e s e r v a t i o n a n d a n n i h i l a t i o n processes respectively. It f o l l o w s f r o m t h e g e n e r a l r e p r e s e n t a t i o n t h e o r y of w h i t e noise on * - b i a l g e b r a s [12] t h a t (1.1) has a u n i q u e s o l u t i o n as a f a m i l y ( £ ; < r ( ) ) i > 0 o f l i n e a r o p e r a t o r s o n t h e O
l
dense linear subspace V = n e R D o m ( a ^ ) ° f ? Q
o p e r a t o r on T.
where N denotes the n u m b e r
+
K . R . P a r t h a s a r a t h y [9] showed t h a t for \q\ < 1 t h e L .^(t) a r e q
b o u n d e d a n d t h a t t h e processes (L (t) q;
+ I „ ( i ) * ) > o for |g| < 1 f o r m a f a m i l y of 9;
(
c o m m u t i n g s e l f - a d j o i n t o p e r a t o r s w h i c h i n the v a c u u m state give rise t o a c l a s s i c a l martingale.
T h e s e m a r t i n g a l e s h a d been i n v e s t i g a t e d b y J . A z e m a [1} a n d M .
E m e r y [3]. T h e y a r e e x a m p l e s of m a r t i n g a l e s w h i c h have t h e chaos
completeness
p r o p e r t y b u t a r e n o t c l a s s i c a l stochastic processses w i t h i n d e p e n d e n t i n c r e m e n t s ; cf.
[8]. H o w e v e r , i t h a s b e e n s h o w n i n [12] t h a t (L ^(t)) >o q]
t
quantum s t o c h a s t i c processes w i t h i n d e p e n d e n t i n c r e m e n t s .
c
a
n
be r e g a r d e d as
318 T h e p r o c e s s e s (L . {t)) q a
are c l o s e l y r e l a t e d to t h e processes (L - (t)) q
t.,M = /V>(2 w h e r e for 5 V
€ M
2
given by
V
;,)(d,w(- ) a,- («•)) ;
we use the n o t a t i o n J(s,t)(B)
+
W
M
for t h e s e c o n d q u a n t i z a t i o n o n
of t h e o p e r a t o r / |
>
B
fX[s,t)
o n C < g > L ( R + ) a n d we p u t J(t)(B) 2
+
/X[o,.)u[«,oo)
= J ( 0 , f ) ( j 3 ) . T h e t w o k i n d s of processes c o m e
2
f r o m t w o different, b u t i n a way d u a l , n o t i o n s of g - i n d e p e n d e n c e .
L e t C(x,x*)
be
the p o l y n o m i a l a l g e b r a i n the n o n - c o m m u t i n g i n d e t e r m i n a t e s x a n d x*.
For an
e l e m e n t a i n a * - a l g e b r a A a n d a p o l y n o m i a l P i n C ( a : , x*) we w r i t e P(a)
for t h e
e l e m e n t i n A o b t a i n e d f r o m P by r e p l a c i n g x b y a a n d a;* b y a * . If $ is a s t a t e o n A we let tp
a
b e the moment functional
o n t h e * - a l g e b r a C(x,x*)
of a i n t h e s t a t e $ , i.e.
g i v e n b y
is t h e s t a t e
F o r a d i r e c t e d set (D, -«) a
f a m i l y ( a . * ) , ^ of elements a , of a * - a l g e b r a ,4 is s a i d to b e q-left independent i n the s t a t e $ on .4. i f
*(^(l)K,„) ' • • ^(«)K „,))
= ¥>.«(-Pi) • • • Va (Pn)
(
for a l l c h o i c e s of n € N , Pi,...,P
(13)
in
£ C ( i , x * ) , p e r m u t a t i o n s 7r of n e l e m e n t s ,
n
a n d l j . , . . . t „ € i ? , i i -< ••• -< i „ , a n d i f (oj)jgx> satisfies the f o l l o w i n g ' d i r e c t e d ' g-commutation
relations a ^ a , , = q a^a*, a
for i ' i -< z . If (a.i)i n 2
€
2
a
M
= 9 < . a
(15)
2
satisfies (1.3), (1.4) a n d
a
r a t h e r t h a n (1.5)
i
(1.4)
:*, i= = 9 i 2 i i a
a
(1-6)
a
we say t h a t (a;) is right q^independent.
A n e x a m p l e of a left
t j - i n d e p e n d e n t f a m i l y is D = {(s,i) : 0 < s
C R+
with (s,t.) -< ( s ' , t ' )
i <
s'
and = L (t) q;
-
i
0 ; c
r(«)'
F o r a n e x a m p l e of a r i g h t (/-independent f a m i l y p u t ai,^
ft
= .£«,(*)/(<, o o ) ( j
9
" x ) .
= B
t
— B,
with
319 C a n o n i c a l e x a m p l e s of left a n d of r i g h t g-independent sequences arise f r o m p a i r s ( x , y) o f elements of a * - a l g e b r a A s u c h t h a t y is s e l f - a d j o i n t , x a n d y g - c o m m u t e (i.e.
iy
= q yx)
a n d 4>(y) = $ ( j / ) = 1 for * a state on A.
F o r s u c h p a i r s the
2
C a u c h y - S c h w a r t z i n e q u a l i t y y i e l d s $ ( j i ) = 1 for a l l n £ N a n d t h e n n
*(P(x)f») =
= ¥>*(P)
(1.7)
for a l l n £ N . F o r m the elements L = y
0
"
(
_
1
)
® 5 ® l ® l ® - . .
(1.8)
I
)
®x® jr®j®...
(1.9)
and f„ = 1 ® " (
of . 4 ® .
-
T h e y generate a s u b - * - a l g e b r a A ^
N
of A
o n w h i c h the state $< > is N
m
defined b y *
(
N
)
(ai
® °2 ® ••• ) = #(ai)#(a ) . . . . 2
It f o l l o w s f r o m the ( / - c o m m u t a t i o n r e l a t i o n s a n d f r o m (1.7) t h a t (Z„) a n d (l )
form
n
a s e q u e n c e of left a n d of right (/-independent elements r e s p e c t i v e l y . N o t i c e t h a t , i n t h e case q ^ ± 1 , for a m o n o m i a l M i n C ( x , x*) we have tpi(M) = 0 unless M is a p r o d u c t of a n e q u a l n u m b e r of x ' s a n d x * ' s . T h i s follows f r o m the g - c o m m u t a t i o n r e l a t i o n s a n d f r o m the first e q u a l i t y of (1.7). If q = — 1 we have tps(M) = 0 unless M is e v e n , a n d i f q = 1 there is no r e s t r i c t i o n like t h a t o n tp . £
In p a r t i c u l a r , the
matrix tp(xx*)
tp(x tp(x *x))
tp{(x*f) is d i a g o n a l unless q = ± 1 . and
y = J(t) ^ J
A n e x a m p l e of a p a i r (x,y)
is g i v e n b y x =
o j ' ^ U
$ t h e state g i v e n b y t h e d e n s i t y m a t r i x ^ J A £ R
+
fV
t
0
For
q
^ ^ for t > 0 a n d g' £ R , g' 7^ 0, if we t a k e for $ the v a c u u m
s t a t e . T h e f u n d a m e n t a l e x a m p l e is as follows. T a k e A = M
and
L '. (t)
2
,
q)
j
denote by [A] the m a x i m u m of the set { m
£ N : m <
A}.
We
f o r m u l a t e our m a i n r e s u l t . C E N T R A L LIMIT T H E O R E M . dent elements nals of a , n
i.e.
Let ( a ) „ N be a sequence of left (of light) n
e
of a *-aIgebra«4 with state
n £ N , are identical. functionals
that the moment
If q = 1 we also r e q u i r e tp = tp
ai
tp(x) = 0, w i i i c i i s automatically T i i e n the moment
a n d suppose
satisfied
of the
if q /
sums j
["<]
s„(i) = -7=y)a* v
*=i
1.
to be
q-indepenfunctiocentralized,
320 for n —• oo converge pointwise to the moment functional
of L ^(t) q]
(of L (i))
in
q;ff
the vacuum state where 2 _
/
- UiV) )
ff
2
To put it in an other way, ( (il,P(L . (t)Q,)
in the left case
q a
hrn^ *(P(s (t)))
= | ^
n
for all P e
p( . (t)Q)
in the right case
Lq a
C(x,x*).
A p r o o f o f t h e c e n t r a l l i m i t t h e o r e m w i l l be g i v e n at t h e e n d of S e c t i o n 2. O u r t h e o r e m i s a g e n e r a l i z a t i o n o f a r e s u l t of [4,5], see a l s o [2], w h i c h c o r r e s p o n d s to t h e case q =
1, a n d of [6,14] w h i c h is t h e case q = — 1 . I n S e c t i o n 3 we also
f o r m u l a t e a c e n t r a l l i m i t t h e o r e m for t h e case q = 0. P r o o f of the central limit theorem.
2.
T h e p r o o f i s d i v i d e d i n t o t w o steps.
F i r s t w e show t h a t t h e m o m e n t f u n c t i o n a l s ip,
converge pointwise to a certain
n
limit. L . (t) q ilr
T h e n we i d e n t i f y t h i s l i m i t as the m o m e n t
f u n c t i o n a l o f L - (t) q
a n d of
a
respectively.
For q € R, q /
0, we i n t r o d u c e t w o c o a l g e b r a s t r u c t u r e s o n C(x,x*}.
The
p o l y n o m i a l * - a l g e b r a C(x, x*) i s N - g r a d e d i f we p u t C( , ') = I
C(x, x*p ' n
0 C (
I
I
,
J
f ) ,
t h e l i n e a r s u b s p a c e o f h o m o g e n e o u s p o l y n o m i a l s o f degree n. W e d e -
n o t e b y e(P) t h e degree of a h o m o g e n e o u s p o l y n o m i a l P.
M o r e o v e r , C(x,x*}
is
a l s o a Z - g r a d e d a l g e b r a i f we set C(x,x*)
C(x,x*)^ ) m
= 0
C(x,x*) , (m)
t h e l i n e a r s p a n of a l l m o n o m i a l s M s u c h t h a t t h e difference
between
t h e n u m b e r o f x ' s a n d t h e n u m b e r of i * ' s o c c u r i n g i s e q u a l t o m. D e n o t e b y d(P) t h e degree of a n element P w h i c h i s h o m o g e n e o u s w i t h r e s p e c t to t h e Z - g r a d u a t i o n . For example, d(x*x ) 2
= d(xx*x) = d ( x V ) = 1.
O n e c h e c k s t h a t (1.4) a n d (1.5) i m p l y P,(a )P (a ) u
2
l2
=
g
d ( P
'
) £ ( P j )
P (a 2
> 2
)P (a 1
I 1
)
a n d t h a t (1.4) a n d (1.6) i m p l y P (a )P (a ) l
ii
2
h
=
9-
d ( P 2 W P
' P (a )
2
i 3
)P (a 1
i l
)
321 for a l l h o m o g e n e o u s P i , P € C(x,x*) a n d a l l i ' i X i . N o w we t u r n t h e v e c t o r s p a c e t e n s o r p r o d u c t C{x,x*) ® C ( x , x * ) i n t o a * - a l g e b r a C(x, x*)® C(x, x*) by setting 2
2
q
(P
® Q ) ( P ' ® Q') = - < " e " M « ) p j > '
(P®QY A n o t h e r * - a l g e b r a C(x,x*)
®
=q- ( d
( p ® C ? ) ( P ' ® c?')
P*
®Q*.
is g i v e n by s e t t i n g
C(x,x*)
q
p)c{Q)
J ' W W ' V P '
=
( P ® < ? ) * = q ( ) (Q)p* d
p
®
QQ'
® Q*.
c
In t h e cases q = ± 1 w e d o n o t need t h e t w o different g r a d u a t i o n s . If q = 1 we set e ( P ) = 0 for a l l P G C ( x , x * ) , a n d i f q = - 1 we p u t e ( P ) = d ( P ) e q u a l to 0 o r to 1 d e p e n d i n g o n w h e t h e r P is a n even or a n o d d p o l y n o m i a l . In b o t h cases ' l e f t ' and ' r i g h t ' m a k e s n o difference a n d we have t h e u s u a l tensor p r o d u c t of *-algebras and o f Z - g r a d e d * - a l g e b r a s . W e define t h e * - a l g e b r a h o m o m o r p h i s m s 2
A : C ( x , x * ) -> A
: C(x,x*}
C(x,x'}® C(x,x*} q
C(x,x')
®,
C(x,x'}
8 : C ( i , x") —> C by e x t e n s i o n o f Aa;
= Aa; =
i®l-rl(gia;
and = 0.
6x
For
example, Ax
= x
2
® 1 + (1
2
+ 9
- 1
)x ® x +
1
® x
2
and Ax 2.1. P R O P O S I T I O N . for A . P R O O F :
2
= x
2
®
1
- f (1 + q)x ® x
+
1
® x . 2
A a n d A a r e coassociative a n d 8 is a counit both for A a n d
W e m a k e t h e r e m a r k t h a t A a n d A b o t h a r e even (i.e. homogeneous o f
degree 0) for t h e N - a n d for t h e Z - g r a d u a t i o n . In t h e s e q u e l , expressions w i t h ® , e t c . a r e to b e r e a d as ® , * , e t c . and as ® , and
etc., a n d t h e y s t a n d for t h e left
f o r t h e r i g h t case r e s p e c t i v e l y . C l e a r l y , i f Ri : B
are e v e n * - a l g e b r a h o m o m o r p h i s m s t h e n even *-algebra h o m o m o r p h i s m .
x
—> Ai a n d R
2
Ri ® R : Bi®„B —> Ai® A 2
2
a
2
: B
2
—• - 4
2
is also a n
W e have t h a t A ® i d a n d i d ® A are * - a l g e b r a
h o m o m o r p h i s m s . Since ((& ® id) o A ) ( x ) = ((id ® A ) o A ) ( x )
322 it follows t h a t A a n d A are c o a s s o c i a t i v e . N e x t n o t i c e t h a t <5 is e v e n for t h e N - a n d for the Z - g r a d u a t i o n i f we e q u i p C w i t h the t r i v i a l N - a n d Z - g r a d u a t i o n s . T h u s the v a l i d i t y of
(
((•5 ® i d )
= ((id ® 8)
o A)(x)
o A)(x)
=
x
i-i
y i e l d s t h a t 8 is a c o u n i t for A . O For a n element a i n a * - a l g e b r a A we denote by
the * - a l g e b r a h o m o m o r p h i s m w h i c h is u n i q u e l y d e t e r m i n e d b y j (x)
= a.
a
2.2.
PROPOSITION.
(i) Let ( a i , . . . , a ) be a sequence of elements in a *-algebra that satisfies (1.4) and (1.5) (or (1.4) and (1.6)). Then n
_ Ja, H
(ii) If (a,!,...
P R O O F :
ai
Jc\
(->• ' • ' ~*J&n '
, a „ ) i s left (or right) q-independent then
W e prove (i) for the r i g h t case. It suffices to show t h a t j
a * - a l g e b r a h o m o m o r p h i s m , because j have j
• <-i
ra-n
A with state $
* ...*j
an
(PQ)
0 l +
...
+ ( l n
= ( j a , ® . . . ® ja )(J2(Plk n
= D I I ?" ' £(P
kl
= Yl
i)d(Q
and j
j
aj
® .-.®
Pnk)(Qll
" m (a )Qu(a )... )
fe
l
is
j
ai
an
agree o n x . W e
an
® ...®
Q )) nl
P (a )Q (a )
1
nk
n
nl
n
T<S
P
^ ( " i ) ••• ^ n
)Qu(a )...
P
Q (a )
1
nl
n
kl
=
tia *...*j J(P)ja *...*ja„(Q)1
a
1
U s i n g (1.3), (ii) follows f r o m (i).<> A s s u m e t h a t the t r i a n g u l a r a r r a y (
f%,j t' • -1 *Pnk„ c o m m u t e for lim
max
\(
nk
n
of l i n e a r f u n c t i o n a l s o n a
each n £ N w . r . t . c o n v o l u t i o n
n—*oo l
(2-1)
- 8)(c)\ = 0 for a l l c £ C
(2.2) *
/
n
P X] K^nfc - * ) ( ) l < oo for a l l c 6 C. c
(2.3)
323 T h e n T h e o r e m 1 o f [10] says t h a t
l i m V(¥>
- 6)(b) = ip(b) for a l l b € C
nfc
(2.4)
71 — ' O O '
/C = l
for a l i n e a r f u n c t i o n a l ip o n C i m p l i e s l i m ((f> i * . . . *
W e a p p l y t h i s t o the s i t u a t i o n o f o u r c e n t r a l l i m i t t h e o r e m w h e r e fc„ = [nt\ a n d *(P(^)).
2.3.
PROPOSITION.
moment functional C(x,x*)
Under
the assumptions
tp is even, i.e.
of the central limit theorem, the
unless d ( P ) = 0.
M o r e o v e r , we have that
L I M
71—»CO
(expitd^)(P)
in the left case
(exp^td )(P)
in the right case
v
for all P € C(x, x*) where for a monomial M in C(x, x")
d
PROOF:
A
M ) = { * 1, 0
{
M
; otherwise.
)
i
f
(
M
)
=
2
L e t Pi, P b e h o m o g e n e o u s elements o f C ( a : , x*) a n d assume t h a t
0 a n d
2
• P ^ M P ^ a , ) ) =
2
= g
d
(
W
l
V(Pl)
2
w h i c h , i n t h e case q ^ ± 1 , m e a n s d ( P j ) = 0 o r e(Pi) = 0. F o r q = - 1 t h e a b o v e e q u a t i o n s h o w s t h a t Pi a n d therefore tp is e v e n . N o w a s s u m e ; /
± 1 . T h e same
a r g u m e n t as before y i e l d s d ( P ) = 0 o r e ( P ) = 0. If e(Pi) ± 0 a n d d ( P ) / 2
2
2
Owe
have * ( P i ( a x ) P ( a ) ) =
2
2
q ^« ^>p(PiMP ) d
p
w h i c h i s a c o n t r a d i c t i o n . S i m i l a r l y , d ( P i ) ^ 0 a n d
s h o w s t h a t
2
0 cannot h o l d .
This
P unless e ( P ) = 0 o r ip(P) = 0 for
P u n l e s s d ( P ) = 0. T h e first s t a t e m e n t m e a n s t h a t ip = 8, t h e
s e c o n d t h a t (p is e v e n . I n b o t h cases (p is e v e n . W e have lim
71-.00
324 and lim
[ni]
71 —• CO
for
a l l P £ C{x,x*)
ip = id , v
For
td (P) v
y/ Ti
w i t h 8(P) = 0 . T h i s m e a n s t h a t ( 2 . 1 ) - ( 2 . 4 ) are satisfied w i t h
a n d the s e c o n d s t a t e m e n t of the p r o p o s i t i o n follows.<>
a p r o o f of the c e n t r a l l i m i t t h e o r e m it r e m a i n s to be s h o w n that (exp itd )(P)
=
(Q,P(L ., {t))a)
( e x p Jd )(P)
=
(Q,P{L .Am)
v
q
r
and v
q
with ip(xx')
xy = q yx
(2.5)
y* = y-
(2.6)
Denote the resulting *-algebra by 2.4.
PROPOSITION.
The
C(x,x*) . q
equations Ax
= x®l
+ y®x
Ax
= x ® y
+ 1 ®
q
q
A y=A y q
(2.7)
x
(2.8)
(2.9)
= y®y
q
6 x = 0;6 y = 1 q
define two *-bialgebia The equations
q
structures (A ,8 ) q
q
and (A ,8 ) q
q
on C(x,
x') . q
f(Py )
= P
(2.11)
r(y P)
= P
(2.12)
n
n
define coalgebra
(2.10)
homomorphisms r:(C(x,x*)„A )^(C(x,x'),A) 9
r : (C(x,x') ,A ) q
g
- » (C(x, x*), A ) .
M o r e o v e r , the adjoint mappings ( r ) * a n d r * m a p a n even positive (conditionally positive) linear functional on C(x, x*) to a positive (conditionally positive) linear functional on C(x, x*) . q
PROOF:
W e r e s t r i c t ourselves t o the r i g h t case. F i r s t define A,
: C(x, x * , y) -> C ( x , x*, y) ® C(x, X*,
2/)
325 b y (2.8) a n d (2.9). W e have ( ( A , ® id) o A , ) ( x ) = ((id ® A , ) o a n d i t follows t h a t A
9
A )(x) q
e x t e n d s to a c o a s s o c i a t i v e * - a l g e b r a h o m o m o r p h i s m . A (xy)
= xy ® y
q
Next
+ y ® xy
1
so t h a t A , respects t h e r e l a t i o n s . It follows t h a t A , gives rise to a c o m u l t i p l i c a t i o n A
q
o n C(x, x*) . q
of C(x,x*) C{x,x*}
q
C l e a r l y , {My
to C{x,x*).
q
a m o n o m i a l ,n £ N } is a v e c t o r s p a c e basis
: M
n
because of r e l a t i o n (2.5). T h u s (2.12) defines a l i n e a r m a p p i n g f r o m O n e proves by i n d u c t i o n on e(P) A (p)
=
q
that
Y, "<®y' ' > < p
(p k)p
i
k if A(P) =
p
ik
®
P
2k
k for P e C(x,x*).
B u t then ((r®r)oA,)(P) =
A(P)
w h i c h s h o w s t h a t r is a c o a l g e b r a h o m o m o r p h i s m .
S u p p o s e t h a t
is
e v e n a n d p o s i t i v e . B y the eveness, r'{y™Py ) n
for a l l m,n
€ N , P G C(x,x*),
=
and
m,n
S i m i l a r l y , for c o n d i t i o n a l l y p o s i t i v e PROOF
f>.§
OF T H E C E N T R A L LIMIT T H E O R E M :
B y P r o p . 2.4 we k n o w t h a t r*(d ) v
c o n d i t i o n a l l y p o s i t i v e o n the * - b i a l g e b r a (C(x,x*) ,
A)
q
q
and that
e x p . l r * ^ ) = r*(exp. id ) 4
for a l l t > 0. W e have
i
v
q '
if M =
y >xy x*y '
q- *ip(x*x)
ii M =
y x*y 'xy *
n
n
0
otherwise.
n
ni
ni
n
n
n
is
326 for M a m o n o m i a l i n x, x* a n d y. If we a p p l y the c o n s t r u c t i o n of [11] to ip =
r*(d ) v
we o b t a i n t h a t D^, C C , 2
1
f
if M =
\o- i v 4
M
)
=
f
{
y x*y * ni
n
)
2
0- l 2
if Af =
y xy ni
n2
\0- 2 2
, o
otherwise
and
Pi>{y) = L e t us d e n o t e by j(t) (C(x, x*) , g
A ,6 ) q
q
q
0
0
g"'
the r e a l i z a t i o n o n the Fock space T of the w h i t e noise on
w i t h g e n e r a t o r ip. T h e n j(t)(P),
P £ C(x, x*) , q
is the s o l u t i o n
of the q u a n t u m s t o c h a s t i c i n t e g r a l e q u a t i o n ,(p)id +
m(p)
u( r ) * , d / ( r ) ) ( P )
[ Jo
(2.13)
with I(i)(P)
= A'(t)(r,^(P))+A(t)( ^(P)
- Sq(P)id) + A{t)( +(P*))
P
V
F o r P = x t h i s is / y(r)(x)d/(r)( ) + f Jo Jo
j(r)(l)dl(r)(x)
2 /
w h i c h m e a n s /(f).(ss) = Lq.„(t).
(Q,P(L .Am) q
M o r e o v e r , for P 6 = r*(exp„ td )(P) q
v
C{x x*) t
=
(expJd^(P)
w h i c h proves the c e n t r a l l i m i t t h e o r e m for the right case. I n t h e left case, (2.13) is r e p l a c e d b y
Jo w h i c h for P = x is
+ ^(P)z.
327 w h e r e we u s e d t h a t the s o l u t i o n of ft
j(t)(y) = i d + j f j(r)(y)dA(r)
is g i v e n b y j(t)(y) =
J(t)
(
9
Q
^_ ° _
1
x
j
q° - ) - <
0
3 . A c e n t r a l l i m i t t h e o r e m f o r t h e c a s e q = 0. W e s t i l l have the * - a l g e b r a C { x , x * ) w h i c h is n o w t h e * - a l g e b r a g e n e r a t e d b y x a n d y w h e r e y is a s s u m e d to b e s e l f - a d j o i n t a n d t h e r e l a t i o n xy = 0 h o l d s . O n e checks t h a t (2.7)-(2.10) a g a i n define t w o * - b i a l g e b r a s t r u c t u r e s on C(x,x*) . H o w e v e r , we cannot define a c o a l g e b r a s t r u c t u r e o n C(x, x*) i n t h e w a y we d i d for q ^ 0. If
0
= i
(3-1)
this d o e s not m e a n , as i n the case q ^ 0, t h a t ip is d e t e r m i n e d by its values on the p o l y n o m i a l s i n x a n d x * . T h e r e are less r e l a t i o n s . F o r e x a m p l e , the y i n s i d e of the m o n o m i a l xx'yxx* c a n n o t be m o v e d to the ends by use of the relations between x,x* a n d y, b u t tp m a y not v a n i s h o n xx'yxx*. M o r e o v e r , m o n o m i a l s of t h i s k i n d a p p e a r i f we a p p l y A Q to m o n o m i a l s i n x a n d x*. O n the other h a n d , we have the following THEOREM.
Let tp be a. state on C(x, x")
satisfying (3.1). Then for P € C(x,
0
l i m pMl««](J»(-*
lim
p-M(P(-J=,»))
where d (M)
=
v
PROOF:
|
=
(
td )(P)
expio
v
(exp..M,)(P)
(p(xx*)
if M
0
otherwise.
A p p l y T h e o r e m 1 of [10] to the N - g r a d e d c o a l g e b r a C(a:,a:*)o where the
N - g r a d u a t i o n is g i v e n b y e(x) = e(x*) of (p* ^ ](P(-j=,y)) a
)) =
x*)
nt
= 1 a n d e(y) = 0. T h i s gives the convergence
to expi ip w h e r e tp vanishes on a l l m o n o m i a l s except those of 0
l e n g t h 2 a n d agrees w i t h ip on m o n o m i a l s of l e n g t h 2. T h e general m o n o m i a l of l e n g t h 2 is of t h e f o r m !
w h e r e x<-
0)
= x a n d x^
r
m i
*
( , ,
V * »y ,ei,e,6{0 l} ,
f ,
,
l
1
= x*. C l e a r l y by (3.1),
y>(2/ a: n i
( e i )
2/ x n 2
( e 3
' i / ) = ^(x^ n s
1
V x s
( e 2 )
)
0
328 a n d this vanishes, a g a i n by (3.1), unless e i = 0 a n d e
2
= I. For
t
n
same reason,
e
O n e m a y ask i f i t is possible to f o r m u l a t e a c e n t r a l l i m i t t h e o r e m i n t e r m s o f ' r a n d o m v a r i a b l e s ' , as we d i d for q jt 0. A n e x a m p l e is a g a i n o b t a i n e d f r o m a p a i r ( S , y) of elements of a * - a l g e b r a A w i t h state * such t h a t iy = 0 a n d <}>(£) = *(y ) 2
= 1. W e f o r m t h e series ( / „ ) a n d (l ) as i n (1.8) a n d (1.9). T h e m o m e n t n
functional on C ( x , x * )
0
of the p a i r (s ,y®
n
n
® 1 ® ...) with s„ = ^
T,k=i ««
i s
equal to p „
/»M(P(
* V "
„))
a
n
d p ~ c^»M(p(
* v
j,)) n
respectively. T h e o t h e r q u e s t i o n is a b o u t a r e a l i z a t i o n of processes w i t h m o m e n t
functionals
e x p i i d ^ a n d exp+td^. T h i s is a g a i n c o n s t r u c t e d w i t h t h e h e l p of t h e t h e o r y of w h i t e noise o n a * - b i a l g e b r a a p p l i e d t o the * - b i a l g e b r a s ( C ( x , x * ) , A n ) a n d t o t h e 0
c o n d i t i o n a l l y p o s i t i v e l i n e a r f u n c t i o n a l td . v
the processes ( i o
; < 7
( i ) ) a n d (Lo (t)) ;lT
T h e w h i t e noises a r e e a s i l y seen t o b e
o n F o c k space over L ( R ) g i v e n b y
Lo-At) =0- ( Jo
2
+
J(r)(0)dA(r)
a n d b y t h e s o l u t i o n of t h e e q u a t i o n
Lo-At) = ~ f LoAr)dA(r) Jo r e s p e c t i v e l y where a =
+ aA{t)
^ ( n ' j .
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H e i d e l b e r g N e w Y o r k : S p r i n g e r 1985 [2] C u s h e n , C D . , H u d s o n , R . L . : A q u a n t u m - m e c h a n i c a l c e n t r a l l i m i t t h e o r e m . J . A p p l . P r o b . 8, 454-469 (1971) [3] E m e r y , M . : O n t h e A z e m a m a r t i n g a l e s . I n : S e m . P r o b a b . bourg.
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stochastic
processes w i t h
independent
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[13]
1989
S c h i i r m a n n , M . : Q u a n t u m 17-white noise a n d a g - c e n t r a l l i m i t
theorem.
C o m m u n . M a t h . P h y s . 1 4 0 , 589-615 (1991) [14]
W a l d e n f e l s , W . v . : A n a l g e b r a i c c e n t r a l l i m i t t h e o r e m i n the a n t i - c o m m u t i n g c a s e . Z . W a h r s c h . v e r w . G e b i e t e 4 2 , 135-140 (1978)
Q u a n t u m P r o b a b i l i t y a n d R e l a t e d T o p i c s V o l . V I I (pp. ©
1992
W o r l d Scientific P u b l i s h i n g
331-336)
331
Company
White Noises Fulfilling Certain Relations R O L A N D
1.
By
SPEICHER
INTRODUCTION
a ( n o n - c o m m u t a t i v e ) p r o c e s s we m e a n a t r i p l e (C, p, ( c r ) j g ) , w h e r e C is a K
• - a l g e b r a , p a s t a t e o n C, SJ t h e r i n g g e n e r a t e d b y a l l s e m i c l o s e d i n t e r v a l s o f t h e f o r m [s,i) for Ii n /
(s,t 2
£ IR, s < i) a n d c j 6 C for e a c h / € R s u c h t h a t ci ^ lKJ
= cj, +
c/
2
= 0- T h u s c / i s t o b e i n t e r p r e t e d as t h e i n c r e m e n t of t h e p r o c e s s i n t h e
time interval / .
N o t e t h a t t h e l a s t e q u a t i o n s a y s t h a t we o n l y c o n s i d e r p r o c e s s e s
w i t h a d d i t i v e i n c r e m e n t s , whereas i n the frame of S c h i i r m a n n [AS W ] m o r e g e n e r a l processes are a l l o w e d . We
are o n l y i n t e r e s t e d i n the d i s t r i b u t i o n of a process,
i.e.
i n the
t i o n o f a l l m o m e n t s o f t h e f o r m ^ ( c j ' ' . . . c * ^ ' ) , w h e r e r £ IN, J i , , , . , I 1
fc(l),...,
k(r) e { 1 , 2} a n d c) := cj, c f := c}.
collecT
£ 3?,
'
W e c o n c e n t r a t e o n p r o c e s s e s w i t h i n d e p e n d e n t i n c r e m e n t s , w h i c h m e a n s t h a t cj and
cj a r e i n d e p e n d e n t f o r d i s j o i n t / a n d J i n t h e sense t h a t p y r a m i d a l l y o r d e r e d
moments factorize (compare [ K u m , B S p l ] ) . W e s h a l l f u r t h e r r e s t r i c t to the s i m p l e s t case, n a m e l y to the case of a g e n e r a l i z e d W i e n e r p r o c e s s w h e r e t h e c o v a r i a n c e m a t r i x (QI),J
: = pi^c'jC ^ is g i v e n b y 3
*-C % F o l l o w i n g t h e p r o o f o f o u r l i m i t t h e o r e m s i n [Spe] we see t h a t pi c a n b e c a l c u l a t e d i f we a r e a b l e t o e x p r e s s m o m e n t s w i t h t h e h e l p o f t h e c o r r e s p o n d i n g e l e m e n t a r y moments.
F o r t i m e o r d e r e d m o m e n t s i t is c l e a r h o w to d o t h i s ( a c c o r d i n g t o t h e
d e f i n i t i o n o f g e n e r a l i z e d i n d e p e n d e n c e ) . T h u s o u r pi i s d e t e r m i n e d i f we i n t r o d u c e r e l a t i o n s f o r t h e o p e r a t o r s c r a l l o w i n g us t o t r a n s f o r m e a c h m o m e n t i n t o a t i m e o r d e r e d o n e . M o t i v a t e d b y t h e r e l a t i o n s of [ P W o ] we w a n t t o c o n s i d e r t h e f o l l o w i n g c a s e : W e t r y t o d e f i n e c({) and
for e v e r y / € L (IR), 2
c{})c{g) = P-c(g)c(f) c ( / ) c * ( < j ) = p.c{g)c(f)
is a n t i l i n e a r i n
/
(Here the notation /
if supp / < supp g i f s u p p p n s u p p / = 0.
< J for t w o sets i n M m e a n s : for a l l t\ £ I a n d <
h a v e
s u c h t h a t c(f)
f u l f i l l s t h e r e l a t i o n s ( f o r s o m e fixed p. € I R )
p r o c e s s i s t h e n d e t e r m i n e d b y c j : = c(\l)-
2
€ J we
332 W e w i l l n o t treat t h e c o r r e s p o n d i n g c e n t r a l l i m i t t h e o r e m , b u t we g i v e a t once a r e a l i z a t i o n o f these r e l a t i o n s .
T h e n we c a n find i n this r e a l i z a t i o n t h e limit
d i s t r i b u t i o n of the central l i m i t theorem ( c o m p a r e
[Sch]).
It s h o u l d b e n o t e d t h a t t h i s e x a m p l e i s a l s o c o n t a i n e d ( i n a d i f f e r e n t
language)
as a s p e c i a l case i n [ L i P ] . T h e connection of this process w i t h the A z e m a m a r t i n g a l e [Par] is revealed i n [Sch].
2.
A REALIZATION OF T H E RELATIONS
W e w i l l w o r k o n t h e F o c k s p a c e T o f L (M)
i n the form
2
F^Q)L {nr)^
I
2
2
( 0 i R " ) ,
n>0
n.>0
w h e r e the u s u a l scalar p r o d u c t is given by < h,g >=
j
- J
h(h,---,t„)g(t ,...,t )dt ...dt . 1
n
1
n
n>0
Our
o p e r a t o r s c * ( / ) s h a l l be t h e left c r e a t i o n o p e r a t o r s ( c o m p a r e
this space.
[Eva,Voi]) on
F o r getting our relations between the o p e r a t o r s we i m p l e m e n t i n the
Fock space the corresponding relations i f s u p p / < s u p p g.
f ® g = ft • g ® f
T h u s , it is sufficient to consider the o r d e r e d p a r t o f the F o c k space: the subset of M
n
L e t i R ^ be
c o n s i s t i n g of t i m e - o r d e r e d n - t u p l e s , i . e .
T h e n we e q u i p Ft,
:= 0
L (IRl)
S i
2
2
( 0 mi)
(f =
©„>„/<">)
7l>0
with the scalar product
< &<»W™> >=
6,
[••• f
n m J
and
d e f i n e c*(f)
on F
h (t ,...,t )g^(t M
J
ti<-
f o r / € L (IR) 2
A
l
n
l l
...,t )dt ...dt n
1
by (the s y m b o l U means that t
{
h a s to be
deleted from the tuple) n
(c'{f)h)(U,...,
t) n
=
H'- f(i,)h(t . I
n
n
t)
u
n
= f{ii)h{t , ...,i ) + nfihMh, h, -..,/«) 2
n
333 1. The adjoint
LEMMA
c(f) of c*(f) is given "
/
(c(/)fc)('i.-.*-) = E
, < i
+
<• : - - — o o , t +i : = oo)
by (with
n
1
/
r ' / l W
( i . M i + i , , . . , ^
,=o H
f(t)h(t,t ..,,t )dt+n
/
u
rf(t)h(t t,t ,..., )dt
n
2
u
J -oo
in
Jtj
PROOF:
=
=
/••• /
^"
/•••/ ff J ./(,<-
+
+
1
1 )
(*I,.--,W)[C (/)A * ](
)
,
(
+ M/(<2)/l
( n )
(
)
( n )
(<2,...,
,...,*.)]*!... dt
h^(t ,...,t )[f
n+
l
f(t)g( (t,t ,.,.,t )dt n+1)
n+1
2
l)
(
+ • • • + /*"/(<»+! = /•••/
+
2
n+l
f'' + n /
+
+
/(0s " (
/
/(0s
( n
+
+ 1 )
1 )
(*2,Ms.•-•>*»+!)$
(<2,--.,
+
l,<)^]^2...^n l +
= < c(/)ff( ),/!<")>. n+ 1
LEMMA
2 . W e n a v e
relations
c(/Wff) = t*c(g)c(f) = Pc(g)c(f) PROOF:
(with
f,g €
L (M)): 2
if s u p p / < s u p p
ff
if s u p p (7 n s u p p / = 0
a) Suppose s u p p / < s u p p p. T h e n
=
E
^i)/(*i)M*i,-.ft,-,^.-.0>» '( ,
1 ) + t
'"
1 )
1<»<;'<» =
E f/(*i)»(*i)M
<
i { i
"
a , + ( i
'
1 }
334 b ) S u p p o s e s u p p 17 n s u p p / = 0. T h e n ( n o t e g(t)f(t)
M / j c W i
0 =^ ^/ f
= X)^ .=0
+i
HWisMh,uu ...,t )dt
+ 1
+u
n
'* '" S( JM' .• • • . * i . • • •.*»'••+!.•••»*»)
wE
^
= 0)
,
1
1
<
J= l + /l'"1S'(<.)M
l'---' ") <
+
n
+ E M ff(%M*X.*.->^l^A-rl.---.^,---.
and
n [c*(gWmiU
t) n
= E
)[c(/)k](
i=o
j=l
/' V~V(0Wi.---.*j-i.M^i.---.O*+ +
V
/
.=7+1
^"7(0^1,
,ti,i,ti+u-•-,<»)*]•
•--,<>>•••
^
Thus c(/)c*( ) = S
pc*(g)c(f).
0 If we d e n o t e b y p t h e v a c u u m e x p e c t a t i o n t h e n i t i s e a s y t o see t h a t
moments
of p y r a m i d a l l y ordered p r o d u c t s factorize. W e c a n s u m m a r i z e o u r results i n the following theorem. THEOREM.
The triple
filling
relations
the
c
c
p, ( c ( x i ) ) j g ! R ) i s a generalized
(B(F&),
( / ) ( f f ) = P (9)c{f) c
c
( / ) * ( f f ) = P-c*(g)c(}) c
W i e n e r process
ful-
i f s u p p / < s u p p .7 if s u p p g - n s u p p / = 0
Note that we have i n general
REMARK:
[ ( c ( / ) c * ( ) - ^ * ( , )c(f))(h)](h S
,...,<„)= [ f > i=0
2
>
f'
i+1
**i
f(t)g(t)dt]
h(
tl
,...,<„),
335 w h i c h r e d u c e s f o r ft = ± 1 t o c(f)c*(g)Tc*(g)c(f)=
I n d e e d , i t c a n b e o b t a i n e d b y a p r o d u c t c o c y c l e as i n the e x a m p l e s of
[LiP]. 3.
A
REMARK
ON T H E G E N E R A L
STRUCTURE OF OUR EXAMPLE
W e c a n look at o u r example of a generalized W i e n e r process i n the f o l l o w i n g w a y : W e a r e w o r k i n g o n t h e f u l l F o c k s p a c e J- a n d w e a r e g i v e n a p o s i t i v e o p e r a t o r P : T —» f'.
T h e n we define as n e w s c a l a r p r o d u c t the f o r m <(,V>P-=<(,PV>
{oi£,T}eF.
T h e o p e r a t o r c * ( / ) i s t h e u s u a l left c r e a t i o n o p e r a t o r l*(f), adjoint
of c*(/)
w i t h respect
to o u r new scalar product
whereas c ( / ) is the
< , >p
(compare
also
[ B S p l , 2 ] ) . I n o u r e x a m p l e we f i n d t h i s g e n e r a l s t r u c t u r e i n t h e f o l l o w i n g w a y : D e f i n e t h e o p e r a t o r i Z : T —• J-\ b y A
{
0,
i f i not time ordered
£ , € S „ h(ir o r V r
w h e r e t = ( < ! , . . . , < „ ) 6 IR T h e exponent
( T )
i f t time ordered,
,
a n d x o i : = ( < ( i ) , . . . , '*•(!>)) f ° a p e r m u t a t i o n x £
71
r
x
S. n
i ( x ) denotes the n u m b e r of inversions of the p e r m u t a t i o n x . T h u s ,
b r i n g s e v e r y e l e m e n t o f !F w i t h t h e h e l p o f t h e r e l a t i o n s f®g
for supp / > supp g
= fi-g®f
into a t i m e ordered f o r m a n d o u r scalar product c a n be written o n T x j F as < h,g > :=< M
RphjRpg
>=<
h^R'^R^g
T h u s t h e p o s i t i v e o p e r a t o r P^ i s g i v e n b y R^R^. (Rlg)(*
°t)
=
(t)
> .
Using
for time ordered i € M
n
and allx €
S
n
w e c a n w r i t e P^ i n t h e f o r m
(p„A)(*ot) =
o<)= £
%otV(*>+•(">
>-es„ for t i m e o r d e r e d t € M
n
and allx € S„.
W e s e e t h a t o u r o p e r a t o r c * ( / ) o n Tcorresponds
indeed to the operator /*(/) o n
336 T
(by using the above mentioned relations which are automatically
by t h e choice of o u r s c a l a r p r o d u c t ) .
implemented
T h u s c(f) i s t h e a d j o i n t o f / * ( / ) w i t h r e s p e c t
to o u r scalar p r o d u c t , i . e . i t is d e t e r m i n e d b y
ll
w h i c h is equivalent to
T h i s g i v e s o f c o u r s e t h e o p e r a t o r o f l e m m a 1.
References [AFL
A c c a r d i L . , Frigerio A . , Lewis J . T . : Q u a n t u m Stochastic Processes.
Publ.
R I M S , K y o t o U n i v . 18, 97-133 (1982) [ASW;
A c c a r d i L . , Schiirmann M . , von Waldenfels increment processes o n superalgebras.
[BSpl]
W.: Quantum
independent
M a t h . Z . 198, 451-477 (1988)
Bozejko M . , Speicher R . : A n E x a m p l e of a G e n e r a l i z e d B r o w n i a n M o t i o n . C o m m u n . M a t h . P h y s . 137, 519-531 (1991)
[BSp2] Bozejko M . , Speicher R . : A n E x a m p l e of a G e n e r a l i z e d B r o w n i a n M o t i o n II. These proceedings [Eva [Kiimj
Evans D . E . : O n O
n
. P u b l . R I M S , K y o t o U n i v . 16, 915-927 (1980)
Kummerer B . : Markov dilations a n d non-commutative
P o i s s o n processes.
Preprint [LiP]
L i n d s a y J . M . , P a r t h a s a r a t h y K . R . : C o h o m o l o g y of P o w e r Sets w i t h A p plications i n Q u a n t u m Probability. C o m m u n .
Math.
Phys.
124, 337-364
(1989) [Par]
Parthasarathy K . R . : A z e m a Martingales a n d Q u a n t u m Stochastic
Calcu-
l u s . P r e p r i n t 1989 [PWo] [Sch]
Pusz W . , Woronowicz
S.L.: T w i s t e d Second Quantization. Preprint
S c h i i r m a n n M . : Q u a n t u m q-White
Noise a n d a g-Central L i m i t
Theorem.
C o m m u n . M a t h . P h y s . 140, 589-615 (1991) [Spe]
Speicher R . : A N e w E x a m p l e o f ' I n d e p e n d e n c e ' a n d ' W h i t e Noise'.
Probab.
T h . R e l . F i e l d s 84, 141-159 (1990) [Voi]
V o i c u l e s c n D . : S y m m e t r i e s o f s o m e r e d u c e d free p r o d u c t
C*-algebras. In:
A r a k i , H . , M o o r e , C . C . , S t r a t i l a , S., V o i c u l e s c u , D . (eds.) O p e r a t o r bras and their C o n n e c t i o n w i t h Topology a n d Ergodic Theory.
Alge-
Proceedings
B u s t e n i , R o m a n i a , 1983 ( L e c t . N o t e s i n M a t h . , v o l 1132, p p . 556-588) B e r l i n H e i d e l b e r g N e w Y o r k : S p r i n g e r 1985
Institut fur Angewandte Mathematik, Universitat Heidelberg, Im Neuenheimer Feld 294, W-6900 Heidelberg, Federal Republic of Germany
Quantum Probability and Related Topics Vol. VII (pp. 337-347) © 1992 World Scientific Publishing Company
337
White Noise Analysis Theory and Applications Ludwig Streit BiBoS - Univ. Bielefeld
Universidade da Madeira
D 4800 B I E L E F E L D 1
P 9000 F U N C H A L
I. - Introduction. Since its beginnings in the Mid-Seventies [Hi80] White Noise Analysis has developed into a viable framework for stochastic and infinite-dimensional analysis [HKPS], [Ku90]. As such it has found various applications in mathematical physics and the question arose whether it might not provide a framework also for quantum probability. As a first step towards such an investigation this lecture intends to present an overview over the present status of White Noise Analysis . We do so in Section II. In 1
Section III we present a small sample of recent quantum physics applications. II. - (Gaussian) White Noise X is a generalized random field; for Schwartz test functions feJ(R) the "smeared out" fields <X,f> are Gaussian random variables with mean zero and covariance oo
E( <X,f> <X,g> ) = | f(t) g(t) dt = (f,g) -OO
In terms of the White Noise measure fi and its characteristic function C we have C(f) . E( e
i < X
'
f >
) = jd„[X] e
i < X
'
f >
= e
^
9
Extending test functions to L (IR) we establish an intimate connection of White Noise with Brownian motion by observing that t B(t) = < X , 1|
[0
t]
> = Jx(s)
ds
0
expresses Wiener's Brownian motion in terms of White Noise. In this sense Brownian functionals are functionals of White Noise. A large collection of the latter is of course provided by those of finite variance, i.e. by the Hilbert space 1
i n the meantime the next step, an actual formulation of quantum probability in terms
of White Noise Analysis, has been undertaken by Zhiyuan Huang [Hu91].
338 (L ) = L (j'(R), % Alt). 2
2
The polynomials in <X,f>, f«J(R) form a dense subset 9 in ( L ) , hence the monomials 2
<X,f>
n
are a total set, but as in the construction of orthogonal polynomials in finite
dimensional analysis it is convenient to orthogonalize them successively,
hence we
introduce :<X,f> : = ( 1| - P n
where P .
n 1
Such
M
) <X,f>"
projects onto the subspace spanned by the polynomials of degree less than n.
monomials
are called
"normal
ordered",
"Wick
ordered",
or
"Hermite
polynomials", as well as nth order Wiener-Ito integrals of the specific form :<X,f> : = J f(t ) ... f(t ) 4B(H) - d B ( t ) . n
x
n
n
R" By linearity and continuity we can extend this expression from product kernel functions to P(tj, ... ,t ) f Sy L^(IR ); as a result we obtain an isomorphism between the White n
n
Noise Hilbert space (L ) and a Boson Fock space 2
(L ) ~ e Sy L (R , n! d"t). 2
2
n
Another very useful characterization of (L ) is through the " S - " and "T-transform" 2
T: *[X] S: *[X] -
E( *[X] e
1 < X
'
f >
) = (T*) (f)
E( *[X+f] )
= (S*) (f).
They are related through S*[f] = C(f) • T(-if) = E( *[X] : e
< X
'
f >
:) .
However, the main interest of these constructions is to go beyond them, to consider functionals such as e.g. (1) the "action integral" for Brownian motion (2) "Donsker's i-function"
f :X (t): dt, or 2
S( B(t) - a ).
In the first case the kernel functions are given by F (t n
lt
... ,t„) = 6
n2
6(t - t ) which x
2
clearly is not in L , in the second example the kernel functions turn out to be square 2
339 integrable, see e.g. [HP88] for an explicit evaluation, but £ n! || F
n
||
2 2
= oo. Thus a
good generalization should relax both the L property and the square summability. 2
A standard procedure in finite dimensional analysis to enlarge Hilbert spaces is through the construction of Gelfand triples such as e.g.
S(R") C L ( I R ) C S*(R ) 2
n
n
with S defined as a countable (intersection of) Hilbert space(s)
S = n S , S = D( H p=i p
where H
o s c
p
o s c
),
is the harmonic oscillator Hamiltonian with spectrum a =
{2k+2 :
k=0,l,2,...}. Here we invoke the so-called "second quantization" [Si74] of operators A that act on L (IR). It is characterized as a linear operator on (L ) by its action on normal ordered 2
2
monomials
r(A):<X,f>":
=
:<X,Af>":.
As in the finite dimensional case we construct test functionals as the intersection of operator domains: (j) =lim (J) , (J) = D( r( E p
osc
p
))
P
and the Gelfand triple
(j) c (L ) c 2
elements of the spaces (3) and (*)*
(*)*.
are called Hida test function(al)s
and Hida
distributions, respectively. Here now are some properties of (J): - (J) is a nuclear space - (f) is an algebra under pointwise multiplication [HPS88] - elements Fe(tf) have partial ("Gateaux"-) derivatives
D F [ X ] = lim h
f
1 (F[X+<:h] - F[X])
in (J), even for distribution valued directions hej'(R) [PY89]. Their images in Fock space are annihilation operators.
340 In particular we may choose for h the Dirac distribution S . The corresponding t
derivative operator D is commonly denoted by d ("Hida derivative") and obeys the t
relations (smeared and on a suitable domain)
[9 ,d *] = «(s-t) s
t
and
X(t) = fl + 3? t
The latter relation is the starting point for the theory of stochastic integration in the White Noise Analysis framework, see e.g. [Ku90]. Consider J*dB(t) F . Formally using t
B(t)=X(t) we would obtain
jdB(t) p, = /dt x(t) F = /dt (e, + af) F . t
t
For the particular case of an Ito integral we are dealing with a forward time differential
d
t
+
0
acting on a non-anticipating integrand, giving zero, hence /dB(t) F
t
= /dt d* F . In fact this approach not only produces a viable approach to Ito t
integration but more importantly, to natural generalizations such as to anticipating integrands. - (J) is Frechet differentiable [HKPS].
V : (S) -» J ( R ) ® ( » ) is given by (VF)(t,X) = 0 F[X] and D t
- Furthermore I V F |
2
2 { R )
h
F =
e (J) for all Hida test functions F .
- Every Hida test function F has a version F[X] = £ < : X ®
n
: , F
n
> with smooth
kernels F cSy :f(R ), extending continuously to all X
n
Now let us turn to the dual space (J)* of Hida distributions. Here we find the following properties: - All Hida distributions are of finite order:
- T- and S-transform extend to (»)*; recall that the second vector is in (i).
Examples:
(1) local Wick powers •[X] = :X"(t):
S«(f) = P(t)
(i)* = (J (J).
p
P=I
T*[f] = < « [ . ] ,
e
1
<
, f >
>, where
341 (2) Donsker's 6-function *[X] = «(B(t) - a)
S*(f) = ( 2 T t ) "
1/2
e
"5t
( f ( t )
"
a ) 2
(3) the White Noise ^-functional, given by < 6 , F > = F[Y]
S< (f) = e Y
Y
< Y
' C(f). f >
(4) normalized Gaussians <X,KX> * M
=
6
<X,KX>
:f S
*( ) = f
e
i r
^f>
"
Note that the normalized exponential of example (4), *[X] = Nexp(<X,KX>) has a well defined S-transform for a much larger class of operators K than that for which the Gaussian and the normalizing constant in the denominator are defined (separately). We shall see below that in this case we can define Nexp(<X,KX>) by its S-transform. In fact we observe quite generally that S-transforms are relatively nice objects. Hence it is natural to characterize Hida distributions through their S- or T-transforms. This is done in the following
Theorem [PS89]: The following three statements are equivalent: (1) U : i(R) -> C such that (i)
U(Af+g), for any i,gel(R), extends to an entire function u(z:f,g) of AeC
(ii) u is uniformly of order two, i.e. _ . *, , , .i i r t n\ i - A l I II "<*<:* H | u(z:f,0) I < A e z
2
i AD J for some A , B and p
(2) U( ) is the T-transform of a Hida distribution (3) U(-) is the S-transform of a Hida distribution *c(J).
Functionals with the above property (1) have been called " U-functionals". For a recent review of this and for further results we should like to point out a paper by Meyer and Yan [MY90]. Applications and consequences of this theorem are many: - T o begin with let us note how easy it is to verify analyticity and growth properties in the examples of S-transforms given above; our theorem assures us almost immediately that these expressions are indeed S-transforms. - U-functionals evidently form an algebra, hence induce two algebraic structures on (j)*.
The corresponding products are convolution (using T " ) , and the normal ordered 1
342 product (with S ) . 1
- There exists a linear relation between pairs of Hida distributions through T * = U = S*. H one replaced the White Noise measure by N(0,1) one would find that $ is nothing but the Fourier transform of
Examples and properties: The Fourier transform of 1 is the White Noise i-function at zero <S: 0
1 = Sg,
6 =1 0
The Fourier transform intertwines derivative and coordinate multiplication: (3 *)- = i X(t) 4 t
(X(t) $)" = i d i t
This makes $—• a natural candidate for an extension of Fourier transform to infinite dimension. Indeed Hida, Kuo, and Obata recently showed [HK91] that Kuo's Fourier transform is the unique continuous linear transformation from (J)* to itself with this intertwining property (up to constant multiples of course). Before moving on to applications we want to emphasize that, as in finite dimensional analysis, the construction of a space of generalized functions is far from being unique. On the contrary different choices of such spaces are not only possible but will be suggested by certain applications that one may have in mind. We mention here as examples the triple of Malliavin, Meyer, Watanabe, Sugita et al. [Wa83] with a larger test function space, and consequently, less distributions. Conversely the triple proposed recently by Meyer and Yan arrives at a larger distribution space by omitting the growth condition (ii) in the characterization theorem given above. As an example of a space of test functionals that was discussed in a quantum probability setting we note the space K = fl D(r(ol|)) of[LM87].
III. - Applications in Quantum Physics.
Canonical Quantum Field Theory.- A non-perturbative definition of quantum field
343 dynamics starts from the consideration [HPS88] of "energy forms" f
(F) = <•,
|VF| >
D( ) =
2
f
(?)
for positive $e(:f)*. By a theorem of Kondratiev [K08O] and Yokoi [Yo90] any such positive Hida distribution $ is a measure v^. We shall call 4> admissible if e(F) J|VF| df^ is a closable form in L (di/^) 2
2
{tp, JT, H}
=
and obtain a canonical quantum field theory
in this Hilbert space by putting ^(f)
=
<X,f>, ir(g) the generator of
translations tp—>
$
is the vacuum vector, and * the (generalized) density of the vacuum
state w.r. to the White Noise measure. Here of course the question arises: is the space (t)* of Hida distributions rich enough? Can the vacua (and e.g. the Gibbs states) be expressed as above in terms of positive Hida distributions? The case of free fields gives rise to Gaussian measures where this can be verified by explicit calculations or constructive quantum field theory models such as P(tp) - It is 2
known that they obey so called "^-bounds"
±V>(f) < a H + p || Else f II +7
a,P,y,p>0
2
as
quadratic forms
on suitable
domains.
Referring to
the
literature
[PS90] for
technicalities we report here that the essential condition for canonical quantum fields, or for Markov fields generated by an energy form, to have a representation in terms of White noise is that the field obeys such a ,t>-bound. Summarizing, the vacua of
d=2
Sine-Gordon, exponential, and P(tp) field theories are all Hida distributions. The same 2
is true for the ground states of their Euclidean counterparts [AHP89,90].
Feynman Integrals. - Mathematically, the Feynman integral is a rather elusive object. Formal expressions such as e.g. [
J] t,
dy(r) exp ( i J L(y,y,r) dr) tj
2
for the quantum mechanical propagator K ( y , t v
V(y)
2
2
| y ,t l
1
), with Lagrangean L = f y
2
do not make sense without a mathematical reformulation which in turn will
impose drastic restrictions on the class of admissible potentials V . We mention [AH 76] as an example and for its extensive list of further references.
-
=
344 Here we shall attempt
a very literal reading of Feynman's ideas, along the lines
proposed in [SH83]. - First of all paths are modelled as Brownian paths starting out at a point y
x
at time
zero: y(t) = y, + %fl J Mr)
ir,
0
We shall set
ft=m=l
in the sequel; and have for (the "free"
part of)
Feynman's
exponential exp(ijy dr) 2
- Secondly we note that Feynman suggests some kind of Lebesgue ("flat") measure of integration, hence we compensate the Gaussian falloff of the White Noise measure by introducing a compensating factor exp(lJX dr). 2
- Thirdly, we note that the combined object
exp(^jy dr)
requires an infinite
2
renormalization to make sense, i.e. we use N e x p ( ^ J y d r ) . 2
- Finally we must ensure the pinning of paths at their destination y . This is done by a 2
Donsker 6-function, so that we are led to the expression
I [X] = Nexp(l±-7y dr) i( y(t ) - y 2
0
2
2
).
Regarding this Feynman integrand the following results have been established recently [FPS90]: (1) I makes sense mathematically - I is a Hida distribution 0
0
(2) It does the right thing physically - is the quantum mechanical propagator 0
K . 0
These results follow from the explicit form of the T-transform
( n
°
) ( f )
=
42^fl
e x p
H
m
i \ M
e
)
~
i
'"frowd)
+
2 T (/fdt
+
y> -
).
by noting that this is a U-functional, and by verification that (TI )(0) = is the 0
0
propagator. (3) More interestingly still, one verifies that also for non-zero argument transform is a propagator; more precisely
f, the T-
345
t o)( ) = J 1
where
f
K ^(y ,t 0
2
2
•
e
e
K
o
(y2' 21 y i ' t i ) 4
| yj.tj) is the propagator for the time dependent Hamiltonian
H = - i A + f(t)y. y
'"in (4) Finally, whenever the potential V is such that the propagator K dependent Hamiltonian
H
v
= -|A + f(t) y +V y
."WfcHtfM)
OVXf) = e
v
^ for the time
is a U-functional, then
K
v
(0
(
y
2
>
t
2
,
y i
,
t i )
can serve to define the Feynman integrand as a Hida distribution. Examples show that new classes of interactions become accessible in this formalism, see also [KS91] for an explicit construction of I . v
References [AH76] Albeverio, S.,
Hoegh-Krohn, R.:
Mathematical
Theory of Feynman
Path
Integrals. L N M no.523. Berlin, Heidelberg, New York: Springer (1976). [AHP 89] Albeverio, S., Hida, T . , Potthoff, J . and Streit, L . : The vacuum of the HoeghKrohn model as a generalized White Noise functional; Phys. Lett. B 217, 511-514 (1989). [AHP 90a] Albeverio, S., Hida, T . , Potthoff, J . , Rockner, M . and Streit, L . : Dirichlet forms in terms of White Noise Analysis I—Construction and Q F T examples; Rev. Math. Phys. 1, 291 (1990) [AHP 90b] Albeverio, S., Hida, T . , Potthoff, J . , Rockner, M . and Streit, L . : Dirichlet forms
in terms of White
Noise Analysis II—Construction of infinite dimensional
diffusions Rev. Math. Phys. 1, 313 (1990) [FPS90] De Faria, M . , Potthoff, J . and Streit, L . : The Feynman Integrand as a Hida Distribution. BiBoS preprint no.432, J. Math. Phys., in print.
346 [Hi 80] Hida, T.: Brownian Motion. Berlin, Heidelberg, New York: Springer (1980). [HK91]
Hida,
T.,
Kuo,
H.-H.,
and
Obata,
N.:
Oral
communication
and
"Transformations for White Noise Functionals", preprint. [HKPS] Hida, T . , Kuo, H,-H., Potthoff, J . , and Streit, L.: White Noise: An Infinite Dimensional Calculus. Monograph in preparation. [HP88] Hida, T . , Potthoff, J . and Streit, L.: White Noise Analysis and Applications. In: Mathematics+Physics. Lectures on Recent Results. Vol. 3 (L. Streit, ed.) Singapore: World Scientific (1988). [HPS88] Hida,T., Potthoff, J . and Streit, L.: Dirichlet Forms and White Noise Analysis. Comm. math. Phys. 116, 235 (1988). [Hu91] Huang, Zh.: Quantum White Noises - White Noise Approach to Quantum Stochastic Calculus. Nagoya lecture and preprint (Math. Dept., Wuhan Univ.) in preparation. [KS91] Khandekar, D . C . and Streit, L.: Constructing the Feynman Integrand. UMa preprint, 1991. [Ku90]: Kuo, H.-H.: Lectures on White Noise Analysis. Soochow Univ. Lectures, 1990. [LM87] Lindsay, J . M . and Maassen, H.: An Integral Kernel Approach to Noise. In: Quantum Probability and Applications III. (Proceedings, Oberwolfach 1987), Accardi, L., v. Waldenfels, W., eds.
[MY90] Meyer, P. A . and Yan, J . A . : Les "fonctions caracteristiques" des distributions sur l'espace de Wiener. Strasbourg preprint 1990. [Ko80] Kondratiev, Ju. G.: Nuclear spaces of entire functions in problems
of infinite-
dimensional analysis. Soviet Math. Dokl. 22, 588 (1980).
[Ku89] Kuo, H.-H.: The Fourier Transform in White Noise Calculus. J . Multivariate Analysis 31, 311 (1989)
347 [PS 89] Potthoff, J . and Streit, L . : A Characterization of Hida Distributions. BiBoS preprint no. 406, to appear in J. Funct. Anal. [PS90] Potthoff, J . and Streit, L . : Invariant States on Quantum and Random Fields: tpbounds and White Noise Analysis. BiBoS preprint no. 422. [PY89] Potthoff,
J . and Yan, J.-A.:
Some Results
about
Test
and Generalized
Functionals of White Noise. BiBoS preprint, to appear in Proc. Singapore Probability Conf., 1989. [SH83] Streit, L . and Hida, T . : Generalized Brownian Functionals and the Feynman Integral; Stock. Processes Appl. 16, 55 (1983). [Si74] Simon, B . : The P(
University Press (1974). [Wa83] Watanabe, S.: Malliavin's Calculus in Terms of Generalized Wiener Functionals. In: Theory and Application of Random Fields. G. Kallianpur, ed., Heidelberg: Springer (1983). [Yo90] Yokoi, Y . : Positive generalized white noise functionals. Hiroshima Math. J. 20, 137 (1990).
Quantum Probability and Related Topics Vol. VII (pp. © 1992 World Scientific Publishing Company
349-381) 349
Large Deviations and Mean-Field Quantum Systems R . F . Werner
F B P h y s i k , U n i v e r s i t a t O s n a b r i i c k , P o s t f . 4 4 6 9 , 4500 O s n a b r i i c k , G e r m a n y
1.
Introduction
T h e t h e o r y of L a r g e D e v i a t i o n s as i n i t i a t e d b y V a r a d h a n [ 4 6 , 4 7 ] has b e c o m e a n i m p o r t a n t t o o l i n t h e s t u d y of p h y s i c a l s y s t e m s , i n p a r t i c u l a r o f t h e i r s t a t i s t i c a l mechanics
[27].
A n e s p e c i a l l y successful series of a p p l i c a t i o n s of L a r g e D e v i a t i o n
t e c h n i q u e s is t h e w o r k o f J . T . L e w i s a n d h i s c o w o r k e r s o n t h e B o s e - E i n s t e i n c o n d e n s a t i o n of i n t e r a c t i n g B o s e G a s e s [ 4 , 5 , 6 , 7 , 2 0 ] .
T h e class of m o d e l s t h e y were
able to t r e a t i s d e t e r m i n e d p r e c i s e l y b y the a p p l i c a b i l i t y o f t h e L a r g e D e v i a t i o n t h e o r y : since L a r g e D e v i a t i o n t h e o r y c o n c e r n s t h e a s y m p t o t i c s of m e a s u r e s o n class i c a l p r o b a b i l i t y s p a c e s , t h e i r a n a l y s i s a p p l i e d o n l y t o m o d e l s w h i c h c a n be to c l a s s i c a l ones ( t h e s o - c a l l e d d i a g o n a l m o d e l s [45]).
reduced
Properly "quantum"
mod-
els a p p e a r e d s u s c e p t i b l e t o L a r g e D e v i a t i o n m e t h o d s o n l y b y a p p r o x i m a t i o n w i t h classical models to w h i c h the Large D e v i a t i o n could then be applied [ 1 6 , 2 2 , 2 3 ] . A g e n e r a l i z a t i o n o f t h e L a r g e D e v i a t i o n t h e o r y d i r e c t l y a p p l i c a b l e to q u a n t u m s y s t e m s thus seemed h i g h l y desirable. A s is o f t e n t h e case i n Q u a n t u m P r o b a b i l i t y , i t is n o t p o s s i b l e to f i n d a g e n e r a l i z a t i o n of a c l a s s i c a l c o n c e p t l i k e t h e L a r g e D e v i a t i o n P r i n c i p l e , w h i c h preserves a l l desirable features of the classical theory.
It is therefore a g o o d s t r a t e g y t o choose
one f i e l d w h e r e t h e c l a s s i c a l t h e o r y h a s b e e n s u c c e s s f u l l y a p p l i e d , a n d to t r y to
find
q u a n t u m t h e o r e t i c a l t e c h n i q u e s , w h i c h a l l o w to t r e a t a n a l o g o u s q u a n t u m p r o b l e m s . I n t h e case o f t h e L a r g e D e v i a t i o n P r i n c i p l e a n i n t e r e s t i n g a r e a of m o d e l a p p l i c a t i o n s is t h e a s y m p t o t i c s o f i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e s [ 1 7 ] , a n d their applications to classical mean-field
systems.
Mean-field q u a n t u m systems h a d been treated i n the physics literature by various t e c h n i q u e s , a n d i n v a r y i n g degrees of r i g o u r . T h e first p a p e r t a c k l i n g m e a n - f i e l d q u a n t u m s y s t e m s i n s e a r c h of a Q u a n t u m L a r g e D e v i a t i o n P r i n c i p l e w a s a s t u d y b y P e t z , R a g g i o , a n d V e r b e u r e [36].
B u i l d i n g on their work G u i d o Raggio a n d myself
d e v e l o p e d a g e n e r a l t h e o r y of m e a n - f i e l d q u a n t u m s y s t e m s , w h i c h c a n be
considered
as a g e n e r a l i z a t i o n of a l l a s p e c t s of c l a s s i c a l L a r g e D e v i a t i o n t h e o r y p e r t a i n i n g to systems of i . i . d .
r a n d o m variables.
Its net r e s u l t r e g a r d i n g m e a n - f i e l d s y s t e m s is
the r i g o r o u s d e r i v a t i o n of a v a r i a t i o n a l f o r m u l a for t h e free e n e r g y p e r p a r t i c l e i n the t h e r m o d y n a m i c l i m i t .
T h e v a r i a t i o n a l expression can usually be derived from
the H a m i l t o n i a n b y i n s p e c t i o n , a n d t h e r e is n o n e e d f o r i n t e r m e d i a t e
approxima-
350 t i o n steps as i n [ 9 , 1 1 , 1 6 ] .
T h e v a r i a t i o n is o v e r t h e o n e - p a r t i c l e s t a t e s p a c e , so
t h a t i n case of s p i n s y s t e m s w i t h t h e i r l o w d i m e n s i o n a l s t a t e s p a c e s , t h e v a r i a t i o n c a n be c a r r i e d o u t o n a P C . T h e present r e p o r t is b a s e d m a i n l y o n t h e r e s u l t s of this collaboration w i t h G u i d o Raggio [37,38,39].
In c o m p a r i s o n w i t h the lecture
g i v e n at t h e D e l h i W o r k s h o p t h e s e c t i o n r e l a t i n g o u r a p p r o a c h t o c l a s s i c a l L a r g e D e v i a t i o n t h e o r y a n d t h e one o n u n b o u n d e d
H a m i l t o n i a n s have been
since t h i s m a t e r i a l has n o t a p p e a r e d i n p r i n t e l s e w h e r e .
expanded,
U n f o r t u n a t e l y t h i s m a d e it
n e c e s s a r y to d r o p t h e s e c t i o n a b o u t m e a n - f i e l d d y n a m i c s a n a l y z e d i n t e r m s of app r o x i m a t e l y s y m m e t r i c sequences, w h i c h w a s b a s e d o n j o i n t w o r k w i t h N i c k D u f f i e l d a n d H a n s j o r g R o o s [ 2 4 , 2 5 ] ( C o m p a r e also [ 1 0 , 1 4 , 4 2 ] . B e f o r e e n t e r i n g i n t o t h e m a i n p a r t of t h e a r t i c l e I w o u l d l i k e t o p o i n t o u t some of the difficulties i n generalizing the L a r g e D e v i a t i o n P r i n c i p l e to Q u a n t u m P r o b a b i l i t y . T h e L a r g e D e v i a t i o n P r i n c i p l e is a p r o p e r t y of a s e q u e n c e K measures on a complete separable
m e t r i c s p a c e X.
n
of B o r e l p r o b a b i l i t y
H e u r i s t i c a l l y , i t is a precise
f o r m u l a t i o n of t h e p r o p e r t y t h a t i n t h e v i c i n i t y of a p o i n t i 6 l t h e m e a s u r e s behave a s y m p t o t i c a l l y l i k e 3K„(d:r) oc exp(—a I(x)) w i t h s o m e " r a t e c o n s t a n t s " n
a
f
n
oo
a n d a p o s i t i v e l o w e r s e m i c o n t i n u o u s " r a t e f u n c t i o n " / w i t h c o m p a c t l e v e l sets. M o r e precisely, the measures K
are s a i d to s a t i s f y t h e L a r g e D e v i a t i o n P r i n c i p l e w i t h
n
r e s p e c t to a. a n d / i f for A C Y N o t e t h a t w h e n / v a n i s h e s o n l y at a s i n g l e p o i n t x £ X,
a n d t h e c l o s u r e of the
o p e n set A does n o t c o n t a i n x we f i n d t h a t K ( A ) t e n d s t o z e r o e x p o n e n t i a l l y fast. n
T h e m o s t i m p o r t a n t c o n s e q u e n c e o f these e s t i m a t e s , k n o w n as V a r a d h a n ' s t h e o r e m , is t h e f o l l o w i n g a s y m p t o t i c f o r m u l a for i n t e g r a l s : w h e n h : X
—> IR is a b o u n d e d
continuous function, then lim
— a
ln /
n
J
TK (dx)e- "^ n
a
x)
= ini {h(x)
+ I(x)\x € X}
.
(*)
x
W h e n X is a c o m p a c t set t h e c o n v e r s e of V a r a d h a n ' s t h e o r e m h o l d s , i.e. t h i s f o r m u l a (for a l l h) is i n fact e q u i v a l e n t t o t h e L a r g e D e v i a t i o n P r i n c i p l e f o r K
n
. N o w i n the
a p p l i c a t i o n s to s t a t i s t i c a l m e c h a n i c s n is u s u a l l y a l a b e l for a n i n c r e a s i n g sequence of s y s t e m s , w h o s e size ( v o l u m e or p a r t i c l e n u m b e r )
grows like a . n
Typically
K„
is t h e G i b b s m e a s u r e of s o m e " u n p e r t u r b e d " s y s t e m w i t h e n e r g y d e n s i t y ( H a m i l t o n i a n p e r size) h , 0
a n d )K (dx) n
oc e x p ( — a h ( x ) ) f i ( d x ) .
i n t e r p r e t a t i o n s for t h e i n t e g r a l i n (*):
n
0
T h e r e are two
possible
o n t h e one h a n d w e c o u l d r e a d t h i s as the
expectation E „ ( e " * " ' ™ ) w i t h r e s p e c t to K „ .
S o after t a k i n g t h e l o g a r i t h m we ob-
t a i n t h e c u m u l a n t g e n e r a t i n g f u n c t i o n of t h e r a n d o m v a r i a b l e h. O n t h e o t h e r h a n d we c o u l d also r e a d i t as t h e partition
function of a p e r t u r b e d s y s t e m w i t h energy
d e n s i t y /io + hW h a t I w a n t to p o i n t o u t is t h a t e a c h of these i n t e r p r e t a t i o n s s u g g e s t s a q u a n t u m g e n e r a l i z a t i o n of t h e i n t e g r a l , b u t t h e t w o g e n e r a l i z a t i o n s are d i f f e r e n t :
when
351 D — e x p ( — aHo)
i s t h e d e n s i t y m a t r i x of a q u a n t u m s t a t e , a n d H is a n o b s e r v a b l e
t h e n t h e e x p e c t a t i o n of exp(—aH) is tr ( e - - e - * ) H
.
H
O n t h e o t h e r h a n d , t h e p a r t i t i o n f u n c t i o n o f a s y s t e m w i t h H a m i l t o n i a n Ho + H would be tr ( e — " " - " )
.
T h e t w o e x p r e s s i o n s a r e c l e a r l y different i n g e n e r a l w h e n Ho a n d H d o n o t c o m m u t e . H e n c e i n t h e q u a n t u m case i n f o r m a t i o n a b o u t g e n e r a t i n g f u n c t i o n s o f p r o b a b i l i t y distributions is n o t freely convertible i n t o i n f o r m a t i o n about p a r t i t i o n functions a n d vice v e r s a . W e t h e r e f o r e h a v e t o s e t t l e for o n e o f t h e t w o p o s s i b i l i t i e s i f we w a n t t o construct a n o n - c o m m u t a t i v e L a r g e D e v i a t i o n theory. I n this paper we exclusively consider the " p a r t i t i o n f u n c t i o n " approach.
T h i s is c l e a r l y t h e m o r e u s e f u l p o i n t
of v i e w i f w e w a n t t o a p p l y t h e t h e o r y t o s t a t i s t i c a l m e c h a n i c s .
W e only mention
that the "expectation value" alternative has been t r i e d out b y A c c a r d i a n d Petz [34]. T h e y f o u n d t h a t e v e n f o r t h e q u a n t u m a n a l o g u e o f i . i . d . r a n d o m v a r i a b l e s t h e a n a l o g u e o f t h e c l a s s i c a l L a r g e D e v i a t i o n r e s u l t f a i l s , a n d therefore c a m e t o f a v o u r the " p a r t i t i o n f u n c t i o n " v i e w as w e l l ( see a l s o [35]).
2. Quantum Statistical Mechanics In this section we w i l l review t h e necessary i n p u t f r o m q u a n t u m s t a t i s t i c a l m e c h a n ics, a n d i n p a r t i c u l a r t h e n o t i o n s o f r e l a t i v e free e n e r g y a n d r e l a t i v e e n t r o p y n e e d e d later o n . W e shall describe each physical system by its observable algebra, w h i c h is a C * - a l g e b r a , s a y A, w i t h u n i t I G A.
A s a s t a n d a r d e x a m p l e i t i s g o o d t o keep i n
m i n d t h e case w h e r e A = Add i s t h e a l g e b r a o f d x d - m a t r i c e s . A n y a s s i g n m e n t o f p r o b a b i l i t i e s t o t h e observables i n A is given b y a " s t a t e " , i.e. a linear f u n c t i o n a l -+ C s u c h t h a t u(A*A)
W : A
space w i l l b e d e n o t e d b y K(A). of i t s " d e n s i t y m a t r i x " D , u
> 0 f o r a l l A G A, a n d w ( I ) = A n y s t a t e w G K(Md)
1.
a n d t h e m a t r i x t r a c e as u>(A) = ti(D A).
T h e only
u
requirements o n density m a t r i c e s are D
T h e state
c a n be w r i t t e n i n terms
> 0, a n d t r D =
1.
T h e dynamics of
the system is g i v e n b y a one-parameter group of a u t o m o r p h i s m s t £ E H Aut(.4).
m
6
T h e a u t o m o r p h i s m p r o p e r t y means that the d y n a m i c s is reversible ( i n a
r a t h e r weak sense), a n d is t y p i c a l of t h e microscopic d e s c r i p t i o n of t h e s y s t e m . I n a m a t r i x a l g e b r a t h e d y n a m i c s i s a l w a y s o f t h e f o r m a (A) t
= e' " tH
Ae~ °, ,iH
where
Ho = Ho € A i d i s t h e H a m i l t o n i a n o f t h e s y s t e m . T h e e q u i l i b r i u m states at i n v e r s e temperature 0 = ( f c T )
- 1
w i t h r e s p e c t t o a d y n a m i c a l g r o u p at, i f a n y e x i s t , a r e
d e t e r m i n e d b y t h e K M S - c o n d i t i o n [12]. I n a m a t r i x algebra, there always exists a u n i q u e e q u i l i b r i u m s t a t e p, w h o s e d e n s i t y m a t r i x is g i v e n i n t e r m s o f t h e H a m i l t o n i a n as D
p
= e~P °/ H
tr(e~P °). H
T h e e q u i U b r i u m t h e r m o d y n a m i c s of a n y system can be
352 d e r i v e d f r o m i t s free e n e r g y as a f u n c t i o n of t h e t e m p e r a t u r e a n d o t h e r e x t e r n a l l y controlled parameters. c o m p u t e d as F(H )
I n t h e s t a t i s t i c a l m e c h a n i c s of
= - 4 ]ntl(e~ ). fiHo
Q
finite
s y s t e m s i t is t o
be
T h e free e n e r g y g r o w s p r o p o r t i o n a l l y t o t h e
size of t h e s y s t e m . T h e r e f o r e t h e u s e f u l q u a n t i t y for t h e s t u d y o f t h e t h e r m o d y n a m i c l i m i t , i n w h i c h t h e s y s t e m size goes t o i n f i n i t y , is t h e free e n e r g y p e r p a r t i c l e , o r , p e r h a p s p e r v o l u m e . E v e n t h e free e n e r g y e n e r g y d e n s i t y m a y n o t b e m e a n i n g f u l i n some systems, since its definition depends on the trace, a n d i n a general C * - a l g e b r a there is n o n a t u r a l t r a c e a v a i l a b l e . W e s h a l l be i n t e r e s t e d i n t h e p e r t u r b a t i o n s of s u c h s y s t e m s .
In the m a t r i x
case these p e r t u r b a t i o n s j u s t c o n s i s t of a d d i n g a n o t h e r t e r m H t o t h e H a m i l t o n i a n , so t h e t o t a l H a m i l t o n i a n b e c o m e s Ho + H.
E v e n i n t h e case w h e n , t h e d y n a m i c a l
a u t o m o r p h i s m ct is n o t i m p l e m e n t e d b y a H a m i l t o n i a n , w e c a n m a k e sense o f t h i s t
p e r t u r b a t i o n by defining the p e r t u r b e d d y n a m i c s a af(A)
=
ds a -s(i[h,
a^(A)}).
t
Hamiltonian
of ot
H
t h r o u g h the integral equation
H
T h e o p e r a t o r H is s o m e t i m e s c a l l e d t h e relative
w i t h respect t o a. It is m o r e d i f f i c u l t t o o b t a i n t h e e q u i l i b r i u m
states of t h e p e r t u r b e d s y s t e m . H e r e we r e l y o n t h e p e r t u r b a t i o n t h e o r y
developed
b y A r a k i [ 1 2 , 1 ] . H e defines for a n y s t a t e p ( p r o v i d e d i t is t h e e q u i l i b r i u m s t a t e of s o m e t i m e e v o l u t i o n ) a n d a n y H = H* (E A a s t a t e fi(p, 0H)
£ K(A),
w h i c h i n the
s i t u a t i o n d e s c r i b e d a b o v e w i l l be a n e q u i l i b r i u m s t a t e for t h e p e r t u r b e d e v o l u t i o n . I n t h e m a t r i x case t h e d e n s i t y m a t r i x of fl(p, BH) is of c o u r s e g o i n g t o b e p r o p o r t i o n a l t o exp(—0(Ho
+ Hj).
A r a k i also defines a n o t h e r i m p o r t a n t c o n c e p t , n a m e l y t h e
relative free energy of t h e p e r t u r b a t i o n , w h i c h i n t h e m a t r i x case b e c o m e s F(p,0H)
= -ilntr(e-"< ° H
+ H
> ) + I l n t r ( - « ° > ) = F(H e
H
+ H) - F(H )
0
0
.
( I n A r a k i ' s n o t a t i o n t h e p e r t u r b e d s t a t e is e x p r e s s e d i n t e r m s of a n o n - n o r m a l i z e d positive linear functional p ~ " , H
F(p,0H)
= - \n -? {l)). p
H
so t h a t f2(p,0H)
=
(p~? (J)) p~? , H
_1
and
H
T h e r e l a t i v e free e n e r g y as a f u n c t i o n of t h e e x t e r n a l
parameters i n the H a m i l t o n i a n contains all the information about the
thermody-
n a m i c s of t h e s y s t e m , so i t is t h e q u a n t i t y we w o u l d l i k e t o c o m p u t e i n a n y g i v e n model.
It is c l o s e l y l i n k e d t o t h e relative entropy of a s t a t e (p w i t h r e s p e c t t o t h e
reference s t a t e p, w h i c h i n t h e case of a m a t r i x a l g e b r a ( w i t h t h e s i g n c o n v e n t i o n s u s e d b y A r a k i ) is d e f i n e d as S(/>, v?) = t ^ / J ^ l n D
V
- l n D )) P
.
F o r m o r e i n f o r m a t i o n o n r e l a t i v e e n t r o p i e s i n C * - a l g e b r a s t h e r e a d e r is r e f e r r e d t o t h e a r t i c l e of D e n e s P e t z i n t h i s v o l u m e [35]. F o r fixed p, F is a c o n c a v e f u n c t i o n a l o n t h e h e r m i t i a n p a r t o f A,
w h e r e a s S is a c o n v e x f u n c t i o n a l o n t h e d u a l s p a c e .
T h e b a s i c f a c t a b o u t these f u n c t i o n a l s is t h a t t h e y are L e g e n d r e t r a n s f o r m s of e a c h other.
W e state this result i n the following L e m m a , u s i n g the notations a n d sign
c o n v e n t i o n s i n t r o d u c e d so f a r . I n t h i s g e n e r a l i t y i t h a s b e e n p r o v e n b y P e t z b u i l d i n g on A r a k i ' s version i n the von N e u m a n n algebra context
[2,3,12].
[32], The
r e a d e r is i n v i t e d t o check i t for m a t r i x a l g e b r a s b y m e a n s of e l e m e n t a r y c a l c u l u s .
353 1 L e m m a . Let p be a separating and any
state of a C*-algebra
Then for any H = H* 6
A.
F( ,0H)-j S(p,tp)>tp(H) holds if and only if tp = f2(p,0H). F{p,BH)
.
j
P
Equality
A
K(A):
In
particular,
= i n f { V i f f ) + $S(p,rp)\ip
G K(A)}
.
I n t h e m a t r i x case we c o u l d h a v e s t a t e d a n a n a l o g o u s r e s u l t for
"absolute"
q u a n t i t i e s : t h e r e a t y p i c a l reference s t a t e is t h e n o r m a l i z e d t r a c e r w i t h d e n s i t y matrix D
T
= d
_ 1
1 , w h e r e d is t h e d i m e n s i o n of t h e u n d e r l y i n g H i l b e r t space. T h u s
Ho = 0, a n d t h e p e r t u r b a t i o n H is t h e f u l l H a m i l t o n i a n . T h e n F{r,H) ^lnd
a n d S(r,tp)
=
l n d — S(tp),
N e u m a n n e n t r o p y of the state. i n f {tp(H) U — TS,
w h e r e S(tp)
+ von
T h e v a r i a t i o n a l f o r m u l a t h e n b e c o m e s F(H)
F o r t h e e q u i l i b r i u m s t a t e tp = S?(p,0H)
— jS(tp)}.
= F(H)
denotes t h e u s u a l " a b s o l u t e "
this becomes F
= =
a f a m i l i a r r e l a t i o n from t h e r m o d y n a m i c s .
T h e v a r i a t i o n a l f o r m u l a is a l s o r e m i n i s c e n t of t h e f o r m u l a i n V a r a d h a n ' s t h e o r e m . T h e r e l a t i v e e n t r o p y S(p, tp) t h e n p l a y s t h e p a r t of t h e r a t e f u n c t i o n . T h i s a n a l o g y is o n e of t h e m a i n t h e m e s of t h i s a r t i c l e , a n d w i l l be i n v e s t i g a t e d i n d e t a i l i n s e c t i o n 5. I n s t a t i s t i c a l m e c h a n i c s we are i n t e r e s t e d i n m a n y - p a r t i c l e s y s t e m s , a n d i n particular i n the t h e r m o d y n a m i c l i m i t , i n which the particle number a n d the system size go t o i n f i n i t y . T h e o b s e r v a b l e a l g e b r a for t h e n - p a r t i c l e s y s t e m w i l l be t h e n - f o l d m i n i m a l C * - t e n s o r p r o d u c t [44] A®A®---A,
w h i c h we w i l l d e n o t e b y A . n
(Other
tensor p r o d u c t s , i n p a r t i c u l a r the W * - t e n s o r p r o d u c t , i f applicable c a n also be used). T h e p a r t i c l e s w i l l b e " d i s t i n g u i s h a b l e " , i.e. we d o n o t c o n s i d e r t h e r e d u c t i o n of t h i s a l g e b r a t o t h e B o s e o r F e r m i sector.
H the particles do not interact, the t i m e
e v o l u t i o n f a c t o r i z e s , i.e. a " ( A i ® ••• A )
= a](Ai)
n
® ••• a\(A ). n
s t a t e for a " w i l l b e t h e n - f o l d t e n s o r p r o d u c t p
n
p G K{A)
:=
is t h e e q u i l i b r i u m s t a t e for a . T h i s s i t u a t i o n w i l l b e our 1
p o i n t of reference.
The equilibrium
p ® ••• p G K(A ), n
where
"unperturbed"
W h e n A is c o m m u t a t i v e we c a n also d e s c r i b e t h e u n p e r t u r b e d
s i t u a t i o n as a set of n i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d r a n d o m v a r i a b l e s w i t h d i s t r i b u t i o n p. A s u s u a l we e m b e d A
n
with A ® l m - n € A . m
into A
m
for m > n b y i d e n t i f y i n g A G
k consecutive factors " 1 " .
T h e C * - i n d u c t i v e l i m i t of t h e a l g e b r a s A
identifications w i l l be denoted by
A°°.
If t h e p e r t u r b a t i o n is of t h e f o r m H
n
=#
1
A
n
H e r e as i n t h e s e q u e l we d e n o t e b y I t a t e n s o r p r o d u c t of
® 1 ® •••1 + 1 ® H
1
® •••! + ••• +
n
w i t h these
354 the particles r e m a i n n o n - i n t e r a c t i n g even i n the p e r t u r b e d s y s t e m . r i u m s t a t e t h e n is
= nF(p,
n
The equilib-
O n e e a s i l y c h e c k s t h a t i n t h i s case
Q^p^H ).
n
1
i.e. J is a n e x t e n s i v e q u a n t i t y g r o w i n g w i t h t h e s y s t e m
0H ), 1
1
size. T h i s is t h e t y p i c a l b e h a v i o u r also for i n t e r a c t i n g s y s t e m s . H e n c e t h e i n t e r e s t i n g q u a n t i t y i n t h e t h e r m o d y n a m i c l i m i t n —» oo is t h e r e l a t i v e free e n e r g y per particle l i m „ -F(p ,0H ). n
W i t h a s u i t a b l e c h o i c e of t h e i n t e r a c t i o n s H
n
n
this framework
describes a n y s y s t e m of i n t e r a c t i n g p a r t i c l e s . M e a n - f i e l d s y s t e m s a r e c h a r a c t e r i z e d b y a specific p r o p e r t y of t h e sequence of p e r t u r b a t i o n s H , w h i c h w i l l b e p r e s e n t e d i n the next section. n
It t u r n s o u t t h a t t h e n o t a t i o n b e c o m e s s i m p l e r , w h e n e x p r e s s i o n s a r e w r i t t e n o u t n o t i n t e r m s of t h e H a m i l t o n i a n s for e a c h n , b u t i n t e r m s o f t h e H a m i l t o n i a n d e n s i t i e s , i.e. t h e H a m i l t o n i a n s d i v i d e d b y n. T h u s w e w i l l w r i t e t h e r e l a t i v e H a m i l t o n i a n as H = nH € A. A f u r t h e r s i m p l i f i c a t i o n is m a d e b y setting t h e i n v e r s e t e m p e r a t u r e 0 = 1. T h i s is p o s s i b l e b e c a u s e a p a r t f r o m a f a c t o r j m u l t i p l y i n g t h e free e n e r g i e s , 0 o n l y a p p e a r s i n t h e c o m b i n a t i o n 0H, so w e m a y a b s o r b t h i s f a c t o r i n t o t h e H a m i l t o n i a n d e n s i t y . T h e o b j e c t o f o u r s t u d y is t h u s the l i m i t l i m „ ^F(p ,nH ), a n d t h e l i m i t i n g b e h a v i o u r o f t h e states fi(p ,nH ). n
n
n
n
n
n
n
3. M e a n - f i e l d H a m i l t o n i a n s I n o r d e r t o s p e c i f y t h e class of m e a n - f i e l d s y s t e m s w e h a v e t o d e s c r i b e h o w the relative H a m i l t o n i a n densities H € A" for different n a r e c o n n e c t e d . U s u a l l y i n s t a t i s t i c a l mechanics t h i s connection is specified i n t e r m s of a n i n t e r a c t i o n potential, w h i c h describes the i n t e r a c t i o n between any two particles of t h e s y s t e m , whatever t h e size o f t h e s y s t e m . F o r m e a n - f i e l d s y s t e m s t h e r e q u i r e m e n t is t h a t t h e sequence n
n H-+ H„ s h o u l d b e a n " i n t e n s i v e o b s e r v a b l e " o f a s p e c i a l k i n d . T h i s c o n d i t i o n c a n be e x p r e s s e d i n t e r m s o f t h e p e r m u t a t i o n p r o p e r t i e s o f t h e H a l o n e . W e use the operator s y m : A —> A o f s y m m e t r i z a t i o n w i t h r e s p e c t t o a l l p e r m u t a t i o n s of t h e n sites. E x p l i c i t l y : n
n
n
n
sym (Ai ® A
2
n
® • • • A) n
= — ]T A
® A*
wl
2
® •••A
nn
,
IT
w h e r e t h e s u m is over a l l p e r m u t a t i o n s n of { 1 , . . . rc}. T h u s s y m o n t o t h e a l g e b r a of p e r m u t a t i o n - i n v a r i a n t e l e m e n t s o f A.
n
is t h e p r o j e c t i o n
n
2 D e f i n i t i o n . A sequence of elements A € A m e t r i c , if A = s y m ( 4 ) f o r a l l rc, a n d form>n n
n
n
n
J
n
l i m l i m s u p \\A
m
- sym (A m
n
is called a p p r o x i m a t e l y :
® J -„)|| = 0 m
sym-
.
T h e set of all such sequences will be denoted by y = y(A). A mean-field intera c t i o n i s a n a p p r o x i m a t e l y symmetric sequence such that H = H* for all n s IN n
355 S l i g h t l y d i f f e r e n t , b u t e q u i v a l e n t d e f i n i t i o n s were g i v e n i n [ 3 7 , 3 9 ] . T h e f o l l o w i n g e x a m p l e s h o w s t h a t m e a n - f i e l d i n t e r a c t i o n s i n t h e u s u a l sense are i n d e e d c o v e r e d by this definition. Example
(Mean-field
pair interaction):
L e t us fix a " o n e - p a r t i c l e
o b s e r v a b l e E € A, a n d a n " i n t e r a c t i o n p o t e n t i a l " V € A® energy of the i i
t h
t
h
o f n s y s t e m s is E^
• • • E ® • • • 1 w i t h the operator E i n the
= 1®
p o s i t i o n . S i m i l a r l y , t h e e n e r g y of i n t e r a c t i o n b e t w e e n t h e i w h e r e e.g. F
(
1
2
)
= 1®V®•
n
n -
h
and j
t
h
p a r t i c l e is
1 f-r?
satisfies t h e e q u a t i o n H
n
l n - 2 ) = sym (H
t
• • 1. T h e n we c o n s i d e r t h e f o l l o w i n g H a m i l t o n i a n :
r-f
T h e H a m i l t o n i a n density H
energy"
T h e n the one-particle
A.
=
n
sym (j5®l„_ )-|-sym ('K® n
1
n
® l n - 2 ) , s i n c e t h e average over p e r m u t a t i o n s s i m p l y b e c o m e s t h e
2
average o v e r a l l p o s s i b l e sites t o w h i c h E or V c a n be p e r m u t e d . It i s easy t o see f r o m this t h a t the n o r m i n the D e f i n i t i o n 2 vanishes i d e n t i c a l l y for m > n > a n d we w i l l c a l l sequences w i t h t h i s p r o p e r t y If we r e p l a c e t h e d e n o m i n a t o r
n — 1 b y n i n t h e d e f i n i t i o n of H
sequence w h i c h is s t i l l a p p r o x i m a t e l y s y m m e t r i c . n
n
n
n
N o t e t h a t since
< \\H - H' \\, i n t e r a c t i o n s H a n d H'
±F(p ,nH' )
-
we o b t a i n a
n
I n f a c t , i f A is a p p r o x i m a t e l y
s y m m e t r i c , a n d l i m „ ||A„ — B \\ = 0, t h e n we also h a v e B e y . ±F(p ,nH )
n
n
2,
" s t r i c t l y s y m m e t r i c of degree 2 " .
for w h i c h t h e
n
n o r m difference goes t o z e r o h a v e t h e s a m e free e n e r g y d e n s i t y i n t h e t h e r m o d y n a m i c limit.
O u r d e f i n i t i o n is s t a b l e u n d e r s u c h t r i v i a l m o d i f i c a t i o n s of t h e i n t e r a c t i o n .
H i s t o r i c a l l y , the t e r m " m e a n - f i e l d " derives f r o m the special interactions w i t h V S®S.
T h e n nH
of t h e " f i e l d "
n
= £i(£
(
+ S{i)
0
• S) + o ( n ) , w h e r e S = £ J2i
s ( i )
i s
t
h
e
m
= e
a
n
S.
O n e c a n o b v i o u s l y generalize this e x a m p l e to "fc-body" interactions b y conside r i n g s t r i c t l y s y m m e t r i c sequences of degree k. W e w i l l n o w give a n e x a m p l e w h i c h is n o t j u s t a t r i v i a l p e r t u r b a t i o n of a n i n t e r a c t i o n of finite degree. E x a m p l e (Mean-field version of a lattice system): o n t h e d - d i m e n s i o n a l l a t t i c e H. x e
at e a c h site
x
is a n i s o m o r p h i c c o p y of a fixed C * - a l g e b r a A.
we t h e n h a v e a l o c a l a l g e b r a A(A) a s u b a l g e b r a of A(A')
Consider a spin system
such t h a t the observable algebra A
D
^
= ® eA x
o r
F o r e a c h finite set A C
b y t e n s o r i n g w i t h f a c t o r s 1. T h e i s o m o r p h i s m a
x
A translation
+ x).
invariant
interaction X
£
IAI" 0£A
1
||V || < 0 0 A
.
c
x
V is given b y a
c o l l e c t i o n o f h e r m i t i a n e l e m e n t s VA G 4 ( A ) s u c h t h a t O- (VA) = V A + Z , a n d
A:
d
t a k i n g the
i s o m o r p h i c c o p y of A a t s i t e y t o t h e c o p y a t site x + y e x t e n d s t o a n o p e r a t o r t a k i n g . 4 ( A ) o n t o A(A
2
A C A ' we i d e n t i f y . 4 ( A ) w i t h
356 It i s c l e a r t h a t f o r l a r g e n t h e m a j o r i t y o f t e r m s w i l l h a v e " o v e r l a p " r — 0, a n d c a r e f u l c o u n t i n g [ 3 7 , L e m m a I V . 1] s h o w s t h a t t h e s u m o f a l l t e r m s w i t h r > 0 is b o u n d e d b y (kijn)
\\F \\ \\G \\. H e n c e k
H
t
= sym {F
n
n
k
® G ® l -h-i) t
n
+ O ^ "
1
)
is a p p r o x i m a t e l y s y m m e t r i c . S i n c e u n d e r t h e s y m m e t r i z a t i o n w e c a n e x c h a n g e Fi a n d Gt, t h i s a l s o s h o w s t h e a s y m p t o t i c c o m m u t a t i v i t y .
It i s c l e a r t h a t t h e s u m of a p p r o x i m a t e l y s y m m e t r i c s e q u e n c e s i s a l s o a p p r o x i m a t e l y s y m m e t r i c . Therefore, i f / is any p o l y n o m i a l i n k n o n - c o m m u t i n g variables, a n d F ,...
G y then F
F
1
h
n
= f{F\,...
F) k
is a l s o a p p r o x i m a t e l y s y m m e t r i c . It
w a s s h o w n i n t h e a p p e n d i x o f [37] t h a t t h e class o f f u n c t i o n s / f o r w h i c h t h i s is valid c a n be extended to the whole non-commutative C * - f u n c t i o n a l calculus. T h e functions of this calculus are defined b y t h e very p r o p e r t y t h a t makes i t possible to talk about "the same f u n c t i o n " evaluated o n a r g u m e n t s i n different C*-algebras: i f rj : A —• B i s a u n i t p r e s e r v i n g ^ - h o m o m o r p h i s m , a n d A ,... 1
m u s t h a v e r]f(A A ) 1
= / ( n A , . . . nA ).
k
k
1
A
k
€ A, t h e n we
T h e d o m a i n o f s u c h f u n c t i o n s consists
of a l l fc-tuples s u c h t h a t ||A'|| < c; w i t h c; i n d e p e n d e n t o f t h e a l g e b r a . A s a s p e c i a l case t h i s c o n t a i n s t h e c o m m u t a t i v e f u n c t i o n a l c a l c u l u s o f a s i n g l e h e r m i t i a n v a r i a b l e . S o i f H £ y i s a m e a n - f i e l d i n t e r a c t i o n , / : IR —> IR i s a c o n t i n u o u s f u n c t i o n , and
K
H,
t h e n K i s also a n a d m i s s a b l e m e a n - f i e l d i n t e r a c t i o n . S i n c e y i s a n a l g e b r a a n d
n
n
= f(H ) n
€ A
n
i s d e f i n e d as u s u a l i n t e r m s o f t h e s p e c t r a l r e s o l u t i o n of
c o n t a i n s a l l " m o n o m i a l s " s y m „ ( A i ® l _ i ) w i t h Ai G A, i t c o n t a i n s t h e " i n t e n s i v e n
p o l y n o m i a l s " i n t h e sense o f [ 2 8 ] , a n d o n e c a n s h o w t h a t t h e i n t e n s i v e p o l y n o m i a l s a p p r o x i m a t e a l l o t h e r a p p r o x i m a t e l y s y m m e t r i c sequences ( u n i f o r m l y f o r l a r g e n ) .
T h i s s h o w s t h a t t h e class o f m e a n - f i e l d i n t e r a c t i o n s i s r a t h e r l a r g e , a n d w e w i l l e x t e n d i t e v e n f u r t h e r l a t e r o n . F o r t h e t h e m o m e n t , h o w e v e r , w e s h a l l seek t o solve t h e s t a t i s t i c a l m e c h a n i c s o f a l l these m o d e l s , b y c o m p u t i n g t h e i r free e n e r g y d e n s i t y lim
n
^F(p ,nH ), n
n
a n d b y d e s c r i b i n g t h e p o s s i b l e l i m i t s o f t h e s e q u e n c e S7(p ,
of e q u i l i b r i u m s t a t e s .
n
nH ) n
357 T h e n f o r a s y s t e m i n t h e f i n i t e v o l u m e A t h e H a m i l t o n i a n ( w i t h "free c o n d i t i o n s " ) i s d e f i n e d as
boundary
SCA
T h i s H a m i l t o n i a n is c e r t a i n l y n o t of m e a n - f i e l d t y p e . H o w e v e r , t h e r e is a c a n o n i c a l way to o b t a i n a mean-field i n t e r a c t i o n f r o m i t , s i m p l y b y forcing every particle to i n t e r a c t w i t h e v e r y o t h e r o n e i n e x a c t l y t h e s a m e w a y : w e set
H e r e we h a v e d i v i d e d b y t h e v o l u m e |A| t o o b t a i n a n i n t e n s i v e q u a n t i t y . O n e c a n s h o w [25] t h a t H
is a n a p p r o x i m a t e l y s y m m e t r i c n e t , w h e n A / *
sense o f v a n H o v e [ 1 2 ] .
W e shall call H
2^ i n the
t h e mean-field version of t h e l a t t i c e
MF
i n t e r a c t i o n V. It is a n o b v i o u s q u e s t i o n h o w m u c h t h i s m e a n - f i e l d " a p p r o x i m a t i o n " c a n t e l l u s a b o u t t h e f u l l i n t e r a c t i o n V. H e u r i s t i c a l l y , t h e a n s w e r t o t h i s i s t h a t t h e " a p p r o x i m a t i o n " b e c o m e s b e t t e r as t h e r a n g e of t h e i n t e r a c t i o n increases [ 1 5 , 3 1 ] , a n d a l s o as t h e d i m e n s i o n d b e c o m e s l a r g e r [ 1 3 , 1 5 ] .
It is n o t easy t o m a k e t h i s
i n t u i t i o n r i g o r o u s , a n d we s h a l l n o t a t t e m p t t o r e v i e w t h e e x i s t i n g a n s w e r s i n t h i s a r t i c l e . I n s t e a d we w i l l f o c u s o n s o l v i n g t h e m o d e l g i v e n b y H
as c o m p l e t e l y as
MF
possible. A n i m p o r t a n t p r o p e r t y of t h e set y
is t h a t i t h a s t h e s t r u c t u r e of a n a l g e -
b r a . M o r e o v e r , t h e m u l t i p l i c a t i o n i n t h i s a l g e b r a is " a s y m p t o t i c a l l y a b e l i a n " i n t h e t h e r m o d y n a m i c l i m i t . T h i s is t h e c o n t e n t of t h e n e x t L e m m a . 3 L e m m a . Let F,G FG n
eA
n
Proof :
G y
for all n€
n
be approximately
symmetric sequences, and let H n
n
B y d e f i n i t i o n a n a p p r o x i m a t e l y s y m m e t r i c sequence F is a p p r o x i m a t e d
u n i f o r m l y for l a r g e m i n n o r m b y t h e s t r i c t l y s y m m e t r i c sequences m i—> s y m ( F m
l _ ). m
n
n
=
F
=
n
I N . Then H € j>, a n d l i m „ \\H - G „ F | | = 0.
n
®
H e n c e w e m a y s u p p o s e t h a t b o t h F a n d G are s t r i c t l y s y m m e t r i c , say sym (F n
k
s y m ^ Gi = Gt.
® ! „ _ * ) , and G„
=
sym (G* ® !„_*)
with sym
n
f c
F
k
=
F
k
and
Then H
n
= sym ((F n
t
® l - )sym (Gt
= sym
n
£ ( F
n
k
®
n
® I - )a„(Gt
® !„_/)
n k
t
!„_/)) ,
TT
where a
K
denotes the a u t o m o r p h i s m of A
n
i n d u c e d b y t h e p e r m u t a t i o n TT o f { 1 , . . . n } .
N o w i t is easy t o see t h a t u n d e r t h e s y m m e t r i z a t i o n o p e r a t i o n a l l t e r m s i n t h e s u m for w h i c h { 1 , . . . fc} f l iv {1,.. c o n t r i b u t i o n to H , n
.£} h a s t h e s a m e n u m b e r r of e l e m e n t s m a k e t h e s a m e
namely
(n!)" syn ((F 1
n
k
® l„_ )(i _ fc
t
r
® G ® l t
n+r
_ _<)) fc
.
358 4. V a r i a t i o n a l p r i n c i p l e s S i n c e for e v e r y n t h e free e n e r g y d e n s i t y ^F(p ,nH ) n
is g i v e n b y t h e v a r i a t i o n a l
n
f o r m u l a of L e m m a 1 i t is n a t u r a l t o e x p e c t t h a t b y t a k i n g n —> oo i n t h i s f o r m u l a one c a n o b t a i n a v a r i a t i o n a l f o r m u l a for t h e t h e r m o d y n a m i c l i m i t . S i n c e t h e m i n i m i z e r o f t h e v a r i a t i o n a l e x p r e s s i o n at f i n i t e n is t h e e q u i l i b r i u m s t a t e fi(p ,nH ) we m a y also e x p e c t t h a t t h e m i n i m i z e r s of t h e l i m i t i n g v a r i a t i o n a l e x p r e s s i o n g i v e us i n f o r m a t i o n a b o u t t h e l i m i t i n g e q u i l i b r i u m states. T h i s i d e a w i l l n o w be c a r r i e d out. n
n
Since sym„ H = H„ for e v e r y n , a n d p" is also p e r m u t a t i o n i n v a r i a n t , i.e. / j " o s y m = p , i t is c l e a r t h a t t h e e q u i l i b r i u m s t a t e f2(p", nH ) is a l s o p e r m u t a t i o n i n v a r i a n t . L e t us d e n o t e b y K (A") t h e set of states (p £ K(A ) such t h a t y>osym = tp. B y K (A°°) w e d e n o t e t h e set of states * s u c h t h a t <&\A £ K (A ) f o r a l l n. E x a m p l e s of p e r m u t a t i o n s y m m e t r i c states are t h e h o m o g e n e o u s p r o d u c t states
n
n
n
n
3
n
n
S
n
s
a
n
n
=
$(H)
lim
$(F ) n
n—>oo
e x i s t s . ( H e r e w e h a v e i d e n t i f i e d A w i t h a s u b a l g e b r a of A°°, o n e l e m e n t s of A ). I n d e e d , for m > n:
so $ m a y b e e v a l u a t e d
n
n
- $ ( f f „ ) | = \9(H )
\$(H ) m
- *(sym (F
m
< \\H
m
m
- sym (H m
n
®
l _„))| m
® l _ )||
n
m
n
.
T h u s t h e d e f i n i t i o n of a p p r o x i m a t e s y m m e t r y g u a r a n t e e s t h a t $(H ) sequence.
is a C a u c h y
n
A n o t h e r l i m i t w h i c h e x i s t s for a l l $ £ K (A°°)
is t h e mean
S
5M(P°°,$)=
l i m -S(p ,$\A ) n
entropy
.
n
n—.oo n
T h a t t h i s l i m i t e x i s t s is a s t a n d a r d a p p l i c a t i o n of t h e a d d i t i v i t y of e n t r o p y [ 3 5 ] . h a v e n o w a l l t h e i n g r e d i e n t s to s t a t e o u r first m a i n r e s u l t .
4 T h e o r e m . Let p £ K(A) interaction. Then l i m -F(p ,nH ) n
n
n—.oo ri
M o r e o v e r , any w*-cluster librium states minimizes
be a s e p a r a t i n g s t a t e , a n d H = i n f U(H)
+
I
* ) | * £ K.(A°°)\ I
point \V £ K (A°°) of the sequence the functional $ (-» §(H) + SM(P°°, a
a
£ y(A)
We
mean-field .
J
S7(p ,nH ) $). n
n
of
equi-
359 P r o o f : F r o m L e m m a 1 we get t h e f o r m u l a /„ := ±F(p ,nH ) n
for a l l tp 6
i n s e r t tp = $\A
S
T h e lower
(*)
n
n
w i t h e q u a l i t y iff ip =
K{A ), n
* € K (A°°),
>
n
F o r t h e upper
C2(p ,nH ). n
n
bound let
i n t o t h i s i n e q u a l i t y , a n d t a k e the l i m s u p .
n
bound relies o n t w o e s t i m a t e s e x p r e s s i n g a k i n d of lower s e m i c o n t i n u i t y
of e n t r o p y a n d e n e r g y r e s p e c t i v e l y .
L e t yj
be a n a r b i t r a r y sequence of
£ K,(A ) n
n
s t a t e s , w h i c h is w * - c o n v e r g e n t a l o n g s o m e s u b n e t AT to a s t a t e
6 K (A°°). a
Then
we n e e d t o s h o w t h a t liminf and
(1)
xp {H )>V(H) n
n
l i m i n f -S(p ,rp )> n
n
0
(2)
n
n£N
G i v e n t h e s e b o u n d s i n s e r t ib
.
Suip" ,
n
i n t o (*), w h i c h is t h e n a n e q u a l i t y , a n d
= fi(p ,nH ) n
n
t a k e t h e l i m i t a l o n g s u i t a b l e s u b n e t s , a l o n g w h i c h fl(p ,nH ) n
is w * - c o n v e r g e n t . I n
n
p a r t i c u l a r , c h o o s i n g s u c h a s u b n e t of a s u b n e t of K , a l o n g w h i c h / „ converges to its l i m i n f we f i n d l i m i n f / „ > * ( # ) + S {p°°, *) > limsup / „ , w h i c h s h o w s t h a t t h e l i m i t of / „ e x i s t s , a n d is g i v e n b y t h e v a r i a t i o n a l f o r m u l a . M
I n s e r t i n g o t h e r s u b n e t s Af we find t h a t a l l c l u s t e r p o i n t s are m i n i m i z e r s . W e w i l l n o t s h o w t h e e n t r o p y e s t i m a t e (2), w h i c h c a n be f o u n d i n [ 3 6 ] , a n d is p r o v e n b y r e d u c t i o n t o t h e l o w e r s e m i c o n t i n u i t y of S M ^
0
0
T h e estimate
,-).
w i l l be s h o w n i n L e m m a 11 i n t h e n e x t s e c t i o n for a l a r g e r class of
(1)
mean-field
interactions.
T h i s t h e o r e m as i t s t a n d s is r a t h e r useless, unless we c a n give a u s e f u l p a r a m e t r i z a t i o n o f i n f i n i t e d i m e n s i o n a l c o n v e x set K (A°°). 3
f u n c t i o n a l $ i-»
It is easy to check t h a t t h e e n e r g y
a n d t h e e n t r o p y f u n c t i o n a l $ >->
are b o t h
affine
f u n c t i o n a l s . H e n c e t h e i n f i m u m c a n be d e t e r m i n e d b y c o n s i d e r i n g o n l y t h e e x t r e m e p o i n t s . It t u r n s o u t t h a t K (A°°)
is e v e n a s i m p l e x , a n d t h a t t h e e x t r e m e p o i n t s are
S
p r e c i s e l y t h e p r o d u c t s t a t e s ip°° w i t h tp G K(A). non-commutative
T h i s is t h e c o n t e n t of S t 0 r m e r ' s
" d e F i n e t t i " - T h e o r e m [ 4 3 ] : every $ € K (A°°)
has a n i n t e g r a l
S
representation * = j
Ma (dtp)
w i t h a u n i q u e p r o b a b i l i t y m e a s u r e M $ o n K(A).
ip°° H e n c e we have to m i n i m i z e t h e
expression
$(H)
+ S (p°°,) M
=
j Mi(dtp)
{tp~(H)
+
S M ^
0
0
^
0
0
) }
360 w i h t respect t o a l l choices of A I * . C l e a r l y , t h e m i n i m u m w i l l b e a t t a i n e d p r e c i s e l y for t h o s e m e a s u r e s w h i c h are s u p p o r t e d b y t h e set of t h o s e tp € K(A)
for w h i c h the
integrand attains its absolute m i n i m u m . W e introduce the n o t a t i o n (jH)(tp) = ip"°°(H) = \imtp (H ) n
,
n
a n d o b s e r v e t h a t d u e t o t h e a d d i t i v i t y of r e l a t i v e e n t r o p i e s S ( °°,v°°) M
= 1™-S(p ,
P
= S(p,
n
.
n XX
T h u s t h e i n t e g r a n d s i m p l i f i e s t o {} = (jH)(
5 Theorem.
Under the assumptions of Theorem
lim
n
n—.oo n
Moreover,
4:
= i n f \(jH)(
-F(p ,nH ) n
A p a r t from some techni-
[37]:
tp € K(A)\
any w*-cluster point $
. J
t
G K (A ) a
co
of the s e q u e n c e fl(p ,nH ) n
u n i q u e integral representation
^ = / My(d
the compact set of minimizers
oftpt-* (jH)(tp) +
has a
n
supported
by
S(p,tp).
I n t h e s p e c i a l case w h e n t h e m i n i m i z e r tp of t h i s v a r i a t i o n a l p r i n c i p l e is u n i q u e , we c a n t h u s assert t h e c o n v e r g e n c e of t h e s e q u e n c e of e q u i l i b r i u m s t a t e s t o tp°°. I n t h e g e n e r a l case e a c h of t h e m i n i m i z e r s r e p r e s e n t s a " p u r e p h a s e " o f t h e s y s t e m , so t h e l a c k of c o n v e r g e n c e of fl(p , n
nH ) n
is r e l a t e d t o t h e p r e s e n c e of p h a s e t r a n s i t i o n s .
O u r a s s u m p t i o n s o n H are s i m p l y n o t s t r o n g e n o u g h t o a d m i t a n y c o n c l u s i o n a b o u t t h e m e a s u r e My
b e y o n d w h a t is s t a t e d i n t h e T h e o r e m . O n e c a n s h o w [37] t h a t b y
m o d i f y i n g H w i t h a p e r t u r b a t i o n g o i n g t o zero i n n o r m ( s u c h p e r t u r b a t i o n s d o n o t c h a n g e jH
o r t h e l i m i t i n g free energy, as n o t e d e a r l i e r ) one c a n m a k e t h e e q u i l i b r i u m
states c o n v e r g e t o a n y of t h e p u r e p h a s e s . T h e o r e m 5 is m u c h m o r e u s e f u l i n a p p l i c a t i o n s t h a n T h e o r e m 4. F o r e x a m p l e , w h e n t h e o n e - p a r t i c l e a l g e b r a A is t h e a l g e b r a for a s p i n - 1 / 2 t h e s t a t e s p a c e is a t h r e e - d i m e n s i o n a l b a l l , a n d p r o v i d e d t h e f u n c t i o n s jH
K(A)
is as e x p l i c i t l y k n o w n as
S(p, •), t h e c o m p u t a t i o n of t h e m i n i m u m is a s i m p l e n u m e r i c a l t a s k . T h e g e n e r a l t h e o r y is c o m p l e t e a t t h i s p o i n t , a n d u n l e s s we c o n s i d e r s p e c i a l cases w e c a n n o t h o p e t o r e d u c e t h e p r o b l e m s t i l l f u r t h e r : we w i l l see t h a t any c o n t i n u o u s f u n c t i o n f r o m K(A)
t o M is of t h e f o r m jH
for a s u i t a b l e m e a n - f i e l d i n t e r a c t i o n H.
Hence the
only general results beyond this point w o u l d be statements about the most general v a r i a t i o n a l p r o b l e m o n K(A), that.
a n d we d o n o t h a v e a n y t h i n g i n t e r e s t i n g t o s a y a b o u t
It is p o s s i b l e t o w r i t e d o w n t h e E u l e r - L a g r a n g e e q u a t i o n f o r t h e v a r i a t i o n a l
p r i n c i p l e i n s o m e g e n e r a l i t y . F o r t h a t we n e e d t h e d e r i v a t i v e o f t h e f u n c t i o n
jH.
361 6 D e f i n i t i o n . Let Hey
a n d tp 6 K(A)
exists. Then H^(tp) determined n i a n i n t h e s t a t e tp.
such that for all ip e K(A)
the
derivative
is called the e f f e c t i v e H a m i l t o -
by this equation
N e a r a m i n i m i z e r tp w e c a n n o w a p p r o x i m a t e (jH)(xp) + S(p,ip) RJ (jH)(tp) + ip(H^(tp)) + S(p, ip). W e k n o w f r o m L e m m a 1 h o w t o c o m p u t e the m i n i m u m o f t h e r i g h t h a n d s i d e w i t h a l i n e a r f u n c t i o n a l i n ip f o r t h e energy, a n d t h a t t h e m i n i m i z e r c a n b e w r i t t e n as a p e r t u r b e d e q u i l i b r i u m state. T h i s i s t h e c o n t e n t of t h e s o c a l l e d " g a p e q u a t i o n " , a n i m p l i c i t e q u a t i o n f o r
7 P r o p o s i t i o n . If tp minimizes
(jH)(tp) + S(p,tp), and H^(tp) exists, ip = n(p,Heg{
then
.
F o r a p p l y i n g P r o p o s i t i o n 7 a n d T h e o r e m 5 we need to c o m p u t e t h e energy f u n c t i o n jH f o r t h e g i v e n m e a n - f i e l d i n t e r a c t i o n H. I n b o t h e x a m p l e s o f s e c t i o n 3 w e c a n c o m p u t e t h e l i m i t (JH)(tp) = l i m „
n
n
n
degree w e get i n t h e e x a m p l e o f p a i r i n t e r a c t i o n s : (jH)(
=
•
T h e effective H a m i l t o n i a n i n t h e s t a t e
v
= E + 2E (V) + At V
,
: A® A —* A is t h e c o n d i t i o n a l e x p e c t a t i o n d e t e r m i n e d b y E ( A i ®A ) v
a n d A i s d e t e r m i n e d so t h a t H$ satisfies t h e c o n v e n t i o n tp(H^(tp))
2
= = 0
w h i c h i s i m p l i c i t i n D e f i n i t i o n 2. F o r t h e m e a n - f i e l d v e r s i o n o f a l a t t i c e m o d e l w e obtain i n the same way: (jH )(tp) MF
=
Y,
|AfV
| A |
(^A)
,
A: oeA
w h e r e t h e s u m c o n v e r g e s a b s o l u t e l y , a n d u n i f o r m l y i n tp b y t h e a s s u m p t i o n o n V. W e s t a t e s o m e p r o p e r t i e s o f jH, a n d refer t o [ 3 7 , s e c t . I V ] f o r f u r t h e r d e t a i l s a n d for proofs.
It i s easy t o see t h a t jH
: K(A)
—• IR i s a l w a y s a c o n t i n u o u s
362 f u n c t i o n , w h e n K(A)
is equipped w i t h its_w*-topology.
is a n i s o m e t r y i n t h e sense t h a t f o r
M o r e o v e r , j : y —>
C(K(A))
Hey:
ll^llc(K(X))=^ll^»IU- • A fundamental property of j it is a h o m o m o r p h i s m , (jH)(tp)
= FG
n
n
into the product
n
O n e corollary of t h i s is t h a t j
the image m u s t be a closed subalgebra, a n d i t contains t h e linear
f u n c t i o n a l s o n K(A), theorem.
H
o f f u n c t i o n s i n C(K(A)).
= (jF)(
is o n t o C(K(A)):
w h i c h follows i m m e d i a t e l y f r o m L e m m a 3 is that
taking the n-wise product
so i t m u s t b e e q u a l t o C(K(A))
b y t h e Stone-Weierstrafi
W e c a n take the h o m o m o r p h i s m property f u r t h e r , a n d e x t e n d i t to a l l
polynomials, a n d to the non-commutative functional calculus described i n Section 3: i f / is a f u n c t i o n o f k v a r i a b l e s i n t h e C * - f u n c t i o n a l c a l c u l u s , F ,...
F
1
F
n
n
k
g j), and
then
= f(F ,...F ), k
fjF)<„) = /({jF
1
)(¥>),...
(jF )(tp))
.
k
H e r e we s l i g h t l y a b u s e d n o t a t i o n s i n c e t h e f u n c t i o n / : IR* —• IR o n t h e r i g h t h a n d side o f t h i s e q u a t i o n is t h e " s c a l a r v e r s i o n " o f t h e e l e m e n t / o f t h e C * - f u n c t i o n a l calculus, defined b y / ( x ! , . . . x * l ) = / ( x , . . . x * ) l . 1
tinuous.
T h i s f u n c t i o n is always con-
1
T h i s complete transformation of the n o n - c o m m u t a t i v e
c a l c u l u s t o t h e c o m m u t a t i v e f u n c t i o n a l c a l c u l u s i n C(K(A)) d e t e r m i n e jH
n-wise functional
often allows one to
from H by inspection.
W h e n F € y is g i v e n i n t e r m s o f s i m p l e r i n t e r a c t i o n s F
i t is sometimes
a
possible
to r e d u c e t h e v a r i a t i o n a l p r i n c i p l e a b i t f u r t h e r . 8 T h e o r e m . Let/ variables, n.
be a f u n c t i o n in the non-comm.uta.tive
and F ,...
G j ) . Suppose
F
1
k
that H
n
= f(F
n
functional
,...F ) k
calculus
is hermitian
of k for all
Then lim
-F(p ,nH ) n
n
= inf (f(x)
+ I(x)\x
eH )
,
k
where Iix ,. 1
with
the convention
..x ) k
= i n f {S(p,
VcUF )(
=
x} a
inf 0 = +oo.
T h e p r o o f is o b v i o u s u s i n g t h e a b o v e f o r m u l a f o r jH
i n the C*-functional cal-
c u l u s , a n d c o m p u t i n g t h e i n f i n T h e o r e m 5 i n stages. S o m e p r o p e r t i e s o f t h e " r a t e f u n c t i o n " I follow i m m e d i a t e l y from the definition, a n d f r o m corresponding ties o f S(p, •). W e h a v e f ( x ) > 0 f o r a l l x G H * , a n d I ( x ) = 0 i m p l i e s x
a
=
proper(jF )(p). a
7 is a l o w e r s e m i c o n t i n u o u s f u n c t i o n o n I R * , a n d is + o o o u t s i d e a c o m p a c t set, w h i c h is t h e i m a g e o f K~(A) u n d e r t h e c o n t i n u o u s m a p jF
t a k i n g tp t o t h e v e c t o r x w i t h
363 x
a
I f e a c h jF"
= (jF )(tp). a
of t h e f o r m F
=
a
i s affine, e.g. w h e n e a c h F
± T%=i(S ) a
with S
ii]
is a o n e - p a r t i c l e o b s e v a b l e
a
€ A, t h e n jF i s affine, a n d I i s c o n v e x .
a
A typical example of this is the following. Example
(Mean-field Heisenberg
s p i n s G |1N a t e a c h s i t e .
model):
W e consider a s p i n system w i t h
Hence t h e one-particle algebra is A is t h e algebra of
d x d - m a t r i c e s w i t h d = 2s + 1. A s a reference s t a t e w e t a k e p = r , t h e n o r m a l i z e d t r a c e . T h e H a m i l t o n i a n c a n b e e x p r e s s e d as a f u n c t i o n o f t h e m e a n m a g n e t i z a t i o n s F* = s y m ( S n
a
® l - i ) , where 5 , 5 , 5 1
n
2
denote the three s p i n operators.
3
The
v e c t o r o p e r a t o r w i t h these c o m p o n e n t s w i l l b e d e n o t e d b y S. T h e H a m i l t o n i a n o f the n - p a r t i c l e s y s t e m t h e n becomes
nH
n
5<'-> - ^
= 0j2Bi=l
E
•
§
U
.
)
i,>=l
w h e r e B d e n o t e s t h e e x t e r n a l m a g n e t i c field, a n d J is a c o u p l i n g c o n s t a n t , w h i c h is p o s i t i v e i n t h e f e r r o m a g n e t i c case. O f course t h e m e a n - f i e l d v e r s i o n o f a n y l a t i i c e s p i n s y s t e m w i t h H e i s e n b e r g t y p e interactions leads t o a m o d e l o f this k i n d . W e c a n w r i t e Hn = 8n(F* F%,F*) r
with u ( x ) = B • x — Jx • x
N o t e t h a t t h i s f u n c t i o n does n o t d e p e n d o n t h e v a l u e o f t h e s p i n s. o n l y enters i n t h e r a t e f u n c t i o n I(x) = i n f { S ( r , tp)\
T h e spin
W e have 7 ( x ) =
a
l n ( 2 s + l ) —7 (|x|), w h e r e t h e f u n c t i o n 7 is d e t e r m i n e d b y e l i m i n a t i n g t h e p a r a m e t e r s
3
t from the relations I,(x) = l n s i n h ( ( 2 s + l ) t ) - l n s i n h t - 2<x x = (2s + l ) c o t h ( ( 2 s + 1)4) - c o t h t
.
S i n c e o u r r e f e r e n c e s t a t e i s t h e n o r m a l i z e d t r a c e w e c a n w r i t e t h e c o n c l u s i o n of T h e o r e m 8 i n "absolute" quantities a n d we o b t a i n Urn ^ l n t r ( e - " n pn
n H
") = influx)-i7 (|xl)} s
'
f
t
K
.
>
T h e same f o r m u l a is v a l i d for a l l mean-field s p i n systems w i t h u depending o n l y o n t h e i n t e r a c t i o n c o n s t a n t s , a n d I, o n l y o n t h e v a l u e o f t h e s p i n .
364 5. C o n n e c t i o n w i t h c l a s s i c a l L a r g e D e v i a t i o n s
I n t h i s s e c t i o n we s h a l l e s t a b l i s h t h e r e l a t i o n b e t w e e n t h e t h e o r y d e v e l o p e d i n t h e p r e v i o u s s e c t i o n , a n d t h e a p p l i c a t i o n s of t h e c l a s s i c a l L a r g e D e v i a t i o n t h e o r y t o t h e asymptotics of independent identically d i s t r i b u t e d r a n d o m variables. In p a r t i c u l a r , we w i l l s h o w h o w t h e t h r e e T h e o r e m s 4,5, a n d 8 i m p l y t h e t h r e e " l e v e l s " [27]
of
L a r g e D e v i a t i o n r e s u l t s for i . i . d . v a r i a b l e s . So let us a s s u m e t h a t A is a b e l i a n , s a y A = C(Y) Y.
for s o m e c o m p a c t
space
T h e fact t h a t Y is a u t o m a t i c a l l y c o m p a c t is a m a j o r s i m p l i f i c a t i o n i n L a r g e
D e v i a t i o n t h e o r y . U s u a l l y t h e t h e o r y is f o r m u l a t e d for m e a s u r e s o v e r a n o n - c o m p a c t P o l i s h space.
It is o f t e n r e l a t i v e l y s i m p l e t o get L a r g e D e v i a t i o n b o u n d s for the
m e a s u r e s of c o m p a c t sets, a n d o f t e n d i f f i c u l t t o e x t e n d these b o u n d s t o a l l closed sets (see t h e d i s t i n c t i o n b e t w e e n
"weak" and "full" Large Deviation Principle in
[18].) I n a c o m p a c t space t h i s d i f f i c u l t y v a n i s h e s . O n t h e o t h e r h a n d , t h e r e is no loss of g e n e r a l i t y i n t a k i n g Y c o m p a c t :
starting from a non-compact
s p a c e Y°
we
c a n t a k e Y as t h e s p e c t r u m s p a c e of t h e C * - a l g e b r a o f b o u n d e d c o n t i n u o u s f u n c t i o n s o n Y°,
t h a t is as t h e S t o n e - C e c h c o m p a c t i f i c a t i o n o f Y°.
Since we want to establish
a non-commutative Large D e v i a t i o n theory, our p r i m a r y object has to be the C o a l g e b r a A — Cb(Y°)
= C(Y),
a n d t h i s a p p r o a c h is m a n d a t o r y f o r u s . It m a y seem
at first t h a t t h i s resolves s o m e of t h e m a j o r d i f f i c u l t i e s of L a r g e D e v i a t i o n t h e o r y , b u t t h e r e is a tradeoff.
I n s t e a d of t h e s u b t l e t i e s of L a r g e D e v i a t i o n t h e o r y i n a
n o n - c o m p a c t s p a c e one n o w h a s t o w o r k w i t h t h e r a t h e r u n w i e l d y c o m p a c t i f i c a t i o n , and
one needs t o e s t a b l i s h c o n d i t i o n s m a k i n g t h e r a t e f u n c t i o n i n f i n i t e n e a r the
c o m p a c t i f i c a t i o n p o i n t s Y \Y°.
W e h o p e t h a t these r e m a r k s f o r e s t a l l t h e false
i m p r e s s i o n t h a t b o o k s l i k e t h a t of E l l i s [ 2 7 ] , t o t h e e x t e n t t h a t t h e y a r e c o n c e r n e d w i t h L a r g e D e v i a t i o n s f o r i . i . d . r a n d o m v a r i a b l e s , a r e m a d e o b s o l e t e b y t h e present paper.
T r a d i t i o n a l l y , t h e set Y is o f t e n t a k e n as a s u b s e t of IR , so we a r e d i s c u s s i n g r a n d o m vectors w i t h k components. b i l i t y m e a s u r e o n Y. T h e points i n Y
n
T h e s t a t e p S K(A)
corresponds to a proba-
T h e n - " p a r t i c l e " system has observable algebra A
n
are c o l l e c t i o n s y
=
n
(j/ *,... j/"') 1
are c a l l e d samples of size n . T h e d i s t r i b u t i o n of t h e s a m p l e s is p , n
vectors
=
C[Y ). n
of n r a n d o m v e c t o r s , w h i c h
are i n d e p e n d e n t , e a c h w i t h d i s t r i b u t i o n p.
i.e. t h e r a n d o m
W e shall denote
expecta-
t i o n s w i t h r e s p e c t t o t h i s m e a s u r e b y E [•]. N o w w e w o u l d l i k e t o s t u d y a s y m p t o t i c p r o p e r t i e s of t h e d i s t r i b u t i o n of t h e s a m p l e s . " L e v e l 1" L a r g e D e v i a t i o n r e s u l t s refer t o t h e a s y m p t o t i c d i s t r i b u t i o n o f t h e average y
n
=
£ Y,i=i
°*
t
n
e
vectors
T h i s is a l s o a r a n d o m v e c t o r , a n d
t h e L a w of L a r g e N u m b e r s t e l l s u s t h a t i t w i l l c o n v e r g e c o m p o n e n t w i s e as n - t oo t o t h e e x p e c t a t i o n of e a c h of t h e y ' ' , i.e. t o p(y) 1
i n t h i s c o n t e x t s a y t h a t t h e d i s t r i b u t i o n of y
n
£ R*.
Large Deviation bounds
converges t o t h e p o i n t m e a s u r e at
365 p(y) e x p o n e n t i a l l y fast. L e t us i n t r o d u c e t h e d i s t r i b u t i o n o f y , i . e . t h e p r o b a b i l i t y n
measure
w i t h f )K (dy)f(y)
= E[/(yJ]
1) n
for /
e C(Y).
establish t h a t these d i s t r i b u t i o n s behave like K ^ ' ( d y )
W e would like to
oc e x p ( - n /
( 1 )
( y ) ) , i n the
sense m a d e p r e c i s e b y t h e L a r g e D e v i a t i o n P r i n c i p l e . B y t h e converse o f V a r a d h a n ' s T h e o r e m t h e L a r g e D e v i a t i o n P r i n c i p l e is equivalent t o t h e f o r m u l a l i m ^ l n JTK (dy)e( 1) n
= lim — I n E
nfM
Je-"'^-']
= inf{/(y) + l( >(y)|y y} 1
.
G
B u t t h i s i s p r e c i s e l y t h e k i n d o f e x p r e s s i o n T h e o r e m 8 i s a b o u t : t h e left h a n d side is s i m p l y ^F(p , n
/ ( y ) ) , a n d w e o n l y h a v e t o i d e n t i f y t h e sequences F ,...
F
1
n
w i t h t h e k s t r i c t l y s y m m e t r i c sequences of c o m p o n e n t s o f y„.
k
G j)
T h e o r e m 8 also
i d e n t i f i e s t h e r a t e f u n c t i o n as I< \z)=w£{S(p,
for z G Y'. B y t h e r e m a r k s after T h e o r e m 8 i t i s clear t h a t vanishes o n l y a t p(y), t h e a l m o s t s u r e l i m i t o f y . D e v i a t i o n s f r o m t h i s l i m i t therefore o c c u r w i t h exponentially decaying probability. n
W e c a n get a n o t h e r e x p r e s s i o n f o r fl ' b y i n s e r t i n g a l i n e a r f u n c t i o n a l f o r / , e.g. / ( y ) = — A • y w i t h A £ IR*. U s i n g t h e i . i . d . p r o p e r t y o f t h e y^ w e c a n c o m p u t e t h e left h a n d s i d e o f t h e l i m i t f o r m u l a a n d get 1
I n E [e ] = s u p | A • y - / X
Hence l ' '
y
( 1 )
(y)}
.
i s t h e L e g e n d r e t r a n s f o r m o f t h e c u m u l a n t g e n e r a t i n g f u n c t i o n A (-»
1
I n E [e ] o f p. T h i s i d e n t i f i c a t i o n is o n e o f t h e s t a n d a r d t e c h n i q u e s o f L a r g e D e v i X
y
ation theory. F o r each sample y Y b y J L (y )(dy)A(y) n
n
n
= ( y ' \ • • • t / ' ) w e c a n define a p r o b a b i l i t y m e a s u r e L J
n
n
= J £ i i A ( y « ) f o r A G A s C(X).
on
T h i s s u m of n point
m e a s u r e s i s c a l l e d t h e empirical measure a s s o c i a t e d w i t h t h e s a m p l e y „ £ Y .
Note
n
t h a t w h i l e f o r L e v e l 1 t h e l i n e a r s t r u c t u r e o f Y C IR* was e s s e n t i a l f o r even d e f i n i n g y , n
the measures
L
n
m a k e sense o n a n a r b i t r a r y c o m p a c t set Y.
s h a l l d r o p t h e r e s t r i c t i o n Y C IR* f r o m here o n w a r d s .
let u s d e n o t e t h e set o f p r o b a b i l i t y m e a s u r e s o n Y b y K(A) L
n
:Y
n
T h e r e f o r e we
A s i n the previous
sections Thus
= K(C(Y)).
—> K( A). T h e f o l l o w i n g L e m m a s h o w s t h a t these m a p s i n d u c e a n inverse of
t h e o p e r a t o r j. W e e m p h a s i z e t h a t t h e e x i s t e n c e of s u c h a n inverse i s a c o n s e q u e n c e of t h e c l a s s i c a l n a t u r e o f A, w h i c h does n o t g e n e r a l i z e t o n o n - c o m m u t a t i v e 9 L e m m a . Let f € C(K(A)), Then foLe
y(A),
a n d define ( / o L ) „ ( y „ ) = f(L (y„))
and j(f o L) = f.
n
for y
A.
n
G Y.
366 P r o o f : C o n s i d e r first t h e case o f a n affine f u n c t i o n a l f(tp)
= ip(A) f o r A € A.
( / o L)(y )
( „).
= L (y )(A)
n
n
= 1 £ £ = i A(y<0)
n
(I
=
E
«
s y m „ ( A ® l _ i ) is s t r i c t l y s y m m e t r i c , a n d j(foL)
=
J
A
(0)
Then
Hence (/ o L )
=
If / is a p o l y n o m i a l i n affine
= f.
n
y
f u n c t i o n a l t h e s a m e f o r m u l a f o l l o w s b e c a u s e j is a h o m o m o r p h i s m f o r t h e n - w i s e product.
B y t h e S t o n e - W e i s e r s t r a f i t h e o r e m t h e a l g e b r a of p o l y n o m i a l s is dense
i n C(K(A)).
H e n c e t h e r e s u l t f o l l o w s f r o m t h e o b s e r v a t i o n t h a t u n i f o r m l i m i t s of
a p p r o x i m a t e l y s y m m e t r i c sequences
a r e a p p r o x i m a t e l y s y m m e t r i c , a n d t h a t j is
continuous w i t h respect to u n i f o r m l i m i t s .
L e t u s d e n o t e t h e d i s t r i b u t i o n of L
n
• b y K ^ \ " L e v e l 2 " L a r g e D e v i a t i o n es2
t i m a t e s refer t o t h e a s y m p t o t i c s of t h i s f a m i l y o f m e a s u r e s , m o r e p r e c i s e l y t o t h e l i m i t s of e x p r e s s i o n s t h e f o r m
an = —
n
w h e r e / : K(A)
I n E \e~ ^A L n
= —
i
ln /
n
J
K(
,
C
K(A)
H-> M is a n a r b i t r a r y c o n t i n u o u s f u n c t i o n . W i t h t h e p r e p a r a t i o n of
t h e p r e v i o u s L e m m a we c a n i m m e d i a t e l y c o m p u t e s u c h l i m i t s u s i n g T h e o r e m 5: we get l i m „ a
= i n f {f(
n
T h i s is t o s a y t h a t t h e m e a s u r e s
t h e L a r g e D e v i a t i o n P r i n c i p l e w i t h r a t e f u n c t i o n I^ \tp) =
satisfy
S(p,(p).
2
T h e o r e m 4 i s t h e " L e v e l 3" L a r g e D e v i a t i o n r e s u l t i n t h i s p a p e r . W h e r e a s o n t h e l o w e r levels o n l y c e r t a i n f u n c t i o n s o f t h e s a m p l e w e r e c o n s i d e r e d , we n o w a l l o w a r b i t r a r y f u n c t i o n s of t h e s a m p l e . H o w e v e r , w e t a k e s a m p l e s t o b e e q u i v a l e n t w h i c h differ o n l y b y a p e r m u t a t i o n .
A convenient q u a n t i t y w h i c h describes the sample
u p t o p e r m u t a t i o n s i s t h e m e a s u r e R (yn)
=
n
n
A
n
<5j/„
0
sym
n
£ K(A").
The
a r e different for e a c h s a m p l e , b u t s i n c e t h e r e a r e c a n o n i c a l
K(A )
»A
for n < m we c a n s t i l l c o n s i d e r t h e a s y m p t o t i c s of t h e e x p e c t a t i o n values
m
for a l l A
(Rm{y ))(A ) m
spaces
embeddings
n
is s i m p l y A
m
m
n
n
lim = i I n E [ e
for a n y A £ U
n €
as s o o n as in > n. A s a f u n c t i o n of t h e s a m p l e t h i s
£ A,
n
= sym (A
® l
m
-
n
)
£ C(Y) .
- ] = inf
m A
H e n c e we c a n c o n s i d e r t h e l i m i t s of
m
+ M ?
0
0
, * ) ! * €
K (A°°)} a
iN
In order to m a k e contact w i t h the u s u a l f o r m u l a t i o n of L e v e l 3 L a r g e D e v i a t i o n s , we h a v e t o f o r c e e a c h R(y ) n
t o b e c o m e a n e l e m e n t o f t h e s a m e m e a s u r a b l e s p a c e for
e a c h n , so t h a t t h e K ^ ' c a n b e m e a s u r e s o n a fixed s p a c e . T h e c u s t o m a r y w a y t o do 3
t h i s is t o use t h e set M t(Y s
m e a s u r e s o n t h e set Y
z
z
) of s t a t i o n a r y ( i.e. t r a n s l a t i o n i n v a r i a n t ) p r o b a b i l i t y
of F - v a l u e d sequences.
consider the p e r i o d i c c o n t i n u a t i o n y
n
£ Y^
F o r each sample y
n
£ Y, n
(i.e. y i ' ^ = yn \ w h e n e v e r i = j
F o r e a c h s a m p l e o n e c o n s i d e r s t h e m e a s u r e R (yn) n
£ M t(Y a
2
we c a n modn).
) , w h i c h is t h e average
367 of t h e n p o i n t m e a s u r e s o n t h e d i s t i n c t t r a n s l a t e s of y . n
on the sample we c a n consider the d i s t r i b u t i o n
T h r o u g h its dependence
of - R „ ( y ) . T h e s t a t e m e n t is n
t h e n t h a t the measures K ^ , ' satisfy the L a r g e D e v i a t i o n P r i n c i p l e w i t h rate f u n c t i o n 3
SM(P°°,-).
A p a r t f r o m t h e t e c h n i c a l difference, t h a t e a c h finite s a m p l e h a d t o b e b l o w n u p a r t i f i c i a l l y t o a " s a m p l e " y „ of t h e i n f i n i t e s y s t e m , t h e i m p o r t a n t difference b e t w e e n t h i s r e s u l t a n d T h e o r e m 4 is t h a t w e d i d n o t c o n s i d e r s t a t i o n a r y , i.e.
translation
i n v a r i a n t s t a t e s , b u t o n l y t h e s u b s e t of f u l l y p e r m u t a t i o n s y m m e t r i c states.
It is
possible to generalize o u r results i n this direction. T h e n a t u r a l setting is t h e n the class o f s y s t e m s o n a n a r b i t r a r y l a t t i c e i n 7L w i t h t r a n s l a t i o n i n v a r i a n t i n t e r a c t i o n s . d
T h e r o l e o f s y m m e t r i c s t a t e s is t h e n p l a y e d b y t h e t r a n s l a t i o n i n v a r i a n t states.
The
L e v e l 3 r e s u l t i n t h i s s e t t i n g i s a g a i n a G i b b s v a r i a t i o n a l p r i n c i p l e , a n d is i n f a c t a w e l l k n o w n r e s u l t f r o m s t a t i s t i c a l m e c h a n i c s [ 1 2 , 3 0 ] . S i n c e t h e e q u i l i b r i u m states of c l a s s i c a l finite r a n g e i n t e r a c t i o n s c a n b e c o n s i d e r e d as M a r k o v i a n r a n d o m t h i s v a r i a t i o n a l p r i n c i p l e c a n also b e c o n s i d e r e d as a n o n - c o m m u t a t i v e
fields,
version of
t h e L a r g e D e v i a t i o n r e s u l t s for M a r k o v c h a i n s . It is i n t e r e s t i n g t h a t a n a n a l o g u e o f t h e L e v e l 2 r e s u l t T h e o r e m 5 does n o t e x i s t . T h e r e a s o n is t h a t t h e r e i s n o u s e f u l p a r a m e t r i z a t i o n o f t h e set o f e r g o d i c (i.e.
e x t r e m a l t r a n s l a t i o n i n v a r i a n t ) states,
w h i c h c o u l d r e p l a c e S t 0 r m e r ' s T h e o r e m . ( C o m p a r e also [40]).
6. U n b o u n d e d
Hamiltonians
T h e v a r i a t i o n a l p r i n c i p l e s of s e c t i o n 4 c a n be e x t e n d e d interactions.
t o e v e n w i d e r classes of
I w i l l d e s c r i b e t w o s u c h e x t e n s i o n s b e c a u s e t h e y i l l u m i n a t e t h e es-
s e n t i a l a s s u m p t i o n s b e h i n d these r e s u l t s , a n d also b e c a u s e t h e y are v e r y u s e f u l i n a p p l i c a t i o n s . I n t h i s s e c t i o n I w i l l c o n s i d e r m e a n - f i e l d i n t e r a c t i o n s H„ w h i c h are n o t b o u n d e d b u t o n l y b o u n d e d b e l o w . T h u s p h y s i c a l l y we a l l o w s t r o n g l y r e p u l s i v e i n t e r a c t i o n s . T h i s d i r e c t i o n of u n b o u n d e d n e s s is u s u a l l y less p r o b l e m a t i c a l i n s t a t i s t i c a l m e c h a n i c s , b e c a u s e d u e t o t h e B o l t z m a n n f a c t o r e~
fiH
h i g h energy corresponds very
s m a l l p r o b a b i l i t y . I n e x t r e m e cases, l i k e a n i n f i n i t e l y r e p u l s i v e h a r d core p o t e n t i a l , t h i s r e s u l t s i n a c o n s t r a i n t o n t h e s y s t e m , i.e. t h e p r o b a b i l i t y for t w o p a r t i c l e s to c o m e c l o s e r t o e a c h o t h e r t h a n t h e h a r d core r a d i u s is e x a c t l y z e r o , a n d t h e c o r r e s p o n d i n g d e n s i t y m a t r i x is n o l o n g e r f a i t h f u l . F r o m t h e L a r g e D e v i a t i o n p o i n t of v i e w t h e i n c l u s i o n of s u c h i n t e r a c t i o n s m e a n s t h a t we h a v e t o p r o v e V a r a d h a n ' s a s y m p t o t i c i n t e g r a l f o r m u l a for f u n c t i o n s , w h i c h are n o t n e c e s s a r i l y c o n t i n u o u s b u t only lower semicontinuous, a n d typically unbounded. T h e s t a r t i n g p o i n t f o r t h e g e n e r a l i z a t i o n we have i n m i n d is L e m m a 1, i.e. t h e formula F( ,H) P
= i n f {ip(H) + S{p,vb)\vb e K{A)}
.
T h i s v a r i a t i o n a l f o r m u l a c a n s t i l l m a k e sense i f t h e e n e r g y t e r m ip(H)
(*) is replaced
368 b y a n u n b o u n d e d f u n c t i o n a l tp i-> H(ip),
a n d L e m m a 1 suggests t h a t w e s h o u l d
define t h e e q u i l i b r i u m state_0(p, H) as t h e m i n i m i z e r o f tp i-» H(tp) + S(p, tp). O u r c o n d i t i o n s o n H : K(A) —• IR = H U { + 0 0 } s h o u l d t h u s e n s u r e a u n i q u e s o l u t i o n for t h e v a r i a t i o n a l p r o b l e m . F o r t h e e x i s t e n c e o f a m i n i m i z e r i t i s sufficient t o p o s t u l a t e t h a t H i s l o w e r s e m i c o n t i n u o u s , w h e n K(A) i s e q u i p p e d w i t h t h e w * - t o p o l o g y . F o r t h e u n i q u e n e s s i t i s sufficient t o i n s i s t t h a t as i n t h e b o u n d e d case H i s affine, i . e . t h a t H((l
- t)tp + tip) = (1 - t)H(ip) + tH(tp).
Since we o n l y a l l o w t h e value
+ 0 0 b u t n o t t h e v a l u e — 0 0 f o r H(tp) t h e r e i s n o a m b i g u i t y i n t h i s r e l a t i o n . N o w r e l a t i v e e n t r o p i e s are s t r i c t l y c o n v e x i n t h e s e c o n d a r g u m e n t [ 3 3 ] , a n d t h i s p r o p e r t y is i n h e r i t e d b y t h e f u n c t i o n a l t o b e m i n i m i z e d i n (*). H e n c e t h e m i n i m i z e r i s u n i q u e . W e d e n o t e b y A} t h e set of l o w e r s e m i c o n t i n u o u s affine f u n c t i o n a l s o n K(A) v a l u e s i n IR U { + 0 0 } .
with
B y c o n s t r u c t i o n , L e m m a 1 w h i c h w a s t h e c o r n e r s t o n e of
o u r t h e o r y r e m a i n s v a l i d f o r s u c h u n b o u n d e d r e l a t i v e H a m i l t o n i a n s . T h e i d e a of t u r n i n g L e m m a 1 i n t o a d e f i n i t i o n o f F a n d f2(p, H) h a s b e e n c a r r i e d o u t i n t h e v o n N e u m a n n algebraic setting b y D o n a l d [19], a n d t h e reader is referred t o that paper for further details. Strictly symmetric mean-field interactions c a n be formed out of a n y H a m i l t o nian H
n
6 (A )\
b y setting H
n
= sym (H
m
m
® l - ).
n
m
T h e operations i n this
n
definition are the n a t u r a l extensions of their " b o u n d e d " b o u n d e d case: i f H
€ A
n
a n d tp 6 K(A )
n
counterparts to the u n -
w e h a v e tp(H ® J
m
n
m
_„)=
(tp\A )(H ), n
n
a n d t h e r e s t r i c t i o n o p e r a t i o n i s w * - c o n t i n u o u s . H e n c e tp 1—• s y m ( i f „ ® l _ „ ) ( < ^ ) := m
H ((tposym )\A ) n
n
m
m
i s affine a n d l o w e r s e m i c o n t i n u o u s . F o r d e f i n i n g " a p p r o x i m a t e l y
s y m m e t r i c " sequences o f u n b o u n d e d H a m i l t o n i a n s t h e n o r m e s t i m a t e i n D e f i n i t i o n 2 is c l e a r l y n o t a d e q u a t e . It t u r n s o u t , h o w e v e r , t h a t i t w a s t o o s t r o n g a n a s s u m p t i o n a n y w a y , a n d o n l y h a l f o f i t is n e e d e d i n o u r t h e o r y . W e define:
1 0 D e f i n i t i o n . Let A be a, C*-algebra. Then a sequence of lower semicontinuous afEne functionals H : K(A) - » H (i.e. H e ( 4 " ) ) is called l o w e r s e m i s y m m e t r i c i f for m>n: n
H
> j m n sym (H
m
where 0 < 7
m
n
T
n
m
< 1, s „
n
® l
m
_„)- e
> 0 w i t h l i m „ l i m i n f -y
m
m
T h e set of such functionals
m
n
l
,
= 1, a n d l i m „ l i m s u p e
mn
m
will be denoted by J>T =
m
n
= 0.
y^(A).
T o see i n w h a t sense t h i s d e f i n i t i o n r e q u i r e s o n l y o n e h a l f o f a p p r o x i m a t e s y m m e t r y s u p p o s e t h a t b o t h H a n d —H s a t i s f y i t , s a y w i t h 7 T h e n each H
n
affine f u n c t i o n a l o n K(A ), n
H
n
€ A. n
e ,e' , mn
mn
= 1 for simplicity.
m 7 l
is b o t h lower a n d upper semicontinuous, i.e. i t is a w*-continuous a n d a l l o f these a r e o f t h e f o r m H (tp) n
Moreover, we have - e
m
„ l < H
m
- sym (H m
n
® I
m
= tp(H ) n
_ „ ) < e' l mn
with
for some
w h i c h w e c a n t a k e t o b e e q u a l w i t h o u t loss o f g e n e r a l i t y . B u t t h i s i s t h e
s a m e as s a y i n g t h a t \\H - sym (H m
m
n
® Im-»)|| < e ,
e t h e n ensures t h a t H is a p p r o x i m a t e l y s y m m e t r i c .
mn
and the above condition on
369 O f course, the m e a n energy * ( i f ) now t u r n out to be infinite. well defined.
=
l i m „ H ($\A )
for *
n
n
G K (A°°)
may
a
T h e following L e m m a shows t h a t i t is
nevertheless
It a l s o s u p p l i e s t h e e n e r g y e s t i m a t e n e e d e d t o c o m p l e t e t h e p r o o f of
T h e o r e m 4. S i n c e t h i s e s t i m a t e is t h e o n l y p r o p e r t y of t h e H a m i l t o n i a n s u s e d i n t h a t p r o o f , w e t h e r e b y e x t e n d t h e v a l i d i t y of T h e o r e m 4 t o a r b i t r a r y l o w e r s e m i s y m m e t r i c Hamiltonians.
1 1 L e m m a . L e t H G IP (A). (1) for $ 6 K (A°°)
G R exists, and $ H->
n
n
is w*-convergent to $ G K (A°°)
n
along a subnet Af of IN t h e n
a
hminf-H (vn) >
.
n
4 ' , 5 ' C o r o l l a r y . Theorems 4 and 5 holds for all lower semisymmetric
Proof : H
n
B y a d d i n g ( — 7 m i i n f H\(tp) + e i ) l t o e a c h H = l i m s u p „ e n , a n d -y
n
h m i n f H (*\A )
> hminf (
m
m
Taking the l i m s u p
n
= liminf 7
n
m
7 m
n
that
Then
m n
„ff„($tA ) - e ) n
sequences H.
we m a y a s s u m e
n
m
> 0. L e t E
$(#)
semicontinuous.
(2) 2Fy>„ G K (A ) a
= l i m „ H ($\A )
the limit $(H)
a
i s w*-lower
Then
= inH ($\A )
mn
n
n
- e
•
n
i n t h i s i n e q u a l i t y y i e l d s a n i n e q u a l i t y of t h e f o r m h m i n f
>
l i m s u p , i.e. t h e l i m i t d e f i n i n g $ e x i s t s . N o w let $
b e a n e t i n K (A°°)
a
converging to
a
T h e lower semicontinuity
$ H-> 4 ( i f ) m e a n s t h a t f o r a n y s u c h net l i m i n f $"(H) $(H)
> y Hn($\A ) n
n
l i m i n f 9"(H)
— e
n
f r o m t h e p r e v i o u s s t e p for e a c h a w e get
> hminf(j H (^ \A ) n
a
of
> $(/J). U s i n g the inequahty
a
n
a
- e„) >
n
7
„ t f ( $ L 4 ) - e, n
n
,
ot
w h e r e at t h e s e c o n d i n e q u a l i t y w e h a v e u s e d t h a t H
is l o w e r s e m i c o n t i n u o u s ,
n
that the r e s t r i c t i o n m a p is w*-continuous.
and
T h e r e s u l t follows b y t a k i n g t h e l i m i t
w i t h respect to n i n this inequality. F i n a l l y , l e t tp
n
—> $ . T h e n o n c e a g a i n
h m i n f H (ip ) m
m€Af
m
> hminf
7 m n
mGAf
if„(¥J f4 ) -e„ m
n
> -y H (\A") - e „ n
n
,
a n d t h e r e s u l t f o l l o w s b y t a k i n g t h e l i m i t o v e r n £ Af.
In order to illustrate how infinitely repulsive interactions become constraints we c o n s i d e r t h e f o l l o w i n g Example
(Mean-field
example. systems
with Bosonic
constraint):
W e have pro-
c l a i m e d f r o m the outset that the particles under consideration are " d i s t i n g u i s h a b l e " ,
370 i.e.
t h a t t h e o b s e r v a b l e a l g e b r a for t h e n - p a r t i c l e s y s t e m is t h e f u l l t e n s o r p r o -
duct A .
F o r " i n d i s t i n g u i s h a b l e " particles only the elements of the s y m m e t r i c part
sym„ A
c a n be " o b s e r v a b l e " .
n
n
W h e n A is a m a t r i x a l g e b r a t h i s s y m m e t r i c p a r t
c a n be d e c o m p o s e d i n t o a d i r e c t s u m l a b e l l e d b y t h e i r r e d u c i b l e s u b r e p r e s e n t a t i o n s of t h e r e p r e s e n t a t i o n of t h e s i t e - p e r m u t a t i o n g r o u p o n t h e u n d e r l y i n g n - f o l d t e n s o r p r o d u c t H i l b e r t space.
P h y s i c i s t s u s u a l l y o n l y look at the sectors
corresponding
t o t h e t w o o n e - d i m e n s i o n a l r e p r e s e n t a t i o n s , i.e. t h e c o m p l e t e l y s y m m e t r i c o r a n t i s y m m e t r i c sectors c o r r e s p o n d i n g t o B o s o n s o r F e r m i o n s , r e s p e c t i v e l y . W e w i l l n o w show how the Bose s y m m e t r y can be i n c o r p o r a t e d i n t o the mean-field scheme. B u t t h e r e a d e r s h o u l d be w a r n e d t h a t we w i l l not o b t a i n a d e s c r i p t i o n o f B o s e gases i n t h e thermodynamic
limit.
R a t h e r we c o n s i d e r a l i m i t i n w h i c h t h e o n e - p a r t i c l e
H a m i l t o n i a n r e m a i n s u n c h a n g e d , i.e. a l l t h e s y s t e m s h v e i n t h e s a m e f i n i t e b o x , b u t t h e p a r t i c l e n u m b e r n goes t o i n f i n i t y . Let
Ti be t h e u n d e r l y i n g H i l b e r t space, a n d let A = B(H)
b e t h e a l g e b r a of
b o u n d e d o p e r a t o r s o n W. F o r i n f i n i t e d i m e n s i o n a l Ti t h e m i n i m a l C * - t e n s o r p r o d u c t A® A is not e q u a l t o t h e W * - t e n s o r p r o d u c t B(H®H),
so i n t h a t case s o m e r e m a r k s
a b o u t t h e W * - v e r s i o n o f o u r t h e o r y w o u l d b e i n o r d e r . W e w i l l s k i p t h e s e , so f r o m h e r e o n we are s t r i c t l y s p e a k i n g c o n f i n e d t o t h e m a t r i x a l g e b r a case d i m H < Let
us d e n o t e b y H"
of TL w i t h i t s e l f .
O n H"
the n sites, a n d b y P
n
in H
n
H ® • • -H
product
we w i l l d e n o t e t h e p r o j e c t i o n o n t o t h e s u b s p a c e o f vectors w
n
the n - f o l d H i l b e r t space tensor
we h a v e u n i t a r i e s 11* r e p r e s e n t i n g t h e p e r m u t a t i o n s of
w h i c h are i n v a r i a n t u n d e r a l l U .
n
= l as A
=
oo.
L a t e r o n we w i l l a l s o use t h e n o t a t i o n
— Pn- T h e o b s e r v a b l e a l g e b r a of t h e n - p a r t i c l e s y s t e m c a n t h e n b e w r i t t e n
= B(Pn'H) C A . n
(Note the disctinction berween upper a n d lower indices n).
A s b e f o r e w e w i l l choose o u r u n p e r t u r b e d reference s t a t e as a s t a t e g i v e n i n t e r m s of one p a r t i c l e d a t a : we t a k e a n o r m a l o n e - p a r t i c l e s t a t e p, i.e. a s t a t e g i v e n b y a density m a t r i x D , p
a n d define
Pn = 0 » V « ) ) " V M » In other words, p
n
'
is t h e n - p a r t i c l e c o m p o n e n t of a q u a s i - f r e e s t a t e . I n p a r t i c u l a r ,
i f p is t h e e q u i l i b r i u m s t a t e of a o n e - p a r t i c l e t i m e e v o l u t i o n w i t h H a m i l t o n i a n h,
p
n
is t h e e q u i l i b r i u m s t a t e of t h e c o r r e s p o n d i n g n o n - i n t e r a c t i n g n - B o s o n s y s t e m . A s a m e a n - f i e l d H a m i l t o n i a n we t a k e a n y sequence H 6 y*,
restricted to A . n
T h u s we
w o u l d l i k e t o c o m p u t e t h e l i m i t of
/„ :=
n
-F(p ,nH \A ) n
n
n
as n —> oo. L e t us w r i t e d o w n t h e v a r i a t i o n a l p r i n c i p l e for t h i s . A n y s t a t e tp £ c a n b e c o n s i d e r e d as a s t a t e tp o n A
n
b y s e t t i n g tp(A)
a r i s i n g i n t h i s w a y are p r e c i s e l y t h o s e w i t h tp(P ) = n
= tp(P AP ), n
1.
n
K(A ) n
a n d t h e states
Since the density m a t r i x
371 of p c o m m u t e s w i t h a l l U , a n d h e n c e w i t h P„, t h e d i r e c t s u m f o r m u l a a n d t h e s c a l i n g p r o p e r t i e s for r e l a t i v e e n t r o p i e s [33] give n
K
S(p ,v)
= S(p*,$)
n
+ ]n(p"(P ))
.
n
N o t e t h a t t h e s e c o n d s u m m a n d is i n d e p e n d e n t o f tp, so t h a t fn = i n f {H (
Cp(P ) = 1} + l n ( p " ( P „ ) )
tp s K(A ), n
n
n
n
.
Before p r o c e e d i n g w i t h t h e s o l u t i o n o f t h e v a r i a t i o n a l p r i n c i p l e , let u s c o m p u t e t h e l i m i t o f t h e a d d i t i o n a l c o n s t a n t . U s i n g a n eigenbasis o f t h e d e n s i t y m a t r i x D i t is easy t o s h o w t h a t p
oo
E^p"(Pn) = exptr(ln((l-^
( )
)- )) 1
.
n=l
T h i s series c l e a r l y c o n v e r g e s for \z\p < 1, w h e r e p is t h e l a r g e s t e i g e n v a l u e o f D. T h i s i m p l i e s t h e b o u n d l i m s u p „ p ( P ) l < p f o r t h e coefficients. O n t h e other h a n d , P is l a r g e r t h a n t h e o n e - d i m e n s i o n a l p r o j e c t i o n o n t o t h e n - f o l d t e n s o r p r o d u c t o f t h e e i g e n v e c t o r b e l o n g i n g t o p , w h i c h i m p l i e s p " ( P „ ) > Q. T a k i n g these b o u n d t o g e t h e r w e find 0
0
n
p
n
x
n
Q
n
P
0
nm-m{p (P )) n
n
=ln
n
P
.
o
W e w o u l d n o w l i k e t o r e w r i t e t h e c o n s t r a i n t i n t h e v a r i a t i o n a l p r i n c i p l e as t h e c o n s e q u e n c e o f a n i n f i n i t e t e r m i n t h e H a m i l t o n i a n . S o l e t G : K(A ) —> R be defined b y n
n
ro
if
" ^ - l + o o
G
= i fc>(P )
•
n
F o r m a l l y w e c a n w r i t e t h i s as G„ = oo • P „ , a n d i t is i m m e d i a t e l y clear t h a t
G„
is t h e i n c r e a s i n g l i m i t as A —» + 0 0 o f A P ^ , a n d h e n c e lower s e m i c o n t i n u o u s . It is also c l e a r t h a t t h e infimum a b o v e is t h e s a m e as t h e i n f i m u m over H + G + n~ S without t h e c o n s t r a i n t
n
m
n
m
m
m
n
L
T
n
2
x
n
2
J
s p a c e . M u l t i p l y i n g w i t h A = 00 w e find t h a t G „ is i n f a c t s t r i c t l y s y m m e t r i c o f degree
372 2. I n p a r t i c u l a r , (jG)(jp) = G (y> ®
> 0
2
for a n y n o r m a l s t a t e o n B{Ti)
w h i c h is n o t p u r e . H e n c e we f i n d
H e n c e t h e l i m i t of / „ is g i v e n a p a r t f r o m t h e e x t r a c o n s t a n t b y t h e s a m e v a r i a t i o n a l e x p r e s s i o n as i n T h e o r e m 5, w i t h t h e a d d i t i o n a l c o n s t r a i n t , t h a t o n l y p u r e states are c o n s i d e r e d i n t h e v a r i a t i o n . F r o m T h e o r e m 4 we f i n d t h a t w e c a n a l s o v a r y {$(H)
a l l states $
+ SM(P°°over
€
whose S t 0 r m e r measures
K,(A ) AO
are
s u p p o r t e d b y t h e p u r e s t a t e s . B y a r e s u l t of H u d s o n a n d M o o r e [29] t h i s i s e q u i v a l e n t t o s a y i n g t h a t a l l r e s t r i c t i o n s $ \A
n
have Bose s y m m e t r y .
L e t us a s s u m e t h a t p is a n e q u i l i b r i u m s t a t e for t h e H a m i l t o n i a n h, i.e. D e~ I h
t r e~ .
p
=
W e can then simplify the entropy t e r m i n the v a r i a t i o n a l principle:
h
S i n c e tp is p u r e , i t is g i v e n b y a u n i t v e c t o r tp 6 H, a n d S(p,tp) = (tp, — l n D ip) p
(tp,htp) + l n t r e ' . -
1
=
T o t h i s we a d d t h e c o n s t a n t Inpo = —ho — l n t r e * , w h e r e ho is -
t h e s m a l l e s t e i g e n v a l u e of h. T o s u m m a r i z e we get:
l i m ±F( ,nE \A ) Pn
n
= inf {(jff )(|x>(xl) + (x,(& - Mx) x e n, llxll
n
7. I n h o m o g e n e o u s m e a n - f i e l d
systems
A n o t h e r e x t e n s i o n of t h e m e t h o d s of s e c t i o n 3 c o n c e r n s so c a l l e d
inhomogeneous
m e a n - f i e l d s y s t e m s . T o e a c h p a r t i c l e i n s u c h a s y s t e m we a s s i g n a " p o s i t i o n " x< 6
X,
w h e r e X is s o m e fixed c o m p a c t s p a c e . A l l c o u p l i n g c o n s t a n t s a r e a l l o w e d t o d e p e n d on this " p o s i t i o n " .
A t y p i c a l H a m i l t o n i a n is t h e i n h o m o g e n e o u s
v e r s i o n of
the
mean-field pair interaction:
nH ( ,... n
where E
: X
—> A
g i v e n t o p o l o g y o n X, considered
x ) = V
Xl
n
and V
: X
£(x,)
x X
( i )
+ i
—> A ® A
V
V(x
h
Xj
)M
,
axe c o n t i n u o u s f u n c t i o n s for the
a n d t h e n o r m t o p o l o g y o n A.
T h e p o s i t i o n s xi,...X
n
as e x t e r n a l p a r a m e t e r s , a n d we are i n t e r e s t e d i n t h e s a m e
a b o u t t h i s s y s t e m i n A"
are
questions
as b e f o r e .
B e f o r e p r o c e e d i n g let u s i n d i c a t e s o m e p o s s i b l e i n t e r p r e t a t i o n s f o r t h e p a r a m e ters x e X.
I n t h e s i m p l e s t case X is a finite set, a n d e a c h x 6 X l a b e l s a s p e c i e s of
p a r t i c l e s [ 2 6 ] . It is c l e a r t h a t t h e o n e - p a r t i c l e e n e r g y E(x) p o t e n t i a l V(x,y)
6 A a n d the interaction
m u s t d e p e n d o n t h e k i n d of p a r t i c l e s i n v o l v e d , a n d t h i s is p r e c i s e l y
the dependence expressed i n the above H a m i l t o n i a n .
373 A n o t h e r i m p o r t a n t e x a m p l e is t h e s o - c a l l e d " f u l l " B C S - m o d e l , i . e . t h e B C S model w i t h o u t the so-called "strong coupling" assumption, w h i c h w o u l d again make it a h o m o g e n e o u s m e a n - f i e l d m o d e l o f t h e t y p e d i s c u s s e d p r e v i o u s l y . F o r t h i s m o d e l X i s a c o m p a c t r e g i o n i n m o m e n t u m space, d e f i n e d b y s o m e u l t r a v i o l e t t cutoff. T h e o n e - p a r t i c l e e n e r g y i s p r o p o r t i o n a l t o \x\ /2m,
a n d V is g i v e n i n t e r m s o f t h e F o u r i e r
2
t r a n s f o r m o f a n i n t e r a c t i o n p o t e n t i a l i n c o n f i g u r a t i o n space. T h e " p o s i t i o n s " X{ of t h e n p a r t i c l e s a r e a n e n u m e r a t i o n o f t h e d i s c r e t e m o m e n t a of a s y s t e m i n f i n i t e v o l u m e (e.g. w i t h p e r i o d i c b o u n d a r y c o n d i t i o n s ) s a t i s f y i n g t h e cutoff c o n d i t i o n . I n c o n n e c t i o n w i t h s p i n glasses m o d e l s , i n w h i c h t h e x ; a r e r a n d o m v a r i a b l e s have recently a t t r a c t e d a lot of a t t e n t i o n . W e c a n treat here models i n w h i c h there is j u s t o n e r a n d o m v a r i a b l e p e r s i t e . T h e s e a r e c a l l e d site random
mean-field models
[21], as o p p o s e d t o " b o n d r a n d o m " m o d e l s l i k e t h e S h e r r i n g t o n - K i r p a t r i c k m o d e l [41].
T h e r a n d o m variables are "quenched"
i n t h e sense t h a t w e seek t o solve
the m o d e l s e p a r a t e l y f o r e a c h s a m p l e , c o n s i d e r i n g t h e v a l u e s o f t h e x - as e x t e r n a l t
parameters.
T h e free e n e r g y d e n s i t y i n t h e t h e r m o d y n a m i c l i m i t t h u s b e c o m e s a
r a n d o m v a r i a b l e , a n d we w i l l s h o w t h a t i t is i n fact i n d e p e n d e n t of t h e s a m p l e a w a y f r o m s o m e n u l l set. T h e m a i n p r o b l e m i n e x t e n d i n g o u r p r e v i o u s r e s u l t s t o t h e w i d e r s e t t i n g is that the techniques of section 4 depend crucially o n the p e r m u t a t i o n s y m m e t r y of a s y m m e t r y w h i c h i s c o m p l e t e l y lost i n t h e H a m i l t o n i a n w r i t t e n a b o v e . T h e
H, n
s o l u t i o n t o t h i s d i l e m m a is t o permute the positions x ; along with the labels i i n a sense w e w i l l n o w d e s c r i b e .
T o o u r k n o w l e d g e t h i s t r i c k w a s first a p p l i e d i n [8].
W e d e f i n e d t h e o n e - p a r t i c l e e n e r g y as a c o n t i n u o u s , 4 - v a l u e d f u n c t i o n o n X, i.e. a n element of C(X, A).
T h i s a l g e b r a is c a n o n i c a l l y i s o m o r p h i c t o C(X) ® A u s i n g t h e
i d e n t i f i c a t i o n ( / ® A)(x) x X,A®A)
C(X
S i m i l a r l y , we c a n c o n s i d e r V as a n e l e m e n t of
= f(x)A.
= C(X)
2
®A
2
T h e a b o v e H a m i l t o n i a n , c o n s i d e r e d as
= C(X, A) . 2
a f u n c t i o n o f t h e n p o s i t i o n s x ; is t h u s a n e l e m e n t o f C(X ,A ) n
C(X,
A) . n
p a i r off o n e f a c t o r C(X) £
( , )
( x i , . . . x ) = B(x f \ i
n
a n d i n A"
n
= C(X)"
® A"
=
T h e g u i d i n g i d e a for a l l this reshuffling of tensor factors is to always i
w i t h o n e f a c t o r A.
W i t h these i d e n t i f i c a t i o n s w e h a v e
where the operation
is t a k e n i n CXA
n
o n t h e left
o n t h e r i g h t . T h e n w e c a n w r i t e t h e d e f i n i t i o n of t h e H a m i l t o n i a n as
a n e q u a t i o n i n C(X",
A ). n
O m i t t i n g the arguments ( x i , . . . i „ ) from this equation
b e t w e e n f u n c t i o n s w e get
n —, i
•=i
d
w h i c h i s f o r m a l l y t h e s a m e as t h e p a i r i n t e r a c t i o n H a m i l t o n i a n i n t h e h o m o g e n e o u s case.
I n p a r t i c u l a r , w e c a n s i m p l y t a k e over t h e v e r i f i c a t i o n t h a t Ti" is a n a p p r o x -
i m a t e l y s y m m e t r i c s e q u e n c e , a l b e i t n o t w i t h one-site a l g e b r a A b u t w i t h one-site a l g e b r a C(X,
A).
374 12 D e f i n i t i o n . A n i n h o m o g e n e o u s
m e a n - f i e l d i n t e r a c t i o n o v e r t h e compact
parameter space X and the one-particle £ (C(X,A) )1
H
(for short: H £
n
n
algebra A is a lower semisymmetric
W e w i l l later treat the parameters
fixed external parameters which
are p a r t of t h e s p e c i f i c a t i o n of t h e m o d e l .
T h e r e f o r e we h a v e t o m a k e s u r e t h a t
£ (C(X,A) )^ defines a H a m i l t o n i a n i n A"
H
for every x € X " .
n
n
H (x) n
£ ( 4 . " ) ^ b y H„(x)(tp)
sequence
y\C(X,A))).
®
= H (6 n
x
X
We
define
denotes t h e u n i t p o i n t m e a s u r e
at x. T h i s e x p r e s s i o n is j o i n t l y l o w e r s e m i c o n t i n u o u s i n x a n d tp b e c a u s e (x,tp) t-t(6 ,tp) >-> S ®tp is c o n t i n u o u s . Needless t o say, t h i s d e f i n i t i o n e x t e n d s t h e m e a n i n g x
x
g i v e n t o H (x)
for H
n
£ C(X,A)
n
=
n
n
c o n s i d e r ^F(p ,nH (x))
W i t h t h i s d e f i n i t i o n we c a n
C(X",A ). n
as a f u n c t i o n o f a; £ X .
It i s c l e a r t h a t t h i s f u n c t i o n
n
n
is p e r m u t a t i o n s y m m e t r i c for e v e r y n .
T h e following P r o p o s i t i o n shows t h a t the
sequence of these f u n c t i o n s i n h e r i t s t h e l o w e r s e m i s y m m e t r y f r o m 13 P r o p o s i t i o n . Let X metric, and define
and H £ yl(C(X, A))
be metrizable,
/„(*)
= -F(p ,nH (x)) n n
for x £
n
H.
be lower semisym-
X. n
Then f £ yT(C(X)), and
07)00
e K(C(X,A)),
= i n f {(jH)(tp) + S(p ® p, )\tp 9
tp\C(X) = p)
.
Proof : W e s h o w first t h a t / „ is l o w e r s e m i c o n t i n u o u s . L e t x be a net i n X c o n v e r g i n g t o x, a n d let tp £ K(A) b e m i n i m i z e r s of H (S ®tp) + ^S(p, tp). T h e n a
a
n
f(x )
= H„(S
a
Xa
® tp ) + -S(p,tp )
X
a
.
a
n W e m a y c o n s i d e r a s u b n e t a l o n g w h i c h f(x ) converges t o i t s i n f e r i o r l i m i t , a n d a l o n g w h i c h tp converges t o s o m e tp £ K(A). U s i n g t h e l o w e r s e m i c o n t i n u i t y of H a n d S w e o b t a i n l i m i n f f(x ) > H (S ® tp) + ^S(p, tp) > f(x). a
a
n
a
a
n
G i v e n m > n, let K £ (C(X, 4 ) " )
K = H®
T
x
b e d e f i n e d as
lm-n + In ® H ® l -n
n
n
w h e r e fc < (m/n) m e t r i c we h a v e mi3"
m
+••' + l(fc-i)„ ® H ® l _ n
is t h e i n t e g e r p a r t o f (m/n). m
> mj
sym (7J ® 771 > T m n y sym (Ar) - e mn
m
T h e n because #
mn
> -ymnn sym (K) m
S i n c e F(pi ® p > i ® I + I ® h ) = P ( p i A ) + F(p ,h ), e n t r o p y , we h a v e F(p ,nK) m
2
= nfn(x( \...x^) 1
,
f c n
n
m
2
m
is lower s e m i s y m -
2
k 1
1
m n
.
b y t h e a d d i t i v i t y of t h e
2
+ ---+nf (x« - ^+ \... ( «)) n
- e
x
k
.
(*)
375 W e n e e d s o m e m o r e b a s i c p r o p e r t i e s of t h e r e l a t i v e e n e r g y f u n c t i o n a l h i-»
F{p,
h)
w h i c h f o l l o w i m m e d i a t e l y f r o m L e m m a 1, n a m e l y t h a t i t is m o n o t o n e a n d c o n c a v e i n h, a n d t h a t F(p,
h + el)
F(p,jh)
+ (1 -
= F(p,jh
e n t r o p y , F(
® p ,h
Pl
2
= F(p, 7
® 1 + 1®
1
h) 2
fm(x )=
M o r e o v e r , f r o m t h e a d d i t i v i t y of t h e
= F{piM)
+ F(p ,
H e n c e for x
h ).
2
2
£
m
X: m
^F(p ,mH (x )) m
m
> ^ F ( p
* ^
m
m
, n ( s y m
^
Y<
m
-
K)(x )) m
m
F i f , m
'
n { a
"
e
mn
K ) { X m ) )
-
£ m n
nk > Imn— sym (/ ® l _ )(z ) - e , m d e n o t e s t h e p e r m u t a t i o n a u t o m o r p h i s m of A associated w i t h the pern
where a
I n p a r t i c u l a r , for 0 < 7 < 1 we h a v e
h) + e.
) 0 ) > yF(p,h).
m
n
m
m
m n
m
w
m u t a t i o n 7r, a n d w e h a v e u s e d (*)
for a K,
o b t a i n i n g k t e r m s f o r e a c h 7r, w h i c h
n
b e c o m e e q u a l u n d e r t h e s u m o v e r n.
S i n c e nk/m
00 t h i s e s t i m a t e
—i-lasm—t
shows t h a t / € 3 > ( C ( X ) ) T
T h e c o m p u t a t i o n o f jf
w i l l be based o n the r e l a t i o n (p®pf(l)
for p. € K(C(X))
and H
= J
p(dx)p ^(l) H
w r i t t e n here w i t h A r a k i ' s n o t a t i o n p .
£ C(X,A)
This
H
was p r o v e n i n [ 3 9 , L e m m a I I I . 2 ] , a n d b y t a k i n g m o n o t o n e l i m i t s i n H, a n d u s i n g t h a t e a c h H € A}
i s t h e s u p r e m u m o f a n i n c r e a s i n g n e t f r o m C(X,
t h i s r e l a t i o n t o H £ C(X, F((p L e t g £ C(X)^ g l n
n
we e x t e n d
A),
I n t h e n o t a t i o n o f t h i s p a p e r w e c a n w r i t e t h i s as
A)^.
= - ln J p ( d x ) e -
® p), H)
F
( P '
H
.
(**)
be arbitrary. Consider a homogeneous mean-field interaction H
+
n
T h e n C o r o l l a r y 5' y i e l d s
£ (C{X,A) )\ n
]im-F((p®p) ,H„+g l ) n
n
n n
n = mi{(jH)(tp)
+ (jg)&)
O n t h e o t h e r h a n d we c a n u s e (**) am-F(p J n n n
n
® p)°°,>p)\
+ S ((p M
•
t o w r i t e t h i s l i m i t as
+ 9n) = i n f { ( j / ) ( v ) + (jg)(u)
+ S(p,u)\
u £ K(C(X))}
.
N o t e t h a t i n t h e f i r s t e x p r e s s i o n t h e t e r m (jg){>p) d e p e n d s o n l y o n t h e r e s t r i c t i o n of ip t o C{X) t h a t (jg)(v)
= C(X)
® I * C C(X)
w i l l r e d u c e t o {jf)(p)i i n t h e P r o p o s i t i o n . If X
H e n c e i f w e c a n f i n d g 6 3> (C(X)) s u c h
® A.
= 0 f o r v = / i , a n d (jg)(v)
T
= 00 for u / p, t h e n t h e s e c o n d e x p r e s s i o n
a n d t h e first w i l l b e c o m e t h e v a r i a t i o n a l e x p r e s s i o n s t a t e d is m e t r i z a b l e , s u c h a s e q u e n c e g is easy t o c o n s t r u c t .
example we can take n
g =E n
s
y
m
" (* ® * ® - ) a
a
In
2
'
For
376 w h e r e ( x a ) a e l N i s a n o r m dense sequence i n C(X), T h u s v {g„)
= J2l=i(I( '( )
n
l
~ Kdx))Xa(x)),
dx
and x
= x « ~ f
a
Kdx)Xa(x).
w h i c h v a n i s h e s for v = fi, a n d
diverges for a l l o t h e r v.
In the r a n d o m context this result has a n i m m e d i a t e i n t e r p r e t a t i o n : suppose that t h e e x t e r n a l p a r a m e t e r s X, are i n d e p e n d e n t w i t h t h e s a m e d i s t r i b u t i o n fi. T h e n H (fn) is t h e e x p e c t a t i o n o f t h e free e n e r g y d e n s i t y o f t h e n - p a r t i c l e s y s t e m . S o t h e P r o p o s i t i o n gives a v a r i a t i o n a l f o r m u l a for t h e l i m i t (jf)(fi) o f t h i s q u a n t i t y (first d e r i v e d b y [8] i n a s p e c i a l case). F o r e a c h x € X c o n s i d e r t h e m e a s u r e Rn(x ) € K(C(X) ), w h i c h is t h e s y m m e t r i z a t i o n o f t h e p o i n t m e a s u r e S . T h e n w i t h p r o b a b i l i t y one R converges t o fi°°. S i n c e s y m / „ = / „ L e m m a 11 i m p l i e s t h a t l i m i n f „ f(x ) = limmi (R (x ))(f ) > ( j / ) ( / i ) - S o P r o p o s i t i o n 13 n o t o n l y gives t h e m e a n o f t h e free e n e r g y d e n s i t y , b u t also a l o w e r b o u n d , w h i c h h o l d s a l m o s t e v e r y w h e r e . T o g e t h e r t h i s m e a n s t h a t t h e free e n e r g y d e n s i t y as a sequence of r a n d o m v a r i a b l e s c o n v e r g e s i n m e a s u r e t o (jf)(fi)n
n
n
n
n
Xn
n
n
n
n
n
n
N a t u r a l l y we w o u l d like t o strengthen this t o a n almost everywhere result. Moreover, i n the multi-species examples a n d i n the B C S m o d e l the external par a m e t e r s are n o n - r a n d o m q u a n t i t i e s , a n d we w o u l d l i k e t o h a v e a d i r e c t c r i t e r i o n for a s e q u e n c e o f e x t e r n a l p a r a m e t e r s , w h i c h ensures c o n v e r g e n c e o f t h e free energy d e n s i t y . T h e case o f a m u l t i - s p e c i e s m e a n - f i e l d s y s t e m m a k e s c l e a r w h a t k i n d o f a s s u m p t i o n is n e e d e d . W e e x p e c t q u i t e different p h y s i c a l b e h a v i o u r f r o m a s y s t e m w i t h 9 0 % p a r t i c l e s o f t y p e x c o m p a r e d t o a s y s t e m w i t h 9 0 % p a r t i c l e s o f t y p e y. W h a t we h a v e t o fix therefore is t h e a s y m p t o t i c r a t i o o f t h e n u m b e r s o f p a r t i c l e s of e a c h species. I n t h e case o f g e n e r a l X , w h i c h c a n b e c o n s i d e r e d as a s y s t e m w i t h a c o n t i n u u m o f " s p e c i e s " t h i s a s s u m p t i o n is f o r m u l a t e d as f o l l o w s : let C\ ,i . . . €X b e t h e n p o s i t i o n s s p e c i f i e d for t h e n - p a r t i c l e s y s t e m . W e c o l l e c t these t o g e t h e r i n t o a n n - t u p l e f 6 X , so t h a t t h e r e l a t i v e H a m i l t o n i a n c a n b e w r i t t e n as nH (f\ ). T h e n we a s s u m e t h a t t h e r e is a m e a s u r e [i o n X, c a l l e d t h e limiting density o f the n
n
n
n
n
sequence £ „ s u c h t h a t for a l l / € C(X)
lim ! £ / ( £ „ , , ) = n—>oo n —' i=l
J
/p(dx)f(x) /
.
A n o t h e r w a y t o p h r a s e t h i s is t o s a y t h a t t h e " o c c u p a t i o n m e a s u r e s " ^ SF=i %> < c o n v e r g e t o fi i n t h e w * - t o p o l o g y . I n t h e m u l t i - s p e c i e s case t h e p r o b a b i l i t y m e a s u r e fi e n c o d e s t h e r e l a t i v e n u m b e r s of p a r t i c l e s o f e a c h species. I n t h e B C S - m o d e l t h e l i m i t i n g d e n s i t y is e a s i l y seen t o b e p r o p o r t i o n a l t o L e b e s g u e m e a s u r e . I n t h e r a n d o m case t h e c o n v e r g e n c e t o fi d e p e n d s o n t h e s a m p l e . F o r e x a m p l e , w h e n £ ; = ^ is a n e r g o d i c s t a t i o n a r y s t o c h a s t i c p r o c e s s w i t h " t i m e " i £ IN a n d s t a t e s p a c e X, t h e l i m i t i n g d e n s i t y e x i s t s n i
377 for a l m o s t a l l s a m p l e s , a n d is e q u a l to t h e ( c o n s t a n t ) d i s t r i b u t i o n \i of L e t us w r i t e e x p l i c i t l y t h e d e p e n d e n c e of f; = £,(w) o n t h e s a m p l e p o i n t a; i n t h e u n d e r l y i n g p r o b a b i l i t y s p a c e , a n d let C d e n o t e t h e set of w for w h i c h t h e l i m i t i n g d e n s i t y e x i s t s a n d e q u a l s (i. W e k n o w t h a t C h a s m e a s u r e one b y e r g o d i c i t y . T h e t h e o r e m b e l o w w i l l s h o w t h a t f o r a l l u e C t h e h m i t of \F(p , nH (£ )) e x i s t s , a n d is g i v e n b y a v a r i a t i o n a l f o r m u l a i n v o l v i n g o n l y p. T o o b t a i n t h i s s t a t e m e n t n o f u r t h e r r a r e e x c e p t i o n s m u s t b e t a k e n i n t o a c c o u n t , so we n o t o n l y h a v e a l m o s t s u r e convergence of ±F(p»,nH (t )) t o a n o n - r a n d o m l i m i t , we c a n also i d e n t i f y a s a m p l e set C of measure one where the result holds pointwise. n
n
n
n
n
A s i n t h e h o m o g e n e o u s case we w i l l also be i n t e r e s t e d i n t h e convergence o f t h e e q u i l i b r i u m s t a t e s fi(p ,n/J (f )). H o w e v e r , t h i s h m i t c a n n o t b e e x p e c t e d to e x i s t , since i t d e p e n d s o n t h e l a b e l l i n g of t h e a n d hence o n the p a r t i c u l a r embeddings A" *—> A u s e d t o d e f i n e t h e i n d u c t i v e l i m i t a l g e b r a A°°. A g a i n we c a n r e s t o r e t h e p e r m u t a t i o n s y m m e t r y b y g o i n g to t h e e x t e n d e d a l g e b r a C(X, A). Introduce the operators n
n
n
m
H „ : C(X,
A)
-
n
A
: F H-> ( s y m F ) ( £ „ )
n
.
n
Since H„ = s y m H w e h a v e H (£ ) = 'E H . Note that the s y m m e t r i z a t i o n here a n d i n t h e a b o v e d e f i n i t i o n is t o b e c o m p u t e d i n C(X, A), i.e. t h e l a b e l s X ; are p e r m u t e d a l o n g w i t h t h e o p e r a t o r s , so t h a t H F a n d E H are n o t n e c e s s a r i l y p e r m u t a t i o n s y m m e t r i c i n A". F o r a n y s t a t e tp € K(A ) we o b t a i n a s y m m e t r i c state tp o H € K (C(X, A) ). N o t e t h a t t h e r e s t r i c t i o n of
n
n
n
n
n
n
n
n
n
n
S
n
n
n
n
p r i n c i p l e s i n t h e i n h o m o g e n e o u s case. A s we h a v e n o t e d a b o v e t h e l o w e r b o u n d follows d i r e c t l y f r o m P r o p o s i t i o n 13. T h e u p p e r b o u n d d e p e n d s o n e n t r o p y e s t i m a t e s applied to a s u i t a b l y "coarse g r a i n e d " system. 1 4 T h e o r e m . ( [ 3 9 ] ) Let H € y(C(X, action, and suppose that the sequence hm
-F(p ,nHAtn))
A)) be a n inhomogeneous mean-Geld inter( 6 X has a limiting density p. Then n
n
= ud{HB)
n
+ S ((p.®p) ,*)}
where *
g
K {C{X,A)°°)
in (**)
a
tp 6 K(C(X,
A))
cluster point
of the s e q u e n c e fi{o ,
in (*)
satisfies
A s i n the homogeneous tp € K(C(X,
A))
the
the constraint n
to c o m p u t e t h e l i m i t .
satisfies nH ((„)) n
+ S(p.®p,
minimizes
n
(*)
$[C(X)°°
constraint tp\C(X)
o E
(**)
00
M
= im* {(jH)(,p)
=
Moreover,
p°°, any
and w*-
(**).
case i t is t h e v a r i a t i o n a l e x p r e s s i o n (*) w h i c h is u s e d
It is u s e f u l t o r e w r i t e (*)
i n the following way: any state
has a direct integral decomposition
Pt(dx)tp
z
378 where p
v
is a p r o b a b i l i t y m e a s u r e o n X,
a n d for e a c h x € X tp i s a s t a t e o n A . B y x
t h i s e x p r e s s i o n we m e a n t h a t for F €
C(X,A):
tp(F) = J
(M (dx)
.
x
F o r t h i s f o r m u l a t o m a k e sense we n e e d t o p o s t u l a t e t h a t x t-* tp (A)
is measurable
X
for a l l A 6 A . C l e a r l y , p the s u b a l g e b r a C(X)
is t h e m e a s u r e t h a t d e t e r m i n e s t h e r e s t r i c t i o n of tp t o
v
C C(X, A).
= C(X)1A
If A is s e p a r a b l e , w h i c h w e w i l l a s s u m e
f r o m n o w o n , tp is / i - a l m o s t e v e r y w h e r e d e t e r m i n e d b y tp. M o r e o v e r , we h a v e t h e x
formula S(p ®p,tp)
= S(p, p ) v
+ J p (dx)S(p,
tp )
v
x
,
w h i c h f o l l o w s b y t h e c o n d i t i o n a l e x p e c t a t i o n f o r m u l a [35] f o r r e l a t i v e e n t r o p i e s , a p p l i e d to the c o n d i t i o n a l e x p e c t a t i o n E o n tp n o w s i m p l y m e a n s p
p
: / ® A i-> / ® 1.4 p(A),
T h e constraint
- p, so w e a r e v a r y i n g o v e r a l l K(A)-valued
v
f u n c t i o n s x t-t tp , w i t h a n e n t r o p y t e r m g i v e n b y J
measurable
p(dx)S(p,tp ).
x
x
A s i n t h e h o m o g e n e o u s case ( T h e o r e m 8) o n e c a n s i m p l i f y t h e v a r i a t i o n a l p r o b l e m a b i t i f t h e e n e r g y t e r m does n o t d e p e n d o n t h e w h o l e s t a t e tp , b u t o n l y o n a x
s m a l l set of e x p e c t a t i o n v a l u e s . T h e r e is a l s o a v e r s i o n of t h e g a p e q u a t i o n ( T h e o r e m 7): i f t h e effective H a m i l t o n i a n for H € y{C(X, e x i s t s , i t is a n e l e m e n t H(g(tp) € C(X,A).
i n t h e s t a t e tp £ K(C(X,
A))
A))
W e c a n e v a l u a t e t h i s f u n c t i o n at x € X
t o o b t a i n t h e "effective H a m i l t o n i a n H^(tp, x) a t t h e p o i n t x".
I n first order around
a m i n i m i z e r tp t h e v a r i a t i o n a l e x p r e s s i o n b e c o m e s
(jH)(y>) + S(p®p,4,)K(jH)(
+ j
p(dx){ip (H (tp,x))+S(p,tp )} x
.
x
M i n i m i z i n g w i t h r e s p e c t t o e a c h xj> s e p a r a t e l y we o b t a i n t h e f o l l o w i n g v e r s i o n of x
the gap equation:
1 5 P r o p o s i t i o n . L e t tp = J p,^(dx)tp and suppose that H^f(tp) exists.
x
be a minimizer
of (jH)(tp) + S(p ® p,tp),
Then
tp = f2(p,Htf(tp,x))
.
x
For a n inhomogeneous mean-field p a i r i n t e r a c t i o n , say of the f o r m
H2(x, ) y
= \
J2( <*w ® £
1
+^y)
1
® °) + E s
a,0
y) ° s
® s"
379 w i t h S°
6 A, a n d s c a l a r f u n c t i o n s e
n i a n i n t h e s t a t e
fltfte,
=
a
6 C(X),
t h e effective H a m i l t o -
V a e C(Xf, a
is r e a d i l y c o m p u t e d t o be
x
^e (x) a
+ J n(dy) £
V ^ ( x , ») »,(5")j ¥
.
T h u s t h e g a p e q u a t i o n is a n e x p o n e n t i a l l y n o n - l i n e a r i n t e g r a l e q u a t i o n for x H->
T h i s is a m u c h m o r e d i f f i c u l t p r o b l e m t h a n t h e g a p e q u a t i o n i n t h e h o m o g e n e o u s case, a n d i t s s o l u t i o n d e p e n d s v e r y m u c h o n t h e p a r t i c u l a r p h y s i c a l p r o b l e m a t h a n d ( see [22] for a n e x a m p l e ) . T h e g e n e r a l t h e o r y p r e s e n t e d i n t h i s p a p e r h a s v e r y l i t t l e to c o n t r i b u t e t o t h i s p a r t of t h e p r o b l e m .
Acknowledgement
F i n a n c i a l s u p p o r t b y t h e D F G i n B o n n is g r a t e f u l l y a c k n o w l e d g e d . References [1] H . A r a k i : R e l a t i v e h a m i l t o n i a n for f a i t h f u l n o r m a l states of a v o n N e u m a n n algebra,
9,
Publ.Res.Inst.Math.Sci.
165-209
(1973)
[2] H . A r a k i : R e l a t i v e e n t r o p y o f states of v o n N e u m a n n a l g e b r a s I , Publ.Res.Inst.Math.Sci.
11,
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QP-PQ Editorial Board Managing Editor
Centro V . Volterra Universita' di Roma-Torvergata V i a Orazio Raimondo Roma 00173 Italia
Filtering, Control and
Institute for Problem in
Quantum Measurement
Mechanics Russian Academy of Science Moscow Russia
Markovian Semigroups
Moscow Institute for Electronic Construction MIEM 109028 Bolshoi Vusovski, Moscow Russia
Connections with Physics
Dipartimento D' Informatica Universita' di Milano Via Moretto Da Brescia 9 20133 Milano Italia
Probability and Geometry
Mathematics
Department
University of Nottingham University Park Nottingham NG7 2RD United Kingdom Dilations, Independent
Mathematisches
Increments
Universitat Tubingen
Processes
Institut
Auf der Morgenstelle 10 D-7400 Tubingen Germany Non Causal Calculus, White Noise
Department of Mathematics King's College London W C 2 R 2 L S England
H . Maassen :
Free Probability and Stochastic Calculus
K . R. Parthasarathy :
Mathematics Institute University o f Nijmegen Toernooiveld 6525 E D Nijmegen The Netherlands
Connections with
Indian Statistical Institute
C l a s s i c a l Probability
7, S. J . S. Sansanwal M r g N e w D e l h i 110016 India
D . Petz :
Operator Theoretical
Hungarian A c a d , of Science
Methods
Mathematical Institute H - 1 3 6 4 Budapest, P F . 127 Hungary
K . B. Sinha
Stochastic Differential
Indian Statistical Institute
Equations
7, S. J . S. Sansanwal M r g N e w D e l h i 110016 India
W . von Waldenfels :
Algebraic Probability
Institut fur Angewandte
Theory
Mathematik U n i v . Heidelberg S F B 123 Im Neuenbeimer feld 294 6900 Heidelberg 1 Germany
QP-PQ Editorial Board Managing Editor
Centro V . Volterra Universita' di Roma-Torvergata Via Orazio Raimondo Roma 00173 Italia
Filtering, Control and
Institute for Problem in
Quantum Measurement
Mechanics Russian Academy of Science Moscow Russia
Markovian Semigroups
Moscow Institute for Electronic Construction MIEM 109028 Bolshoi Vusovski, Moscow Russia
Connections with Physics
Dipartimento D' Informatica Universita' di Milano Via Moretto Da Brescia 9 20133 Milano Italia
Probability and Geometry
Mathematics
Department
University of Nottingham University Park Nottingham NG7 2RD United Kingdom Dilations, Independent
Mathematisches
Increments
Universitat Tubingen
Processes
Institut
Auf der Morgenstelle 10 D-7400 Tubingen Germany Non Causal Calculus, White Noise
Department of Mathematics King's College London W C 2 R 2 L S England
H . Maassen :
Free Probability and Stochastic C a l c u l u s
Mathematics Institute University of Nijmegen Toernooiveld 6525 E D N i j m e g e n T h e Netherlands
K . R. Parthasarathy :
Connections w i t h
Indian Statistical Institute
C l a s s i c a l Probability
7, S. J . S. Sansanwal M r g N e w D e l h i 110016 India
D . Petz :
Operator Theoretical
H u n g a r i a n A c a d , o f Science
Methods
M a t h e m a t i c a l Institute H - 1 3 6 4 Budapest, P F . 127 Hungary
K . B. Sinha
Stochastic Differential
Indian Statistical Institute
Equations
7, S. J . S. Sansanwal M r g N e w D e l h i 110016 India
W . von Waldenfels :
A l g e b r a i c Probability
Institut fur Angewandte
Theory
Mathematik U n i v . Heidelberg S F B 123 I m Neuenbeimer feld 294 6900 H e i d e l b e r g 1 Germany
