F o r the c h a r a c t e r i s t i c of functions in 91, we r e m a r k that f o r x _< 0 we h a v e qo(x; Xa) = ~p(0; X2) cos Xx +
~0'(0; X ~) s i n X x / X , and a l s o t h a t f o r a r e d u c e d s t r i n g S we have by [6] ~0 (0; k S) = (Q(X) + Q I - X ) ) / 2 , q0'(0; x 2 ) / x =
(Q(X) - Q ( - X ) ) / 2 i . T h e r e f o r e , f o r an a r b i t r a r y F ~ 91 (F = srf, ] ~ ~X), Eq. (3) t a k e s the f o r m (1) if we put h(x) =
(f~l-x) - if2(-x))/2 f o r x > 0 and h(x) = (fl(x) +f2{x))/2 f o r x ___0, s o that h(x) E L2(-oo, ¢¢). C o n v e r s e l y , f o r an
a r b i t r a r y h(x) E L2(-oo, co), it is e a s i l y v e r i f i e d t h a t the f u n c t i o n (1) is the F o u r i e r t r a n s f o r m of s o m e f ~ ~
Since f o r a r e d u c e d s t r i n g S, the v e c t o r functions Cj(x) = (~0(x; z~), ~0'(x; z~)/zj) (sj e Z) f o r m a s y s t e m
which is c o m p l e t e in # s (cf, [6]), ~ = ~ # s c o i n c i d e s with the c l o s u r e of the l i n e a r s p a n of the functions Qj(X) =
~ ¢ j (x) . A s i m p l e c a l c u l a t i o n shows that up to a c o n s t a n t f a c t o r , Qj(X) c o i n c i d e s with QlX)/iX - zj).
We have t h e r e b y obtained a new j u s t i f i c a t i o n and, at the s a m e t i m e , g e n e r a l i z a t i o n (for the c a s e of an even
weight) of the r e s u l t of G u r a r i i .
A k h i e z e r and G u r a r i i have a l s o obtained a g e n e r a l i z a t i o n of the P a l e y - W i e n e r t h e o r e m f o r the t r a n s f o r m s which they c o n s i d e r e d - f o r an e n t i r e f u n c t i o n ~ ~ ffa of d e g r e e g ( P ) + c~, the p r o j e c t i o n F of G onto 9l
has a r e p r e s e n t a t i o n (1) in which h(x) = 0 f o r [ x I > o~.
This r e s u l t can be e x t e n d e d in the c a s e when the
s t r i n g S is r e g u l a r , Joe., 5~ < co . In this c a s e , Q(X) t u r n s out to be a function of finite d e g r e e (~ = a(Q) < ~)
and thus s a t i s f i e s the conditions of G u r a r i i . A r e p r e s e n t a t i o n can now be obtained f o r a ~ ea a l s o when
0 < ~ (G) < cr (P). U n d e r this condition, C ~ ~ and G can be r e c o v e r e d u s i n g a s u i t a b l e g e n e r a l i z a t i o n of the
P a l e y - W i e n e r t h e o r e m in the t h e o r y of s p e c t r a l functions of a r e g u l a r s t r i n g [9] (cf. a l s o [10]).
LITERATURE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
N . I . A k h i e z e r , Dokl. Akad. Nauk SSSR, 96, No. 5, 889-892 (1954).
V . P . G u r a r i i , Mat. Sb., 5__8,439-452 (1962).
L. de B r a n g e s , H i l b e r t S p a c e s of E n t i r e F u n c t i o n s , P r e n t i c e - H a l l , New Y o r k (1968).
I . S . Kats and M. G. K r e i n , Izv. V y s s h . U c h e b m Zaved., Mat., No. 2, 1 3 6 - 1 5 3 (1958).
I. So Kats, Dokl° Akad. Nauk SSSR, 157, No. 1, 34-37 (1964).
M. G° K r e i n and A° A. N u d e l ' m a n , Dokl. Akad. Nauk SSSR, 247, No. 5, 1046-1049 (1979).
I . S . Kats and M. G. K r e i n , Appendix II to: F . Atkinson, D i s c r e t e and Continuous B o u n d a r y P r o b l e m s
[ R u s s i a n t r a n s l a t i o n ] , Mir, Moscow (1968).
M. G° K.rein, Dold. A k a d . Nauk SSSR, 8__7,No. 6, 881-884 (1952)°
M . G . K r e i n , Dokl. Akad. Nauk SSSR, 94_, No. 1, 13-16 (1954)0
H. D y m and H° P . McKean, G a u s s i a n P r o c e s s e s , F u n c t i o n T h e o r y , and the I n v e r s e S p e c t r a l P r o b l e m ,
A c a d e m i c P r e s s , New Y o r k (1976).
RATIONAL
FOR
CITED
SOLUTIONS
YANGI.
M.
MILLS
OF
THE
DUALITY
EQUATIONS
FIELDS
Krichever
UDC 517.43
In this note we p r o p o s e a c o n s t r u c t i o n f o r a l a r g e c l a s s of r a t i o n a l solutions of the duality equations [1].
T h e s e e q u a t i o n s a r e equivalent t o the condition t h a t the o p e r a t o r s (cf. [2])
c o m m u t e , w h e r e X is the s p e c t r a l p a r a m e t e r , Bi = Bi(zl, z2, ~1, z2) the c o n n e c t i o n c o e f f i c i e n t s in the c o o r d i n a t e s z 1 = i/2(x 2 + ix1); z 2 = 1/2(x4 + ixs); ~z; ~1 r e s p e c t i v e l y ; (x 1, .... x~) ~ R 4. F o r S U ( n ) - c o n n e c t i o n s , B 3 =
- B +, B 4 = --B +. We r e s t r i c t o u r s e l v e s below to the c a s e n = 2.
I. C o n s i d e r the l i n e a r s p a c e L 2N+2 of p a i r s of functions (fi(X), f~(X)) r a t i o n a l in having poles at Xl, .... XNA s s u m e given functions ai(z, 5 , X) = ai0 + ail(X - A i) and bi(z, ~, A) =bi0 + bii(A - Xi) s u c h that
M o s c o w State U n i v e r s i t y . T r a n s l a t e d f r o m F u n k t s i o n a l ' n y i Analiz i E g o P r i l o z h e n i y a , Vol. 13, No. 4,
pp. 81-82, O c t o b e r - D e c e m b e r , 1979. O r i g i n a l a r t i c l e s u b m i t t e d D e c e m b e r 28, 1978.
0016-2663/79/1304-0303507.50
© 1980 P l e n u m Publishing C o r p o r a t i o n
303
(~a I --~ ~2)ai :
(~,01 ~- O~)b i = ( ) ~
( ~ 2 - - O1)ai = 0 (~, - - ~i) "~,
- - 01)hi = 0 (~, - - ~,i) ~,
(~)
z = (zl, z~).
The s y s t e m of 2N l i n e a r equations depending on z, ~ as p a r a m e t e r s is e q u i v a l e n t to the conditions
(2)
and defines a t w o - d i m e n s i o n a l subbundle Z in R ~ × L ~l+~.
We w r i t e the v e c t o r ffl, f~) in the f o r m
N
(f. b.) = L (b~°' a~o)~
÷ (9/v÷i, ~N+~.),
SO t h a t f o r e a c h i, one of the equations in (2) h o l d s . T h e r e m a i n i n g equations f o r the ~0i take the f o r m
2
a l o b j o - bloa.io
)~i - - ~'.i
~J 4Z di~i = biotiN+2 - - aioq'N+l'
di = bi°ail - - ai°bii'
i = t, . . . , N .
(3)
We denote by ~ ( z , ~, ?~) the (2 x 2) m a t r i x with r o w s f o r m e d by t h e b a s i s s o l u t i o n s of E q s . (2). ~I, is d e fined uniquely up t o m u l t i p l i c a t i o n by a n o n s i n g u l a r m a t r i x g(z, z-) not depending on X.
T H E O R E M 1. Let ~1 = ~" 6, ~, ~), R~ = ~" ~, e, 0) and B~ = --0~R~.R~~, B, = --0~R,.R,: ~,
B~ = _ ~ . / ~ t
. Then ~ s a t i s f i e s the equations L~ ~I"(z, ~, M = 0, L~ ~I"(z, ~, X) = 0
C O R O L L A R Y . The o p e r a t o r s L 1 and L2 c o m m u t e .
B~ = - - ~ . R ~ t,
T h e i r c o e f f i c i e n t s Bi define a self--dual c o n n e c t i o n .
R e m a r k 1. C h a n g i n g the b a s i s s e c t i o n s , i.e., the r e p l a c e m e n t ~ ( z , ~, X) = g ( z , z-),l~(z, ~, X), is e q u i v a l e n t
to a g a u g e t r a n s f o r m a t i o n of the f~elds.
II. A v e r y s i m p l e s e t of functions a i, bi s a t i s f y i n g E q s . (1) is g i v e n by
(~+,, b+~)=
- ,..~+ t---~ i -~+a 'I
~ + ~ / A +
H e r e ~i, fib ci a r e c o n s t a n t s , A i E SL(2, C ) a c o n s t a n t m a t r i x .
a r e equal.)
+
~o' o .
(These e q u a l i t i e s m e a n that the row v e c t o r s
In this c a s e , the m a t r i x e l e m e n t s of ~ ( z , ~, X) a r e r a t i o n a l functions of d e g r e e 2N in the v a r i a b l e s z 1,
D
z 2, z 1, z 2. The fields B i a r e a l s o r a t i o n a l functions of z, ~.
E x a m p l e . Let N = 1, A 1 = f , X~ = 0. We then obtain a o n e - i n s t a n t o n s o l u t i o n of the duality equations,
f o r which ~, fl define the c e n t e r , and the absolute value of the r e a l negative c o n s t a n t c in the d i m e n s i o n of
the instanton.
R e m a r k 2. It c a n be shown that f o r points in g e n e r a l position, the above c o n s t r u c t i o n c o i n c i d e s with the
" m u l t i p l i c a t i o n " p r o c e d u r e (cf. [2]) and t h e r e f o r e c o n t a i n s at a m i n i m u m the (hN + 6 ) - p a r a m e t e r f a m i l y of N i n s t a n t o n solutions obtained in [3].
R e m a r k 3. An a l g e b r o g e o m e t r i c c o n s t r u c t i o n of all i n s t a n t o n s o l u t i o n s was obtained in [4]. H o w e v e r ,
s o f a r it has not g i v e n an explicit p a r a m e t r i z a t i o n of the m a n i f o l d of " m o d u l f f of N - i n s t a n t o n fields. The p o s s i b i l i t y of obtaining all N - i n s t a n t o n fields, i.e., s m o o t h , a s y m p t o t i c a l l y longitudinal fields in the f r a m e w o r k of
the above c o n s t r u c t i o n and its g e n e r a l i z a t i o n s (one of which will be g i v e n below) r e m a i n s u n a n s w e r e d . In o r d e r
to obtain s m o o t h fields on R 4, it is n e c e s s a r y and s u f f i c i e n t that the m a t r i x in the l e f t - h a n d s i d e of Eq. (3) be
n o n s i n g u l a r f o r all z, ~. In this c a s e , the d e t e r m i n a n t of ~ ( z , ~, X) (which d o e s not depend on )t, as follows
f r o m the definition of ~) is n o n z e r o . The l a s t condition d e t e r m i n e s a s u b m a n i f o l d in the s p a c e of p a r a m e t e r s
)~i, ~i, fli, ci, Ai, i = 1, ..., N, the d i m e n s i o n of which is unknown.
R e m a r k 4. In o r d e r to obtain S U ( 2 ) - c o n n e c t i o n s , the d i m e n s i o n a l i t y of the p r o b l e m c a n be doubled by
adding the points ?~"i+N =-l/-Xi, f o r which we put ai+N(k) = b i ( - 1 / ~ ) , bi+N0~) = - ~ i ( - 1 / ~ ) .
In this c a s e h i + N
and bi+ N a u t o m a t i c a l l y s a t i s f y E q s . (1).
HI. The c o n s t r u c t i o n p r o p o s e d a d m i t s a n a t u r a l g e n e r a l i z a t i o n which p e r m i t s c o n s t r u c t i o n of " e i g e n f u n c t i o n s " of the o p e r a t o r s L 1 and L 2 which a r e not r a t i o n a l in ?~ as b e f o r e , but a r e r a t h e r defined on h y p e r elliptic c u r v e s .
304
The s i m p l e s t g e n e r a l i z a t i o n of a one-instanton solution (ef. Example 1) c a n b e obtained as follows. Con~N+2
s i d e r the hyperelliptic c u r v e ~, defined by ~ = 1] (~ - ~J). It is a twofold c o v e r i n g of the ~. plane. The d i j=l
mension of the space L of rational functions on ~, having poles of multiplicity N + 1 at two points 1~ on diff e r e n t sheets over a fixed point k 0 is equal to 2N + 2 - N + 1 = N + 3 by the R i e m a n n - R o c h t h e o r e m .
Let
N
a (z, ~, ~.) = Z as (z, ~) (~ - - ~o)~, b (z, ~, ~) = 2 bs (z, ~) (~ - - ~o)s
s~O
s~J
satisfy Eqs. (1)
in which
the
right-hand side O(X - ~ 0)2
is r e p l a c e d by O(k - X0)N+I. The solutions of the equations
a (z, Z, ~,)~ (z, ~, X+) - - b (z, ~, ~,), (z, ~, ~,-) = 0 (t) I;~=M,
where X~ are the i n v e r s e images on different sheets of z, , (z, ~, p) ~ L, P ~ ~, f o r m a two-dimensional subbundle 5~ in R 4 × L. We denote by ~b~(z, ~, P) and ~ ( z , ~, P) basis sections of 5~ .
THEOREM 2. Let the Bi(z, z-) be given by the f o r m u l a s in T h e o r e m 1, where
!
/
\,~ (~, ~, P;) ,~ (:, :, p;) / '
Then the fields Bi(z, z--') satisfy the duality equations.
Here, P ~ , P0~ are the i n v e r s e i m a g e s on ~
of the points X = ~ and X = 0, r e s p e c t i v e l y .
R e m a r k 5. If the functions as(Z, z-) and bs(z, z-) are chosen to be rational, the fields B i will also be r a tional in the Variables z, ~.
R e m a r k 6. In [5], the inverse p r o b l e m method was used to c o n s t r u c t finite-zone solutions of the duality
equations. In this c a s e , the eigenfunctions of the o p e r a t o r s L l and L~ are defined on a hyperelliptie curve, as
in this section, but in c o n t r a s t to our construction, they have e s s e n t i a l singularities on the c u r v e .
LITERATURE
1.
2.
3.
4.
5.
CITED
A. Belavin, A. Polyakov, A. Shwartz, and Y. Tyupkin, Phys. Lett., 59B, 85-89 (1975).
A . A . Belyavin and V. E. z a k h a r o v , P i s ' m a Zh. Eksp. Teor. F i z . , 25, No. 1 2 , 6 0 3 - 6 0 7 (1977).
A. I~. Arinshtein, Yad. Fiz., 29, No. 1, 249-256 (1979).
M. Atiyah, N. Hitchin, V. Drinfeld, and Yu. Manin, P h y s . Lett., 65A, 185-187 (1978).
I . V . Cherednik, Dokl. Akad. Nauk SSSR, 246, No. 3, 575-577 (1979).
CLASSIFICATION
M.
OF VON
NEUMANN
J-ALGEBRAS
UDC 513.88
M. M e l ' t s e r
In v a r i o u s works on quantum field t h e o r y (cf. [1, 2]), the problem is d i s c u s s e d of the utility of introducing
an indefinite m e t r i c ( J - m e t r i c ) into an initial Hilbert s p a c e ~ and considering, in addition to the usual duality
operation, duality with r e s p e c t to the J - m e t r i c . The concept of a yon Neumann J - a l g e b r a is t h e r e f o r e natural,
i.e., a yon Neumann a l g e b r a of o p e r a t o r s on the space -~ (cf. [3, 4]) with an additional involution induced by
the J - m e t r i c .
The purpose of this note is to d e s c r i b e yon Neumann J - a l g e b r a s .
Let ~ be a Hilbert space with a J - m e t r i c (cf. [5, 6]), .~ = .% ~ 9-, dim.¢~+ dim S~_, ~ a Hilbert space
i s o m o r p h i c to 9+ and ~_, and J ± , i s o m e t r i e s of 9~ onto s? . The J - m e t r i c in ~ is defined with the aid of
0 ' where I is the identity o p e r a t o r in 9 . Namely, [~, y] =
an o p e r a t o r ] given in m a t r i x f o r m by ~ = (0I -!)
=
(J~, y). The o p e r a t o r ~0 defined by [A~, ~1 = [~, ~I%], ~, ~ ~ ~ (cf. [5, 6]) is called the J - d u a l of A. An o p e r a t o r
" U z r e m d o r p r o e k t w Planning and P r o s p e c t i n g Institute of the Uzbek Republic. T r a n s l a t e d f r o m Funkts i o n a l ' n y i Analiz i Ego P r i l o z h e n i y a , Vol. 13, No. 4, pp. 83-84, O c t o b e r - D e c e m b e r , 1979. Original article submitted March 20, 1978.
0 0 1 6 - 2 6 6 3 / 7 9 / 1 3 0 4 - 0 3 0 5 5 0 7 . 5 0 © 1980 Plenum Publishing Corporation
305