Local and unitary approximation to commutators

Volume 35B, number 4

PHYSICS LETTERS

7 June 1971

L O C A L AND U N I T A R Y A P P R O X I M A T I O N TO C O M M U T A T O R S H. D. DAHMEN, D. GROMES, K. D. ROTHE and B. STECH

Institut fi~r theoretische Physik der Universitiit Heidelberg and Kernforschungszentrum Karlsruhe, Germany Received 17 April 1971

The saturation of commutator matrix elements is extended to include two particle cuts. For the case of vacuum one particle matrix elements,a complete solution fulfilling locality and elastic unitarity is given.

A consistent saturation of commutator matrix elements is known in pole approximation [i]. The r e sulting expressions are local. The detailed local properties - the equal time limits, for instance - can be studied in their dependence on the particle pole p a r a m e t e r s of the contributing form factors and scattering matrix elements. At this stage unitarity is still badly violated as in the tree approximation obtained from effective Langrangians t. In the present paper we proceed with the local saturation scheme taking continuum contributions into account. We r e s t r i c t ourselves to the discussion of two particle cut contributions and fulfil elastic unitarity for the simplest set of matrix elements. For simplicity the commutator of a scalar spin zero field A(x} with itself is considered. This field is supposed to have a Hang expansion [3] in t e r m s of asymptotic fields Ain(X) corresponding to a single particle of mass m > 0. The following goals have been achieved: i) the approximation to the matrix element of the commutator is local. ii) the corresponding retarded commutator obeys elastic (off-shell) unitarity. These statements hold, of course, only for the matrix element considered for which we can give an explicit solution. The method used in the following is not based on perturbation theory but is more r e miniscent of a local form of the Tamm-Dancoff [4] approximation. The Haag expansion for the scalar hermitian field

A(x) -

1 f d4k exp (- ikx) A(k) (2~)3/2

h a s the f o r m 2~(k) =Ain(k) + n ~=2 ~ f "

(i~-I=1=d 4 k ) f 4 ( k

n -/~--1 k i ) h n ( k l " ' ' k n ) : A i n ( k l ) ' " A i n ( k n ) :

,

(1)

where Ain(k) = ~(m 2 _ k 2) [0 (ko)ain(k) + 0(- ko)a~n(- k)] . The functions h n a r e L o r e n t z i n v a r i a n t , t o t a l l y s y m m e t r i c and defined for o n - s h e l l v a l u e s k 2 = m 2. T h e y have the p r o p e r t y hn(k 1. . .k n) = h ~ ( - k 1. . . - k n ) . In o r d e r to be able to e x p r e s s t h e s e functions in t e r m s of r e t a r d e d c o m m u t a t o r s we a s s u m e that they a r e b o u n d a r y v a l u e s of a n a l y t i c functions in each t i m e component k ° with cuts on the r e a l axis. M o r e o v e r , we take f 2 ( ( k l + k2 + i~ )2)

f 3 ( ( k l + k 2+iE) 2, (k l + k 3+i£) 2, (k l + k 2 + k 3+iC) 2) h3(kl,k2,k3) = lim ¢~0

m 2 - (k 1 + k 2+ k 3+i£) 2

(2)

There are recent attempts towards a unitarisation of the method of chiral Lagrangians and the hard pion technique [2]. 335

Volume

35B, number

4

PHYSICS

LETTERS

7 June

1971

• with fn(zi) = [fn( Z *i )] * " T h e 1" £ - p r e s c m p" t i o n r e f e r s to the t i m e c o m p o n e n t s . T h e f o r m (2) a s s u r e s u n d e r r a t h e r w6ak c o n d i t i o n s for the f u n c t i o n s f n the weak c o n v e r g e n c e of A(x) to the field Ain(X). At the s a m e t i m e it g u a r a n t e e s the weak c o n v e r g e n c e of A(x) to an o p e r a t o r Aout(X) d e f i n e d by

2niC(ko)6(rn 2 - k 2) y(k) w h e r e the s o u r c e f u n c t i o n j(x) = (Z~+m2)A(x) is o b t a i n e d f r o m eq. (1).

Aout(k) = Ain(k) +

(3)

The o p e r a t o r s Aout(k) d e f i n e d in t e r m s of " i n " - o p e r a t o r s by eq.(3) a r e in g e n e r a l not local and do not; obey the f r e e field c o m m u t a t i o n r e l a t i o n s . In p a r t i c u l a r the TCP r e l a t i o n b e t w e e n "in" and " o u t " - f i e l d s d o e s not hold and h a s to be i m p o s e d l a t e r on for the m a t r i x e l e m e n t s of i n t e r e s t . H o w e v e r , the r e l a t i o n s aou t [0} = 0 and a~u t [0} = a h ]0} a r e s a t i s f i e d . I n s e r t i n g the s e r i e s (1) into the c o m m u t a t o r

K(q) = 1 fd4x e x p ( i q x ) [ A ( x ) , A(0)]

(4)

we a r r i v e at the e x p a n s i o n n

K(q) : c°(q) + n=l ~ l

(i~___l=d4ki)cn(q, kl . . . . . hn):Ain(kl)...Ain(kn):

(5)

w h e r e the cn(q, k 1, • •., kn) a r e given in t e r m s of the f u n c t i o n s hn. The c o r r e s p o n d i n g e x p a n s i o n of the F o u r i e r t r a n s f o r m s of the r e t a r d e d c o m m u t a t o r and the c o m m u t a t o r of the " o u t ' - f i e l d with i t s e l f is of t h e s a m e f o r m , with the c o e f f i c i e n t s c r e t ( q , k l , . . . , kn) and c°Ut(q, k l , . . . , kn) , r e s p e c t i v e l y . T h e s e c o e f f i c i e n t s a r e c l o s e l y r e l a t e d to each o t h e r and - v i a r e d u c t i o n f o r m u l a e - a l s o to s i n g l e fn'S, e.g. f 2 ( q 2) = •(m 2 _ q2)(rn2 _ (p _ q)2)c~et(q,

p) (p_q)2=rn2

(6)

.

It f o l l o w s d i r e c t l y f r o m o u r d e f i n i t i o n s that csUt(q) = E(qo)6(m2 _ q2) ,

C~lut(q , k) = 0 .

F u r t h e r m o r e , ~ co(q), the v a c u u m e x p e c t a t i o n v a l u e of K(q), is a u t o m a t i c a l l y the F o u r i e r t r a n s f o r m of a local function. We a r e now in a p o s i t i o n to i n v e s t i g a t e the c o n s t r a i n t s which a r e i m p o s e d by l o c a l i t y on t h e c o e f f i c i e n t s cn(q, k l , . . . , kn). L e t u s c o n c e n t r a t e on t h e s i m p l e s t c o e f f i c i e n t cl(q,p) in the s e r i e s of eq. (5):

cl(q' p) = ~1 fd4x exp (iqx) <0 [[A(x), A(0)] IP) = ~1

fd4x exp(iqx)

(0

I[A(x)[A(O), a~n(p)]] [ 0 ) - (q;_P-pq)

(7) Expressing

cl(q , p) in t e r m s of the f n ' s one has:

¢l(q ' p) = £(q0)6(m2 _ q2) f2((_P_- q+ie)2_~) rn2 - (p - q+ ie)2 1 f 2 ( ( q + ie)2) + E(q0) m2 _ (q+ i£)2 m 2 . ( P - q+ iE)2

fd4kl d4k 2 ½(o(kO)o(k~)

(8)

+ 0(- k~)O(- k~))

-

+

"

"

We d e n o t e the i n t e g r a l in t h i s e q u a t i o n by l t ~ ( ( _ q + i c ) 2, ( P - q + i E ) 2) . 77

T h e t e r m c o n t a i n i n g t~ a r i s e s f r o m 2 p a r t i c l e i n t e r m e d i a t e s t a t e s in eq. (7) a n d has a t h r e s h o l d at q2 = 4rn 2. T h e n e g l e c t i o n of t h i s p a r t and the c o n t r i b u t i o n s with s t i l l h i g h e r t h r e s h o l d s would b r i n g u s b a c k to 336

Volume 35B, number 4

PHYSICS

LETTERS

7 J u n e 1971

t h e p o l e a p p r o x i m a t i o n t r e a t e d in r e f . [1]. A s we know f r o m this investigation the locality requirement f o r cl (q,p ) then gives the unique solution f2 = const. By the i n c l u s i o n of t h e 2 - p a r t i c l e c o n t i n u u m , the q - d e p e n d e n c e of f 2 and the d e p e n d e n c e of the s w a v e p a r t of f3 on q and P e n t e r the p r o b l e m fully. It is not at all e v i d e n t that a l o c a l f o r m f o r Cl(P , q) c a n s t i l l be a c h i e v e d , in p a r t i c u l a r in v i e w of t h e f a c t that TCP r e l a t i o n s h a v e to be i m p o s e d on the f u n c t i o n s f2 m~d f3 and that t h e H a a g e x p a n s i o n s h a l l now be t r u n c a t e d by s e t t i n g f4 = f5 = • • • = 0. T h e TCP r e l a t i o n s a r e

(Olj(o) ipl,P2,in~ = <01j(0) l P l , P 2 , o u t ~ * ,

(oIj(o) lpl,P2,P3, in) = (Otj(o) l P l , P 2 , P 3 , o u t } * .

(9)

T h e y h a v e to hold f o r p o s i t i v e a s w e l l as n e g a t i v e f o u r m o m e n t a . If t h e s e c o n d i t i o n s (with the o u t - s t a t e s g e n e r a t e d by the o p e r a t o r d e f i n e d in eq. (3)) a r e f u l f i l l e d the vanishing of the coefficient c~Ut (q, k 1 , k2) and the unitarity of the 2-particle S - m a t r i x ~ t in the e l a s t i c r e g i o n is g u a r a n t e e d . F r o m now on we c o n s i s t e n t l y put f4 = f5 = • "" = 0 and r e s t r i c t o u r s e l v e s in eq. (9) to 2 - p a r t i c l e i n t e r m e d i a t e s t a t e s i n c l u d i n g of c o u r s e , the c o r r e s p o n d i n g p a r t i a l l y d i s c o n n e c t e d m a n y - p a r t i c l e s t a t e s . U n d e r t h e s e c o n d i t i o n s we o b t a i n f r o m e q s . (9) and (3) f o r f 2 and t O the e x p e c t e d r e l a t i o n s I m f 2 ( ( q + iE) 2) = •(qo)f2((q+ i • ) 2 ) t 0 ( ( q + i•) 2, m 2) (10) I m to((q+ i•) 2 , (p - q + i¢) 2) = ~(qo)to((q + i • ) 2 , ( p - q + i•)2)t~((q + iE) 2 ,m 2) . T h e q u a n t i t y t O can now be i d e n t i f i e d with the s - w a v e s c a t t e r i n g a m p l i t u d e - off s h e l l in the m a s s of the f o u r t h p a r t i c l e . A s a c o n s e q u e n c e of eq. (10) it has to h a v e the f o r m t0(q2,(,b - q)2) = exp {i5(q2)} R(q 2,(p - q)2) ,

q2 >/ 4m2 ,

(11)

with R(q2,(P _q)2) r e a l and e q u a l to sin 5(q 2) at (P _q)2 = m2. T h e s e c o n d i t i o n s a r e in a g r e e m e n t with t h e g e n e r a l c o n s e q u e n c e of TCP which r e q u i r e s cl(q,p) to be r e a l ~1'~. At t h i s s t a g e one can o b t a i n a l o c a l e x p r e s s i o n f o r cl(q,p) and e~et(q,P) if one m a k e s u s e of a d i s p e r s i o n r e l a t i o n f o r f 2 and t a k e s the o f f - s h e l l s - w a v e s c a t t e r i n g a m p l i t u d e t0(q2 , 09 _q)2) to be i n d e p e n d e n t of i t s s e c o n d a r g u m e n t . In r e a l i t y t h i s l a t t e r c o n d i t i o n will c e r t a i n l y not hold, it i s , h o w e v e r , to b e l o o k e d at a s an a p p r o x i m a t i o ~ l i k e t h e r e q u i r e m e n t f 2 = c o n s t in the c a s e of the p u r e p o l e a p p r o x i m a t i o n . T h e l o c a l s o l u t i o n f o r c~ e (q,P) o b t a i n e d t h i s way w i l l be p r e s e n t e d in eq. (l~r)6t I n s t e a d of d e s c r i b i n g in d e t a i l t h e a b o v e p r o c e d u r e we p r e f e r to u s e a c a u s a l r e p r e s e n t a t i o n f o r c 1 (q,p) which, e v e n though m o r e q u e s t i o n a b l e , s e e m s m o r e i n t e r e s t i n g : c ~ e t ( q , p ) = n1 f d q ' 2 d V'2

~(q,2, Q ,2) (q ,2 _ (q + i •)2)(Q ,2 _ (p _ q _ i •)2) .

(12)

The double s p e c t r a l f u n c t i o n ~ then f u l f i l l s (r(q'2, V,2) = ~ . ( q , 2 , Q,2) = ~(q,2, q,2) .

(13)

A l t h o u g h we know r~o r i g o r o u s p r o o f f o r t h i s r e p r e s e n t a t i o n , it s e e m s v e r y h a r d to find any o t h e r : the p o l e r e s i d u e of c[e~(q,P) at (P _q)2 = m2 m u s t g i v e f2((q + i • ) 2) (eq. (6)) with the c o r r e c t a n a l y t i c p r o p e r t i e s , Cl (q,P) s h o u l d e x h i b i t the • (q0) and • (P0 - q0) s t r u c t u r e and p o s s e s s the r e a l i t y p r o p e r t i e s c o n t a i n e d in eq. (8), etc. In any c a s e , f o r the p r o b l e m at hand the r e p r e s e n t a t i o n (12) is i n t e r n a l l y c o n s i s t e n t with o u r r e q u i r e m e n t s . C o n s i d e r i n g the c o n s e c u t i v e a b s o r p t i v e p a r t s in t h e v a r i a b l e s q and ( P - q) the f o l l o w i n g e q u a t i o n d e termining ~is plausible:

e(q~) e(qo)~(q ,2, V ,2) _ (2~) 7 /12 " fd4z d4y e x p ( i q ' z ) e x p ( i Q ' y ) (01 [A(z)[A(y),j(O)]] [O>[(q,+Q,)2=m2

(14)

A c a r e f u l d i s c u s s i o n of c o n n e c t e d and d i s c o n n e c t e d c o n t r i b u t i o n s on the r i g h t - h a n d s i d e of t h i s e q u a t i o n s h o w s c o n s i s t e n c y with t h e l e f t - h a n d s i d e and t h u s with the r e p r e s e n t a t i o n (12). T h i s r e p r e s e n t a t i o n J" We leave possible polynomials in the variable q out of the discussion. ~'J" The S-matrix is defined as with "out" operators from the expansion (3). J-'~ The inclusion of three particle contributions would lead to complex values o f f 2 tO in certain regions, whereas cl(q,p ) still remains real. 337

Volume 35B, number 4

PHYSICS LETTERS

7 June 1971

+

...

Fig. 1. Graphical representation of the contributions to the spectral function ff(q,2,Q,2). t r e a t s continuum s t a t e s as the l i m i t of a l a r g e n u m b e r of d i s c r e t e states. Now we have in p a r t i c u l a r ,

O(q~)O(_Q~)~(q,2,Q,2)

_

1 f d 4 z d 4 y exp(iq'z) (2~)7/2 °

exp(iQ'y)
(15)

The g r a p h i c a l i n t e r p r e t a t i o n of eq. (15) leads to the" ( n o n - F e y n m a n and n o n - p e r t u r b a t i v e ) d i a g r a m s of fig. 1. Since we have put f4 =f5 = • • • = 0 and only t r e a t the two p a r t i c l e cut c o n t r i b u t i o n s here, the fourth and higher d i a g r a m s will be omitted. The f i r s t t h r e e d i a g r a m s a r e easily calculated and give ~(q,2,V,2) = 5(m2 _q,2)5(m2 _Q,2)f2(m2 ) + l l

f2(q 72)~0(q

2)

ir - ~ q ' 2 ~

5(m2

-q'2)O(q'2-4m2)

+

(q'~-*q')l " (16)

An identification of f 2 with the help of eqs. (8) and (12) finally yields

1--

1

c~et(q'P) :v (m2_(q+i¢)2)(m2 _(p _q _ iE)2)

{f2(m2) + [f2((q +i~)2) +/2((P - q -i£)2) - 2f2(m2)]}

(17)

as well as

f2((q+ie)2)

m2 _q2 : f 2 ( m 2) +

~

f 2(q'2)to(q'2,m 2)

dq,2

f (m2_q,2)(q,a_(q+ie)2) 4 m2

"

The l a t t e r equation has the solution [5] f2 ((q +iE)2) : f 2 ( m 2 ) e x p

1 q2-m2~ f ( q , 2

_ m-2~)(qq~:~ i E ) 2 ) 5 ( q ' 2÷) d q ' 2 1"

(18)

We note that only o n - s h e l l values of the s c a t t e r i n g amplitude wittl phase 5(q '2) c o n t r i b u t e to the m a t r i x e l e m e n t of the c a u s a l c o m m u t a t o r . A c o m p a r i s o n of the r e s u l t i n g c 1(q,p) with the g e n e r a l form (8) then shows as we have stated before that locality - in the approximation used - n e c e s s i t a t e s the independence of t0(q2, ( P - q)2) of its second a r g u m e n t . This is compatible with eq. (10). The r e s t r i c t i o n onc~et(q,p) which was a product of two poles with constant r e s i d u e in the local one p a r t i c l e a p p r o x i m a t i o n is now lifted by the a p p e a r a n c e of the additional t e r m in eq. (17) r e p r e s e n t i n g the two p a r t i c l e cut c o n t r i b u t i o n s . The s t i l l r e m a i n i n g c o n s t r a i n t on t o would be r e m o v e d by the higher graphs c o r r e s p o n d i n g to t h r e e p a r t i c l e cut c o n t r i b u t i o n s with t h r e s h o l d s at 9m 2. Obviously, the eq. (lq) fulfills our r e q u i r e m e n t s , at l e a s t for a suitable phase 5(q2). The asymptotic condition can be satisftect and the r e l e v a n t coefficients in the e:Tpansion of the c o m m u t a t o r of the "out"field with itself, 1.e. c out I (q,P) and the s - w a v e p r o j e c t i o n o f c °2u : (q . .,P.l ,P . 2 ) vanish, c~et(q,p) obeys an offs h e l l u n i t a r i t y equation in which the s u m over i n t e r m e d i a t e states includes the 2 p a r t i c l e continuum. The c o m m u t a t o r m a t r i x e l e m e n t is c e r t a i n l y local if f2((q + iE) 2) is a s y m p t o t i c a l l y bounded by a constant. Since no p o l y n o m i a l behaviour has been taken into account the v a c u u m - o n e p a r t i c l e m a t r i x e l e ment of the equal t i m e c o m m u t a t o r is zero. A local r e p r e s e n t a t i o n analogous to the one we have suggested can, of c o u r s e , be c o n j e c t u r e d for m a t r i x e l e m e n t s including m o r e p a r t i c l e s . Of c o u r s e , the u n i t a r i t y c o n s t r a i n t s a r e m o r e complicated in these cases. We take p l e a s u r e in thanking K. Hepp for his c o m m e n t s during his visit in Heidelberg.

338

Volume 35B, n u m b e r 4

PHYSICS

LETTERS

7 June 1971

References [1] B. Stech, Z. P h y s i k 239 (1970) 387; H. D. Dahmen, K. D. Rothe and B. Stech, P h y s . L e t t e r s 34B (1971) 83. [2] T. L. B a s d e v a n t and B. W. Lee, P h y s . Rev. 2D (1970) 1680; H. T. Schnitzer, Phys. Rev. 2D (1970) 1621. [3] R. Haag, Dan. Mat. F y s . Medd. 29 no. 13 (1955); V. Glaser, H. Lehmann and W. Zimmermann, Nuovo Cimento 6 (1957) 1122. [4] I. Tamm, J. Phys. (USSR) 9 (1945) 449; S. M. Daneoff, Phys. Rev. 78 (1950) 382. [5] R. Omn~s, Nuovo Cimento 8 (1958) 316.

339