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Some astrophysical speculations about scaling in very high energy collisions
Volume 37B, number 5
PHYSICS
LETTERS
27 December 1971
SOME A S T R O P H Y S I C A L S P E C U L A T I O N S ABOUT SCALING IN VERY HIGH ENERGY C O L L I S I O N S
H. A. K A S T R U P
Universititt M~2nchen, Sektion Physik, Munich, Germany Received 23 November 1971 The paper consists of two parts: First, equilibrium distribution functions are discussed under the assumption that binary collisions are dilation on conformally invariant. Second, supposing that 'scaling' in several high energy collisions is associated with dilation or {and) conformal invariance at short distances, some speculative astrophysical implications for a hot expanding universe are presented. When a s y m p t o t i c s c a l i n g of c o l l i s i o n a m p l i t u d e s , a s a c o n s e q u e n c e of d i l a t a t i o n and c o n f o r m a l i n v a r i a n c e at s h o r t d i s t a n c e s , w a s s u g g e s t e d [1] s o m e y e a r s ago, no c l e a r c u t e x p e r i m e n t a l i n f o r m a t i o n w a s a v a i l a b l e in o r d e r to t e s t t h i s h y p o t h e s i s and t h e r e w a s a l s o the p r o b l e m of s e l e c t i n g the r i g h t s c a l i n g v a r i a b l e s . In the m e a n t i m e , due to the t h e o r e t i c a l w o r k by B j o r k e n , F e y n m a n , Mack, W i l s o n and m a n y o t h e r s [2, 6], m u c h m o r e i s known about the d y n a m i c a l b a c k g r o u n d of s c a l i n g and about the r i g h t c h o i c e of the s c a l i n g v a r i a b l e s . T h e m a i n e x p e r imental support comes from deep inelastic elect r o n - n u c l e o n s c a t t e r i n g [3] and f r o m ' i n c l u s i v e ' h a d r o n i c r e a c t i o n s [4]. Though t h e r e a r e s t i l l m a n y open p r o b l e m s , it s e e m s that s c a l i n g and d i l a t a t i o n and c o n f o r m a l i n v a r i a n c e at s h o r t d i s t a n c e s w i l l p l a y an i m p o r t a n t p a r t in h i g h - e n e r g y p h y s i c s [5]. S u p p o s e now that s c a l i n g of high e n e r g y c o l l i s i o n a m p l i t u d e s i s a s s o c i a t e d with a s y m p t o t i c d i l a t a t i o n and c o n f o r m a l i n v a r i a n c e at s h o r t d i s t a n c e s - that t h i s i s so in d e e p i n e l a s t i c e - N s c a t t e r i n g w a s s h o w n [6] by M a c k - and s u p p o s e f u r t h e r that t h i s i m p l i e s , in a s e n s e to be s p e c i f i e d b e l o w , the a p p r o x i m a t e c o n s e r v a t i o n of d i l a t a t i o n and (or) c o n f o r m a l m o m e n t a [7] in the r e a c t i o n s in q u e s t i o n . T h e n , a c c o r d i n g to o u r k n o w l e d g e about r e l a t i v i s t i c B o l t z m a n n e q u a t i o n s [8], t h e r e s h o u l d be e q u i l i b r i u m s o l u t i o n s of t h i s e q u a t i o n in f l a t s p a c e s u c h that the e x p o n e n t of the d i s t r i b u t i o n f u n c t i o n i s l i n e a r l y r e l a t e d to these momenta. Now, if the v e r y e a r l y u n i v e r s e w a s a v e r y hot o n e , with t e m p e r a t u r e s in the high e n e r g y r e g i o n (1013 d e g r e e K e l v i n and h i g h e r ) and if a p proximate scaling for different processes was p r e s e n t (which i s l i k e l y ) and if it i n d e e d l e d to d i s t r i b u t i o n f u n c t i o n s of the kind j u s t m e n t i o n e d
(which is not so obvious~), then t h i s s c a l i n g would h a v e and m i g h t s t i l l h a v e an t r e m e n d o u s i n f l u e n c e on the e x p a n s i o n of the u n i v e r s e which c o u l d h a v e gone m u c h b e y o n d i t s g r a v i t a t i o n a l one'. O b v i o u s l y t h e r e a r e many ' i f s ' and a c c o r d i n g l y o u r p a p e r is s e p a r a t e d into two p a r t s : We f i r s t d i s c u s s s o m e s t a t i s t i c a l p r o p e r t i e s of s y s t e m s with b i n a r y d i l a t a t i o n and c o n f o r m a l l y i n v a r i a n t c o l l i s i o n s . We s h a l l s e e that c o n s e r v a t i o n of d i l a t a t i o n m o m e n t u m l e a d s to M i l n e ' s u n i v e r s e [9] and c o n s e r v a t i o n of the c o n f o r m a l m o m e n t a to a new i s o t r o p i c but not h o m o g e n e o u s m o d e l . We then d i s c u s s s e v e r a l r e a c t i o n s with s c a l i n g , which, a f t e r the 'big b a n g ' , might h a v e c a u s e d at l e a s t a p a r t of the d i s t r i b u t i o n and e x p a n s i o n of m a t t e r in the u n i v e r s e . T h e f i r s t p a r t is not s p e c u l a t i v e , but the s e c o n d one is: 1. In the f i r s t p a r t we can u s e the g e n e r a l r e s u l t s [10] of E h l e r s , G e r e n and S a c h s , a p p l y ing t h e m to the flat M i n k o w s k i s p a c e . A c c o r d i n g to t h o s e r e s u l t s , a l o c a l l y i s o t r o p i c d i s t r i b u t i o n f u n c t i o n f ( x , p) f o r p a r t i c l e s with v a n i s h i n g o r n e g l i g i b l e r e s t m a s s e s has the f o r m *
f(x,p) =g[~P(x)p#]
,
w h e r e ~/Z(x) is a c o n f o r m a l K i l l i n g v e c t o r d e f i n ing a v e l o c i t y f i e l d u #(x), uPut~ = c 2, in t e r m s of ~/~ = (exp U(x))ulz w h e r e U(x) is s o m e function. In o r d e r to s i m p l i f y our d i s c u s s i o n t e c h n i c a l l y , we s h a l l t a l k e x p l i c i t l y only about B o l t z m a n n d i s t r i b u t i o n s , the g e n e r a l i z a t i o n s to E i n s t e i n - B o s e and F e r m i - D i r a c s t a t i s t i c s a r e s t r a i g h t f o r w a r d , u s i n g the r e s u l t s of r e f . [8]. 1.1. D i l a t a t i o n s . T h e t r a n s f o r m a t i o n s D(a):x/x ~#
= eC~x/x, a r e a l ,
g i v e the c o n f o r m a l K i l l i n g v e c t o r • We use the metric x2=xtlxll = (xO) 2 - (x)2,x o= ct. 521
Volume 37B, number 5
PHYSICS
+ ~ ( x ) = (ok~/+c~)a=o = xU
(1)
and we o b t a i n the e q u i l i b r i u m d i s t r i b u t i o n f u n c tion
f(x,p)
[ pgug(x)] = FDexp, - k T ( x )
LETTERS
27 December 1971
r e q u i r e m e n t that the d y n a m i c s of b i n a r y c o l l i sions is invariant under dilatations. 1.2. S p e c i a l c o n f o r m a l t r a n s f o r m a t i o n s : SC(c): xlZ ~ ~
= (x~.-c~x2)/~(x;c)
~(x;c):
(2)
1-2c.x+c
,
2x 2
(7)
H e r e we get the f o u r c o n f o r m a l K i l l i n g v e c t o r s
u~ (x ) = c x ~
(x2)1/2
x2
'
>
0
~a = ( ~ )
,
~,g
kT(x) -
c
FD, 3 3 = c o n s t .
,
~(x)
= 2xt~xa - g ~ x 2
,
= 0,1,2,3
T h e i n v a r i a n t d e n s i t y n (x), d e f i n e d by _ . ~ 3 ~.
L e t b be a t i m e - l i k e c o n s t a n t unit v e c t o r : b2 = 1, t h e n we c a n f o r m (for X2 ¢ 0) the t i m e - l i k e v e c t o r ~ c = ~ b ~ , 42 = (x2)2. In t h i s c a s e e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n , v e l o c i t y f i e l d and t e m p e r a t u r e a r e g i v e n by
ng(x) : J~oo P~f(x, p) =n(x) u~(x)
f ( x , p) = F c e x p [ - u g t ( x ) p g / k T ( x ) ]
3 3 ( x2)1/2
'
i s c a l c u l a t e d to be
(3)
8~F D nD(x=O,t)-c2([3Dt) 3"
n o ( x ) = 8~ FD(kT(x))3; The energy-momentum d3p
t e n s o r T P V(x), d e f i n e d by
_
= (~ + k / c 2 ) u g u V_ggVD
,
w h e r e h(x): p r o p e r e n e r g y d e n s i t y , p(x): p r o p e r p r e s s u r e and 3/~ = ~ c 2 f o r v a n i s h i n g r e s t m a s s e s , i s d e t e r m i n e d by
])D(X) = 8~c 2 F D (kT (x)) 4
(4)
c 1 ug;u ~ Ovug(x) - (X2)1/2 (ggtv - ~ U l j . Uv)
0 ~ 0
,
~p ~
(p
,
(5)
, ~"
Fc, ~c : c o n s t .
(8)
,
~c(X ) = 8 ~ c 2 F c ( k T ) 4 !9)
D i f f e r e n t i a t i o n with r e s p e c t to x v g i v e s
u~z;v(x)__ 1 i* u v + 2 x ' b ( g -wl c2 ~ x2 ~v - c~ UgUv)
hn = u~;vuV : ; ( x "
b u~ - cxg)
,
(10)
p = r / x 2,
0-~ O,
~
~,
(11)
l e a d s to c o m o v i n g c o o r d i n a t e f r a m e s :
u~-*~
:c(T2 -o2)go ~
,
(12)
ds 2 - 1 [d1_2 _ do 2 _ p2(d 0 2 + sin2O dq~2)] (~-2 _p2) 2 (6)
= (X2) 1/2
,
ds 2 = d~-2 _ 1-2[dw 2 + s i n h 2 w ( d 0 2 + sin20 d(p2)] In t h e c o m o v i n g s y s t e m we h a x e ~ z = cg~o. We s e e that the s y s t e m so o b t a i n e d i s j u s t M i l n e ' s u n i v e r s e [9], e x t e n s i v e l y d i s c u s s e d in the l i t e r a t u r e [11]. In o u r c a s e we o b t a i n it f r o m t h e 522
,
,
P r o p e r d e n s i t y n c (x) and p r o p e r p r e s s u r e / ~ c (x) a r e , c a l c u l a t e d to be
T = x O / x 2,
and the e x P a n s i o n v e l o c i t y 0 h a s the v a l u e 3 c / ( x 2 ) 1/2. T h e a c c e l e r a t i o n ut~ = ul~;vuV vanishes. T r a n s f o r m a t i o n to c o m o v i n g c o o r d i n a t e s i s o b t a i n e d by the s u b s t i t u t i o n r = Tsinhw
2)
i . e . in t h i s c a s e the a c c e l e r a t i o n d o e s not vanish'. C h o o s i n g b = (1,0,0,0), we h a v e u ~ = = - c x 2 0 g ( x O / x 2 ) and the i n v e r s i o n
B e c a u s e u g (x) = c Og(x2) 1/2, the f l o w a s s o c i a t e d with the v e l o c i t y f i e l d i s h y p e r s u r f a c e o r t h o g o n a l . We f u r t h e r h a v e
,
kT(x) : C/~cX2
nc(X ) = 8 u F c ( k T ) 3
T gzv(x ) = J ~ - o p~" PVf(x, P) =
x° = Tcoshw
u~(x) = -cx2~(b.x/x
,
We s e e f r o m eq. (12) that the s y s t e m o b t a i n e d i s i s o t r o p i c , but not h o m o g e n e o u s , i . e . it i s not a F r i e d m a n u n i v e r s e with a R o b e r t s o n - W a l k e r metric L e t us d i s c u s s a s i m p l e but i m p o r t a n t c o n s e q u e n c e : Suppose t h e r e i s an o b s e r v e r at (t = to, x =0) ~ (*o = l / t o , P =0). S u p p o s e f u r t h e r that light w a s e m i t t e d at ( t , x ) ~ (1-, p) w i t h w a v e l e n g t h X, a r r i v i n g at x = 0 at t i m e t o = t + r / c .
Volume 37B, number 5
PHYSICS
T h e n the r e d shift z = (;t o - ;t)/;t is c a l c u l a t e d to be 1
p =~T oz
1
,
~- = ~ T o ( z +
2)
,
(13)
o r , in t e r m s of the M i n k o w s k i c o o r d i n a t e s r , t and to: r = ~1c i ~ t z
o
t - ~ +I 2 + zz to
,
(14)
F o r z < < l the f i r s t one of the e q s . (14) r e d u c e s to H u b b l e ' s l a w r = ( c / H o ) z , H o = 2/1 o. F o r z ~ 1 , however, Hubble's law would give distances l a r g e r than the e x a c t r e l a t i o n (14) does'. B e c a u s e of 1,/x 2 = (1 + z ) / ' c 2 t 2
,
(15)
which f o l l o w s f r o m eq. (14), we c a n , in p r i n c i p l e , d e t e r m i n e the t e m p e r a t u r e T(x), the d e n s i t y n c (x), the p r e s s u r e p c (x) and the e n e r g y d e n s i t y (x) by r e d s h i f t m e a s u r e m e n t s a l o n e , if we know t o and the c o n s t a n t s of F c and tic (which m a y be g o t t e n by l o c a l m e a s u r e m e n t s ' . ) ' . F o r a b l a c k body g a s with two i n t e r n a l d e g r e e s of f r e e d o m we h a v e F c = 2/(27r~)3. 2. A s t r o p h y s i c a l s p e c u l a t i o n s . T h e two e x p a n d i n g s y s t e m s of u l t r a r e l a t i v i s t i c p a r t i c l e s d e s c r i b e d a b o v e s u g g e s t v e r y m u c h to c o m p a r e t h e m with o u r e x p a n d i n g u n i v e r s e . A s the f e a t u r e s of the f i r s t one ( a s s o c i a t e d with d i l a t a t i o n s ) a r e w e l l - k n o w n , let us c o n f i n e the d i s c u s s i o n to the s e c o n d one: 2.1. L o c a l l y , i . e . f o r r 2 << c 2 t 2, the s y s t e m is i s o t r o p i c a n d h o m o g e n e o u s and H u b b l e ' s l a w is v a l i d a s m e n t i o n e d a b o v e . T h e l o c a l R o b e r t s o n W a l k e r m e t r i c has an e x p a n s i o n f u n c t i o n R = = Rlt2 , R 1 = const. This function is not a solut i o n of F r i e d m a n ' s e q u a t i o n [11], but c a n be shown to be a s o l u t i o n of the c o r r e s p o n d i n g B r a n s - D i c k e q u a t i o n [12] with ¢o - -~3 ( s e e r e f . [13]), v a n i s h i n g s p a t i a l c u r v a t u r e and the Brans-Dicke scalar
LETTERS
27 December 1971
o t h e r hand, b e c a u s e we h a v e ~ ( r , t ) cc (1 + z) 4 f o r the i n v a r i a n t e n e r g y d e n s i t y , the e n o r m o u s e n e r g i e s and the d e v i a t i o n s f r o m u n i f o r m d i s t r i bution [14] a s s o c i a t e d with q u a s a r s m a y h a v e s o m e t h i n g to do with o u r m o d e l and the p h y s i c a l m e c h a n i s m s which p r o d u c e it'. We a r e not s u g g e s t i n g that the p r o p e r t i e s of q u a s a r s can be u n d e r s t o o d c o m p l e t e l y in the f r a m e w o r k of o u r m o d e l , n e v e r t h e l e s s t h i s f i e l d s e e m s to be an interesting candidate for further investigations. 2.4. The m o s t i m p o r t a n t p h y s i c a l p r o b l e m in the c o n t e x t of o u r c o n s i d e r a t i o n s is the q u e s t i o n : which s p e c i f i c d y n a m i c a l m e c h a n i s m s c o u l d be a b l e to g i v e r i s e , in a r e a s o n a b l e a p p r o x i m a t i o n , to the type of d i s t r i b u t i o n f u n c t i o n s we h a v e b e e n d i s c u s s i n g . In o t h e r w o r d s , a r e t h e r e high e n e r gy p r o c e s s e s which a r e a p p r o x i m a t e l y d i l a t a t i o n o r (and) c o n f o r m a l l y i n v a r i a n t and l e a d to the a b o v e s y s t e m s ? P r e s e n t l y we a r e not a b l e to give a clearcut answer. However, there are s e v e r a l t h e o r e t i c a l and e x p e r i m e n t a l hints which s u g g e s t s that one should t a k e the a b o v e s t a t i s t i c a l s y s t e m s s e r i o u s l y . O u r p r o c e d u r e in t h i s c o n t e x t is the f o l l o w i n g : we l i s t s e v e r a l i m p o r t a n t r e a c t i o n s with c r o s s s e c t i o n s of a f o r m w h i c h c a n be r e l a t e d to an e f f e c t i v e d i l a t a t i o n and (or) c o n f o r m a l l y i n v a r i a n t ' p o t e n t i a l ' l i k e , e.g. the C o u l o m b p o t e n t i a l , o r a B o r n a p p r o x i m a t i o n in k i n e m a t i c a l r e g i o n s w h e r e all r e s t m a s s e s a r e n e g l i g i b l e and w h e r e we c a n h a v e a p p r o x i m a t e c o n s e r v a t i o n of d i l a t a t i o n and (or) c o n f o r m a l momenta: 2.4.1. P u r e l y e l e c t r o m a g n e t i c i n t e r a c t i o n s in lowest order: a) C o m p t o n s c a t t e r i n g off e l e c t r o n s f o r s , It I>> m2e: dcr
27r~ 2 f..ss + ~ ' ~
dlt I-
s2
ltl
sJ
'
_
1/~(t) : (1/¢1)t6
,
(Ol : 287r2Fc/9C9f14
2.2. W h e r e a s the E i n s t e i n - D e S i t t e r m o d e l [11] of the u n i v e r s e (which t a k e s g r a v i t a t i o n into a c count) h a s a t i m e d e c r e a s e of the d e n s i t y n ( x ) g i v e n by n ( x ) t 2 = c o n s t . , a c c o r d i n g to eqs. (8) and (9) the d e n s i t i e s in o u r m o d e l d e c r e a s e m u c h f a s t e r , o v e r t a k i n g the g r a v i t a t i o n a l d e c r e a s e . T h i s m a y be one r e a s o n f o r the m o d e l b e i n g of some relevance'. 2.3. F o r r e d s h i f t s of the o r d e r 1, the m o d e l i s no l o n g e r h o m o g e n e o u s and H u b b l e ' s law i s m o d i f i e d . If the m o d i f i e d v e r s i o n (14) h a s a n y t h i n g to do with, e.g. q u a s a r s , t h e y w o u l d not be so f a r away a s H u b b l e ' s law s u g g e s t s and, on t h e
s:
(Pc +PT) 2
,
b) E l e c t r o n - e l e c t r o n
t:
(pc-Pc) 2
s c a t t e r i n g f o r s, It] >>m2:
d~ 2vc~ 2 d l t l - t 2 s 2 (t 2 + 2 s t + 2 s 2) Higher order will introduce 'non-scaling' correct i o n s , but we h a v e at l e a s t an a p p r o x i m a t e scaling. 2.4.2. I n c l u s i v e s e m i h a d r o n i c p r o c e s s e s . T h e p r o c e s s e + N ~ N p l u s anything d o e s not only s h o w s c a l i n g of the s t r u c t u r e f u n c t i o n s [2, 3] but the t o t a l d i f f e r e n t i a l c r o s s s e c t i o n i s C o u l o m b l i k e [15, 16]: d ~ / d [ t I ~ 4~ o l 2 / t 2
This process
- at v e r y l a r g e i n i t i a l e n e r g i e s
523
Volume 37B, n u m b e r 5
PHYSICS
may be looked at as an effective binary electron s c a t t e r i n g off a C o u l o m b p o t e n t i a l . H o w e v e r , i n s u c h a c a s e we h a v e c o n s e r v a t i o n of t h e d i l a t a t i o n m o m e n t u m [1] a n d t h e t i m e - c o m p o n e n t of t h e e o n f o r m a l m o m e n t a *'. T h u s we do h a v e i n t h i s c a s e a k i n d of ' q u a s i ' - b i n a r y c o l l i s i o n we need for establishing the distribution function in q u e s t i o n . In w e a k i n t e r a c t i o n s we c a n a p p l y t h e s a m e r e a s o n i n g o n l y [17] if t h e r e i s a n i n t e r m e d i a t e b o s o n W a n d if I t I >> m 2w. 2.4.3. P u r e l y h a d r o n i c i n t e r a c t i o n s . In i n c l u sive hadronic reactions a + b ~ c + anything ' F e y n m a n ' s e a l i n g [4] i s o b s e r v e d f o r s ~ ~o: T h e production cross section for the particle c with l o n g i t u d i n a l c . m . m o m e n t u m PH a n d t r a n s v e r s m o m e n t u m p± i s s u p p o s e d to b e of t h e f o r m
d2~/p± dPxdY =f ( p x , y ),
y : 2pH /'~s ,
w h i c h i s s u p p o r t e d by e x p e r i m e n t s
[4]. E i t h e r if
f ( p ~ , y) f a c t o r i z e s [4]: f ( P l , Y) = f l ( P ± ) f 2 ( Y ) o r if we i n t e g r a t e o v e r p±, we g e t
dcr/dy : g (y). This production cross section may be interpreted i n t h e f o l l o w i n g way: A c r o s s s e c t i o n i n o n e s p a c e d i m e n s i o n , d e f i n e d by t h e d e c r e a s e Aj = -~jn (x3)Ax 3 of t h e c u r r e n t d e n s i t y j a l o n g A x 3 i s d i m e n s i o n l e s s . T h i s m e a n s : if t h e d y n a m i c s do not c o n t a i n a n y f i x e d l e n g t h s l i k e r e s t m a s s e s or coupling constants with non-vanishing dimensions etc., then the production cross section for one particle in the final state, after summing over all other secondaries, can only be a funct i o n of t h e r a t i o y . In t h i s s e n s e p u r e l y h a d r o n i e interactions exhibit one-dimensional (longitudinal) d i l a t a t i o n i n v a r i a n c e [1,18]'. 2.5. It i s not c l e a r , w h e t h e r t h e h i g h e n e r g y r e a c t i o n s j u s t l i s t e d a c t u a l l y d i d l e a d to d i s t r i b u t i o n f u n c t i o n s in a s t r o p h y s i c s a s s u g g e s t e d a b o v e , because the physical situation is so complex. T h e y o r o t h e r s mighl h a v e , b u t o b v i o u s l y f u r t h e r s t u d i e s a n d t h e i n c l u s i o n of g r a v i t a t i o n a r e necessary'. I a m v e r y m u c h i n d e b t e d to J. E h l e r s f o r a c r i t i c a l r e a d i n g of a f i r s t v e r s i o n of t h i s p a p e r and for very helpful discussions about its statist i c a l e l e m e n t s . P a r t s of t h i s w o r k w e r e d o n e during a stay as Visiting Professor at the Univers i t y of H a m b u r g i n t h e s u m m e r 1971 a n d d u r i n g a v i s i t at C E R N i n f a l l 1971. I a m v e r y g r a t e f u l * The conformal momentum here has the form h ° = 2 x ° ( x p ) - p ° x 2 - 2 g r , w h e r e g r -1 is the Coulomb potential.
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27 D e c e m b e r 1971
to m a n y c o l l e a g u e s f o r d i s c u s s i o n s a n d f o r t h e v e r y k i n d h o s p i t a l i t i e s of t h e U n i v e r s i t y of H a m b u r g , of DESY a n d of t h e T h e o r y D i v i s i o n at CERN.
References [1] H.A. Kastrup, Phys. Lett. 4 (1963) 56; Nucl. Phys. 58 (1964) 561; H a b i l i t a t i o n T h e s i s , Munich, 1964. [2] J.D. Bjorken, Phys. Rev. 179 (1969) 1547; R . P . Feynman, Phys. Rev. Lett. 23 (1969) 1415; K.G. Wilson, Phys. Rev. 179 (1969) 1499; S. Ciccariello, R. Gatto, G. Sartori and M. Tonin, Ann. Phys. (N.Y.)65 (1971) 265; D. G r o s s a n d J . Wess, Phys. Rev. D2 (1970) 753; Y. F r i s h m a n , Ann. Phys. (N.Y.) 66 (1971) 373; R. Brandt and G. P r e p a r a t a , Nucl. Phys. B 27 (1971) 541 ; H. F r i t s c h and M. Gell-Mann, Contribution to the Coral Gable Conf. 1971. [3] E.D. Bloom et al., Phys. Rev. Lett. 23 (1969) 930. [4] F o r a review see: D. Horn. Many-particle production, P h y s i c s Reports, to be published. [5] Of specific significance is the following paper: G. Mack and I. Todorov, Conformal invariant Green function without ultraviolet d i v e r g e n c e s , to be .published ; see also: A. A. Migda[, Conformal invariance and bootstrap, Landau Institute (Moscow), P r e p r i n t 1971. [6] G. Mack, T r i e s t e P r e p r i n t IC/70/95, Nucl. Phys., to be published. [7] H.A. Kastrup, Phys. Rev. 150 (1966) 1183. [8] J. Ehlers, in General relativity and cosmology, Proc. Intern. School of P h y s i c s ' E n r i c o F e r m i ' , Course 47 (Academic P r e s s , 1971); J . M . Stuart, Lecture Notes in Physics, Vo[.1O (Springer-Verlag, Berlin, Heidelberg, N.Y. 1971). [9] E. A. Milne, Kinematic relativity (Oxford University P r e s s , 1948). [10] J. Ehlers, P. Geren and R.K. Sachs, J . M a t h . Phys. 9 (1968) 1344; G. E. T a u b e r and J. W. Weinberg, Phys. Rev. 122 (1961) 1342. [11] H. P. Robertson and T. W. Noonen, Relativity and cosmology (W.B. Saunders Comp., Philadelphia, London, Toronto, 1968); W. Rindler, E s s e n t i a l relativity (Van Nostrand Reinhold Comp., N.Y., 1969). [12] C. B r a n s and R.H. Dicke, Phys. Rev. 124 (1961) 925. [13] J. L. Anderson, Phys. Rev. D3 (1971) 1689. [14] M. Schmidt, Astrophys. J. 162 (1970) 371. [15] J . D . Bjorken, Proc. 1967 Intern. School of P h y s i c s ' E n r i c o F e r m i ' (Academic P r e s s , N.Y., London, 1968). [16] H.A. Kastrup, Phys. Rev. 147 (1966) 1130; Schladming L e c t u r e s in: P a r t i c l e s , c u r r e n t s , symm e t r i e s , ed. P. Urban (Springer-Verlag, Wien, N.Y., 1968) p. 407. [17] H.A. Kastrup, Nuovo Cimento 48 A (1967) 271. [18] M. Bandor, Phys. Rev. D3 (1971) 3173.
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27 December 1971
SOME A S T R O P H Y S I C A L S P E C U L A T I O N S ABOUT SCALING IN VERY HIGH ENERGY C O L L I S I O N S
H. A. K A S T R U P
Universititt M~2nchen, Sektion Physik, Munich, Germany Received 23 November 1971 The paper consists of two parts: First, equilibrium distribution functions are discussed under the assumption that binary collisions are dilation on conformally invariant. Second, supposing that 'scaling' in several high energy collisions is associated with dilation or {and) conformal invariance at short distances, some speculative astrophysical implications for a hot expanding universe are presented. When a s y m p t o t i c s c a l i n g of c o l l i s i o n a m p l i t u d e s , a s a c o n s e q u e n c e of d i l a t a t i o n and c o n f o r m a l i n v a r i a n c e at s h o r t d i s t a n c e s , w a s s u g g e s t e d [1] s o m e y e a r s ago, no c l e a r c u t e x p e r i m e n t a l i n f o r m a t i o n w a s a v a i l a b l e in o r d e r to t e s t t h i s h y p o t h e s i s and t h e r e w a s a l s o the p r o b l e m of s e l e c t i n g the r i g h t s c a l i n g v a r i a b l e s . In the m e a n t i m e , due to the t h e o r e t i c a l w o r k by B j o r k e n , F e y n m a n , Mack, W i l s o n and m a n y o t h e r s [2, 6], m u c h m o r e i s known about the d y n a m i c a l b a c k g r o u n d of s c a l i n g and about the r i g h t c h o i c e of the s c a l i n g v a r i a b l e s . T h e m a i n e x p e r imental support comes from deep inelastic elect r o n - n u c l e o n s c a t t e r i n g [3] and f r o m ' i n c l u s i v e ' h a d r o n i c r e a c t i o n s [4]. Though t h e r e a r e s t i l l m a n y open p r o b l e m s , it s e e m s that s c a l i n g and d i l a t a t i o n and c o n f o r m a l i n v a r i a n c e at s h o r t d i s t a n c e s w i l l p l a y an i m p o r t a n t p a r t in h i g h - e n e r g y p h y s i c s [5]. S u p p o s e now that s c a l i n g of high e n e r g y c o l l i s i o n a m p l i t u d e s i s a s s o c i a t e d with a s y m p t o t i c d i l a t a t i o n and c o n f o r m a l i n v a r i a n c e at s h o r t d i s t a n c e s - that t h i s i s so in d e e p i n e l a s t i c e - N s c a t t e r i n g w a s s h o w n [6] by M a c k - and s u p p o s e f u r t h e r that t h i s i m p l i e s , in a s e n s e to be s p e c i f i e d b e l o w , the a p p r o x i m a t e c o n s e r v a t i o n of d i l a t a t i o n and (or) c o n f o r m a l m o m e n t a [7] in the r e a c t i o n s in q u e s t i o n . T h e n , a c c o r d i n g to o u r k n o w l e d g e about r e l a t i v i s t i c B o l t z m a n n e q u a t i o n s [8], t h e r e s h o u l d be e q u i l i b r i u m s o l u t i o n s of t h i s e q u a t i o n in f l a t s p a c e s u c h that the e x p o n e n t of the d i s t r i b u t i o n f u n c t i o n i s l i n e a r l y r e l a t e d to these momenta. Now, if the v e r y e a r l y u n i v e r s e w a s a v e r y hot o n e , with t e m p e r a t u r e s in the high e n e r g y r e g i o n (1013 d e g r e e K e l v i n and h i g h e r ) and if a p proximate scaling for different processes was p r e s e n t (which i s l i k e l y ) and if it i n d e e d l e d to d i s t r i b u t i o n f u n c t i o n s of the kind j u s t m e n t i o n e d
(which is not so obvious~), then t h i s s c a l i n g would h a v e and m i g h t s t i l l h a v e an t r e m e n d o u s i n f l u e n c e on the e x p a n s i o n of the u n i v e r s e which c o u l d h a v e gone m u c h b e y o n d i t s g r a v i t a t i o n a l one'. O b v i o u s l y t h e r e a r e many ' i f s ' and a c c o r d i n g l y o u r p a p e r is s e p a r a t e d into two p a r t s : We f i r s t d i s c u s s s o m e s t a t i s t i c a l p r o p e r t i e s of s y s t e m s with b i n a r y d i l a t a t i o n and c o n f o r m a l l y i n v a r i a n t c o l l i s i o n s . We s h a l l s e e that c o n s e r v a t i o n of d i l a t a t i o n m o m e n t u m l e a d s to M i l n e ' s u n i v e r s e [9] and c o n s e r v a t i o n of the c o n f o r m a l m o m e n t a to a new i s o t r o p i c but not h o m o g e n e o u s m o d e l . We then d i s c u s s s e v e r a l r e a c t i o n s with s c a l i n g , which, a f t e r the 'big b a n g ' , might h a v e c a u s e d at l e a s t a p a r t of the d i s t r i b u t i o n and e x p a n s i o n of m a t t e r in the u n i v e r s e . T h e f i r s t p a r t is not s p e c u l a t i v e , but the s e c o n d one is: 1. In the f i r s t p a r t we can u s e the g e n e r a l r e s u l t s [10] of E h l e r s , G e r e n and S a c h s , a p p l y ing t h e m to the flat M i n k o w s k i s p a c e . A c c o r d i n g to t h o s e r e s u l t s , a l o c a l l y i s o t r o p i c d i s t r i b u t i o n f u n c t i o n f ( x , p) f o r p a r t i c l e s with v a n i s h i n g o r n e g l i g i b l e r e s t m a s s e s has the f o r m *
f(x,p) =g[~P(x)p#]
,
w h e r e ~/Z(x) is a c o n f o r m a l K i l l i n g v e c t o r d e f i n ing a v e l o c i t y f i e l d u #(x), uPut~ = c 2, in t e r m s of ~/~ = (exp U(x))ulz w h e r e U(x) is s o m e function. In o r d e r to s i m p l i f y our d i s c u s s i o n t e c h n i c a l l y , we s h a l l t a l k e x p l i c i t l y only about B o l t z m a n n d i s t r i b u t i o n s , the g e n e r a l i z a t i o n s to E i n s t e i n - B o s e and F e r m i - D i r a c s t a t i s t i c s a r e s t r a i g h t f o r w a r d , u s i n g the r e s u l t s of r e f . [8]. 1.1. D i l a t a t i o n s . T h e t r a n s f o r m a t i o n s D(a):x/x ~#
= eC~x/x, a r e a l ,
g i v e the c o n f o r m a l K i l l i n g v e c t o r • We use the metric x2=xtlxll = (xO) 2 - (x)2,x o= ct. 521
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+ ~ ( x ) = (ok~/+c~)a=o = xU
(1)
and we o b t a i n the e q u i l i b r i u m d i s t r i b u t i o n f u n c tion
f(x,p)
[ pgug(x)] = FDexp, - k T ( x )
LETTERS
27 December 1971
r e q u i r e m e n t that the d y n a m i c s of b i n a r y c o l l i sions is invariant under dilatations. 1.2. S p e c i a l c o n f o r m a l t r a n s f o r m a t i o n s : SC(c): xlZ ~ ~
= (x~.-c~x2)/~(x;c)
~(x;c):
(2)
1-2c.x+c
,
2x 2
(7)
H e r e we get the f o u r c o n f o r m a l K i l l i n g v e c t o r s
u~ (x ) = c x ~
(x2)1/2
x2
'
>
0
~a = ( ~ )
,
~,g
kT(x) -
c
FD, 3 3 = c o n s t .
,
~(x)
= 2xt~xa - g ~ x 2
,
= 0,1,2,3
T h e i n v a r i a n t d e n s i t y n (x), d e f i n e d by _ . ~ 3 ~.
L e t b be a t i m e - l i k e c o n s t a n t unit v e c t o r : b2 = 1, t h e n we c a n f o r m (for X2 ¢ 0) the t i m e - l i k e v e c t o r ~ c = ~ b ~ , 42 = (x2)2. In t h i s c a s e e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n , v e l o c i t y f i e l d and t e m p e r a t u r e a r e g i v e n by
ng(x) : J~oo P~f(x, p) =n(x) u~(x)
f ( x , p) = F c e x p [ - u g t ( x ) p g / k T ( x ) ]
3 3 ( x2)1/2
'
i s c a l c u l a t e d to be
(3)
8~F D nD(x=O,t)-c2([3Dt) 3"
n o ( x ) = 8~ FD(kT(x))3; The energy-momentum d3p
t e n s o r T P V(x), d e f i n e d by
_
= (~ + k / c 2 ) u g u V_ggVD
,
w h e r e h(x): p r o p e r e n e r g y d e n s i t y , p(x): p r o p e r p r e s s u r e and 3/~ = ~ c 2 f o r v a n i s h i n g r e s t m a s s e s , i s d e t e r m i n e d by
])D(X) = 8~c 2 F D (kT (x)) 4
(4)
c 1 ug;u ~ Ovug(x) - (X2)1/2 (ggtv - ~ U l j . Uv)
0 ~ 0
,
~p ~
(p
,
(5)
, ~"
Fc, ~c : c o n s t .
(8)
,
~c(X ) = 8 ~ c 2 F c ( k T ) 4 !9)
D i f f e r e n t i a t i o n with r e s p e c t to x v g i v e s
u~z;v(x)__ 1 i* u v + 2 x ' b ( g -wl c2 ~ x2 ~v - c~ UgUv)
hn = u~;vuV : ; ( x "
b u~ - cxg)
,
(10)
p = r / x 2,
0-~ O,
~
~,
(11)
l e a d s to c o m o v i n g c o o r d i n a t e f r a m e s :
u~-*~
:c(T2 -o2)go ~
,
(12)
ds 2 - 1 [d1_2 _ do 2 _ p2(d 0 2 + sin2O dq~2)] (~-2 _p2) 2 (6)
= (X2) 1/2
,
ds 2 = d~-2 _ 1-2[dw 2 + s i n h 2 w ( d 0 2 + sin20 d(p2)] In t h e c o m o v i n g s y s t e m we h a x e ~ z = cg~o. We s e e that the s y s t e m so o b t a i n e d i s j u s t M i l n e ' s u n i v e r s e [9], e x t e n s i v e l y d i s c u s s e d in the l i t e r a t u r e [11]. In o u r c a s e we o b t a i n it f r o m t h e 522
,
,
P r o p e r d e n s i t y n c (x) and p r o p e r p r e s s u r e / ~ c (x) a r e , c a l c u l a t e d to be
T = x O / x 2,
and the e x P a n s i o n v e l o c i t y 0 h a s the v a l u e 3 c / ( x 2 ) 1/2. T h e a c c e l e r a t i o n ut~ = ul~;vuV vanishes. T r a n s f o r m a t i o n to c o m o v i n g c o o r d i n a t e s i s o b t a i n e d by the s u b s t i t u t i o n r = Tsinhw
2)
i . e . in t h i s c a s e the a c c e l e r a t i o n d o e s not vanish'. C h o o s i n g b = (1,0,0,0), we h a v e u ~ = = - c x 2 0 g ( x O / x 2 ) and the i n v e r s i o n
B e c a u s e u g (x) = c Og(x2) 1/2, the f l o w a s s o c i a t e d with the v e l o c i t y f i e l d i s h y p e r s u r f a c e o r t h o g o n a l . We f u r t h e r h a v e
,
kT(x) : C/~cX2
nc(X ) = 8 u F c ( k T ) 3
T gzv(x ) = J ~ - o p~" PVf(x, P) =
x° = Tcoshw
u~(x) = -cx2~(b.x/x
,
We s e e f r o m eq. (12) that the s y s t e m o b t a i n e d i s i s o t r o p i c , but not h o m o g e n e o u s , i . e . it i s not a F r i e d m a n u n i v e r s e with a R o b e r t s o n - W a l k e r metric L e t us d i s c u s s a s i m p l e but i m p o r t a n t c o n s e q u e n c e : Suppose t h e r e i s an o b s e r v e r at (t = to, x =0) ~ (*o = l / t o , P =0). S u p p o s e f u r t h e r that light w a s e m i t t e d at ( t , x ) ~ (1-, p) w i t h w a v e l e n g t h X, a r r i v i n g at x = 0 at t i m e t o = t + r / c .
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PHYSICS
T h e n the r e d shift z = (;t o - ;t)/;t is c a l c u l a t e d to be 1
p =~T oz
1
,
~- = ~ T o ( z +
2)
,
(13)
o r , in t e r m s of the M i n k o w s k i c o o r d i n a t e s r , t and to: r = ~1c i ~ t z
o
t - ~ +I 2 + zz to
,
(14)
F o r z < < l the f i r s t one of the e q s . (14) r e d u c e s to H u b b l e ' s l a w r = ( c / H o ) z , H o = 2/1 o. F o r z ~ 1 , however, Hubble's law would give distances l a r g e r than the e x a c t r e l a t i o n (14) does'. B e c a u s e of 1,/x 2 = (1 + z ) / ' c 2 t 2
,
(15)
which f o l l o w s f r o m eq. (14), we c a n , in p r i n c i p l e , d e t e r m i n e the t e m p e r a t u r e T(x), the d e n s i t y n c (x), the p r e s s u r e p c (x) and the e n e r g y d e n s i t y (x) by r e d s h i f t m e a s u r e m e n t s a l o n e , if we know t o and the c o n s t a n t s of F c and tic (which m a y be g o t t e n by l o c a l m e a s u r e m e n t s ' . ) ' . F o r a b l a c k body g a s with two i n t e r n a l d e g r e e s of f r e e d o m we h a v e F c = 2/(27r~)3. 2. A s t r o p h y s i c a l s p e c u l a t i o n s . T h e two e x p a n d i n g s y s t e m s of u l t r a r e l a t i v i s t i c p a r t i c l e s d e s c r i b e d a b o v e s u g g e s t v e r y m u c h to c o m p a r e t h e m with o u r e x p a n d i n g u n i v e r s e . A s the f e a t u r e s of the f i r s t one ( a s s o c i a t e d with d i l a t a t i o n s ) a r e w e l l - k n o w n , let us c o n f i n e the d i s c u s s i o n to the s e c o n d one: 2.1. L o c a l l y , i . e . f o r r 2 << c 2 t 2, the s y s t e m is i s o t r o p i c a n d h o m o g e n e o u s and H u b b l e ' s l a w is v a l i d a s m e n t i o n e d a b o v e . T h e l o c a l R o b e r t s o n W a l k e r m e t r i c has an e x p a n s i o n f u n c t i o n R = = Rlt2 , R 1 = const. This function is not a solut i o n of F r i e d m a n ' s e q u a t i o n [11], but c a n be shown to be a s o l u t i o n of the c o r r e s p o n d i n g B r a n s - D i c k e q u a t i o n [12] with ¢o - -~3 ( s e e r e f . [13]), v a n i s h i n g s p a t i a l c u r v a t u r e and the Brans-Dicke scalar
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27 December 1971
o t h e r hand, b e c a u s e we h a v e ~ ( r , t ) cc (1 + z) 4 f o r the i n v a r i a n t e n e r g y d e n s i t y , the e n o r m o u s e n e r g i e s and the d e v i a t i o n s f r o m u n i f o r m d i s t r i bution [14] a s s o c i a t e d with q u a s a r s m a y h a v e s o m e t h i n g to do with o u r m o d e l and the p h y s i c a l m e c h a n i s m s which p r o d u c e it'. We a r e not s u g g e s t i n g that the p r o p e r t i e s of q u a s a r s can be u n d e r s t o o d c o m p l e t e l y in the f r a m e w o r k of o u r m o d e l , n e v e r t h e l e s s t h i s f i e l d s e e m s to be an interesting candidate for further investigations. 2.4. The m o s t i m p o r t a n t p h y s i c a l p r o b l e m in the c o n t e x t of o u r c o n s i d e r a t i o n s is the q u e s t i o n : which s p e c i f i c d y n a m i c a l m e c h a n i s m s c o u l d be a b l e to g i v e r i s e , in a r e a s o n a b l e a p p r o x i m a t i o n , to the type of d i s t r i b u t i o n f u n c t i o n s we h a v e b e e n d i s c u s s i n g . In o t h e r w o r d s , a r e t h e r e high e n e r gy p r o c e s s e s which a r e a p p r o x i m a t e l y d i l a t a t i o n o r (and) c o n f o r m a l l y i n v a r i a n t and l e a d to the a b o v e s y s t e m s ? P r e s e n t l y we a r e not a b l e to give a clearcut answer. However, there are s e v e r a l t h e o r e t i c a l and e x p e r i m e n t a l hints which s u g g e s t s that one should t a k e the a b o v e s t a t i s t i c a l s y s t e m s s e r i o u s l y . O u r p r o c e d u r e in t h i s c o n t e x t is the f o l l o w i n g : we l i s t s e v e r a l i m p o r t a n t r e a c t i o n s with c r o s s s e c t i o n s of a f o r m w h i c h c a n be r e l a t e d to an e f f e c t i v e d i l a t a t i o n and (or) c o n f o r m a l l y i n v a r i a n t ' p o t e n t i a l ' l i k e , e.g. the C o u l o m b p o t e n t i a l , o r a B o r n a p p r o x i m a t i o n in k i n e m a t i c a l r e g i o n s w h e r e all r e s t m a s s e s a r e n e g l i g i b l e and w h e r e we c a n h a v e a p p r o x i m a t e c o n s e r v a t i o n of d i l a t a t i o n and (or) c o n f o r m a l momenta: 2.4.1. P u r e l y e l e c t r o m a g n e t i c i n t e r a c t i o n s in lowest order: a) C o m p t o n s c a t t e r i n g off e l e c t r o n s f o r s , It I>> m2e: dcr
27r~ 2 f..ss + ~ ' ~
dlt I-
s2
ltl
sJ
'
_
1/~(t) : (1/¢1)t6
,
(Ol : 287r2Fc/9C9f14
2.2. W h e r e a s the E i n s t e i n - D e S i t t e r m o d e l [11] of the u n i v e r s e (which t a k e s g r a v i t a t i o n into a c count) h a s a t i m e d e c r e a s e of the d e n s i t y n ( x ) g i v e n by n ( x ) t 2 = c o n s t . , a c c o r d i n g to eqs. (8) and (9) the d e n s i t i e s in o u r m o d e l d e c r e a s e m u c h f a s t e r , o v e r t a k i n g the g r a v i t a t i o n a l d e c r e a s e . T h i s m a y be one r e a s o n f o r the m o d e l b e i n g of some relevance'. 2.3. F o r r e d s h i f t s of the o r d e r 1, the m o d e l i s no l o n g e r h o m o g e n e o u s and H u b b l e ' s law i s m o d i f i e d . If the m o d i f i e d v e r s i o n (14) h a s a n y t h i n g to do with, e.g. q u a s a r s , t h e y w o u l d not be so f a r away a s H u b b l e ' s law s u g g e s t s and, on t h e
s:
(Pc +PT) 2
,
b) E l e c t r o n - e l e c t r o n
t:
(pc-Pc) 2
s c a t t e r i n g f o r s, It] >>m2:
d~ 2vc~ 2 d l t l - t 2 s 2 (t 2 + 2 s t + 2 s 2) Higher order will introduce 'non-scaling' correct i o n s , but we h a v e at l e a s t an a p p r o x i m a t e scaling. 2.4.2. I n c l u s i v e s e m i h a d r o n i c p r o c e s s e s . T h e p r o c e s s e + N ~ N p l u s anything d o e s not only s h o w s c a l i n g of the s t r u c t u r e f u n c t i o n s [2, 3] but the t o t a l d i f f e r e n t i a l c r o s s s e c t i o n i s C o u l o m b l i k e [15, 16]: d ~ / d [ t I ~ 4~ o l 2 / t 2
This process
- at v e r y l a r g e i n i t i a l e n e r g i e s
523
Volume 37B, n u m b e r 5
PHYSICS
may be looked at as an effective binary electron s c a t t e r i n g off a C o u l o m b p o t e n t i a l . H o w e v e r , i n s u c h a c a s e we h a v e c o n s e r v a t i o n of t h e d i l a t a t i o n m o m e n t u m [1] a n d t h e t i m e - c o m p o n e n t of t h e e o n f o r m a l m o m e n t a *'. T h u s we do h a v e i n t h i s c a s e a k i n d of ' q u a s i ' - b i n a r y c o l l i s i o n we need for establishing the distribution function in q u e s t i o n . In w e a k i n t e r a c t i o n s we c a n a p p l y t h e s a m e r e a s o n i n g o n l y [17] if t h e r e i s a n i n t e r m e d i a t e b o s o n W a n d if I t I >> m 2w. 2.4.3. P u r e l y h a d r o n i c i n t e r a c t i o n s . In i n c l u sive hadronic reactions a + b ~ c + anything ' F e y n m a n ' s e a l i n g [4] i s o b s e r v e d f o r s ~ ~o: T h e production cross section for the particle c with l o n g i t u d i n a l c . m . m o m e n t u m PH a n d t r a n s v e r s m o m e n t u m p± i s s u p p o s e d to b e of t h e f o r m
d2~/p± dPxdY =f ( p x , y ),
y : 2pH /'~s ,
w h i c h i s s u p p o r t e d by e x p e r i m e n t s
[4]. E i t h e r if
f ( p ~ , y) f a c t o r i z e s [4]: f ( P l , Y) = f l ( P ± ) f 2 ( Y ) o r if we i n t e g r a t e o v e r p±, we g e t
dcr/dy : g (y). This production cross section may be interpreted i n t h e f o l l o w i n g way: A c r o s s s e c t i o n i n o n e s p a c e d i m e n s i o n , d e f i n e d by t h e d e c r e a s e Aj = -~jn (x3)Ax 3 of t h e c u r r e n t d e n s i t y j a l o n g A x 3 i s d i m e n s i o n l e s s . T h i s m e a n s : if t h e d y n a m i c s do not c o n t a i n a n y f i x e d l e n g t h s l i k e r e s t m a s s e s or coupling constants with non-vanishing dimensions etc., then the production cross section for one particle in the final state, after summing over all other secondaries, can only be a funct i o n of t h e r a t i o y . In t h i s s e n s e p u r e l y h a d r o n i e interactions exhibit one-dimensional (longitudinal) d i l a t a t i o n i n v a r i a n c e [1,18]'. 2.5. It i s not c l e a r , w h e t h e r t h e h i g h e n e r g y r e a c t i o n s j u s t l i s t e d a c t u a l l y d i d l e a d to d i s t r i b u t i o n f u n c t i o n s in a s t r o p h y s i c s a s s u g g e s t e d a b o v e , because the physical situation is so complex. T h e y o r o t h e r s mighl h a v e , b u t o b v i o u s l y f u r t h e r s t u d i e s a n d t h e i n c l u s i o n of g r a v i t a t i o n a r e necessary'. I a m v e r y m u c h i n d e b t e d to J. E h l e r s f o r a c r i t i c a l r e a d i n g of a f i r s t v e r s i o n of t h i s p a p e r and for very helpful discussions about its statist i c a l e l e m e n t s . P a r t s of t h i s w o r k w e r e d o n e during a stay as Visiting Professor at the Univers i t y of H a m b u r g i n t h e s u m m e r 1971 a n d d u r i n g a v i s i t at C E R N i n f a l l 1971. I a m v e r y g r a t e f u l * The conformal momentum here has the form h ° = 2 x ° ( x p ) - p ° x 2 - 2 g r , w h e r e g r -1 is the Coulomb potential.
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to m a n y c o l l e a g u e s f o r d i s c u s s i o n s a n d f o r t h e v e r y k i n d h o s p i t a l i t i e s of t h e U n i v e r s i t y of H a m b u r g , of DESY a n d of t h e T h e o r y D i v i s i o n at CERN.
References [1] H.A. Kastrup, Phys. Lett. 4 (1963) 56; Nucl. Phys. 58 (1964) 561; H a b i l i t a t i o n T h e s i s , Munich, 1964. [2] J.D. Bjorken, Phys. Rev. 179 (1969) 1547; R . P . Feynman, Phys. Rev. Lett. 23 (1969) 1415; K.G. Wilson, Phys. Rev. 179 (1969) 1499; S. Ciccariello, R. Gatto, G. Sartori and M. Tonin, Ann. Phys. (N.Y.)65 (1971) 265; D. G r o s s a n d J . Wess, Phys. Rev. D2 (1970) 753; Y. F r i s h m a n , Ann. Phys. (N.Y.) 66 (1971) 373; R. Brandt and G. P r e p a r a t a , Nucl. Phys. B 27 (1971) 541 ; H. F r i t s c h and M. Gell-Mann, Contribution to the Coral Gable Conf. 1971. [3] E.D. Bloom et al., Phys. Rev. Lett. 23 (1969) 930. [4] F o r a review see: D. Horn. Many-particle production, P h y s i c s Reports, to be published. [5] Of specific significance is the following paper: G. Mack and I. Todorov, Conformal invariant Green function without ultraviolet d i v e r g e n c e s , to be .published ; see also: A. A. Migda[, Conformal invariance and bootstrap, Landau Institute (Moscow), P r e p r i n t 1971. [6] G. Mack, T r i e s t e P r e p r i n t IC/70/95, Nucl. Phys., to be published. [7] H.A. Kastrup, Phys. Rev. 150 (1966) 1183. [8] J. Ehlers, in General relativity and cosmology, Proc. Intern. School of P h y s i c s ' E n r i c o F e r m i ' , Course 47 (Academic P r e s s , 1971); J . M . Stuart, Lecture Notes in Physics, Vo[.1O (Springer-Verlag, Berlin, Heidelberg, N.Y. 1971). [9] E. A. Milne, Kinematic relativity (Oxford University P r e s s , 1948). [10] J. Ehlers, P. Geren and R.K. Sachs, J . M a t h . Phys. 9 (1968) 1344; G. E. T a u b e r and J. W. Weinberg, Phys. Rev. 122 (1961) 1342. [11] H. P. Robertson and T. W. Noonen, Relativity and cosmology (W.B. Saunders Comp., Philadelphia, London, Toronto, 1968); W. Rindler, E s s e n t i a l relativity (Van Nostrand Reinhold Comp., N.Y., 1969). [12] C. B r a n s and R.H. Dicke, Phys. Rev. 124 (1961) 925. [13] J. L. Anderson, Phys. Rev. D3 (1971) 1689. [14] M. Schmidt, Astrophys. J. 162 (1970) 371. [15] J . D . Bjorken, Proc. 1967 Intern. School of P h y s i c s ' E n r i c o F e r m i ' (Academic P r e s s , N.Y., London, 1968). [16] H.A. Kastrup, Phys. Rev. 147 (1966) 1130; Schladming L e c t u r e s in: P a r t i c l e s , c u r r e n t s , symm e t r i e s , ed. P. Urban (Springer-Verlag, Wien, N.Y., 1968) p. 407. [17] H.A. Kastrup, Nuovo Cimento 48 A (1967) 271. [18] M. Bandor, Phys. Rev. D3 (1971) 3173.