Centre Siciliano di Fisica iVucleare e di Struttura della Materia - Catania ... Recently a ehronotopical (quadrivcctor) co-ordinate operator has been proposed ..... --if we call interaction duration the time spent by the wave packet center (4) inside.
LETT~R~ AL NU0VO CI~NTO
COL. IV, N. 24
12 D i e e m b r e 1970
A b o u t a S p a c e - T i m e Operator in Collision D e s c r i p t i o n s . V. S. 0LKHO~rSKY
K i e v State University - K i e v
E. RECAMI (*) Institute ]or Theoretical Physics, U k r a i n i a n Academ 9 el Sciences - K i e v Istituto di Fisica Teoriea dell' Universith - Catania Istituto £Vazionale di Fisiea Nucleate - Sezione di Catania Centre Siciliano di Fisica iVucleare e di Struttura della Materia - Catania (ricevuto il 28 0 t t o b r e 1970)
1.
-
Introduction.
R e c e n t l y a e h r o n o t o p i c a l ( q u a d r i v c c t o r ) c o - o r d i n a t e o p e r a t o r has b e e n p r o p o s e d for r e l a t i v i s t i c p a r t i c l e s (1), w h o s e H e r m i t i a n ( L o r e n t z - e o v a r i a n t ) p a r t is
(1)
^
i
_o__
~ . ~ (x, - - ~) = 2 ~ p . ;
let us n o t i c e t h a t a ~ r a d i a n c e ~ o p 6 r a t o r e n t e r s in (1). T h e m e t r i c e h o o s e n is ( + - - - - - - ) . Since PAUL1 (3) e x p r e s s e d his o p i n i o n s a g a i n s t H e r m i t i a n t i m e o p e r a t o r s a c t i n g o n t o H i l b e r t s p a c e s ( w h i t h i n nonrelativistic theories), m a n y discussions h a v e b e e n d e v e l o p e d a b o u t p o s i t i o n a n d p a r t i c u l a r l y t i m e o p e r a t o r s , also in t h e r e l a t i v i s t i c f r a m e w o r k a n d also in r e c e n t t i m e s (a). W e shall here confine ourselves only t o analising explicitly if a n d h o w t h e o p e r a t o r s (1) w o r k w i t h r e g a r d t o r e l a t i v i s t i c spin-zero free particles. Clearly our h e u r i s t i c aim, for t h e m o m e n t , claims b y no m e a n s to c o m p l e t e n e s s or to ]ull m a t h e m a t i c a l rigour.
(') On leave of absence from the Istjtuto di Fisica dell'Universitk, Catania, under an exchange supported by the Istituto Nazionaie di Fisiea Nucleare, Sezione di Torino, and the Ukrainian Academy of Sciences, Kiev. (1) See, e.g., M. BAL~)O and E. R E c ~ I : LeiL Nuovo Cimento, 2, 643 (1969), and references therein, (J) W. PAULI: HandSook der Physik, Vol. 24/1, p. 1~3. (') See A. A. BROyLES: PAys. ~ v . D, t, 979 (1970), and references therein; particularly A. O. BARUT and S. M~t,IN: Rev. Meal. Phys., 40, 632 (1968); D. ~ROSENBA~JM:Journ..¢[agh. Phlts., 10, 1127 (1969); M, RAZ~.VT: NUOVO Cimento, 63 B, 271 (1969).
program
1165
1166
v.s.
OLKtIOVSKY and E. RECAMI
We base ourselves on previous works dealing with plane-wave packets (~). N a t u r a l units are generally adopted. W e w a n t at this point to underline the canonical correspondence existing for a free Klein-Gordon (K-G) particle (in the configuration and quadrimpulse spaces, respectively) between the probability-density (, tetraflux operator ,) and the c h r o n o t o p i c a l opera tor: i c
(21~>
,,,03
1; =
i
i ~ ?~V '
c'
i
(2,~)
~-~e'
;' . . . .
where we have also indicated the proportio~mlity between the o p e r a t o r s tetraflux and t e t r a m o m c n t u m (*). On the other fiend, the co-ordinate o p e r a t o r s (2e), (2d) m a y be re,Qardcd as energy-impulse density operators. Let us r e m e m b e r (~t t h a t (2d) is nothin~ but the N e w t o n - W i g n e r position o p e r a t o r (in the timelike ease). Within our confined work. w(~ shall take into account the a n t i - H e r m i t i a n operators (containing symbols (~) ~ instead of ~), which m a y be added to the H e r m i t i a n operators (2), only briefly, in S~,ct. 3. In fact, on a ]ormaI ground, their m e a n i n g - - p a r t i c u l a r l y in the K-G ease---seems to us not wholly clarified. In any ease see since now also ref. (x,s). Notice explicitly that the eonsid¢:red tetracectors, as well known, are respectively ( x , - - t ) and ( J , - - o ) •
2. - Practical behaviour of the chronotopical
operator.
Let us now i n t r o d u c e the positive-energy p l a n e - w a v e packet for a free K-G particle, len~ before or lorez after an interaction (~):
=]+
2P0 G(p) exp [ i ( p . x - - p o t ) j ,
where the squ~re-integrable weight function G(p) m a y depend also on the scattering m a t r i x (besides on the initial and, eventually, final e x p e r i m e n t a l conditions). F o r m u l a (3) can be i m m e d i a t e l y generalized for the m a n y - p a r t i c l e ease• We can clearly define the mean values of the chronotopical co-ordinate operators (in the space-time representation), averaged over any a r b i t r a r y w a v e packet, as follows: (4)
^ of ['xo d~x x ) ~ ~ "f:d3 x .
(~>~ d~ fto dt dS fO d t d S '
(') V . . q • OLKHOVSRr trod E• I~I.:CA.MI: N ~ o r o Cim, ento, 5 3 A , 610 (I968); 6 3 A , 814 (1969); E, RIgO_~I[: l~e~dic..lccad. N a z . L i n t e l (to a p p e a r ) . See ~dso: V. S. OLKHOV,qtEY: .Ygo'00 Cimenlo, 48 13, 178 (1967). (*) See Sect. 3, (s) See: A. J. KXLNAY et aL: Boletin deI f J i , . i F (C6rdoba), 2, 41 (1966); JVuovo Cimento, 4 8 A , .393 (1967); P h y s . R e v . , i 5 8 , 1484 (1967); P h y s . ltev. D, i, 1092 ( 1 9 7 0 ) ; p r e p r i n t 1C/69/134 (Trieste, 1969); I n l e t . J o u r n . Theor. P h y s . , Letters (to ~ppear, 1970).
ABOUT
A SPACE-TIME
OPERATOR
IN
COLLISION
DESCRIPTIONS
1167
dS being the element of a wave p a c k e t (~t r a n s v e r s a l , section. A n d (by using Fourier transformations, one integration by parts and delta-function properties, i.e. by following the same technique as in ref. (4)) we obtain the same results as by making our operators (1)
to act into momentum space:
(5a)
X"
=
^
(5b)
i y(d3p/2po) ~*(e/~Po) ~
_ 2
,
where (*) (z being the direction of the wave packet average velocity)
(6)
Jc(t, p) ~ G(p) exp:[--ipo t],
~(Po, x) ~ v-[1/2 G(p) exp [/p~z] •
F o r obtaining eqs. (5a), (5b), while integrating b y parts we supposed the weight functions G(p) to be zero at the integration limits (besides to have there derivative zero, for avoiding singularities in using the (( radiance ,~ operators). co
co
If we preferred to define of e i g e n f u n e t i o n s ( p l a n e w a v e in t h e i m p u l s e s p a c e , a n d t h e c o m p l e x c o n j u g a t e one) c x p [5= i x ' p ] . L a s t l y , let us recall (1) in p a r t i c u l a r t h a t for tile ( t t e r m i t i a n ) t i m e o p e r a t o r (**)
(I bis)
"~ 2 ~P0
(*) see sect. 3. (**) Jus~ as for ~ pasition % no time ix sometimes a vt~riablc parameter and sometimes is to be am~si(lered a-a operalor. Besides ~nd in parflieular, ( i s p o s i t i o n measuremeTlls a.rc performed (' in time % so time measurements (by a ('lock) are always dependent Oll ~ position variable.
ABOUT A SPACE-TIME OPERATOR IN COLLISION D~SCRIPTIONS
1169
we have (11)
[?, E ] = -
ih,
so t h a t w i t h i n the limits of our w o r k - - a rigorous formulation m a y be i m m e d i a t e l y given to the u n c e r t a i n t y correlation: h At.AE> ~.
(12)
3. -
A n t i - H e r m i t an p~rts a n l
interpretation.
If we consider also the a n t i - H e r m i t i a n (i) operator Y~H a n d define 1 ~+
1^
(13) we could conversely obtain, e.g.,
(14)
f(d'j,/2po). (v/2po)I P (Y>v'~ ~ \ 2 ~ p / ~ . ,
I(dZp/2po) [~-(p)]2
=
'
i n accordance with ref. (6). I n this reference, n o n n o r m a l operators z were suggested for a n extended.type position, h a v i n g reyions as eigenvalues. A n d the m e a n values (for a considered state) of the (~a n t i - H e r m i t i a n ~>p a r t (~) of ~ were interpreted as forwarding the dimensions of the region. The philosophy is t h a t relativistic particles could n o t correctly suffer p u n c t u a l localization, owing for instance to pair creation (*). If, consequently, we associate (~) to every (relativistic) free particle, with proper m a s s m o a n d m o m e n t u m p , the characteristic q u a n t i t y ~x -------(~x, ~t) : (15)
/~ v ~x ~ - - - - , c 21Oo
~t
8x
//
1
v
c 2pc'
we might, into eq. (14), before all rewrite (6) j u s t
(16)
1 V
-'--
~- 8x ,
"" ~ . v "
2Pc
(e) j . A. GALLARDO, A. J. ]~kLI~AY, B. A. STEC a n d B. P. TOLEDO: IVuOVOCi~ento, 49 A, 393 (1967); A. J. K&LNAY: Boletln del I M A F (C6rdoba), 2, 41 (1966). (*) JAUCH (7) has also s h o w n t h a t p u n c t u a l localizability w o u l d be i n c o n t r a s t w i t h ~unim o d u l a r i t y ~ (s). -4 p r / o r i a n i n t r i n s i c n o n l o c a l i z a b i l i t y m i g h t be realized t h r o n g h t t h e use e i t h e r of n o n n o r m a l o p e r a t o r s , w i t h c o m m u t a t i n g oompononts, or of n o r m a l operators, w i t h n o n c o m m u t i n g c o m p o n e n t s (~). (7) j . lV[. JAUCH: Fundations o] Quantam 3lerhanics (London, 1968). (s) M. BALDO: p r i v a t e c o m m u n i c a t i o n .
1170
v. s, OLKIKOVSKY and z. RECAMI
Besides, it m a y be noticed that, in eorrispondence with the known relation E . , o ( p ) = ~/c2p "- -_ c~'m~0, wc might formally h a v e
(17)
t,(x)
~ + ~:~. tSt)~
with x " - u '(~x) 2 and k n - - n . (2m0/m) -°. 0 b v i o u s l s , if S t y 0 , t h e n q*. (~t)2~x/v 2. If one takes thesc consideration seriously, it is possible to observe that, when applying, e.9., the e x t c a d c d - t y p c position openttor ~ to represcnt formally the e x p e r i m e n t a l determination of the (~localization region ,) of a relativistic p~rticle, one would h a v e in general
(Is) F o r the position m a y be tian (x)
= - - 2 n i ( ~ ( E - - E o )
exp [ i ( E - - Eo) t] E + ~,7--Eo is the total scattering-state for our case:
['t'"+'> =
Li> +
1
E.+ + T
"'
(*) See, e.g., V. S. 0LKHOVSKY a n d E. RECAMI: p r e p r i n t I T P / 7 0 , K i e v (to a p p e a r ) ; A. A~oI)I, ,~¢I. BALDO a n d E. RECAMI: p r e p r i n t I T P , K i e v (to a p p e a r ) . (*) T h e f u n c t i o n s (25) a r e obviously corresponding to t h e * a n t i - H e r m i t i a n ~) p a r t s of t h e t o t a l ( n o n n o r m a l ) * t e t r a f l u x * operator. (le) M. L. GOLDBERGER a n d K . M. WATSON: Collision Theory (New Y o r k , 1963). (~t) V. S. OLKHOVSKY: C h a p t e r 9 i n : ,dome Problems i n F a s t - N e u t r o n P h y s i c s , e d i t e d b y V. I . STRIZHAK ( K i e v , 1970), in Russian.
1172
V. S. O L K H O V S K Y
altd
E. R E C A 3 [ I
The states !i>, !/> are initial and final plane waves, respectively. The operators V a n d T ar(~ the potential and the kinetic enel~y. The quantities E 0, E arc respectively the (kinetic) energies associated with the initial and final plane waves. For instane(', let us analyse (without limiting the generality) the scattering of one wave packet by a ~(potcntial,). The final wave packet m a y be written (~.~x) (p2 ~ 2/,1"o, q2 : 2/lE)
(27)
(2.~)-@~paqa(p).exp[i(q.r-- Et)]
~m(t t h ( ' l t
yb(r, t) = V',(r~, t~) + A,/,1(r, t) ,
(2s)
wl~¢,rc lh(, asymptotic part is
,f
y,,(ro~, l~)
__
i %/2.~
(.)9)
7,' dq~ dplI dp, dq±dE o G{p) -c,xp [i(q .r --/~o t)]- ( ~ - d£o- T,, ,
-
and the remaining part (which, of course, wmislles asymptotically) is
_~,,,,
exp [iq' .r]
f
~t dqi~E'+ iu--Eo
z ~,
w]le~
z ~
z' ,
ABOUT
A SPACE-TIME
OPERATOR
IN COLLISION
DESCRIPTIONS
1173
where z' varies within the potential region (andz (~within ~>the final wave p a c k e t here considered). By using results (31), expression (30) becomes
(32)
AV/=/~
1
A. d-~°-~exp E "
[iq.r]'
