On the reduction formulae for the S-matrix elements

Nuclear Physzcs 1 S (1960) 6 5 7 - - 6 6 3 , ( ~ North-Holland Publishing Co, .4 msterdam Not to be reproduced by photoprmt or tmca'ofdm without written pernasstoa from the pubhsher

ON THE REDUCTION FORMULAE FOR THE S-MATRIX ELEMENTS j. HILGEVOORD

Inst, tute [or Theoretical Phys$cs, Umvers,ly o[ Amsterdam R e c e i v e d 18 N o v e m b e r 1959 R e d u c t i o n f o r m u l a e for t h e S - m a t r i x e l e m e n t s , g i v e n b y L e h m a n n , S y m a n z l k a n d Z l m m e r m a n n , a r e d e r i v e d w i t h p a r t m u l a r e m p h a s t s o n a p r o p e r u s e of t h e a s y m t o t m conchtlon a n d o n t h e o r d e r of m t e g r a t l o n w h i c h xs n o t a l w a y s i r r e l e v a n t . I t xs also s t r e s s e d t h a t local c o m m u t a t l v l t y is n o t n e e d e d for t h e derxvatton of m o s t of t h e s e formulae.

Abstract:

1. Introduction Some years ago, Lehmann, Symanzik and Zlmmermanrl 1) have given so-called reduction formulae, m which a general element of the S-matrix of field theory is expressed in terms of the interacting operator field. In their derivation, they make only use of the general postulates for a relativistm quantum field theory and of an asymptotic condition which relates the field asymptotically to the free fields describing the ingoing and outgoing particles. The interaction need not be specified. Such formulae have been the starting point for a rigorous derivation of dispersion relations, which m a y be considered as special consequences, for the S-matrix elements, of the postulate of local commutativity. This postulate states that any two field operators, taken at points x and y such that (x--y) is space-hke, commute (or anticommute) The reduction formulae are not unique. In fact, several forms are possible, which in some cases seems almost paradoxical. It is the purpose of this note to derive, for the special case of the S-matrix element between two-partmle states, some alternative forms, in a httle more careful manner than usual, and to make clear that they are not contradictory. It is also stressed that the postulate of local commutativity is not needed for a derivation of the reduction formulae of L S.Z. In a recent paper, F. Kaschluhn 2) has raised some doubts with respect to this point He bases himself on a comparison of L S Z.'s and of Bogoliubov's formulation of the theory. We shall show that the expression for the S-matrix element, used b y him, is not equivalent to one of the forms of L.S.Z. and can be derived from them only if local commutatlvity is assumed. 657

(~58

J

HILGEVOORD

2. D e r i v a t i o n of R e d u c t i o n F o r m u l a e

For convenience, we summarize the formulation of a relativistic quantum theory of a real, scalar field, given by Lehmann, Symanzik and Zimmermann 1) as far as is needed for the following. Two fields, Ain(x) and Aout(x) are introduced, satisfying the free field equations = o,

= 0,

(1)

and the commutation relations [.4 m (x), A m (y)]_-~ iA(x--y). out

out

These fields describe the asymptotically free mgomg and outgoing particles. We assume that no bound states exist. Let/~(x) represent a complete set of positive frequency solutions of the Klein-Gordon equation (1), orthonormal with respect to the Klein-Gordon in-product

'

Ot

Ot [p dsx = ~p"

and

Aout ---~(/~, Aout(X)) ,

Then the operators Am = (/~, Ain(X))

(2)

which on account of equation (1) do not depend on time, create lngoing and outgoing particles, respectively, in the state/~. With the help of these operators, two complete sets of states in our Hflbert space m a y be constructed: ~m~'"a"~-~ A ~ ' ' "

attt Ain[0),

~1 " 0~t ~1 • all ~)out : Aout " " Aout[0)"

The scattering operator is defined as the unitary operator that connects these two sets __

%

or also

Aout ~- StAinS. The dynamical aspect of the theory is now introduced by the assumption that there exists an invariant field A (x), called the interpolating field, which connects the in- and outgoing fields according to the following asymptotic conditions:

lira (#[A~(t)[~) ----(~b[Aout]kg) $---~+co

for all pairs of normalized states ~b and ~. The field A~(t) is defined analogously to (2), and will in general depend on

ON

THE

REDUCTION

FORMULAE

FOR

THE

S-MATRIX

ELEMENTS

(]59

time. Here we obviously assume that the operators A'(t) are defined, i.e. that they lead to states with a fimte norm, which is not required of the operators A (x). It is essential to have normalized states when applying the asymptotic condition (3). Lehmann, Symanzik and Zimmermann now show how one can derive from these assumptions so-called reduction formulae, b y whmh the elements of the S-matrix are expressed in terms of the interpolating field. We here derive again these formulae, in the special case of the S-matrix elements between two incoming states, each containing two particles, in a slightly more careful manner, making only use of the asymptotic condition between states with finite norm. We shall fred that several forms of the final result are possible, and that one has to be careful when interchanging orders of integration, in order to avoid paradoxical results Consider the S-matrix element m

.

s

(4)

=

Accordmfi to the asymptotic condition we m a y write this out I

m

/

~

\

out

out

m~

=

m/

lim

Jim

f z---~--oo

fw--~+oo

,

(5)

where the order in whmh the hmlts are taken is irrelevant. Now, introduce two kinds of time-ordered products

T±A (x)A (y) = O(x--y)A (x)A (y) 4-O(y--x)A (y)A (x), where 0 (x) = I if t > 0, and -----0 if t < 0. For equal times we suppose T+ to be defined in some way, which we do not specify here because it will turn out to be not important for our present purpose Expression (5) m a y then be replaced b y hm (q~ao~t[T~:AP**(t,)A~,(t®)[~)

lim ~d--~--oo

(6)

fv--~-I-oo

where both kinds of T-product are allowed. Now, due to the time-ordering

(~o~tJT±AP~*(t~)A~'(t®)[q~n*) = O,

hm ftd--~. --O0

if (/al,/-.) = 0, e.g. From now on, we suppose that both/=1 and/~, are orthogonal with respect to b o t h / a l a n d / p , . Thus, (6) m a y be written lim

~°dt * ~*,(q~°~t[T±Aa** (t*)A=*(t*)Jq~)" I-oo

fm'-"~ - - 0 0 '

On the other hand, b y performing the steps in the opposite direction, hm

I dt, ~,,(~o~t[T±A P** (t,)A~t(I,)[O~n* )

f z . - . ~ +o o -

=

hm fd--~+oo

hm (q~o~tIAal*(t,)A~'l(t,)lq~ff) ~---*+co

= X/q~Pxatt#~l~,\ : 0 out I out /

660

j HILGEVOORD

Here we used the fact that for one-particle states # ~ = # o u t , and the orthogonality of the in- and out-states as assumed above. We get

(~&a,,~,'~ out I m /

---

_ fdt®O~,f dt~O,,(~o~tlT±A&.(t~)A~(t~)[~).

(7)

I n this expression, the order in which the integrations are performed is irrelevant; the origin of this lies in the fact that in (6) the hmits m a y be interchanged, irrespective of the presence of the time-ordering. We can now perform the well-known transformation to integrals over spacetime, by substituting the definition (2) of the operators and using the fact that the ]~ are normalized solutions of the Klein-Gordon equation. Treating A a~* (t~) first, we get

i fdt~, fdy/~(y) (Dy--m ~)(~o~tlT~A (Y)A~'(t~)[~'~).

(8)

The Klein-Gordon operator in this expression also acts on the time-dependence contained ill the time-ordering. We notice that an extra term, f dr. O,. f dy 0,, 1/~, (y)(~oa~tl'i'+A (y)A~'(t.)[~,~)}, appearing when going from expression (7) to (8), is zero, because for fixed t® the expression in curly brackets vanishes at the boundaries of the tw-integration. The operator "F±A(y)A~,(t®) denotes all the terms of the derivative of the Tproduct that contain derivatives of the 0-function. Treating next A ~, (t®) in the same way, we obtain -

-

f f dx dy t~, (x)l~, (y) (D.--m') (Wlu--m') (#ao~tlT~A (y)A (x) l#~) -- f f d~ dy /~, (y) (['~,--$vB')~,. (/% (~) (OPo~t]~F±A(y)A (x)[O~) }.

The second term vanishes again. We remark, however, that to conclude this, it is not necessary to interchange the time integrations: the term on which the Klein-Gordon operator acts, vanishes for fixed t, at the boundaries of the t,integration, so we m a y integrate over y by parts, whereby all boundary terms, both of space (on account of the normalized character of ]~,(y)) and of time vanish. The result is then zero because ]~,(y) satisfies the Klein-Gordon equatmn. The final result, in accordance with ref. 1) is therefore, #'l#21 o5%aix, (d) --out I~ln /

--~ -- f f dx dy /% (;~)/~x(y ) ([~z--m 2) (E~y--~7~2)(~oP~t[T~ - A (y)A (~) I ~ ) ~ ,

(9)

where the order of integration is irrelevant, although we did not use this fact in our derivatmn. A completely analogous result m a y be derived by using the retarded commu-

ON THE REDUCTION FORMULAE FOR THE S-MATRIX ELEMENTS

661

tator or anticommutator instead of the T . products. Define

R . A (x)A (y) -~ --iO (x--y) [A (x)A (y)]~:. Then = ( o.tl out

\~'out l~m

In]

in) = (¢o.tl [Ao,,t A,~]~lO,nb

The extra term, introduced in the second step, is zero e.g. on account of ([Pl'/=~) = 0. The order of the limits is irrelevant. B y going through the same steps as in the former derivation we now get (O/tlJl I~balal\ out

I

rn

/

= i / / d x dy l:,(x)l$,(y)([],-m")([:],--m")(~o~tlR~A (y)A (x)lO~nt). (10) We have thus obtained four equivalent expressions for the matrix element (4), with the two kinds of T-product and the retarded commutator and anticommutator respectively, irrespective of the boson or fermlon character of the field, if we had started from a fermion field, we should have got similar results That several answers m a y come out is not surprising, because in the procedure of obtaining these formulae, boundary values of integrals have been replaced by complete integrals, which is of course not a unique procedure. This arbitrariness m a y give rise, however, to an apparent contradiction, if we take out the two m-particles, instead of one in- and one out-particle. This leads, via (!~ i~tljtl ~ =I=I~, Oll[ In /

=

IqSJlJllA'I1A =l 0"~ ", out[ 112 In /

=

lira

hm

(qM&Pt,IT±A*,(t=)A~,(t~)IO),

(11)

f~--¢---cO flt-+--oo

to the result

~= otuStl f l I

i'fialal

X"m

",,

/

= --.[/dx dy

") (rm,-m,)(OPo~,IT.A (x)A (y)10~, (l~)

which seemingly leads to the contradiction t h a t it is symmetric or antisymmetrlc in c¢1 and ,¢~, according to our choice of T+ or T_. Such a conclusion is not allowed, however, for in this case the order of Integration ~s relevant. This may be seen from (11), where an interchange of the limits results in an interchange of the two in-particles. Consequently, because in our case we have bosons, the order of integration in (12) m a y be interchanged if we choose T+, but m a y not if we choose T_ If we have fermlons, just the opposite is true The above derivation of formulae (9) and (12) m a y be directly generalized if an arbitrary number of particles are present in the lngomg and outgoing states We get expressions where an arbitrary number of A (x) operators is taken out

662

J

HILGEVOORD

from both sides. Integrals which correspond to particles from different redes can be freely interchanged. Integrals which correspond to particles from the same side can be interchanged if the choice of the sign in T . agrees with the statistics of the asymptotic particles; in the other case, a change of sign is introduced, which is equal to the sign of the permutation involved. The form (lO) does not permit such a generalization without the introduction of extra terms. Still another version of (9) and (10) is obtained if we shift one of the KleinGordon operators to the right of the time ordering or retardation. Introducing ](X) = (D®--m2)A(x), we obtain out I

m

/

out

In

/

[

- ff =

¢f f

dy /,,,(x)/$,(y) ([[],--m2)(q~o~t]T±A (y)i(x)]q~.~j),

dy /,,,(x)/$t(y) (D,--m*)(q~oo~tlR±A

(9')

(10')

These are indeed equivalent to expressions (9) and (10), although the mtegrands differ; but this difference contains the function/$1 (Y) together with the operator ([:]y--m*) operating on a function with support only at ty -----t.. Therefore, a partial integration with respect to space and time is allowed and yields zero because (V-l,--m*)/~ (y) = O. We conclude this section by remarking once more that in this derivation of the reduction formulae we assumed the interpolating field A (x) to be such that operators like A~(t) give rise to states with finite norm. As far as I know, it is not clear at the moment whether this assumption leads to a wide enough framework for a field theory. Less restrictive schemes have been formulated, in which one only assumes that averages of the field over space-time of the form fA (x)/(x)dx, /(x) being a test function, gave rise to well-defined operators a), It is not difficult to gave a modified formulation of the asymptotic condition in this scheme; the main difference consists in the replacement of our operators by time-averaged operators. However, I do not see how one can derive the above reduction formulae in this case; the concept of time ordering does not seem to fit well into a scheme where one considers time averages of the fields only. 3. C o m p a r i s o n t o a n o t h e r R e s u l t

Finally, we want to compare the above results with still another from of the S-matrix element, given by Bogohubov, Medvedev and Polivanov 4), and used by them in their proof of dispersion relations:

,rip,p, ,n=,=,,~ / f dx dy /=l(x)/$l (y) ( q~o~t[ ~SAtn(y)~Atn(X) O2S ~"out ~"m / S*l°~)" (13)

ON T H E R E D U C T I O N F O R M U L A E FOR T H E S - M A T R I X E L E M E N T S

663

It can be shown that, for all points x ~= y,

~*S ~A m (y)(~Am (x)

(14)

s* =

where

l(x)=i--

~S

St,

if we assume the causality condition as stated b y Bogoliubov, namely i(x)

= o,

(15)

0Atn(Y) if y IS earlier than x, or if (x--y) is space-like The definition of i(x) given here can be shown to be equivalent to that of the preceding section. The condition (15) implies local commutativity for the operators j(x), b u t is in fact stronger z). To obtain a comparable result, we start from (9') with the T+-product and take the Klein-Gordon operator to the right of the time ordering. This, however, introduces an extra term which is not zero. Formally we get

(qbP,a,,O~x~,,~ out I m / = _ f f dx dy /~, (x)/$, (y) (~boa~t[T+f(y)] (x)[~n*)

(16)

+ f f dx dy l=, (x)l$, (Y) {~ (t,--tz)[A (y), l(x)]_--20 (t,--t=)[A (y), ?(x)~_}. In contrast to section 2, we now meet a situation where the definition of the time ordered product at x ~- y becomes important We see that, if local commutativity is assumed for all operators, the extra term in (16) vanishes, except for a possible contribution from the points x ---- y. Therefore, for a formula like (16) (without the extra term) to hold t, local commutativity is required, while this is not the case for the expressions of section 2 The result then agrees with (13), (14), up to contributions from the integrand at x = y. A fuller comparison of both results requires a knowledge of the definition of the quantities appearing in (14), at x = y . t Thin formula has been used by Kasvhluhn, ref ~)

References 1) H Lehmann, K Symanzlk and W Zlmmermann, Nuovo Clmento I (1955) 205; 6 (1957) 319 2) F Kaschluhn, Nuovo Clmento 12 (1959) 541 3) See e g A S Wlghtman, Quelques probl~mes mathdmatlques de la thdorm quantlque relatlvlste, Lille Conference 1957 4) N. N Bogohubov, B. V Medvedev and M K Pohvanov, Problems in the theory of dispersion relatmns (Flzmatglz, Moscow, 1958), German translation m Fortschmtte der Phymk 6 (1958) 169