and r-Functions
0 . S T E I N M A N N
Eidgenoseieche Technieche Hochschule, Zurich, Switzertond
I. Statement of the Problem
We consider a field theory satisfying the usual assumptions, namely, invariance under the inhomogeneous proper Lorentz group, existence and stability of the vacuum Ω, and locality. For the sake of simpli
city we restrict ourselves to the case of a single scalar field A (x).
The Wightman function Ψ~η is defined by
(1) (*„ · · · , * · ) = <A(x0) · · · A(xn)\.
I t has the following important properties (1):
(a) iTn depends only on the variables ηί = ΧΊ — Χ4-1 (2) irn{x9, · · · , xn)=Wn(Vu ···, ηη).
Wn is invariant under the homogeneous proper Lorentz group L*+: (3) Ψη(ηΐ9 -·;ηη) =Ψη(Αηι, -·',Αηη) for all AeL*+.
(b) The Fourier transform
(4) Wn{qt, · · ·, qn) = j d4» q e x p [ - i 2 M J % , -, ηη)
of Wn vanishes if one of the arguments q\k does not lie in the forward light cone V+.
(e) The equation
(5) irn{-··,xkrxk+1, ···) = * ^ , ( · · · · ) holds for all space-like separations (xk+1 — xk).
We shall not take into account the set of inequalities also given in ref. (1), The r-functions, on the other hand, are defined to be the
Ο. S T E I N M A N N
vacuum expectation values of the operators
(6) Rn(x0; xx · · · xn) = 2 θ(χο — ®ι) θ(Χι — oo2) · · · θ(χη-χ — xn) ·
· [ · · · [ ^ o W W ] • • • , Α Ο ] .
The sum Σ Η* this definitions goes over all permutations of the variables xt · · * xn. Obviously rn can be expressed in terms of the corresponding Wightman function Wn. We are now going to ask ourselves whether this connection can be reversed or not, i.e., under what conditions there exists a function Wn satisfying (a;, (b), (c), such that the cor
responding rn assumes a prescribed value.
The following properties of rn follow immediately from the de
finitions :
(A) rn is a function (which we shall call rn likewise) of the va
riables
f * = #o — ®k , * = 1 · • · Λ
only. Furthermore,
(7) RN(£L9 · · · , £ . ) = τη(Αξι, · · ·, Λξη) for all AEL\.
(B) rn is totally retarded:
(8) rn( £ , · · · , £ „ ) = 0 if one f ^ V + .
( # ) Ρη(£ι? • • ' , ί η ) is symmetrical in all variables.
Another simple property (which Ave shall give here for the case η = 3) can be derived from the operator identity (2)
(9) B(x0; Χχ,χ2, xz) — ϋ ^ ; x09x29 %z) = [B(x0, x2, xz\ A(xx)] —
— [B{xu x29 xz\ A(x0)] — [B(xl9 x2\ B(x0J xz)] — [B(x19 xz), B(x0, x2)], by considering the Fourier transform of its vacuum expectation value and taking into account the spectral conditions imposed on the theory.
I t reads Φ)
(10) r3(Pi, p2 > Pa) = rz(Po, P2, Ps) , Po = — Pi — P2 — Pa if the four vectors p09 pu (/>i + p3), ( P i+ p2) are all space-like.
Similar equations can be given for general n. The proof is quite straightforward and will not be given here.
Now we return to our question, whether the defining Eq. (6) of RN can be solved for Wn. I t has been shown by E. Jost (3), that
in the case η = 2 the conditions (A)-(D) are necessary and sufficient for the existence of W2. (For n = l this obviously holds too). He has been able to give an algebraic expression for W2 in terms of r2. Here we shall look into the case of the four point function (n = 3).
We shall see, that in this case we have to impose some additional con
ditions of a non-trivial kind.
II. The Multiple Commutator
We shall divide the problem into two parts by introducing the multiple commutator
(11) K(x0, · · ·, = < [ [ [ A( 0O) , A(x1)l A(x2)l A(xz)])0 as an intermediate function. Κ may evidently be expressed as a sum of Wightman functions.iT of the arguments x0, xz. From this fact we can derive the following properties of K:
(α) Κ is invariant under the inhomogeneous proper Lorentz group. ^
(β) The Fourier transform K(p0, · · ·, pz) [which of course contains a factor δ(ρ0+ f-p3)] has the following support properties:
(12) K(p0J · · ·, pz) = 0 if pz is space-like, (13) K{p0, px, p2, — K(p0, px, pz, p2) = 0 if (p2 + pz) is space-like.
(?)
(14) Χ(α?ο, · · ·, xz) = 0 if (#o — «χ) is space-like.
(<5) Κ satisfies the identities
(15) K(x0, x17x2, xz) = — Κ(χχ ,oo01x2, xz) ,
(16) K{x0, ^ , #2, a?,) + J£(a?!, x2, #0, #s) + # ( # 2 # 0X l y ®z) = 0 , (17) iC(a?0, ^ , x2, a?,) — ir(#0, xx, a?3, a?2) =
= J l( #2, #3, #ι> #o) — ^ ( # 2 , # 3 , »i) ·
These conditions on Κ are not only necessary but also sufficient for the existence ol W(x0J - - xz). (From now on we shall omit the index η = 3 in Wn and r„). The proof of the sufficiency is rather cumbersome and shall only be sketched here.
We consider a function Κ with the properties (a) to (<5). Then we can construct the corresponding W by an induction process in
Ο. S T E I N M A N N
momentum space. The Fourier transform of the function
<l[A(xQ\ A{x1)]A(x2)]A(xz)\
may be denoted by ([PoP^lPs) and similar notations may be used in an obvious way for other expressions of that type. Thus,
(18) K(p0J · · ·, p3) = ([P0P1P2YP3) — (p3[PoPiP2]) ·
Now we notice that according to condition (b) of the first section (rewritten for iT instead of W) ([PoPiPziPs) bas to vanish if pz$V+, while (pdPiPzPz)) vanishes if p3$V-. Thus,
ί ([PoPiP2]Pa) = θ(ρζ)Κ(ρ0, · · ·, Pa) ;
(19) \ ~ [ (ΡιίΡοΡιΡ*]) = — θ(—ρ*)Κ(ρ0, · · ·, ρ ) .
In virtue of (12) these functions have in fact the mentioned p3-supports.
Furthemore,
ί ([PoPiPjPa) = 0(P»){(#(PoPiP*P«-- ^(PoPiPaP«))+^(PoPiP8Pa)}=0 1 if p2 2< 0 and (p2 + p3)2< 0 , and analogously,
(21) (p3[PoPiP2]) = 0 if p2 2< 0 and (p2 + p3)2< 0 . Because of (12) the 0-functions in (19) can be replaced by the cha
racteristic function of V+ so that the Lorentz invariance is not de
stroyed by these 0's. Going back to the #-space one sees immediately that also the locality property (14) is unchanged by the removal of the outermost commutator bracket.
By a similar procedure we can remove the remaining two com
mutators and get eventually the -function with all properties (a), (b), (*).
III. The Function r(~k\ \ fc2; ^3)
We have yet to clear up the connection between Κ and r. In order to do this we have to make heavy use of analytical methods.
It is well known that as a consequence of the retardation of r in all variables, the function (or rather distribution) r(p1,p2Jp3) is a boundary value of an analytic function r(kly fc2, fc3), kf = pj + iq^
which is regular in the domain
(22) « : & e r + , 7 = 1,2,3.
The above mentioned properties of τ(ξχ, f2, ξζ) render, together with the first lemma of Hall and Wightman (4), the following conditions
(A1) rCkj) is invariant under the complex Lorentz group L+(C).
(Β') r(kj) is analytic in the domain
For q, going to oo within ^ , r has to increase slower than any ex
ponential exp [2*a|&|].
(Gf) r is symmetric in all variables.
(D
)
T(1CXJA/
2,
lcz)= i"(A/
0, fc
2, &
3),
Ίί^ = Jcx Jc2 — Jcz. The same symmetry holds of course also between k0 and Jczr or TcQ and Tc2. I t is clear that these symmetries give an extension of the domain of regularity of r, since they allow an analytic continuation of f into the regularity domains of r(k01 hu Jc2) and so on. This is possible because the respective domains of regularity of r(JclrJc2, h9) and r(Jcor Jc2, kz) have some points in common.We have now to study some consequences of the properties of Κ derived in Section I I . For this purpose we introduce a new function once more, namely,
(23) s(x0, · · ·, a?a) = <\B(x0, a?x, a?,), A(xz)]}0
Because of Eqs. (14) to (17) this function vanishes outside the region
I (x» — x2)eV+
{Xo — Xz)2 > 0 or (xx — xz)2 > 0 or (x2 — xz)2 > 0 . From the definitions of s and r and the properties of Κ we get by a simple but tedious calculation which shall not be repeated here (25) s{x0, = r{x0, - · ·, xz) if {x3 — Xi)$V+ and (xz — x2) $V+ . (This equation holds obviously if x\ is the smallest of the four time components.)
The function
(26) g(x0, · · ·, xz) = s(x0, · · ·, xz) — r(x0, · · ·, xz) , for r(fc,).:
= θ(χ0 — xx)θ(χχ — X2) K(X0jXl9X2, xz) + (Xx<-» x2).
(24)
Ο. STEINMAtfN therefore, has the support
(27) G • (x0-xt)er+
fe-»i)e7+ or (a?8 — x2) e Γ + .
Now we consider r, *, (/ again as functions of the differences £; and have a look at their Fourier transforms. In the same way as with r we find that g(pu p2, p3) is a boundary value of an analytic function g(ku k2, fc3), which is regular in
(28) @1 2 3: & e F _ , (qi + q^)eV+, (& + q*)eV+
and fulfills similar growth conditions as those mentioned in (Bf).
From Eq. (12) we get
s(Pu P*> Pz) = 0 for p\< 0.
Thus,
(29) g(px, p«, p.) = — r(Pi , PZ > Pa) if ΡΪ < 0.
Now we know the real points in ffl to be the Jost-points defined by the condition
(Σ hPi)2< ° f o r a 1 1 λ< > o, 2 ^ > °- A similar condition we get in the same way for the real points in © i2 3, which domain is defined from ©1 2 3 analogously to The two domains
© i2 3 and &tf have real points in common, e.g., a neighborhood of the point px = p2 = (0, 10, 0, 0), p3 = (0, 0, 1, 0). In these points we have p j < 0 , i.e., Eq. (29) is valid. Therefore, g(ks) is an analytic conti
nuation of r(kj) into the domain ® i2 3:
(30) g(kx, fc2, *,) = — r(kx, k2J k3)
and r is still regular in ® [2 3. In the same way we obtain, by making use of the symmetry properties of r and g, the analyticity of r in all the domains Qb'uk, where ijk is any set of three numbers out of 0, · · ·, 3 in any ordering. We notice further that by virtue of (Ar) we have
r(kx, k2, kz) = r( kx, k2, k3) .
By taking into account all these symmetries we get eventually the following:
Theorem 2. The function r{Jcl9k2^kz)^ k, = p3 + iq, is analytic in
all the points (k,) with (q} + qh)2>0 for all A = 0, 1, 2, 3 (includ
ing j = A).
Of course this is not the entire domain of regularity of r, in fact it is no domain at all. The region given here is a union of tubes of the kind considered in the paper by Hall and Wightman, and thus we get a wider domain of regularity by applying the first lemma given there. As yet there is nothing known about the envelope of holomorphy of this domain, which might be interesting in connection with the derivation of dispersion relations for the scattering amplitude. I t can be shown that even this hull of holomorphy is not yet the entire domain of analyticity of r (see the concluding remarks given later on).
We have to give yet another condition for r in order to be sure of the existence of the corresponding Whigtman function. I shall only mention this condition without giving the proof:
Theorem 2. The function g defined by (26) satisfies the following identity:
(31) g(x0, a?i, 0,, a?,) + 9(®2, #3, #ι, ®o) =
= 9( ®11 X0 J X2 J ®z) ~f~ ff( ®Z J ®2 J X0 J ^ l ) ·
(The proof makes use of the identity (17) with which (31) is essentially equivalent). This condition can of course be rewritten in terms of boundary values of the function ?(&,·), g being such a boundary value.
IV. The Existence of Κ
In this Section I want to prove that the conditions given in Sec
tion I I I are sufficient for the solubility of our problem, i.e., that these conditions contain all the information that can be gained from the properties of the Whigtman functions. (At least I want to sketch this proof.)
Let us assume then that there is given a function r(Jclr Jc2, fe3) with all the properties derived in Section I I I . We define
r(Pu P*J Pz) = lim r(kxj fc2, fc3),
Qj-*0
(32) {
9iPi, P*> Pz) = — r(ki, *i, *i) -
Ο. STEINMANN
of distributions, which is of course an additional assumption about the function r.
It is evident, that r and g thus defined have the right symmetry and invariance properties. Furthermore it follows from the work of L. Schwartz (5) on the Laplace transform of distributions, that the Fourier transforms r and g of f and g have the supports (8) and (27), respectively [here we have to make use of condition (31)]. The defi
nition of the corresponding functions of the four variables xk is straight
forward. Then we define
s{x0, xlr x2, xz) = r(x0, · · ·, xz) + g(x0, · · ·, #3);
8 is Lorentz-invariant and has the desired support (24) in #-space. In momentum space we have
(33) i(pl9 p2, p3) = una r(fcx, fc2, Jcz) — lim r(Tc19 fc2, Jcz) .
The two prescriptions for going to the limit entering into this equation differ only by the sign of the vector qz, all other vectors (£. + #*) going to zero in the same half cone in both terms. For space-like p3 the point (Jcs) with ql9 q2eV+, qz = 0 lies within 0t' as can be seen by applying a suitable infinitesimal transformation out of L+(C). In this case we can therefore let g3 tend to the limit without leaving the domain of regularity. Thus the difference in sign of qz is of no importance, i.e., the two terms are equal:
(34) s(pl9 p2, p3) = 0 if p3 is space-like From 8 we can now get to Κ by purely algebraic means, using the identities (15) and (16) and the fact that under our assumptions the CTP-theorem holds. We obtain
(35) ί( Λ? ο , " ' # a ) = 8(Xo, # i > ®2, #3) — 8{XX} X0, X29 Xz) —
— 8(— Xor —Xlr — X2y —Xz) + 8(—Xlr —X0, — X2, —Xz) .
This expression should satisfy the conditions (a) to (δ) in Section I I . The only ones which are difficult to check are the Eq. (13) and (17).
(17) holds as a consequence of (31). (13) can be proven by the same method we have used for the derivation of (34). We define
(36) G(x0, xux29x3) = g(—xu —x0, — # 2 , — #3) — 9(®*i #u #2) ·
Thus
#(Pi> p2, P2) = Km r(ku fc2, fc3) — Urn r(Tcu fc2, fc3) = 0 if (ρ2 + Ρ* )2< 0 . Now
#(Ρο, Pi, Pt, Pz) — K(Po, Pi, Ps, P»)
= # (Po, Px, P2, Pa) — #(Pi, Po, P2, Ρ») — #(Po, Pi, Pa, P « ) ( P i , Po, P*, P«) = 0 if p2 + p3)2< 0 which completes the proof.
V. Concluding Remarks
1. The signification of the identity given in Theorem 2 is as yet not understood very well. It can be shown in a rather involved way that this identity blows up the domain of regularity of the function r:
?(Pi,p2,Ps) can be expressed as a function of the variables η{=ρ\
« = 0, · ' · , 3 ) ,
Δ2 = - [ ( Ρ ι +
Po)
2M],
ω = [(Pi - Po)(p2 - P3)/2V(p2 - Pz)2].By putting u{ = 1 we get a function Τ (ω. Δ2) which is nothing else than the scattering amplitude for the scattering of two particles with mass 1. It has therefore to satisfy a dispersion relation in ω for all z J2< 2 . There exists now an example of a r-function satisfying all conditions except Theorem 2 (including of course the stronger spectral conditions due to the non-vanishing mass) such that the corresponding Τ(ω, Δ2) does not fulfill a dispersion relation for values of Δ2 smaller than 2. There are some hints as to where this blowing up of the re
gularity domain could originate but still the situation is very un
satisfactory.
2. For general η it can be proved that ΊΓη is determined uniquely (up to terms of a very special kind) by rn if it exists at all. The pro
cedure given in this paper for the derivation of necessary and suffi
cient, conditions seems to be generalizable to higher n. However, the calculations become very awkward in the general case and it would be desirable to find an easier way of doing the job.
NOTE ADDED IN PROOF: The general case η > 3 has since been treated independently by D. Buelle and the present author (6).
Ο . S T E I N M A N N
B E F E R E N C E S
1. Α. S. Wightman, Phys. Rev., 101, 860 (1956).
2. V. Glaser, H. Lehmann and W. Zimmermann, Nuovo Cimento, 6,1122 (1957).
3. K. Jost, Helv. Phys. Acta, 31, 263 (1958).
4. D. Hall and A. S. Wightman, Kgl. Danske Videnskab. Selskab Mat.-jys.
Medd., 31, no. 5 (1957).
5. L. Schwartz, Medd. Lunds Univ. Mat. Seminar. Suppl. (1952).
6. T>. Ruelle, Thesis, U n i v e r s e libre de Bruxelles (1959). 0. Steinmann, Helv. Phys. Acta, 33, 347 (1960).
