Properties of Wightman Functions *
E E S J O S T
Physikalisches Institut der E.T.H., Zurich, Switzerland
I. Introduction
According to a fundamental theorem of A. Wightman (1) a field theory can be completely characterized by the vacuum expectation values of products of field operators
( L I ) 2B„(#o, <&!,···
x
N) =
<ψνο(οο0) ψνι{χ,) · · · ψΡΝ{(ΡΝ))ο ·Here
ip
Vk(x)
stands for one of the finite number of fields that describe the theory. vk represents a set of dotted and undotted indices (<*!" ' <*mk β " - βη) and describes the spinor character of the field.ν finally stands for the set v0, · · · vN . The Wightman-functions 993, (^o* satisfy certain properties, which originate from the axioms of field theory.
1. From translational invariance one has
(L2) 93&,0»o, Xu - - - χΛ) = W,^, |2, · · · ξΝ) h = xk — Xk-i.
2. The invariance under the restricted Lorentz group L\ leads to (1.3) w,{A£i, · · ; ΛξΝ) = 2 8^(Λ)^μ(ξ19 · · ·, !,),
for AeL\. S(A) is a finite dimensional representation of L\. Since W^fx, · · ·, ξΝ) is one valued, it follows from the above equation that Wv vanishes for a double valued representation S(A) of L\.
3. The absence of states with negative energy allows for an ana
lytic continuation of Wv(Cly · · ·, ζΝ) into the domain ΐϋ^: lmCkeV+,
* I, I I , I I I , part of IV, and V are a shortened version of part of the author's contribution to the W . Pauli memorial volume to be published by Interscience, New York.
127
R E S JOST
where V+ represents the forward cone V+ = [η; η0> 0, η2> θ ] , (2) still holds for these points. Simultaneously 2B„ has a continuation Ή&ΜοιΖυ ' * *? zn) in t o a domain characterized by zk — z^efRx.
4. There are important restrictions on 2Β„(#0 · · · xN) due to the positive definite scalalar product defined in Hubert space. We will not make explicit use of the corresponding inequalities.
5. In a local theory the fields either commute or anticommute for space-like separations. F o r ( #t— xk)2<0, i Φ h we have therefore
/ τ η /( 0 ) / χ ( 1 ) / \ ι \\ /(* « > / \ / v\
(Ι Λ) <ψν0(χο) ψΨι{?ι) ' · · Vvn(xn)\ = σ< W^Sxk) ''' v W^ ) > o >
or
(1.5) SBv(^o, Xi,--,xv) = <*28?K,> ^ > · ' X*N) j
where a denotes the signature of the permutation, which the anti- commuting fields undergo in (1.4) and gg- is the Wightman function for the fields taken in the new order.
II· Some Properties of the Lorentz Group
Our aim is to prove the Lemma 1 of Hall and Wightman (2), but before we do this, I have to recapitulate a few well-known properties of the homogeneous Lorentz group L.
1. The real homogeneous Lorentz group L has four components (11.1) L = (L\ + PTL\) + {PL\ + TL\),
where Ρ: ζ°' = £°, = — and Τ : ζ°' = — £°, £*' = C*. ί ζ , the res
tricted Lorentz group, is the identity component of L. (L\ +PTlJ+) =L+
contains all the Lorentz transformations Λ with det A= + l and is again a group.
2. The complex homogeneous Lorentz group L(G) (i.e., the group of complex linear transformations of ζ = (£°, ζ1, ζ2, ζ3), which leaves
3
£2 ( f0)2 — ^ (£*)* invariant) has two components (11.2) HG) = £+( 0 ) + M + l ^ ) ·
·£+(<?) contains all the Λ with d e t ^ = + l . It contains the group (L\ + PTL{).
3. A simple analytic parametrization of L and L\(C) can be given
P R O P E R T I E S O F W I G H T M A N F U N C T I O N S
λί λ.
Κ 0 λ3 -λ,
Κ 0 λι
Α2 -Κ 0 as follows:
(ΙΙ.3) Β =
Then,
i o r A . {
c o™ y ,
(ΙΙ.4) Λ = (Ε + Β)(Ε - Β)-1
{
complex] -^orentz transformation, provided the |Areal I fc|'s are small enough. On the other hand(11.5) R = (A — Ε)(Λ + Ε)-1
{
complex] ^real 1 0 Γ θ η*ζ transformation is of the form (II.3) with{
complex] ^* provided Λ is close enough to Έ. real 1 The proof of these facts is elementary.4. We will need a normal form for a complex Lorentz transfor-
0
mation. Let us call Λ equivalent to Λ if
(11.6) Λ = ΑΧΑΛ2 and AxeL\ , A2eL\ .
Then we have: Λ is always equivalent to one of the following normal forms
Μι = Λ _ . . \ ψ, X r e a l
or
M2= ±
' 1 0 X ix
0 1 X ix
r — r 1 0
ix — ix 0 1
R E S J O S T
III. The Theorem of Bargmann, Hall, and Wightman Theorem. If SSv(Ci * * ' , ζΝ) is analytic and single valued in diy and if for AeL\
( I I I . l ) ψ9(ζ19 • · ·, Cir) = Σ Βζ(Λ-ηΨμ{Αζΐ9 · · ·, ΛζΛ),
μ
where 8 (Λ) is a single valued finite representation of L\ then Wv(Ci
has a single valued analytic continuation into 9l'v = 17*^13t^ and ( I I I . l )
holds for AeL+(C). A E L + ( C )
Remark. Since W^ C i , belongs to a finite and single-valued representation of £+(0) it is essentially a tensor. 8(A) is therefore a polynomial in the matrix-elements of Λ and has an obvious analytic continuation into L+(C).
Proof. A. The theorem states in part, that for AeL+(C) ( I I I . l ) can be used to define Wv and in the points of iRfs that do not belong to dt# and that this continuation is unique. That this is so, follows from the definition of every point of dt'N is the image of a point in 9t^ under a suitable A~leL+(C). Necessary and sufficient for the uniqueness is the condition, that ( I I I . l ) holds if ( ί ι , ' · ·, ζ#) ediN and
(ACU · · ·, ΛζΝ) edljf, where AGL+(G). To prove this is the main diffi
culty. The analyticity of the continuation by ( I I I . l ) is trivial, since one is dealing with a linear transformation of the variables only.
B. From now on we will keep (£1 ? · · ·, ζΝ) ejft^ fixed and inter
pret the right-hand side of ( I I I . l ) as a function F:(A) on L+(G).
F^A) is defined as long as (Λζχ, · · ·, ΛζΝ) edlN. This condition de
fines a certain domain ©c of L+(C). All one has to show is that FC(A) is constant in ©c.
Now we will make use of the parametrization of J0+(G) given in par. I I . Fc(A(kl9 · · ·, λ3)) is regular in a neighbourhood |Xk\< α which is chosen so that firstly Α(λ)Ε®ζ, secondly the parametrization is regular. But FC(A(X)) stays constant for real λ, which correspond to real Lorentz transformations. Therefore it stays constant also for complex λ in our neighbourhood.
C. A simple generalization of Β leads to the conclusion that FC(A) stays constant in a neighbourhood of every point in ©c, and from this we have, that FC(A) is constant in every component of ©c. The with τ real and, e.g., equal 1. Let me skip the proof of this simple statement.
P R O P E R T I E S O F W I G H T M A N F U N C T I O N S
real difficulty is therefore to show that ©c is connected. In other words we have to show that we can find to every ( d , · · · , CA) e 9 lA and to every AeL+(C) such that (Λζ19 · · ·, ΛζΝ) GdtN a path A(t),
0 < i < 1 , such that {Λ(β)ζ1, · · ·, yl(J)C*) Ε 9 Ϊ * and Λ(0) = E, A(l) = A.
D. At this point we make use of the normal form of A discussed in par. I I . Let A have the properties in C and let A=A1Mlf2A2, then we can evidently leave Ax and A2 out of our discussion. I t is however very fortunate that the normal forms M1 and Jf2, which have the property to transform a fixed point in 9%^ into a point in diN, consti
tute themselves a connected set. To see this we start with the normal form M2(r) and with N = l. Let ζ = ξ-\-ίη and ηΕΥ+ and write ζ(τ)= ξ(τ) +ίη(τ) = ΜΖ(χ)ζ then we have
(III.3) (r/(r))2 = (η)2 + 2τ [ξ*(η"-ηη _ η*(ξο_ξι)γ_
_Τ2 | " ^ ο _ ^ ι) 2 + ( ίο _ | ΐ) 2 ]4
The τ values for which both expressions (III.2) and (III.3) are posi
tive form evidently an interval which contains the origin. If we have several vectors ( d , · · ·, ζΝ) the allowed r values lie in the intersection of all the intervals belonging yo the different ζι. They form again an interval containing the origin.
E. The case of the first normal form Μχ(χ, φ) is somewhat more involved. Starting again with Ν = 1 we put ζ(χ, φ) = ΜΧ(χ} ψ)ζ =
= f ( & ψ) + ir)(X> Ψ)and w e find
(III.4) η°(χ, φ) = η° cos φ + ξ1 sin φ > 0 ,
where
(ΙΙΙ.6) Α = (η° cos φ + ξ1 sin φ)2 — (η1 cos φ + ξ0 sin φ)2, and
(ΙΙΙ.7) Β = (η2 Ch χ + ξ* Sh χ)2 + (η* Οϊιχ-ξ2 Sh χ)2 .
We limit ourselves to values of \<ρ\<π. Equation (III.4) allows an interval with length π around φ = 0, since η°>0. A can be trans
formed to principal axes
(III.8) A = oc2 cos2 (φ — φ0) — β2 sin2 (φ — φ0), (ΙΙΙ.2) η°(τ) = η°+ τ(η2 + ξ*)
and
(ΙΙΙ.5) (η(χ, φ))2 = Α-Β>0,
R E S J O S T
and φ0 can be chosen \φ0\< π/2. For the allowed values of φ we have A > 0, but also cos (φ ~φ0)> 0 because φ = 0 is allowed. Similarly Β can be transformed to diagonal form
(111.9) Β = γ2 Ch2 (χ - χ0) + γ2 Sh' (χ - χ0)} .
If not, η2 = εξ3 and η* = — ε£2, with e = ± 1 in which cases Β is of the form
( Ι Ι Ι Λ α ) B = y « e x p [ ± 2Z] . Substituting (III.8) and (III.9) into (III.5) and letting φ'=φ — φ07 X'=X — Xo, we have
(111.10) cos φ' > (οί2+β2)^νβ2+γ2+(γ2+δ2)8η2χ', with \φ'\<π/2 in the normal case, and
(ΙΙΙ.ΙΟα) cos φ'> (<χ2+β2)-*\/β2 + γ2 exp [ ± 2χ] , with \φ' |<π/2 in the case (IIL9a).
I t is a matter of simple computation to show, that the allowed do
mains are convex sets in (%'r φ') and therefore in (χ, φ). They contain obviously χ = φ — 0. If we have several vectors (ζι"'ζΝ) we get as allowed set in (χ, φ) the intersection of convex sets, which have a point in common. This is again a convex and therefore a connected set. This finishes the proof. I t is a consequence of our theorem that the Wightman Wv functions are regular analytic in di'N. Similarly 2Β,(20 " 'ζν ) will be regular in a domain <&'N chraacterized by the con
dition (zx -*0, · · ·, zN—zN_x) edt[w.
IV. The Real Regularity Points of and of the Permuted Domain The real regularity points in dt'N are denoted by ( ρη ρ2, · · ·, ρ#).
They are characterized by the following property (3): for any choice of Xk ^ 0 and 2,Xk> 0 the vector 2AkQk is space-like.
Proof, (a) the condition is necessary because there is a AeL+(C) such that Qk = ACk and Ckedli. Therefore,
(IV.1) ( Σ 2 = ( Σ* * ί * )2 = ( 02
and ζ e fRx since 9^ is convex. But ζ2 is real and the only real squares of vectors in SRi are negative.
(b) The condition is sufficient. Let I be the convex cone
P R O P E R T I E S O F W I G H T M A N F U N C T I O N S
ί=Σλλρ*> ΛΛ =0 . Let V+ and F_ be the forward and backward cone respectively. Evidently ϊ π V+ = 0 and i n 7 _ = 0. Let (<χξ) =
= α„ £v = 0 be a tangential plane to V+ separating Ϊ and V+ and (βξ) = βνξν = 0 be a tangential plane to Y- separating ! and Y- such that
(IV.2) (IV.2a) and (IV.3) (IV .3a)
(«f) > 0
0»f) < ο
for f
for
ξ el
or f e 7 _ , for f 6 F _ ,for f e l or £ e V+. Evidently we have α2 = β2 = 0 and (α/S) < 0. We normalize α and β by the requirement (a/3) = — 2. In a suitable coordinate system we can achieve that α = (1, 1, 0, 0) and β = (—1, 1, 0, 0). But then (IV.2a) and (IV.3a) give for ξ el: ξ°— ξχ<0 a n d —f ° — Γ< 0 or ξ1>\ξ°\. The Qk satisfy, therefore,
(IV.36) But
(IV.4)
Ql>\Qt\ * = 1, 2, 2Γ.
Im
0 0 0
t 0 0 0 0 0 1 0 0 0 0 1
The matrix belongs to a Lorentz transformation from L+(C).
Since it is entirely possible that is no domain of regularity, i.e., that a function regular in dtj, is necessarily also regular in certain points outside it is good to convince oneself, that the real points of regularity are not affected by analytical completion. In fact: if the real point · · ·, ξΝ) φ ffl'NJ then there are λ * ^ 0 with 2 Afc == 1 such that (2Af cff c)a = 0 or (2Af cff c)f —1 = 0 depending on the real point. Neither of these functions vanishes if one substitutes a point ( f i , - ; - , C, ) e 8 i ^ . [(Σ^ ί * )2] -1 or [ ( £ - l ] " 1 is therefore regular in dl'x but singular in the given point (f,, · · ·, ξΝ). W e denote the real regularity points of by (r0, r1 ? · · ·, rA) . These are evidently to
tally spacelike, since for any Jc > I the difference rk—rl = Qk+Qk-i + + · · * is spacelike.
R E S JOST
The permuted domains ©jj are defined as PPP © ^ , where Ρ runs over all permutations of z0, zx, · · ·, zN. In a local theory the Wightman function 3Siv(z29 zl9 - -zN) is regular in © * . This is an immediate consequence of Eq. (1.5) of par.-I taken for the points (r0, r19 · · ·, r^).
Since the left-hand side of (1.5) is regular, (1.5) can be analytically continued and yields an enlargement of the domain of regularity of 28-. The real points of ©£ are the points (rf c e, rki, · · ·, rkjr). They are totally space-like but for Ν ^ 3 not every totally spacelike point is in ©£ (O. Steinmann). An example for Ν = 3 is given by the points (1 —ε, ± 1 , ± 1 , 0), (— 1 + ε , ± 1 , ± 1 , 0) where ε>0 is small enough. I t is however a result of Ruelle (4), that all the totally space
like points are contained in the holomorphy envelope of ©£ *.
Proof of Ruelle's Theorem. Let (#0, xl 9- · ·, xN) be totally space-like i.e. (xk — Xi)2< 0 for k Φ I. Let zk = xk + i(tk, 0). The point ( z0> Z i >
• • · , 2Λ) is regular, as long as tk^tt for k Φ I. To see this we order the points zk according to increasing values of tk(zko, zki, · · ·, zkN) =
^ (zoi z'u "' > zn) 8 U Ch that x'k = tk — ^ _ ! > 0 . Evidently £k = z'k —
— z'^edl and therefore ( 20, s1 ? · · ·, z„) e ©£.
Singularities can only occur if some of the x'k vanish. They are therefore contained in certain hyperplanes in <-space. A singularity is called m-singularity, if exactly m of the x'k vanish. An m-singu- larity is contained in an JV+l — m-dimensional subspace and the m- singularities in this subspace form an open set. Now we remove the m-singularities by induction. If m = 1 we have no singularity because then all the ClefRu except one, which is real and spacelike. A suit
able small complex Lorentz transformation will send this vector into dtL without removing the other from dtx **. Let us assume there
fore that we have removed the singularities for m < m0. Let (z0, zx, · ··, zN) be an m0-singularity and let τ" and x"' be two of the vanishing x'k, and ωχ = ω2 = ί°'" be the zero-components of the corresponding difference vectors. Now we let ωχ and ω2 vary, keeping all the remaining variables fixed. If we keep the variation of ωχ and ω2 small, then Im ωχ = Im ω2 = 0 will be the only singularity that occurs. But such a singularity can be removed by applying the
« Kantensatz » (5).
* F. J. Dyson was the first to prove this. His proof remained unpub
lished, however.
0
+ isC)
Take ε > 0.PROPERTIES OF WIGHTMAN FUNCTIONS
(IV.6) r ί
J
f(x) j(x) ά*χ Ω
A further application of the real points in is the following:
Lemma: 8Β,,(#0, a^,' * '> xn ) a r e> for ^ 3 completely determined by their values for equal times. I t is evidently sufficient to prove this for Ν = 3. Due to Lorentz invariance W ^ d , f2, f3) is known for all points f2, £3) which lie in a spacelike 3-plane through the origin.
Such a point in e.g., ξχ = (0, 1, 0, 0), ξ2 = (0, 0, 1, 0), |3 = (0, 0, 1, 0) but also any point in a sufficiently small neighbourhood of this point.
These are however, all real regularity points. Ψρ(ξ19'ξ%, ξ9) is there
fore known in a real neighbourhood of a real regularity point. By this Wv(fi> ξ29 f3) is uniquely determined
The following theorem, which has some bearing on Wightman's form of Haag's theorem (2), is stated here only for the case of one real scalar field.
Theorem: If 2δ(#ο> xly · · ·, xN) = (A(x0) · · ·Α(ΧΛ)\agree for 3 with the corresponding functions of a free field, then this is true for all N.
Proof: Let the free field A°(x) belong to the mass m such that (• + m2)A° = 0. We first claim that also ( • + m2)A = j = 0 . Kow we have
(IV.5) <Kx)j{y)\ = (a + ™2) (•, + m2KA(x)A(y)\ =
= (•,+ m2){Uu + m2)<A»(x)A»(y)>0 = 0 . Multiplying (IV.5) by f*(x)f(y), where f(x) is an arbitrary test function, yields (Ω stands for the vacuum state)
and (using the positive metric) this gives (IV.7) j{x)Q = 0 .
If we use now the locality of the theory we evidently have (IV.8) <A(r0)A(ri) · · -j(rk) · · · Α(τ,)\ = 0
and by analytic continuation
(IV.9) ζΑ(Χ0)Α(Χ1) · · -j(xk) · • · A(xN)\ = 0 .
If the theory is complete, then all the matrix elements of j(x) vanish and we have
(IV.10) ( • + m2)A(x) = 0 .
R E S J O S T
Let us note that the derivation of (IV.10) uses only (XV.11) (A(x)A(y)>0= <A°(x)A°(y)\.
In order to finish our proof we have also to show that (IV.12) [A(x)A(y)] = [Α*(χ)Α'&)] =iA(x-y) . We first derive the weaker result
(IV.13) [A(x)A(y)]Q = iA(x-y)Q.
We have in fact
(IV.14) ||([A(x)A(y)] — iA(x-y))Ω\\2 = 0 .
To see this one only has to remark that (IV.14) is true for the free field A°(x) and that the left-hand side of (IV.14) can be completely expressed in terms of 2B(#0, · · ·, xN) with Ν <ί 3. But these agree with the free field values. Using locality again
if only (x — yk)2< 0 and (y — yk)2< 0 for Jc = 0, 1, · · ·, M. Now the left-hand side of (IV.15) can be continued analytically to points (zo >' *' j zN > ζι ω, o)0, · · ·, ωΜ) such that Im (zk— zk-x) e V+, Im (z — zN) G Imz = Im ω, Im (ω0 — ω) Ε V+, Im (ωΛ — ω^) e V+. The same is of course true for the right-hand side. Therefore (IV.15) holds for (zoi'' Ί zut zy ωι ωο,'' mj ω*) with the above imaginary parts but with arbitrary real parts and thus for the corresponding boundary values.
Equation (IV.15) holds therefore for all real values of (x0,
xi V} Voi'''? VM) a n (i we have, using completeness,
I t is well known, that (IV.10) and (IV.J6) together with the existence of a vacuum state define a free field theory. I will close this section by a few additional remarks:
1. From (A(x) A(y)}0 = (A°(x) A°(y)}0only, we concluded ( D+ m2) ·
• A(x) — 0. If one assumes for A(x) an asymptotic condition (6), then one finds immediately from the above equation that A(x) = An(x) =
= Aont(x). This result was derived differently by Schroer in Hamburg (unpublished).
(IV.15) (A(x0) - · · A(xN)[A(x)A(y)]A(y0) · · · A(yM)\ =
= iA(x — y)<A(x0) · · · A(xN)A(y0) · · · A(yM)\
(1V.16) [A(x)A(y)] =iA(x-y).
P R O P E R T I E S O F W I G H T M A N F U N C T I O N S
2. The application to Haag's Theorem (7) is immediate. We take one of the fields of Hall and Wightman to be a free field. From the assumptions if this theorem (canonical commutation relation for both fields, momenta taken to be Euclidian scalars, the same repre
sentation of the commutation relations for both fields at a given time t0, one and only one Euclidian invariant normalizable state Si in each theory) one first concludes that 2B(#0, · · ·, xN) have the free field value for the time t0. From there on we have all the steps derived, which lead to the conclusion that the theory is in fact a free field theory (8).
V. The CTP-Theorem and the Connection between Spin and Statistics
As we have seen locality enlarges the domain of regularity of (*o> · · *> ZM) from <&'N to ©£. More precisely: any relation of the form
<0> (N) (k9) {kN)
(V.l) <yr,(re) · · · y)vJry)>0 = <r< y>Vko(rkJ · · · y>VkN(rky)}0,
which holds in the neighbourhood of a point (r0, · · ·, r^) enlarges the domain of regularity of 28v(z0, * · ·, zN) with exactly one non-trivial ex
ception. This exception corresponds to the permutation (0, 1, 2, · · · -> (JV, Ν — 1, Ν — 2, · · ·, 0). That this is true follows from the fact that (rer rM„lr · · ·, r0) e <&'N1 or (—ρ*, —Qy.ly · · ·, Qt)edt'N. To see this we use the fact that iR'N is invariant under L+(G) and therefore under PT.
From this (— ρ1 ? — ρ2, · · ·, — ρ,ν) eiRy [and since the definition of fR'N is asymmetric in (£χ, · · ·, £Λ)] we have (— ρ,, — ρΝ_„ · · ·, — ρχ) e?R'N. This suggests the following weak form of locality {weak locality): in
stead of requiring (V.l) for all permutations we require (V.l) only for the. special permutations (0, 1, 2, · · ·, Ν) (N, N — l, N—2, · · ·,0), (V.l') <vl/r<,). ·. ψ^{τΜ)\ = σ<ψνΝ(τΝ) · · · vle(r0)>0,
or (in obvious notations)
(V.2) SB,(re, · · ·, rM) = σΒτ(τ„ · · ·, r.) .
I t is a remarkable consequence of our assumptions that (V.2), which is only postulated for the real points in <Β'Ν leads to a relation, which holds for all real points (#0, · · ·, xN). The derivation of this result is simple. From the covariance of 2SV under L+(C) we have as conse
quence of (V.2)
(V.3) 28v(r0, · · ·, rN) = ο τ · ( - 1 ) " ® τ ( - ^ , · · ·, - rQ) ,
R E S J O S T
where η is the number of dotted (or undotted) indices contained in v.
By analytic continuation
(V.4) 3B,(*0, · • · , * ) = σ(-ΐγ^[-ζ^ - · ·, -*β) .
The mapping (z0, · · · , * * ) - • (—*N, y ·, — «e) or ( d , · * ·, f*) -> "' *, Ci) does not only map ©*(8tj) onto ©*(81*), but also ©^(9t*) onto ©^(9?*) as is evident. We can therefore go in (V.4) to the real boundary points and obtain, using the definition of 2BV and (9),
(V.5) <ψη(χο)' ·' fVy(xx)\ = <r · (— l )w<(^ ( - · " xo)>o . A simple argument shows that (V.5) and (V.l) are completely equi
valent. From (V.5)
(V.5a) <ψη{χ9) · · · ψ,,Μϊο = <*·(—1)·<?*.(— » β ) · " ψ*Α-χΛ>*ο · The question whether (V.5) leads to an anti-unitary transformation, which leaves the theory invariant, depends evidently on the value of o*. As we will see there are restrictions on a if we impose locality and a positive metric The most important choice of a corresponds to the normal commutation-relations, a is then the signature of the permutation, which the fields belonging to a two valued representation of i t . undergo in (V.l). In fact a= ± 1 depending on whether this number is congruent to 0 or 2 modulo 4. For this case one has the anti-unitary transformation
ψν(χ) -> (— 1)ηψ*(— χ) for one valued representations, ψΛχ) -> i(— l )ny * l —x) f °r two valued representations.
We remember that ν stands for (ax, · · ·, a m , · · ·, βη) and that η ==• m(2) for one valued, w=£m(2) for two valued representations. Let's also remember that the conjugate complex of a sx>inor (m9n) transforms like a spinor (w, m). This last fact necessitates the factor i in the second case above (10). The most important consequence of the above ana
lysis of the OTP-theorem for mathematical field-theory is the result that, if we replace strong locality by weak locality, the axioms of field theory are compatible with an ^-matrix different from 1. In fact we get a field theory to every S, which is OTP-invariant by a suitable choice of an interpolating field between incoming and out
going fields (11). The only restriction on this interpolation is that it be OTP-invariant. It seems however evident, that the analysis of
ordinary (strong) locality necessarily leads to all the analytic compli-
P R O P E R T I E S OF W I G H T M A N F U N C T I O N S
ψμ(χ) f"(x) ά*χΩ ψ*μ(χ)ςμ(χ) ά*χΩ
cations of which we will hear from J. Toll. The analysis of the « con
nection of spin and statistics » which we will interpret as usual as the problem of finding compatible local commutation relations for fields with a given transformation character does not seem to me to be in a satisfactory state. The results which one has (in our framework) are due to ΒΓ. Burgoyne (12). They have to do with the commu
tation relations between a field rpv(x) and its conjugate complex field Theorem: The weak commutation relations
(V.6) <ψ»ψμ(ν)>ο= (-1)η+^Κψμ(ν)ψ»)ο for (x-yY< 0 have as consequence
(V.7) φ,(χ)Ω = ΰ and γ*(χ)Ω - 0 . Here again ν = (<xu · · ·, aw, βχ, · · ·, βΗ).
Proof: From (V.5) we get as consequence of (V.6) (V.8) <Ψ»Ψμ(ν)>ο = - <Ψμ(-ν)ψ*Λ-Φο,
or (using an argument similar to the step from (IV.5) to (IV.6) in par. I V )
= 0, which is (V.7).
The above theorem does not say that a field satisfying (V.6) cannot exist. If one wants to get this conclusion (and thereby the connection between spin and statistics) one is bound to make further assumptions.
One can, e.g., impose an asymptotic condition on ψν(χ)\ then (V.7) is a contradiction. Or one can invoke strong locality and get y>v(x) = 0 by an argument used in part I V (IV.8) and (1V.9), risking that one talks only about physically trivial theories. It is clear that the above discussion is not complete but it might show that the discussion of spin and statistics lies deeper than the OPT-theorem. I t is true that we have in the latter assumed the « r i g h t » commutation relations.
But now it is the problem to show that the «wrong » commutation relations lead necessarily to a contradiction.
Finally I want to point out that the results of G. Luders (13) about the commutation relations between different fields have not yet been derived in the Wightman formalism. About the role of the posi
tive metric, see W . Pauli (14).
R E S J O S T
VI. On the Structure of
What follows is intended to form an introduction to the lectures by J . Toll. I t is to a large extent dependent on the Appendix I I of Kallen and Wightman (15). Let me first start with the case of one time and only one space dimension. "We will write the two vectors (w, v) and chose a coordinate system, in which the scalar product becomes
(VI.l) ((%, vx\ (u2, v2)) - \(uxv2 + u2vx).
For V+ we have u > 0, ν > 0 and a point (u, v) is in 9^ if Re u > 0 and Re ν > 0. A point ( d , — , CJF) is i n $tx i f R e ui > 0 a n d R e ^ > °>
i.e., if all the components of the N-vectors are in the upper half plane.
Under a Lorentz transformation from Z+(C), (u, v) transforms as follows:
(VI.2) u'=Xu, v' = λ~ιν.
For a further discussion it is convenient to introduce
(VI.3) = * = , 1 . · . , 2 Γ
for the i?t's. (uu · · ·, u2N) ed{y if for all k lmuk>0. Under a Lo
rentz transformation all the components now multiply by the same factor. A point (ulf · · ·, u2N) edi^ if there is a half plane Im ei<pu> 0 such that Im ei(p uk > 0 for all k. This can also be formulated as the condition that
(VIA) 2 kk uk Φ0 for Xk ^ 0 with £ ^ > 1 ·
This last formulation shows immediately that for our case diy is a domain of regularity. I t is also bounded by analytic hypersurfaces.
Both these properties depend however critically on the number of dimensions (see report by J . Toll).
For the further discussion (in 4 dimensions) I have to quote at least the main result of D. Hall and A. Wightman (2) for the cases ^ = 1,2,3, and for scalar functions W ( k , · · ·, ζΝ) (cor
responding to a discussion of scalar fields only): Ψ(ζ19 · · ·, ζ„) is a regular analytic function of the scalar products 2 « =(£<,£*) in a domain 233^ defined by {zik = ( C . C * ) ; dem, I 1, · · ·, JV}. The first problem to be solved concerns the boundary 32B^. SB, is a plane cut along the positive real axes. That this is so can be seen by squaring
P R O P E R T I E S OF W I G H T M A N F U N C T I O N S
the special vectors Ζ = (Ξ°+ΊΗ°1 ~Ζ)Η°> 0. 39EB, is therefore the cut 2η = ρ > 0 real. More generally: the cut 2ί 1= : ρ > 0 is in 32B^.
To find the rest of the boundary 3823^ one discusses the inter
section of the boundaries ddl* and 331^ of dl* and di'N, respectively.
I t is clear that 93 = 39ΐ*Λ 33ϊ^ contains at least one representative of every equivalence class of boundary vectors of dt'N under L+(C). It can be shown that the scalar products of the vectors in 93 contain all the boundary points of 28^. A point ( & , · · · , £ * ) € 38ΐ^ lies in οϊΆ'Ν if there is no AeL+(G) such that (Λζ19 · · ·, Λζ#) efRy. We will re
strict ourselves to infinitesimal Λ and this also can be justified in all the relevant cases, ( d , · · ·, ζΝ) e 39ΐΛ if for some indices lc' C^edfRx.
If Λ = 1+εΜ, Μ = Mx + iM2, ε real, is an infinitesimal Lorentz trans
formation, then Λχ = 1 +εΜ1 and Λ2 = 1 +εΜ2 are also infinitesimal (real) Lorentz transformations. The question is, for what Μ a given Ceddti is mapped into a £e9ii, i.e., for what Μ the (Im (1+εΜ)ζ)*
becomes positive. Now with ζ = ζ +ΐη : δη= ε(Μχη+Μ2ξ), and (VI.5) δη* = 2ε((Μιη+ Μ2ξ)-η) = 2ε(Μ2ξ·η) = - 2β(£· Μ%η), or
(VI.6) δη* = ελμν{ημ/\ξν) = ελ·{ηΛξ),
where ημ/\ξν=-ημξν—ηνξμ and Α^ = — λνμ is in a simple way related to Mz. Our condition is therefore λ-(η/\ξ) > 0. If d , * ' ' ί η ' each are in 39?!, then we expect that ( d > " " v f* )e8 3 if the inequalities
(VI.7) A - ( i j * A { r) > 0
contain a contradiction. This is actually only true if none of the (*7*'Λέν) = 0. It is natural to introduce the convex cone
(VI.8) Ϊ == a = 2 O f r' A f. V ; «*' ^ 0, £ a*' > 0 .
Then (VI.7) describes a supporting plane λ·α> 0. It is evident that (VI.7) cannot be satisfied if and only if α = 0 is a point in Ϊ: ( d , · · · ,ζΝ)β 93 if there exist a* > 0 such that
(VI.9) 2 (^, Λ !*<)<**' = 0.
We are evidently only interested in the case afc' > 0 because other
wise already a subset of the inequalities (VI.7) contains a contra
diction. We will discuss now the cases n = 2, 3.
R E S J O S T
Case η = 2:
(VI.10) forAfr)*1' +
(rjrhM*
2' =
0 .This implies that d* a re in a 2-plane. This case can there
fore be completely discussed in 2-dimensions and this has already been done.
Case η = 3 (=N):
(VI.11) O j i A f i ^ - f (ι?,Λί,)αβ + 0?.Λί,)α» = 0
with af c> 0 . This implies that the vectors η2, ηζ, fu ξ21 ξζ are in a 3-plane. W e can therefore limit the discussion to 3 dimensions.
Then ( V I . l l ) says in part that the planes through η1ξ1*, η2ξ2', η*ξζ cut in a line. Let us finally give a representation of those boundary points of 2S3, for which η19 η2, η3 are linearly independent and which satisfy ( V I . l l ) : \\zik\\ denotes the matrix ||(£<£*)II·
(VI.12) \\zik\\= DANAD,
where D is diagonal whith positive diagonal elements, AT= A is sym
metric, has real off-diagonal elements and complex diagonal elements with positive imaginary part and
Proof: Since η2, η3 are linearly independent one can always find 0*>Ο such that η1ύ = β1^1 and (ηΐ)2 = 0 (ηΙ,ηϊ)=1 for k^l Further
(Vi.13) f t = l M ! -
ι
Kow condition ( V I . l l ) implies the existence of a * > 0 such that (VL14) Σ2**β**ΜΛηϊ)=0,
A: I
which is only possible if
(VI.15) \\akl\\ = \\K^*bkl\\ = A,
is a symmetric matrix. Thus the matrix 2? = ||6fcI|| can be written as B = BA1, where D is diagonal and has positive diagonal elements dk.
P R O P E R T I E S O F W I G H T M A N F U N C T I O N S
Appendix to part IV
We have shown in par. I V that a theory of a real scalar field A(x) (satisfying all the axioms of Wightman, including locality), for which the Wightman functions 28 (#0, xu · · ·, xN) for ^V<4 agree with the corresponding functions of a free field, is identical with a free field theory.
We want to show here that the same conclusion can be derived from the weaker assumption
(A.1) (A(x)A(y)}0 = - iA+(x - y),
where Α+(ξ) is the familiar singular function belonging to a certain mass m > 0.
As in part I V we conclude from (A.l) first
(A.2) ( • + m2)A = 0 .
Equation (A.2) allows an invariant decomposition of A(x) into a posi
tive and a negative frequency part:
(A.3) A{x) = A+{x) + A_(x) .
The state Α+(χ)Ω would be a superposition of states with negative energies. Since these should not exist we have
(A.4) Α+(χ)Ω = 0 . Next we analyze the states A+(x) Α_^)Ω.
These states would be superpositions of states to space like energy- momentum vectors. Since the vacuum is the only state of this kind A+(x) Α^)Ω must be a multiple of Ω itself. But from (A.4) we have (A.5) (A\x)A{y)y0 = <A+(x) A4y)\,
We have therefore arrived at the following representation of £*:
(VI.16) £*= Σ dk{akl + i f l A O C >
ι
which leads directly to (VI.12) with
(VI.17) Ώ = \\dkdkl\\ and A = \\akl +
ίβΑιΙΙ .
R E S J O S T
therefore,
(A.6) A+{x)A.(y)Q = — iA+(x — y)Q, and
(A.7) [A+(x), A.(y)]D = - iA+(x - . Equation (A.7) and the trivial equation
(A.8) [A+(x), A+(y)]D = 0 lead to
(A.9) [A(x), A(y)]Q =
iA(x-y)Q
+[A.(x),
A4y)\Q .L e t χ be an arbitrary state and
(A.10) F(x, y) = (χ, [A_(x), A4y)]Q) . Equation (A.9) and the postulate of locality imply
( A . l l ) F(x, y) = 0 for (x — y)2< 0, but F(x, y) has evidently an analytic continuation F(z, w) into ImzeV-, I m y e V _ , because the Fourier transform X-(p) has its sup
port in y _ . The vanishing boundary values ( A . l l ) then imply Fix, y) = 0 and
(A.12) [A{x), A(y)]Q - iA(x — y)Q .
By this we have reached the essential point in the proof of part I Y (*).
* The result of this appendix has been given in the lectures. The proof, however, was not given. It is clear that I have left out all distribution- theoretic subtleties. I would also like to acknowledge my discussions with the group of theoriticians in Zurich.
Note added in proof. The problem mentioned at the bottom of p. 139 has recently been solved by H. Araki (oral communication). He was able to derive the main results of Gr. Luders within the framework of Wightman's formalism.
P R O P E R T I E S O F W I G H T M A N F U N C T I O N S
R E F E R E N C E S
1. A. Wightman, Phys. Rev., 104, 860 (1955).
2. I). Hall and A. Wightman, Kgl. Danske Videnskab. Selskab Mat.-fys.
Medd., 34, no. 5 (1957).
3. R. Jost, Helv. Phys. Acta, 30, 409 (1957).
4. D. Ruelle, Helv. Phys. Acta, 32, 135 (1959).
5. H. Behnke and P. Thullen, Ergeb. Math., 3, no. 3, 52 (1934).
6. H. Lehmann, K. Symanzik and W . Zimmermann, Nuovo Cimento, 1, 205 (1955).
7. See ref. (2), p. 35.
8. This result has been independently derived by O. W. Greenberg Phys. Rev., 115, 706 (1959).
9. See ref. (2). The slight simplification used is due to H. Stapp. I would like to thank him for his remark.
10. This very crucial fact was first noted by W . Pauli, Phys. Rev., 58, 716 (1940).
11. H. Lehmann K. Symanzik and W . Zimmermann, Nuovo Cimento, 6, 319 (1957).
12. N. Burgoyne, Nuovo Cimento, 8, 607 (1958).
13. G. Luders, Z. Naturforsch., 43a, 254 (1958).
14. W. Pauli, Progr. Theoret. Phys., Kyoto, 5, 526 (1950).
15. G. Kall&n and A. Wightman, Kgl. Danske Videnskab. Selskab Mat.-fys.
Skrifter, 4, no. 6, 48 ff. (1958).
