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Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric
24 July 1997
PHYSICS
LETTERS B
Physics Letters B 405 (1997) 243-248
Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric Sergio Albeverio a,b,c, Hanno Got&chalk a, Jiang-Lun Wu a*b*d a Fakultiit und Institut fir Mathematik der Ruhr-Universitiit Bochum, D-44780 Bochum, Germany b SFB237 Essen-Bochum-Diisseldodorf; Germany c BiBoS Research Centre, Bielefeld-Bochum, Germany and CERFIM, Locamo, Switzerland d Institute of Applied Mathematics, Academia
Sinica, Beijing 100080, PR China
Received 2 April 1997; revised manuscript received 30 April 1997 Editor: PV. Landshoff
Abstract We study models of self-interacting massless spin 1 local relativistic quantum fields with indefinite metric in space-time dimension four. We prove that these models for large times converge to free fields and we derive explicit formulae for their (nontrivial, gauge invariant) scattering amplitudes. These scattering amplitudes have properties expected for S-matrix theory. @ 1997 Elsevier Science B.V.
1. Introduction
Since the early seventies, the construction of interacting local quantum fields has been related to the construction of Euclidean random fields. Several models( especially in space-time dimension two) have been constructed from such Euclidean fields. From the 80’s on, a program was started to construct interacting Euclidean random vector fields by solving stochastic partial differential equations driven by a non-Gaussian white noise(cf. [3] ). In [4] the Wightman functions of a local relativistic quantum field model were derived by performing the analytic continuation of the corresponding Schwinger functions (moments) of these random fields (described in terms of a first order stochastic partial differential equation, see [ 3,4] ) . The following explicit formulae for the truncated Wightman functions were obtained (here and in the following, we use the Einstein convention of summation: equal upper and lower Greek indices are running from 0 to 3 and Latin indices are running from 1 to 3):
(1) 0370-2693/97/$17.00
0 1997 Elsevier Science B.V. All rights reserved
PI1 s0370-269?l(97)00574-1
244
S. Alhrurrio
et d/Physics
Letters
B 405 (1997)
243-248
(2) for n 2 3, where Xj = (.$,xj) E R x R3,j = I,. . . , n, c, CO,c~,,,, are constants depending on the law of the noise, gap is the Minkowski metric and (XI - ~2)~ := -(.$ - x~)* + /xt - ~21~. Moreover, d is the exterior derivative on R4 and DM = DM( f-, &) is the differential operator defined by D&
L) ax] ’ ax2
:= -gjl*M(e’A(*~(eo*e~)+ieo~e’))~*M(ePh(*~(eo*e’)+ieohe’))~~
(3) I
2
with *M the Hodge star operator. {e’, e’, e2,e3} is the dual of an orthogonal (with respect to gas) frame on R4. G,,, n > 3, in Elq. (2) is a tempered distribution defined as the inverse Fourier transform of (cf. [ l] ) n j-1 &(k,,...
,kn) :={~l+Tj-(k&$ j=l
1=I
6
$(kl)}&kr)
’ l=J+l
(4) I=1
with S:(k) := l~p~o>(ko)6(k2) = h&k0 F lkl) (h ere we take the convention on the Minkowski metric as in [ 41 which carries the opposite sign as in [ 1 ] ) . G,, n 2 3, are basic quantities from which the Wightman function can be computed. We call G,, n > 3, the structure functions. If not mentioned otherwise, singularities like $ have to be tacitly understood in the sense of Cauchy principal value. Recently we proved in [2] that the Wighiman functions given by Eqs ( 1) and (2) fulfil the modified Wightman axioms (i.e., Poincare invariance, locality, hermiticity, spectral condition and Hilbert space structure condition) of Morchio and Strocchi [ lo]. Thus, these Wightman functions are associated with a local relativistic quantum field theory with indefinite metric (this is different from the free electromagnetic potential field unless c,,, = 0). In this letter, we calculate the scattering amplitudes for the above mentioned relativistic local quantum field models. We show they are nontrivial and have properties expected from S-matrix theory. The underlying local relativistic quantum field models thus constitute the first known models in four space-time dimensions with nontrivial (gauge invariant) S-matrix (models with nontrivial S-matrix, however without locality of the interpolating fields, have been discussed in [ 1l] and [ 51). To this aim, we first calculate the scattering amplitudes related for the distributions Gn, n > 3, as in the next section. 2. Scattering amplitudes associated with the structure functions For I= 1, . . . , n, let $1 be a smooth fast falling function on R4 with support in (0, co) x lR3. We then define wave packets 1 4$(x) := w
s
eik”@l(kl) e ‘(@-lkll)td,&
(5)
w4
where - denotes the Minkowski inner product. The complex conjugate * is given by
(6) It is well known that the essential support of the wave packets (DF’(*) is concentrated near the hyperplane x0 = ft for t + +co. The following statement is the crucial mathematical result of this letter
S. Albeverio et al. / Physics Letters B 405 f 1997) 243-248
Theorem 1.
245
Let G,,, n 2 3, and @f be given as above. Then
...(prt,*(xr)sp:+,(xr+l) -qf,(~n)dx~ ---hi
, .,x,)p,t+*(x~)
&(a,.
fp,t r =
s,f(kl)G(Ckl-
2Ti
I=1
e
kl)$:(ki)
.**@;(kr)&+l(kr+l)
. ..&(k.)dkl...dk,
(7)
l=r+l
and G,(xI,.
2% J
(xl)...rp~‘,*(x,)(o,f:l(x,+l)...~~t(~,)dx,...dx,=O. ..~n>sp~‘**
(8)
hf. We will prove Bq. (8) only for the plus sign, the proof of Eq. (8) for the case of the minus sign is done in the same manner. By Parseval theorem and Eqs (5) and (6), we have
I Gn(~~,...,~,)~~t,*(x,)... =
J
P;~~*(x~)~P;+,(x,+,) ~-(o;(.ddxl . ..dx.,
. . . , kn)~::(_k,)e~i(k~+lkll)t..
&(k,,
.&j;( _krJe&+ikdt
IIp4” x ~r+I(kr+,)ei(k~+,-lk,+ll)f..
.~n(k,)ei(kRI-lk,~l)‘dkl..
.dk,, .
(9)
We now put Eq. (4) in Eq. (9). By the support properties of ~$1,I = 1,. . . ,n, we get that only those summands in Eq. (4) with j = r or j = r + 1 give a nonzero contribution. Thus, the right hand side of Eq. (9) is equal to
.c$,(k,)dkl
. ..dk.
(10) But in the sense of tempered distributions, cf. [6], we have
= &i&(x)
using this, $ = i
,
Cq-lkjD:q+lk,l,'
(11)
and the definition of S,f, we get from Eq. ( 10) in the limit t --+ +cc
S. Albeuerio et d/Physics
Letters B 405 (19971243-248
Changing now variables kl + -7~ for 2 = 1,. . . , r we get the statement of the theorem. n We call the quantities given by Eq. (7) scattering amplitudes associated with the structure functions G,, n > 3.
3. The scattering amplitudes of the models From now on we want to work in the Feynman gauge, i.e., we assume that co = c (this can be achieved by adequately choosing the law of the noise). In this case, the Fourier transform of IIJ~,~ simply is &&Jkiw-I
+ k2> ’
(12)
Let us choose vector wave packets @I,1 = 1, . . . , n, which are smooth fast falling functions from IR4 to C4 with support in (0, +co) x lR3. Furthermore, we impose the following gauge condition on the &: g&,P(k),@=O
forZ=1,...,nandkER4.
(13)
We now want to define the scattering amplitudes of the model in the usual way (see, e.g. [ 8 3) : Let vi(x) defined in the manner of Eq. (5). For n E N, 1 5 r 5 n, we set (+9p1 . . .cpy/qr+J.. .py
:= pm
W$,,...,,,(x1,.
. . ,x”)qJf?-“*(XI).
be
. .py*(&)
s Iw x ‘P;$.~(x,.+,)
(cpr. . .(p:“lq,+l..
. ..dx.
y~~+~)dx, 4p”)’
:= ,
w~a,...&Jx~,.
(14) . . ,x,)p;“-“*(x1).
. .py’*(&>
s p’
x c$$‘*-~(x,+~)
.q~9-t(~,)dx,
. ..dx.,
(15)
and let (4pr. . . ~#~[cp,+1~. . ~pp’)~ be defined analogously to Eq. (15) with --t replaced by t. Furthermore, for n, m E N, 0 5 m 5 n, and m is even, we define the ( C4) @‘“-valuedpolynomial Pn,nr(kl,. *. ,k,) := (-i>“(2~)3-2”Sym((k’;~.~.~k~_,~D~(kn-m+~,kn-m+2)~.~.~~~(kn-~,k,)) (16)
where ki is the one form dual to the vector kl with respect to g,p. We are now ready to state our main result: Theorem 2. Let the truncated Wightman functions of the model be given by Eqs. ( 1) and (2) and let , n, be vector wave packets which fulfil the on-shell gauge condition Eq. ( 13). Then the truncated &,Z=l,... scattering amplitudes for the model are given by the following formulae: +PYl&Y = ;
$(kt)&kr / iPxR4
- k~)ga~~;l~*(~M$?~2)
dkt&
>
(17)
S. Albeverio
et ul./ Physics Letters B 405 (1997)
241
243-248
where the two ex stand for all combinations of in and out. For n 2 3, n is even,
(~1. . .sp?J(Dr+, ..+P~“~)’= --2rh
X
f&(k&k, I=1
I=1
s
p,,,(-kl,.
. . , -k,, k+l,. . . , k,), ,...a,,
2 kl)#?3* (kl) . ..~~.*(k,)~~~~(k,+,) l=r+1
. ..c$?(k.,)
dk,...dk,
(18)
while for n 2 3, n is odd, the left hand side of Eq. ( 18) is zero. Furthermore, for I + +co, the field converges to a free field, i.e., for n 2 3, we get ($01. . * pplp,+l.~.(D”)’= ($7, . . .pl)“tIcpril..
‘q7;“ty = 0.
(19)
Proof. Eq. (17) can directly be derived from the definition of the scattering amplitudes and Eq. (12) making use of the Parseval theorem. To prove Eqs (18) and (19) we notice that the Fourier transform of Wtm,...n,, is of the form
2 m=o:m
(;) --&c,,,&.,(k,,. ..,k,)a ,,,&h.. I...
.,kn)
9
even
where e,, n 2 3, are the Fourier transform of the structure functions, given by Eq. (4). If the distribution defined by Eq. (20) is smeared out with the wave packets gF’*‘( &kt)e*‘(@*lki)‘, then, by duality, this yields the same result as e,, (kl , . . . , kl) smeared with these wave packets multiplied by some polynomial in kl , . . . , k,. But this product again can be written as a finite sum of products of wave packets as in Eq. (5) or Eq. (6), respectively. Thus, Eq. ( 18) and Eq. ( 19) follow directly from Theorem 1 (the fact that only the term m = n in Eq. (20) gives a nonzero contribution is a consequence of the gauge condition). W The S-matrix constructed here satisfies important properties of S-matrix theory (see e.g. [ 7,9] ), e.g. crossing symmetry. A question which is so far not settled is concerning a physical Hilbert space for our model, i.e., a subspace of the state space which consists of gauge invariant states and on which the (indefinite) inner product induced by the Wightman functions is positive semidefinite. However, the results of Theorem 2 indicate that the local fields A(x”, x) of our model converge to free fields Ainlout(x0, x) for x0 -+ foe. Moreover, at least for the free fields (if these can be defined as operators on the state space) there exist XF$“’ consisting of gauge invariant states (in the sense of [ 121) on which the inner product is positive semidefinite. In fact, these states are singled out by the Gupta-Bleuler gauge condition which in our case is hidden in Eq. ( 13). Thus, the gauge tranformation A(x’, x) H A(x”, x) + d,y(x’, x), where x is any scalar quantum field, in the limit x0 --+ &co leaves the inner product of states from ?$$y’ invariant. Furthermore, by Eqs (3), (16) and (18), we have kyP,,,,(kl,...,
kj ,...,
k,), ,... a,... (1,,=O for j=
l,...,
n
and kj in the (lightlike) support of the scattering amplitude. This implies that the S-matrix maps ‘7-$&_to ‘FIi&. Since in scattering experiments only states from 7-1F$yt can be observed, the S-matrix of our model can be considered as gauge invariant. The financial support of SFB 237 is gratefully acknowledged. We thank the referee for a stimulating remark.
248
S. Alheuerio
ef ul./Physic.v
Letters
B 405 (19971
243-248
References [ I] S. Albeverio, H. Gottschalk and J.-L. Wu, Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions, Rev. Math. Phys. 8 ( 1996) 763. [ 21 S. Albeverio, H. Gottschalk and J.-L. Wu, Models of Local Relativistic Quantum Fields with Indefinite Metric (in All Dimensions), Commun. Math. Phys. 184 (1997) 509. ]3] S. Albeverio and R. H@egh-Krohn, Euclidean Markov fields and relativistic quantum fields from stochastic partial differential equations, Phys. Lett. B 177 (1986) 175. [4] S. Albeverio, K. Iwata and T. Kolsrud, Random fields as solutions of the inhomogeneous quatemionic Cauchy-Riemann equation. I. invariance and analytic continuation, Commun. Math. Phys. 132 (1990) 555. ]5] H. Baumgiirtel and M. Wollenberg, A class of nontrivial weakly local massive Wightman fields with interpolating properties, Commun. Math. Phys. 94 (1984) 331. [ 61 E Contantinescu, Distributionen und ihre Anwendung in der Physik (Teubner, Stuttgart, 1973). [7] R.J. Eden, I? Landshoff, D.I. Olive and J.C. Polkinghome, Analytic S-Matrix (Cambridge University Press, Cambridge, 1966). [ 81 K. Hepp, On the connection between the LSZ and Wightman quantum field theory, Commun. Math. Phys. 1 ( 1965) 95. [ 91 D. Iagolnitzer, Scattering in Quantum Field Theories. The Axiomatic and Constructive Approaches (Princeton University Press, New Jersey, 1993). [lo] G. Morchio and E Strocchi, Infrared singularities, vacuum structure and pure phases in local quantum field theory, Ann. Inst. H. Poincam A 33 ( 1980) 251. [ 111 W. Schneider, S-Matrix und interpolierende Felder, Helv. Phys. Acta 39 ( 1966) 81. [ 121 F. Strocchi and A.S. Wightman, Proof of the charge superselection rule in local relativistic quantum field theory, J. Math. Phys. 15 (1974) 2198.
PHYSICS
LETTERS B
Physics Letters B 405 (1997) 243-248
Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric Sergio Albeverio a,b,c, Hanno Got&chalk a, Jiang-Lun Wu a*b*d a Fakultiit und Institut fir Mathematik der Ruhr-Universitiit Bochum, D-44780 Bochum, Germany b SFB237 Essen-Bochum-Diisseldodorf; Germany c BiBoS Research Centre, Bielefeld-Bochum, Germany and CERFIM, Locamo, Switzerland d Institute of Applied Mathematics, Academia
Sinica, Beijing 100080, PR China
Received 2 April 1997; revised manuscript received 30 April 1997 Editor: PV. Landshoff
Abstract We study models of self-interacting massless spin 1 local relativistic quantum fields with indefinite metric in space-time dimension four. We prove that these models for large times converge to free fields and we derive explicit formulae for their (nontrivial, gauge invariant) scattering amplitudes. These scattering amplitudes have properties expected for S-matrix theory. @ 1997 Elsevier Science B.V.
1. Introduction
Since the early seventies, the construction of interacting local quantum fields has been related to the construction of Euclidean random fields. Several models( especially in space-time dimension two) have been constructed from such Euclidean fields. From the 80’s on, a program was started to construct interacting Euclidean random vector fields by solving stochastic partial differential equations driven by a non-Gaussian white noise(cf. [3] ). In [4] the Wightman functions of a local relativistic quantum field model were derived by performing the analytic continuation of the corresponding Schwinger functions (moments) of these random fields (described in terms of a first order stochastic partial differential equation, see [ 3,4] ) . The following explicit formulae for the truncated Wightman functions were obtained (here and in the following, we use the Einstein convention of summation: equal upper and lower Greek indices are running from 0 to 3 and Latin indices are running from 1 to 3):
(1) 0370-2693/97/$17.00
0 1997 Elsevier Science B.V. All rights reserved
PI1 s0370-269?l(97)00574-1
244
S. Alhrurrio
et d/Physics
Letters
B 405 (1997)
243-248
(2) for n 2 3, where Xj = (.$,xj) E R x R3,j = I,. . . , n, c, CO,c~,,,, are constants depending on the law of the noise, gap is the Minkowski metric and (XI - ~2)~ := -(.$ - x~)* + /xt - ~21~. Moreover, d is the exterior derivative on R4 and DM = DM( f-, &) is the differential operator defined by D&
L) ax] ’ ax2
:= -gjl*M(e’A(*~(eo*e~)+ieo~e’))~*M(ePh(*~(eo*e’)+ieohe’))~~
(3) I
2
with *M the Hodge star operator. {e’, e’, e2,e3} is the dual of an orthogonal (with respect to gas) frame on R4. G,,, n > 3, in Elq. (2) is a tempered distribution defined as the inverse Fourier transform of (cf. [ l] ) n j-1 &(k,,...
,kn) :={~l+Tj-(k&$ j=l
1=I
6
$(kl)}&kr)
’ l=J+l
(4) I=1
with S:(k) := l~p~o>(ko)6(k2) = h&k0 F lkl) (h ere we take the convention on the Minkowski metric as in [ 41 which carries the opposite sign as in [ 1 ] ) . G,, n 2 3, are basic quantities from which the Wightman function can be computed. We call G,, n > 3, the structure functions. If not mentioned otherwise, singularities like $ have to be tacitly understood in the sense of Cauchy principal value. Recently we proved in [2] that the Wighiman functions given by Eqs ( 1) and (2) fulfil the modified Wightman axioms (i.e., Poincare invariance, locality, hermiticity, spectral condition and Hilbert space structure condition) of Morchio and Strocchi [ lo]. Thus, these Wightman functions are associated with a local relativistic quantum field theory with indefinite metric (this is different from the free electromagnetic potential field unless c,,, = 0). In this letter, we calculate the scattering amplitudes for the above mentioned relativistic local quantum field models. We show they are nontrivial and have properties expected from S-matrix theory. The underlying local relativistic quantum field models thus constitute the first known models in four space-time dimensions with nontrivial (gauge invariant) S-matrix (models with nontrivial S-matrix, however without locality of the interpolating fields, have been discussed in [ 1l] and [ 51). To this aim, we first calculate the scattering amplitudes related for the distributions Gn, n > 3, as in the next section. 2. Scattering amplitudes associated with the structure functions For I= 1, . . . , n, let $1 be a smooth fast falling function on R4 with support in (0, co) x lR3. We then define wave packets 1 4$(x) := w
s
eik”@l(kl) e ‘(@-lkll)td,&
(5)
w4
where - denotes the Minkowski inner product. The complex conjugate * is given by
(6) It is well known that the essential support of the wave packets (DF’(*) is concentrated near the hyperplane x0 = ft for t + +co. The following statement is the crucial mathematical result of this letter
S. Albeverio et al. / Physics Letters B 405 f 1997) 243-248
Theorem 1.
245
Let G,,, n 2 3, and @f be given as above. Then
...(prt,*(xr)sp:+,(xr+l) -qf,(~n)dx~ ---hi
, .,x,)p,t+*(x~)
&(a,.
fp,t r =
s,f(kl)G(Ckl-
2Ti
I=1
e
kl)$:(ki)
.**@;(kr)&+l(kr+l)
. ..&(k.)dkl...dk,
(7)
l=r+l
and G,(xI,.
2% J
(xl)...rp~‘,*(x,)(o,f:l(x,+l)...~~t(~,)dx,...dx,=O. ..~n>sp~‘**
(8)
hf. We will prove Bq. (8) only for the plus sign, the proof of Eq. (8) for the case of the minus sign is done in the same manner. By Parseval theorem and Eqs (5) and (6), we have
I Gn(~~,...,~,)~~t,*(x,)... =
J
P;~~*(x~)~P;+,(x,+,) ~-(o;(.ddxl . ..dx.,
. . . , kn)~::(_k,)e~i(k~+lkll)t..
&(k,,
.&j;( _krJe&+ikdt
IIp4” x ~r+I(kr+,)ei(k~+,-lk,+ll)f..
.~n(k,)ei(kRI-lk,~l)‘dkl..
.dk,, .
(9)
We now put Eq. (4) in Eq. (9). By the support properties of ~$1,I = 1,. . . ,n, we get that only those summands in Eq. (4) with j = r or j = r + 1 give a nonzero contribution. Thus, the right hand side of Eq. (9) is equal to
.c$,(k,)dkl
. ..dk.
(10) But in the sense of tempered distributions, cf. [6], we have
= &i&(x)
using this, $ = i
,
Cq-lkjD:q+lk,l,'
(11)
and the definition of S,f, we get from Eq. ( 10) in the limit t --+ +cc
S. Albeuerio et d/Physics
Letters B 405 (19971243-248
Changing now variables kl + -7~ for 2 = 1,. . . , r we get the statement of the theorem. n We call the quantities given by Eq. (7) scattering amplitudes associated with the structure functions G,, n > 3.
3. The scattering amplitudes of the models From now on we want to work in the Feynman gauge, i.e., we assume that co = c (this can be achieved by adequately choosing the law of the noise). In this case, the Fourier transform of IIJ~,~ simply is &&Jkiw-I
+ k2> ’
(12)
Let us choose vector wave packets @I,1 = 1, . . . , n, which are smooth fast falling functions from IR4 to C4 with support in (0, +co) x lR3. Furthermore, we impose the following gauge condition on the &: g&,P(k),@=O
forZ=1,...,nandkER4.
(13)
We now want to define the scattering amplitudes of the model in the usual way (see, e.g. [ 8 3) : Let vi(x) defined in the manner of Eq. (5). For n E N, 1 5 r 5 n, we set (+9p1 . . .cpy/qr+J.. .py
:= pm
W$,,...,,,(x1,.
. . ,x”)qJf?-“*(XI).
be
. .py*(&)
s Iw x ‘P;$.~(x,.+,)
(cpr. . .(p:“lq,+l..
. ..dx.
y~~+~)dx, 4p”)’
:= ,
w~a,...&Jx~,.
(14) . . ,x,)p;“-“*(x1).
. .py’*(&>
s p’
x c$$‘*-~(x,+~)
.q~9-t(~,)dx,
. ..dx.,
(15)
and let (4pr. . . ~#~[cp,+1~. . ~pp’)~ be defined analogously to Eq. (15) with --t replaced by t. Furthermore, for n, m E N, 0 5 m 5 n, and m is even, we define the ( C4) @‘“-valuedpolynomial Pn,nr(kl,. *. ,k,) := (-i>“(2~)3-2”Sym((k’;~.~.~k~_,~D~(kn-m+~,kn-m+2)~.~.~~~(kn-~,k,)) (16)
where ki is the one form dual to the vector kl with respect to g,p. We are now ready to state our main result: Theorem 2. Let the truncated Wightman functions of the model be given by Eqs. ( 1) and (2) and let , n, be vector wave packets which fulfil the on-shell gauge condition Eq. ( 13). Then the truncated &,Z=l,... scattering amplitudes for the model are given by the following formulae: +PYl&Y = ;
$(kt)&kr / iPxR4
- k~)ga~~;l~*(~M$?~2)
dkt&
>
(17)
S. Albeverio
et ul./ Physics Letters B 405 (1997)
241
243-248
where the two ex stand for all combinations of in and out. For n 2 3, n is even,
(~1. . .sp?J(Dr+, ..+P~“~)’= --2rh
X
f&(k&k, I=1
I=1
s
p,,,(-kl,.
. . , -k,, k+l,. . . , k,), ,...a,,
2 kl)#?3* (kl) . ..~~.*(k,)~~~~(k,+,) l=r+1
. ..c$?(k.,)
dk,...dk,
(18)
while for n 2 3, n is odd, the left hand side of Eq. ( 18) is zero. Furthermore, for I + +co, the field converges to a free field, i.e., for n 2 3, we get ($01. . * pplp,+l.~.(D”)’= ($7, . . .pl)“tIcpril..
‘q7;“ty = 0.
(19)
Proof. Eq. (17) can directly be derived from the definition of the scattering amplitudes and Eq. (12) making use of the Parseval theorem. To prove Eqs (18) and (19) we notice that the Fourier transform of Wtm,...n,, is of the form
2 m=o:m
(;) --&c,,,&.,(k,,. ..,k,)a ,,,&h.. I...
.,kn)
9
even
where e,, n 2 3, are the Fourier transform of the structure functions, given by Eq. (4). If the distribution defined by Eq. (20) is smeared out with the wave packets gF’*‘( &kt)e*‘(@*lki)‘, then, by duality, this yields the same result as e,, (kl , . . . , kl) smeared with these wave packets multiplied by some polynomial in kl , . . . , k,. But this product again can be written as a finite sum of products of wave packets as in Eq. (5) or Eq. (6), respectively. Thus, Eq. ( 18) and Eq. ( 19) follow directly from Theorem 1 (the fact that only the term m = n in Eq. (20) gives a nonzero contribution is a consequence of the gauge condition). W The S-matrix constructed here satisfies important properties of S-matrix theory (see e.g. [ 7,9] ), e.g. crossing symmetry. A question which is so far not settled is concerning a physical Hilbert space for our model, i.e., a subspace of the state space which consists of gauge invariant states and on which the (indefinite) inner product induced by the Wightman functions is positive semidefinite. However, the results of Theorem 2 indicate that the local fields A(x”, x) of our model converge to free fields Ainlout(x0, x) for x0 -+ foe. Moreover, at least for the free fields (if these can be defined as operators on the state space) there exist XF$“’ consisting of gauge invariant states (in the sense of [ 121) on which the inner product is positive semidefinite. In fact, these states are singled out by the Gupta-Bleuler gauge condition which in our case is hidden in Eq. ( 13). Thus, the gauge tranformation A(x’, x) H A(x”, x) + d,y(x’, x), where x is any scalar quantum field, in the limit x0 --+ &co leaves the inner product of states from ?$$y’ invariant. Furthermore, by Eqs (3), (16) and (18), we have kyP,,,,(kl,...,
kj ,...,
k,), ,... a,... (1,,=O for j=
l,...,
n
and kj in the (lightlike) support of the scattering amplitude. This implies that the S-matrix maps ‘7-$&_to ‘FIi&. Since in scattering experiments only states from 7-1F$yt can be observed, the S-matrix of our model can be considered as gauge invariant. The financial support of SFB 237 is gratefully acknowledged. We thank the referee for a stimulating remark.
248
S. Alheuerio
ef ul./Physic.v
Letters
B 405 (19971
243-248
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