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Current definition and a generalized federbush model
ANNALS
OF PHYSICS
Current
115,
136152
Definition
(1978)
and a Generalized
Federbush Model*
L. P. S. SINGH Physics Department, Pahlavi University, Shirar, IRAN AND
C. R. HAGEN Department of Physics and Astronomy,
University of Rochester, Rochester, New York, 14627
Received August 4, 1977
The Federbush model is studied, with particular attention being given to the definition of currents. Inasmuch as there is no a priori restriction of local gauge invariance, the currents in the interacting case can be defined more generally than in Q.E.D. It is found that two arbitrary parameters are thereby introduced into the theory. Lowest order perturbation calculations for the current correlation functions and the Fermion propagators indicate that the theory admits a whole class of solutions dependent upon these parameters with the closed solution of Federbush emerging as a special case. The theory is shown to be locally covariant, and a conserved energy-momentum tensor is displayed. One finds in addition that the generators of gauge transformations for the fields are conserved. Finally it is shown that the general theory yields the Federbush solution if suitable Thirring model type counterterms are added.
INTRODUCTION The two-dimensional theory of Federbush [l] is essentially unique, being the only model with massiveFermions for which an exact nontrivial solution is known [l, 2, 31.
The model involves the current-current coupling h E,,j@‘JYwhere j,, and J, are the vector currents associatedwith the Dirac fields $ and Y, respectively. The study presented here begins with a perturbation calculation, to lowest order, of the fermion propagators and the current correlation functions. Schwinger’s method of external sources [4] is used, and the currents in the interacting case are constructed by a well-defined limiting procedure. As noted in a previous treatment of several two dimensional models [5, 61 there are no a priori restrictions required by local gauge invariance, a fact which allows a more general definition of the currents than that of Schwinger for the case of the electromagnetic current [7]. Thus two arbitrary parameters q and H (unrelated to those in the Lagrangian), corresponding * Research supported in part by the U.S. Energy Research and Development under contract number EY-76-C-02-3065.000. 136 OOO3-4916/78/1151-0136$05.00/0 Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
Administration,
GENERALIZED
FEDERBUSH
137
MODEL
to j, and J, , respectively, are introduced into the theory. In particular, 7 = 0 (respectively, H = 0) corresponds to the case where j, (respectively, J,) is conserved in the presence of interactions. The perturbation calculations are found to be in agreement with the exact solution of Refs. [l-3] only for the case 7 = H = 0. In particular there is no mass renormalization, and the amplitude for particle-antiparticle scattering is found to vanish. On the other hand a class of new solutions is generated when either or both of q, H are nonzero. One finds then that the mass renormalization is logarithmically divergent although the wavefunction renormalization remains unchanged. The particleantiparticle scattering amplitude is also found to be nonzero. However, the Federbush solution is reproduced if additional Thirring model [S] type terms are included in the Lagrangian. We subsequently proceed to analyze the operator formalism of the theory. A conserved energy-momentum tensor is obtained and the theory is shown to be locally covariant. As in the case of Refs. [.5, 61 it turns out that the canonical energy density does not suffice and noncanonical terms have to be added. One finds that there are always two conserved currents in the theory whose corresponding charges generate gauge transformations of the first kind on the Dirac fields $J and Y. Finally we seek the relation between the general solution for arbitrary values of 7 and Hand the Federbush solution which corresponds to q =z H = 0. It is found that upon including suitable Thirring model type counterterms the former theory becomes equivalent to the Federbush model with conserved vector currents. The model is outlined and the notation established in Section 2, which also includes a summary of the external source method. In Section 3 the current correlation functions and the particle-antiparticle scattering amplitude are obtained to lowest order. Section 4 contains the lowest order calculation for the Fermion propagators while the operator properties are discussed in Section 5. ln Section 6 the connection between theories corresponding to different values of 7 and H is established. We end with a few concluding remarks.
2. The Federbush
model is formally
defined by the Lagrangian
27 = (i/2)(&Q,t4 + k”“juJv
PRELIMINARIES
+ Fy’a,Y) +,j,,a”
- &?1tJ# -
+ J,,A”
;-MFY (1)
where 4 = #y”, and a,, and A, are external sources which will be set equal to zero in the end. The notation used here is such that the metric is gl'
=
-.-go0
=
1, go1 =
0
138
SJNGH
while the Levi-Civita
tensor P = -F
AND
HAGEN
is specified by
Hermitean fields are used so that the y” are pure imaginary antisymmetric and symmetric respectively. These satisfy y”y” = -g””
+ cLL”y5)
a relation which also defines the real symmetric obtains the useful identity
In order to construct currents makes use of the charge matrix
the Lagrangian
pseudoscalar
matrix y5 . One readily
one doubles the number of field components
in the associated charge space. If the currents .i”(X) = Mx)
with /3 = y” and y1 being
4YU$4”),
are formally
J”(x)
= $F(x)
and
defined as qyY(x),
(2)
(I) leads to the equations of motion [Y”(P,
- vu) + ml+ = kv,d@‘Jv
(34
and
[Y”(P, - !A) + WY = --x~Y,~%~,
(3b)
where pu = -3, and all operator products such as t,bJuare understood to be symmetrized. The action principle also implies the equal time anticommutators iK4> KJJ>> = VW3 KY)> = (xx - VI, {?&a KY>> = 0.
(4)
In order to obtain the solution of the coupled theory one makesuseof the dynamical aspect of the action principle. This principle assertsthat the variation in a vacuum expectation value (0 j R(x) 10) arising from the variation of any parameter in the theory is given by
where 66p and 6R are the resulting variations in Sand R respectively, and T denotes time ordering. Thus variation of the coupling constant yields
GENERALIZED
FEDERBUSH
139
MODEL
where the subscripts denote the dependence of the matrix elements on various parameters. Variations with respect to the external sources result in &
(0 i R(x) I o>A,n,~ = Q I WW.L(.d)
I @A.~,A
(6)
and &
.
(0 I NJ4 I Oh,n,A = 0
I W(x)
J,(Y)) I O)A.O.A
(7)
provided R does not explicitly contain a, or A,, . Thus one has
a relation which is trivially
integrated to yield
(0iR(.u) IO~A.~.A = exp (--iAs4 cuv&&)
*(0I R(x) IOh=, . (8)
The problem thus reduces to the determination of matrix elements as functions of a, and A, for h = 0, i.e., in the limit in which the field equations (3) decouple. In the case of massless fields these equations can be explicitly solved, so that closed form expressions for the matrix elements of the coupled theory can be obtained. In the massive case, however, an iterative procedure has to be employed thus reproducing the ordinary perturbation expansion. The following two sections are devoted to such calculations for the current correlation functions and the propagators.
3. THE
CURRENT
CORRELATION
We begin with the lowest order calculation j, which is defined by TU”(.u, Y> = &
FUNCTION
for the current correlation
(j”(Y)>
function for
(9)
where =
(0 IL(x) I 0) (0 IO> .
(10)
To carry out this calculation the current j, must be suitably defined in the presence of the external field. As is well known, the electromagnetic current should be defined as
where the limit E + 0 is to be taken in a symmetric fashion along a spacelike surface, usually taken to be the surface x0 = constant.
140
SINGH
AND
HAGEN
In electrodynamics the prescription (11) is required by local gauge invariance. However, in cases such as the present one, where there is no gauge field, and consequently no a priori restriction arising from gauge invariance of the second kind, a more general definition can be employed. This fact has been used to generate new solutions for the Thirring model [5], and for a model involving a massless Fermion current coupled to a massive vector Boson [6]. In the same spirit, we define the current to be
6 and 71 being arbitrary real parameters. It will subsequently be seen that Lorentz invariance imposes the condition
!$+7=1.
(13)
In view of Eq. (12), one has (L(x)>
= f 1:~ Tr y,@%x + E, x) exp[ --iq j”” x
d&(5gaB + yys@) Q&J)], (14)
with the Fermion propagator G* defined as (15)
which, for h = 0 satisfies h4~~ - q4x))
+ m I W-T v) = %x9 v>.
(16)
Since Eq. (16) cannot be solved exactly for m f 0, an iterative procedure must be used. We restrict ourselves to a calculation to lowest order closely following that of Johnson [9] in Q.E.D. In the present case, vU, is given by the first functional derivative in the limit uU --f 0, and it is sufficient to compute (j,) up to terms linear in arr . The result is (j,(x)>
= f 1~5 Tr yU [s dx’ G,*(x + E, x’) yVaV(x’) G,,@‘(x’,x) - iG,“(x + E, x) jxz+’ 4&28
+ rl@w
as(u)] 9
where GoG(x, v) is the free propagator. In view of the symmetric employed it is more convenient to write Eq. (17) as
integration
= k 1:~ Tr yU [I dx’ Go*(x + 42, x’) y’ay(x’) GOti(x’, x - 42)
(17)
to be
GENERALIZED
which, in momentum
FEDERBUSH
141
MODEL
space reads
(18)
+ G~(p)(W+ yYny6) s_‘ltd.u,eig’z\dd.
In the second term, the x integral can be evaluated along a straight line, yielding r/2 dx s -E/z
e*g’s
2~Y
=
y
sin $4 . E q-6
= E, + O(8),
This is multiplied by GOG(e),which in the limit E - 0 behaves as (I/E) + finite terms. Thus it is sufficient to keep the first term in (19). After straightforward manipulation (18) can then be brought to the form
from which a straightforward n,,(q)
= ljz i s &
trace evaluation leads to the result at - 7) I @2 + In2)2
eip.r
m2gw
+ 4 (PUPY - ; &“P2) ( ,(P-~)2+m2;[(P+I).+m.]
)
(20)
Considering first the last term, one finds that its contribution l$i (Y
must be of the form
- ; gw) (5 + 7 - 1>f(E2, m2),
where f (c2, m2) = f
s
&
eip’E ‘W2-+2”
m22
jP)“l
-
(21)
As shown in the Appendix, f has a nonzero limit as E-+ 0. On the other hand, the limit of E~E,/E~ depends on the direction along which the limit is taken, thereby
142
SINGH
AND
HAGEN
leading to a noncovariant result. The situation can be satisfy Eq. (13). The remaining terms in the integrand of Eq. (20) corresponding integral therefore, converges uniformly limit inside the integral. A straightforward calculation ~w(d
= ;
1; &“(f
- 77-
remedied if and only if [ and 71 fall off as p-’ for large p. The in E, allowing one to take the then leads to the result
1) + (g,, - 7)
[l -
I(m,
q)]/
(22)
where I(m, q) = m2 lo1
cl’ 1112+ u(1 - 21)q* *
(23)
It should be noted that conservation of the vector current which would imply 4”~w‘ = 0, is possible only with t - 7 = 1, which in view of (13) would require [=l,q=O.Onth e other hand, a conserved axial current j,u = l “jV in the zero mass limit would imply qUEw”n-,, = 0 and correspondingly yield 5 = 0, 7 = 1. These conditions on 4 and 17 are directly obtained in Section 5 from the operator structure of the theory. It may be noted that the treatment of J, is identical to that carried out here for the currentj, . One has two additional parameters E, H corresponding to [, 77respectively, and all the preceding results are reproduced with the replacements ([, 71) -+ (E, H) andm+M. Having now obtained +‘, the lowest order amplitude for particle-antiparticle scattering is readily calculated. Let e and E be the particles associated with the fields fields $ and Y, respectively. Considering the process e B -+ e .5 with the respective incoming and outgoing momenta being p, p and q, 4, the second-order amplitude is readily seen to be
where 17(q) is the current correlation function for the current J, . Upon substituting the expression corresponding to (22) this becomes
For completeness it may be noted that the amplitude for E E to second order in X is given by an identical expression with H -+ 7, i.e.,
GENERALIZED
FEDERBUSH
143
MODEL
We recall that the explicit solution of Refs. [I, 2, 31 predicts that both these amplitudes vanish, a result which is seen here to be possible only for the case 17= H = 0. We thus conclude that the known explicit solution is reproduced only when the vector currents are defined so as to be conserved in the interacting case. This is not surprising, of course, in view of the fact that the derivation of that solution crucially depends upon current conservation. It is clear that the general definition of the currents leads to new solutions of the model. Also, it can easily be shown that the difference between the Federbush solution and the general solution for the processesunder consideration (to lowest order at least) is of the sameform asthat generated by a massiveThirring type coupling. Thus, at least to the lowest order, a modification of the Lagrangian by
in the general case reproduces the Federbush solution. The possibility of such a correspondence to all orders in h is discussedin Section 6.
4. THE FERMION PROPAGATORS
We next turn to the lowest order calculation for the Fermion propagator defined as
d(d)1o? G$(x >v) = j(’ 1T(#(x) (0I0)
(27)
(the calculations for the propagator of the Y field proceed along identical lines). From the field equations (3a) one has
Upon using Eq. (7), the second term on the right-hand side becomes
kvw~“”((J,(x))- i &)
G”k Y)
with (J,) defined by an equation analogous to Eq. (10). The functional derivative with respect to Ati is transformed to one with respect to (J”) by using the current correlation function 17,” via the equation -= 6 GA’+) jgj,/IIj/I-IO
6 s dr’ QL”(X, x’> qJ&,)>
3
144
SINGH
AND
HAGEN
while the remaining functional derivative can be eliminated function P as defined by hP(x,
y; z) = -
&
by introducing
the vertex
KWx, YII-l.
Equation (30) can also be written as
G*(x,v) = hjj dx'dy'G&(x, x’)
&
r“(x), J”; z) G$( y’, y).
(31)
With the use of Eqs. (29) and (31), the propagator is finally seen to satisfy
(Y”P,+ m>G’(x,u>- ~2w&‘y !u dx’
dy’ dz lTvb(x, z) G(x, x’) T”(x’, y’; z) G”( y’, y)
= 6(x - y),
(32)
where the external sources have now been set equal to zero. This is the DysonSchwinger equation for the Federbush model. From Eq. (32) the mass operator is identified as “@cc, v) = m&- - v) + SJqx, v>
(33)
with the Fourier transform of 6&Z being given by &A!(p)
= --X2qyyE”~
s &17,&d
GYP - 4) WP
- 4, da
(34)
In order to compute S&Y to lowest order one subsitutes the zeroth-order expressions for the quantities in the integral (L’ is given by Eq. (22) with m, 5, r] replaced by M, 8, H, respectively), thereby obtaining
~J@~)(P)= g j (p _ $ + m2Wfm + 11- WA dl x PO - 4L - m + 2q-2yu~“q*q0E,,E,B1}. A straightforward but lengthy calculation terms results in the expression 8~@~)(p)
where
=
$$
{A
+
(y”p,
followed by a suitable rearrangement
+
m)B
+
(y”pIL
+
m)”
Z,(p)}
of
(35)
GENERALIZED
FEDERBUSH MODEL
145
and Cr (p) is a finite function ofp which does not contain H. It is seenthat all the H dependence is contained in the mass renormalization term A which is in general logarithmically divergent. The contribution to the wavefunction renormalization constant given by B is found to be logarithmically divergent. The results for the Y field are trivially obtained from Eqs. (35) and (36) by making the replacement H + 77 mt,M. Once again one has a situation in which agreement with the Federbush solution which predicts no mass renormalization, is found only in the case 77= H = 0. For arbitrary values of 7 and H the recovery of the Federbush solution again requires the counterterms (26). 5. OPERATOR
FORMALISM
At this point we turn to an examination of the operator structure of the model paying particular attention to general features such as covariance and current conservation in a nonperturbative framework. The procedure is the sameas that of Ref. [S], and some of the results obtained are quite similar to ones obtained there. In order to make use of the canonical commutation rules (4), all operators must be expressedas well-defined functions of the field operators on a given spacelike surface. To this end we note that in view of the definition (12) the current operators in the coupled theory (with the external sourcesequal to zero) have the form
l
Noting that the singularity of the product $(x + )$(X) as E--f 0 is contained in the vacuum expectation value, we take the limit on the surface ~0 = constant, obtaining
It is sufficient therefore to consider only the first two terms coming from the exponential. A straightforward calculation then yields
.j”(-x) = (1 - $ ET)-l(,j”(x)
+ $7$(.x)),
j’(x) = (I - $ SH)-‘(g’(x)
- $ (p(x)).
J’(x) = (1 - ;
- + Hjl(x)),
@(j”(x)
(38)
146
SINGH
AND
HAGEN
and
J’(x)= (1 - $ sq) (p(x)+ $ @O(x)). where the quantities
WY are well-defined products of the field operators. Equations (38) and (39) enable one to evaluate all the equal time commutation relations involving currents. Thus one has
[j”(4 #(Y)I = - (1 - $ Eq)-l q+(x) 6(x - y),
(40)
[J’(x),
z)(y)] = - $3
and other commutators,
(1 - ;
&$I
q+(x) 8(x - .v)?
obtained from these by the replacement:
(41) One notes that these differ from the canonical form and furthermore that both j” and Jso generate gauge transformations on z,k However, it is of interest to note that the charges corresponding to the currents
generate canonical gauge transformation for the fields # and Y, respectively, each leaving the other field invariant. It is shown below that these currents are conserved
GENERALIZED
FEDERBUSH
147
MODEL
and we merely note before preceeding to that proof that the current-current tors of the model are [jO(x),j’(x’)]
= C-l (1 - ; &)( - ; qqx-, - x;,);
[JO(x), 51(x’)] = c-1 (1 - ; q[jO(x>, J”(x’>]
commuta-
$ +3(x, - x;,),
= g (7 - H) c-1 (-
+ iilS(Xl - x;,)
(43) = [j](x),
Jl(x’)],
where c = (1 - $z)(l
- $q),
with all the remaining commutators vanishing. A straightforward application of the action principle to the Lagrangian (1) fails to yield the correct energy-momentum tensor. However, it can be verified that the tensor Tuv given by TlO = TO1 = -;
-
(#a,# + ?mlY)
(44)
he,,,juJY
- $
@jo2 + Hj12 + tJ$ + 7J12)
does yield the correct PoincarC group generators. In other words the operators P” and Jo1defined by P” = 1 dx, TO@(x)
and Jo1 = x”P1 -
s
dx, xlToo(x)
(45)
satisfy
P, x(x>1= @xc4 [Jol, x(x)] = i(x”8 - xlao - &y”yl) x(x),
(46)
148
SINGH
AND
(x being zj or Y) as required of infinitesimal Lie algebra for the group, the relations
HAGEN
generators. As to the verification of the
[Pp, P”] = 0, [P’, Jo11 = iPo
immediately
follow from (46) and the Dirac-Schwinger [Too(x), Toa(
= -i(TOl(x)
+ P(x’))
covariance condition 8,6(x, - xi)
(47)
which can be verified directly provided the conditions (+7=1=6+H
(48)
are imposed, leads to [PO, Jo11 = iP1.
Furthermore
it can be demonstrated
(49)
that the conservation equation aLLTuY = 0
holds, and that Tuv transforms as a second-rank tensor. A few remarks are in order concerning the origin of the “extra” term - g
(Ejo” + frj12 + CJO” + T)Jl”)
(W
in the above expression for Tuv. In view of Eqs. (37) for the currents the formal Lagrangian (1) does not yield the field equations (3). Insofar as the latter are taken as the basis for the discussion of the dynamics, extra terms must be added to the formal Lagrangian if the field equations are to be obtained via the variational principle. It can be verified that these additional terms are precisely what is needed to generate the energy density given above. Turning to the current operators themselves it is seen thatju and Ju transform as vectors. Furthermore, one has
(51)
with the corresponding equations for J,, obtained by the replacement (41). Thus one finds that the conservation of j@ (J”) requires 77 = 0 (H = 0), whereas 5 = E = 0 leads to the canonical expressions for the divergence of the axial currents, conclusions already indicated by the perturbation treatment of Section 3. We also note that the case m = 0 = M is a special one for which the vector and axial-vector currents are both conserved for arbitrary values of 77and H. Finally the conservation of the currents $$ and jY referred to earlier may be shown to hold for all values of 7 and H, a result which is readily obtained by application of (51).
GENERALIZED
6. EQUIVALENCE
FEDERBUSH MODEL
149
TO A CONSERVED CURRENT MODEL
It was noted in Sections 3 and 4 that although the lowest order results for particleantiparticle scattering amplitudes and for mass renormalizations depend on the parameters 77and H, the results for arbitrary values of these parameters can be made to coincide with those obtained for conserved vector currents (7 = H = 0) by including Thirring model type counterterms. We now ask whether such a correspondence can be established quite generally to all orders in the coupling constant. Specifically we seek the answer to the following question: do the results of the theory given by the formal interaction
involving conserved currents j, u and J,” coincide with those obtained from 2, = X~u,,juJ'
(53)
where the currents are defined in a general way, for a suitable choice of (II, p, y, and A, . The answer is affirmative if the equations of motion corresponding to (52), namely (W and Wb) are identical to the equations and (y"pu + WY
= -hqy%,,Yj
(55b)
obtained from (53) when explicitly written in terms of the fields. This requires that the conditions (56)
be satisfied. By following the same procedure as in the previous section, the currents can be expressed in terms of the bilinear IimitsjY, p. Thus one has f" =.ico, j’ = .ic’ + ;
i”
=
J?,
(hoJcof ajcl + y Jc’),
r’ = Jcl - + (X,jco - /?J:-
yj:),
(57)
150
SINGH AND HAGEN
andjU, J’” are given by Eq. (38). Upon eliminatingp, p between Eqs. (57) and (38), and comparing the resulting equations with (56), one obtains the conditions
h -I!= h
t =
(
l-i!,-,
)-l(l
1 - $IZ~j-l(l
- $7)
= (1 - ;fH)-l(l
+ J-j 7r
= (1 - 2 [H)-l(l
01TX@” a - A,) (1 - ;Q-’ 97
= -;
H’(l
- $H) + zj, 7r
+$)(l
-z(H)-‘,
and /3 = ; (A[ - &,) (1 - ;
q (1 + +j(
fH)-1 = - ;
1 - ; +)-I.
It is easily seen that Eqs. (58) are satisfied if and only if one has &, = x (1 + ; a=-?H’ p = - ;
77
qH)-‘,
(
I+“2
H-l, n-217 >
(59)
7 (1 + $7H)-‘.
and y = 0. The reciprocal correspondence can be established in the same manner. Thus it can be shown that the Federbush model with generalized currents but modified by the inclusion of suitable Thirring model type terms is equivalent to the usual Federbush model with conserved vector currents. Specifically equivalence of the theories corresponding to the Lagrangians and can be established provided the parameters are related by Eq. (59). This result, to second order in A, yields the counterterms (26) obtained in Sections 3 and 4. 7. CONCLUSION
The work presented here has largely consisted in an application to the Federbush model of some ideas developed within the framework of the Thirring model [5]. In particular it has been found that currents can (as in the Thirring model) be defined
151
GENERALIZED FEDERBUSH MODEL
more generally than previously allowed, thereby leading to new classes of solutions of the Federbush Lagrangian. While only the conventional Federbush model is at present known to be exactly soluble, considerations of the type presented in the preceding section together with recent progress in the study of the massive Thirring model seem to offer some promise for obtaining solutions of the more general version. On the other hand the nonvanishing of the mass renormalization in the general theory is perhaps an indication that the conserved current case is indeed a very special limit. In either event it would appear that the enrichment of the class of interesting twodimensional field theories which has been accomplished here could provide a new laboratory for the theoretical examination of hypotheses in particle physics. This is particularly likely if nonconserved currents come to play a more important role in future models of the real world.
APPENDIX
We show that the function f(G, m”) defined by Eq. (21) has a nonzero limit as E tends to zero. Recall that the limit has to be taken along a spacelike direction. We can, therefore, set EO= 0 before taking the limit 8 -+ 0 without any loss of generality. By making use of the fact that the poles of the integrand in thep” plane are shifted off the real axis one can rotate the path of the p” integration in Eq. (21) from the real to the imaginary axis, thereby obtaining f(e
, nz ) =
-$
j
dp, dp, eialrl .
(P22
(PI2 +
-
P12)
P22 +
rn2j2
A transformation to polar variables and subsequent integration variable results in
-
over the angular
2i m f(e2, m”) = - s dp 77 0 =- 2i m 7.r s0 dp
(p2
+p;2e12)2
Jz(P)*
As the last integral is uniformly convergent in Ed, the limit can be evaluated under the integral sign. One immediately has
Q+$lf(E2, my = ; . ACKNOWLEDGMENT One of the authors &.P.S.S.) is grateful for the hospitality of the Department of Physics and Astronomy, University of Rochester, during the summers of 1976 and 1977.
152
SINGH
AND
HAGEN
REFERENCES
1. P. FEDERBUSH,Phys. Rev. 121 (1961), 127; Progr. Theor. Phys. 26 (1961), 148. 2. A. S. WIGHTMAN, in “High Energy Interactions and Field Theory” (M. Levy, Ed.), Cargese Lectures in Theoretical Physics 1964, Gordon and Breach, New York, 1966. 3. B. SCHROER,T. T. TRUONG, AND P. WEISZ, Ann. PfzJx (N.Y.) 102 (1976), 156. 4. J. SCHWINGER, Lectures on Coupled Fields, Harvard University, 1954 (unpublished). 5. C. R. HAGEN, Nuovo Cimento 51B (1967), 169. 6. C. R. HAGEN, Nuovo Cimento 51A (1967), 1033. 7. J. SCHWINGER, Phys. Rev. 128 (1962), 2425. 8. W. THIRRING, Ann. Phys. (N.Y.) 3 (1958), 91. 9. K. JOHNSON,in “Lectures on Particles and Field Theory,” Vol. 2, (S. Deser and K. W. Ford, Eds.), 1964, Brandeis Summer School in Theoretical Physics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.
OF PHYSICS
Current
115,
136152
Definition
(1978)
and a Generalized
Federbush Model*
L. P. S. SINGH Physics Department, Pahlavi University, Shirar, IRAN AND
C. R. HAGEN Department of Physics and Astronomy,
University of Rochester, Rochester, New York, 14627
Received August 4, 1977
The Federbush model is studied, with particular attention being given to the definition of currents. Inasmuch as there is no a priori restriction of local gauge invariance, the currents in the interacting case can be defined more generally than in Q.E.D. It is found that two arbitrary parameters are thereby introduced into the theory. Lowest order perturbation calculations for the current correlation functions and the Fermion propagators indicate that the theory admits a whole class of solutions dependent upon these parameters with the closed solution of Federbush emerging as a special case. The theory is shown to be locally covariant, and a conserved energy-momentum tensor is displayed. One finds in addition that the generators of gauge transformations for the fields are conserved. Finally it is shown that the general theory yields the Federbush solution if suitable Thirring model type counterterms are added.
INTRODUCTION The two-dimensional theory of Federbush [l] is essentially unique, being the only model with massiveFermions for which an exact nontrivial solution is known [l, 2, 31.
The model involves the current-current coupling h E,,j@‘JYwhere j,, and J, are the vector currents associatedwith the Dirac fields $ and Y, respectively. The study presented here begins with a perturbation calculation, to lowest order, of the fermion propagators and the current correlation functions. Schwinger’s method of external sources [4] is used, and the currents in the interacting case are constructed by a well-defined limiting procedure. As noted in a previous treatment of several two dimensional models [5, 61 there are no a priori restrictions required by local gauge invariance, a fact which allows a more general definition of the currents than that of Schwinger for the case of the electromagnetic current [7]. Thus two arbitrary parameters q and H (unrelated to those in the Lagrangian), corresponding * Research supported in part by the U.S. Energy Research and Development under contract number EY-76-C-02-3065.000. 136 OOO3-4916/78/1151-0136$05.00/0 Copyright All rights
0 1978 by Academic Press, Inc. of reproduction in any form reserved.
Administration,
GENERALIZED
FEDERBUSH
137
MODEL
to j, and J, , respectively, are introduced into the theory. In particular, 7 = 0 (respectively, H = 0) corresponds to the case where j, (respectively, J,) is conserved in the presence of interactions. The perturbation calculations are found to be in agreement with the exact solution of Refs. [l-3] only for the case 7 = H = 0. In particular there is no mass renormalization, and the amplitude for particle-antiparticle scattering is found to vanish. On the other hand a class of new solutions is generated when either or both of q, H are nonzero. One finds then that the mass renormalization is logarithmically divergent although the wavefunction renormalization remains unchanged. The particleantiparticle scattering amplitude is also found to be nonzero. However, the Federbush solution is reproduced if additional Thirring model [S] type terms are included in the Lagrangian. We subsequently proceed to analyze the operator formalism of the theory. A conserved energy-momentum tensor is obtained and the theory is shown to be locally covariant. As in the case of Refs. [.5, 61 it turns out that the canonical energy density does not suffice and noncanonical terms have to be added. One finds that there are always two conserved currents in the theory whose corresponding charges generate gauge transformations of the first kind on the Dirac fields $J and Y. Finally we seek the relation between the general solution for arbitrary values of 7 and Hand the Federbush solution which corresponds to q =z H = 0. It is found that upon including suitable Thirring model type counterterms the former theory becomes equivalent to the Federbush model with conserved vector currents. The model is outlined and the notation established in Section 2, which also includes a summary of the external source method. In Section 3 the current correlation functions and the particle-antiparticle scattering amplitude are obtained to lowest order. Section 4 contains the lowest order calculation for the Fermion propagators while the operator properties are discussed in Section 5. ln Section 6 the connection between theories corresponding to different values of 7 and H is established. We end with a few concluding remarks.
2. The Federbush
model is formally
defined by the Lagrangian
27 = (i/2)(&Q,t4 + k”“juJv
PRELIMINARIES
+ Fy’a,Y) +,j,,a”
- &?1tJ# -
+ J,,A”
;-MFY (1)
where 4 = #y”, and a,, and A, are external sources which will be set equal to zero in the end. The notation used here is such that the metric is gl'
=
-.-go0
=
1, go1 =
0
138
SJNGH
while the Levi-Civita
tensor P = -F
AND
HAGEN
is specified by
Hermitean fields are used so that the y” are pure imaginary antisymmetric and symmetric respectively. These satisfy y”y” = -g””
+ cLL”y5)
a relation which also defines the real symmetric obtains the useful identity
In order to construct currents makes use of the charge matrix
the Lagrangian
pseudoscalar
matrix y5 . One readily
one doubles the number of field components
in the associated charge space. If the currents .i”(X) = Mx)
with /3 = y” and y1 being
4YU$4”),
are formally
J”(x)
= $F(x)
and
defined as qyY(x),
(2)
(I) leads to the equations of motion [Y”(P,
- vu) + ml+ = kv,d@‘Jv
(34
and
[Y”(P, - !A) + WY = --x~Y,~%~,
(3b)
where pu = -3, and all operator products such as t,bJuare understood to be symmetrized. The action principle also implies the equal time anticommutators iK4> KJJ>> = VW3 KY)> = (xx - VI, {?&a KY>> = 0.
(4)
In order to obtain the solution of the coupled theory one makesuseof the dynamical aspect of the action principle. This principle assertsthat the variation in a vacuum expectation value (0 j R(x) 10) arising from the variation of any parameter in the theory is given by
where 66p and 6R are the resulting variations in Sand R respectively, and T denotes time ordering. Thus variation of the coupling constant yields
GENERALIZED
FEDERBUSH
139
MODEL
where the subscripts denote the dependence of the matrix elements on various parameters. Variations with respect to the external sources result in &
(0 i R(x) I o>A,n,~ = Q I WW.L(.d)
I @A.~,A
(6)
and &
.
(0 I NJ4 I Oh,n,A = 0
I W(x)
J,(Y)) I O)A.O.A
(7)
provided R does not explicitly contain a, or A,, . Thus one has
a relation which is trivially
integrated to yield
(0iR(.u) IO~A.~.A = exp (--iAs4 cuv&&)
*(0I R(x) IOh=, . (8)
The problem thus reduces to the determination of matrix elements as functions of a, and A, for h = 0, i.e., in the limit in which the field equations (3) decouple. In the case of massless fields these equations can be explicitly solved, so that closed form expressions for the matrix elements of the coupled theory can be obtained. In the massive case, however, an iterative procedure has to be employed thus reproducing the ordinary perturbation expansion. The following two sections are devoted to such calculations for the current correlation functions and the propagators.
3. THE
CURRENT
CORRELATION
We begin with the lowest order calculation j, which is defined by TU”(.u, Y> = &
FUNCTION
for the current correlation
(j”(Y)>
function for
(9)
where
(0 IL(x) I 0) (0 IO> .
(10)
To carry out this calculation the current j, must be suitably defined in the presence of the external field. As is well known, the electromagnetic current should be defined as
where the limit E + 0 is to be taken in a symmetric fashion along a spacelike surface, usually taken to be the surface x0 = constant.
140
SINGH
AND
HAGEN
In electrodynamics the prescription (11) is required by local gauge invariance. However, in cases such as the present one, where there is no gauge field, and consequently no a priori restriction arising from gauge invariance of the second kind, a more general definition can be employed. This fact has been used to generate new solutions for the Thirring model [5], and for a model involving a massless Fermion current coupled to a massive vector Boson [6]. In the same spirit, we define the current to be
6 and 71 being arbitrary real parameters. It will subsequently be seen that Lorentz invariance imposes the condition
!$+7=1.
(13)
In view of Eq. (12), one has (L(x)>
= f 1:~ Tr y,@%x + E, x) exp[ --iq j”” x
d&(5gaB + yys@) Q&J)], (14)
with the Fermion propagator G* defined as (15)
which, for h = 0 satisfies h4~~ - q4x))
+ m I W-T v) = %x9 v>.
(16)
Since Eq. (16) cannot be solved exactly for m f 0, an iterative procedure must be used. We restrict ourselves to a calculation to lowest order closely following that of Johnson [9] in Q.E.D. In the present case, vU, is given by the first functional derivative in the limit uU --f 0, and it is sufficient to compute (j,) up to terms linear in arr . The result is (j,(x)>
= f 1~5 Tr yU [s dx’ G,*(x + E, x’) yVaV(x’) G,,@‘(x’,x) - iG,“(x + E, x) jxz+’ 4&28
+ rl@w
as(u)] 9
where GoG(x, v) is the free propagator. In view of the symmetric employed it is more convenient to write Eq. (17) as
integration
(17)
to be
GENERALIZED
which, in momentum
FEDERBUSH
141
MODEL
space reads
(18)
+ G~(p)(W+ yYny6) s_‘ltd.u,eig’z\dd.
In the second term, the x integral can be evaluated along a straight line, yielding r/2 dx s -E/z
e*g’s
2~Y
=
y
sin $4 . E q-6
= E, + O(8),
This is multiplied by GOG(e),which in the limit E - 0 behaves as (I/E) + finite terms. Thus it is sufficient to keep the first term in (19). After straightforward manipulation (18) can then be brought to the form
from which a straightforward n,,(q)
= ljz i s &
trace evaluation leads to the result at - 7) I @2 + In2)2
eip.r
m2gw
+ 4 (PUPY - ; &“P2) ( ,(P-~)2+m2;[(P+I).+m.]
)
(20)
Considering first the last term, one finds that its contribution l$i (Y
must be of the form
- ; gw) (5 + 7 - 1>f(E2, m2),
where f (c2, m2) = f
s
&
eip’E ‘W2-+2”
m22
jP)“l
-
(21)
As shown in the Appendix, f has a nonzero limit as E-+ 0. On the other hand, the limit of E~E,/E~ depends on the direction along which the limit is taken, thereby
142
SINGH
AND
HAGEN
leading to a noncovariant result. The situation can be satisfy Eq. (13). The remaining terms in the integrand of Eq. (20) corresponding integral therefore, converges uniformly limit inside the integral. A straightforward calculation ~w(d
= ;
1; &“(f
- 77-
remedied if and only if [ and 71 fall off as p-’ for large p. The in E, allowing one to take the then leads to the result
1) + (g,, - 7)
[l -
I(m,
q)]/
(22)
where I(m, q) = m2 lo1
cl’ 1112+ u(1 - 21)q* *
(23)
It should be noted that conservation of the vector current which would imply 4”~w‘ = 0, is possible only with t - 7 = 1, which in view of (13) would require [=l,q=O.Onth e other hand, a conserved axial current j,u = l “jV in the zero mass limit would imply qUEw”n-,, = 0 and correspondingly yield 5 = 0, 7 = 1. These conditions on 4 and 17 are directly obtained in Section 5 from the operator structure of the theory. It may be noted that the treatment of J, is identical to that carried out here for the currentj, . One has two additional parameters E, H corresponding to [, 77respectively, and all the preceding results are reproduced with the replacements ([, 71) -+ (E, H) andm+M. Having now obtained +‘, the lowest order amplitude for particle-antiparticle scattering is readily calculated. Let e and E be the particles associated with the fields fields $ and Y, respectively. Considering the process e B -+ e .5 with the respective incoming and outgoing momenta being p, p and q, 4, the second-order amplitude is readily seen to be
where 17(q) is the current correlation function for the current J, . Upon substituting the expression corresponding to (22) this becomes
For completeness it may be noted that the amplitude for E E to second order in X is given by an identical expression with H -+ 7, i.e.,
GENERALIZED
FEDERBUSH
143
MODEL
We recall that the explicit solution of Refs. [I, 2, 31 predicts that both these amplitudes vanish, a result which is seen here to be possible only for the case 17= H = 0. We thus conclude that the known explicit solution is reproduced only when the vector currents are defined so as to be conserved in the interacting case. This is not surprising, of course, in view of the fact that the derivation of that solution crucially depends upon current conservation. It is clear that the general definition of the currents leads to new solutions of the model. Also, it can easily be shown that the difference between the Federbush solution and the general solution for the processesunder consideration (to lowest order at least) is of the sameform asthat generated by a massiveThirring type coupling. Thus, at least to the lowest order, a modification of the Lagrangian by
in the general case reproduces the Federbush solution. The possibility of such a correspondence to all orders in h is discussedin Section 6.
4. THE FERMION PROPAGATORS
We next turn to the lowest order calculation for the Fermion propagator defined as
d(d)1o? G$(x >v) = j(’ 1T(#(x) (0I0)
(27)
(the calculations for the propagator of the Y field proceed along identical lines). From the field equations (3a) one has
Upon using Eq. (7), the second term on the right-hand side becomes
kvw~“”((J,(x))- i &)
G”k Y)
with (J,) defined by an equation analogous to Eq. (10). The functional derivative with respect to Ati is transformed to one with respect to (J”) by using the current correlation function 17,” via the equation -= 6 GA’+) jgj,/IIj/I-IO
6 s dr’ QL”(X, x’> qJ&,)>
3
144
SINGH
AND
HAGEN
while the remaining functional derivative can be eliminated function P as defined by hP(x,
y; z) = -
&
by introducing
the vertex
KWx, YII-l.
Equation (30) can also be written as
G*(x,v) = hjj dx'dy'G&(x, x’)
&
r“(x), J”; z) G$( y’, y).
(31)
With the use of Eqs. (29) and (31), the propagator is finally seen to satisfy
(Y”P,+ m>G’(x,u>- ~2w&‘y !u dx’
dy’ dz lTvb(x, z) G(x, x’) T”(x’, y’; z) G”( y’, y)
= 6(x - y),
(32)
where the external sources have now been set equal to zero. This is the DysonSchwinger equation for the Federbush model. From Eq. (32) the mass operator is identified as “@cc, v) = m&- - v) + SJqx, v>
(33)
with the Fourier transform of 6&Z being given by &A!(p)
= --X2qyyE”~
s &17,&d
GYP - 4) WP
- 4, da
(34)
In order to compute S&Y to lowest order one subsitutes the zeroth-order expressions for the quantities in the integral (L’ is given by Eq. (22) with m, 5, r] replaced by M, 8, H, respectively), thereby obtaining
~J@~)(P)= g j (p _ $ + m2Wfm + 11- WA dl x PO - 4L - m + 2q-2yu~“q*q0E,,E,B1}. A straightforward but lengthy calculation terms results in the expression 8~@~)(p)
where
=
$$
{A
+
(y”p,
followed by a suitable rearrangement
+
m)B
+
(y”pIL
+
m)”
Z,(p)}
of
(35)
GENERALIZED
FEDERBUSH MODEL
145
and Cr (p) is a finite function ofp which does not contain H. It is seenthat all the H dependence is contained in the mass renormalization term A which is in general logarithmically divergent. The contribution to the wavefunction renormalization constant given by B is found to be logarithmically divergent. The results for the Y field are trivially obtained from Eqs. (35) and (36) by making the replacement H + 77 mt,M. Once again one has a situation in which agreement with the Federbush solution which predicts no mass renormalization, is found only in the case 77= H = 0. For arbitrary values of 7 and H the recovery of the Federbush solution again requires the counterterms (26). 5. OPERATOR
FORMALISM
At this point we turn to an examination of the operator structure of the model paying particular attention to general features such as covariance and current conservation in a nonperturbative framework. The procedure is the sameas that of Ref. [S], and some of the results obtained are quite similar to ones obtained there. In order to make use of the canonical commutation rules (4), all operators must be expressedas well-defined functions of the field operators on a given spacelike surface. To this end we note that in view of the definition (12) the current operators in the coupled theory (with the external sourcesequal to zero) have the form
l
Noting that the singularity of the product $(x + )$(X) as E--f 0 is contained in the vacuum expectation value, we take the limit on the surface ~0 = constant, obtaining
It is sufficient therefore to consider only the first two terms coming from the exponential. A straightforward calculation then yields
.j”(-x) = (1 - $ ET)-l(,j”(x)
+ $7$(.x)),
j’(x) = (I - $ SH)-‘(g’(x)
- $ (p(x)).
J’(x) = (1 - ;
- + Hjl(x)),
@(j”(x)
(38)
146
SINGH
AND
HAGEN
and
J’(x)= (1 - $ sq) (p(x)+ $ @O(x)). where the quantities
WY are well-defined products of the field operators. Equations (38) and (39) enable one to evaluate all the equal time commutation relations involving currents. Thus one has
[j”(4 #(Y)I = - (1 - $ Eq)-l q+(x) 6(x - y),
(40)
[J’(x),
z)(y)] = - $3
and other commutators,
(1 - ;
&$I
q+(x) 8(x - .v)?
obtained from these by the replacement:
(41) One notes that these differ from the canonical form and furthermore that both j” and Jso generate gauge transformations on z,k However, it is of interest to note that the charges corresponding to the currents
generate canonical gauge transformation for the fields # and Y, respectively, each leaving the other field invariant. It is shown below that these currents are conserved
GENERALIZED
FEDERBUSH
147
MODEL
and we merely note before preceeding to that proof that the current-current tors of the model are [jO(x),j’(x’)]
= C-l (1 - ; &)( - ; qqx-, - x;,);
[JO(x), 51(x’)] = c-1 (1 - ; q[jO(x>, J”(x’>]
commuta-
$ +3(x, - x;,),
= g (7 - H) c-1 (-
+ iilS(Xl - x;,)
(43) = [j](x),
Jl(x’)],
where c = (1 - $z)(l
- $q),
with all the remaining commutators vanishing. A straightforward application of the action principle to the Lagrangian (1) fails to yield the correct energy-momentum tensor. However, it can be verified that the tensor Tuv given by TlO = TO1 = -;
-
(#a,# + ?mlY)
(44)
he,,,juJY
- $
@jo2 + Hj12 + tJ$ + 7J12)
does yield the correct PoincarC group generators. In other words the operators P” and Jo1defined by P” = 1 dx, TO@(x)
and Jo1 = x”P1 -
s
dx, xlToo(x)
(45)
satisfy
P, x(x>1= @xc4 [Jol, x(x)] = i(x”8 - xlao - &y”yl) x(x),
(46)
148
SINGH
AND
(x being zj or Y) as required of infinitesimal Lie algebra for the group, the relations
HAGEN
generators. As to the verification of the
[Pp, P”] = 0, [P’, Jo11 = iPo
immediately
follow from (46) and the Dirac-Schwinger [Too(x), Toa(
= -i(TOl(x)
+ P(x’))
covariance condition 8,6(x, - xi)
(47)
which can be verified directly provided the conditions (+7=1=6+H
(48)
are imposed, leads to [PO, Jo11 = iP1.
Furthermore
it can be demonstrated
(49)
that the conservation equation aLLTuY = 0
holds, and that Tuv transforms as a second-rank tensor. A few remarks are in order concerning the origin of the “extra” term - g
(Ejo” + frj12 + CJO” + T)Jl”)
(W
in the above expression for Tuv. In view of Eqs. (37) for the currents the formal Lagrangian (1) does not yield the field equations (3). Insofar as the latter are taken as the basis for the discussion of the dynamics, extra terms must be added to the formal Lagrangian if the field equations are to be obtained via the variational principle. It can be verified that these additional terms are precisely what is needed to generate the energy density given above. Turning to the current operators themselves it is seen thatju and Ju transform as vectors. Furthermore, one has
(51)
with the corresponding equations for J,, obtained by the replacement (41). Thus one finds that the conservation of j@ (J”) requires 77 = 0 (H = 0), whereas 5 = E = 0 leads to the canonical expressions for the divergence of the axial currents, conclusions already indicated by the perturbation treatment of Section 3. We also note that the case m = 0 = M is a special one for which the vector and axial-vector currents are both conserved for arbitrary values of 77and H. Finally the conservation of the currents $$ and jY referred to earlier may be shown to hold for all values of 7 and H, a result which is readily obtained by application of (51).
GENERALIZED
6. EQUIVALENCE
FEDERBUSH MODEL
149
TO A CONSERVED CURRENT MODEL
It was noted in Sections 3 and 4 that although the lowest order results for particleantiparticle scattering amplitudes and for mass renormalizations depend on the parameters 77and H, the results for arbitrary values of these parameters can be made to coincide with those obtained for conserved vector currents (7 = H = 0) by including Thirring model type counterterms. We now ask whether such a correspondence can be established quite generally to all orders in the coupling constant. Specifically we seek the answer to the following question: do the results of the theory given by the formal interaction
involving conserved currents j, u and J,” coincide with those obtained from 2, = X~u,,juJ'
(53)
where the currents are defined in a general way, for a suitable choice of (II, p, y, and A, . The answer is affirmative if the equations of motion corresponding to (52), namely (W and Wb) are identical to the equations and (y"pu + WY
= -hqy%,,Yj
(55b)
obtained from (53) when explicitly written in terms of the fields. This requires that the conditions (56)
be satisfied. By following the same procedure as in the previous section, the currents can be expressed in terms of the bilinear IimitsjY, p. Thus one has f" =.ico, j’ = .ic’ + ;
i”
=
J?,
(hoJcof ajcl + y Jc’),
r’ = Jcl - + (X,jco - /?J:-
yj:),
(57)
150
SINGH AND HAGEN
andjU, J’” are given by Eq. (38). Upon eliminatingp, p between Eqs. (57) and (38), and comparing the resulting equations with (56), one obtains the conditions
h -I!= h
t =
(
l-i!,-,
)-l(l
1 - $IZ~j-l(l
- $7)
= (1 - ;fH)-l(l
+ J-j 7r
= (1 - 2 [H)-l(l
01TX@” a - A,) (1 - ;Q-’ 97
= -;
H’(l
- $H) + zj, 7r
+$)(l
-z(H)-‘,
and /3 = ; (A[ - &,) (1 - ;
q (1 + +j(
fH)-1 = - ;
1 - ; +)-I.
It is easily seen that Eqs. (58) are satisfied if and only if one has &, = x (1 + ; a=-?H’ p = - ;
77
qH)-‘,
(
I+“2
H-l, n-217 >
(59)
7 (1 + $7H)-‘.
and y = 0. The reciprocal correspondence can be established in the same manner. Thus it can be shown that the Federbush model with generalized currents but modified by the inclusion of suitable Thirring model type terms is equivalent to the usual Federbush model with conserved vector currents. Specifically equivalence of the theories corresponding to the Lagrangians and can be established provided the parameters are related by Eq. (59). This result, to second order in A, yields the counterterms (26) obtained in Sections 3 and 4. 7. CONCLUSION
The work presented here has largely consisted in an application to the Federbush model of some ideas developed within the framework of the Thirring model [5]. In particular it has been found that currents can (as in the Thirring model) be defined
151
GENERALIZED FEDERBUSH MODEL
more generally than previously allowed, thereby leading to new classes of solutions of the Federbush Lagrangian. While only the conventional Federbush model is at present known to be exactly soluble, considerations of the type presented in the preceding section together with recent progress in the study of the massive Thirring model seem to offer some promise for obtaining solutions of the more general version. On the other hand the nonvanishing of the mass renormalization in the general theory is perhaps an indication that the conserved current case is indeed a very special limit. In either event it would appear that the enrichment of the class of interesting twodimensional field theories which has been accomplished here could provide a new laboratory for the theoretical examination of hypotheses in particle physics. This is particularly likely if nonconserved currents come to play a more important role in future models of the real world.
APPENDIX
We show that the function f(G, m”) defined by Eq. (21) has a nonzero limit as E tends to zero. Recall that the limit has to be taken along a spacelike direction. We can, therefore, set EO= 0 before taking the limit 8 -+ 0 without any loss of generality. By making use of the fact that the poles of the integrand in thep” plane are shifted off the real axis one can rotate the path of the p” integration in Eq. (21) from the real to the imaginary axis, thereby obtaining f(e
, nz ) =
-$
j
dp, dp, eialrl .
(P22
(PI2 +
-
P12)
P22 +
rn2j2
A transformation to polar variables and subsequent integration variable results in
-
over the angular
2i m f(e2, m”) = - s dp 77 0 =- 2i m 7.r s0 dp
(p2
+p;2e12)2
Jz(P)*
As the last integral is uniformly convergent in Ed, the limit can be evaluated under the integral sign. One immediately has
Q+$lf(E2, my = ; . ACKNOWLEDGMENT One of the authors &.P.S.S.) is grateful for the hospitality of the Department of Physics and Astronomy, University of Rochester, during the summers of 1976 and 1977.
152
SINGH
AND
HAGEN
REFERENCES
1. P. FEDERBUSH,Phys. Rev. 121 (1961), 127; Progr. Theor. Phys. 26 (1961), 148. 2. A. S. WIGHTMAN, in “High Energy Interactions and Field Theory” (M. Levy, Ed.), Cargese Lectures in Theoretical Physics 1964, Gordon and Breach, New York, 1966. 3. B. SCHROER,T. T. TRUONG, AND P. WEISZ, Ann. PfzJx (N.Y.) 102 (1976), 156. 4. J. SCHWINGER, Lectures on Coupled Fields, Harvard University, 1954 (unpublished). 5. C. R. HAGEN, Nuovo Cimento 51B (1967), 169. 6. C. R. HAGEN, Nuovo Cimento 51A (1967), 1033. 7. J. SCHWINGER, Phys. Rev. 128 (1962), 2425. 8. W. THIRRING, Ann. Phys. (N.Y.) 3 (1958), 91. 9. K. JOHNSON,in “Lectures on Particles and Field Theory,” Vol. 2, (S. Deser and K. W. Ford, Eds.), 1964, Brandeis Summer School in Theoretical Physics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965.