An integral representation as a function of invariants is found for the Fourier transform of the matrix element between the vacuum and a one-particle state of the retarded commutator of two currents. A special case is a spectral representation for the vertex as a function of momentum transfer. The threshold in this representation is lower than that found in the usual perturbation theory

Deser, Stanley

Gilbert, Walter

Sudarshan, E. C. G.

Caltech Authors - Main

EUCLIDEAN QUANTUM ELECTRODYNAMICS 731
it takes the form
{0I w)G[ft I/g', (1/i)b/8A')8[i@]ft [A']=e ~«'"'I
with the aid of the following consequences of the
canonical transformation 8 -+ A, 2 ~ —&; /f/ ~f:
Q /g / I p/g/) elf'+ tBIA I
(&/~/
I
&//~") =~L&/ —&//]b&~/ —~"]
It should also be mentioned that an integration concept
can be devised for both types of field variable, such that
d [//b']d [A']ee'&' —'B'"',
d[&']dÃ']b[&']b[&'] =1.
The generating function, for example, thereby acquires
the integral representation
fd[P/]d$g/]e //r[P'A— 'I+e'$' iB—'A'.
d[y']d[Z']e- «' 't
The discussion of Euclidean Green's functions will be
continued in another publication.
Note added il proof It h.—as not been suKciently emphasized
in this paper that the term "Lorentz gauge, " as descriptive of a
gauge in which there is no distinguished time-like vector, refers
to a class rather than just the special gauge used in the paper. One
can also introduce, for example, the transverse Lorentz gauge
{here "transverse" has a four-dimensional space-time significance),
characterized by
(8)g G~'L'(x, g)=0, n=1. . . v
together with appropriately modified Maxwell differential equa-
tions. The radiation gauge functions constructed from Lorentz
gauge Green's functions are clearly independent of the specific
Lorentz gauge employed. This subject will be discussed further in
a later paper.
P H YSI CAL REVI EW VOLUME 11$, NUMBER 3 AUGUST 1, 1959
Structure of the Vertex Function
STANLEY DESER, WALTER GILBERT' AND E. C. G. SUDARsHAN$
Lymaw Laboratory of Physics, ffart/ard U/t&/ersity, Cambridge, Massach/tsetts
(Received October 15, 1958; revised manuscript received April 6, 1959}
An integral representation as a function of invariants is found for the Fourier transform of the matrix
element between the vacuum and a one-particle state of the retarded commutator of two currents. A special
case is a spectral representation for the vertex as a function of momentum transfer. The threshold in this
representation is lower than that found in the usual perturbation theory.
INTRODUCTION
E shall study the structure of the matrix element
of the commutator, the retarded commutator,
and the time-ordered product of two Geld operators
taken between the vacuum and a single-particle state.
Our technique will be to manipulate these functions in
the physical region in such a way as to obtain an
integral representation for the function in terms of the
invariant momentum parameters that characterize its
Fourier transform.
We shall discover that, for a special case of this
representation, one can continue analytically certain of
the invariant parameters out of the physical region and
automatically obtain a spectral relation for the vertex
function as a function of the momentum transfer.
*Now at Brandeis University, Kaltham 54, Massachusetts.
f National Science Foundation Postdoctoral Fellow.
f. On leave of absence from Tata Institute of Fundamental
Research, Bombay, India.
The basic representation we shall 6nd for the Fourier
transform of the retarded commutator of two currents is
e"*II(x)dx&[J (x),i (0)] I p)
H(p, P,P')
dpdp (1)h'+ 2PPk —p+is (Pk+PP')
in which the variables are limited by
0&P& —1, p&max{ f(P), —P'P'},
where f(p) is a function, which we will specify later,
determined by the mass spectra of the intermediate
states, and H(p, p,p') is uniquely determined by the
Fourier transform in the physical region.
THE REPRESENTATION
We sha11 begin by deriving a representation for the
matrix element of the commutator of two currents. We
DESER, GILBERT, AN D SU DARSHAN
H(X' P) = (P)fH (X',P)
h(x, Px) = (Lj,(x),j,(O)PIP),
shall discuss only boson fields, the extension to the to remove the e(xo) behavior produced by (7):
general case being trivial. We consider a matrix element
of the form
in which ji(x) and j2(x) are the currents associated
with two fields and the state
I p) is a state of energy
momentum p„; p'=a'; po&0. This state may be either
a single-particle state or the state produced by operating
on the vacuum with a single Fourier component of a
field. As a consequence of I,orentz invariance we know
that this matrix element depends on the coordinate
only through the variables x', px. It is also a function
of p' and of other eigenvalues; the weight functions
will depend implicitly on these parameters.
We introduce the Fourier transform of the matrix
element with respect to the invariants:
The result of these substitutions is the expression
f
h(x', px) = )' dkdp e(pk) exp( —ikx)
)&exp(iPPx) 8 (O' —X')H (X',P), (9)
or, for the Fourier transform of the commutator,
f(k', pk) = I e"*dx h(x', px)
dV dp e(pk+pp')
0 —oo
&&~((k+PP)' —1~')HP ' P) (1o)
h(x', px) =, dndp exp(inx') exp(ippx) g(n, p) (3.) For this representation for the commutator, the
retarded Green's function may easily be derived; it is
The condition that the commutator vanish for space-
like intervals, whenever x'&0, is equivalent to the
requirement that g(n, p) be ana. lytic as a function of n
in the lower half n-plane. The part of the Fourier
transform, g(n, p), that is determined by values of the
invariants in the region p'x'~&(px)' is arbitrary but,
does not enter into the momentum-space Fourier
transform that we shall take since this is extended only
over real values of the coordinate-space vector. We
express the exponential exp(inx') as the Fourier trans-
form of a Gaussian, absorbing the numerical factors
into the weight function:
h(x', px) = I dkdndp exp( —ikx)
g(n, p)
&&exp( —ik'/4n) exp(iPPx) . (4)
Since g(n, p) is analytic in the lower half n-plane, we
represent it as
~ e'"'0(x)dx h(x' px)"a
H( ',p)
dp . (11)J, a „(kypp+i, p)~
LIMITS ON THE PARAMETERS
Ke turn now to the problem of restricting the ranges
of the parameters in the representation. We have, as
yet, imposed only the condition that the commutator
vanish for space-like intervals. Further restrictions
arise from the mass-spectra of the intermediate states.
If we expand the commutator over a complete set of
eigenvectors of the energy-momentum vector, the
coordinate-space integration yields a 8-function and
we have
f(k', pk) = (2+)' p 8(k —&i)(jil &i)(+il Al p)
—(2~)'2 ~(k+~2 —p)02I~2)(~2l jil p) (»)
Thus the Fourier transform of the commutator will
vanish unless e~ther
Then the n-integration can be performed, yielding
exp( ikx) exp(ip—px)
h(x2P*) =O, tdk Il(VP)d~'dP (6)
k2 —X2
We use the identity
f(k' pk) Wo if
t|,'&~ Mg'
2pk&~ 2~Mi
k=e& or p —k=n2,
where e~ and e2 are possible intermediate momenta.
In terms of invariants,
~
—ill
6' dk = —im e(xo) ~dk e (ko) e—i'~8 (k' —g') (7)
or
2pk & —&22+p'+ k'
2pk~& 2~(~—M2),
(13)
where M~ and M2 are the lowest masses occurring in
and introduce the Hilbert transform of H(X',P) in order each ordering of the commutator. We know the Fourier
STRUCTURE OF VERTEX FUNCTION
transform only for real vectors k„; thus the function
f(k', pk) is given only for values of the invariants that
lie in the physical region (pk) ' &~p'k'. In the nonphysical
region, for those real values of the invariants that
correspond to a momentum vector k„with imaginary
components, the function will be determined by the
representation (10). In the region (it, which consists of
all of the (k',pk)-plane outside the nonphysical region
and outside region (13), f(k'p, k) vanishes.
For convenience we introduce a new variable p, =)'
—p'p' in the representation. Then
f(k', pk) = di1dp H(I2,p) e(pk+pp')
&&5(k2+2pkp —p). (14)
We allow p and p to run from —~ to ~ and incorporate
the causality condition by having H(p, ,p) vanish in the
region p( —p'p'. Each point in the (k',pk) plane corre-
sponds to a line integral of the function e(pk+pp2)H(p, p)
over the straight line @=k'+2PkP in the (p, ,P) plane.
The parabolic boundary of the nonphysical region in
the (k',pk) plane, (pk)'=p'k', generates a set of lines
that have as their envelope the curve A: p= —p'p',
and touch the envelope at the point p= —pk/p' at
which the e-function changes sign. All other points in
the physical region correspond to lines on which the
&-function changes sign in the interior of the region in
which H(12,p) is zero. The boundary O'= M1', 2pk) sM1
of (R corresponds to the set of lines of positive slope
passing through the point P=O, p, =M12 and lying
between the line p= 0 and a line tangent to the curve A.
The other boundary —2pk= M22 —p' —k' corresponds
to a set of lines produced by the clockwise rotation of a
line through the point P= —1, p=M22 P' from the-
line p= —1 to a line tangent to the curve A. The
other points in the (k',pk) plane at which f(k', pk)
vanishes correspond to all other lines lying entirely in
the region swept out by these two sets of lines. If
H(p, p) vanishes sufEciently rapidly for large p, then a
necessary condition for all these line integrals to vanish
is that the function H(p, p) vanishes in the region
covered by these lines. The support of H(p, P) lies in
the region S, the complement of the region covered by
Fze. 1. The Fourier transform of the commutator as a function
of k' and 2pk. The part of the physical region in which the function
is nonzero is cross-hatched; the nonphysical region is shaded.
&ly/ll&l/&i&/+Pl/&pi~
PIG 2. T. he (p,p)-plane. The weight function H(11,p) is
nonzero in the shaded region S.
lines generated by points in S. Only in this region can
H(p, p) be arbitrary. An alternative derivation of a
necessary condition involving a different characteriza-
tion of H(p, p) is given in the Appendix.
A typical situation is illustrated in Fig. 1, and the
corresponding domain for H(p, p) is shown in Fig. 2.
The two physical regions produced by the two
orderings of the commutator may or may not be
disjoint.
Case A: If M1+M2~& a, the two regions are disjoint.
The region S will touch the boundary of the non-
physical region, and the region S will include only
values of p bounded below by the lines arising from the
vertices of the region (iL Then H(p, p) is nonvanishing
only if) {M,'+2 M1P, (M —)'+2 ( -M )P) (15)
0)P) -1.
Case 8: If M1+M2(~, the two physical regions
overlap. Then the region N, is bounded by two straight
lines. Their intersection generates a line which is a
lower bound for p, but which always cuts the parabola
p= P'P'. The regi—on g is then
p~~ max{M12+p(M12 M22+a2) p2p2) (16)
0)p) —1.
Case A is the normal case. For a vertex arising from
the interaction of stable particles, the stability condition
will make the sum of the intermediate masses greater
than any of the external masses. In this case the sign
of the e-function associated with a point (k', pk) depends
only on which physical region contains the point. The
support of f(k', pk) generated by the representation is
bounded by straight lines, all of which lie wholly in
the complement of region R. Thus in case A we have
shown that f(k', pk) vanishes in the nose of the non-
physical region, up to the line defined by
p, =M1(M2 —z), p~ —(Mt+~ —M2)/2~, (1/)
which corresponds to the tip of the region S. Further-
more, for extreme values of the relative masses, the
representation demands that the contributions from
DESER, GlLBERT, AND SUDARSHAN
certain matrix elements vanish. If 3E2—~&M~, then the
term P(j1~22)(22I j2(p) in which the sum is over all
eigenvalues consonant with a fixed e„, vanishes for
M1'
~& 22' & (M2 —K)'. Similarly, if M 1)M2+K, then
p(j2 ( 22)('0 ~ j1t p) VaIllslles fol M2 ~& I ~& (M1—K)'.
An analogous representation can easily be derived
for the matrix element of a commutator between two
single-particle states with the same momentum.
e'2'*e'2»dxdy((j 1(x)&j 2(y))+
~
p)
=5(k1+k2—p)G+(k12, k22, p2), (18)
then, changing the limits on p to incorporate the term
—PP' arising from the change in the momentum
variables,
I' &u&6db &($1+6—1)II(4 6 W P')
G+(k12,k22, p') =,
p,k12+ &2k22 @+ie— (19)
p ~wmaX{$1M1 +$2(M1 K) ) $2M2 +$1(M2—K) },
$1, )2&~0; p2&~ (M1+M2)'.
For case 8 we use the similar retarded function: the
Only Change in (19) iS that ie beCOmeS ie{k12 k22—
+P'($1—$2)} and the hmits are
p&~ max{$1M1 +$2M2 y p $152}y
6, 6&~ o; p'~& (M1+M2)' ~ (20)
SPECTRAL REPRESENTATIONS
If we specialize the representation, putting a second
particle on the energy shell, we obtain a spectral
representation for the vertex as a function of momentum
transfer. Consicler two identical particles: let k= p —p'
where (p')'= p'=K' (then 2pk=k'), and introduce a new
variable o =p/(1+p) in (1); then, in the physical
region where k'&0, the vertex is
where
H(o.)= tdP(1+P)II(o(1+P), P), .
(21)
(22)
GREEN'S FUNCTIONS
We have derived the representation (1) with limits
(15) or (16). Now we shall write more symmetric
representations, in terms of the momenta k~ and k2
associated with each current. In case A, we shall write
the time-ordered Green s function. Since in this case
the time-ordered Green's function differs from the
retarded function only by the sign of the i e in the
denominator of (1) associated with the physical region
arising from the reversed order of operators in the
commutator, the function we want is given by dropping
the sign factor from the ie term. If we define
and the value of 0-0 is the minimum value of o- in the
region g and is determined by (17):
oo=2KM1(M2 —K)/(K+M2 —M1). (23)
Here M~ is the lowest mass that the odd-particle
current can couple to the vacuum, and M~ is the lowest
mass that can be created by the current of the particle
of mass I(:. For stability SI2&~. In the pion-nucleon
vertex, M1 3222—, M2=K+m, and 0o=3'm 'K/(K m—)
In the first-order electromagnetic vertex, M&=2',
0,=2m 2K/(2K 222 —) The. relation (21), which has been
derived for negative momentum transfer, defines an
analytic function for all values of the momentum
transfer with a cut along the positive real axis. Ke can
observe that the weight function is real and that,
above 0= (2K)', it is related to the amplitude for pair
creation in an external held.
The thresholds for these spectral or dispersion
relations are lower than those of the usual perturbation
theory. However it has been shown by Karplus,
Sommerfield, and tA'ichmann, ' Nambu, ' and Oehme'
that the use of a wider set of intermediate masses in
perturbation theory will lower the threshold. Our
thresholds are lower bounds to the true threshold. It is
quite possible that use of the fact that the intermediate
states are made up of several particles will raise the
threshold s.
ANALYTICITY IN THE MASSES
tA'e can use this representation to study another
aspect of the analytic behavior of the vertex. Let us
treat p' as a time-like momentum transfer and consider
the function (19) or (20) as a function of the mass
ky =k2 =M . Then the function is analytic for negative
3P and can be continued as a function of 3P up to the
minimum value of p. In the case of equal intermediate
masses, this minirnurn value occurs in (19) when
p2=M12=M22 and is —2'M12. Thus in the meson-nucleon
vertex one can continue only up to M'&~(K+222 )'/2.
This would be on the mass shell only if 222 ~)(v2 —1)K,
which is the condition obtained by Sremermann,
Oehme, and Taylor. '%e do not need a continuation of
this type to obtain (21), since we have worked in the
physical region and have obtained the spectral repre-
sentation directly.
ACKNOWLEDGMENTS
We wish to thank Professor Julian Schwinger for
suggesting such representations and for several discus-
sions. One of us (W.G.) thanks the University of
California Radiation Laboratory, Berkeley for its
hospitality where this work was completed.
' Karplus, Sommer6eld, and IA"ichmann, Phys. Rev. 111, 1187
(1958).
2 Y. Nambu, Nuovo cimento 9, 610 (1958).
3 R. Oehme, Phys. Rev. 111, 1430 (1958).
4Bremermann, Oehme, and Taylor, Phys. Rev. 109, 2178
(1958l.
S T R 'lJ C T U R E 0 F V E R T E X F U iX C T I 0 iX 735
APPENDIX
Ke shall give an alternative derivation of the repre-
sentation (1) and prove that the function H(ti, p)
vanishes outside the region S if f(k,pk) vanishes in (R.
It is convenient to give the proof in two stages: First
we show that if the function f(k', pk) vanishes in the
region (Rp. (k —P)'&0, ks&0, then H(/4, P) vanishes
outside the region Sp .. /t4~& —P'P', 0&~P &~—1. No
assumption concerning the mass spectrum, beyond the
axiom of the completeness of positive-energy states, is
involved in requiring the vanishing of f(k,pk) in (Rp.
It is now a simple matter to incorporate the detailed
ma, ss-spectrum restrictions and to show that if f(k', pk)
vanishes in (R, then H(/4, P) vanishes outside S.
Since the Fourier transform f(x) of the function
If we now use the identity
goo
Ag(q) = s(qp) d)~s Ai(Q) ')5(q' —X')
and introduce the parameters
p= —(n+-') / =) '—p'p'
it follows that
f(k', pk) = dp d/4 H(/4, p) e(pk+pp')J i a! p2 2
Xb (k'+2pkp —/t4),
f(k ',p) =—f-(k', pk)
d~e"" '"'(Lji(~l2), js(—*l2)J
~
P)
~oo
~
y&/4
H(/4, P) =4r dt' db' A,L(/4+P'P')'$
ut 0
Xl.(tt b'- (P+-') P'3'}0(b'-(P+-') P')
vanishes for space-like x and the function itself vanishes
in (Rp, according to Theorem 2 of Jost and Lehmann'
it has a representation
f(k' Pk) = O'I dt' yi(u)t')+ yp(u ts)
~~o
X~4(k sP u)— —
with suitable weight functions qj and y2, where 5& is
the invariant commutator function for a "mass" t. In
this case, the weight functions p~ and q2 depend only
on u'. Ke may now make use of the following identity
given by Dyson':
b
(q u) 5(up bs)dp44 prJ Ipt t (bs ns);$6 i(q 8)ds
where v is a four-vector with vanishing space compo-
nents, and time component equal to v. Since in the
Lorentz frame chosen p has only a time component,
~, we may put
and obtain
)ps/4 /I 5/gf(k' Pk) =pr dt' i dbs)' dn lpLt(b' —n'P')~j
0 0 —5/tt
8
X /4q i(b', t')+ q s(b', t')—hi(k —(n+-', )p).
BQ
6 R. Jost and H. Lehmann, Nuovo cimento 5, 1598 (1957).
4 F. J. Dyson, Phys. Rev. 111, 1717 (1958), Eq. (14).
X(s ppi(b', t') —q s (b', t') pj/BP }.
This concludes the first stage of the proof and yields a
completely general representation including causality
and the positive-definite nature of the energy spectrum.
For arbitrary functions ppi and pps, H(p„p) will, in
general, not be a distribution in the Schwartz sense.
It is here that we must con6ne ourselves to the narrower
class of Schwartz distributions. (This corresponds to
an implicit restriction on q~ and p2, which in the
Jost-Lehmann representation could be taken as arbi-
trary functions. ) For these distributions we can apply
the arguments of Jost and Lehmann to incorporate
explicit mass restrictions and show that the function
H(/4, p) which vanishes outside Sp vanishes in the region
outside S, provided f(k',pk) vanishes in (R. For any
point in (R, the line integrals of p(pk+pps)H(/4, p) over
all lines that do not intersect the come@ region S
vanishes if f(k', pk) vanishes in (R; but for all the lines
generated by points belonging to (R, p(pk+pp') changes
sign inside the parabola where H(/4, p) vanishes. Conse-
quently we can replace p(pk+pp')H(/4, p) by an appro-
priately modified H(/4, p) for suitable sets of these lines,
which cover the entire region between S and Sp. A
lemma of Jost and Lehmann' which states that for a
function which is nonvanishing only in a bounded
domain, a necessary and sufFicient condition that all
line integrals over straight lines that do not intersect a
convex region vanish is. that the function vanishes
outside the convex region, then permits us to conclude
that H(/4, P) itself vanishes in the region between S and
Sp. Consequently H(/4, P) vanishes everywhere outside S.


Structure of the Vertex Function

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Structure of the Vertex Function

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