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On a convergent model of quantum electrodynamics III
Ferwerda, H. A. 1965
Physica 31 6 13-628
ONACONVERGENTMODELOFQUANTUM ELECTRODYNAMICS
III
by H. A. FERWERQA*) Afdeling
voor theoretische
natuurkunde
van het Natuurkundig te Groningen, Nederland
Laboratorium
der Rijksuniversiteit
Synopsis This paper contains two applications of a regularized model of quantum electrodynamics which has been discussed in the two preceding papers 1). The regularization consists of the introduction of an elementary length for the electron. The effects calculated in this paper are the anomalous magnetic moment of the electron and the Compton effect for free electrons both in lowest order of perturbation theory. It turns out that an elementary length of about 6.2 x IO-13cm would give rise to an anomalous magnetic moment which does no longer agree with the experimental value within the limits of accuracy. An elementary length of about 6.2 x lo-13 cm would give corrections of the order of 0.2 x0 to Compton scattering of free electrons. The paper concludes with a discussion of the photon mass. It is shown that one has to introduce a counterterm in the Lagrangian for the photon.
1. Intro&.&ion. This paper deals with two elementary applications of a regularized model of quantum electrodynamics, which has been the subject of two preceding papers 1) quoted as I and II in the following. The regularization consists of the introduction of elementary lengths for photon and electron, called ar and a, respectively. In the greater part of the calculations given in I and II it turned out to be advantageous to choose al = 0, a choice which will be adhered to also in this paper. It was shown in I and II that this regularized model of quantum electrodynamics does not suffer from ultraviolet divergences and may be renormalised in the same way as conventional quantum electrodynamics, The purpose of this paper is to investigate the dependence”of observable quantities on the regularizing parameter a, (because we put a; ti 0 we may omit the subscript e in a,). We choose for this investigation the anomalous magnetic moment in lowest order of perturbation theory and the lowest order Compon scattering. Because the theory is renormalizable the observable quantities only weakly depend upon the parameter a. Because the anomalous magnetic moment and lowest order Compton scattering are in very good agreement with the experimental values, we can find an u@er bound *) Now at Cornell
University,
Laboratory
for Nuclear
-
613 -
Studies,
Ithaca
(N.Y.),
U.S.A.
614
H. A. FERWERDA
for the length a, such that every value of a which exceeds this upper bound gives rise to values of observable quantities which do not agree with the values experimentally found within the experimental errors. The anomalous magnetic moment is a quantity which is very well suited to provide such an upper bound, because it has very accurately been measured 2). It turns out that a value of a which is greater than 6.18 x 1O-13 cm gives rise to an anomalous magnetic moment which falls outside the limits set by the experimental uncertainty. In the case of the Compton scattering of photons at free electrons we may compare with experimental values for the intensity of the scattered photons found by Friedrich and Goldhaber 3). Unfortunately, these authors do not state the accuracy of their measurements, so that we cannot derive an upper bound for a from their values. Forced by this situation, we restrict ourselves to a comparison of the differential cross section of the scattered photons as calculated in conventional quantum electrodynamics (Klein-Nishina formula) and the differential cross section as calculated in the regularized model. For the energy of the primary photon we take 0.173 m,, the same value as in the Friedrich-Goldhaber experiment, for the length a we take the “critical” value 6.18 x lo-13 cm. The difference between conventional quantum electrodynamics and regularized model is about 0.2x,. The value 6.18 x lo-13 cm for the critical value of a is of the same order of magnitude as in other attempts to obtain finite expressions, such as the introduction of a cutoff in momentum space 4).
2. Calculation of the anomalous magnetic moment of the electron in lowest order. The way in which the anomalous magnetic moment of the electron may be calculated in lowest order in the regularized model proceeds in exactly the same way as in conventional theory. Therefore, it suffices to indicate how the formula for the magnetic moment is obtained. For the details the reader is referred to any textbook on quantum electrodynamics (e.g.5)). We assume a static external electromagnetic field ylu, the field strengths of which are given by qpv = a,qv -
avqfi.
(2.1)
In order to obtain the anomalous magnetic moment, it is necessary to work in the nonrelativistic limit. If we assume a magnetic moment ,u, the term in the interaction Hamiltonian is H’(T) = -&u / q opv yq~y do
(2.2)
where T is the label of the space-like integration surface n .x = T (n is the normal of this surface). agv denotes, as usual, the spin-density operator (2.3)
615
CONVERGENTMODEL
0 denotes
the surface
element.
(2.2)
also contains
the electric
dipole in-
teraction, but this vanishes in the nonrelativistic limit. If we choose for the integration surface 3dimensional space at a certain time 7, we may write for the contribution of (2.2) to the iteration expansion of the S-matrix: co S’ = 1 -i
s -co
H’(T)dT...
= 1 ++/A .
s
If we go over to the description in momentum tribution of (2.4) may be written
$%,,yq+‘d%
...
space, the first-order
(2.4) con-
(2.5) We used in (2.5) the normalization of Jauch and Rohrlich, a normalization to which we shall adhere throughout this paper. Further in (2.5) we made use of the fact that the energies of the ingoing electron and outgoing electron (E and E’ respectively) are equal, because the external electromagnetic field is supposed to be static. Further it must be borne in mind that the momentum K of the external field, occurring in (2.5)) is considered as outgoing. Let us next consider the two diagrams of fig. 1. The diagrams are the lowest order diagrams contributing to the magnetic moment. The contribution of the diagram in fig. la is denoted by M(ii), the diagram in fig. lb by M(ar) (the first index denotes the total number of vertices, the second index the number of vertices to which the external field is attached). The 0
i
-
P'
p’
-L_L-!+
k
la)
(b)
Fig. 1. (a) and (b) are the lowest order diagrams contributing to the magnetic moment of the electron.
external field has been drawn as a wavy line, the quantized photon field as an intermittent line. If zt(fi) d enotes a Dirac spinor belonging to a free electron just as in (2.5), we have
(2.6b)
$1 is the lowest order radiative correction from the vertex diagram. Af,,&‘, +) has been introduced in II, section 3c. It is well-known that the contribution of M(ii) to the magnetic moment of the electron is given by ,~a = e/2m, the value according to Dirac’s theory. For the calculawhere4,&f,
616
H. A. FERWERDA
-
tion of the contribution of M(ai) to the magnetic moment we expand zZ(+‘) x x 4,&‘) fi) 4fi) in Powers of the momentum k of the external field. If we go up to first order in k, the expansion may be written as
which is immediately clear from covariance considerations. The first term in the right hand side G@‘)yaD U(P) = G(p) A,&, p) u($) = 0 by definition (see section 3c of II). The second term is effectively zero because ka contracted with @(k) gives zero on account of the subsidiary condition. So, in fact, G(fi’) Ai,&‘, p) U(P) reduces to C(fi’)o,,kYJ u(p). Inserting this in M(a1) we find up to first order in k: M(31) =
’ Gw
%@‘) &
a&‘J~(P)
g+(k).
Comparing (2.8) and (2.5) we find for the contribution netic moment of the electron:
(2.8)
of (2.5) to the mag-
A,u = eJ.
(2.9)
The rest of this section is devoted to the calculation of J. J may be calculated from the vertex diagram A&‘, p) calculated in II, section 2. Note that the momentum k was taken there as incoming. In using the formulae for A,@‘, p) given in II, we shall first reverse the sign of k. J may be found by calculating of E(p’) A&p’, rp) .u(fi) the term linear in k. On account of the relation A&‘, 9) = Lye + A&S, p) we see that A&‘, 9) and A,,,(fl’,+) contain the same term linear in k. For the determination of J we may just as well work with A&‘, p). In II A&‘, $) has been separated in two parts, $). We will now a total part _A:“($‘, $) and a longitudinal part A?($‘, show that the longitudinal fiart does not contribute to the magnetic moment (compare the analogous discussion in II, section 6). From the diagram (b) of fig. 1 we write down the contribution of the longitudinal part as: (*’ = + -
4
a@‘) I?S,($
-
I) y&&5
S($‘) @‘-($‘-Q} = A+‘)
-
I) +
u(fi) =
S&‘-z)y&(p-z)
{i(fi’-Z,+m>
Jq$
s&‘-z)y&(+)
{fi-
(a-01
49)
~{q-~)+m)
= z@) (2.9a)
where we made use of (ifi + m) U(P) = 0; In our regularized model we have SC($) =
l 2 Kl(z) ia + m
(ifi’ + m) z&J’) = 0.
with z = (az(fi2 + m2))f
(2.10)
CONVERGENT
(see I, section 2). Abbreviating using (2. IO) : Q’)
z S&’
-
J)ybsc(p -
617
MODEL
zKi(z)
= Xi(z)
I) i h(l)
zt($) =
= -fi(p’)
-xl(P'
we may simplify
(2.9~) by
DC(J) - ~)ycT.xl(p- 4 --,-u(P).
This term can never give a contribution to the magnetic moment, because there are no terms proportional to oovk9. So for the calculation of J it suffices to consider _4pt(p’, $), which has been calculated in II, formula (2.10) :
(2.11) where Rs is the triangle
Ei 2 0; 5s 2 0; 0 5 li + 62 5
1
ba”(-$+$) B Pa&,
E2) =
=
W2
+
m2)t,
-_YvWpl($ +
-
4
~yvy%&5(
(2.1 la) +
(P2 +
+
($’
)P(
)p -
m2)E2 -
-
(ftl
~)yl7yqyY(
++fi’
-
+
(2.1 lb)
$t2)2
)a +
m)pJ($
-
m) 7.u
(2.1 lc)
where ( )# = EiPi + &&; p’ = p - k. The first term of the integrand in (2.11) has no terms proportional to uaVkvand so does not contribute to the magnetic moment. So we may forget about that term. Of the rest of (2.11) we are only interested in the freeparticle value. The free-particle value of AB is ABfree
=
a2
(El +
E2)3
El52
-
P(51
+
(2.12)
t2)
where we abbreviated p = 2p*k/m; CC= am. The free-particle P,(&, 62) gives after a long but straightforward computation P:free)(L
t2)
=
2im2yA2
+ 2im2y&
+
-
iyop-k)
4q%
+
+ Es(6im2ya -
4@:
+ 4iyop .k) +
+ 4 mpi) + 4M2(--im2y~
+ #-2imsr0
of
4+0 p-k) +
-
+ E?(--2im2r0
51(6im2yo
value
+ m(pG + PL) +
+ 4m#a)-44im2y,-44iy,
p*k.
(2.13)
Formula (2.13) may considerably be simplified if we only keep those terms which will give rise to a contribution to the magnetic moment. To this end
618
H. A. FERWERDA
we need the following formulae G(;b’)&u(P) = MY)
(Yufi + fiY0) u(P) =
= &qfi’) (imy
+ $70) u(p) =
= 9Q’)
+ #‘ya + &7) U(p) =
(imy
= Q’)
(2.14)
(iq%Y + Bkyo) a(P).
In an analogous way we derive rQ’)fi,:
u(P) = Q’)
(imy, -
(2.15)
&ok) u(P)
Another term which occurs in (2.13) is: f4fi’)yoP.k
~($4 = ~W+4~fl
+ &,
.u(fi) =
= +q$‘)
(r&
+ imy&)
Q)
=
= +(;b’)
(yofi’k + yok2 + imy&)
u(9).
This last form may further be simplified, because we may omit the term containing k2 (k2 operating on q gives zero). So we are left with QQa
fi.k u(P) = &ii@‘) (y&k = *ii
(-$‘r&
= zqfi’) &($
-
+ imy&) ~(9) = + 2&f; + imy&) N(p) = (2.16)
$) U(p) = 0.
We now insert (2.14), (2.15) and (2.16) in (2.13) and discard all those terms which do not give a contribution to the magnetic moment. Finally we are left with: p:fre”‘(sl,
E2) -
--k
cwkY L-(51 +
E2)2 --mkc
(61 +
E2)l +
l-(51 +
$2) -
(El +
62)21
(2.17)
where the N sign denotes that all terms not contributing to the magnetic moment have been omitted. The term containing k, may be omitted too, because on account of the Lorentz-condition for the external field k,,pc(k) =O. So (2.17) is reduced to : p(freeyEl, 52) 0
--2h
uavk” [(51 +
52)2 -
(51 +
t2)].
(2.18)
Let us now return to (2.11). The problem was to extract from (2.11) the term proportional to agvkv. Because (2.18) is already linear in k, we need only take k = 0 in the remaining factors. This implies that AB may be replaced by (2.19) and B by ms(& + 52)s. So the constant J in (2.9) becomes: J = _f-
8n2m
oovk” R1
CONVERGENT
619
MODEL
The problem is now to calculate the integrals (2.21) for n = - 1, 0. These integrals may be calculated in II, section 4~. So we substitute
This substitution
ti =
a2(51
6% =
cr2(5!1 +
transforms
+
=
52)
(2.22)
62)x
(2.21) into: cc2
MnP2)
along the lines set out
in (2.21):
--&
s
d6i
1
s
dx t?+i
6’Kr(C’)
(2.23)
0
0
with 5’ given by (2.24) Instead of (2.23) we may also calculate: N,(as)
= idli;dx 0
(2.25) is most easily evaluated
EF+’ C’Ki([‘)
(2.25)
0
by considering 1
dNn(a2)
dcts
=
,2n+2
r dx zKr(z) 0
(2.26)
J
where 2= Using the techniques
(x(1
”
discussed in II, appendix
dNn(a2) da2
=
(2.27)
x)}” .
a2”+4[KT(a) -
we get K:(a)].
(2.28)
It is quite easy to compute from this formula N-r(as) and Ne(a2). Namely, from (2.25) it is clear that N,(G) = 0 for a = 0. If we do not differentiate with respect to a2 but with respect to a (2.28) becomes
an(a2) da
= 2a2n+5 [KY(a) - K:(a)].
(2.29)
The integration of (2.29) with respect to a for the cases n = - 1, 0 leads to evaluation of Lommel integrals which have been discussed in II, appen-
620
H. A. FERWERDA
dix. Finally we arrive at the results: N-1(a2)
= -$aQa2Ki(a)
+ iazKi(a)
+ ta2Ko(aJKz(a) No(a2) = (Q -
-
Aa”) a4Ki(4
+ ($ + +2)a3Ko(a)K1(a) + (-$
-
+
aKo(a)Kl(a)]
+ ($ + &a2 + $a4)azKf(a)
&ae)a4Ko(a)Kz(E)
+ t-g-
(I + 2a2) K;(a)
+ a Kl(a)Kz(a)
$a2) adKi(a)
+ (-i
-
+ Q
(2.30)
+ +
ga2)a3Kl(a)Kz(a)
+
+$Y (2.31)
Combining the formulae (2.20), (2.21), (2.23), (2.25), (2.30) and (2.31) we arrive at the following formula for the anomalous magnetic moment in lowest order: 1
Ay=?_ 8n2WZ
82
a2
+
a2)a2Ki(a)
+
A(20
+
2a2 -
+
&J--6 + a2)a2Ki(a) + &(20 + a2)aKo(a)Kl(a)
+
&J--6 +
a2)a2Ko(a)&(a)
-
Jad)KT(a)
+
+
A(12 + a2)aKl(a) &(a) +
t-&-g]*
(2.32)
For small values of a A,u reduces to (,DO= e/2m) :
AP=PO; $ -(QC2 -
-
[l -
*a2 ln2(&a) -
++C + $&)a2
&(16OC2
-
12%
(SC -
-
&a* lna (&a) -
+
*)a4
_..I
+$)a2
In (+a) +
(&C -
$&)a4 In&a) + (2.33)
where C is Euler’s constant: C = 0.577215665 . . . . It is clear from (2.33) that for a -+ 0 (2.33) reduces to the radiative correction of the conventional theory, namely (e2/4n) (1/2n),~e. In table I we have calculated d,u/,ua for a series of values of a. The numbers have been plotted in a graph sketched in fig. 2. It is seen from this graph, that if a exceeds a value of about 0.016, the anomalous magnetic moment does not any longer agree with the value experimentally found. Though we are not completely entitled to deduce from this an upper bound for a because we are working in perturbation theory, we hope that the value a = 0.016 gives the order of magnitude of the real bound. a = 0.016 corresponds to an elementary length 6.18 x lo-13 cm. This is of the same order of magnitude as the values one has previously assumed in calculations working with a cutoff in momentum space 4).
CONVERGENT
In this connection
621
MODEL
it is instructive
to remark that the classical electron
radius, YO= e2/4nm, gives ram = & order of magnitude as tc = 0.016.
= 0.007, a value which is of the same
0.0011633
+ “J%J.
Fig. 2. Anomalous magnetic moment of electron. The experimental value lies between 0.0011633,~0 and 0.0011585p0. - - - - is the lowest order value according to standard quantum electrodynamics. -*-.-.-* is the value up to order e4 in the conventional theory. TABLE Anomalous
I
-
magneticmoment
of the electron OL
-4
x 106
PO
0.000 0.001
1161.40 1161.38
0.002
1161.32
0.003
1161.24
0.004
1161.13
0.005 0.006 0.007 0.008
1161.01 1160.87 1160.71
0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019
1160.54 1160.36 1160.16 1159.95 1159.72 1159.49 1159.24 1158.99 1158.72 1158.44 1158.16 1157.86
I
622
H. A. FERWERDA
3. Lowest order Compton scattering. For the discussion of lowest order Compton scattering we have to consider the diagrams of fig. 3. The calculations are just as elementary as in conventional theory because
Fig. 3. Diagrams
we do not need to integrate standard notations
for lowest order Compton
over virtual
momenta.
scattering.
Introduce
the following
$50 = e; 9’0 = &I; K0 = w; K’O = 0’.
(3.1)
If e describes the polarization of the initial photon and e’ the polarization of the final photon, we find for the sum of the two diagrams in fig. 3: (fk’e’
1M(2)
4P’) x
X
+e” On account
~l({a2((p’
i(j - P) - m (p -
k’)2 + ms
of momentum
xi({as((p
+ k’)2 + m2)}Y
- k’)2 + rn2))*) E’] u(p).
conservation
(3.3)
9, $‘, k and k’ refer to external
lines we have
fis + ms = p’s + ms = k.2 = k’s = 0. So the quantities
(3.2)
we have
jS+k’=#+k Because
+
(3.4)
in (3.2) may be expressed in the scalars K = -p.k K’ = -p-k’
= +‘.k’
(3.5a)
= -p’.k.
(3.5b)
To simplify the calculations we assume the initial electron to be at rest: in this special Lorentz frame we have _ fi = (0, . 0, 0, m). Another important relation, the proof of which may be found in any textbook of quantum electrodynamics, is 0’
1
-=
w
1 + -J
(1 -
(3.6) cos e)*
623
CONVERGENTMODEL
where 0 is the angle between the directions of the ingoing and the outgoing photon. We may now write down the formula for the cross section for the scattering of photons at electrons, the outgoing photons emerging in a solid angle d.Q with polarization e’. Because we do not observe the initial and final spin of the electron we average over the former and sum over the latter. do p_ cl!2 -(
e2 2 1 dafi’uPd& 457> 2ms’ww~ s
S($’ + k’ -
fi -
k)X
(3.7)
with X given by (cf. ref. 5) : x = g
&({--2wma”}*)
+ J$
.xr({2Ic’us}“)
-
ms __ Zr({-2oma2}*)
-
;K;
xT({-2C0ma2}~) ~~((2AZs)“)
3-,({2K’U2)*)
B +
S-;({2Ic’as}*)
KK’
A +
,X3{-2CUWZU2}‘)
c + D
(3.7a)
where msA =
+KK’
+
~(e' -k) (e’ *+‘)
m2B =
&KK’
f
K’(e
m2C = m2D = Formulae (3.7), (3.7~) = 0. We still have to argument of %I. The pagator which means
&KK’
(3.7b)
*k’) (e*p’) -
KK’(e*e’)2
+
iK’(e’*k)(e’.fi’)+&K(e*k’)(e-f).
and (3.7b) are only valid in a gauge for which ea = e’a= give an interpretation of (-20.~2~2)~ occurring in the answer is given by the fact that we use the cuusul prothat we have to go in a specified way around the
Fig. 4. Path for the causal Gropagators A,($) and S,(p).
singularities. This has been sketched for the propagator d,(k) in fig. 4. From this path we infer that we have to interpret (-22mw&)+ = --i12wmua/*. From well-known formulae of Bessel-functions 6) we know that Kr(--ix)
= -&zH:l)(x)
where Hi”(x) is a Hankel-function. The integration in (3.7) may be performed
x>o
(3.8)
as usual and gives: (3.9)
624
H. A. FERWERDA
where we have taken into account the relations (3.6) and (fi’O)Z = (j.)s+ U’s + ms p’ = k Introducing
(3. IO)
2c0w’ cos 8
k’.
(3.11)
the cIassica1 radius of the eIectron YO= ea/4nwz (3.9) becomes (3.12)
In order to obtain the correction due to the introduction of an elementary length we expand X in d. An elementary but long calculation gives:
+ u2 (W-W’) I
In (40.~0’) + $(2C-1)
-L$In2wr+f(l-2C)$) __:(1_2C)
”
X
(e”‘lz’*p’) (e*h;t.fi’)
WI
(O-W’)+
--In 20.+$(1-2C)+
+ (In2~‘--_1(1--2C)+$ln2o+
+ (w’ In 2~‘--w
)
(
In 2~--$(l-2C)
X
(w’-w)) (e-et)2} . ..I
(3.13)
where u and w’ have been measured in unit mmC is Euler’s constant C = 0.577216. From (3.13) one infers immediately that in the limit Ed-+ 0 one gets back the well-known Klein-Nishina formula. If we do not observe the polarization of the initial and final photon, we have to average (3.13) over e and to sum over e’. This leads us to (again measuring w and W’ in units m) :
x ln(4mw’) + t(l-2C) x (w’ In 2w’-
w In 2~) +
x $+*(ln2o’-$(l--2C)+
(-11+cos28)(o-~‘)+~(1+c0s20)X
(
~ln2~+~(l-~C)~~ln2w’+~(l~2C)~ 0
~~ln2rU_(l--2C)
$)&fl}
0 >
X
. ..I. (3.14)
625
CONVERGENTMODEL
We will calculate the right hand side of (3.14) for the values o = 0.173, CL= 0.016 as a function of 8. The value w = 0.173 for the energy of the primary photon is the same as in the experiment of Friedrich and Goldhaber 3) which is claimed to be in very good agreement with formula (3.14) for a = 0 7). Unfortunately, these authors do not give the experimental uncertainties of their results. So we cannot deduce an upper bound for CL from this experiment. Therefore we calculate (3.14) for a = 0.016, the upper bound deduced from the magnetic moment. The results have been tabulated in table II. The limit of (3.14) for CL= 0 has been denoted by Klein-Nishina formula. It is seen from these values that to the accuracy of the values in the second column (which is certainly not exceeded by the FriedrichGoldhaber values) the correction due to the introduction of an elementary length corresponding to CL= 0.016 remains unobservable. So values of a which are smaller than 0.016 will not come into conflict with the experimental values of Compton scattering at free electrons. TABLE
II
Compton-scattering
e
I
do
Y$
d0
--
according
correction value
of -
to Klein-Nishina
to Kl.-N. 1
do Yc? dQ
from
correction dl
in 0i~~
00
1.000
-11
x
10-s
0.1
100
0.973
-11
x
10-h
0.1
20” 30” 40”
0.919 0.836 0.732
-11 -11 -10
x x x
10-e 10-s 10-5
0.1 0.1 0.1
50”
0.629
-10
x
10-5
0.2
60’
0.532
-8
x
10-S
0.2
70”
0.453
-8
x
10-S
0.2
80”
0.384
-7
x
10-5
0.2
90”
0.370
-6
x
10-5
0.2
1000
0.363
-5
x
10-5
0.1
110”
0.360
-5
x
10-5
0.1
120”
0.410
-5
x
10-5
0.1
130” 1400
0.449 0.490
-4 -4
x 10-5 x 10-5
0.1 0.1
150”
0.524
-3
x
0.0
160” 1700 180’
0.555 0.572 0.572
-3 -3 -3
x 10-5 x 10-Z x 10-5
to-5
0.0 0.0 0.0
4. Discussion. The effects calculated in this paper form a rather random selection out of the set of all processes which may give an upper bound for CL.The reason why we chose the anomalous magnetic moment and Compton-effect is that these effects may rather easily be calculated. Another process which will presumably give a good upper bound for u is the Lambshift, because this effect has been measured very accurately. Because the calculation of the correction to the Lamb-shift due to the introduction of cc
626
H. A. FERWERDA
gives rise to laborious
calculations,
we left this effect out of our program.
We may now also answer a question raised in II, section 3b, concerning the mass of the photon. In II we mentioned that the upper bound for the photon mass 6,~ is lo-65 g. This bound has been deduced by de Broglie 8) from cosmological considerations : assuming the de Sitter model for the description of the universe and taking the value 1026 cm for the radius of the universe de Broglie arrives at the value, mentioned above. Needless to say that this bound is very speculative. We will now derive an upper bound for 6,~ for experimental values. In ref. 9 de Broglie suggests that one may obtain an upper bound for 6,~ from data about double stars (eclipsing variables). The underlying idea is the following: if the photon has a mass 6~ # 0, the velocity of light waves depends upon their frequency according to the dispersion formula (SP)2c4 FWC
,
kb)2c4 2h2v2 >
(4.1)
where v is the velocity of the light waves, c is a constant, “the invariant velocity of light”, the value of which is about 3 x 1010 cm/s and v is the frequency. Let us now consider a double star which emits simultaneously two light signals with wavelengths ill and As. Let the signal with wavelength Li reach the earth after tl seconds, the signal with wavelength ils after t2 seconds. From (4.1) one immediately infers that, if I denotes the distance between the double star and the earth, t1 -
t2
=
s,,
-
II;,.
I
In particular, if one chooses ill < 12, (4.2) predicts tl - t2 < 0. This prediction may be tested with astronomical observations. In the beginning of this century the astronomers have measured such a time difference at double stars. They found a difference in the time of eclipse for the various colours of the eclipsing variables. This effect has been called NordmannTikhov effect. Recently Szafraniec 10) *) discussed this effect again and added to the old observations new ones which had been obtained by improved experimental techniques. The result of the analysis of Szafraniec is that one cannot conclude to the existence of the N.T. effect. The colours which have been compared were blue (jlbi M 4500 AU) and yellow (&w5500 AU). From ref. 10 we derived for the mean value of (tbl - ty)/r the value
rblrtyL
= +13
*) The author is indebted
to Dr. L. Plaut
x 1O-2o s cm-i
for bringing ref. 10 to his attention.
CONVERGENT
MODEL
627
with root mean square deviation 0 = 46 x lo-20 s cm-l.
(4.36)
We are now faced with the problem how to interpret this time difference. There may be several effects which contribute to the N.T. effect, for instance, interstellar matter will cause dispersion of light waves in interstellar space. Astronomers also have thought at tidal waves which the components of the double star induce on each other. Our aim is to find a bound for 6,~ from this effect. At first sight it seems that 6,~ # 0 cannot be reconciled with the positive value of (4.3~). Namely, from (4.2) we would conclude that (tbr - ty)/r is negative. The root mean square deviation of (4.3~) is so large, however, that there are negative values of (tbi - ty)/r which are tolerable with the value mentioned in (4.3~). The smallest negative value for (tbr - ty)/r which may still be admitted is Lb1 -
t,
= -33
x 1O-20 s cm-l.
Y
Suppose we could correct (4.3~) for such effects like dispersion from interstellar dust, tidal waves, etc. If after inclusion of these corrections there are still negative values of (i&r - ty)/r which can be tolerated, we might try to calculate a bound for S,u from the value of (?$l - ty)/r. Such a correction is probably very difficult. If we may neglect tidal waves, dispersion of interstellar dust, etc. so that we consider (tbi - &)/Y as solely due to S,U # 0, we arrive after substitution of (4.4) in (4.2) at the bound S,u 5 lo-36 g. We have encountered two bounds for 6,~ by now. We next calculate the bound for a, the regularizing parameter, corresponding to these bounds. Omitting calculational details we find: 6,~ 5
1O-65 g implies a > 84 (see II, section 3b)
6,~ 5
lo-36 g implies a > 23
The “critical” value a = 0.016 derived from the magnetic moment gives 6,~ N 2.7 x lo-27 g. This photon would be about three times as heavy as the electron ! So the question posed in II, whether a counterterm for the photon is inevitable in order to keep the photon mass within physically acceptable limits, must be answered in the affirmative. One must be aware, however, that all these statements are founded upon perturbation theory in lowest order. We believe, as far as observable effects are concerned, that this lowest order is a reasonable representative for the exact value, at least as long as we are working in quantum electrodynamics. This is no longer true for such quantities which are “renormalised away” such as 6,u, for instance. Therefore, it is very well possible that the bounds for a derived from bounds for 6,~ have to be completely revised if we calcu-
628
CONVERGENT
MODEL
late 6~ in higher order of perturbation theory. For the observable effects, it is believed that higher order corrections will not appreciably alter the results obtained thus far. Therefore, the bound o! = 0.016, derived from the magnetic moment, is regarded as a reliable one. We have seen that cc= 0.016 corresponds to an elementary length of 6.2 x lo-13 cm. Instead of an elementary length one may also introduce an elementary time. An elementary length of 6.2 x lo-13 cm corresponds to an elementary time of 1.8 x x 1O-2 cm c-l in our natural system of units (about lo-23 s). It is interesting to note that this is of the same order of magnitude as the lifetime of resonances. Further it would be desirable to compare the results for the Compton scattering and similar processes with more accurate experimental values than were used here in order to get an improved bound for CLFor instance, we would expect such an improvement if we had data on Compton scattering at free electrons for various energies of the primary photon. Especially high energies for the primary photon (in the lab. system) are of interest, because in this region quantum electrodynamics in its conventional form may break down. One may hope (but this is as yet a sfieculatiort) that the regularized model will retain its validity up to somewhat higher energies. In this connection one may think of a being a phenomenological constant which summarizes the effects of virtual particles which do not occur in quantum electrodynamics such as 7c-mesons, etc. As soon as the energy becomes sufficient for these particles to be produced really the model will also break down. We have not yet tried to find an answer on these questions. Compton scattering seems to be a rather simple (from the standpoint of the theoretical physicist) process to check these speculations. Unfortunately, the necessary experimental data were not available at the time of writing of this paper. Acknowledgement. The author owes much to Professor Groenewold for his continuous advice and encouragement. Received
18-2-64 REFERENCES
1) 2)
Ferwerda,
8)
De Broglie,
H. A., Physica
29 (1963) 999; ibid. 30 (1964) 665, quoted
as I and II respectively.
Schupp, A. A., Pidd, R. W. andcrane, H. R., Phys. Rev. 121 (1961) 1. W., Goldhaber, G., 2. Phys. 44 (1927) 700. 3) Friedrich, S. D., Ann. Physics 4 (1958) 75. 4) cf. Drell, J. M. and Rohrlich, F., Theory of photons and electrons, Addison-Wesley, Cambridge 51 Jauch, (Mass.). G. N., Theory of Bessel functions, Cambridge 1952. 6) Watson, W., Quan$um Theory of Radiation, Oxford Clarendon Press, 1954, p. 219. 7) Heitler, L., Mecanique
Ondulatoire
du Photon
et Theorie
Gauthier-Villars (1949) especially chapter V. L., Une Nouvelle Theorie de la Lumiere 9) De Broglie, pp. 39 aud 40. R., Acta Astronomica 12 (1962) 181. 10) Szafraniec,
Quantique
I, Hermann,
des Champs,
Paris,
Paris,
1940 particularly
Physica 31 6 13-628
ONACONVERGENTMODELOFQUANTUM ELECTRODYNAMICS
III
by H. A. FERWERQA*) Afdeling
voor theoretische
natuurkunde
van het Natuurkundig te Groningen, Nederland
Laboratorium
der Rijksuniversiteit
Synopsis This paper contains two applications of a regularized model of quantum electrodynamics which has been discussed in the two preceding papers 1). The regularization consists of the introduction of an elementary length for the electron. The effects calculated in this paper are the anomalous magnetic moment of the electron and the Compton effect for free electrons both in lowest order of perturbation theory. It turns out that an elementary length of about 6.2 x IO-13cm would give rise to an anomalous magnetic moment which does no longer agree with the experimental value within the limits of accuracy. An elementary length of about 6.2 x lo-13 cm would give corrections of the order of 0.2 x0 to Compton scattering of free electrons. The paper concludes with a discussion of the photon mass. It is shown that one has to introduce a counterterm in the Lagrangian for the photon.
1. Intro&.&ion. This paper deals with two elementary applications of a regularized model of quantum electrodynamics, which has been the subject of two preceding papers 1) quoted as I and II in the following. The regularization consists of the introduction of elementary lengths for photon and electron, called ar and a, respectively. In the greater part of the calculations given in I and II it turned out to be advantageous to choose al = 0, a choice which will be adhered to also in this paper. It was shown in I and II that this regularized model of quantum electrodynamics does not suffer from ultraviolet divergences and may be renormalised in the same way as conventional quantum electrodynamics, The purpose of this paper is to investigate the dependence”of observable quantities on the regularizing parameter a, (because we put a; ti 0 we may omit the subscript e in a,). We choose for this investigation the anomalous magnetic moment in lowest order of perturbation theory and the lowest order Compon scattering. Because the theory is renormalizable the observable quantities only weakly depend upon the parameter a. Because the anomalous magnetic moment and lowest order Compton scattering are in very good agreement with the experimental values, we can find an u@er bound *) Now at Cornell
University,
Laboratory
for Nuclear
-
613 -
Studies,
Ithaca
(N.Y.),
U.S.A.
614
H. A. FERWERDA
for the length a, such that every value of a which exceeds this upper bound gives rise to values of observable quantities which do not agree with the values experimentally found within the experimental errors. The anomalous magnetic moment is a quantity which is very well suited to provide such an upper bound, because it has very accurately been measured 2). It turns out that a value of a which is greater than 6.18 x 1O-13 cm gives rise to an anomalous magnetic moment which falls outside the limits set by the experimental uncertainty. In the case of the Compton scattering of photons at free electrons we may compare with experimental values for the intensity of the scattered photons found by Friedrich and Goldhaber 3). Unfortunately, these authors do not state the accuracy of their measurements, so that we cannot derive an upper bound for a from their values. Forced by this situation, we restrict ourselves to a comparison of the differential cross section of the scattered photons as calculated in conventional quantum electrodynamics (Klein-Nishina formula) and the differential cross section as calculated in the regularized model. For the energy of the primary photon we take 0.173 m,, the same value as in the Friedrich-Goldhaber experiment, for the length a we take the “critical” value 6.18 x lo-13 cm. The difference between conventional quantum electrodynamics and regularized model is about 0.2x,. The value 6.18 x lo-13 cm for the critical value of a is of the same order of magnitude as in other attempts to obtain finite expressions, such as the introduction of a cutoff in momentum space 4).
2. Calculation of the anomalous magnetic moment of the electron in lowest order. The way in which the anomalous magnetic moment of the electron may be calculated in lowest order in the regularized model proceeds in exactly the same way as in conventional theory. Therefore, it suffices to indicate how the formula for the magnetic moment is obtained. For the details the reader is referred to any textbook on quantum electrodynamics (e.g.5)). We assume a static external electromagnetic field ylu, the field strengths of which are given by qpv = a,qv -
avqfi.
(2.1)
In order to obtain the anomalous magnetic moment, it is necessary to work in the nonrelativistic limit. If we assume a magnetic moment ,u, the term in the interaction Hamiltonian is H’(T) = -&u / q opv yq~y do
(2.2)
where T is the label of the space-like integration surface n .x = T (n is the normal of this surface). agv denotes, as usual, the spin-density operator (2.3)
615
CONVERGENTMODEL
0 denotes
the surface
element.
(2.2)
also contains
the electric
dipole in-
teraction, but this vanishes in the nonrelativistic limit. If we choose for the integration surface 3dimensional space at a certain time 7, we may write for the contribution of (2.2) to the iteration expansion of the S-matrix: co S’ = 1 -i
s -co
H’(T)dT...
= 1 ++/A .
s
If we go over to the description in momentum tribution of (2.4) may be written
$%,,yq+‘d%
...
space, the first-order
(2.4) con-
(2.5) We used in (2.5) the normalization of Jauch and Rohrlich, a normalization to which we shall adhere throughout this paper. Further in (2.5) we made use of the fact that the energies of the ingoing electron and outgoing electron (E and E’ respectively) are equal, because the external electromagnetic field is supposed to be static. Further it must be borne in mind that the momentum K of the external field, occurring in (2.5)) is considered as outgoing. Let us next consider the two diagrams of fig. 1. The diagrams are the lowest order diagrams contributing to the magnetic moment. The contribution of the diagram in fig. la is denoted by M(ii), the diagram in fig. lb by M(ar) (the first index denotes the total number of vertices, the second index the number of vertices to which the external field is attached). The 0
i
-
P'
p’
-L_L-!+
k
la)
(b)
Fig. 1. (a) and (b) are the lowest order diagrams contributing to the magnetic moment of the electron.
external field has been drawn as a wavy line, the quantized photon field as an intermittent line. If zt(fi) d enotes a Dirac spinor belonging to a free electron just as in (2.5), we have
(2.6b)
$1 is the lowest order radiative correction from the vertex diagram. Af,,&‘, +) has been introduced in II, section 3c. It is well-known that the contribution of M(ii) to the magnetic moment of the electron is given by ,~a = e/2m, the value according to Dirac’s theory. For the calculawhere4,&f,
616
H. A. FERWERDA
-
tion of the contribution of M(ai) to the magnetic moment we expand zZ(+‘) x x 4,&‘) fi) 4fi) in Powers of the momentum k of the external field. If we go up to first order in k, the expansion may be written as
which is immediately clear from covariance considerations. The first term in the right hand side G@‘)yaD U(P) = G(p) A,&, p) u($) = 0 by definition (see section 3c of II). The second term is effectively zero because ka contracted with @(k) gives zero on account of the subsidiary condition. So, in fact, G(fi’) Ai,&‘, p) U(P) reduces to C(fi’)o,,kYJ u(p). Inserting this in M(a1) we find up to first order in k: M(31) =
’ Gw
%@‘) &
a&‘J~(P)
g+(k).
Comparing (2.8) and (2.5) we find for the contribution netic moment of the electron:
(2.8)
of (2.5) to the mag-
A,u = eJ.
(2.9)
The rest of this section is devoted to the calculation of J. J may be calculated from the vertex diagram A&‘, p) calculated in II, section 2. Note that the momentum k was taken there as incoming. In using the formulae for A,@‘, p) given in II, we shall first reverse the sign of k. J may be found by calculating of E(p’) A&p’, rp) .u(fi) the term linear in k. On account of the relation A&‘, 9) = Lye + A&S, p) we see that A&‘, 9) and A,,,(fl’,+) contain the same term linear in k. For the determination of J we may just as well work with A&‘, p). In II A&‘, $) has been separated in two parts, $). We will now a total part _A:“($‘, $) and a longitudinal part A?($‘, show that the longitudinal fiart does not contribute to the magnetic moment (compare the analogous discussion in II, section 6). From the diagram (b) of fig. 1 we write down the contribution of the longitudinal part as: (*’ = + -
4
a@‘) I?S,($
-
I) y&&5
S($‘) @‘-($‘-Q} = A+‘)
-
I) +
u(fi) =
S&‘-z)y&(p-z)
{i(fi’-Z,+m>
Jq$
s&‘-z)y&(+)
{fi-
(a-01
49)
~{q-~)+m)
= z@) (2.9a)
where we made use of (ifi + m) U(P) = 0; In our regularized model we have SC($) =
l 2 Kl(z) ia + m
(ifi’ + m) z&J’) = 0.
with z = (az(fi2 + m2))f
(2.10)
CONVERGENT
(see I, section 2). Abbreviating using (2. IO) : Q’)
z S&’
-
J)ybsc(p -
617
MODEL
zKi(z)
= Xi(z)
I) i h(l)
zt($) =
= -fi(p’)
-xl(P'
we may simplify
(2.9~) by
DC(J) - ~)ycT.xl(p- 4 --,-u(P).
This term can never give a contribution to the magnetic moment, because there are no terms proportional to oovk9. So for the calculation of J it suffices to consider _4pt(p’, $), which has been calculated in II, formula (2.10) :
(2.11) where Rs is the triangle
Ei 2 0; 5s 2 0; 0 5 li + 62 5
1
ba”(-$+$) B Pa&,
E2) =
=
W2
+
m2)t,
-_YvWpl($ +
-
4
~yvy%&5(
(2.1 la) +
(P2 +
+
($’
)P(
)p -
m2)E2 -
-
(ftl
~)yl7yqyY(
++fi’
-
+
(2.1 lb)
$t2)2
)a +
m)pJ($
-
m) 7.u
(2.1 lc)
where ( )# = EiPi + &&; p’ = p - k. The first term of the integrand in (2.11) has no terms proportional to uaVkvand so does not contribute to the magnetic moment. So we may forget about that term. Of the rest of (2.11) we are only interested in the freeparticle value. The free-particle value of AB is ABfree
=
a2
(El +
E2)3
El52
-
P(51
+
(2.12)
t2)
where we abbreviated p = 2p*k/m; CC= am. The free-particle P,(&, 62) gives after a long but straightforward computation P:free)(L
t2)
=
2im2yA2
+ 2im2y&
+
-
iyop-k)
4q%
+
+ Es(6im2ya -
4@:
+ 4iyop .k) +
+ 4 mpi) + 4M2(--im2y~
+ #-2imsr0
of
4+0 p-k) +
-
+ E?(--2im2r0
51(6im2yo
value
+ m(pG + PL) +
+ 4m#a)-44im2y,-44iy,
p*k.
(2.13)
Formula (2.13) may considerably be simplified if we only keep those terms which will give rise to a contribution to the magnetic moment. To this end
618
H. A. FERWERDA
we need the following formulae G(;b’)&u(P) = MY)
(Yufi + fiY0) u(P) =
= &qfi’) (imy
+ $70) u(p) =
= 9Q’)
+ #‘ya + &7) U(p) =
(imy
= Q’)
(2.14)
(iq%Y + Bkyo) a(P).
In an analogous way we derive rQ’)fi,:
u(P) = Q’)
(imy, -
(2.15)
&ok) u(P)
Another term which occurs in (2.13) is: f4fi’)yoP.k
~($4 = ~W+4~fl
+ &,
.u(fi) =
= +q$‘)
(r&
+ imy&)
Q)
=
= +(;b’)
(yofi’k + yok2 + imy&)
u(9).
This last form may further be simplified, because we may omit the term containing k2 (k2 operating on q gives zero). So we are left with QQa
fi.k u(P) = &ii@‘) (y&k = *ii
(-$‘r&
= zqfi’) &($
-
+ imy&) ~(9) = + 2&f; + imy&) N(p) = (2.16)
$) U(p) = 0.
We now insert (2.14), (2.15) and (2.16) in (2.13) and discard all those terms which do not give a contribution to the magnetic moment. Finally we are left with: p:fre”‘(sl,
E2) -
--k
cwkY L-(51 +
E2)2 --mkc
(61 +
E2)l +
l-(51 +
$2) -
(El +
62)21
(2.17)
where the N sign denotes that all terms not contributing to the magnetic moment have been omitted. The term containing k, may be omitted too, because on account of the Lorentz-condition for the external field k,,pc(k) =O. So (2.17) is reduced to : p(freeyEl, 52) 0
--2h
uavk” [(51 +
52)2 -
(51 +
t2)].
(2.18)
Let us now return to (2.11). The problem was to extract from (2.11) the term proportional to agvkv. Because (2.18) is already linear in k, we need only take k = 0 in the remaining factors. This implies that AB may be replaced by (2.19) and B by ms(& + 52)s. So the constant J in (2.9) becomes: J = _f-
8n2m
oovk” R1
CONVERGENT
619
MODEL
The problem is now to calculate the integrals (2.21) for n = - 1, 0. These integrals may be calculated in II, section 4~. So we substitute
This substitution
ti =
a2(51
6% =
cr2(5!1 +
transforms
+
=
52)
(2.22)
62)x
(2.21) into: cc2
MnP2)
along the lines set out
in (2.21):
--&
s
d6i
1
s
dx t?+i
6’Kr(C’)
(2.23)
0
0
with 5’ given by (2.24) Instead of (2.23) we may also calculate: N,(as)
= idli;dx 0
(2.25) is most easily evaluated
EF+’ C’Ki([‘)
(2.25)
0
by considering 1
dNn(a2)
dcts
=
,2n+2
r dx zKr(z) 0
(2.26)
J
where 2= Using the techniques
(x(1
”
discussed in II, appendix
dNn(a2) da2
=
(2.27)
x)}” .
a2”+4[KT(a) -
we get K:(a)].
(2.28)
It is quite easy to compute from this formula N-r(as) and Ne(a2). Namely, from (2.25) it is clear that N,(G) = 0 for a = 0. If we do not differentiate with respect to a2 but with respect to a (2.28) becomes
an(a2) da
= 2a2n+5 [KY(a) - K:(a)].
(2.29)
The integration of (2.29) with respect to a for the cases n = - 1, 0 leads to evaluation of Lommel integrals which have been discussed in II, appen-
620
H. A. FERWERDA
dix. Finally we arrive at the results: N-1(a2)
= -$aQa2Ki(a)
+ iazKi(a)
+ ta2Ko(aJKz(a) No(a2) = (Q -
-
Aa”) a4Ki(4
+ ($ + +2)a3Ko(a)K1(a) + (-$
-
+
aKo(a)Kl(a)]
+ ($ + &a2 + $a4)azKf(a)
&ae)a4Ko(a)Kz(E)
+ t-g-
(I + 2a2) K;(a)
+ a Kl(a)Kz(a)
$a2) adKi(a)
+ (-i
-
+ Q
(2.30)
+ +
ga2)a3Kl(a)Kz(a)
+
+$Y (2.31)
Combining the formulae (2.20), (2.21), (2.23), (2.25), (2.30) and (2.31) we arrive at the following formula for the anomalous magnetic moment in lowest order: 1
Ay=?_ 8n2WZ
82
a2
+
a2)a2Ki(a)
+
A(20
+
2a2 -
+
&J--6 + a2)a2Ki(a) + &(20 + a2)aKo(a)Kl(a)
+
&J--6 +
a2)a2Ko(a)&(a)
-
Jad)KT(a)
+
+
A(12 + a2)aKl(a) &(a) +
t-&-g]*
(2.32)
For small values of a A,u reduces to (,DO= e/2m) :
AP=PO; $ -(QC2 -
-
[l -
*a2 ln2(&a) -
++C + $&)a2
&(16OC2
-
12%
(SC -
-
&a* lna (&a) -
+
*)a4
_..I
+$)a2
In (+a) +
(&C -
$&)a4 In&a) + (2.33)
where C is Euler’s constant: C = 0.577215665 . . . . It is clear from (2.33) that for a -+ 0 (2.33) reduces to the radiative correction of the conventional theory, namely (e2/4n) (1/2n),~e. In table I we have calculated d,u/,ua for a series of values of a. The numbers have been plotted in a graph sketched in fig. 2. It is seen from this graph, that if a exceeds a value of about 0.016, the anomalous magnetic moment does not any longer agree with the value experimentally found. Though we are not completely entitled to deduce from this an upper bound for a because we are working in perturbation theory, we hope that the value a = 0.016 gives the order of magnitude of the real bound. a = 0.016 corresponds to an elementary length 6.18 x lo-13 cm. This is of the same order of magnitude as the values one has previously assumed in calculations working with a cutoff in momentum space 4).
CONVERGENT
In this connection
621
MODEL
it is instructive
to remark that the classical electron
radius, YO= e2/4nm, gives ram = & order of magnitude as tc = 0.016.
= 0.007, a value which is of the same
0.0011633
+ “J%J.
Fig. 2. Anomalous magnetic moment of electron. The experimental value lies between 0.0011633,~0 and 0.0011585p0. - - - - is the lowest order value according to standard quantum electrodynamics. -*-.-.-* is the value up to order e4 in the conventional theory. TABLE Anomalous
I
-
magneticmoment
of the electron OL
-4
x 106
PO
0.000 0.001
1161.40 1161.38
0.002
1161.32
0.003
1161.24
0.004
1161.13
0.005 0.006 0.007 0.008
1161.01 1160.87 1160.71
0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019
1160.54 1160.36 1160.16 1159.95 1159.72 1159.49 1159.24 1158.99 1158.72 1158.44 1158.16 1157.86
I
622
H. A. FERWERDA
3. Lowest order Compton scattering. For the discussion of lowest order Compton scattering we have to consider the diagrams of fig. 3. The calculations are just as elementary as in conventional theory because
Fig. 3. Diagrams
we do not need to integrate standard notations
for lowest order Compton
over virtual
momenta.
scattering.
Introduce
the following
$50 = e; 9’0 = &I; K0 = w; K’O = 0’.
(3.1)
If e describes the polarization of the initial photon and e’ the polarization of the final photon, we find for the sum of the two diagrams in fig. 3: (fk’e’
1M(2)
4P’) x
X
+e” On account
~l({a2((p’
i(j - P) - m (p -
k’)2 + ms
of momentum
xi({as((p
+ k’)2 + m2)}Y
- k’)2 + rn2))*) E’] u(p).
conservation
(3.3)
9, $‘, k and k’ refer to external
lines we have
fis + ms = p’s + ms = k.2 = k’s = 0. So the quantities
(3.2)
we have
jS+k’=#+k Because
+
(3.4)
in (3.2) may be expressed in the scalars K = -p.k K’ = -p-k’
= +‘.k’
(3.5a)
= -p’.k.
(3.5b)
To simplify the calculations we assume the initial electron to be at rest: in this special Lorentz frame we have _ fi = (0, . 0, 0, m). Another important relation, the proof of which may be found in any textbook of quantum electrodynamics, is 0’
1
-=
w
1 + -J
(1 -
(3.6) cos e)*
623
CONVERGENTMODEL
where 0 is the angle between the directions of the ingoing and the outgoing photon. We may now write down the formula for the cross section for the scattering of photons at electrons, the outgoing photons emerging in a solid angle d.Q with polarization e’. Because we do not observe the initial and final spin of the electron we average over the former and sum over the latter. do p_ cl!2 -(
e2 2 1 dafi’uPd& 457> 2ms’ww~ s
S($’ + k’ -
fi -
k)X
(3.7)
with X given by (cf. ref. 5) : x = g
&({--2wma”}*)
+ J$
.xr({2Ic’us}“)
-
ms __ Zr({-2oma2}*)
-
;K;
xT({-2C0ma2}~) ~~((2AZs)“)
3-,({2K’U2)*)
B +
S-;({2Ic’as}*)
KK’
A +
,X3{-2CUWZU2}‘)
c + D
(3.7a)
where msA =
+KK’
+
~(e' -k) (e’ *+‘)
m2B =
&KK’
f
K’(e
m2C = m2D = Formulae (3.7), (3.7~) = 0. We still have to argument of %I. The pagator which means
&KK’
(3.7b)
*k’) (e*p’) -
KK’(e*e’)2
+
iK’(e’*k)(e’.fi’)+&K(e*k’)(e-f).
and (3.7b) are only valid in a gauge for which ea = e’a= give an interpretation of (-20.~2~2)~ occurring in the answer is given by the fact that we use the cuusul prothat we have to go in a specified way around the
Fig. 4. Path for the causal Gropagators A,($) and S,(p).
singularities. This has been sketched for the propagator d,(k) in fig. 4. From this path we infer that we have to interpret (-22mw&)+ = --i12wmua/*. From well-known formulae of Bessel-functions 6) we know that Kr(--ix)
= -&zH:l)(x)
where Hi”(x) is a Hankel-function. The integration in (3.7) may be performed
x>o
(3.8)
as usual and gives: (3.9)
624
H. A. FERWERDA
where we have taken into account the relations (3.6) and (fi’O)Z = (j.)s+ U’s + ms p’ = k Introducing
(3. IO)
2c0w’ cos 8
k’.
(3.11)
the cIassica1 radius of the eIectron YO= ea/4nwz (3.9) becomes (3.12)
In order to obtain the correction due to the introduction of an elementary length we expand X in d. An elementary but long calculation gives:
+ u2 (W-W’) I
In (40.~0’) + $(2C-1)
-L$In2wr+f(l-2C)$) __:(1_2C)
”
X
(e”‘lz’*p’) (e*h;t.fi’)
WI
(O-W’)+
--In 20.+$(1-2C)+
+ (In2~‘--_1(1--2C)+$ln2o+
+ (w’ In 2~‘--w
)
(
In 2~--$(l-2C)
X
(w’-w)) (e-et)2} . ..I
(3.13)
where u and w’ have been measured in unit mmC is Euler’s constant C = 0.577216. From (3.13) one infers immediately that in the limit Ed-+ 0 one gets back the well-known Klein-Nishina formula. If we do not observe the polarization of the initial and final photon, we have to average (3.13) over e and to sum over e’. This leads us to (again measuring w and W’ in units m) :
x ln(4mw’) + t(l-2C) x (w’ In 2w’-
w In 2~) +
x $+*(ln2o’-$(l--2C)+
(-11+cos28)(o-~‘)+~(1+c0s20)X
(
~ln2~+~(l-~C)~~ln2w’+~(l~2C)~ 0
~~ln2rU_(l--2C)
$)&fl}
0 >
X
. ..I. (3.14)
625
CONVERGENTMODEL
We will calculate the right hand side of (3.14) for the values o = 0.173, CL= 0.016 as a function of 8. The value w = 0.173 for the energy of the primary photon is the same as in the experiment of Friedrich and Goldhaber 3) which is claimed to be in very good agreement with formula (3.14) for a = 0 7). Unfortunately, these authors do not give the experimental uncertainties of their results. So we cannot deduce an upper bound for CL from this experiment. Therefore we calculate (3.14) for a = 0.016, the upper bound deduced from the magnetic moment. The results have been tabulated in table II. The limit of (3.14) for CL= 0 has been denoted by Klein-Nishina formula. It is seen from these values that to the accuracy of the values in the second column (which is certainly not exceeded by the FriedrichGoldhaber values) the correction due to the introduction of an elementary length corresponding to CL= 0.016 remains unobservable. So values of a which are smaller than 0.016 will not come into conflict with the experimental values of Compton scattering at free electrons. TABLE
II
Compton-scattering
e
I
do
Y$
d0
--
according
correction value
of -
to Klein-Nishina
to Kl.-N. 1
do Yc? dQ
from
correction dl
in 0i~~
00
1.000
-11
x
10-s
0.1
100
0.973
-11
x
10-h
0.1
20” 30” 40”
0.919 0.836 0.732
-11 -11 -10
x x x
10-e 10-s 10-5
0.1 0.1 0.1
50”
0.629
-10
x
10-5
0.2
60’
0.532
-8
x
10-S
0.2
70”
0.453
-8
x
10-S
0.2
80”
0.384
-7
x
10-5
0.2
90”
0.370
-6
x
10-5
0.2
1000
0.363
-5
x
10-5
0.1
110”
0.360
-5
x
10-5
0.1
120”
0.410
-5
x
10-5
0.1
130” 1400
0.449 0.490
-4 -4
x 10-5 x 10-5
0.1 0.1
150”
0.524
-3
x
0.0
160” 1700 180’
0.555 0.572 0.572
-3 -3 -3
x 10-5 x 10-Z x 10-5
to-5
0.0 0.0 0.0
4. Discussion. The effects calculated in this paper form a rather random selection out of the set of all processes which may give an upper bound for CL.The reason why we chose the anomalous magnetic moment and Compton-effect is that these effects may rather easily be calculated. Another process which will presumably give a good upper bound for u is the Lambshift, because this effect has been measured very accurately. Because the calculation of the correction to the Lamb-shift due to the introduction of cc
626
H. A. FERWERDA
gives rise to laborious
calculations,
we left this effect out of our program.
We may now also answer a question raised in II, section 3b, concerning the mass of the photon. In II we mentioned that the upper bound for the photon mass 6,~ is lo-65 g. This bound has been deduced by de Broglie 8) from cosmological considerations : assuming the de Sitter model for the description of the universe and taking the value 1026 cm for the radius of the universe de Broglie arrives at the value, mentioned above. Needless to say that this bound is very speculative. We will now derive an upper bound for 6,~ for experimental values. In ref. 9 de Broglie suggests that one may obtain an upper bound for 6,~ from data about double stars (eclipsing variables). The underlying idea is the following: if the photon has a mass 6~ # 0, the velocity of light waves depends upon their frequency according to the dispersion formula (SP)2c4 FWC
,
kb)2c4 2h2v2 >
(4.1)
where v is the velocity of the light waves, c is a constant, “the invariant velocity of light”, the value of which is about 3 x 1010 cm/s and v is the frequency. Let us now consider a double star which emits simultaneously two light signals with wavelengths ill and As. Let the signal with wavelength Li reach the earth after tl seconds, the signal with wavelength ils after t2 seconds. From (4.1) one immediately infers that, if I denotes the distance between the double star and the earth, t1 -
t2
=
s,,
-
II;,.
I
In particular, if one chooses ill < 12, (4.2) predicts tl - t2 < 0. This prediction may be tested with astronomical observations. In the beginning of this century the astronomers have measured such a time difference at double stars. They found a difference in the time of eclipse for the various colours of the eclipsing variables. This effect has been called NordmannTikhov effect. Recently Szafraniec 10) *) discussed this effect again and added to the old observations new ones which had been obtained by improved experimental techniques. The result of the analysis of Szafraniec is that one cannot conclude to the existence of the N.T. effect. The colours which have been compared were blue (jlbi M 4500 AU) and yellow (&w5500 AU). From ref. 10 we derived for the mean value of (tbl - ty)/r the value
rblrtyL
= +13
*) The author is indebted
to Dr. L. Plaut
x 1O-2o s cm-i
for bringing ref. 10 to his attention.
CONVERGENT
MODEL
627
with root mean square deviation 0 = 46 x lo-20 s cm-l.
(4.36)
We are now faced with the problem how to interpret this time difference. There may be several effects which contribute to the N.T. effect, for instance, interstellar matter will cause dispersion of light waves in interstellar space. Astronomers also have thought at tidal waves which the components of the double star induce on each other. Our aim is to find a bound for 6,~ from this effect. At first sight it seems that 6,~ # 0 cannot be reconciled with the positive value of (4.3~). Namely, from (4.2) we would conclude that (tbr - ty)/r is negative. The root mean square deviation of (4.3~) is so large, however, that there are negative values of (tbi - ty)/r which are tolerable with the value mentioned in (4.3~). The smallest negative value for (tbr - ty)/r which may still be admitted is Lb1 -
t,
= -33
x 1O-20 s cm-l.
Y
Suppose we could correct (4.3~) for such effects like dispersion from interstellar dust, tidal waves, etc. If after inclusion of these corrections there are still negative values of (i&r - ty)/r which can be tolerated, we might try to calculate a bound for S,u from the value of (?$l - ty)/r. Such a correction is probably very difficult. If we may neglect tidal waves, dispersion of interstellar dust, etc. so that we consider (tbi - &)/Y as solely due to S,U # 0, we arrive after substitution of (4.4) in (4.2) at the bound S,u 5 lo-36 g. We have encountered two bounds for 6,~ by now. We next calculate the bound for a, the regularizing parameter, corresponding to these bounds. Omitting calculational details we find: 6,~ 5
1O-65 g implies a > 84 (see II, section 3b)
6,~ 5
lo-36 g implies a > 23
The “critical” value a = 0.016 derived from the magnetic moment gives 6,~ N 2.7 x lo-27 g. This photon would be about three times as heavy as the electron ! So the question posed in II, whether a counterterm for the photon is inevitable in order to keep the photon mass within physically acceptable limits, must be answered in the affirmative. One must be aware, however, that all these statements are founded upon perturbation theory in lowest order. We believe, as far as observable effects are concerned, that this lowest order is a reasonable representative for the exact value, at least as long as we are working in quantum electrodynamics. This is no longer true for such quantities which are “renormalised away” such as 6,u, for instance. Therefore, it is very well possible that the bounds for a derived from bounds for 6,~ have to be completely revised if we calcu-
628
CONVERGENT
MODEL
late 6~ in higher order of perturbation theory. For the observable effects, it is believed that higher order corrections will not appreciably alter the results obtained thus far. Therefore, the bound o! = 0.016, derived from the magnetic moment, is regarded as a reliable one. We have seen that cc= 0.016 corresponds to an elementary length of 6.2 x lo-13 cm. Instead of an elementary length one may also introduce an elementary time. An elementary length of 6.2 x lo-13 cm corresponds to an elementary time of 1.8 x x 1O-2 cm c-l in our natural system of units (about lo-23 s). It is interesting to note that this is of the same order of magnitude as the lifetime of resonances. Further it would be desirable to compare the results for the Compton scattering and similar processes with more accurate experimental values than were used here in order to get an improved bound for CLFor instance, we would expect such an improvement if we had data on Compton scattering at free electrons for various energies of the primary photon. Especially high energies for the primary photon (in the lab. system) are of interest, because in this region quantum electrodynamics in its conventional form may break down. One may hope (but this is as yet a sfieculatiort) that the regularized model will retain its validity up to somewhat higher energies. In this connection one may think of a being a phenomenological constant which summarizes the effects of virtual particles which do not occur in quantum electrodynamics such as 7c-mesons, etc. As soon as the energy becomes sufficient for these particles to be produced really the model will also break down. We have not yet tried to find an answer on these questions. Compton scattering seems to be a rather simple (from the standpoint of the theoretical physicist) process to check these speculations. Unfortunately, the necessary experimental data were not available at the time of writing of this paper. Acknowledgement. The author owes much to Professor Groenewold for his continuous advice and encouragement. Received
18-2-64 REFERENCES
1) 2)
Ferwerda,
8)
De Broglie,
H. A., Physica
29 (1963) 999; ibid. 30 (1964) 665, quoted
as I and II respectively.
Schupp, A. A., Pidd, R. W. andcrane, H. R., Phys. Rev. 121 (1961) 1. W., Goldhaber, G., 2. Phys. 44 (1927) 700. 3) Friedrich, S. D., Ann. Physics 4 (1958) 75. 4) cf. Drell, J. M. and Rohrlich, F., Theory of photons and electrons, Addison-Wesley, Cambridge 51 Jauch, (Mass.). G. N., Theory of Bessel functions, Cambridge 1952. 6) Watson, W., Quan$um Theory of Radiation, Oxford Clarendon Press, 1954, p. 219. 7) Heitler, L., Mecanique
Ondulatoire
du Photon
et Theorie
Gauthier-Villars (1949) especially chapter V. L., Une Nouvelle Theorie de la Lumiere 9) De Broglie, pp. 39 aud 40. R., Acta Astronomica 12 (1962) 181. 10) Szafraniec,
Quantique
I, Hermann,
des Champs,
Paris,
Paris,
1940 particularly