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The magnetic moment of the electron
ANNALS
OF
PHYSICS:
The
6,
26-57
(1958)
Magnetic
Moment
Ck4R~m Department
of Physics,
RI.
of the
Electron*
S~MMERFIELD~.
Universit!J
of California,
Berkeley,
California
The spin magnetic dipole moment of t’he electron is computed to fourth order in perturbation theory. The procedure is to consider a system of one electron moving in a constant external magnetic field. The Green’s function and mass operator for the electron are computed as functions of this field. The expectation value of the mass operator is the self-energy of the electron and the magnetic moment is identified from that part of the self-energy linear in the external field. The techniques used are illustrated in a calculation of the second-order moment. The derivation of the fourth-order moment is sketched. It is found to be ~(4) = -0.328 (LY/?T)? in Bohr magnetons, where OL is the fine-structure constant. This differs from the result of a previous calcu-
lation. 1. IXTRODUCTIOX
In a previous note (I), we reported the results of a new calculation of t’he anomalous magnetic moment of the electron to fourth order in perturbation theory, which result differed from the earlier value obtained by Karplus and Kroll (2). In addition, we discussed the effects of the new moment on various aspects of atomic physics. It is the object of the present paper t,o exhibit the calculation itself, in some detail. The theory of the spin magnetic dipole moment of the electron had it,s bcginnings in 1928, when, on the basis of his equation for a relativistic elect,ron interacting with an external magnetic field H, Dirac (3) found that the energy of the electron contained a term - (efi/2mc)d.H. (The components of d are the three Pauli spin matrices). The identification of p = efi/2mc as the magnet,ic* moment followed immediately. The subsequent quantization of the electromagnetic and electron fields required that this be corrected to take into account the possibility of radint#ion of electromagnetic quanta by the electron and polarization of the electron vacuum by t#he elect’romagnetic field with electron-positron pairs. These so-called radi* This work is based, in part, on a Ph.D. thesis submitted to the Department of Physics, Harvard University (May, 1957) and was supported, in part, by a grant from the National Science Foundation. t National Science Foundation predoctoral fellow at Harvard (1956-57) and postdoctoral fellow at University of California (1957-58). 26
THE
MAGNETIC
MOMENT
OF
THE
ELECTRON
27
ative corrections had to be treated by perturbation theory. The second-order correction of Schwinger (4) and the fourth-order correction of Karplus and Kroll changed Dirac’s moment to ,u = (efi/2mc)[l
+ >h(a/r)
- 2.973((r/7f)‘].
Karplus and Kroll used the techniques of Feynman and Dyson for constructing matrix elements of observable quantities. As pioneers in the hitherto unexplored fourth order, they were interested in seeing if these techniques remained valid and if the renormalization program could be carried through in a consistent way. Their results indicated that everything behaved as expected. The major difficulty was the length of the calculation. To provide an independent procedure and in an attempt to simplify the structure of the calculation, we here use the massoperator formalism of Schwinger (5). The mass operator may be thought of as describing the mass or self-energy of an electron in interaction with the electromagnetic field. If we impose a constant (in space and time) external magnetic field on the system of one electron, the expectation value of the mass operator, computed to the desired order of perturbation theory, will contain a term which is the self-energy of the electron in such a field. The magnetic moment is identified from the coefficient of that term in the energy which is linear in the magnetic field. In Section 2, after deriving the normal Dirac moment, we define the mass and polarization operators in terms of the differential equations for the Green’s functions of the electron and photon fields. The usual procedure is to define these Green’s functions in the limit of a vanishing external electromagnetic field. The presence of this external field is then taken into account in the same way as the radiative corrections due to the virtual photon field. On the other hand, our Green’s functions are defined to describe the propagation of electrons and photons in the full presence of the external field we have imposed. The inclusion of effects due to the field’ is accomplished by replacing the operator (&/i)(a/&r,) for the mechanical momentum of the electron by (fi/i)(a/&rJ - (e/c)&(z) when the electron moves in a field characterized by the potential A,(x). With the definition of a vertex operator the theory is cast in a form amenable to perturbation calculations. There follows a discussion on the extraction of the field-dependent self-energy from the mass operator. The renormalization of the various operators is carried out so that they correspond to physically observable quantities. Finally, by a series of iterations on the unperturbed and second-order operators, an expression is generated for the mass operator in fourth-order perturbation theory. In order to illustrate the method of calculation, we spend some time in Section 1 When magnetic
we say field.
“field”
without
any
qualifying
adjectives,
we mean
the external
electro-
28
SOMMERFIELD
3 deriving the second-order moment. This involves an integration over the possible four-momenta and a sum over the possible polarizations of a virtual intermediate photon. A method for carrying out the integration is described in detail since it is also used in the fourth-order calculation. If we ignore terms which are of higher degree than first in the external field, the integral turns out to be only slightly more complicated than it would have been in the absence of t,his field. Section 4 is devoted to a description of the second-order Green’s functions and vertex operator and their contributions to the fourth-order mass operator. The resultant effects on the self-energy are invariably in the form of complicated integrals over auxiliary variables which had been introduced in order to simplify the momentum-space integrals. Changes of these variables are specified which serve to bring the integrals into manageable forms. The tot)al fourth-order moment is presented in closed form. In an Appendix we present another method of deriving the moment-which has the advantage that the divergences are eliminated at’ the beginning, but which seems to involve more work. 2. THE
MASS-OPERATOR
The Dirac equation for an electron electromagnetic potential A,(z) is MP, The notation
- 4l(x)l
FORMALISM
moving
in the presence
+ mM.-i(x)
of an external (2.1)
= 0.
is as follows: n=c=1; 2, = (x, ix,) = (x, it), (p = 1, 2, 3, 4); a&,, = a.b - a& p, = -iid/dx,
=
ab;
;
Iys, ~“1 = yII-yy + yvyr = $(x1
=
--CL,
Y,,+ =
-Ye,
~4
=
ir0
=
28;
#+(4P.
The mass and charge of the electron are represented by m and e, respectively. We shall demonstrate that the magnetic moment may be thought of as contributing to a field-dependent mass or self-energy of the electron. For simplicity we impose a constant (in space and time) external field, FPy , given by
In particular,
F,u = dA,/dxp - dA,/dx,. if we take A, = - j@,,~, , Eq. (1) reads (rp + %rFx
+ m)tia(x:)
= 0.
THE
MAGNETIC
MOMENT
OF
THE
29
ELECTRON
Interpreting $$eyFx as an operator which represents an addition to the mass of the electron due to the action of F, we may evaluate the effect to first order in F by taking its expectation value in a state 1$0) of the free electron whose wave function, (Z 1&) = &(z), satisfies (rp + m)#&r) = 0. That is, (AF~)‘~’
= ($0 I MerFx
I $0)
=
- (e/4m)@O I (7~) (rF4
+
WW(rp)
I $0)
= - (e/4m)(& 1 - iy~;y~F&~ - 26pvFp~~~p, I$o) =
-
(e/24
($0
I ( fhmF,w)
+
MLvFd
I #oh
where Lpy = x,,p, - x,p, and u,, = 35 i [ys, ~~1.This is just the change in mass to be expected of a particle with spin $4 and relativistic orbital angular momentum L,, , and hence we say that the spin magnetic moment of the electron is 1 Bohr magneton. We want to find the additional field-dependent change in the self-energy of the electron, in interaction not only with a constant external field, but also with the virtual fluctuations in the electromagnetic field. Schwinger (5) has shown that the Dirac equation may then be modified to read
dp - eoA(x)l#h) + 1 (dx’)Wx, x’Mh’>
= 0,
(2.2)
where M(x, z’) is called the mass function and where $A(x) is still a numerical wave function. The parameter eorepresents the “bare charge”. The correspond ing causal one-electron Green’s function satisfies x")G(x#, rip - eoA(xE)lG(x,x’) + I (dxN)M(x,
x') = 8(x - x').
(2.3)
In the Lorentz gauge the causal one-photon Green’s function equation is - k3~/a~&.,,(~, 0 + J wwdf,
.f)sd5”,
0 = W(E - 0,
(2.4)
where P,,“(,$ s’) is called the polarization function. It describes the interaction of photons with virtual electron-positron pairs. We shall distinguish between the electron and photon coordinates by using x and .$,respectively; in momentum space, p and 12.It is convenient (6) to think of a function O(z, x’) as the matrix element of an operator 0: (x 10 [ x’). The matrix elements of 0 in momentum space may be obtained from the Fourier transformation O(p, p’) = (27r)4 / (dx>(dx’>e+20(x,
x’)eip’z’,
30
SOMMERFIELD
which
with
is equivalent
to the matrix
transformation
(z / p} = (2?r)-2eip’. Some special matrices
are
(x 1 1 ] x’} = 6(x - x’), (x 1p, 1x’) = -i(a/ax,)s(x (x 1A, 1x’) = A,(x)G(x
If we suppress
- x’), - x’),
(x I Y,(E) lx’>
= YdiX - 0%
(x 1y,(k)
= ypCik”6(x
1x’)
(2.5)
- xl,
- x’).
all indices, Eqs. (3) and (4) assume the abstract [fn
+
M]G
=
[I? + P]$j
forms (2.6)
1,
(2.7j
= I(
in which II
The introduction
of a “vertex”
=
p -
or “interartion”
r,it) = -Wh4d~)lG-1
(2.8)
e&4.
operator
= ~~(0 - W~ed,(~)l~~
(2.(3)
allows one to write M = mo + iez PC&, .t’) = -ieo2 tr
s
s
(ds>(ds’)r,i~>Gr.(~‘)~“~(~‘,
t>,
(dx>(~x')(dx">r,(s>Gix,x’)L(x’,
xN; .$‘)G(x”, xj,
(2.10) (21.1)
where mo is t,he “bare-mass” parameter and where “tr” indicates a sum over diagonal matrix elements. The operator M is a relativistic scalar matrix in the same algebra as the y-matrices and is an operator function of the operator II. ,411suchscalar operators not explicitly dependent on the constant external field F can be expressed in terms of rTI. Scalar matrix operators linear in Fare rFr and ~FII, while F’, yFFH and IIFFn are quadratic in F; and so on. We write Al as follows:” M = m. + Mo[ylI] ’ Boldface
brackets
will
be used to exhibit
+ Mp[ylI,
functional
FjJ
dependence
(2.12) on operators.
THE
MAGNETIC
MOMENT
OF
THE
31
ELECTRON
so that
G = {-ia + mo+ Mo1-A + M,[-,a, 81I-‘. The symbol quire
8 represents
all scalar operators
explicitly
dependent
(2.13) on F. We re-
lj? MPF-yq31 = M&p, 01 = 0. In the absence of an external field, G must possess a pole at ?II = -m, in order that it represent the physical electron of mass m. We expand fM,[yII] in a Taylor series about -yII = -m:
Mo[$I] = M0[-ml + (-ill + m)JZ0+ MhN.
(2.14)
The number a0 is given by
I@0= (a/am>M0trIIl I?II+~=o. In the limit F = 0, ylI --f -m,
(2.15)
we require that
Mo[-m]
= m - m. ,
whence G[F = 0, -/II + -m]
-
{YP + m + (rp +
m>B0}-’
= (1 +
JZo)-‘(rp +
m)-I.
(2.16)
The Green’s function of the physical electron should behave near the pole like pa; ml-‘. A ccordingly, we define the renormalized electron Green’s function, > G’ = (1 + &,)G
= {YU + m + (1 + ~0)-‘WhW
+ Jfd~& 51))-’
(2.17)
= &II + m + M’)-l. We renormalize the photon Green’s function in a manner similar to our treatment of the mass operator and electron Green’s function. The polarization operator must be a function of k and F only:
P = POW+ P&k Fl such that P&k,
(2.18)
F] -+ 0 as F ---f 0. We write P,[k]
= k2P’o + P,[k].
The term P&k2 = 01, which would correspond absent since the principle of gauge invariance
(2.19)
to a physical photon mass, is guarantees that both the bare
32
SOMMERFIELD
and physical photon Green’s function
masses are zero. Thus we define a renormalized
photon
9’ = $21 + PO) = [Ii’ + (1 + P&‘(P,[k]
+ P&,
(2.20)
F]))-’
= (k” + p’)-’ so that 6’ - l/k2 for k2 -+ 0 and F -+ 0. Since $j is a vacuum expectation value quadratic renormalized electromagnetic potential A,’ = (1 + &)1’28, In order to preserve the gauge-invariant we identify the physical charge e as
in A,, , we must define a
.
structure
(2.21 j
of II as it appears in Eq. (17),
e = (1 + PO)-l’neo . If the external field were made vanishingly to yield G-’
=
YP
+
mo
+
where rro is the field-independent
Mo[vI
-
(2.22)
small, Eq. (9) could be integrated
e
s
(2.23)
~W,OW,‘W,
part of rP , possessing the general structure r,o
=
Y,wlP,
74
(in which the ylr may appear imbedded among the other factors). Near the pole yp = -m, and in the limit of interaction with a very low energy (X = 0) photon, we have rpo- -yph[ - m, 0] so that G-l -
(yp + m)(l
In order that gauge invariance preserved, which implies
L@O)
- eyA’h[-m,
be maintained,
h[-m, By defining a renormalized
+
01.
the structure
interaction
for G’-l
operator
corresponding
G ‘+ = rP + m + MJrpl We may now express
erl’ must be
0] = 1 + Sio.
rr’ = r,(i + ATo)-l, we obtain an expression
p -
(2.24)
to (23):
- e / (dS)r,‘(@A,‘([).
the reduced mass operator,
(2.25)
M’, in terms of the renor-
THE
malized quantities
MAGNETIC
MOMENT
OF
THE
33
ELECTRON
e, G’, $j’, and I”:
M’ = ie”( 1 + itTo)-’ 1 (~s)(~~‘)Y~(C;)G’~UI(~‘)S~~(~‘,
r
(2.26)
t).
The subscript T indicates the removal of the first two terms of the expansion in powers of (~II + m) of the F-independent part of the integral. Likewise, the reduced polarization operator, P', is given by Ppyl(,$, 4’) = -ie” (1 + BJ’
tr / (ch)(ch’)(~zn)r&9 +
(2.27)
x G'(x, x')r,'(x', x"; (‘)G’(x”, x), where the subscript r now indicates the removal of the first two terms of the expansion in powers of h?. In momentum space, these formulas are
M' = i(a/45~~)(1 + iI&)-’ 1 (dk)(dk’)r,(
- k)G’r,‘(k’)S,‘(
+
- Ic’, -k)
P' = --i(ar/47r3)(1 + @J-l tr j (dx)(dx’)(dz”)r,(k) 7
(2.28)
(2.29)
x G'(x, x')I'y'(x', xv; -k')G'(xn, x), where ~1 = e2/4r is the fine-structure constant. We consider the quantities we have defined to various orders in e. To lowest order : M r(O) = 0 ,
P I(O)=o, (2.30)
G‘(O) = (TV + m)-’ = Go,
rp,(O)(k)= y,(k) = yrewiks, gw‘(“)(k, k’) = (6,,/k2)6(k- k’). To second order:
M ‘(2)
=
i(cy/4?r3) sr
(dk)(dk’)r,(-Ic)(rTI
+
m>-‘r”(k’>(6,,l~2C2)g(k
-
k’)
(2.31)
= i(a/4?r3) s (dk)(lc2)-'~,M~ - 76)+ 43, r P ‘(2) = -i(a/47r3) tr s (dx)(dx’)~~e-ik”G”o’(x, r
x’)y,eik’2’G’(0)(x’,
x).
(2.32)
34
SOMMERFIELD
In deriving Eq. (31) we have used t,he fact t,hat eiks acts like a displacement operator. That is qiklG,[n]e-i%l = G’[~I - X]. (L7.X~j The fourth-order
mass operator may be written I(?) = MJf’ + M,’ + Afr’ + AI,‘. N
(Lx4‘)
Each term contains one of the quant,ities A??,, G’, l”, and 9’ t,o second ordrl and the others to lowest order. The four second-order ingredients are as follows: [(l + i7fo)-1](2) = -j$()-oi2’, G ‘W = (~II + m + icf’(2))-1 p The derivation have rrc2)(i)
of I”(‘)
r,‘“‘(k)
= -i((r/47r3)~-ik”
,
= -(l/p)p’(“(l/]i~)~
s
(2.:sci;l (2.3i‘i
From Eqs. (9) and (SJ), WC’
[ (dlz’)(k”)-‘y,[6/GeA:(~)lGo[~
= -&/4n3) so that in momentum
(yII + nz) = -Gadf”2’G3
is a bit more complicated.
= -i(cx/47r3)
(2.30 i
(dk’)(k’2)-‘y,G,,[~
- l&p
- k’]y,([)G,,[r~
- lily,,
space 1 (dk’)(k’2)-‘y,G&I
- 1; - I&G&I
- k’]yy.
(2.38)
We have made use of the fact, that [6/6eA,‘(t)]II,
= -6,”
.
(2.39)
To renormalize I’,(k) we must divide by 1 + .!a,, , which, to second order, means subtracting Be’“‘r,(k). In terms of these second-order ingredienk, the four terms of the fourt,h-order mass operator are JIM’ = - *jo(?)L7p’ , 34,’ = i(a,‘47r3) I’ (dlc)(k2)-*~,J’2’[IT r n/l,’ = i(cf/47r3) 1 (dk)(k2)-‘y,[y(lI I MO’ = i(cr/4rr3)
/ (dk)(dk’)y,[y(II 7
(3.40) - k]y, ,
- k) + m]-1eik”r,‘2’(k)
- k) + m]-lyyei~‘t-k”~yp”“(
-k’,
(2.41) + f&f,
(2.42)
-k).
(2.43j
THE
MAGNETIC
MOMENT
OF
THE
ELECTRON
35
To find the magnetic moment to second and fourth orders, we examine the expectation value of MP(2) + MpC4) in a state whose wave function satisfies Eq. (1). 3. SECOND-ORDER
CALCULATION
We introduce some simplifications in notation by writing A, and PPpyy for eA,’ and eF,,v’. From the definition of II in Eq. (2.8) we infer the operator commutation relation
[II,, El = iF,v .
(3.1)
We have taken F,,, to represent a constant field so that L
, F,AI = 0.
(3.21
We define 5 = ?&,wF,w = (i/2)-&7,tv The two relativistically tion relation
invariant
.
(3.3)
scalar matrices, 5 and yII, obey the commuta[.ylI, 51 = 2iyFI-I.
The second-order calculation contains, in short form, practically all of the features of the more lengthy fourth-order calculation. We consider the secondorder mass operator as given in Eq. (2.31). Without the subscript r the integral would diverge for large k. Upon making the subtractions indicated by r, we would find that the integral would now converge for large k, but would diverge in the region of small k. To handle these difficulties in a consistent manner, we introduce two cutoffs: the ultraviolet cutoff Xm2(X -+ a), and the infrared cutoff em2(e + 0). These are incorporated into the photon Green’s function, 6, so that em’ behaves like a photon mass and Xm2/(IC2 + hm2) multiplies g. That is, we use a new 6(X, E) given by (ii2 + em” f P)S(X, 6) = km2/(k2 + Am*),
where lim,+,. x+, g(X, E) = 6. Then (2.31) takes the form
This type of integral will occur throughout the calculation. It is characterized by an integration over momenta k, and a sum over polarizations ~1. The major complication is the presence of the y-matrices and the variable k in the several denominators.
36
SOMMERFIELD
To eliminate nominator: [y(lI
the y-matrices
in favor
of the F-matrix,
we rationalize
the de-
= [m - y(rI - k)] (m’ - [y(rl - k)]“)y.
- k) + ml-’
By virtSurc of the anticommutativity
of the y’s and Eqs. (1) and (3) we hare (3.5)
py(Il - liq2 = 5 - (rl - x.y so that
J!l ‘(2) = $ [ (dk) k2 ; Em2k2 yirn2 rlh - -a - k)l r 1 X n&z+ (n - /02 - 5 -Yp.
(3.6)
A simple method of doing the k-integration involves the introduction of integrals over auxiliary variables, by means of which the various denominat,ors can be combined into a single factor. The causal Green’s functions G(z, 2’) and ~(5, .$‘) are such that their Fourier expansions contain only positive frequencies for r. > CC,,’ and fo > to’; and only negative frequencies for xo < x0’ and 4‘0 < .$O’. These boundary conditions are made explicit, in the momentum representat,ion of the Green’s functions 1/(7II + m) and l/k”, by a.ssigning an infinitesimal negative imaginary part to the denominators, which may be thought of as an addition to the electron massm. It is seenfrom (6) t,hat each of the denominators is thereby given the appropriate imaginary part’. We make use of the identity m dse-isa, l/a = i (3.7) s0 which is valid provided that a possesses a negative imaginary part. We apply (7) 60 the three denominators of (6) and obtain iVf’(2) = (a/4n3)hm2 l
ds l*
X l (dk) exp I-iit(k*
dt lm dq
+ Em”) - iq( k2 + Am”)] r,,[m -
-/(II -
IL)] (3*8)
X exp ( -is[m2 + (II - k)’ - 51]y,. The exponential containing the sum, m” + (n - k)” - 5, may be decomposed into a product of the individual exponentials, since the terms in this sum commute with one another. We are interested in terms linear in F and therefore may disregard any expressions of higher degree. We write, accordingly, exp is5 = 1 + is5
(3.9)
THE
MAGNETIC
MOMENT
OF
THE
37
ELECTRON
to obtain M’@) = (a/4,r3)Xmz
l*
(dk) exp [-
ds l*
dt im dq
im”(s + te + qx)lr,[m + y(JI exp [--is@
k)l(l
+ isF)y,
(3.10)
- k)2 - i(t + g)k2].
The indicated sum over polarizations can now be carried out. The only machinery needed is the anticommutation relation of the r-matrices. It follows that YNYP = -4, (3.11)
The relations (3.12) are consequences of the fact that F,,” is antisymmetric. The sum over polarizations for any expression of first degree in the field may be derived from these formulas. In the present example, we have r,b
- rm
- k)](l
+ iti)rN
= -4m
- 2y(II
- k) + 2 isFy(II
- k).
(3.13)
The k-integration has been reduced to an essentially Gaussian integral. Such an integral is normally computed by completing the square in the exponent with respect to k. Since we are dealing with noncommuting operators (for example and -&I2 &I), it is necessary to investigate the process of integraPI - k112 tion in some detail. It will turn out, that to first degree in the field, only slight modifications must be made in the normal procedure. We consider the integral I = 1 (dk)[l To complete
+ sk,
+ c2k,kx + c3k,k,kpl
the square in the exponent
exp [-is(k2
- 2akII + bU2)].
(3.14)
we write
k2 - 2ukII + bIf = (k - &‘I)’
+ (b - a2)II12.
We note that in any expression in which F,,,, appears or in which there is a commutator [II, , II,], the remaining II’s can be freely commuted with one another
38
SOMMERFIELD
(to first degree in F). Thus we may apply the formula @4+B) =eee A B -l.'Z[A,B] (in which it is assumed that [A, [,4, B]] = [B, [il, B]] = 0), t,o our exponentid and obtain exp( -is(k
- aH)2 + I12(b - a’)) = js(exp [-is(k
- an)‘], exp [-is@
- a”)II’J).
We have used the fact that the exponential of t#hecommutator can be expanded t’o first degree in F. In the same vein we may write exp ( -is[(lz
- an)’ + II’@ - a”)]1
= !<{exp [-isII’(l,
- a2)] exp [-is@,
- alIl)‘] exp [-is(kz
X exp [-2$X,
- dI,j’]
- ul&)‘] exp [is(ko - a&)‘]}+n
.
where the notation ( )+n indicates that an average is to he taken of t.he quantity in braces and the same quantity with t,he order of II-factors reversed. (Any y-matrices that appear are not to be symmetrized in this manner. ) Each of the factors k, , k, , . . is now placed next to the exponential con&~ing the samecomponent of k. The int)egral can then he expressed as a product of four integrals of the form I’ =
mdkl(l + cl’kl + cZ’kl’) + c3’k13)exp [-is(kl s-co
-
dhYJ,
(x15?
each of which involves only a single component of II, which, in turn, has no other operator with which not to commute. We can then apply ordinary methods of integration to 1’. We let k, - an1 = 5, so that, the range of 5, also extends from - m to m. Terms of odd degree in ,& will integrate to zero since they transform to their negatives under the reflection EL+ -& . We are left with I’ =
- dfGJl + C1’UrIl+ c2’(u21112 + f,“) + c3’(u3r113 + 3u&12111)] s-cc
(3.16)
X exp ( --is&t).
Integrating by parts, we obtain
so that I’ = [I + Cl’UrIl +
c2’(u2rL2
- ibis-‘) + C3’(u3r113 - ~&xIls-‘)]
x 1: d&iexp(-id;).
(3.17)
THE
Still to be determined
MAGNETIC
MOMENT
is the product
OF
THE
39
ELECTRON
of the four Gaussian
integrals
The first three are all the same while the fourth is their complex conjugate. the value of their product is (~/is)“2(?r/is)1’2(?r/is)1’2[(?r/iS)1’2]* When we put together I = -i(7r2/s2)~~{[[I
= (a/is)(?r/s)
the four integrals
= -i(?f2/s2).
of type I’, we obtain
+ clan, + c*(u2111,11~- ~~is-‘s,,)
+ c3(a3mLJ-&3 - ~~~s-1a[n,6,~
(3.18)
+ II,&4 + q?S,,l)]
X exp [-GI12(b We have shown the k-integrations, except that the which is of first The square is
Thus
- u”)])+n
that in the presence of the noncommuting to first degree in F, are the same as for result must be symmetrized in the manner degree in F before integration need not be completed in the exponent of (10) by writing
.
operators lI,, , II, , ordinary numbers, indicated. A term symmetrized at all.
s(n - w2 + (t + qP = (s + t + q)P - sW(s The translation yield
i = k -
M ‘(2) = -&/4?r)X?rL2
[sn/(s
+
t
sm ds’ I’d&
+ t + q>J” + n24t + q)/(s
i- ~$1, and subsequent
+ t + 4).
integration
over E
- U) l1 dw
r0
X exp { -is’m”[u x K-4m
+ (1 - u)we + (1 - u)(l
- 2yII + 2is’u( 1 - u)FyII)
exp [ -Ls’I12u(l
+ +52~{2-yII, exp [-is’II”u(l where the new variables
- w)X]]
(3.19)
- u)] - u)])]
s’, U, and w are given by s’=s+t+q, u = s/(s + t + a>,
(3.20)
w = t/(t + 9). Since we intend to take matrix elements of M’(2) in a state for which +ylI has a definite value, we use (5) to replace II2 by (m)” in the exponentials of (19).
40
SOMMERFIELD
Also, to first degree in F we have exp [-is’u(1 - IL)S] = 1 - is’u(l that m .1 1 ‘(21 dw esp { -im’s Af = -i(cY/47r)Xm2 ds &L(l - 2L) I0 s0 Ts O x X [-4m
- u)we + (1 - u)(l - w)Xl}
[u+(l
- 2(1 - u)yII
-
+ 4isu(l
-
(3.21)
+ iszc(1 - ~)(2 - u)(yII,
u)mS
X exp [isu(l
a)5 so
5)]
- u>(yII)“].
We have used the fact that [5, (yII)*] = 0 and have dropped the prime from s’, Equation (7) together with its generalization m dssne-ias = 7I! a-‘“+” 2.(n+l) (3.22) s0 are used to obtain M ‘(2)
1 =
4:
’ -
Am2
+s 0
1
du(f
- u)
dw
.I*0
4m + 2(1 - u)yII mz[u + (1 - u>we + (1 - u)(l - w)h] 4mu(l - u)5 + ~(1 (m2[u + (1 - u)w~ + (1 - u)(l [4m + 2(1 - u)ylI]
- u){5,yII} - w)X] - ~(1 - u)(~II)~}~, i
24)(2
log m2[u
- ~(1 - u)(m)”
+
(1 - u)hm2 (1 - U)E] - ~(1 - u)(~II)~
_ 4mu(l - u>F + ~(1 - u)(2 - u)jF, yIIj m2[u + (1 - u)~] - ~(1 - u)(~II)~ The second-order magnetic dependent part of (23): Jf,‘2’
= - -El 4rr o ’ du(1 s
moment,
- u)u
= &)sl 4nwl 0
+ 0(1/x).
pC2), is to he extracted
from
4m5 + (2 - u)lS, -III) mz[u + (1 - u)a] - U(l - U)(YrV
duu(1 - U)
(3.23)
2u 22 + (1 - u>e
t,he field-
(3.24)
(3.25)
= --cr(s)/4?mL + O(&) where (5) = ($A 15 ) +A). Thus the second-order moment is cr@)= a/2*.
(3%)
THE MAGNETIC
MOMENT
OF THE ELECTRON
41
We list the steps we have taken with the mass operator in obtaining this result, for they are to be repeated in the fourth-order calculation. 1. Rationalization of the denominators. 2. Exponentiation of the denominators. 3. Summation over polarizations P. 4. Integration over momenta Tcby completing the square in a Gaussian integral and symmetrizing the result appropriately. 5. Introduction of new variables of integration such that only one integral extends from 0 to 00, while the others run from 0 to 1. 6. Integration over the variable running from 0 to 00, to convert the exponent back into a denominator. 7. Evaluation of the expectation value for a state with -rlI = --m. 8. Integration over the remaining auxiliary variables. 4. FOURTH-ORDER
A. CONTRIBUTION
CALCULATION
FROM THE SECOND-ORDER
ELECTRON
GREEN'S
FUNCTION
To find the second-order electron Green’s function as given in (2.36), we must perform the subtractions indicated by r on the integral in (3.23). Integration by parts yields MO
(2)
=z
(4m+yrI)logx
-4m
+I
i
(4.1)
4mu + yrI(2u - u”) + 61dum2[u + E(l - U)] - U(1 - u)(yrI)2
[m2(1 - E) - (1 - 2U)(yII)2]
1
for the field-independent term. The expansion in powers of (yR + m) is then Mot2’
zz
2
3mlogh-;m+(7rI+m(logh-;) .,g’dU
(2;2;t;;2u
--) ‘) + (~II + m) I’
du (4.2)
X
-4u3 + 2u* - 2~” + e(4u - 2u2 - 8u3 + 5u4) + :(-2u
+ 3u2 - u”)
[u” + ((1 - U)]2
+ Mlc2’. We place E = 0 in those integrals which converge in this limit. The forms taken by the integrals which diverge or are indeterminate as 6 + 0 are found by substituting u = IL’&, and dropping terms of higher order in di. The result is
MlC2’ = MO”’ - Bm - jiiT~‘~‘(yII + m),
(4.3)
42
SOMMERFIELD
where 3 = (3a/47r)(log
x + $6)
(4.4
and @o@) = (a/47r) (log x + 95 + 2 log e). The Green’s function
is then
G ‘C2) = (.fD + m + J/p’
- Bm - (yrI + m)il&(2) 1-l - (-frI + m)-’
1 1 lGoC2) += i ylI + m + M(2) - yrI + m I yn + m 1 + -a + m(l i
1
= G'@)a+ ,y'(%b+
- B) - m+ G’(2)C.
Except for a multiplicative to a fourth-order moment
p b
factor,
G’(*)* is the same as Go and thus gives rise
- (2)
(4)
G
(4.5)
\
(2)
=
MO
P C-l.61
=
(a”/T”)(g
log
x +
3fs
+
$22 log
The first term in G’(‘)’ is the same as Go except that m(1 - B). Equation (3.24) then tells us
‘m2(1
Ej.
has been replaced by
m
4mS(l - B) + (2 - u)(5,yII} - B)2[u + (1 - u)~] - u(l - U)(YII)~
- MpC2) $A)
4 + 4u[u + t(l - u)l cfw9 l duu(1 - U) I u2 + t(l - U) 4rrm 0 [u” + E(1 - u)]” I i = cdl(5)/27rm J4y = (&r”>(-3/4 log x - 36).
zzz-
We rationalize follows : G”2)a[~]
and exponentiate
the denominator
= (m - -yII -I- Mc2’) (m” - (m)” =2
-1 0
+ [yII, M”‘]
(m - -yII)(m” - (7lI>“)-’ m ds( Mc2) - is(m - ylI)(2mM’2’
X exp ( -is[m’
-
(m)“]).
of G”“‘[II]
by writing
(4.7)
it as
+ 2mM’“’ 1-l
+ [yII, M’2’])l
(4.8)
THE
MAGNETIC
We place this expression Mg
MOMENT
THE
43
ELECTRON
into Eq. (2.41), using Eq. (3.19) for &I@), and obtain
= --i((r2/16?r4)X2m4 6’ ds f x \ (dk)O
OF
-
ds’ lm dt lrn dg 6’ du 6’ dw
24) exp {-im2[s
+ s’u + ~‘(1 - u)we
(4.9)
+ s’ (1 - u)( 1 - w>Xl - it(k2 + em”) - i&c2 + Am”) ) X 7pN~-fp exp ( -$II
- Ic)~[s + s’u(1 - u)] ).
The expression NI is a polynomial function of the various operators, matrices, variables and constants, the details of which will not be reproduced here. The basic structure of the integral is contained, not in N, , but in the accompanying exponentials. Performing the indicated sum over polarizations ~.r, we obtain a new polynomial N2 . The exponent whose square is to be completed involves k2[s + s’u(1
- U) + t + fJ] - 2[s + s’u(1
- U)]krI
= [s + s’u(1 - u) + t + q]{k - n[s + s’u(1 - u)]/[s
+ [s + s’u(1 - u)]rI” + s’u(1 - u) + t + Cj])”
+ r12[s+ S’UO - u)l(t + Q)/[S+ The integration
proceeds as indicated
f = k - lI[s + s’u(1
s’u(1 - u) + t + g].
in (3.18). The translation
- u)]/[s
+ s’u(1
is
- U) + t + q].
We obtain MGa = -(ar2/16c)X2m4
1 ds ladi
lmdt
d dp 6’ du 1’ dw(1
x [s + S’UO - 24)+ X exp ( -im2[s
+ SIU + et + e/(1 -
x (N3, exp [-iII”[s
-I- s’u(1 -
where the parentheses indicate tial with terms of Na . We change variables :
u)](t
i- Q)/[s -I- s’u(1 - u) -k
the appropriate
t
t/(t
+ q).
(4.10)
symmetrization
- W)]) -I- aI)>
t
of the exponen-
+ 9,
- u>]/[s + s’u(1 - U) + t + q],
v = S’U(1 - U)/[S + s’u(1 w’ =
+ q1-”
+ hq + XS’(1 - U)(l
U)W
p = s + s’u(1 - U) + 2 = [s + s’u(1
t
- U)
- U)],
(4.11)
44
SOMMERFIELD
In order to express everything exp [-ipz(l
in terms of -/II and 5, \ve use
- 2)1112]= [l - 2$x(1 - 2)5] exp [ipx(l
and unsymmetrise into NP :
the quantity
in parentheses,
- x)(yll)‘]
thereby introducing
extra terms
Maa = -(a2/16?r2)X2m4 bmdp 1’ du 1’ dv 1’ dx l’ dw X d’ dw’pu-‘x(1
- x) exp
+ EXF + xx v(1 ; w)
+ E(1 - r)w’
+ h(1 - x)(1
X We are interested (A~v#~)/
in the field-dependent
II
- w’)
exp [@x(1 - z)($l)“].
N4
part evaluated
for yll
= -m:
= (4~ I MFOa I 1c/za) = -(~2/16?r2)(S)X2m4 s0 1 dw’p2U1x2(1
l-dp
1’ dtL i1 dv 1’ de 1’ dw
- x)
1
x-v+
X exp --ip(l
- z>m2[ew’ + X(1 -
X (m[-16i(l
- x) - 4i(l
-
16i(l
- u)(l
f
lSi(l
- v)x(2
- 8(1 - v)(l + 16(1 - v)(l
w’)]
- z&)(1 - s)’ f 32i(l
- v) - 4i(l
- u)(l
- z&)(1 - v)z(l
- 2) + 4iv(2 - u)(l
- x)(1 - u)] -i- m3px(1 - x)[-32(1
- x)(1 - u) + 16(1 - v)(l - x)*(1 - u) - 8(1 + 8v(l
The integrations
(4.13) - .c) - 2)
- v)
- x)
z~)(l - U) + 16(1 - U)
- u)(2 - U) -
16v(l - a)]).
with respect to w, w’, and p yield
(AFm)‘4’$
= (a2/4a2m)(5)(A,,
- -‘lhr - ‘IA + AXA),
(4.14)
THE
x
MAGNETIC
MOMENT
OF
THE
45
ELECTRON
2U2(1 - U)222(1 - X)(1 - V>[UV + x(5 - 3U - 22 + 2UX)] i Ml - u)x2 + u2xv + (1 - 4xdJ + 41 - UN - XM”
(4.15)
+ u(1 - z&)x(1 - 2) X
-10+6~+132-13~~+v(6-55u-122+12z~r 41 - u)x2 + u2xv + (1 - u)xv+ + $1 - U)(l
- z)# ) *
We are interested in the leading terms of this expression in the limits c -+ 0 and X + co. The presence of X in the denominators of A,x , Axe , and AXX would seem to indicate that these terms are of order l/X. But in the region of integration where the coefficient of X is small, the X term in the denominator may no longer predominate. In Ahe , if v - l/X, dv - l/X, then Axe - dv/v - 1 and not -l/X. On the other hand, in A,x if (1 - z) or (1 - u) - l/X, there are sufficient powers of (1 - U) and (1 - z) in the numerator to cause A,x to vanish at least as fast as l/x. The same applies to Axi . Thus we have (AFm)‘4’g
= (a2/4r2m)(5)(Are
v = v’/X in Axe and the dropping
By means of the substitution higher order in l/h, we find
A,, - Axe = 6’ dub’
- Ax,).
(4.16) of terms of
d~~(l - X)
- t&)x” + E(1 - x>( -10 + 6~ + 132 - 13~~) [x” + E(1 - X)1”
x 1%[ux +
Xz(1 - U) + 8(1 - U)][U + x - UX]
2u(l - U)&5 - 3U - 2x + 2?Lr) [x2 + 4 - x>][u + x - ux][ux + e(1 - $1 1
+ ~lduj)+h{ + 0 - 4(l
-
2XZU3(1 - U)” (u + x - ux>qux + ((1 - U)]2
- x)(6 - 5u - 12x + 12~~) x(1 - u> + vu
2 2UV(l - x) + 2UX + 22(1 - X)(5 - 3U - 2x + 2Uz) x(1 + ?m . u> (1 - U) 1
(4.17)
SOMMERFIELD
46
We have set t = 0 in terms which converge in that limit. For those terms which are divergent or indeterminate when C= 0 we use the substitution x = 4; X' and reject terms of higher order in 4. The integrations are straightforward and yield A,, - A4X.= -?i
(4.18)
log x - 1 - f.J log C.
Using Eqs. (16), (6) and (7), we obt*ain (4.19)
IJ(4)LIa = (c+r”)(~‘s log x + $% + 14 log C) and pL(4)a
=
B. CONTRIBUTION
pcc(4)ca
+
$4)
b G +
/14)oc =
(a"/hr')(l+j
FROM THE SECOKD-ORDER
PHOTON
+
!i
GREEN'S
log
c).
(4.20)
FLTNCTIOX
The second-order photon Green’s function is given by Eq. (2.37). The fact that s is vacuum expectation value quadratic in A, implies that $, and hence I” are even functions of the external field. In particular, P’ possesses no term linear in F so that for our purposes it suffices to look at the zero-field part of P’. That is, me use the zero-field electron Green’s function (yp + m)-’ in Eq. (2.32). This may be written as an integral over intermediate electron momenta P ‘py@)(k,k’) = -i(a/47r3) tr / (dp)(dp’)y,(p’ 7
j e--iks I P>
(4.21)
le jklz’ / p’)(yp’ + WI-‘.
x (YP + C%(P Noting that (p’ 1e--ikz1p) = (2#
1 (dx)(dx’)e-ip’Ze-ik26(x - x’)eiPZ = 6(p’ - p + k)
(p I eikPZ1p’) = s(p - p’ - k’) we may write P’p”(2) = --6(k - k’)(ia/47ra) tr [ (dp)r,(rp r
+ m)-‘rJr(p
- Ic) + m]--‘.
(4.22)
Since Ppy’ describes the polarizat,ion current we must require that it satisfy the conservation condition k,P,,’ = 0.
(4.23)
The p-integration in Eq. (22) is similar to the h--integrations that we have already discussed, and in fact simpler. For there are no noncommuting operators. To maintain the validity of (23) a proper cutoff procedure must be used. Two such
THE
procedures
MAGNETIC
are described
MOMENT
by Feynman
P ‘*i2) = 6(/k - Id) (&
OF
THE
2 Icy-;;
6’ du(1 - U”)
X exp [ -ism” The projection
[6,, -
operator
(4). The result is
(7) and Schwinger
- g)
(k,k,/k2)]
47
ELECTRON
- ~krc2(1
guarantees
P,y’. The lower limit SO(+ 0) is the cutoff. Subtracting
(4.24)
- U’>l.
the transverse character of the first two powers of k2
we find
P’$Q
= -(&
- %)6(k
- k’)(k2)y-&-+ (4.25) x I’
g&v(2y-k’,
-k)
= (b
du(3u2 - u”> [l
+ 4$
(1 - u’)]-l
- fg)“(k-k’)g-& (4.26)
4
x I’
d” 1 + (;&);1
- U”) *
The longitudinal, k,ky/k2, part of $y,,(2) does not affect the evaluation of any observable quantities. Thus the part of the fourth-order mass operator contributed to by s’(‘) is
M, = i($/48?r4m2)
1’ du(3u2 - u”) s, (dk)y,[y(II x r,[l
- k) + ml-’ + (k2/4m2)(1
We do not need any cutoffs here because we are interested pendent part of M, for which we obtain 1
Mpg=
-&
J0
&3U2 1 -u*
u4
(4.27) - u”)l-‘. only in the F-de-
-&
f0
1
s0
dx(1 - x)x exp [-im”sz
- 4im2s(l
- x)/(1
X [4mS + (2 - 2) {-ylI, 511 exp [isz(l The field-dependent (A~vz)‘~‘,
self-energy
- u”)] - s)
(4.28)
(~II>“l.
is then
= @A 1 Mm 19-4) a2(5) l = --($-&I dx(1 - z) 4’ du(3u2 - U”)
[-+J
- aI-’
(4.29)
48
SOMMERFIELD
so that (4) CL 0 = (a2/a2)(l’~&
- 7773).
(4.30)
C. CONTRIBUTIONFROMTHESECOND-ORDEKVERTEXOPERATOR The vertex part of the mass operator contains two terms, as given in Eq. (2.42). The contribution of M,’ to the magnetic moment is just /pM = -&(2y (t.31 j Using Eq. (4), we have (4X)
$*)M = (cz2/7r2)[-;~~ log x - yis - ,r,i log Cl.
The unrenormalized second-order vertex operator is given by Eq. (2.38) t.ogether with the cutoff modifications :
The details of the integration are quite similar to those we have already discussed. We shall content ourselves here with describing the choice of a set of variables which will make the final parameter integrations as simple as possible. To this end we must examine the development of the exponentials in which we complete squares. Upon introducing an auxiliary variable for each of the denominators in (xs), we obtain an exponent (apart from a factor -i): m”(r + s + tr + qx) + r(n - 6 - ky
+ s(n - k’y + (t + q>Y2
= m”(r + s + It + qX)
+(r+s+t+q)
[
(T + s)rI
k’-
+
r+s+t+q
7-k r+s+t+q
1 2
(4.34)
With the introduction of the new variables p
=T+s+t+q,
‘U
=
(T
V
=
T/(T
+
s),
w
=
t/tt
+
Y),
+
S>/(T
+
9 +
1 +
q),
(4.33
THE
MAGNETIC
MOMENT
OF
THE
49
ELECTRON
the exponent may be written m2p(u + (1 -
U)[CW + X(1 - w)])
+ p(k’
+ pu(1 - U)l12 - 2pu(l
- Url + UUk)2 - u)?Jkll + puv(1 - UV)k2.
(4.36)
The translation k’ = L’ + uII - kuv and subsequent integration over 5’ leaves the exponent without its second term. The expression for eikzI’P(2)(Ic) thus obtained is inserted into Eq. (2.42). Three additional parameters, s’, t’, and Q’, are needed for the new denominators. The additional contribution to the exponent of (36) will be m2[s’ + t’E + q’h] + s’( IIAfter we complete into the form
the square with
respect
k)2 + (t’ + q’)k2.
(4.37)
to Ic, the entire exponent
is brought
m”(pu + s’ + E[P(l - u>w + t’l + Qdl - do - w> + q’l] [s’ + pU(1 - U)V]lI 2 + [puv(l - uv) + s’ + tl + q’] k puv(1 - UV) + s’ + t’ + q’ 1 i
(4.38)
[s’ + pu(1 - U)V]” + l-I2 s’ + pu(1 - U) PUV(l - uv) + s’ + t’ + q’ 1 * i If we introduce
the variables, r’
= puv(1 - UV) + s’ + t’ + q’, II: = bUV(l
- UV) + s’]/buv(l
y = puv(1 - uv)/[puv(l w’ = t’/(t’
over k expression
m2r’
-
v(l
“-“.,)
+ dl + x
Y> + E [$(l-~u~
Y(1 1
(4.39)
- UV) + a’],
+ q’),
then after integration
1
- uv) + s’ + t’ + q’],
ddl
- w>
UV(1 - UV)
(38) becomes + W’O - 41
(4.40)
+ (1 - w’)(l - X)]}
II
Y(l - 4 Y(l - 4 2 + r12r’x (1 - Y) + v(l _ uv) - x (1 - Y> + l-WV * [ i
The next steps are to set II2 = 5 - (yIQ2, take the expectation value in a state for which yll = -m and integrate over r’. If we confine ourselves to the F-dependent part of MY(~), the integrations over w and w’ can be performed and we obtain four terms, which arise from w and w’ = 0 or 1, just as in Eq. (14). Methods
50
SOMMERFIELD
similar to these we used before must be called upon t’o extract the leading terms in Eand X. We are then left with the following structure, now as a denominator: Y [ V(1 - UV) + l - y
1i -
Y(l
-d
l - y + v(l - *cy)
Ii =v(lyuv)+z [1 - 41z
-x
[
1-y+
y(1 - u) 2 1 - uv
(4.41)
YO
-y+
l-uv
.
At this point our integral is a sum of terms of rational form. The numerators are polynomials in u, v, x, y and the denominator is the one we have exhibited either as it stands, or squared, together with various factors like (1 - uv) or 21. The whole works, however, is explicitly finite. All integrations go from 0 to 1. The best procedure is to make additional changes of variables, such t,hat expression (41) becomes factorable. Consider I(%4
(4.42)
= ~l~Y~lduf(u,Y,v,z).
We make the substitution y(l - v)u = (1 - y’)(l
- MU)
[4.-r.?)
so that
I(v, 2) = 6’ du J1:.,,,,,,, s0
(4.44)
1
1
=
dy’ ,L’,l--u;j f [u, Y(Y’, u, v), v,xl
dd
du u;l-mu;l fh,
sa-u')l(l-uu')
Y(Y’,
u, v>,v, 4.
Our denominator is then
v;l--y;) + qi2 = v(l
y v) [XY’“V(l - v>-
Y’
+ 11
(4.45)
and the u-integration can be done forthwith. Finally, the substitutions [ = 1 - vxy’; Tl = (1 - vy’>/(l
r =0 -
Y’Ml
- vxy’); -
2
NY’, 2, v> =
act, 71,!?I
(1 - $;1
- &J{)
(4.46)
VY’L
bring the denominator into the form (1 - W2(1
- t + G-)(1 - i-)-Y1 - w
(4.47)
THE
MAGNETIC
MOMENT
OF
THE
51
ELECTRON
a product of factors each of which is linear in f, I], and [. The q integral is evaluated first and then, by means of several integrations by parts, the !: and 4 integrations are performed. An advantage of this method is that each term is finite at every step. Some of the E integrals are Spence functions which are evaluated using the results of Sandham (8). For example, we have 1
-2 s0
NC2 - a-’ log E + t-’ log (2 - [II log (1 - E) = - 2 1’ d&$-l log ] 1 - 4 1 log (2 - 0 = -2$Q(3)
+ 7r2log 2,
0
where l(3) is the Riemann-zeta The result is (4) l-w
= (a2/7r2)[~
function
log x - 38 + %2r2
of 3. + 9i5-(3)
= (cd2/7r2)[% log x - s3.ie + x27r2 + 9&(3) The total fourth-order moment (20), (30), (32), and (48): (4) IJ
_
a2
-
g&r’ log 21 + /kL4)
is found by adding
the contributions
197
R2 144 -[ + g + ; r(3) = - 0.328 (a”/~“>. APPENDIX-ANOTHER
(4.48)
- >$r” log 2 - x log E].
- ; 7r2log 2
1
from
(4.49) (4.50)
METHOD
The reader, after observing how the infrared and ultraviolet divergences have cancelled, may wonder if it is not possible to compute Mp + Ma directly. Indeed, it is possible and furthermore one can do the entire calculation without introducing infrared or ultraviolet cutoffs. Unfortunately, there apparently is more labor involved in this procedure than in extracting the effects of the cutoffs. We have M. + Mr = i(a/47r3) $ (dk)(k2)-‘r,e”“[Gor,“2’(Ic)
+ G’(2)ype-isz]. (A.l)
Considering
r,‘(k) = /- (ds) e-i”=I$(d = = -
s s
= - /- (d~)e-““[s/sA,(~)]G’-’
(dx) (dk’)eik”[6A,(k’)/6A,(~)1
([6/6Av(k’)lG’-‘1 (A.3
(dx) (dk’)e-“““e-ik’“( [6/6A,(k’)]G’-‘1
= - (Z?r)“[s/SA,(-k)lG’-’
52
SOMMERFIELD
we have, in terms of the notation (2a)4[S/SA,(-K>]
(xl‘,
= 8A )
rfp@)(x;) zz -&@-1)(2) = L[Go-‘G’@‘Go-‘1 = (L[Go-‘G”“]
( A.-ii
1Go-l + G,‘G”“SAG;-I
= - ( &[jjfl@)Go] } Go-’ - (&-1(=‘(2)yre--ikB, When this expression leaving
is placed in (Al)
the second term will cancel with
(A.3)
Ma + Mr = - i((u/4n3) 1 (dk) (k2)-1yreikZGo (6A [M’(*)G,,] } Go-‘. We have already computed an expression more. For consider the derivative
A/, ,
for &I’(“’ in Eq. (3.23). But n-c need
The effect is one of reducing the degree of F. Hence we would have to compnt,c M’@’ to second degree in F to obtain a derivative correct to first degree. Furthermore, in such a calculation we could no longer make use of the fact that F is :k constant field. For we observe that even though [II,, Fir-] = 0, the same is not, true of the derivative: GaPI, , FL,] = [III, , 6,FxJ
= -ii~-~~‘k,[G,Ji,
- 6,,kx] f 0.
We note that as long as BA does not act directly on the field F, we may correctly use Eq. (3.23) for M’@). Thus we may write M I@) = jjflW + M F(“1 + j@(2) (A.7:1 where the effect of a@) appears only when 6a acts directly on F. Let us consider the contribution of 41,“’ + MF(” in Eq. (5). We compute MI” by making the subtractions indicated in Eq. (1.3). Letting c = 0, we find
M1@‘Go = Ml’“‘(7II
+ m)-l =-
(A.8)
a 4n
+ 1
; [
+
(yrI)“u
1
+ mgI(2 + u) - 2?nS m” - (1 - u)(-yrp .
THE
MAGNETIC
The first term of the integral so that
MOMENT
OF
is infrared
&JMl@)Go] = ad 2 I’ du
THE
divergent.
(4Vu
53
ELECTRON
But it is independent
+ myII(2 + u) - 2m2 m2 - (1 - u)(~II)~
of II
(A.9)
which is finite. We may take the derivative and insert the resulting expression into (5). When 6a acts on II it replaces it with eCikzy, . The displacement property of this operator implies that each II standing between the two y,,‘s will be shifted to II - Ic. After the k-integration is done we set yB = -m. The term Go’ = ~II + m, on the far right of (5), is not affected by the L-shift and integration so that it will cause the whole expression to vanish unless the integral standing to the left of it has a pole at yII = -m. In the present case the k-integral turns out to be quite finite at -yII = -m so there is no contribution at all to the magnetic moment. We next consider Mpc2) as given by Eq. (3.24): 4m5’ + (2 - U>lS’, 7II) m2 - (1 - u)(rII)2 *
MF
(A.lO)
The prime on 5’ signifies that we are to treat it as being unaffected by the operation 6a . Then MF(‘) commutes with Go and we may write {8A[Mp(2)G,,])Go-1
= (s~[G,,M~(2)])Go-1 = Goy,,e-ikzMp(2) + Go[GAMac2)]Go-‘.
Placing the quantity -&/4?r3)
s
(A.ll)
Ggype-ZksMF(2) in (5) we obtain
(dk) (k2)-1e”ZY,GoGo^lre-ik2MF’2’ = -i(a/47r3)
s
(dl~)((k~)-~y~[y(Il
-
k) + mlw2y,
(A*12)
= [(a/am> Moc2)lMpc2). When we set -/II = -m this will just cancel the contribution to the self-energy from M, as given in (2.40), if we use the definition (2.15) of 20~~). The other term, G&iAMF(2)]Ga-1 will produce no contribution to the moment because of the factor yII + m on the right. In this case, it turns out that the k-integral behaves like log [I - (rJJ)“/m”] as ylI -+ -m, but this is not sufficiently singular to counteract the zero of (YII + m). After all these terms have been disposed of, what remains is simply Ma + Mr + M,
= --i(a/47r3)
/ (dlc)(k2)-‘r,eik”Go[6,~‘2’l.
(A.13)
54
SOMMERFIELD
As we have said, it is extremely difficult, if not impossible to calculate a’” directly. But we can find 8afi@‘. We shall indicate how this may be done. The second-order mass operator is MeI0 = -i(a/4n3)
jrn dss I1 du j” (dl&[m 0
- T(n
0
-
k’)]
(A. 14)
X exp ( -isu[m2 + (II - k’)’ - 51]yy exp (-itk”). We may not use the fact that 5 commutes with IYIuntil after ~3~has been applied. Instead, following Newton (9), we write more generally, correct t.o second degree in F exp ( -isu[(H
- k’)’ - 51) = exp [-is@
- k’)“]
1
+ isu
s
n
dv exp ]-isu(1
- v)(n - k’)‘]~ exp [---isuv(~z _ k’)“] (A.15)
-I- (isu)”
I’
dv exp [-isu(1
- v)(n - k’)‘]-Fl’dw
X exp [--isu(v - zo)(II - k’)“]S exp [-isuw(II Using the fact that aa 6‘4M ~2) =
-i(a/47r3)6
= e-ikzia,hkx,
X rY[m - y(n
we obtain (dk’)
.l-sds[du/
- k’)‘].
exp [ - isum’]
- k’)](l + isuS’)y, exp ] -is[(k’
- un)”
(AJ6)
+ 41 - u)r121) + Kl, where KI = i(a/4*3)e-ik” I= szdsllu
dul’dv
/ (dk’)rJm
- ~(II
-
1~ - k’)]
(A.17)
X [a,&~ + isuva&~5 + isu(1 - v)Fu,Ji+yV X exp [-isu(l
- v)(II - k - Jc’)~]exp [-isuv(n
- k’)‘].
If we can extract the terms 6,[Mo’*’ i- M,‘*‘] explicitly from 6A[M(2’], then we will be left with SA[SP]. TO this end we note that the v-summation in (A-l(j) is the same as that which occurred in (3.13). If we write -4m
- 2y(II =
-4m
-
Ii’) + 2isuS’T(II -
- k’)
2yH(l - u) + 2isu(l - ,td)SSyIT+ 2(1 - iszLs’)y(k’ - un),
THE
and, following exp ( -is[(k’
MAGNETIC
Karplus, - uII)”
Klein,
MOMENT
OF
and Schwinger
+ ~(1 -
- @I)*
from 6,&T’*’ the following
- K1 - K2 - Ka = -i(a/4&A X exp (-isi%‘* + 2isu(l
(IO),
4n”11 + I1 $Y@lay)
X exp (-is[(k’
6,M’*’
55
ELECTRON
+ u(1 - u)II”] 1
= exp { -is[k’*
then we may extract
THE
- u)S’ylI]
- isum*)[-4m - u)II*l
(-isum*>[-4m
+ 2isu(l
+ ~(1 - u)II*]},
part:
jm s ds 1’ du / (dk’) 0 0
exp [isu(l
uexp
(A.18)
- 27TI(l
U)
= -(a/4&i,, - 2-yII(l
- u)S’yII]
-
lrn dss-’
-
(A.19)
u)
exp [--isu(l
- u)II”],
where
Kz =
-~(w’~T~PA
irn
s
ds I,’ du f
(dk’) exp (-;m&)
(A.20)
x 2(1 - isuS’)#
- urI)P(l)
and KS = -i(d4r3)64
ia
x I-4m
s ds 1’ du 1’ dy 1’ dw / (&‘) - 2$I(l
- u) + 2isu(l
exp (-isum’)
- u)s’yrI](--is)
(A.21)
e-i6(l--1D)g(y)[(a/dy)~(Y)]e-i~g(Y). We have used the abbreviation 4(Y) = (k - uyIQ* + u(1 - u)l12
(A.22)
and ‘,he formula (A.23)
56
SOMMERFIELD
We may write exp [-isu(l
- u)II’]
= exp (--ku(l
= exp [isu(l exp [isu(l
- u)[Cf -
- u)(yII)‘]
- u)(l
(-fII)‘J)
-isu(1
- v)(-yII)‘]T
- ,f(> 6’ dv
exp [z’su(l - 2l)v(yII)‘]
(A.24)
11)
+[-isu(l
- u)]”
dv
s0
X exp [is(v - w)u(l in order t’o replace II” by (7II)‘. 5 explicitly, we obtain 6,kf’“’
dw exp [is(l
s0
- u)(yII)‘]
5
+ 2isu(l
-
4m -2yII(
- u)“-JIS’
+ 2isu(l
lrn dss-’
1-
- u)(yII)‘];7
[iswu(l
-
Using this in (19) and
- h-1 - K, - K, - K4 = -(a,i4a)6, exp ( -isum’)[
exp
- u)u(t
u)(yn)']
the derivative
d&g
I’ du
w) + 4isntzc( 1 - u)5’
.?L):T’$T]
exp
of
[i.su(l
-
(A.25)
u)(yn,"]
where
X [-4m
- 2y(II - k)(l - u) + 2isu(l - u)T’y(II X exp (iS(1 -
V)U(l
-
U)[r(II
-
- k)] (A.26)
Ii)]‘]
x [a,J;x - isu(1 - ?L)VU,XbXS’ - isu( 1 - u) (1 - 2’)3’U,XkJ X exp [isuu(l - u>(-fII)‘]. Finally, we write 6,M”’
- K1 - K2 - h-3 -
K4
-
K5
= - ((~/47r)6* lrn dss-’ l1 du exp ( -isum2) X [ - 4m - 2rII(l
(A.271
- u) + 4isu(l - u)5’ + isu(1 - 2~) X (2 - u)(5’, rII)]
exp [isu(l - ,u)(yII)“],
1
duu’(1 - ze)exp (-isum*)[V,
rn]
X exp [isu(l - u)(rII)‘].
(A.%)
THE
MAGNETIC
MOMENT
OF
THE
ELECTRON
57
We recognize that after the subtractions of the first two powers of (7JI + m) from the field-independent part of (27), the quantity remaining to he differentiated is precisely 1MIC2’ + MpC2) whose effects we have already taken into account. Thus we may conclude &Jz(2)
= KI + K2 + Ka + &
+ KS.
(A.29)
All that remains is to substitute this into (13) and carry out the integrations. As a concluding remark we present a formula to be used in differentiating the exponentials of (2O), (21), and (28) :
s 1
6*ex =
0
due (l--v)x6Axe*x*
(A.30)
ACICNOWLEDGMENT It is a pleasureto thank Professor Julian and advice during the course of this study. RECEIVED:
Schwinger
for his continual
May 26, 1958 REFERENCES
1. C. M. SOMMERFIELD, Phys. Rev. 107,328 (1957). d. R. KARPLUS AND N. M. KROLL, Phys. Rev. 77,536 (1950). 3. P. A. M. DIRAC, Proc. Roy. Sot. A117, 610 (1928). 4. J. SCHWINGER, Phys. Rev. 73,416 (1948) and 76,790 (1949). 6. J. SCHWINGER, Proc. Nat. Acad. Sci. 37, 452,455 (1951). 6. J. SCHWINGER, Phys. Rev. 82, 664 (1951). 7. R. P. FEYNMAN, Phys. Rev. 76, 769 (1949). 8. H. F. SANDHAM, J. London Math. Sot. 24,83 (1949). 9. R. G. NEWTON, Phys. Rev. 94, 1773 (1954). IO. R. KARPLUS, A. KLEIN, AND J. SCHWINGER, Phys. Rev. 36, 288 (1952).
encouragement
OF
PHYSICS:
The
6,
26-57
(1958)
Magnetic
Moment
Ck4R~m Department
of Physics,
RI.
of the
Electron*
S~MMERFIELD~.
Universit!J
of California,
Berkeley,
California
The spin magnetic dipole moment of t’he electron is computed to fourth order in perturbation theory. The procedure is to consider a system of one electron moving in a constant external magnetic field. The Green’s function and mass operator for the electron are computed as functions of this field. The expectation value of the mass operator is the self-energy of the electron and the magnetic moment is identified from that part of the self-energy linear in the external field. The techniques used are illustrated in a calculation of the second-order moment. The derivation of the fourth-order moment is sketched. It is found to be ~(4) = -0.328 (LY/?T)? in Bohr magnetons, where OL is the fine-structure constant. This differs from the result of a previous calcu-
lation. 1. IXTRODUCTIOX
In a previous note (I), we reported the results of a new calculation of t’he anomalous magnetic moment of the electron to fourth order in perturbation theory, which result differed from the earlier value obtained by Karplus and Kroll (2). In addition, we discussed the effects of the new moment on various aspects of atomic physics. It is the object of the present paper t,o exhibit the calculation itself, in some detail. The theory of the spin magnetic dipole moment of the electron had it,s bcginnings in 1928, when, on the basis of his equation for a relativistic elect,ron interacting with an external magnetic field H, Dirac (3) found that the energy of the electron contained a term - (efi/2mc)d.H. (The components of d are the three Pauli spin matrices). The identification of p = efi/2mc as the magnet,ic* moment followed immediately. The subsequent quantization of the electromagnetic and electron fields required that this be corrected to take into account the possibility of radint#ion of electromagnetic quanta by the electron and polarization of the electron vacuum by t#he elect’romagnetic field with electron-positron pairs. These so-called radi* This work is based, in part, on a Ph.D. thesis submitted to the Department of Physics, Harvard University (May, 1957) and was supported, in part, by a grant from the National Science Foundation. t National Science Foundation predoctoral fellow at Harvard (1956-57) and postdoctoral fellow at University of California (1957-58). 26
THE
MAGNETIC
MOMENT
OF
THE
ELECTRON
27
ative corrections had to be treated by perturbation theory. The second-order correction of Schwinger (4) and the fourth-order correction of Karplus and Kroll changed Dirac’s moment to ,u = (efi/2mc)[l
+ >h(a/r)
- 2.973((r/7f)‘].
Karplus and Kroll used the techniques of Feynman and Dyson for constructing matrix elements of observable quantities. As pioneers in the hitherto unexplored fourth order, they were interested in seeing if these techniques remained valid and if the renormalization program could be carried through in a consistent way. Their results indicated that everything behaved as expected. The major difficulty was the length of the calculation. To provide an independent procedure and in an attempt to simplify the structure of the calculation, we here use the massoperator formalism of Schwinger (5). The mass operator may be thought of as describing the mass or self-energy of an electron in interaction with the electromagnetic field. If we impose a constant (in space and time) external magnetic field on the system of one electron, the expectation value of the mass operator, computed to the desired order of perturbation theory, will contain a term which is the self-energy of the electron in such a field. The magnetic moment is identified from the coefficient of that term in the energy which is linear in the magnetic field. In Section 2, after deriving the normal Dirac moment, we define the mass and polarization operators in terms of the differential equations for the Green’s functions of the electron and photon fields. The usual procedure is to define these Green’s functions in the limit of a vanishing external electromagnetic field. The presence of this external field is then taken into account in the same way as the radiative corrections due to the virtual photon field. On the other hand, our Green’s functions are defined to describe the propagation of electrons and photons in the full presence of the external field we have imposed. The inclusion of effects due to the field’ is accomplished by replacing the operator (&/i)(a/&r,) for the mechanical momentum of the electron by (fi/i)(a/&rJ - (e/c)&(z) when the electron moves in a field characterized by the potential A,(x). With the definition of a vertex operator the theory is cast in a form amenable to perturbation calculations. There follows a discussion on the extraction of the field-dependent self-energy from the mass operator. The renormalization of the various operators is carried out so that they correspond to physically observable quantities. Finally, by a series of iterations on the unperturbed and second-order operators, an expression is generated for the mass operator in fourth-order perturbation theory. In order to illustrate the method of calculation, we spend some time in Section 1 When magnetic
we say field.
“field”
without
any
qualifying
adjectives,
we mean
the external
electro-
28
SOMMERFIELD
3 deriving the second-order moment. This involves an integration over the possible four-momenta and a sum over the possible polarizations of a virtual intermediate photon. A method for carrying out the integration is described in detail since it is also used in the fourth-order calculation. If we ignore terms which are of higher degree than first in the external field, the integral turns out to be only slightly more complicated than it would have been in the absence of t,his field. Section 4 is devoted to a description of the second-order Green’s functions and vertex operator and their contributions to the fourth-order mass operator. The resultant effects on the self-energy are invariably in the form of complicated integrals over auxiliary variables which had been introduced in order to simplify the momentum-space integrals. Changes of these variables are specified which serve to bring the integrals into manageable forms. The tot)al fourth-order moment is presented in closed form. In an Appendix we present another method of deriving the moment-which has the advantage that the divergences are eliminated at’ the beginning, but which seems to involve more work. 2. THE
MASS-OPERATOR
The Dirac equation for an electron electromagnetic potential A,(z) is MP, The notation
- 4l(x)l
FORMALISM
moving
in the presence
+ mM.-i(x)
of an external (2.1)
= 0.
is as follows: n=c=1; 2, = (x, ix,) = (x, it), (p = 1, 2, 3, 4); a&,, = a.b - a& p, = -iid/dx,
=
ab;
;
Iys, ~“1 = yII-yy + yvyr = $(x1
=
--CL,
Y,,+ =
-Ye,
~4
=
ir0
=
28;
#+(4P.
The mass and charge of the electron are represented by m and e, respectively. We shall demonstrate that the magnetic moment may be thought of as contributing to a field-dependent mass or self-energy of the electron. For simplicity we impose a constant (in space and time) external field, FPy , given by
In particular,
F,u = dA,/dxp - dA,/dx,. if we take A, = - j@,,~, , Eq. (1) reads (rp + %rFx
+ m)tia(x:)
= 0.
THE
MAGNETIC
MOMENT
OF
THE
29
ELECTRON
Interpreting $$eyFx as an operator which represents an addition to the mass of the electron due to the action of F, we may evaluate the effect to first order in F by taking its expectation value in a state 1$0) of the free electron whose wave function, (Z 1&) = &(z), satisfies (rp + m)#&r) = 0. That is, (AF~)‘~’
= ($0 I MerFx
I $0)
=
- (e/4m)@O I (7~) (rF4
+
WW(rp)
I $0)
= - (e/4m)(& 1 - iy~;y~F&~ - 26pvFp~~~p, I$o) =
-
(e/24
($0
I ( fhmF,w)
+
MLvFd
I #oh
where Lpy = x,,p, - x,p, and u,, = 35 i [ys, ~~1.This is just the change in mass to be expected of a particle with spin $4 and relativistic orbital angular momentum L,, , and hence we say that the spin magnetic moment of the electron is 1 Bohr magneton. We want to find the additional field-dependent change in the self-energy of the electron, in interaction not only with a constant external field, but also with the virtual fluctuations in the electromagnetic field. Schwinger (5) has shown that the Dirac equation may then be modified to read
dp - eoA(x)l#h) + 1 (dx’)Wx, x’Mh’>
= 0,
(2.2)
where M(x, z’) is called the mass function and where $A(x) is still a numerical wave function. The parameter eorepresents the “bare charge”. The correspond ing causal one-electron Green’s function satisfies x")G(x#, rip - eoA(xE)lG(x,x’) + I (dxN)M(x,
x') = 8(x - x').
(2.3)
In the Lorentz gauge the causal one-photon Green’s function equation is - k3~/a~&.,,(~, 0 + J wwdf,
.f)sd5”,
0 = W(E - 0,
(2.4)
where P,,“(,$ s’) is called the polarization function. It describes the interaction of photons with virtual electron-positron pairs. We shall distinguish between the electron and photon coordinates by using x and .$,respectively; in momentum space, p and 12.It is convenient (6) to think of a function O(z, x’) as the matrix element of an operator 0: (x 10 [ x’). The matrix elements of 0 in momentum space may be obtained from the Fourier transformation O(p, p’) = (27r)4 / (dx>(dx’>e+20(x,
x’)eip’z’,
30
SOMMERFIELD
which
with
is equivalent
to the matrix
transformation
(z / p} = (2?r)-2eip’. Some special matrices
are
(x 1 1 ] x’} = 6(x - x’), (x 1p, 1x’) = -i(a/ax,)s(x (x 1A, 1x’) = A,(x)G(x
If we suppress
- x’), - x’),
(x I Y,(E) lx’>
= YdiX - 0%
(x 1y,(k)
= ypCik”6(x
1x’)
(2.5)
- xl,
- x’).
all indices, Eqs. (3) and (4) assume the abstract [fn
+
M]G
=
[I? + P]$j
forms (2.6)
1,
(2.7j
= I(
in which II
The introduction
of a “vertex”
=
p -
or “interartion”
r,it) = -Wh4d~)lG-1
(2.8)
e&4.
operator
= ~~(0 - W~ed,(~)l~~
(2.(3)
allows one to write M = mo + iez PC&, .t’) = -ieo2 tr
s
s
(ds>(ds’)r,i~>Gr.(~‘)~“~(~‘,
t>,
(dx>(~x')(dx">r,(s>Gix,x’)L(x’,
xN; .$‘)G(x”, xj,
(2.10) (21.1)
where mo is t,he “bare-mass” parameter and where “tr” indicates a sum over diagonal matrix elements. The operator M is a relativistic scalar matrix in the same algebra as the y-matrices and is an operator function of the operator II. ,411suchscalar operators not explicitly dependent on the constant external field F can be expressed in terms of rTI. Scalar matrix operators linear in Fare rFr and ~FII, while F’, yFFH and IIFFn are quadratic in F; and so on. We write Al as follows:” M = m. + Mo[ylI] ’ Boldface
brackets
will
be used to exhibit
+ Mp[ylI,
functional
FjJ
dependence
(2.12) on operators.
THE
MAGNETIC
MOMENT
OF
THE
31
ELECTRON
so that
G = {-ia + mo+ Mo1-A + M,[-,a, 81I-‘. The symbol quire
8 represents
all scalar operators
explicitly
dependent
(2.13) on F. We re-
lj? MPF-yq31 = M&p, 01 = 0. In the absence of an external field, G must possess a pole at ?II = -m, in order that it represent the physical electron of mass m. We expand fM,[yII] in a Taylor series about -yII = -m:
Mo[$I] = M0[-ml + (-ill + m)JZ0+ MhN.
(2.14)
The number a0 is given by
I@0= (a/am>M0trIIl I?II+~=o. In the limit F = 0, ylI --f -m,
(2.15)
we require that
Mo[-m]
= m - m. ,
whence G[F = 0, -/II + -m]
-
{YP + m + (rp +
m>B0}-’
= (1 +
JZo)-‘(rp +
m)-I.
(2.16)
The Green’s function of the physical electron should behave near the pole like pa; ml-‘. A ccordingly, we define the renormalized electron Green’s function, > G’ = (1 + &,)G
= {YU + m + (1 + ~0)-‘WhW
+ Jfd~& 51))-’
(2.17)
= &II + m + M’)-l. We renormalize the photon Green’s function in a manner similar to our treatment of the mass operator and electron Green’s function. The polarization operator must be a function of k and F only:
P = POW+ P&k Fl such that P&k,
(2.18)
F] -+ 0 as F ---f 0. We write P,[k]
= k2P’o + P,[k].
The term P&k2 = 01, which would correspond absent since the principle of gauge invariance
(2.19)
to a physical photon mass, is guarantees that both the bare
32
SOMMERFIELD
and physical photon Green’s function
masses are zero. Thus we define a renormalized
photon
9’ = $21 + PO) = [Ii’ + (1 + P&‘(P,[k]
+ P&,
(2.20)
F]))-’
= (k” + p’)-’ so that 6’ - l/k2 for k2 -+ 0 and F -+ 0. Since $j is a vacuum expectation value quadratic renormalized electromagnetic potential A,’ = (1 + &)1’28, In order to preserve the gauge-invariant we identify the physical charge e as
in A,, , we must define a
.
structure
(2.21 j
of II as it appears in Eq. (17),
e = (1 + PO)-l’neo . If the external field were made vanishingly to yield G-’
=
YP
+
mo
+
where rro is the field-independent
Mo[vI
-
(2.22)
small, Eq. (9) could be integrated
e
s
(2.23)
~W,OW,‘W,
part of rP , possessing the general structure r,o
=
Y,wlP,
74
(in which the ylr may appear imbedded among the other factors). Near the pole yp = -m, and in the limit of interaction with a very low energy (X = 0) photon, we have rpo- -yph[ - m, 0] so that G-l -
(yp + m)(l
In order that gauge invariance preserved, which implies
L@O)
- eyA’h[-m,
be maintained,
h[-m, By defining a renormalized
+
01.
the structure
interaction
for G’-l
operator
corresponding
G ‘+ = rP + m + MJrpl We may now express
erl’ must be
0] = 1 + Sio.
rr’ = r,(i + ATo)-l, we obtain an expression
p -
(2.24)
to (23):
- e / (dS)r,‘(@A,‘([).
the reduced mass operator,
(2.25)
M’, in terms of the renor-
THE
malized quantities
MAGNETIC
MOMENT
OF
THE
33
ELECTRON
e, G’, $j’, and I”:
M’ = ie”( 1 + itTo)-’ 1 (~s)(~~‘)Y~(C;)G’~UI(~‘)S~~(~‘,
r
(2.26)
t).
The subscript T indicates the removal of the first two terms of the expansion in powers of (~II + m) of the F-independent part of the integral. Likewise, the reduced polarization operator, P', is given by Ppyl(,$, 4’) = -ie” (1 + BJ’
tr / (ch)(ch’)(~zn)r&9 +
(2.27)
x G'(x, x')r,'(x', x"; (‘)G’(x”, x), where the subscript r now indicates the removal of the first two terms of the expansion in powers of h?. In momentum space, these formulas are
M' = i(a/45~~)(1 + iI&)-’ 1 (dk)(dk’)r,(
- k)G’r,‘(k’)S,‘(
+
- Ic’, -k)
P' = --i(ar/47r3)(1 + @J-l tr j (dx)(dx’)(dz”)r,(k) 7
(2.28)
(2.29)
x G'(x, x')I'y'(x', xv; -k')G'(xn, x), where ~1 = e2/4r is the fine-structure constant. We consider the quantities we have defined to various orders in e. To lowest order : M r(O) = 0 ,
P I(O)=o, (2.30)
G‘(O) = (TV + m)-’ = Go,
rp,(O)(k)= y,(k) = yrewiks, gw‘(“)(k, k’) = (6,,/k2)6(k- k’). To second order:
M ‘(2)
=
i(cy/4?r3) sr
(dk)(dk’)r,(-Ic)(rTI
+
m>-‘r”(k’>(6,,l~2C2)g(k
-
k’)
(2.31)
= i(a/4?r3) s (dk)(lc2)-'~,M~ - 76)+ 43, r P ‘(2) = -i(a/47r3) tr s (dx)(dx’)~~e-ik”G”o’(x, r
x’)y,eik’2’G’(0)(x’,
x).
(2.32)
34
SOMMERFIELD
In deriving Eq. (31) we have used t,he fact t,hat eiks acts like a displacement operator. That is qiklG,[n]e-i%l = G’[~I - X]. (L7.X~j The fourth-order
mass operator may be written I(?) = MJf’ + M,’ + Afr’ + AI,‘. N
(Lx4‘)
Each term contains one of the quant,ities A??,, G’, l”, and 9’ t,o second ordrl and the others to lowest order. The four second-order ingredients are as follows: [(l + i7fo)-1](2) = -j$()-oi2’, G ‘W = (~II + m + icf’(2))-1 p The derivation have rrc2)(i)
of I”(‘)
r,‘“‘(k)
= -i((r/47r3)~-ik”
,
= -(l/p)p’(“(l/]i~)~
s
(2.:sci;l (2.3i‘i
From Eqs. (9) and (SJ), WC’
[ (dlz’)(k”)-‘y,[6/GeA:(~)lGo[~
= -&/4n3) so that in momentum
(yII + nz) = -Gadf”2’G3
is a bit more complicated.
= -i(cx/47r3)
(2.30 i
(dk’)(k’2)-‘y,G,,[~
- l&p
- k’]y,([)G,,[r~
- lily,,
space 1 (dk’)(k’2)-‘y,G&I
- 1; - I&G&I
- k’]yy.
(2.38)
We have made use of the fact, that [6/6eA,‘(t)]II,
= -6,”
.
(2.39)
To renormalize I’,(k) we must divide by 1 + .!a,, , which, to second order, means subtracting Be’“‘r,(k). In terms of these second-order ingredienk, the four terms of the fourt,h-order mass operator are JIM’ = - *jo(?)L7p’ , 34,’ = i(a,‘47r3) I’ (dlc)(k2)-*~,J’2’[IT r n/l,’ = i(cf/47r3) 1 (dk)(k2)-‘y,[y(lI I MO’ = i(cr/4rr3)
/ (dk)(dk’)y,[y(II 7
(3.40) - k]y, ,
- k) + m]-1eik”r,‘2’(k)
- k) + m]-lyyei~‘t-k”~yp”“(
-k’,
(2.41) + f&f,
(2.42)
-k).
(2.43j
THE
MAGNETIC
MOMENT
OF
THE
ELECTRON
35
To find the magnetic moment to second and fourth orders, we examine the expectation value of MP(2) + MpC4) in a state whose wave function satisfies Eq. (1). 3. SECOND-ORDER
CALCULATION
We introduce some simplifications in notation by writing A, and PPpyy for eA,’ and eF,,v’. From the definition of II in Eq. (2.8) we infer the operator commutation relation
[II,, El = iF,v .
(3.1)
We have taken F,,, to represent a constant field so that L
, F,AI = 0.
(3.21
We define 5 = ?&,wF,w = (i/2)-&7,tv The two relativistically tion relation
invariant
.
(3.3)
scalar matrices, 5 and yII, obey the commuta[.ylI, 51 = 2iyFI-I.
The second-order calculation contains, in short form, practically all of the features of the more lengthy fourth-order calculation. We consider the secondorder mass operator as given in Eq. (2.31). Without the subscript r the integral would diverge for large k. Upon making the subtractions indicated by r, we would find that the integral would now converge for large k, but would diverge in the region of small k. To handle these difficulties in a consistent manner, we introduce two cutoffs: the ultraviolet cutoff Xm2(X -+ a), and the infrared cutoff em2(e + 0). These are incorporated into the photon Green’s function, 6, so that em’ behaves like a photon mass and Xm2/(IC2 + hm2) multiplies g. That is, we use a new 6(X, E) given by (ii2 + em” f P)S(X, 6) = km2/(k2 + Am*),
where lim,+,. x+, g(X, E) = 6. Then (2.31) takes the form
This type of integral will occur throughout the calculation. It is characterized by an integration over momenta k, and a sum over polarizations ~1. The major complication is the presence of the y-matrices and the variable k in the several denominators.
36
SOMMERFIELD
To eliminate nominator: [y(lI
the y-matrices
in favor
of the F-matrix,
we rationalize
the de-
= [m - y(rI - k)] (m’ - [y(rl - k)]“)y.
- k) + ml-’
By virtSurc of the anticommutativity
of the y’s and Eqs. (1) and (3) we hare (3.5)
py(Il - liq2 = 5 - (rl - x.y so that
J!l ‘(2) = $ [ (dk) k2 ; Em2k2 yirn2 rlh - -a - k)l r 1 X n&z+ (n - /02 - 5 -Yp.
(3.6)
A simple method of doing the k-integration involves the introduction of integrals over auxiliary variables, by means of which the various denominat,ors can be combined into a single factor. The causal Green’s functions G(z, 2’) and ~(5, .$‘) are such that their Fourier expansions contain only positive frequencies for r. > CC,,’ and fo > to’; and only negative frequencies for xo < x0’ and 4‘0 < .$O’. These boundary conditions are made explicit, in the momentum representat,ion of the Green’s functions 1/(7II + m) and l/k”, by a.ssigning an infinitesimal negative imaginary part to the denominators, which may be thought of as an addition to the electron massm. It is seenfrom (6) t,hat each of the denominators is thereby given the appropriate imaginary part’. We make use of the identity m dse-isa, l/a = i (3.7) s0 which is valid provided that a possesses a negative imaginary part. We apply (7) 60 the three denominators of (6) and obtain iVf’(2) = (a/4n3)hm2 l
ds l*
X l (dk) exp I-iit(k*
dt lm dq
+ Em”) - iq( k2 + Am”)] r,,[m -
-/(II -
IL)] (3*8)
X exp ( -is[m2 + (II - k)’ - 51]y,. The exponential containing the sum, m” + (n - k)” - 5, may be decomposed into a product of the individual exponentials, since the terms in this sum commute with one another. We are interested in terms linear in F and therefore may disregard any expressions of higher degree. We write, accordingly, exp is5 = 1 + is5
(3.9)
THE
MAGNETIC
MOMENT
OF
THE
37
ELECTRON
to obtain M’@) = (a/4,r3)Xmz
l*
(dk) exp [-
ds l*
dt im dq
im”(s + te + qx)lr,[m + y(JI exp [--is@
k)l(l
+ isF)y,
(3.10)
- k)2 - i(t + g)k2].
The indicated sum over polarizations can now be carried out. The only machinery needed is the anticommutation relation of the r-matrices. It follows that YNYP = -4, (3.11)
The relations (3.12) are consequences of the fact that F,,” is antisymmetric. The sum over polarizations for any expression of first degree in the field may be derived from these formulas. In the present example, we have r,b
- rm
- k)](l
+ iti)rN
= -4m
- 2y(II
- k) + 2 isFy(II
- k).
(3.13)
The k-integration has been reduced to an essentially Gaussian integral. Such an integral is normally computed by completing the square in the exponent with respect to k. Since we are dealing with noncommuting operators (for example and -&I2 &I), it is necessary to investigate the process of integraPI - k112 tion in some detail. It will turn out, that to first degree in the field, only slight modifications must be made in the normal procedure. We consider the integral I = 1 (dk)[l To complete
+ sk,
+ c2k,kx + c3k,k,kpl
the square in the exponent
exp [-is(k2
- 2akII + bU2)].
(3.14)
we write
k2 - 2ukII + bIf = (k - &‘I)’
+ (b - a2)II12.
We note that in any expression in which F,,,, appears or in which there is a commutator [II, , II,], the remaining II’s can be freely commuted with one another
38
SOMMERFIELD
(to first degree in F). Thus we may apply the formula @4+B) =eee A B -l.'Z[A,B] (in which it is assumed that [A, [,4, B]] = [B, [il, B]] = 0), t,o our exponentid and obtain exp( -is(k
- aH)2 + I12(b - a’)) = js(exp [-is(k
- an)‘], exp [-is@
- a”)II’J).
We have used the fact that the exponential of t#hecommutator can be expanded t’o first degree in F. In the same vein we may write exp ( -is[(lz
- an)’ + II’@ - a”)]1
= !<{exp [-isII’(l,
- a2)] exp [-is@,
- alIl)‘] exp [-is(kz
X exp [-2$X,
- dI,j’]
- ul&)‘] exp [is(ko - a&)‘]}+n
.
where the notation ( )+n indicates that an average is to he taken of t.he quantity in braces and the same quantity with t,he order of II-factors reversed. (Any y-matrices that appear are not to be symmetrized in this manner. ) Each of the factors k, , k, , . . is now placed next to the exponential con&~ing the samecomponent of k. The int)egral can then he expressed as a product of four integrals of the form I’ =
mdkl(l + cl’kl + cZ’kl’) + c3’k13)exp [-is(kl s-co
-
dhYJ,
(x15?
each of which involves only a single component of II, which, in turn, has no other operator with which not to commute. We can then apply ordinary methods of integration to 1’. We let k, - an1 = 5, so that, the range of 5, also extends from - m to m. Terms of odd degree in ,& will integrate to zero since they transform to their negatives under the reflection EL+ -& . We are left with I’ =
- dfGJl + C1’UrIl+ c2’(u21112 + f,“) + c3’(u3r113 + 3u&12111)] s-cc
(3.16)
X exp ( --is&t).
Integrating by parts, we obtain
so that I’ = [I + Cl’UrIl +
c2’(u2rL2
- ibis-‘) + C3’(u3r113 - ~&xIls-‘)]
x 1: d&iexp(-id;).
(3.17)
THE
Still to be determined
MAGNETIC
MOMENT
is the product
OF
THE
39
ELECTRON
of the four Gaussian
integrals
The first three are all the same while the fourth is their complex conjugate. the value of their product is (~/is)“2(?r/is)1’2(?r/is)1’2[(?r/iS)1’2]* When we put together I = -i(7r2/s2)~~{[[I
= (a/is)(?r/s)
the four integrals
= -i(?f2/s2).
of type I’, we obtain
+ clan, + c*(u2111,11~- ~~is-‘s,,)
+ c3(a3mLJ-&3 - ~~~s-1a[n,6,~
(3.18)
+ II,&4 + q?S,,l)]
X exp [-GI12(b We have shown the k-integrations, except that the which is of first The square is
Thus
- u”)])+n
that in the presence of the noncommuting to first degree in F, are the same as for result must be symmetrized in the manner degree in F before integration need not be completed in the exponent of (10) by writing
.
operators lI,, , II, , ordinary numbers, indicated. A term symmetrized at all.
s(n - w2 + (t + qP = (s + t + q)P - sW(s The translation yield
i = k -
M ‘(2) = -&/4?r)X?rL2
[sn/(s
+
t
sm ds’ I’d&
+ t + q>J” + n24t + q)/(s
i- ~$1, and subsequent
+ t + 4).
integration
over E
- U) l1 dw
r0
X exp { -is’m”[u x K-4m
+ (1 - u)we + (1 - u)(l
- 2yII + 2is’u( 1 - u)FyII)
exp [ -Ls’I12u(l
+ +52~{2-yII, exp [-is’II”u(l where the new variables
- w)X]]
(3.19)
- u)] - u)])]
s’, U, and w are given by s’=s+t+q, u = s/(s + t + a>,
(3.20)
w = t/(t + 9). Since we intend to take matrix elements of M’(2) in a state for which +ylI has a definite value, we use (5) to replace II2 by (m)” in the exponentials of (19).
40
SOMMERFIELD
Also, to first degree in F we have exp [-is’u(1 - IL)S] = 1 - is’u(l that m .1 1 ‘(21 dw esp { -im’s Af = -i(cY/47r)Xm2 ds &L(l - 2L) I0 s0 Ts O x X [-4m
- u)we + (1 - u)(l - w)Xl}
[u+(l
- 2(1 - u)yII
-
+ 4isu(l
-
(3.21)
+ iszc(1 - ~)(2 - u)(yII,
u)mS
X exp [isu(l
a)5 so
5)]
- u>(yII)“].
We have used the fact that [5, (yII)*] = 0 and have dropped the prime from s’, Equation (7) together with its generalization m dssne-ias = 7I! a-‘“+” 2.(n+l) (3.22) s0 are used to obtain M ‘(2)
1 =
4:
’ -
Am2
+s 0
1
du(f
- u)
dw
.I*0
4m + 2(1 - u)yII mz[u + (1 - u>we + (1 - u)(l - w)h] 4mu(l - u)5 + ~(1 (m2[u + (1 - u)w~ + (1 - u)(l [4m + 2(1 - u)ylI]
- u){5,yII} - w)X] - ~(1 - u)(~II)~}~, i
24)(2
log m2[u
- ~(1 - u)(m)”
+
(1 - u)hm2 (1 - U)E] - ~(1 - u)(~II)~
_ 4mu(l - u>F + ~(1 - u)(2 - u)jF, yIIj m2[u + (1 - u)~] - ~(1 - u)(~II)~ The second-order magnetic dependent part of (23): Jf,‘2’
= - -El 4rr o ’ du(1 s
moment,
- u)u
= &)sl 4nwl 0
+ 0(1/x).
pC2), is to he extracted
from
4m5 + (2 - u)lS, -III) mz[u + (1 - u)a] - U(l - U)(YrV
duu(1 - U)
(3.23)
2u 22 + (1 - u>e
t,he field-
(3.24)
(3.25)
= --cr(s)/4?mL + O(&) where (5) = ($A 15 ) +A). Thus the second-order moment is cr@)= a/2*.
(3%)
THE MAGNETIC
MOMENT
OF THE ELECTRON
41
We list the steps we have taken with the mass operator in obtaining this result, for they are to be repeated in the fourth-order calculation. 1. Rationalization of the denominators. 2. Exponentiation of the denominators. 3. Summation over polarizations P. 4. Integration over momenta Tcby completing the square in a Gaussian integral and symmetrizing the result appropriately. 5. Introduction of new variables of integration such that only one integral extends from 0 to 00, while the others run from 0 to 1. 6. Integration over the variable running from 0 to 00, to convert the exponent back into a denominator. 7. Evaluation of the expectation value for a state with -rlI = --m. 8. Integration over the remaining auxiliary variables. 4. FOURTH-ORDER
A. CONTRIBUTION
CALCULATION
FROM THE SECOND-ORDER
ELECTRON
GREEN'S
FUNCTION
To find the second-order electron Green’s function as given in (2.36), we must perform the subtractions indicated by r on the integral in (3.23). Integration by parts yields MO
(2)
=z
(4m+yrI)logx
-4m
+I
i
(4.1)
4mu + yrI(2u - u”) + 61dum2[u + E(l - U)] - U(1 - u)(yrI)2
[m2(1 - E) - (1 - 2U)(yII)2]
1
for the field-independent term. The expansion in powers of (yR + m) is then Mot2’
zz
2
3mlogh-;m+(7rI+m(logh-;) .,g’dU
(2;2;t;;2u
--) ‘) + (~II + m) I’
du (4.2)
X
-4u3 + 2u* - 2~” + e(4u - 2u2 - 8u3 + 5u4) + :(-2u
+ 3u2 - u”)
[u” + ((1 - U)]2
+ Mlc2’. We place E = 0 in those integrals which converge in this limit. The forms taken by the integrals which diverge or are indeterminate as 6 + 0 are found by substituting u = IL’&, and dropping terms of higher order in di. The result is
MlC2’ = MO”’ - Bm - jiiT~‘~‘(yII + m),
(4.3)
42
SOMMERFIELD
where 3 = (3a/47r)(log
x + $6)
(4.4
and @o@) = (a/47r) (log x + 95 + 2 log e). The Green’s function
is then
G ‘C2) = (.fD + m + J/p’
- Bm - (yrI + m)il&(2) 1-l - (-frI + m)-’
1 1 lGoC2) += i ylI + m + M(2) - yrI + m I yn + m 1 + -a + m(l i
1
= G'@)a+ ,y'(%b+
- B) - m+ G’(2)C.
Except for a multiplicative to a fourth-order moment
p b
factor,
G’(*)* is the same as Go and thus gives rise
- (2)
(4)
G
(4.5)
\
(2)
=
MO
P C-l.61
=
(a”/T”)(g
log
x +
3fs
+
$22 log
The first term in G’(‘)’ is the same as Go except that m(1 - B). Equation (3.24) then tells us
‘m2(1
Ej.
has been replaced by
m
4mS(l - B) + (2 - u)(5,yII} - B)2[u + (1 - u)~] - u(l - U)(YII)~
- MpC2) $A)
4 + 4u[u + t(l - u)l cfw9 l duu(1 - U) I u2 + t(l - U) 4rrm 0 [u” + E(1 - u)]” I i = cdl(5)/27rm J4y = (&r”>(-3/4 log x - 36).
zzz-
We rationalize follows : G”2)a[~]
and exponentiate
the denominator
= (m - -yII -I- Mc2’) (m” - (m)” =2
-1 0
+ [yII, M”‘]
(m - -yII)(m” - (7lI>“)-’ m ds( Mc2) - is(m - ylI)(2mM’2’
X exp ( -is[m’
-
(m)“]).
of G”“‘[II]
by writing
(4.7)
it as
+ 2mM’“’ 1-l
+ [yII, M’2’])l
(4.8)
THE
MAGNETIC
We place this expression Mg
MOMENT
THE
43
ELECTRON
into Eq. (2.41), using Eq. (3.19) for &I@), and obtain
= --i((r2/16?r4)X2m4 6’ ds f x \ (dk)O
OF
-
ds’ lm dt lrn dg 6’ du 6’ dw
24) exp {-im2[s
+ s’u + ~‘(1 - u)we
(4.9)
+ s’ (1 - u)( 1 - w>Xl - it(k2 + em”) - i&c2 + Am”) ) X 7pN~-fp exp ( -$II
- Ic)~[s + s’u(1 - u)] ).
The expression NI is a polynomial function of the various operators, matrices, variables and constants, the details of which will not be reproduced here. The basic structure of the integral is contained, not in N, , but in the accompanying exponentials. Performing the indicated sum over polarizations ~.r, we obtain a new polynomial N2 . The exponent whose square is to be completed involves k2[s + s’u(1
- U) + t + fJ] - 2[s + s’u(1
- U)]krI
= [s + s’u(1 - u) + t + q]{k - n[s + s’u(1 - u)]/[s
+ [s + s’u(1 - u)]rI” + s’u(1 - u) + t + Cj])”
+ r12[s+ S’UO - u)l(t + Q)/[S+ The integration
proceeds as indicated
f = k - lI[s + s’u(1
s’u(1 - u) + t + g].
in (3.18). The translation
- u)]/[s
+ s’u(1
is
- U) + t + q].
We obtain MGa = -(ar2/16c)X2m4
1 ds ladi
lmdt
d dp 6’ du 1’ dw(1
x [s + S’UO - 24)+ X exp ( -im2[s
+ SIU + et + e/(1 -
x (N3, exp [-iII”[s
-I- s’u(1 -
where the parentheses indicate tial with terms of Na . We change variables :
u)](t
i- Q)/[s -I- s’u(1 - u) -k
the appropriate
t
t/(t
+ q).
(4.10)
symmetrization
- W)]) -I- aI)>
t
of the exponen-
+ 9,
- u>]/[s + s’u(1 - U) + t + q],
v = S’U(1 - U)/[S + s’u(1 w’ =
+ q1-”
+ hq + XS’(1 - U)(l
U)W
p = s + s’u(1 - U) + 2 = [s + s’u(1
t
- U)
- U)],
(4.11)
44
SOMMERFIELD
In order to express everything exp [-ipz(l
in terms of -/II and 5, \ve use
- 2)1112]= [l - 2$x(1 - 2)5] exp [ipx(l
and unsymmetrise into NP :
the quantity
in parentheses,
- x)(yll)‘]
thereby introducing
extra terms
Maa = -(a2/16?r2)X2m4 bmdp 1’ du 1’ dv 1’ dx l’ dw X d’ dw’pu-‘x(1
- x) exp
+ EXF + xx v(1 ; w)
+ E(1 - r)w’
+ h(1 - x)(1
X We are interested (A~v#~)/
in the field-dependent
II
- w’)
exp [@x(1 - z)($l)“].
N4
part evaluated
for yll
= -m:
= (4~ I MFOa I 1c/za) = -(~2/16?r2)(S)X2m4 s0 1 dw’p2U1x2(1
l-dp
1’ dtL i1 dv 1’ de 1’ dw
- x)
1
x-v+
X exp --ip(l
- z>m2[ew’ + X(1 -
X (m[-16i(l
- x) - 4i(l
-
16i(l
- u)(l
f
lSi(l
- v)x(2
- 8(1 - v)(l + 16(1 - v)(l
w’)]
- z&)(1 - s)’ f 32i(l
- v) - 4i(l
- u)(l
- z&)(1 - v)z(l
- 2) + 4iv(2 - u)(l
- x)(1 - u)] -i- m3px(1 - x)[-32(1
- x)(1 - u) + 16(1 - v)(l - x)*(1 - u) - 8(1 + 8v(l
The integrations
(4.13) - .c) - 2)
- v)
- x)
z~)(l - U) + 16(1 - U)
- u)(2 - U) -
16v(l - a)]).
with respect to w, w’, and p yield
(AFm)‘4’$
= (a2/4a2m)(5)(A,,
- -‘lhr - ‘IA + AXA),
(4.14)
THE
x
MAGNETIC
MOMENT
OF
THE
45
ELECTRON
2U2(1 - U)222(1 - X)(1 - V>[UV + x(5 - 3U - 22 + 2UX)] i Ml - u)x2 + u2xv + (1 - 4xdJ + 41 - UN - XM”
(4.15)
+ u(1 - z&)x(1 - 2) X
-10+6~+132-13~~+v(6-55u-122+12z~r 41 - u)x2 + u2xv + (1 - u)xv+ + $1 - U)(l
- z)# ) *
We are interested in the leading terms of this expression in the limits c -+ 0 and X + co. The presence of X in the denominators of A,x , Axe , and AXX would seem to indicate that these terms are of order l/X. But in the region of integration where the coefficient of X is small, the X term in the denominator may no longer predominate. In Ahe , if v - l/X, dv - l/X, then Axe - dv/v - 1 and not -l/X. On the other hand, in A,x if (1 - z) or (1 - u) - l/X, there are sufficient powers of (1 - U) and (1 - z) in the numerator to cause A,x to vanish at least as fast as l/x. The same applies to Axi . Thus we have (AFm)‘4’g
= (a2/4r2m)(5)(Are
v = v’/X in Axe and the dropping
By means of the substitution higher order in l/h, we find
A,, - Axe = 6’ dub’
- Ax,).
(4.16) of terms of
d~~(l - X)
- t&)x” + E(1 - x>( -10 + 6~ + 132 - 13~~) [x” + E(1 - X)1”
x 1%[ux +
Xz(1 - U) + 8(1 - U)][U + x - UX]
2u(l - U)&5 - 3U - 2x + 2?Lr) [x2 + 4 - x>][u + x - ux][ux + e(1 - $1 1
+ ~lduj)+h{ + 0 - 4(l
-
2XZU3(1 - U)” (u + x - ux>qux + ((1 - U)]2
- x)(6 - 5u - 12x + 12~~) x(1 - u> + vu
2 2UV(l - x) + 2UX + 22(1 - X)(5 - 3U - 2x + 2Uz) x(1 + ?m . u> (1 - U) 1
(4.17)
SOMMERFIELD
46
We have set t = 0 in terms which converge in that limit. For those terms which are divergent or indeterminate when C= 0 we use the substitution x = 4; X' and reject terms of higher order in 4. The integrations are straightforward and yield A,, - A4X.= -?i
(4.18)
log x - 1 - f.J log C.
Using Eqs. (16), (6) and (7), we obt*ain (4.19)
IJ(4)LIa = (c+r”)(~‘s log x + $% + 14 log C) and pL(4)a
=
B. CONTRIBUTION
pcc(4)ca
+
$4)
b G +
/14)oc =
(a"/hr')(l+j
FROM THE SECOKD-ORDER
PHOTON
+
!i
GREEN'S
log
c).
(4.20)
FLTNCTIOX
The second-order photon Green’s function is given by Eq. (2.37). The fact that s is vacuum expectation value quadratic in A, implies that $, and hence I” are even functions of the external field. In particular, P’ possesses no term linear in F so that for our purposes it suffices to look at the zero-field part of P’. That is, me use the zero-field electron Green’s function (yp + m)-’ in Eq. (2.32). This may be written as an integral over intermediate electron momenta P ‘py@)(k,k’) = -i(a/47r3) tr / (dp)(dp’)y,(p’ 7
j e--iks I P>
(4.21)
le jklz’ / p’)(yp’ + WI-‘.
x (YP + C%(P Noting that (p’ 1e--ikz1p) = (2#
1 (dx)(dx’)e-ip’Ze-ik26(x - x’)eiPZ = 6(p’ - p + k)
(p I eikPZ1p’) = s(p - p’ - k’) we may write P’p”(2) = --6(k - k’)(ia/47ra) tr [ (dp)r,(rp r
+ m)-‘rJr(p
- Ic) + m]--‘.
(4.22)
Since Ppy’ describes the polarizat,ion current we must require that it satisfy the conservation condition k,P,,’ = 0.
(4.23)
The p-integration in Eq. (22) is similar to the h--integrations that we have already discussed, and in fact simpler. For there are no noncommuting operators. To maintain the validity of (23) a proper cutoff procedure must be used. Two such
THE
procedures
MAGNETIC
are described
MOMENT
by Feynman
P ‘*i2) = 6(/k - Id) (&
OF
THE
2 Icy-;;
6’ du(1 - U”)
X exp [ -ism” The projection
[6,, -
operator
(4). The result is
(7) and Schwinger
- g)
(k,k,/k2)]
47
ELECTRON
- ~krc2(1
guarantees
P,y’. The lower limit SO(+ 0) is the cutoff. Subtracting
(4.24)
- U’>l.
the transverse character of the first two powers of k2
we find
P’$Q
= -(&
- %)6(k
- k’)(k2)y-&-+ (4.25) x I’
g&v(2y-k’,
-k)
= (b
du(3u2 - u”> [l
+ 4$
(1 - u’)]-l
- fg)“(k-k’)g-& (4.26)
4
x I’
d” 1 + (;&);1
- U”) *
The longitudinal, k,ky/k2, part of $y,,(2) does not affect the evaluation of any observable quantities. Thus the part of the fourth-order mass operator contributed to by s’(‘) is
M, = i($/48?r4m2)
1’ du(3u2 - u”) s, (dk)y,[y(II x r,[l
- k) + ml-’ + (k2/4m2)(1
We do not need any cutoffs here because we are interested pendent part of M, for which we obtain 1
Mpg=
-&
J0
&3U2 1 -u*
u4
(4.27) - u”)l-‘. only in the F-de-
-&
f0
1
s0
dx(1 - x)x exp [-im”sz
- 4im2s(l
- x)/(1
X [4mS + (2 - 2) {-ylI, 511 exp [isz(l The field-dependent (A~vz)‘~‘,
self-energy
- u”)] - s)
(4.28)
(~II>“l.
is then
= @A 1 Mm 19-4) a2(5) l = --($-&I dx(1 - z) 4’ du(3u2 - U”)
[-+J
- aI-’
(4.29)
48
SOMMERFIELD
so that (4) CL 0 = (a2/a2)(l’~&
- 7773).
(4.30)
C. CONTRIBUTIONFROMTHESECOND-ORDEKVERTEXOPERATOR The vertex part of the mass operator contains two terms, as given in Eq. (2.42). The contribution of M,’ to the magnetic moment is just /pM = -&(2y (t.31 j Using Eq. (4), we have (4X)
$*)M = (cz2/7r2)[-;~~ log x - yis - ,r,i log Cl.
The unrenormalized second-order vertex operator is given by Eq. (2.38) t.ogether with the cutoff modifications :
The details of the integration are quite similar to those we have already discussed. We shall content ourselves here with describing the choice of a set of variables which will make the final parameter integrations as simple as possible. To this end we must examine the development of the exponentials in which we complete squares. Upon introducing an auxiliary variable for each of the denominators in (xs), we obtain an exponent (apart from a factor -i): m”(r + s + tr + qx) + r(n - 6 - ky
+ s(n - k’y + (t + q>Y2
= m”(r + s + It + qX)
+(r+s+t+q)
[
(T + s)rI
k’-
+
r+s+t+q
7-k r+s+t+q
1 2
(4.34)
With the introduction of the new variables p
=T+s+t+q,
‘U
=
(T
V
=
T/(T
+
s),
w
=
t/tt
+
Y),
+
S>/(T
+
9 +
1 +
q),
(4.33
THE
MAGNETIC
MOMENT
OF
THE
49
ELECTRON
the exponent may be written m2p(u + (1 -
U)[CW + X(1 - w)])
+ p(k’
+ pu(1 - U)l12 - 2pu(l
- Url + UUk)2 - u)?Jkll + puv(1 - UV)k2.
(4.36)
The translation k’ = L’ + uII - kuv and subsequent integration over 5’ leaves the exponent without its second term. The expression for eikzI’P(2)(Ic) thus obtained is inserted into Eq. (2.42). Three additional parameters, s’, t’, and Q’, are needed for the new denominators. The additional contribution to the exponent of (36) will be m2[s’ + t’E + q’h] + s’( IIAfter we complete into the form
the square with
respect
k)2 + (t’ + q’)k2.
(4.37)
to Ic, the entire exponent
is brought
m”(pu + s’ + E[P(l - u>w + t’l + Qdl - do - w> + q’l] [s’ + pU(1 - U)V]lI 2 + [puv(l - uv) + s’ + tl + q’] k puv(1 - UV) + s’ + t’ + q’ 1 i
(4.38)
[s’ + pu(1 - U)V]” + l-I2 s’ + pu(1 - U) PUV(l - uv) + s’ + t’ + q’ 1 * i If we introduce
the variables, r’
= puv(1 - UV) + s’ + t’ + q’, II: = bUV(l
- UV) + s’]/buv(l
y = puv(1 - uv)/[puv(l w’ = t’/(t’
over k expression
m2r’
-
v(l
“-“.,)
+ dl + x
Y> + E [$(l-~u~
Y(1 1
(4.39)
- UV) + a’],
+ q’),
then after integration
1
- uv) + s’ + t’ + q’],
ddl
- w>
UV(1 - UV)
(38) becomes + W’O - 41
(4.40)
+ (1 - w’)(l - X)]}
II
Y(l - 4 Y(l - 4 2 + r12r’x (1 - Y) + v(l _ uv) - x (1 - Y> + l-WV * [ i
The next steps are to set II2 = 5 - (yIQ2, take the expectation value in a state for which yll = -m and integrate over r’. If we confine ourselves to the F-dependent part of MY(~), the integrations over w and w’ can be performed and we obtain four terms, which arise from w and w’ = 0 or 1, just as in Eq. (14). Methods
50
SOMMERFIELD
similar to these we used before must be called upon t’o extract the leading terms in Eand X. We are then left with the following structure, now as a denominator: Y [ V(1 - UV) + l - y
1i -
Y(l
-d
l - y + v(l - *cy)
Ii =v(lyuv)+z [1 - 41z
-x
[
1-y+
y(1 - u) 2 1 - uv
(4.41)
YO
-y+
l-uv
.
At this point our integral is a sum of terms of rational form. The numerators are polynomials in u, v, x, y and the denominator is the one we have exhibited either as it stands, or squared, together with various factors like (1 - uv) or 21. The whole works, however, is explicitly finite. All integrations go from 0 to 1. The best procedure is to make additional changes of variables, such t,hat expression (41) becomes factorable. Consider I(%4
(4.42)
= ~l~Y~lduf(u,Y,v,z).
We make the substitution y(l - v)u = (1 - y’)(l
- MU)
[4.-r.?)
so that
I(v, 2) = 6’ du J1:.,,,,,,, s0
(4.44)
1
1
=
dy’ ,L’,l--u;j f [u, Y(Y’, u, v), v,xl
dd
du u;l-mu;l fh,
sa-u')l(l-uu')
Y(Y’,
u, v>,v, 4.
Our denominator is then
v;l--y;) + qi2 = v(l
y v) [XY’“V(l - v>-
Y’
+ 11
(4.45)
and the u-integration can be done forthwith. Finally, the substitutions [ = 1 - vxy’; Tl = (1 - vy’>/(l
r =0 -
Y’Ml
- vxy’); -
2
NY’, 2, v> =
act, 71,!?I
(1 - $;1
- &J{)
(4.46)
VY’L
bring the denominator into the form (1 - W2(1
- t + G-)(1 - i-)-Y1 - w
(4.47)
THE
MAGNETIC
MOMENT
OF
THE
51
ELECTRON
a product of factors each of which is linear in f, I], and [. The q integral is evaluated first and then, by means of several integrations by parts, the !: and 4 integrations are performed. An advantage of this method is that each term is finite at every step. Some of the E integrals are Spence functions which are evaluated using the results of Sandham (8). For example, we have 1
-2 s0
NC2 - a-’ log E + t-’ log (2 - [II log (1 - E) = - 2 1’ d&$-l log ] 1 - 4 1 log (2 - 0 = -2$Q(3)
+ 7r2log 2,
0
where l(3) is the Riemann-zeta The result is (4) l-w
= (a2/7r2)[~
function
log x - 38 + %2r2
of 3. + 9i5-(3)
= (cd2/7r2)[% log x - s3.ie + x27r2 + 9&(3) The total fourth-order moment (20), (30), (32), and (48): (4) IJ
_
a2
-
g&r’ log 21 + /kL4)
is found by adding
the contributions
197
R2 144 -[ + g + ; r(3) = - 0.328 (a”/~“>. APPENDIX-ANOTHER
(4.48)
- >$r” log 2 - x log E].
- ; 7r2log 2
1
from
(4.49) (4.50)
METHOD
The reader, after observing how the infrared and ultraviolet divergences have cancelled, may wonder if it is not possible to compute Mp + Ma directly. Indeed, it is possible and furthermore one can do the entire calculation without introducing infrared or ultraviolet cutoffs. Unfortunately, there apparently is more labor involved in this procedure than in extracting the effects of the cutoffs. We have M. + Mr = i(a/47r3) $ (dk)(k2)-‘r,e”“[Gor,“2’(Ic)
+ G’(2)ype-isz]. (A.l)
Considering
r,‘(k) = /- (ds) e-i”=I$(d = = -
s s
= - /- (d~)e-““[s/sA,(~)]G’-’
(dx) (dk’)eik”[6A,(k’)/6A,(~)1
([6/6Av(k’)lG’-‘1 (A.3
(dx) (dk’)e-“““e-ik’“( [6/6A,(k’)]G’-‘1
= - (Z?r)“[s/SA,(-k)lG’-’
52
SOMMERFIELD
we have, in terms of the notation (2a)4[S/SA,(-K>]
(xl‘,
= 8A )
rfp@)(x;) zz -&@-1)(2) = L[Go-‘G’@‘Go-‘1 = (L[Go-‘G”“]
( A.-ii
1Go-l + G,‘G”“SAG;-I
= - ( &[jjfl@)Go] } Go-’ - (&-1(=‘(2)yre--ikB, When this expression leaving
is placed in (Al)
the second term will cancel with
(A.3)
Ma + Mr = - i((u/4n3) 1 (dk) (k2)-1yreikZGo (6A [M’(*)G,,] } Go-‘. We have already computed an expression more. For consider the derivative
A/, ,
for &I’(“’ in Eq. (3.23). But n-c need
The effect is one of reducing the degree of F. Hence we would have to compnt,c M’@’ to second degree in F to obtain a derivative correct to first degree. Furthermore, in such a calculation we could no longer make use of the fact that F is :k constant field. For we observe that even though [II,, Fir-] = 0, the same is not, true of the derivative: GaPI, , FL,] = [III, , 6,FxJ
= -ii~-~~‘k,[G,Ji,
- 6,,kx] f 0.
We note that as long as BA does not act directly on the field F, we may correctly use Eq. (3.23) for M’@). Thus we may write M I@) = jjflW + M F(“1 + j@(2) (A.7:1 where the effect of a@) appears only when 6a acts directly on F. Let us consider the contribution of 41,“’ + MF(” in Eq. (5). We compute MI” by making the subtractions indicated in Eq. (1.3). Letting c = 0, we find
M1@‘Go = Ml’“‘(7II
+ m)-l =-
(A.8)
a 4n
+ 1
; [
+
(yrI)“u
1
+ mgI(2 + u) - 2?nS m” - (1 - u)(-yrp .
THE
MAGNETIC
The first term of the integral so that
MOMENT
OF
is infrared
&JMl@)Go] = ad 2 I’ du
THE
divergent.
(4Vu
53
ELECTRON
But it is independent
+ myII(2 + u) - 2m2 m2 - (1 - u)(~II)~
of II
(A.9)
which is finite. We may take the derivative and insert the resulting expression into (5). When 6a acts on II it replaces it with eCikzy, . The displacement property of this operator implies that each II standing between the two y,,‘s will be shifted to II - Ic. After the k-integration is done we set yB = -m. The term Go’ = ~II + m, on the far right of (5), is not affected by the L-shift and integration so that it will cause the whole expression to vanish unless the integral standing to the left of it has a pole at yII = -m. In the present case the k-integral turns out to be quite finite at -yII = -m so there is no contribution at all to the magnetic moment. We next consider Mpc2) as given by Eq. (3.24): 4m5’ + (2 - U>lS’, 7II) m2 - (1 - u)(rII)2 *
MF
(A.lO)
The prime on 5’ signifies that we are to treat it as being unaffected by the operation 6a . Then MF(‘) commutes with Go and we may write {8A[Mp(2)G,,])Go-1
= (s~[G,,M~(2)])Go-1 = Goy,,e-ikzMp(2) + Go[GAMac2)]Go-‘.
Placing the quantity -&/4?r3)
s
(A.ll)
Ggype-ZksMF(2) in (5) we obtain
(dk) (k2)-1e”ZY,GoGo^lre-ik2MF’2’ = -i(a/47r3)
s
(dl~)((k~)-~y~[y(Il
-
k) + mlw2y,
(A*12)
= [(a/am> Moc2)lMpc2). When we set -/II = -m this will just cancel the contribution to the self-energy from M, as given in (2.40), if we use the definition (2.15) of 20~~). The other term, G&iAMF(2)]Ga-1 will produce no contribution to the moment because of the factor yII + m on the right. In this case, it turns out that the k-integral behaves like log [I - (rJJ)“/m”] as ylI -+ -m, but this is not sufficiently singular to counteract the zero of (YII + m). After all these terms have been disposed of, what remains is simply Ma + Mr + M,
= --i(a/47r3)
/ (dlc)(k2)-‘r,eik”Go[6,~‘2’l.
(A.13)
54
SOMMERFIELD
As we have said, it is extremely difficult, if not impossible to calculate a’” directly. But we can find 8afi@‘. We shall indicate how this may be done. The second-order mass operator is MeI0 = -i(a/4n3)
jrn dss I1 du j” (dl&[m 0
- T(n
0
-
k’)]
(A. 14)
X exp ( -isu[m2 + (II - k’)’ - 51]yy exp (-itk”). We may not use the fact that 5 commutes with IYIuntil after ~3~has been applied. Instead, following Newton (9), we write more generally, correct t.o second degree in F exp ( -isu[(H
- k’)’ - 51) = exp [-is@
- k’)“]
1
+ isu
s
n
dv exp ]-isu(1
- v)(n - k’)‘]~ exp [---isuv(~z _ k’)“] (A.15)
-I- (isu)”
I’
dv exp [-isu(1
- v)(n - k’)‘]-Fl’dw
X exp [--isu(v - zo)(II - k’)“]S exp [-isuw(II Using the fact that aa 6‘4M ~2) =
-i(a/47r3)6
= e-ikzia,hkx,
X rY[m - y(n
we obtain (dk’)
.l-sds[du/
- k’)‘].
exp [ - isum’]
- k’)](l + isuS’)y, exp ] -is[(k’
- un)”
(AJ6)
+ 41 - u)r121) + Kl, where KI = i(a/4*3)e-ik” I= szdsllu
dul’dv
/ (dk’)rJm
- ~(II
-
1~ - k’)]
(A.17)
X [a,&~ + isuva&~5 + isu(1 - v)Fu,Ji+yV X exp [-isu(l
- v)(II - k - Jc’)~]exp [-isuv(n
- k’)‘].
If we can extract the terms 6,[Mo’*’ i- M,‘*‘] explicitly from 6A[M(2’], then we will be left with SA[SP]. TO this end we note that the v-summation in (A-l(j) is the same as that which occurred in (3.13). If we write -4m
- 2y(II =
-4m
-
Ii’) + 2isuS’T(II -
- k’)
2yH(l - u) + 2isu(l - ,td)SSyIT+ 2(1 - iszLs’)y(k’ - un),
THE
and, following exp ( -is[(k’
MAGNETIC
Karplus, - uII)”
Klein,
MOMENT
OF
and Schwinger
+ ~(1 -
- @I)*
from 6,&T’*’ the following
- K1 - K2 - Ka = -i(a/4&A X exp (-isi%‘* + 2isu(l
(IO),
4n”11 + I1 $Y@lay)
X exp (-is[(k’
6,M’*’
55
ELECTRON
+ u(1 - u)II”] 1
= exp { -is[k’*
then we may extract
THE
- u)S’ylI]
- isum*)[-4m - u)II*l
(-isum*>[-4m
+ 2isu(l
+ ~(1 - u)II*]},
part:
jm s ds 1’ du / (dk’) 0 0
exp [isu(l
uexp
(A.18)
- 27TI(l
U)
= -(a/4&i,, - 2-yII(l
- u)S’yII]
-
lrn dss-’
-
(A.19)
u)
exp [--isu(l
- u)II”],
where
Kz =
-~(w’~T~PA
irn
s
ds I,’ du f
(dk’) exp (-;m&)
(A.20)
x 2(1 - isuS’)#
- urI)P(l)
and KS = -i(d4r3)64
ia
x I-4m
s ds 1’ du 1’ dy 1’ dw / (&‘) - 2$I(l
- u) + 2isu(l
exp (-isum’)
- u)s’yrI](--is)
(A.21)
e-i6(l--1D)g(y)[(a/dy)~(Y)]e-i~g(Y). We have used the abbreviation 4(Y) = (k - uyIQ* + u(1 - u)l12
(A.22)
and ‘,he formula (A.23)
56
SOMMERFIELD
We may write exp [-isu(l
- u)II’]
= exp (--ku(l
= exp [isu(l exp [isu(l
- u)[Cf -
- u)(yII)‘]
- u)(l
(-fII)‘J)
-isu(1
- v)(-yII)‘]T
- ,f(> 6’ dv
exp [z’su(l - 2l)v(yII)‘]
(A.24)
11)
+[-isu(l
- u)]”
dv
s0
X exp [is(v - w)u(l in order t’o replace II” by (7II)‘. 5 explicitly, we obtain 6,kf’“’
dw exp [is(l
s0
- u)(yII)‘]
5
+ 2isu(l
-
4m -2yII(
- u)“-JIS’
+ 2isu(l
lrn dss-’
1-
- u)(yII)‘];7
[iswu(l
-
Using this in (19) and
- h-1 - K, - K, - K4 = -(a,i4a)6, exp ( -isum’)[
exp
- u)u(t
u)(yn)']
the derivative
d&g
I’ du
w) + 4isntzc( 1 - u)5’
.?L):T’$T]
exp
of
[i.su(l
-
(A.25)
u)(yn,"]
where
X [-4m
- 2y(II - k)(l - u) + 2isu(l - u)T’y(II X exp (iS(1 -
V)U(l
-
U)[r(II
-
- k)] (A.26)
Ii)]‘]
x [a,J;x - isu(1 - ?L)VU,XbXS’ - isu( 1 - u) (1 - 2’)3’U,XkJ X exp [isuu(l - u>(-fII)‘]. Finally, we write 6,M”’
- K1 - K2 - h-3 -
K4
-
K5
= - ((~/47r)6* lrn dss-’ l1 du exp ( -isum2) X [ - 4m - 2rII(l
(A.271
- u) + 4isu(l - u)5’ + isu(1 - 2~) X (2 - u)(5’, rII)]
exp [isu(l - ,u)(yII)“],
1
duu’(1 - ze)exp (-isum*)[V,
rn]
X exp [isu(l - u)(rII)‘].
(A.%)
THE
MAGNETIC
MOMENT
OF
THE
ELECTRON
57
We recognize that after the subtractions of the first two powers of (7JI + m) from the field-independent part of (27), the quantity remaining to he differentiated is precisely 1MIC2’ + MpC2) whose effects we have already taken into account. Thus we may conclude &Jz(2)
= KI + K2 + Ka + &
+ KS.
(A.29)
All that remains is to substitute this into (13) and carry out the integrations. As a concluding remark we present a formula to be used in differentiating the exponentials of (2O), (21), and (28) :
s 1
6*ex =
0
due (l--v)x6Axe*x*
(A.30)
ACICNOWLEDGMENT It is a pleasureto thank Professor Julian and advice during the course of this study. RECEIVED:
Schwinger
for his continual
May 26, 1958 REFERENCES
1. C. M. SOMMERFIELD, Phys. Rev. 107,328 (1957). d. R. KARPLUS AND N. M. KROLL, Phys. Rev. 77,536 (1950). 3. P. A. M. DIRAC, Proc. Roy. Sot. A117, 610 (1928). 4. J. SCHWINGER, Phys. Rev. 73,416 (1948) and 76,790 (1949). 6. J. SCHWINGER, Proc. Nat. Acad. Sci. 37, 452,455 (1951). 6. J. SCHWINGER, Phys. Rev. 82, 664 (1951). 7. R. P. FEYNMAN, Phys. Rev. 76, 769 (1949). 8. H. F. SANDHAM, J. London Math. Sot. 24,83 (1949). 9. R. G. NEWTON, Phys. Rev. 94, 1773 (1954). IO. R. KARPLUS, A. KLEIN, AND J. SCHWINGER, Phys. Rev. 36, 288 (1952).
encouragement