QIJANTTIN{ ELTCTRODYNAMICS
AUANTUM ELECTRODYNAMICS ?3 486
.
".
r.ilrr*
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C. K. IDDIN...
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QIJANTTIN{ ELTCTRODYNAMICS
AUANTUM ELECTRODYNAMICS ?3 486
.
".
r.ilrr*
Transla.ted trom the German by
C. K. IDDINGS & M. MIZUSHIMA
LIBRARY MONTANA COLLEO€ OI MINERAL SCIENCE ANDTECHruNL(}&V BUTTE
Springer-VerlagNew
E I9rk. 1972
Heidelberg. Berlin
Gunnar I(?illdn Late Protessorin the Uniuersity ol Lund'
Prof. Carl I(. Iddings and Prof. MasatakaMizushima U nitsersitY of C olorado Deparfinent of Physics and,Astrophysics
All rights reserved No part of this book may be translat-edor reproduced in arry fo'tm without written permission from Springer'Verlag' @I972 by Springer-Verlag New York Inc. Library of Cottgre.. Citalog Caid Number:76'172529
Printed in the United Statesof America. . Berlin ISBN 0-387-05574-6springer.verlag New York . Heidelberg ' New York ISBN 3-540-05574'6Springer'Verlag Berlin ' Heidelberg
TRANSLATOR'SPREFACE
Kei115n's Quantenelektrodynamik provides a concise treatment of the subject. Its strong points are the careful attention to explanatory detail, the methodical coverage of alI the major results and the straightforward, lucid style . Certainly it wilt be a valuable reference for one learning the subject or for one who requires the details of the practical results. Of course modern quantum field theory has now grown far beyond its dramatic beginnings in electrodynamics and we have therefore included some references to introduce the reader to the more recent and more specialized literature . We have corrected some minor errors: wewould appreciate it if readers would inform us of any others which they find. We thank Professors paul Urban and C. trztdler for permissron to use the biographical material on Kii115n.We also wish to thank Springer-Verlag for undertaking pubtication of this edition by an unorthodox method, but one which will reduce the cost to the reader. In particular, we are gratefui to Dr. H. Mayer-Kaupp and Mr. Herb Stillman for their kind cooperation. FinaIIy, we t h a n k M r . M i c h a e l r e a g u e f o r r e a d i n g a n d c o m m e n t i n go n t h e f i r s t d o z e n s e c t i o n s a n d w e t h a n k M r s . J o a n n eD o w n s f o r e d i t i n q a n d typing the final manuscript. May i972
C. K. lddings M. Mizushima Department of Physics and Astrophysics University of Colorado Boulder, Colorado 80302
IN MEMoRIAM pRoFESSoR GUNNAR KliLrfN
On October 13, 1968,Professor Gunnar Kiill6n of the University of Lund, Sweden, died in an alrplane accident near Hanover. Undoubtedly Europe lost one of its most prominent theoretical physicists. Let me flrst briefly state the significant dates in his remarkable scientific career: B o r n o n F e b r u a r y 1 3 , 1 9 2 6 a t K r i s t i a n s t a d , K e 1 1 5 ns t u d i e d physics in Sweden and completed his doctorate at the University of Lund in 1950. Starting as an assistant professor at Lund from 1 9 5 0t o i 9 5 2 , h e g o t t o t h e T h e o r e t i c a l S t u d y D i v i s i o n o f C E R Na t C o p e n h a g e n0 9 5 2 - 1 9 5 8 ) , w o r k e d a t N O R D I T A( 1 9 5 7 - I 9 S B u) n t i l h e was offered a professorship for theoretical physics at the University of Lund. Moreover, Kiill6n made several journeys for the purpose of research and took part in numerous scientific conferences in about15countries, includingthe U.S.A. and the U.S.S.R. Let me now mention some details about his scientific work: Initiaily he studied electrical engineering but soon he changed over to physics, especially to the problems of quantum elecrrodynamics; in this field he achieved most important results in the years 1949 to 1955. His principal aim was the treatment of the theory of renormalization using, unlike other authors, the Heisenberg picture instead of the interaction picture and the relations now known as Yang - Feldman equations. Considering spectral representations for two- and three-point functions, he succeeded in separating the renormalization constants of quantum electrodynamics and in expressing them as integrals over certain weight functions; thus he could precisely formulate and try to solve the problem of the value of renormalization constants. Indeed, other authors are in doubt about his famous proof that at least one of the renormalization constants has to be infinite, but so far no definite answer to this question has been found. K?ill6nts authority at that time in the field of quantum electrodynamics is well illustrated by the fact that it was he who was requested to write the article on this topic in the Handbuch der Physik. In connection with his work on quantum electrodynamics, he began to study closely the analyticity properties of three- and four-point functions and obtained a number of important results, partially cooperating with Wightman and Toll .
vllr
We must not forget a treatment of the Lee model, which Kli116n did together with Pauli and where they discovered and discussed the possibility of "ghost-states" . K6116nwas not only interested in the development of the general theory; he also treated many difficult concrete problems, such as in his works on vacuum polarization of higherorder. During the last years KdII6n performed fundamental work in the field of radiative corrections to weak decays. He took up the idea proposed by Berman and Sirlin in 1962, namely to take into account strong interactions by the introduction of suitable form factors. But there is one crucial difference from Berman and Sirlin: K2il15nuses on-mass-shelI form factors; that means quantities which are experimentally measurable in principle and can therefore be used as phenomenological parameters of the theory. By suggesting for the unknown form factor abehaviorsimilar to the usual ones, one obtains higher powers of the photon momentum in the denominator; infinite integrals do not occur any more. There are two important features in thls method: first we get finite radiative corrections (although no exact numerical results can be expected because of the approximative character of the formalism), and secondly, an estimate of the cutoff is possible. This estimate shows that the assumption A**pin the bare particle calculation was a very good approximation. Kiill6nts result -- finite radiative corrections by means of strong interactions --is in striktng disagreement with works of other authors, who included the modern concept of current algebra in the calculation of radiative corrections in 6-decays. Though K61t5n did not solve the problem of the influence of strong interactions on the convergence of radiative corrections in weak decays (by means of his form factor method), it turned out to 'an important controversial question in this way, still lacking be a final satisfactory solution. Kei116nwas one of the first who used reduction formalism, dispersion calculations and spectral representations in alI his works , methods which became standard tools in modern physics. Surely Kdll6nts works have contributed much to the fact that field theory is applied in elementary particle physics more than ever. Furthermore, Kiill5n has earned considerable merit in the field of elementary particte physics as the author of an excellent book in which many problems of strong and weak interactions are treated. Here, just as in his conference lectures, K61I5nproved his outstanding pedagogical talent. In his book he has shown excellently how much about mathematical methods and detailed calculations should be presented, enough to clear upthe connection between theory and experiment, but not so extensively that the presentation could be spoiled. r hnno ihat trl a certain extent I have been successful in doing r
rrvyv
lx
justice to the personality of Gunnar Kiill6n and his position in science. The reader will certainly agree if I emphasize again that his early death undeniably has left a gap among the most out_ standing theoretical physicists of Europe. Paul Urban
These remarks are a condensation of those appearlng 1n ',particle Physics", Acta Physica Austriaca, Supplementum 6, Vienna, New York; Springer-Verlag (1969) .
GUNNAR KALLEN IN MEMORIAM
G u n n a r K i i l l 5 n w a s b o r n F e b r u a r y1 3 , 1 9 2 6 , s o h e w a s o n l y 4 2 years old when he died in the fatal airplane crash on October 13, 1968. In spite of the short span of time in which he was active tn physics he left behind him a large number (about 60) of original papers, conferenoe reports, lecture notes, and monographson many different subjects of modern physics, in particular in the domains of quantum electrodynamics, quantum field theory in qeneral, and elementary particle physics. It wilt not be possible, a n d i n t h i s c i r c l e a l s o n o t n e c e s s a r y ,t o m e n t i o n a l l t h e s e p a p e r s here, but I shall try to give an outline of his main contributions to our science in the different stages of his nineteen years of activity in physics. As so many other physicjsts he started his career as an engi_ neer. Twenty-two years old, he came to Lund to pursue hi.s studies of theoretical physics at the University. With amazing speed he caught up wlth the problems and soon he was working at the front line of our knowledge at that time. The main subject of interesr among the theoretical physiclsts in Lund and elsewhere at that time was the new method in quantum electrodynamics which was initiated by Kramersin 1947,and which seemedto make it possible to evade the disturbing divergence difficulties, inherent in the 'of quantum formalism electrodynamics, by a renormalization procedure. In the following years this program was successfully carried through by Tomonaga, Schwinger and Feynman by makinq use of the so-called interaction picture. r6tt5n was fascinated by this difficult subject and by the challenge it represented. His first p a p e r a p p e a r e di n t h e H e l v . P h y s . A c t a i n 1 9 4 9 . I t c o n t a i n e d a treatment of the higher approximations in the vacuum polarizatron. This problem was suggested to him, during a short visit in Zirich, b y W o l f g a n g P a u l i w h o w a s m u c h i m p r e s s e db y t h e y o u n g s t u d e n t , s quick and independent mind. Kiill6n, on the other side, admired Pauli immensely and took him as a model for his future work. The relations between the older and the young physicist developed into a Iife-long warm friendship, which also led to a fruitful collaboration between them in the later years. A-fterhis return to Lund, Kiill5n set himself the task to carry through the renormalization program without the use of the inreraction representation which he regarded as an unnecessary math_ ematical complication. In a series of papers leading up to his
x1r Inaugural Dissertation in 1950,he was able to show that the ideas of renormalization can easily be formulated in the original Heisenberg picture, and that many of the calculations are simpler and their physical interpretation more transparent in this picture. In these papers the notions of free "in"- and "out"-fields were defined clearly for the first time, and a method was developed which ln the literature often has been called the Yang-Feldman method. The reason for this.is probably that Yang's and Feldman'spaper appeared in the Phys. Rev., while K6iI15n'sfirst paper on the sub* ject was published in fuk. f . Fysik. Since these papers appeared nearly simultaneously and were produced independently, there is no room for any priority claims (and Gunnar would have been the Iast to make such claims), but one thing is certain: K5115nmade much more extensive use of his method for practical calculations, and soon he was also recognized by his colleagues everywhere as a master in his field. His briliiant appearance at international conferences, starting with the Paris Conference in the spring of 1950, contributed much to this. His elegant way of presentinghis points of view and his sharp and witty dialogue in the discussions made him an excellent advocate for his ideas, which evoked the admiration of his older and younger colleagues. One of the latter wasA.S. Wightman who later wrote about the early work of Kdll5n: 'At that time I was trying to puzzle out the qrammar of the language of quantum field theory, and here was Kiill6n already writing poetry i n t h e l a n g u a g e ." Gunnar f5:Il6n's connectlon with CERN dates back to the very first years of this organization. Already in 1952, when the site in Meyrin still consisted of a collection of deep holes in the ground and a few shacks , Kiill6n became a Fellow of CERN's Theoretical Study Group, which at that time was placed at the Niels Bohrlnstitute in Copenhagen. I remember vividly his appearance there, which brought exciting new life to our group. He gave a series of admirable Iectures on quantum electrodynamics, which clearly showed his superior mastery of the field and his exceptional gifts as a lecturer. Simultaneously, he pursued with characteristic energy a plan which he had conceived after the completion of his Doctor'sthesis. The current renormalization theory was based on a series expansion in powers of the fine structure constant and, although each term in this expansion was finite and showed a surprisingly good agreement with the experimental results, the convergence of the series had not been proved. Thus, it was still an open question whether renormalized quantum electrodynamics could be regarded as a consistent physical theory or whether it only represented a handy cookery-book prescription for getting useful results. The answer to this question was of great principle importance, but also so difficult to obtain that it required all the courage and tenacity of a Kdll5n to attack and finally solve the problem.
xiii By means of the formulation of the renormalization theory he had given in his thesis, he was able to define the renormalization constants without making use of perturbation theory. Tn a series of papers in Helv. Phys. Acta and in physica he showed how this can be done and, in a final paper in the proceedings of the Danish Academy (which later was reprinted in special collections both in Japan and U. S, A.) he proved that.at Ieast one of the renormalization constants had to be infinite. Thus, he had come to the conclusion that renormalized quantum electrodynamics could not be regarded as a completely satisfactory physical theory, in spite of the success of the perturbation theory version of the theory in accounting for the experimental results. On the other hand, the latter circumstance gives good reason for believing that thepresent formalism may be regarded as a limiting case of a future more complete theory. Krill6n was a Fellow at CERN's Theoretical Study Group from October I, 1952 to June 15, 1953. During this period, his professional ability and his personality had impressed us so much that we naturally tried to get him on the permanent staff of the Study Group. AJter he had finished a second longer stay in Zi.irich, he joined our staff in October I954, where he remained until CERN's Theoretical Study Group finally moved to Geneva in September 1957. Thereafter, he accepted a chajr as professor at the simultaneously established NORDITA in Copenhagren, where he stayed until a personal professorship was created for him at the University of Lund at the end of 1958. Thus we had the privilege of having Gunnar with us as collaborator in Copenhagen during more than five of his perhaps most productive years. It is impossible in a few words to describe how much we owe him as a constant source of inspiration, as a teacher, and last but not least as an always alert critic. The ruthless honesty and objectivity of his criticism, which soon became legendary, recalled that of the young pauli. It has even been said that Gunnar modelled his style on pauli, but this was only partly true. I rather think that the similarity in their reactions was due to an inherent kinship of these two original personalitres. In Ziirich KtiiISn and Pauli had started a fruitful collaboration which was continued after Gunnarrs arrival in Copenhagen. It resulted in a paper "On the mathematical structure of T. D. Lee's model of a renormalizable field theory', which was published in the Proceedings of the Danish Academy in 1955. AJthough the Lee model is non-relativistic, it is of great interest as an illustration of what might be hidden in the more complicated formalism of quantum electrodynamics. The advantage of the model is that it contains a renormalization of both the coupling constant and the mass, and still is so simple that it allows exact solutions. The main result in the just mentioned paper was the surprisinq discovery that the renormalized Lee model contains an unphysical state--the "ghost" state--which has a negative probability. It is
xrv quite possible that the formalism of renormalized quantum electrodynamics also contains such unphysical states, which further supports the view that quantum electrodynamics, when taken 1iterally, does not represent a consistent physical theory. A fortiori, this holds for the current meson theories which, for obvious reasons , do not lend themselves to a perturbation treatment. Therefore, in the following years and in particular after his retwn to Lund, Kdll6n joined in the trend of research, which has been called the axiomatic way, and which was being pursued at several places in Europe and America. Instead of investigating the properties of a definite formalism, the idea was to see how far one can go by startingl from a few general physically necessary requirements, such as relativistic invariance, causality and positive energty. With his usual energy, terrrts+v
if fhc
rrnii
of
lcnnfh
fhc
ccntimpicr
iq iaken
as
the
basic unit, then the unit oftime is the time required for light to travel a distance of I cm. and the unit of mass is that mass with a Compton wavelength of 2n cm. These units are often referred to in the literature as "natural units". E^r
!,dr,, lrrra
-,n1116pq
1z
a 1l tha
rosrrlts
of
the
are of physical interest must tend to finite limits. pens that a sum of the form
iLhr r ee ovrr v J
w vvh i C h
It often hap-
s: + Ztte)
( 1 .3 )
occurs in a result where either l(d i" independentof V or tends to a f i n i t e l i m i t a s V - > a . W e s e e i m m e d i a t e l y f r o m E q . ( 1. 2 ) t h a t the number of states with spatial vectors p lying between p and p+dp is given by (
"d . ' b' . \ .^2t, 1\ '*ar 'b^"rdj ^hr ,zd b " : l, 2^ nv l.t^ t Thus it follows that as V-->a,
(r.4)
the sum S tends to the limit
t*o;,n I o'ottnl.
0 .s )
T h e s y m b o l 6 t . p i n ( l . 4 ) a n d ( 1. 5 ) m e a n s t h e t h r e e - d i m e n s i o n a l
Different
Sec.2
Pictures in Hilbert
3
Space
Later we shall also use four-dimenvolume element in 1-space. sional volume elements and designate these simply as dP (nq! as d,ap). Different Pictures in Hilbert Space It is well known that in ordinary quantum mechanics the time evolution of the system can be treated in several ways. For ex2.
amnio ertrP.Lv
nno
aan
o v rm r r Pn . Llvn f rz
a
nicfrrro u
s
ytvLsr
uzhoro
tho
n n o r aet nL rv < r vvvr
aro
fjry1g-
independent quantities, and aIl motion of the system is described by means of the state vector. Alternatively, one can regard the state vectors as constants and describe the evolution of the system by a time variation of the operators. In the first case, we have the "Schroedinger picture" and in the second, the "Heisenberg picture'r. These two possibilities are also present in field theory. In a way, they are extreme examples of the more general possibility in which the operators as well as the state vectors are treated as time-dependent quantities, and the precise t'distribution" of the time dependence between the two is a matter of convenience. In the Schroedinger picture, the time variation of the state vector is given by the Schroedinger equation
(2.1)
o!]#):HWU))
Here lg(l)) is the state vector, and the Hamiltonian operator H is a Hermitian operator depending on the dynamical variables. Ina f i e l d t h e o r v . F f d e n e n d s r r n o n f h c f i c l d o n c r a f o r s . T T n d c r. e r t a i n conditions, 17 can be time-dependent, but then the time must occur explicitly and not implicitly in the field operators. In this case, the system is not closed but is interacting with external sources. v y v l s u v r u r
T.afor vze qha lI
freorrcnilrz
cnnsider
su grvrr cl h
q vstcms: J f
ir rnr
C v r rhg a y |n
- T ! , ,
fOf
simplicity, we shall limit ourselves to closed systems, i.e. , to time- independent Hamiltonian operators. of The generalization the results of this chapter to systems which are not closed can usually be made without difficulty. We may therefore assume
!,ls!\")) : 0 .
(2.2)
at
In a field theory, .H usually contains the field operators at all points of three-dimensional space and can be written asan integral of a Hamiltonian density #:
H:ld3x,tr(E"@)).
Q.3)
in this picture, Although the operators are time-independent "time derivative" of an operator F(n) can be defined by
F(u) : i lH, F(n)): i(H F(r) - F(n)u) . This operator is
obviously
not the actual
derivative,
a
Q.4) but does
G . K i i t l 6 n , Q u a n t u mE l e c t r o d y n a m i c s
Sec. 2
have the property that its expectation value is the time derivative of the expectation value of the original operator:
(,pQ)| h @)tv Q)>: * ;#rrlff, ts(r'), v"?)t ,,
"
E.(")],
_
( 3 .l o )
Ar,o_
g4! : - u?".{,r,,1|!34'_, r /s- Jf 'a",,' s",(")1. (3.11) " In,,,', ' +9+ b,'o c.yo o)'o l'"0'-' ,08p0') ll P x!:/o
L
"
7xi
I
a'p:(*\ appears on the right-hand^side, Axn 28"\x) it is to be regarded as the same"function of no(x) ,9o(x)and # as one has in the classical theory. Equation (3.1I) must then determine the corresponding function in the quantized theory. It obviously has the solution that the classical and quantum mechanical functions are equal. That is, (3.11) is equivalent to Eq. (3.3) conIt then follows that (3.8) is the sidered as an operator equation. This concludes the operator equation corresponding to Eq . (3.2). proof. The difficulties which arise from the order of the operators Since it in the Hamiltonian have not been taken into account. will lead to no serious difficutties in the applications, we shaII not pursue this question further. Trlct as in the cla qsica I iheorv - ther:e are also conservation We laws for energy and momentum in the quantized field theory. have already shown that the conservation of energy in quantum In Eqs. (3.6) to (3 .lI) , when
J
qp
L
u
r
Sec. 3 Lagrangian,
Equations of Motion,
Canonical euantization
9
theory is a trivial consequence of the fact that H always commures with itself, and therefore that the time derivative of the total energy must vanish. As in the classical theory, this conservation law can also be proved by direct calculation, starting from the operator equations (3.2) and (3.3). Also as in the classical theory, the three spatial components of the momentum are defined bv
pa:- I d'r4r"@)W
(1,1r\
and, in the usual way, we have
lPa,v"@)): i I d.'x, 6(a _ n,)3!!y1 : ; aq_,(,) oxh ?xn ro:z'o
( 3. 1 3 )
n,(x)l: -,r{,!', n.(*') lPe, #oi 6(t - u,): , 9W, .
(3.14)
From these Iast two equations, it follows that any operator F(r) which is constructed from gn@)and zo(r) satisfies (3.1s) lp".F(x\j:;!r?) 0'n I n t h e H e i s e n b e r g p i c t u r e E q . ( 3 . 1 5 ) c a n b e u n d e r s t o o da s a c o u n t e r p a r t o f E q . ( 2 . 6 ) . I n t r o d u c i n g t h e f o u r t h c o m p o n e n to f t h e m o mentum by the definition
Pt:iPo:iH,
( 3. 1 6 )
we then have
l P , , F ( r \ 1 : ; a ox" !(x)
( 3. 1 7 )
for lt:I.to 4. The Eqs. (3.17) are relativistically covariant , and will be very important in what follows. In Sec. 4 we shall derive them in another way. lfwe insert the energy density ff in (3.15) and use the periodic boundary condition, we obtain
l P oH, '1J d: x ni I a "x - 2 * : 0 .
( 3. i 8 )
As in the classical theory, the components of momentum are conserved quantities under the quantum mechanical motion of the system. In a similar way, any two spatial components of the momentum commute with each other. Thus it follows in qeneral that
lP,,1): o.
( 3. 1 e )
In principle it is therefore possible to choose a representation ln which all four quantities We shall designate { are diagonal.
10
G . K a 1 I 6 n , Q u a n t u mE l e c t r o d y n a m i c s
the state vectors of this representation by the corresponding eigenvalues of \ show that {or an arbitrary F(*) in (3.17),
t
a+n
h(a) rp
.a
_ py)l(alF(x)lb) : ?plf, (ol lp,,F(x)llb> In this representation the fore given by
lt a \ --/
pI
ox"
lt /" r/ \
(an
t
i)na
a e uf .v . , aan vgrr
L
a s .nr rs ' l
rr au:sdvi *l rr jr
( a l F ( x ) I b ) . ( 3 .2 0 )
x - d e p e n d e n c e o f e a c h o p e r a t o r is there-
"' ( a l F ( x ) l b ): ( a P 1 u S, o | @ ' i ' - P ' ?,) where the quantities (alFlb) we shall refer to this remark.
(3. 21)
are independent of r.
In Chap.VII
Transformation Properties of the Theory the original In the canonical formalism developed so far, Certainly the relativistic invariance of the theory has been lost. equations of motion (3.2) and (3.3) have the desired covariant 4.
form
aq
lona
as
fhc
T.aaranaian
iq
an
inrrariant
Hnwerrcr, r r v v ! v
v v r
/
the
canonical commutation relations (3.5), which allow the proper transition to the quantized theory, are not covariant because the two times ,0 and ri are assumed equal. In this section we shall further investigate the properties of the quantized theory under In particular, it will appear that Eqs . Lorentz transformations. ( 3 ^ 1 7 ) n o t o n l v F o r ma c n \ / A r i ^ n t s r z s f c m h r r t a l s n t h a t t h c v h a v e for the transformation properties of a fundamental significance the whole theory. We assume that we have two Lorentz frames, n and r'and that ifa point P has coordinates * in the first system, its coordinates in the other system are given by \w
.
,1,
/
rrvL
vrrrl
av1
rrr
rYrr!
xp:
v
1
f i p | . e , , , , x 1, 6 , , ,
( 4. 1 )
Here , and in the following, we sum over repeated indices . For simplicity we further assume that the quantities F,, and bu are so that their higher powers may be neglected infinitesimal, Since the Lorentz transformations form a group, it is sufficient to study the tansformation properties of the theory under infinitesimal In order that (4.1) describe an actual Lorentz transformations. transformation without stretching the coordinate axes r rp, rflust be chosen antisymmetric. The transformation properties of the classical fields are known and have the form
VL(x'): L sp,S,,,oB Ve@@'))I q"(x (x')).
(4.2)
The left side of this equation gives the field funptions in the new On the coordinate system as functions of the new coordinates. right side are the field functions in the old coordinate system,
Sec. 4
Transformation Properties of the Theory
evaluated at the same point Rrr
cnlrlinc
/1
l)
f^.
P which is present
1l
on the
lAff
cid6
x",
xu:xL-ep,x:-6p,
(a 7,\ \
4
.
v /
one can likewise regard the right side of Eq. (4.2) as a function of the new coordinates. We shall now consider two points p and P' which are chosen so that the point P in the first system has the same coordinates as the point P' in the second system. The field functions g,(x) at the two points therefore differ by amounts dV"@) which are given by
: l, uu,Sr,.oBvB@) 6v,@) * ry: @,- ,L): 1
:
^
7o-(l\
tu,Su,,oBqP@)
( e u , x ,l 6 r ) . )
(4.4)
\I
\
In a similar way we find the change in the quantities a.g
(4.s)
a4"VL 0r, in going from one system to another: c
/ \
I
-
o x t d p \ x ):
;
^
t
l-Tho -^*o.+^ L rl rv
An-,.(x)
E b s t , , p a n B u \ x )-
( e t , x , * 6 ^ )* e * n , , ( x ) .
/ ? 3 ) a r e s p e c i a l c a s e s o f t h e s e z r - . .. l
. . . v . r r v .
v/
vr L
r yvvr
vr
Lr r Er c
J04l
.J
(+.a)
Thtr last Iltg
t e r m i n ( 4 .6 ) a p p e a r s b e c a u s e t h e v e c t o r i n d e x p r i n ( a . 5 ) m u s t a l s o be transformed. In particular, for the canonical momenta,we obtain r-
/--\
o x r d \ x ):
-
I
z
.
E ) . v5 t , , B u n p \ x ) -
)n.(xl
;;
( e ^ ,x , l
6 ^ )- i e n , n o , ( x ) . ( 4 . T )
Now if we first carry out the canonical quantization in the original coordinate system and then subsequently quantize in the new coordinate system, it is necessary that g*(z) and no@) as well as q"(x)*6q"(x) and n"(x)16n"(x) satisfy the commutation relations (3.5). The two procedures of quantizing are only equivalent if there exists a Hermitian matrix 7 for which q"(x) *.68"(x) : n"(x) i
6n"(x) :
eir gn(x) e-dr , e;r vo1a1t-n,
(4 9,\
(4.e)
with the same matrix Z in both Eqs . (4 . 8) and (4 .9) . Since we have assumed that the Lorentz transformation (4.1)is infinitesimal, we can also take T infinitesimal and obtain
6q"(*) : i lT, q"(x)),
( 4. 1 0 )
Sec. 4
G. Kdll5n, Quantum Electrodynamics
(4.11)
6n"(x):ilT,n"(x)l'
-he existence of this matrix can be shown most simply by giving :r explicit expression for it. We wish to prove that Eqs' (4'I0) and (4.11) are satisfied if we choose the following form for T:
r:
gp(x) * x,€,n""(4ry! + I Ia'xlt up5,,,"an"(x)
-1-
dr)(""@) * i (ennxn* W
6en,(x) ry!)
Iv,,t
runs only from I to 3' The summation over the index hin(4.I2) We shall introduce the general convention that Latin characters like ft , I , etc. can take only the values I to 3, while Greekchar acters like pl , 'v , elc. can also take the value 4. The proof f.or(4.I0) and (a.li) is simple in principle, but somewhat laborious . We first give the calculation f.or 9,@). With the help of (3.5), one has
ilT,q"(x)l:
t )x , a , n L t l : * i @ n n x u t 6 n ) x ) ro,su,,,avp@
From (3.3) , we conclude that the last two terms in the sguare b r a c k e t s c a n c e l . T h e o t h e r t e r m s c a n b e g r o u p e dt o g i v e t L t , P q \ x ) J:
1^Ao-(x\"?o-(x\
T
u u u J p , , " p V p \tx )r , a , p
r;,
- o,,-
V.I4)
bli,
T h i s p r o v e s ( 4 . 1 0 ) .T h e c a l c u l a t i o n f o r z o ( r ) p r o c e e d s i n t h e s a m e way: i lT,n*(x)l=|
,u" Su,,fonp@)+ e,enn(x))* i (e+axu{ 6n)x *?, ^ ova@) , app@)\ - n B \x) 0 x ^ + A E -+;WA- r9*T " A x o l _
a s" G l
a q ,1 ,1
)-
- Ju,o1'n' xt lo r 6t)#;1
uu-t
- au4!
:
(4.15)
: - + ep,sp,,BonB(x) * x,quW - 6 rw 'i : ,n ,n n )(x)
* i bnnxe* 6a) ry!
: -
- orff
t *, r,u#? + ep,Sp,,pone@)
-
- i en,n,,(x) .
Sec. 4
Transformation
Properties of the Theory
The last form for the right side of (4.15) is identicat side of (4.7) , and this proves (4. ll) . The matrjx Z can be written in the following form:
r : fasx { 1e,,, x,. | 6,,\ P' | J l'"" I
",
I3 fn
fho
rinh+
(x1a!,"(*)- t' -g a^. - " * J| + Axu * * r , S r , , o p n n ( xv)' t 4 j . ]""'
Because of the antisymmetry of -
V
ep, (5,a, op n"r(x) np(x) |
is equal to zero. fni
l nrarina
-
e,n , the expressionl Spr,,Bn",(x) gp@))
(4 .r7)
We can rewrj.te the last term in (4.16) in the
ura rr.
; n"n@)9B@)* S,+,"p nou@)qe@)+ Z to, (Sp,,op pu@))= - i er,1r,: * Spa,oBn',(x) o , . , : t Z q L f o t , @ , n x o* 6 ) l - i ( e , nx n * 6 , ) a ftln^,: , A ., :i;;llnn,@,nxnt 6,)f- i (e"o xn| 6)
n
I I I \/ 4 a "t"ni l [
I
fttr^,.J
Ohesymbol fr", is antisymmetricin the first two indices.) From ( 4 . 1 6 )a n d ( 4 . 1 8 )w e o b t a i n T - - i I d . 3 x T n , 6,x ,
(4.19)
9lra? + 6t,gTr": - nnr@) f, l^u,,
(4.20)
6xu: errxr* Otr,
@.ZI)
with and
Fromthe equations ofmotion (3.2) it follows that the tensor {,, onrraf i nn satisfies
the continuitrr
!*,':-#qhp,:o
(4.22)
The three -dimens iona i inte gra ls
(4.23)
Pr:-ifdsxTn,
]-
F. l. Belinfante, Physica, Haag 6,887
( 1 9 3 9 .)
14
C. fdtt6n,
Quantum Electrodynamics
Sec' 4
(4.20) are therefore constants of the motion. since the last term in d i v e r g e n c e ' the a t h r e e d i m e n s i o n a l a s b e r e g a r d e d p : 4 c a n for with the momenta defined in (3 '12) and ( 4 . 2 3 ) i d e n t i c a l a r e o f 1, (3 . 16) . If we now put tr,: 0 in (4 .2I), then with the help of (a ' a) a n d ( 4 . 7 ) w e c a n s i m p l i f v E q s . ( 4 . I 0 ) a n d ( 4 ' 1 1 )t o
o!';,? , lP,,E*@)7: -- 'i,444 . lP, n,(x)l
(A 'A\
lA
1\\
them as Here we have recovered Eqs. (3.i7) and can now regard transi n f i n i t e s i m a l u n d e r t h e o r y o f t h e i n v a r i a n c e t h e expressing For this reason we shali also refer to the operators Po lations. as displacement operators . The conservation laws (4.22) and (4.23) can also be deduced under in other ways. since the operator 11 is obviously invariant ( 4 ' 1 1 )t h a t every translation , it follows from (4 .10) and
: 0. 6H : i V, Hl : i lPp6r,,l7l
(4.26)
from (4'26) The quantities D, are arbitrary, and we can conclude derivatives time vanishing have so that all P, commute with I/ and differenp o s s i b l e t h e i f i s o n l y T h i s s y s t e m . in every Joordinate This consideration is partictial conservation law (4.22) holds. but ularly important because it is useful not only for translations i n f o u r d i m e n s i o n al f o r " r o t a t i o n s " f o r m , also, in slightty modified h a v e w e t r a n s f o r m a t i o n s s u c h space. For
aH : +
le, 1,,,H) : - i e4,P,,
(4.27)
where
Ir,:'i
f d ' ' x( T n rx , T n ,x , ) ,
(4.28)
or
lH,I,,l-
-
6arP,* 6a,Pr.
(4.2e)
operators Since the operators !u, are formed not only from the we have gd and zo but also c'Jntain the coordinates r explicitly'
- Ta,6r) : o' 0", (Tnu6,n lld : ilH, I,f I ff
(4.30)
must If Eq. (4.30) is to hold in every coordinate system, there also be a differential conservation law here:
-f, {r^, xo- T^,xr): o.
(4.31)
Sec. 4
Transformation Properties of the Theory
15
T h r o u g hc o m p a r i s o n o t ( 4 . 2 2 ) a n d ( 4 . 3 1 ) w e o b t a i n Tur-T,u:0, i
a
+ha +6hc^r
T r ur
ic
crrmmafrin
symmetric energlF-momentum density the tensor
r+ ig
(4.32) Often
feferred
tO aS
in order to distinguish
@,,:- nn,@) s, ry! + ap,
the
itfrom
(4.33)
w h i c h i s k n o w n a s t h e " c a n o n i c a l " e n e r g y - m o m e n t u mt e n s o r . T h i s Iatter quantity usually gives the same dlsplacement operators as T, and therefore can also be interpreted as an energy-momentum density. For the construction of the angular momentum fu, ,however, it is essential to employ the symmetricaltensor Tun.
CHAPTER THE
FREE
ELECTROMAGNETIC
Iaqranqe Function and Canonical Formalism The Lagrange function of classical electrodynamics from
FlELD
5.
is obtained
g--lFpF,,.
( s. 1 )
Here .{,, is the electromagnetic field tensor: Fta: - 44: 4.2: - 4r:
i E h'
H a , a n d c y c l i c p e r m u t a t i o n s'
( s. 2 a ) ( s. 2 b )
In this Lagrangian we introduce the potentials Ar(x) as the dynamical variables, rather than the field strengths. The field strengths are glven bY -pt ' r -_ a A , Q \ 0*u
_AAu@) 0n,
tc.J,
Regarding the potentials as the variables in the lagrange function 7 we obtain the equations of motion according to (3 .2):
.
- t"!,1"!,\ :0. - !A-,(,\) : I- e,, -a ( aA-,(t) P.ul axpottt 0x, I Axu
7xu \
(s.4)
This formulation of the classical theory is obviously Lorentz inThat is, it is invariant under varlant as well as gauge invariant. the transformations
Au@)-->A,(4 + 4+ Neither the field strengths nor the equations of motion are changed In the classical theory this invariance by these transformations. under general gauge transformations is often reduced by requiring that the potentials satisfY
a 4 : @:)o . ofrp
With an appropriate choice of the gauge function A(x) in is always possible to achieve this; however, the gauge
( s. 6 ) ( s. 5 ) , i t function
Sec. 5
lagrange
Function and Canonical
Formalism
is still not uniquely determined by this condition. possible to carry out further gauge transformations, functions which satisfy the wave equation
Iz
It is always but only with
Z A(x): o.
/c
7l
In this gauge, the equations of motion (5 .4) simplify to
Z A , ( x ): s . If we attempt to go to the Hamiltonian form of the theory, starting from this Lagrangian, we find that the canonical momenta (3.3) are Tvok):iFnu@). The momentum conjugate to Aa(x) vanishes /\ v( . J9/ ) n ea s rnrnr rnvi, h - ,a
identically
and Eqs.
oA"(zl
-c-n. vr e d
for
all
Ar"
d e v e l o p e di n C h a p . I i s u s e l e s s . with the following Lagrangian: g:-LP 4 The equations momenta are
(5.9)
of motion
.
Thus the
whole
method
Instead of (5.I) we can start
a A ' ( t t ')-3- 4 A rar@ - ) 'ttt P ^ttt - r Z A_, (5.8) are obtained,
(5 .10)
and the canonical
n e @ ):
i Fno@),
( S. I l )
na\x) :,
. --;; 0A..(x\ .
(5 .12)
In the Hamiltonian formulation of this theory, we do not obtain Eq. (5,6) as an equation of motion, but only the weaker condltion n
04,(x) ofro
\c ..r\t.l
If we do not consider the most general solution of these equatrons but only those which satisfy the initial conditions
- 0a4; ,:( x )
02A,(x)
Z;a-"
:0'
for all c'
( s. 1 4 )
and for some fixed time (e.g., xo:O), then from (5.13) it follows 0- 4 " ( x \ that the quantity a';,, must vanish for all times . As desired , we then recover the usual theory of electromagnetism. In the quantization of the electromagnetic field, we shall first ignore completelythe inltiai conditions (5.14) and use only the Iagrangian (5.10). For 5o: x'o the canonical commutation relations become
18
G. Kli116n,Quantum Electrodynamics
Sec.5
A,(x')l: g, 14,,(x),
(s.is)
I 93# - t aL;:?): .l, lA,(*),', 4(x')l: lo,t,l, [o,f 2P] : I ( s . 1 6 ) r =t6pn6(u-n') )
4#] :fo utu), :lo u@),, r#l no(x1l u@), LA :fon(i, r#l na@,)f lAn@),
:l#, n1(x')l lno@), :[#!, nn(x,)f lnu@),
:,
(s. r7)
: i 6(a_ &,),
( s. 1 8 )
o, \fl: aff]: o.
(s. Ie) (s.20)
T h e E q s . ( 5 . 1 5 )t h r o u g h ( 5 . 2 0 ) c a n b e s u m m a r i z e d :
yAr(x), A,(x')f: g, | 0Au@)
ltif
(c . z.r./
, -.,,1 - i 6p,6(n- n'), , d,(''))"
| 0Au@)
1A,(x')l
(5.23)
a t o1 : " '
t--a;'
(5.22)
It must be emphasized that the commutation relations (5.15) to (5.23) are valid ina Heisenberg picture only for equal times ro and ri. For what follows, it will be of interest to carry out the quantlzation in momentum space. We therefore introduce the expansion ( t . I ) f o r A u @ )z
AP, (.x ) : + t yv
+
eih"Ar(k).
$.24)
From (5 .8) it follows that in (5 . 24) only those ft can be present for which
hz:kz_ kE:0.
tc. rc,
Equation(5.25) has two solutions: ho: la
,:
,
*lF
i
/tr
, A\
and we can write (5.24)as
Au@):*
yrT
> \eh"tr(rt)| e-ih'Al (Ql,
(s.27)
where ho:a
(s.27a)
Sec. 5
Lagrange Function and Canonical Formalism
lg
From the requirement that ,4u(r) is a Hermitian operator and z{^(,') is an anti-Hermitian operator, it follows that Ao(k) , iAn(k) and Ai(k) , iAt(It), respectively, are Hermitian conjugate operators. Moreover, since Ar(x) is a vector, for everylc therearefour independent "possible polarizations", which are conveniently described by using a set.of orthonormal, but otherwise arbitrary, " p o l a r i z a t i o nv e c t o r s " , l ! r , ) , : I , . . . , 4 . I f ( 5 . 2 7 ) i s t o b e t h e m o s t general possible solution of the equations of motion (5.g), we must sum over the four possible directions of polarization:
Au@): #ZZ vv
4
t)\
(k)-y e-ta,e*a)(k)), (s.28) .,+^ leihraQ)
h ^ _ L l l2 a
: 6u(. 4) e('x')
(s.2e)
In (5.28) we have introducedan additional factor llZa in the cie_ nominator. This is only a matter of notation with ho special sigTha ahni sives the commutationrelations for aa)(k) a very ,r*or."ioJl'.28) It is not necessaryfor the vectors af;) to be independentof.h,
nifinanna
and it is even advantageous choice:
to make the following
lc-dependent
ef,t: sf,): ef):o ,
(5.JUal
et')k,: ejDk,: g ,
(J.JUDl
e/ ar\ ,: ;
hr
elto):
,
( s. 3 0 c ) ( 5. 3o d )
O '
pl4) v4 -,. a
(s.3oe)
In this case we shall refer to transversely ar
I , t)! L
*n
nryi+r,Ai-rl L U .l nr O n g l L U O t n A . L yh ^vl .^.ru. ir?r^af u e r vi rnrn
rf nv-r
^7-. 3 ,
polarized light for i:t and
to
scalar
polarrza_
tion for )":4. We now see that if we were to have Eq. (S.6) as an operator equation, only transversely polarized liqht could be present. Since Eq. (5.6) has been ignored so far, we have totake into account all four polarizations. with this choice ofunit vecrors e@ , in addition to (5.29) , we have
, P ( i l 'Pvl ^ ) _ L-u I
A uuy' '
f\ vq
'
?ll
v+/
Because of Eq. (5.30e) where the right side is I (rather than z, for example), Eq. (5.31) has formal Lorentz invariance. The r e a l i t y c o n d i t i o n s f o r 4 t i r ( l c ) a n d o , r \ A \ ( k )a r e t h e s a m e a s t h o s e
t?,
^ !,
\J.zdl,
(lc) and Atr (1"). we geI
FromEq. (5. 2l) , with the hetp of Eq.
G . K 6 1 1 6 n ,Q u a n t u m E l e c t r o d y n a m i c s
20
.
^(,1)^{.1')
lA, (x),4, (x'))"": "r - # Z Z #' k,k' 1,1'
V@u
Sec.6
r e')-i (ur a')*6 y { ei\k +t''
i (hr + tt'u'J+ i (- t a')zo @ (It), ox lt') (k' * x lal^) (k), 6l]') (tc')I + ela* )l *n r" y tt') (h,)l or" s-i(t'r-Ht\ri(a-u') eiltsa-IE'n')*i\a*a') I + VA) Qt),
(s.32)
x la*tt\ (It), a0')(tc')l) : 0. Since all
a,l are positive and since ( 5 . 3 2 ) i s t o h o l d f o r a r b i t r a r y
l-.ima
:Ir r0). (6.12) nh n ft t'-l'lg,#(*) ^:r k A:r V t l 4 )( k ) I In (6.12) and all subsequent discussion, l0) is to be the eigenvector of the coinplete Hamiltonian which represents the state The arbitrary phases which could have apwith "no particles". peared in (6.12) have been taken as zero, since they have no significance. we have found that the quantized electromagTo recapitulate, netic field can be described by means of a system of oscillators
Sec. 6
The Hilbert
Space of Free photons
23
and that the energy of such an oscillator isquantized in the usuat way. This is the old (light) quantum hypothesis which is in a sense the source of the quantum theory and which we have re_ covered here as a consequence of our formalism. From (6.12), matrjx elements of the operators aV)(l&) can be simply determined. obviously these operators are diagonal in all quantum numbers other than Ml^J(li . we again suppress alt indrces with other quantum numbers and so obtain
( n l a t \ 1 +n t ) : ( n * t l a * t t ) 1 n S : l n + i for A+4, (6.r3) ( n a t l a t a t l e : - f u l a , F t 4 ) l n +r ) : (6.14) l;+1. The other matrix elements are zero.
The operators
lgrr)(rc)_ d6(^)(k) a0)(Ir) 1,rtal(k) :
for ,t g 4 ,l
_ a(4)(k) e* t4)(k)
|
are therefore diagonal in the representation (6.12) and have the eigenvaiues n@(lt). They are the operators whichgive the number of light quanta of given polarization and given momentum and are usually designated as the ',number operators". In a similar way, we call the operators 4lA)(k) for ,tr{ 4 and - a,t(4)(tc) tfre "annihilation operators" and the quantities oxttl(k) for 1{ 4 and ala)(k) the "creation operators ". By (6.15) we can write the enersy as
H: >,{i 1ur{rr (k) 1- (,r]r
lurtar 1k)}a.r,
to . .Lo/
w h e r e t h e z e r o - " p o i ne t nergy isagain neglected. This distinguishes ( 6 . 1 6 ) f r o m ( 6 . 1 ) a n d ( 6. O ) l u t , a s a l r e a d y n o t e d , t h e r e i s n o physical significance to this difference. As an illustration of the interpretation of these results in terms vo rf
f ru^ qn rf r^L q , l ri ng hr rt L tf 1
Li tL !i *q
infarocf
inn
t,--n a , -n, ,*t p g t e
the
tgtal
m6menlgm
and the angular momentum. For the first quantity, we obtain from ( 3 . 1 2 ) , ( s . 1 1 ) ,( s . 1 2 ) ,a n d ( 5 . 2 8 ) ,
a!'\ ?4L a!" u!'\ :* p^: ' R - - [ a'* * ' " {t(a!L +i Ax1 7xp , " 0*, 0*nJ J
\" \Ara
: - [ a" ul- z,: ox60xt J -
J
\ |
- f tu.rtl (rl ke{at]'t (k)t a*a)Qqat^I4q \ ' axtt) 1 z f> ,i" \
a(^)(Is)a0) (- k) ,-ziaro-
I
o,*lt)(k) a*Q) (- k) ezi.,,\.
)
Since the operators a{,t)(Ic) and aV')(-Id commutewith each other, the last two terms in brackets are symmetric in lc . The sum over both terms vanishes because of the antisymmetry of Ao , and Ee.
G . K l i t l 5 n , Q u a n t u mE l e c t r o d y n a m i c s
24 th
I /t
clmntrfrd
+ [ + eih I 1arit,a*(r'))l0) | e-ih l - l s - i h * - ; e ' z('O ; 1 a * t r ta, * ( / ) )l 0 ) ) . F r o mt h e r e p r e s e n t a t i o n s ( 6 . 1 3 )a n d ( 6 . 1 4 )w e o b t a i n u .10)
( o 1 1 4 t aa,a ' ) j l o > : ( o l { a * Q \ ,a x $ l } l 0 ) : 0 , ( o l { a * @ ( k ) , a v 0' )c ) } 1 0 ) : 6 t t , 6 * u ,
A a n d1 ' * 4 ,
for
(o | {o* G\(k), a\^\(tc')}| 0) : - 6**,6^n ,
(7.11) (T .rz)
and we can write (7.9) as
I,'.", 4,1
i j'rli) et^t - eLat (ol{Ao@), estfx A,(x')}lo): r " / - v Ii r t. l ! " , I I l(r) (yt : x leih("-"')| e-ih(*-t')1: (dr, - 26u&,a,) where the function p1x)- I D o@ - x,) l @,)d,x,. (7.r8)
I n ( 7 . 1 8 )w e h a v e i n t r o d u c e d t h e r e t a r d e d D - f u n c t i o n the definition D11@):{_;,,
for xolo ,I for
xolo
. )
Do@) by
,r.r$
If no limits of integration are written, for example, as in the last term of (7.18), the integration is to be taken over the whole fourdimensional space. By analogy to (7.19), we can also introduce an "advanced" D^(x) by the definition D-function
for D^@):{^? . lD(") for
xo) O
,]
( 7. 2 0 )
x 0< - 0 .
Half the sum of D*(x) and Dn@) i s o f t e n d e s i g n a t e d b y D ( x ) i n the literature:
D (*) : $lDo(*)-f Da@)1.
l'7
C1\
30
G. Kdlt6n, Quantum Electrodynamics
Sec.7
The last function is therefore related to D(r) by
D ( " ) : - + -# D @ ) :
-
t'ol
) e@)D(x).
(7.22)
It will often be useful to have the Fourierrepresentations for D(r), Do(r) , and Dn@) , analogousto (7.6) and (7.i4). Startingwith D(x\ , we write 6+@
e(x\: ,\:'
lxol
I
o'
n J
z
, i n f' t x ^ \ : "'
t rn
P [ !! r','o. J
r
(7.23)
0-@
From this we obtain
D(*):-rnt,p ,',,"# Io: I6prihx6(k)e(k):
l
-: : [ 6* p" "r 'i n , p I o z' 6 " \('t-r - ( A \ "^u*'z ") /2t) . l . o ! ' , : = ] -( pz n ) n J-[ * n , ", u' )u , . l ( 7 ' 2 4 ) en)aJ J lho+rl Tn (7 .23) and (7 ,24) the letter P in front of the lntegral sign indicates that the principal value is to be taken. With this, we have the integral representation for D@) . From the equatlons Dp(x) : D (*) - *D (*) ,
(7.2sa)
D 1 @ ) :D @ )+ * D ( * ),
(7.zsb)
we obtain representations for D^(z) and
Do@): '
Dn@) z
( 7, 2 6 a ) dkeih, # * on 6(h,)eft)|, {n D A ( x ) :- ^ 1 _I,ono r , u , { ohi ,' _ i n 6 ( k ) e ( k ) ) . (7 .26b)
rr',, I
l2n)"J
For completeness,
")
\
we give a few definitions:
D @ ( x ) :l , t o ( x )- i D < 1 ) 6 ) : #
[
a n , o u " a 1 n ) ] l r ( f rt)l+i 0 . 2 7 )
a n r " ' 6 ( h \+ t u ( A ) -1 ] t ( 7 . 2 8 ) D-(x): ) P@)-liD S :)
n\s),ntr)
a,,"nn,!IHl1to>.
n\,.,tu\t)
(8.6)
Yn?)ln@l
S u b s t i t u t i o ni n ( 8 . 3 ) a n d ( 8 . 4 ) g i v e s
- 1,nQ))+i{86 + 1 lo,u,,,ull@ln'.t1 l n @ , n @+ i ) ] : 0 , Z o,*,,,rl/n@+T I n- i lW
(8.7)
pG)- I )] : o. (8. 8) In@t,
The most general solution of (8. 7) and (8.8) is d,nl)ntt\:
where the constant requirement
c 6nril, rtnt (-
i)""'
,
/R
q\
c is to be determined from the normalization
({D4l 9 t, h e r i g h t s i d e o f ( 8 . 3 7 ) b e c o m e s i n f i n i t e . W e s e e , h o w ever, that this singularity occurs only in a "gradient". Consider, for example, a gauge invariant expression such as the integral [[ d,xdx' Fu,(x, x') Qpol{At,(x),A,(x')}lrtto)
(8.38)
with 0 \,,(x, -_ oKp
x''1 _
0 Fr,(*, x') _ n o1,
l|a ?q)
We can integrate by parts before taklng the limit p+0, and the last term in (8.37) then contributes nothing.r With these requirements we can compute as if the limit of the expectation value of the anticommutator were qiven bv
(\)ol{A,(x),A,(x')}l,po): 6,,,D\r)(r'- *).
(8.40)
We shall derive this result inSec.9 by another, more systematic, method. We have seen that we may work with only the transverse photons for gauge invariant expressions and that the longitudinal and scalar degrees of freedom of the electromagnetic field may be completely ignored. Obviously the gauge invariance of the theory
1. F. J. Dyson, Phys. Rev. 77, 420(1950).
3B
G. Kii115n,Quantum Electrodynamics
Qan
q
has been lost by the special choice of the subsidiary condition, and clearly an infinite gauge function was chosen in (8.37). At best, this is an inelegant point in the theory, and it would be preferable to have a formulation where there is always the possibility of different gauges and where we could avoid the transition to unnormalizable state vectors and infinite gauge functions. Second Method The Subsidiary Condition: The procedure given above for treating the Lorentz condition (5.1) in the quantized theory is not the only possible one_. Many other methods have been suggested by different authors.r In this section we shall concentrate on a method which has been developed by Gupta and Bleuler.2 with unnormalizable state vectorsarise because The difficulties are present in (8.3) and both creation and annihilationoperators (8.4). A condition like (8.2) can be satisfied without difficulty operator; however, if creation operators are for an annihilation also present, one is led to a system of coupled equations like (8.7) and (8.8) . The solution of such a system necessarily entails many particles, and these are not state vectors with infinitely seen above. It is clear always normalizable, as was explicitly that the vanishing of the expectation value 9.
: (nl atdI n I t'2: lfn { r,
(e.2)
instead of (6 . l ) . By this choice , all a* (i)are made Hermitian conjugates of oll) , and hence all the operators A,(x) are made selfadjoint. Certainly this contradicts the reality requirements^ for the classical electromagnetic potentials, but the introductionr of H e i s e n b e r ga n d w . P a u l i , z . P h y s i k , @,w. 5 6 . I ( 1 9 2 - 9a)n d 5 9 , 1 6 8 ( 1 9 3 0 ) ; F . ] . B e l i n f a n t e , P h v s. R " Y ' 8 , ! , e a i O g ' s i i l . c - . v a i a t i n , D a n . M a t . F v s . M e d d . ' 2 6 , N o . 1 3( l 9 5 T J . 2 . S . G u p t a , P r o c . P h y s . S o c . L o n d .A 6 ! , 6 8 1( 1 9 5 0 )a n d 6 4 , 8 5 0 ( 1 9 5 i ) ;K . B l e u l e r , H e I v . P h y s .A c t a . 2 3 , 5 6 7 ( 1 9 5 0 ) .S e e a l s o W . H e i t ler, Quantum Theory of Radiation, Third Ed., Oxford, 1954, pp . 9 0-103. 3. Such an operator was introduced by P. A. M. Dirac in a q u i t e d i f f e r e n t c o n n e c t i o n . S e e P r o c .R o y .S o c . L o n d . A ! 8 0 , 1 ( I 9 4 2 ) .
Sec. 9
The Subsidiarv Condition:
39
Second Method
a "metric operator" 4 enables us to get around this objection. With the use of the metric, the norm of a state vector isdefined as
( e. 3 )
Normlrp) : (rpl rllrtt). In order that the norm always
be realr
must be Hermitian:
{
( e. 4 )
n* : tl' So that we can ignore some trivial qulre
numerical
q':rtrl*The expectation
As a consequence of this, qarilrr
h r ra v r z e
a
we also re-
1'
/q
('l
value of an operator F is now defined by
F:
rq r4 rl
factors,
rtral
a Hermitian
trync.fafion vJ:yvvLssr vr r
(e.6)
QplqFllt)). rralrrc
operator does not necesthiS is just what we
and
want. With these new definitions, the norm of a state vector is not aiways positive . We can divide the state vectors into three classes. The first class contains those vectors of positive norm, and these can be normalized to I . The usual probability interp r e t a t i o n o f t h e q u a n t u mt h e o r y c a n b e g i v e n f o r t h e s e v e c t o r s . The second ciass contains vectors of negative norm,which we can take as -1, and the third class contains the null vectors. For the iast two classes of state vectors there is no probability interpretation, and we must therefore require that every physically realizable state belong to the first class. Fortunately, it will be seen later that our form of the subsidiary condition is compatible with t h i s r e q uj r e m e n t . In order that the expectation values for Ap(r) be real and the corresponding quantities for An(x) be pure imaginary, we require
Q t l qA n @ ) l ' p: ) k t l A t @ ) r t * l , t: ) Q p l A u @ n l), p ),
( s. 7 )
l A u @ ) , r )o: ,
( e. 8 )
Q t l q A n @ ) l v ): - ( . p l A n @ ) r t l y,t )
( 9. 7 a )
and
(e. 8a)
{An@),ri:0. G o i n g o v e r t o m o m e n t u m s p a c e , f r o m ( 9 . B ) a n d ( 9 . B a ) w e have
l a ( 4( k ) , T ) : o , {a@(Ic),rt}:o.
(e.e)
1+4 , ,
( e. 1 0 )
40
C . f i i t t 6 n , Q u a n t u mE l e c t r o d y n a m i c s
Sec.9
From (9.9) it follows immediately that 4 is diagonal in the quantum numbers ntz\(k) f.or .l$ 4 , and has matrix elements I. Tr rnr lL' hr ar v rr av Pnr r 6 v s e n t a t i o n ( 9. 2 ) , E q . ( 9. l 0 ) i s s a t i s f i e d b y (nltl 1r, n' @)>: d;$ o,o,t (- 1),,n,.
( e. 11 )
For ? we therefore have the solution (alrtlh:1-
( e. l 2 )
r 1 " L 6n o' o,
where the quantity nLo)is equal to the sum of all the s c a l a r p h o tons in the state l4) :
nf,r:Znf,i(k).
(e.13)
tc
In this metric, a few of the previous formulas are changed a little because we now have to put N(4):a*G) aQ) . For example, instead of (6.7) and (6.19), we obtain 4
H : Zar ) lftir (lc; ,
( e .r 4 )
Pu:ZAe X N(.1)(ft).
/q rcl
,t
k
l:L
t:t
We now return to the subsidiary eondition. In momentum space it has the form [4(3)(tc)! i a(at(k)f lrp) : o.
( e. r 6)
I n t - s p a c e ( 9 . 1 6 )b e c o m e s aAL+\ @) '
/q r7l
-;lv):o'
In general the symbol F(rr(x) denotesthat part of the operator F(r) which has only positive frequencies, i.e. , wlth r-dependence g i v e n b y e i h' . w i t h t h e u s e o f ( 9 .8 ) a n d ( 9 .8 a ) i n ( 9 . 1 7 ), w e h a v e
, , t a A r ) @_ ?aA l t @ \ n: \ :ee t l n l P 9 : 0 , ---a
\?t\-iA
1,1
ont
(e.tB)
and therefore hAt+t l y\ t^ , (x\ < v t n0 4f" (rx \f l D : Q t t r0 Avl ;h : x t r p*) ( v t r t x f i # l y ) : 0 . ( e ' r e )
I n o u r n e w f o r m a l i s m ( 9 . 1 7 )r e p l a c e s t h e c o n d i t i o n ( 8 . 2 ) . I n t h i s w a y (9.1)is satisfied, and the connection with classical theory ensured.I also be shown that several factors +* @ have the expectation value zero, It is simple to make the necessary revisions . For example, for two factors , aAlY) ( u , t, z A l , j , ( x ) a A t f , @ ' lV t)^ . ._ \ e p l n 4 90x, ): A*i. It i, p':) : dr, AxlaAti)(x) -a,nf) ; i ' l l t1x'11 p): , , - \,, t t, ,.-l t ( , p l , t l , t ) . D e , Q , _x ) : 0 , e t c . tL-;,
Sec. 9
The Subsidiary Condition: SecondMethod
4I
Equation (8.2) had only the single solution (8.9), but the new subsidiary condition (9.16) has many independent solutions. As before, we write an arbitrary state vector as
lrp):lrl)n@*) h
r
e.Zl)
and consequently Eq. (9.16) is only a condition [as (8.2) was previously] on the quantities l@p) . As a fundamental system of solutions, we can choose
ip1>, il,,l:li,l; ,
:
( 9. 2 t a )
( s.2r b)
-r..-
_ >.(i),ll (:) ln _ r,r) t@(")> ' . Y\//' --'
( 9. 2 l c )
:
Obviously all the vectors (9.2I) are mutually orthogonal, ( o t t 1 , l @ t r ' ) :; 6 .
ia
lo
)t\
rfo
t2\
More important is the norm of the vectors l@t't);
( o a t , ,| @ a t:sI
( - r ) '( : ) : u " r , f---O
\ r /
None of these norms is negative, and only the vector l@(0)) has a non-zero norm. Because we require that a phystcally realizable state satisfy (9.16) and have norm I, the allowed vectors are of the form
(s.24) with arbitrary coefflcientr ,(r)(k) . If we now caiculate the expectation value for the electromagnetic potentials ln the states (9.24), we obtain (assuming for simplicity that there are no transverse photons present)
I ry lv
f,^ lr.-
(k)a e-in,o*(a(k)l, (rr.20) ao)
-' ) l (, 1. "1. 2 I ) ' rs\ - i n , f i * (\ r- -)l(rk Ll s ' i xvr' f i\(-t-)t!g
From these equations and (Ii.19) we have aa)(ri - 6at(k) :.-r*,r,
(k, o) i"^a.
(r1.22)
Here we have lntroduced the Fourier components of the current density by setting
(lc,AJf e-ih'i*(i)(rc, frJ];d,".[r.23) oor-,LZt,!)(tr)leih"iQ\ ir@):+ ' ' i ^ u[' o Vv t"'it
The complex conjugate of. i@ (fc,fto) is always |'*lt)(k, fto), and the factor id,tr'ensures the correct reality properties f.or the i,@). Simtlarly, for the particle number operators we find
- N(D (rg) fr(i)(rc) +
(k,r) o* tt)(k)* 5#in ^)l, (rl + f^ vo (rc, + HV# i* @(k,a) at^\
l
gL.z,)
with
N(4 (k) : a*Q')(g att)1ts1 ,
(il. 2s)
(rc): A* 0)(k) a0)(k) . fr/(,r)
( l r .26 )
and
As an example, we shall consider the state which has no inparticles. Thls is the state which we denote as the in-vacuum A question of physical interest is the calculation of the l0). number of outgoing particles of given momentum vector k and given polarization i. We shall confine ourselves to a single momentum
50
G. Kiill6n, Quantum Electrodynamics
lc and a single polarization f,, for simplicity. In our notation an in-state of z particles is la), and one of ,c out-particies is lZ). The probabilitythat n such particlesare emitted by our system is clearly
wf,i(tt): l(tl0)1,.
(rr.27)
li,>:+y n l @\"t6> .
0 1 .28 )
From the equation
we find
: (6ra' (##)"o>: 0, the switching-off is done adiabatically. In this process .E is not constant in time as ln b u t . 4 3 ) , v a n i s hes for xo-->- @ t as is obvious from (tl.t0). 01 R a t h e rt h a n ( 1 1 , 4 3 ) , E n o w b e c o m e s
E: L I I !#( H#l* x Jr@') I,@")-
t}x + ala-*,;]e-"{tn-' ""'|+t'.-'r.."
II +#
r-a{trnt*t'o-t,,'t) Ir@)Ir@').)u'.,0
The first term in brackets is of the order of magnltude or (and not oc2!) and vanishes if the switching-on is adiabatic. The other t e r m s g o o v e r t o ( 1 1. 4 3 ) . I n t a k i n g t h e l i m i t , , 0 i s t o b e h e l d constant as o( goes to zero. For the time-dependent current (II.44), the total energy (Il. 8) contains one more term of the following f orm:
t II 0,..r_i-#lrh?6(,..,- x.oix,o) t't'): + A l | ) @f). 6 @ , " ' - x o * t ; ' o ) ] u e - "It,' @
: + I I !#!lW
(ll. 4 6)
t qA@ @)]'t",-r".'t I,@')
The integrals appearing here converge for a=0;therefore the whole term vanishes in the adiabatic limit. By this explicit calculation, we have verified a special case of the adiabatic theorem of quantum mechanics. We can summarize the moSt important result of this example as follows: If a state is an eigenstate of the Hamiltonian and if a parameter in the Hamiltonian is adiabatically changed (in this case the current), then the same state is also an
Interaction with Classical Currents
Sec. l1
53
eigenstate, after the Hamiltonian is changed, but with a different eigenvalue. If the currents vanish, then the Hamiltonian is just gtol 1,4jPr)with well known eigenstates which can be characterized by the particle numbers. From Eq. (tl .46) it follows that the nondiagonal terms in the Hamiltonian vanish in this representation, and therefore that these same states are also eigenstates of the complete energy operator H(Ar). If the eigenvalue of the operator 1 1 t 011A f ) ) i s Z n r o f o r a s t a t e , t h e n , b y ( 1 1. 3 9 ) , t h e e i g e n v a l u e o f the operator H(A*) is lna*E for the same state. Here E is s i v e n b y ( 1 1 . 4 3 ). we should note that there are other ways to In conclusion, introduce free particles into this problem. For example, we can introduce a free field ,4jft (x, T) which coincides with the complete field A,,(z) at some arbitrary time I. The field AI?)@,I) is defined by A I P ( x , T ') :
-
[ a t * ' l n ( 'x J I ,l: T
This field satisfies
x ' '\
aAp\x') bD(x -:'t * A (,,\l ht a7\ "tt\-ll'\rf,'='/ A*6 ' Axn
the homogeneous wave equation
aAf)(,{,r) : o.
rir 1A'l
ln this way particles have been constructed which would appear if the current iu@) were suddenly (not adiabatically) switched off at xo:T In general, they are not of much interest. If we were to use a Schroedinger picture which coincided with the Heisenberg picture for xo: I then, in a sense, these particles would be the "natural" ones. For simplicity, we take T = 0 and write the a!t*!,o) in thuirFourier represchroedinger operators Ar(r,o) ana "'o sentations A , ( r , o') : + > W [ b ( 4 Vvo"^ll,.
( 1 0 )e i l . r+ 6 * t t )( r t' ) e - . i krt lf
(rr.49)
aer@ : =f ell)UdllT lba)1d eil"..- b*t^\0c\ -iro | \ - - l e-iktl r , 1 tt. 11.50) 1,"_o-FhLi'\tu) V 2lu"\tu)o F o r t h e o p e r a t o r s b ( / () t c )i n ( 1 1. 4 9 ) a n d ( I l . 5 0 ) , w e c a n u s e t h e m a t r i c e s ( 6 . 1 3 )a n d ( 9 . 2 ) . f t r e s e s a t i s f y t h e c o m m u t a t i o nr e l a t r o n s in the Schroedinger pictwe. The Hamiltonian is not diagonal. By means of a simple calculation, we find that the state of lowest energy (the vacuum) has the expansion
t0):c['o''',l],7rlW)" r'^4tn@)(^)(r,))1 , (r1.sr) where
lntb)>: fil4"to@s.
ur. c 2,,
trA
G . K i i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s
From the condition (010):1, to be
Sec' 1l
the normalization constant c is found _ 1 ,
trt-a'i^
li(i)(k)l' 203
(ll.s3)
The physical vacuum is therefore a mixture of "particles at time zero" given by (11.51). Particularly in the older literature, this result is often described by saying that the physical vacuum With the conventions we contains a mixture of "free" particles. employ, we must be more cautious with this term because we will often make use of both the incoming and outgoing free fields. Acare identical cording to the results found above, the in-particles with the physical ones, yet because they are described by a free field, they can well be called "free" particles. In particular, if the current is a point source, then i('z)('c) is In independent of lc, and therefore the sum in (1I.53) diverges' this case the physical states cannot bg expanded in terms of the Nevertheless, the two states of free particles at time zero.' but cannot be expanded in kinds of states exist simultaneously In a strictly mathematical sense, they do terms of one another. This shows only that no not belong to the same Hilbert space. free particles at time to the be attached to physical meaning is p o i n t s o urces. w i t h m o d e l i n o u r l e a s t n o t a t zero, given here of the adiabatic theorem and the The discussion for introducing free particles has been spevarious possibilities field interacting with an electromagnetic with cific to a model The adiabatic theorem is very general given classical currents. and holds under quite weak conditions in the ordinary quantum A general proof for a system with mechanics of point particles.z For has not been given yet. freedom of degrees many infinitely calcuour model we have verified the hypothesis by an explicit lation. Later we shall often make use of the adiabatic hypothesis. Although the various free fields have been constructed only for a special example here, the discussion can be taken over, almost unchanged, to the general case of two interacting fields.
particularly emphasized by van Hove, @en Physica, Haag 18, 145 (1952). In this connection, he has introduced the word "orthogonality". (If the constant c is equal to zero, then every term in (ll.5l) vanishes and, according to van Hove, the vector l0) is therefore "orthogonal"to every vector lnlt\S !) See a l s o R . H a a g , D a n . M a t . F y s . N 4 e d d . ? 9 , N o . 1 2 ( 1 9 5 5 )a s w e l l a s A. S. Wightman and S. S. Schweber, Phys. Rev. 98, 812 0955)' 2. M. Born and V. Fock, Z. Physik 51, 165 (1928).
CHAPTER THE
FREE
]]1
DIRAC
FIELDX
1 2 . E q u a t i o n s o f M o t i o n , L a g r a n g eF u n c t i o n , a n d a n A t t e m p t a t a Canonical Quantization The Dirac equation for a free electron is
:0. ? * + m)v@)
0z.r)
Here tp designates a quantity with four components Vr,@)...tpt@), and the terms / are matrices (T)rB, which obey the anticommutation relations {y, y,J : 2 6p,. (lZ ,2) lf we write out explicitly a1I the matrix multiplicatlons in (12.1) and (I2.2), r,r,'eobtain
$ 1 $ ( n , \ ^ a , .+" ^m)d"Pj : o' vP(x) A\ A\ril"na;; i, lW) "u 0) po* Q) "BQ) Bu1: 2 6o,6na.
p-r
\tt.J)
(r2,4)
Usually the short notation of 02.1) and (12.2)is most convenient and leads'to no confusion. An expliclt representation for the matrices y is not necessary for most calculations; however, one can be constructed in the following manner:
I o - i o " "\ Tu:\ioo o ), tI
(rz . D/
0 \
zn:\o _r)'
( r 2. 6 )
Here.0 is the two-row zero matrjx, .I is the two-row unlt matrjx, and the quantities d' dre the Pauli spin matrices: o, , : t / 0 { \_ 1 , o . , : l / 0 \lo/'"'-\;
-r\ I o )'
t,t "o'"- :\ lo
0\ -r)' I
r ^z-' ./ ) \r L
* See also Chap. B of the article byW. Pauli in the Handbuch der Physik, edited by S. Fliigge, Springer-Verlag, Heidelberg, VoI. V, part I.
G. Kdll6n, Quantum Electrodynamics
56
Sec. 12
The matrices 7 are therefore four by four, and one readily verlfies that they satisfy Eq. (12,2). With the use of this representation for the T t wa can attempt to flnd the plane wave solutions of (12.1). It is stralghtforward to show that for every wave lt,(x)
:
(t2. 8)
uo(q) ei(an-eo't
of given spatial momentume , there are two possible values of. qoz
(12. e)
Qo:*E-*Wr+?nr,
. and that for each 4o there are two independent solutions We shall enumerate these solutlons in the following table:"9\@)
x
4
12'
! -
1
0
-
O
1
_q,+,r!r_ m*E
d"
-q,li8y
4'= mlE
mlE
*:" 4t\
d--x0^,
rr+E
mlE
o.*io^.
a,
m* E
t/t+ E
J!__:__:_:,L
\ \
r/
l"
m*E
I
The two solutions with z:l and r:2 go wlth the value {o:E,and These solutions are the two others go wtrth the value llo:-E. normallzed so that
lwlvt (q)uf\(q): d,,.
a:1
02.ro)
Moreover, one can show from this table that lu{t,t (q) "tir (q) :
6oF ,
(r2.1r)
I:T
,
uf)(a): - ftttrd.,-*)pn , zay,@)
(1,2,r2)
r_L 4
Za!, @)utp(q):
/:z
tiT rr, - m)po. "',
( 1 2. 1 3 )
Here we have used the notation
a@): w*(8)'yE ,
(r2.r4)
and qt+t :
(q, i E) ,
qt-) :
(q, -;' E) .
( 1 2. 1 5 )
Equation (12.10) is the orthogonalityrelation for the solutions of the wave equation, and (12.11)shows the completeness of the set of solutions. Physically the two different solutions for each value
Sec. 12
E q u a t i o n s o f M o t i o n , L a g r a n g eF u n c t i o n , E t c .
ST
of qo correspond to the two possible orientations of the elecrron spin. Obviously the total spin is then one-half. The transformation properties of the Dirac theory have been discussed in various textbooks.t We shail not enter into a discussion of them here except to note that V @) : tt* @)yE ,
,l @)',p(x); is an invariant and that
i rp(x) yrtp (x) has the transformation properties of a four-vector. In particular, ,p*(x)rp(x) is not an invariant but is the time component of a vector. Formally the equation of motion can be obtained from the Laqranglan
s: -v,@) (r* * *)v@)
( 1 2. 1 6 )
by allowing variations of rp(*)and rp(x)as if they were independent fields. The canonical momenta can be obtained directlyfrom (12.16): n r ( x ) : i r l @ ) y n : i r p * ( x )'
(r2.r7)
nq(x) : s.
/rt
ro\
The momentum conjugate to rp(r)vanishes identically, and the time d e r i v a t i v e s o f t h e s e f u n c t i o n s d o n o t e n t e r ( 1 2 . 1 7 )a n d 0 2 . 1 8 ) . Thus it is impossible to express the time derivatives as functrons of the momenta. Despite this, we can construct a Hamiltonlan which is a function of. tp(x), its spatial derivatives, and the momentum nr(x). This Hamiltonian is
tr:i,'@)v'w .::, o, ,nr ,y:^:"-1,,;. ) av{.) In this form the function g(r) has been eliminated, and the Hamiltonian contains only the independent field tp(x). The CUrrent denS jt.'
fnr fha
Triran olan+rOn
iS well
knOwn:
i,(*):ietp(x)TpV@).
(r2.20)
These quantities transform Like a vector underLorentz transformations and, as a consequence ot (12. I), satisfy the continuity equation
+#: t'fo{irr\
+ e-Wy,,t'@)l:
: i e l- *rp (x)Ip@)+ m'p(x)rp(x)l : o.
(12.2r)
l. See P. A. M. Dirac, The Principles of Quantum Mechanjcs, T h i r d E d . , O x f o r d , 1 9 4 7, p . 2 5 7 .
G. K51l6n, Quantum Electrodynamics
58
Sec. 13
Previously we regarded the electromagnetic potentials as operNow ators which satisfied the canonical commutation relations. This we shalt attempt to interpret the field tp(x) as an operator. procedure is often called the "second quantization" of the electron In this language, "the ordinary Dirac equation" should be field. the first quantization, but we shaII refer instead to the "classical Dirac theory',. ln this theory the field rp(x) is a classical quantity After the second which is also the state vector of the theory. dynamical varithe but is quantization, v@) is not a state vector longer be interrp(r) no particular,lt)*Pc) can theory. In able of the preted as a probabilitY densitY. Previously we expanded the electromagnetic field in plane Now we do the same for the field 9(r) : waves.
u| (q) nt't(g)+ ,'rr" FE'o)>luy (q)e\')@)l .(tZ'ZZ'l ] ,i ta*-r'.15' Z- *' ) -_
tt^z41
(" t uY a \ x\:+, 1 ,1
Y'q
/)|
\
t:r
a(')(q) are therefore operators which we now regard The quantities The Hamiltonian can be expanded in as the dynamical variables. t h e m : terms of
H : l d' , x . i ( ("x \ : I E I i V
l':-r
Q2.23) (q)- f a*ot(q)o(,r a*?)(q)a(,) @)f j,
as can the charge: 4
a : * i I ir@)d}x: eIq ) o*t't(S)o?)(q). /:7
(r2.24)
Here we could attempt to use the representation (6. 13) for the matrices a(') . This is equivalent to the requirement
lnrQ\, tP(x')),,:r" :
- i 6 (n - n') .
(r2.25)
This does bring the energy into dl.agonal form; however, this apAccordinq to (12.23) the eigenproach is scarcely satisfactory. as well as positive values. take negatirre can energty the of values unlike the electromagnetic field, there is no subsidiary condition Furthermore, here which excludes the states of negative energy. whereas limited, is not in same state the the number of electrons e x clusion t h e o b e y e l e c t r o n s t h a t e x p e r i m e n t a l . l y k n o w n i s it principle. Hence the orthodox method of canonical quantization cannot be employed, and we have to develop another method in Sec. 13 for quantizing the Dirac field. Quantization of the Dirac Field by Anticommutators modified quantization method, we sha11 require that the 1" - i "* commutator of H and an arbitrary field operator be equal to we if holds clearly times the time derivative of the operator. This other any since rp(x), property operators for the only require this operator in the theory can be expressed in terms of tp\x) ' From ,J2.22 ) ancl (12.23) it follows that a sufficient prescription for 13.
Sec. 13
of the Dirac Field by Anticommutators
Quantization
quantization
59
is
la*o:1n,o(,)(q), a\')(q,)l : - ao (q) 6,, 6ro,, (') la*o)(q\ o\,)(q),o* ts)(q')l : a* (g)d".dnq,.
( 1 3. 1 ) 03.2)
of which the T h e E q s . ( 1 3. I ) a n d ( 1 3. 2 ) h a r r e m a n y s o l u t i o n s , canonical quantization is only one. They are obviously also sat'i c fiad
i f
razo ronrr
ira
t
{a* r,t1Ol, oG)(q,)} : 6,, 6.0,{; t".0,(s)- n-t"r I 0r.,, :Zou(lr*t't10,*N-t'r10,1.
ru = - ;[ asxtp1x)yn
I
Q,f
The particles
N-,
which
were taken to represent positrons,
have
1 . W . P a u t i , P r o g r . T h e o rP . hys.5,526 (1950). References to the older literature are given here. 2 . W . H e i s e n b e r g ,Z . P h y s i k , 9 0 , 2 0 9 ( 1 9 3 4 ) .
Sec. I3
Quantization
of the Dirac Field byAnticommutators
6I
a m o m e n t u m q a c c o r d i n g t o t h e d e f i n i t i o n s ( 1 3 . 6 ) , ( 1 3 . 7 ), a n d (13.10). This is in complete agreement with the general result ( 3 . 2 1 ) , s i n c e t h e l a s t t e r m i n ( 1 2. 2 2 ) i s a c r e a t i o n o p e r a t o r f o r positrons according to (13.6) and (13.7). Because of this , the xdependence of thisterm must be s-'p' if p is the energy.momentum of the one-positron state. As a final topic, we compute the angular momentum for these particles. The symmetrical energy-momentum tensor can be constructed by the method given in Sec. 4. The result is
7,,(x):v(x)r,"ff+ i fi O'Al(y^ly,, y,)t T,ly,, y^l+ I trr.,rl f - y , U p , y i l r l @ D).
' o r . L ^ , ,d Lfo 1 c o mp l oner F rr th, r t 1nt( i tegri t i o n l Jrat:
r. 1. . : trt _ -
Ii'{9): 1
{l) I(l)
J 7 '1
(9)J- 7.(1 )
7(0j ! - l J 1 1i l
'r.[o s x l l u (*) l yn d3 (*),
1,
/l
2,
\L' '
i t^ -1" 8,
"x \ty)\ X ) , T E
with
I n ( 1 3 . 1 9 )a n d ( 1 3 . 2 0 ) t h e p r o d u c t o t t p ( x ) a n d g ( r ) h a s b e e n r e p l a c e d by the commutator Llrl@),rl@)l so that the vacuum expectation value of the angular momentum will vanish. In momentum space t h e t e r m j ( , 9 )c o n t a i n s a f a c t o r and is conseu t r ? ) @ )u f , ) ( q ) : 6 , , quently independent of the state of polarization of the particles. We shall therefore interpret this as the orbital angular momentum. We now consider the component of the other term Jf(l)in the direction of propagation of a particle. For a single electron state, if we take the direction of propagation as the z-axis, from 0g.ZO) and (I2 .22) we readlly obtain
lll)lq): +(- r)'+'lq). In a similar
fashion,
for a single positron
( 1 3. 2 r )
state we have
l|',)lq'): j (- r)/-rtto',
\rJ.
LL)
The complete similarity of (I3.2I) and (13.22) has been obtained by the choice of index in (13.6) and (13.7) . In working out (13.21) and (13.22) we have used the matrix representation for o;; which is given in (I2.5) and (12.6):
OL
G . K A l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s
o'i:
/o* 0\ , und cyclic permutations. \O o^,)
/re
t?\
The Charqe Symmetrv of the Theory of the in the interpretation There is a certain arbitrariness theory, as has already been noted in connection with Eq. (13.12). Either we take the particles -l{+ as electrons , .Iy'- as positrons, and give a negative value tothe quantity e of (I3.12) and (13.13), or we interchange the roles of electrons and positrons and then a More precisely, the theory is invariant must be taken positive. 14.
under the transformations A/+ -
(r4. l)
AI_ I
(r4.2)
e
(ls.2)
-(iy q'-'-m)pneiqt-'@'-x)) q'r'-m)uneioitt"'-'t I +#l(iy -r) (i y - m)f (q" dq eio@' + mr)u(q). S I "6
The right slde of (15.2) will appear quite frequently in what follows, and it will be convenient to have a special notation for it. Accordingly, we shall write { ' , p " ( * ) , r p p ( x ' ) }-: i S p n ( x -' x ) ,
/ r( a )
wnere S ^r'r't- =+
"qq\"t-
enlt
l ' d q e i e \,"(/ i' y1 .a - r"u' t\d,F^ 6 ( q 2 * m 2 \ e ( q \ .
1*2"
/r( /r
\rJ.r,t
Thls function S(r) is obviously quite closely related to the functions / (r) which we studied earlier. From the remark following e q . ( 7 . 3 7 )a n d f r o m ( 1 5 . 4 ) , ( 7 . 6 ) , w e h a v e
s(,):0+-*)t1,1 In particular, for ro:9,
(rc. ci
S(r) becomes S(r)1,.:o:iyn6(n).
(rs. 6)
Equation (15.6) can be shown directly from the integral represent a t i o n ( I 5 . 4 ) . A l t e r n a t i v e l y , i t f o l l o w s f r o m ( 1 5 . 5 ) b yu s i n g ( 7 . 3 5 ) . Just as the function A(x) can be used to solve the wave equatlon (Z - m')u(x) : 6 (r5. 7) wlth given lnitial conditions , the S-function can be u s e d t o s o l v e the Dirac equation /2\-
with the inltial
\ yz ; + m ) v @ ) :o
(rs. 8)
y(r) : u(u)
05 .9)
condition for xo: T.
From (15.4)or (15.5)it follows that the S-function satisfies the equation
0***)s1,1:0, and thus that
( r s. 1 0 )
66
Sec. 15
G . K d t l 5 n , Q u a n t u mE l e c t r o d y n a m i c s I d z x 'S ( r -
,t(x):-i
x')ynu(n')
/iR
ll\
., _T
is always a solution of the Dirac equation (15.8). Furthermore, according to (15.6) this solution satisfies the initial conditions ( 1 5 . 9 )f o r x o : T , and hence is the solution of the specified probIem. In a similar way, we can also solve the inhomogeneous equation l - . a ' . . . \. . . , . . r r t - \ t \y ar+n)v\x):l\x) with the same initial
{\ rrFJ . rr .z\, ,
as above by the use of
condition
- i t dsx'S(x - x')ynu(n'), (15.13) v@):"1 3@ x')f(x')dx' ro-T ,'o:T
As in Sec. 7 we can go over to the limit I+sumption that the last term tends to the limit )s w r i t e ( 1 5 . 1 3a rP(x) : tProt (x) - / s " ( ' -
oo. With the as,l\o)(x), we can re-
Here the retarded S-function
S" (r) is
rlaf inad
I- tt'l
for
sa(r):
,o) 0rl
for
to (
Io
(tS. 14)
x') | (x')d.x'. hrr
)
/'r( 1(l
0.
Just as for the functions D(r) and A(x) ,we can also introduce an advanced S-function S,a(z)anda function S(*) by f
O
s'(t):is1ry S(r):
for xo>Q, I
f o rx o < o , i
- g e ( x )s ( x ) .
05.16) / ' rR 1 r \
These functions are used in a manner simllar to the corresponding They satisfy the differentlal equations /-functlons.
-d(x). (rs.iB) s*(x\:(, b4 + *\s(*):(, ! + m\ ! + *\I so@): ^'', "', Y O1 ) " \'. dx I \', Ox and have the integral representationsl
s-(,): i*n I a.per'ffi ' 5 6 ( r ) :O + I o Oei p'(i yf-*l l rV l a
(1s. re) t int( p)6( p' + *' ) \,Qs.zo\
l. As in Eq. (7. 24) , the symbol P refers to the principal va1ue.
Sec. 15
Anticommutators, Commutators, S-Functions
S n @ ) : T r a roJ p r ' o (' i y f - m ) p I
f
.f-
,,;Ay,.
67
- i n t ( p ) 6 ( P ' + * ' ) )I . 0 s . z t )
I
na*
The proof of these equations is not trivial, but requires some care because of the differentiation of the function e(x). As an example, we give the calculation for the integral representation (I5.19):
s ( ' ) : - ) z, @ )"( ra+x - * )I t 6 1 + - * \I 2 1 r 12+' - l y n ab(^ ioa.l((rAs . z z ) ' :\ '6 ax " The time derivative of the function e(x) is just 2.6(xo), and so the last term in (15,22) contains the function A(x\on the surface xo- 0, From (7.35) we therefore find /
2
\-
S \( 'r' /) : [ ^ t J - - m l / h \ \t ax
I
/ 1(
,'
??\
F r o m t h i s , ( 1 5 . 1 9 f) o l l o w s b y a s i m p l e c a l c u l a t i o n , u s i n g t h e i n t e g r a l r e p r e s e n t a t i o n o f . Z 1 x ) . T h e o t h e r e q u a t i o n s ( 1 5 . 2 0 ) a n d( I 5 . 2 1 ) can either be obtained in a similar way or can be proved from the relation
(rs.24)
So,o@):S(')T is(r) .
T h e d i f f e r e n t i a l e q u a t i o n s ( 1 5 . 1 8 )f o l l o w m o s t e a s i l y f r o m t h e i n tegral repre sentations . W e n o w r e t u r n t o t h e a n t i c o m m u t a t o r( 1 5 . 3 ) . F r o m ( 1 5 . 6 )w e s e e that if the two times are equal, the anticommutator takes the value
{rp"(x),tpB(x,)},,:,6: en)8"6(n_ r,) . This equation can be written .
in the following
{r*(x), p (x')} "":4:
i 6 (n -
rlR
,q\
form: n') .
(rs.26)
The above equation has a certain formal similarity to the canonical commutation relations, the sole difference being the sign. I n a s i m i l a r w a y , f r o m ( I 2 , 2 2 ) , ( I 3 . 3 ) , a n d ( 1 3 . 4 ), w e h a v e
{rp(x), ,p(*')\ : {rp(r) , ,p(x')} : o .
( r s. 2 7 )
In (15.27) it is not assumed that the two points r and r'have s p a c e- l i k e s e p a r a t i o n . The commutator lrl"@),,pp(*')l
(rs.28)
is not a c-number but we can take the vacuum expectation value of (I5.28), just as we did previously for the anticommutatorof the electromagnetic potentials. After a straightforward calculation we obtain
(oll,y"1x1,,t'p(x')lj o) : sll(2,- r)
(15.2e)
68
C . f a i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s
Sec. 15
s ( , ) ( r ) :( , i - * ) 1 , , 1 "r 1 : , " 1 , ,I A p r , r , ( i y f- * ) 6 ( p r f z z r )(.l s . 3 o ) \,a.r.r J l2n)".1 *, since the The function S(l)(t) does not vanish for space-like function 7tr)(x) does not vanish there. Ifwe attempted a discussion of the measurement of the Dirac field in a manner similar to that which we previously gave for the electromagnetic field, we would find that disturbances propagate with velocity greater than that of liqht. This would be in contradiction to the basic postulates of There are no experirelativity and therefore is not admissible. ments , however-- even gedanken experiments--by means of which Consequently the field rp(x) we can study the field g(x) directly. cannot be used to transmit a signal between two observers . By We do i t s e l f , E q . ( 1 5. 2 9 ) i s n o t i n c o n t r a d i c t i o n t o r e l a t i v i t y . have to check that the commutator of two components of the current, which are observables, vanishes for space'Iike separations. For these quantities we have
liu@),i, @')) : - t; ll,p(r),vr,tt(x)1,lrp(x'),v,v (x')l) : : - e2lrl @)y (*) , rp(x') y,tp (x')) : ,rp :e2 (ttt(x')y,v@') rp(*)y,,vQ)-rp (r)yr',t@)rp(r')y,rtt@'))'|
,,,. ',,
The first transformation in 05.31) is verified by noting that the difference between Elry"(*),rye@')land y"(x) yp(x') is a c-number which vanishes in the commutator. By means of the equation , t t " @ ) r l t p @ ' ) -: i S * p ( x-
x ' ) - ' , l t p @ ' ) r t " @,)
(r5.32)
we can rewrite 0S.St) in the following way:
(x'-x)vrrl,@)) lir@),i,@')l:i e'z(r1t(x)Trs(x-x'1y,v@')-',t@')v,s ! Ior.,rt ' (x\) (y"rl tp"(*) (y 1x'1) 1 ez(y,"(x) vo@') rtp "(y,rtt(x')) r(yrrp(x)),) . | B-vB@') B e c a u s eo f ( 1 S . 2 7 ) , t h e l a s t t w o t e r m s i n 0 5 . 3 3 ) c a n c e l , a n d u p o n us lng
: ') v*Q)rt'p@ * l ' p * ( r ) , r p a @ ' )|l s r " { r ' x ) ,
0s.sa)
we obtain the result
(*),r,t(x- x')y;p (x')l-lrp (*'),y,s t*'- 4yrrp li"@),i,@')l='t{1, @)l}l * t {t, ly,s (*- *')y,s (*'-r)l - sply,s (*'- x)v,"s (x-"')ll O,.rO I : * {f, Ol,Tus (* - x')v,rt(x')l- W@'), v,s (x'- x)v,rt'@)l}.J appears in each term in 05.35), the comSinie a factor S("'-r) mutator vanishes for all non-zero, space-Iike separations.
Sec. 16
The Dirac Equation wjth an External Field
69
16. The Dirac Esration *ith a Ti*u-Irdependent, E*ternal E1e..romaqnetic FieId The classical Dirac equation for an erectron in an external electromagnetic field is I
tA
\
r
(16.r)
ly\;;-ieA)*m]v?):o.
If the external field is time-independent, we can expect solutrons of the form ,1"@) : The function
u(n) satisfies
uo(n) e*ia,'" .
the eigenvalue
(16.2)
equation
(ro. s)
,f u(a): ?ou(r), with the Hermitian operator ff, f f : a * ( -.
J
' a I eA * ( t ) ) ' f m Y n l eA o @ ) ai ,
06.4)
and the Hermitian matrices ap , (X.b:
1'Y'at\
:
o / ",). \or o/
T h e r e f o r e w e c a n a s s u m e t h a t t h e s o l utions orthonormal and complete :
0 6 .s ) up(n) of. (16.3) are
! dsxulr"t (u) u!'t (u) : 6nn,, lulot (n) (n'): 6op6(n - n) . "?
( 1 6 .6 ) (167 . )
The eigenvalue po in (16.3) can be either positive or negative. From the charge symmetry of the theory, we see immediately that if fo=E , E) 0 , is an eigenvalue, and with the corr e s p o n d i n g e i g e n f u n c t l o n d e n o t e d b y u , ( n , E )' o, f . t, h t re n t h e f u n c t i o n u'"(n,E):C"pk@,-E) is an eigenfun.iior,l withthe eigen_ value ps:-8. Here the matrix C is the one used in Sec. 14. In the quantized theory we write ttt"(x):
Z {"!\ (n) e-iE","at")* u'J"t(n) eiE"'"6*@)y ,
(16. B)
E"> o
and, as we did previously, we require {rl"k), tp,(x')},,:,t,: 0r) p (n - n,), "6
( 1 6e. )
1. (Translator's note) In general, the functionu, is an eigenfunction (with negative energy) of a Hamiltonian equal to that of Eq. (I6.4), except that the sign of the four-vector potential must be reversed. consideration of the probrem of an electron :.n a coulomb field makes this clear. only if there are additional svmmetries can the change of sign of A be ignored.
Sec' 16
G . K i i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s
70
(16.10)
{rl"@),tpp(x')}"":,t:{rl"@),pp(x')},":q:o. F r o m E q s . ( 1 6 . 6 ) t h r o u g h ( 1 6 . 1 0 )i t f o l l o w s d i r e c t l v t h a t Ie*
(n) , e@')j :
: {a@, a\"')tt
{b*
(n) b\,t:l} : ,
:... {a\"), b(tu')}
(r6. tI)
6tutu, I :
(16.12)
0.
Instead of (16.9), we obtain for two arbitrary points x and - - i S p o Q ' ,x ) , { r p " ( * )r, t p \ ' ) }
06.13)
(r') eiEn\';-'dl'06'14) sp*(x',t)=i Z ln!) @)up (u') ,-iEo@i-rtaa'{") (n) u'u@) E"> o
Clearly
the
function
S(x, x,)
l, (* with the initial
satisfies
the
equation
differential
- i ed @))n *) t (x'x'): s'
(rb.rci
condition
(16' 16) s (x, x'): i Ynd(n - n') for xo: xL ' ( 16 ' 1 5 ) In fact, the function S(x, x') can be taken as defined by and (16.I6). This S-function differs from the corresponding funcdetion for the free field in several respects. In particular, it just of a function is not and quantities and *' r pends upon two xx' difference the In analogy to (IS.5), let us try to express S(r, r') as
- i eA(r)l s (x,x') ' x'). . , r *)) A (x, , : {rl , ( t a"q=
(16.17)
The function A(x, r') then satisfies the differential equation (r a :. t, \'2 jlaT-teA\n))
o
e - ,r)I /(x,x'\:g, 26rurF,\.,
(16.Ig)
with the initial conditions for xo:
Ylo.
( r 6. r e) (16.20)
Actually, these transformations do not really help too much, since the function A(x, x') still contains two spin indices, owing to (16'i8) is the last term in 0O.fg). In general the solution of Eq' ( 1 6 .15). As E q . o r i g i n a l o f t h e s o l u t i o n almost as involved as the constant of a case the first consider us J.et example, a simple *ugr,"ti" fi"ld H . In this problem the transformation (16'I7)does turn out to be useful . With no loss of generalitY, we can assume that H is in the direction of the z-axis and that eH> 0' Then we may take the vector potential ,4(r) as A(n) : (0, H r, 0) .
(ro. zU
Sec. 16
The Dirac Equation with an External Field This brings the differential equation (l6.Ig) into the form (
1o
-
22
,;T
- 2 i e H r ; -a
7I
-) e z H z x- 2 m zI e o r r U l Z @ x, , )- o , ( 1 6 . 2 2 )
First, let us considerthe simpler equation tAzAl L A T -s- 2 i x 6 , *'- mzl cl x,x' ): o,
and the corresponding IA1
( 16. 23)
eigenvalue problem
L-t +zix,
* xz*m,lu@):nzu@).
(16.24)
The eigenfunctions for (16,24) are obviously u n t , ( u ) : J - d t U t + a " I 1 g - , ( x- l . ) ,
(16.2s)
Eznn:m2+k2+2n+1.
06.26)
Here the function H,(x) in (16.2s)is the normalized eigenfunctlon for a harmonic oscillator with frequency l. For this function, we shall use the integrat representation
:w -# #_fi?,,,-#,,. H*(x) "+ Thus the function G(x, x,) in (16.23)
\ro.zt)
can be wrltten
o.!Gk.x,\=v IJP-! .,u - u-t')th(z-z')ly1,('- l)H,'(x'-:;sin lEnr'Q'o-*o)l ' en1z ,\-')-----E;*.(16.28) ,?-uJ
Both the integration over r and the summation over n canbe done explicltly. It is then possible to transform the integration on & so that 06.28) becomesl G " \ ("x' .. x" t, -\ -
t(x'- z) ] i - , +-4' q ' , l : _ _r #_(_2_" -_*_t *"J : l t 1 o o r c n , " . - | t ' - r t t , ' + * fJf 6; tnr t ,(16.29)
). : (x' - n), + (y' - y), * (r, - z), _ (xL_ xo)r, o:il@,-x)z+(y,_y)rl.
(16.30) (ro. Jti
F r o m t h i s s o l u t i , o no f E q . ( 1 6 . 2 3 ) , w e c a n immediately find the corresponding solutionof Eq. IA.ZZ1. We introduce the notation M : eorzH, tro.JzJ so that M2: szp1z. JJ,f uo.
l . I . c 6 h 6 n i a u ,p h y s i e a ,H a a g l 6 , g 2 2 0 9 5 0 ) . S e ea l s o J . G 6 h 6 n r a u and M. Demeur, physica, Haag ILTI(1951); M" Demeur, physica V,933 (lesl),
Mem.Acad. Roy.Bets.l[rvo. s (irjssl; t. K.;;;ulFronr. T h e o rP. h v s .6 , 3 0 9( l g s r )i;. s d r r . * s " r , t n u . . R e v .8 2 , 6 6 4( r 9 5 1 ) .
Sec' 16
G. Kaitl5n, Quantum Electrodynamics
72 We then find
:!L - 1t * "']*ao rlrl-" ^* t;a)' r- 1v'v1
A (x,x,) :'#
i6
.,1 2d
2e
eH
t-) =*;4 ot.,li; "ftl-"^+ v,*- # ""*l
^^. ,H \
" ;-|:::7'
,i6.-|2d
^^.,H\
06.34)
=-eH\ZT-""'-ti) 2d. -slneH
eH 2d.
llere we have defined (D(x,x'):s
-J!L
2 6" , ' - ! ) ( x ' +
r)
(16.3s)
introduce a retarded' an advanced' Just as we did inSec. 7, we can These are obtained directly from /-function. and an "barred " or (16.34) by multiplication bv -; t{ + e(x x')), lrlt e(x x')) , -$e(x- x') we obtaln' for exif *e tet H tend lo zero in these functlons ample for Z1x, x') ' _ ;,,_ " ; + L ) . , r (16.36)
Z ( x , x ' 1 - i-F
J
dae2t
7' since from This is the same result which we obtained in Sec' +o
PFia
(16.37)
: * et'rP'+*') lY,i Io*
it follows that
-# a'P *iaf ,"^' : o' ao I + ;d,I ft r, u;e I ffi "'V I ou. u4)'l ^') : ;" I #'t- s *" : -!- [ an'j'-l-"
In this expression, we have used the integral
'i"it'fA:l)]' ' "
:l_t\r,(cos(lzol ,'*0" f',)+'r'6 ! d.p .
I f:*
'
'l
iq2
(tutfi)- t #[sin (lzol eil)): ffi dpo(cos | _t |
^.
.
u
L, l(to'sg) l'
]
(x, x,) bv In a similar way we can define a function 7{D
: (o|[v,(*),pp(x'))to> {rl*
- ; ee @')]- *\ t(o'J{*"') . uo
Obviously this function can be written
(16'40)
Sec. 16
The Dirac Equation with an External Field
73
@
ntt)@,x,) :
-1 s.Lt(t-t)).h(z-2,)t L f f Y:! n l)n\2
n:o"
"
coslEhL\.r'ox H*(x - t) H,(x' - l) -&fr
and the subsequent calculation that for A(x, x') . The result is
proceeds along lines
I
,I
xo\)
(r6.41)
similar
to
xtr)@, x,): @@, x,)Ji*o*+ rtl--^*#), =+ , 8nz lql
x fcos *-#'r#] At this point it is possible
e
loeH l2q 2 \aH
_rH\ ---2dJ
( 1 6. 4 2 )
2" . -"' "H-' eH 2a
to define a function
Ao(x, x') : - 2i tr(x, x')4 /{1)(x, x,) , and so forth. factor @(x,x') in (16.34)and (16.42)which was definedby . T,h". (16.35) can also be written in the followingway: (r6.43) T h e l i n e i n t e g r a l i n ( 1 6 . 4 3 )i s n o t i n d e p e n d e n t o f t h e p a t h , a t r e a s t not if the electromagnetic field is not identically zero, but is defined as the integral along the straight line between x and, x,. In our case, with a constant fierd, the other factor in (16.34) and (16.42) is a function only of x-x, . In general, this is not the case: it is possible to show that after splitting off the factor @(x, x') the remainder is independent of the gauge which is used. If the external fierd is not constant, but depends upon the spatial coordinates, the calculation is very much more complicated than that of the previous example. As yet, no explicit examples for the singular functions have been given in a spatiaily varying external field. Recently, however, wichmann and Kroill have evaluated the expression
p ( r ): ( o l l , t @ ) , y n v @ ) l l o- ) [ d . x(,o l l , p ( * , ) , y o , p @o, ;) .] ld ( r ) ( 1 6 . 4 4 ) for a Coulomb field. This expression is quite important for the so-called "vacuum polarization,, (c.f . Sec. 29) . Their result is €1
Q(p) 2vt'_:e:s _l a-,trr 1laPltl+u")l' 1 . E . H . W i c h m a n na n d N . M . K r o l l, p h y s . R e v . 9 6 , 2 3 2 ( 1 9 5 4 ) ; 1 0 1 , 8 4 3( 1 9 s 6 ) .
AA
G. Kiill5n, Quantum Electrodynamics
, the charge of the external Here Q:Ze breviations have the following definitions:
14 yt
and the other ab-
field,
( 1 + u z \ ( 1 - z ( 1 - 1 t- \ )
,
t
, -. (1-z)(t+uz)
c:7rr=LIog
Sec. 16
(16. aGa) (r6.46b)
1-20-o-'
V,r,
(16.46c)
u:--3:,
2yr+t2
s\R):VR"-7" f
-
Zez 4n'
,
0 6. 4 6 d ) (16. 46e)
summation over The summatlon over A in (16.45) corresponds to a integratlons are The eigenfunctions of varlous angular momenta' of the representations integral the of obiained by transformations . '--llrutio" radial eigenfunctions (16.45) represents an analvtic functlon of / in the the use of ,"gionlZl< I . A powei series in 7, which results from ordirruryperturbationtheoryintheexternalfj'eldproblem'therefore gives a series which is convergent for lyl< I '
CHAPTER THE
DIRAC
FIELD
IN
FlELD
AND
INTERACTION.
THE
IV ELECTROMAGNETIC
PERTURBATION
THEORY
1 7 . L a g r a n g i a na n d E q u a t i o n s o f M o t i o n we are now ready to attack our major problem: the interaction of a quantized electron field and a quantized erectromagnetic field. As equations of motion, we will continue to use the Dirac equation (16.1), except that Ar(r) is now the operator of the quantized electromagnetic field, and not an externat field. In c e r t a i n p r o b l e m si t i s n e c e s s a r y t o u s e n o t o n i y t h e q u a n t i z e d field, but also to add an external field. Then Ar(z) becomes the sum of both these terms. In order not to complicate the formalism too much, we shall initiatty confine ourselves to the quantized field only. As a second system of operatorequations, we wilt use Eq. (11.2) with the current given by the Djrac operators of Eq. (13.I3j. We obtain all of these equations if we choose the Lagrangian to be t h e f o l l o w i n g e h a r g e - s y m m e t r i ce x p r e s s l o n : 9:9*l9e*
9w ,
(17.I)
e, : - i[o ot,(,* * *) v{4]- - !Y, + mrp(x), Qz.z) e 1x11, +t ee: '-
aAt,@)\ - !(!4!-L - aAr'@\(a4,@- -6i-l _ - 1 j4!!L !4@\ n- ^, 4 \ 'axp \t/.rl 0ro 7ru / \
T--dr"--;t
s* : + A*@)lrp(x),yoy@)l .
g7.4)
terms g* and ga are formally identicat with (14.12) and Tl"-iyo (5.10) and the term g* can be written as
9w:
A,(x)iu@)-
(17. s)
In the usual way, we obtain the desired equations of motion from this Lagrangian:
(, *
* * ) v @ ) : i e y A ( x )y tQ c ) ,
D Ar(x) : -
- ir@). + l',p(x),y,rt @D-
(r7.6) (17 7\
The canonical quantization is considerably simplified because the "new" term -9* does not contain any time derivatives of the
G . K 6 1 1 6 n ,Q u a n t u m E l e c t r o d y n a m i c s
76
Sec. 17
same The canonical momenta are therefore the field operators. can we and previously, functions of the dynamical variables as recommutation canonical immediately write down the equal time lations:
:lW, A,(x')1,":"i, iA,(x),
34#),":",":0, (17'B)
laA l v \ , A , ( 4 )) z o : x o , : - i 6 r , 6 ( r - n ' ) OxO
,
(r7. e)
I
-tp(x')}, :,'" : o, {ry(r), rp(*' )}, ":, r, {rp(x), " - n') , l e 6 ( n {rp(*),',1@')}"0:xL:
: o, y (r')f,,:";: lY#?,'P@)),,:,' SA u@), yA tp(x')f ,":,":lY#!,,t (r)f,,:,r:o' r(x),
(17. Io) (17. II) (r7.r2) ( 1 7. 1 3 )
the commutators for For free fields we were able to construct is now imposarbitrary times from those for equal times' This equations of simple satisfy not sible, since the commutators do are not even they separations' time-Iike for motion. In particular, expressions for give explicit to able not are we and c-numbers them. InSec.Ilwewereabletosolvetheequationofmotion(II.2) (ll'5)' In the present by means of a retarded singular function in motion by this case, we are not able to solve the equation of to an a transformation make to method; however, we can use it integralequation.Insteadofthedifferentialequations(I7.6)and (17.7) we can write (J/ . .t{, ,p(x) -- V(ot(x) - / So(" - .x')i e Y A (x') Y (x') d'x' , Ar(x) :
Aft (x) *
I
d' ' ) f o ^O- *')# l rp( *' ) ,yur t' @'
(I/.r-c/
I\\o)(x) and Here it is necessary to introduce two new operators soluobviously are Af)@) lnto the equations of motion' These tions of the free-field equations
(r!+*\rtot1v1:0, \'
0x
(.r/ . ro,
/'
f l , 4 1 9 ) ( r:) 0 .
(r7.r7)
(17'15)contain the reBecause the integral equations (17 '14) and the initial values formally are '4!o)(r) and ,ura"O functions ,ttt@) prinxo--.f'.In f o r a n d A r ( x ) o p e r a t o r s 9(r) of the Heisenberg to us allowing as motion ciple, we can regard these equations of c a l c u l a t e t h e f i e l d o p e r a t o r s a s f u n c t i o n s o f t h e i r i n i t i a l v a l uproblem es. to the At first sight this is quite a different approach the eigenvalues from that studied previously (where we looked for
Sec. 17
Lagrangian and Equations of Motion
77
of a Hamiltonian operator) . Here the analogous problem would be to find such a representation of the operators tp(x) and ,4,(r) that the Hamiltonian operatoris diagonal: /t
H (A ,',t) : H$) (A , ,p) * H\') (A , ,p)
I a\ "
0 7. r e )
H6(A,,i : H9 (A)* HIP@),
Hp@): ) I o'.14#Y#!+ !!#Y#?), 07.zo) : t I o".fot-t,(,,, HP@) , fa + *)y'@))
(17.2r)
: - -oi n'*o,(i l,t@),yuv@)l Htl)(A,y,) . I
(r7.22)
statement of the problem of quantum elecThis is the "classical" trodynamics. If we were to attempt such a calculation in detail , we would soon find difficulties. In fact, it would turn out that the theory formulated here is not well-defined mathematically and contains several infinite quantities. Great progress has been made in recent years with the realization that it is sufficient to regard these infinite quantities as a physically unobservable "reof the constants m and e. As we shall see later, normalization" it is quite important to have the formal relativistic covariance apparent at all stages, for otherwise the infinite quantities cannot be uniquely identified. The diagonalizatlon of the Hamiltonian 07.18) is formally not a covariant problem and it wouid be quite to arrive at a unique interpretation of the infinite parts difficult of the theory in this way. It is therefore preferable to attack the problem only by the use of the covariant equations of motion (I7.14) and (17.15), rather than by means of the Hamiltonian directly. A previous example has shown that it is possible to diagonalize the complete Hamiltonian (11.8) by solving the equations of motion for the field operators (11.5) with the use of "adiabatic switching". given of adiabatic switching Although the proof of the validity above does not hold here, it is straightforward to attempt a similar method. The justification of the method will presumably follow when we know the results of the calculation. We therefore write the equations of motion (I7.I4) and (17.15) in the form ,l @, u) :
Ar(x, a) :
yrot(x) - i t I So (ry-
el:r(*)+ *
[
x') e-al''ol y A(x' , a) \t (x' , u) dx' ,
(17 .23)
o o(, - x') e-ot'it lrp(r', o),T, v (x',a)l d.x', (r7.24)
and regard Tp and A: ^" functions ically interesting quantities are
of
g\ot, A(ol, and or .The phys-
,t@):hrytp(x,a) , A,(r) ::tylAt"@, d).
(17
' tr\
( r 7. 2 6 )
78
G . K i i 1 1 5 n ,Q u a n t u m E l e c t r o d y n a m i c s
Sec. t7
W e e x p e c t t h a t t h e t w o ] i m i t i n g v a l u e s ( L 7 . 2 5 )a n d ( 1 7 . 2 6 )e x i s t and diagonalize the Hamiltonian operator 07.18). At present, we cannot give a general proof of these conjectures but must test explicitly whether or not they hold when we find a particular solution. The operators rp(o)(r)and Af)@) are therefore to be chosen so that they diagonalize the Hamiltonian operator lor xo-->- oo; i.e. , H(A,rt)|""--€ : Il',(o)(1(0), ?(0)): Hf)(1o)) + HJo)(?(0))
(17.27)
is to be diagonal. Finding two such operators is just the problem which we have solved in Chap. II and IlI , so we can use those solutions here. From now on wb regard the operators gtot(r) and A , P@ ) a s k n o w n . T h e y s a t i s f y
(r), rrot1x,)}: - i s (*' - r) , {rp(ot (0 | [9(o) (r' - %) , (x), tptot (r')] lo) : S(1) IAIP@),Alot(x')): - i 6p,D(*'- ,) , (o 11/lot(x), trtot(r')) lo) : 6t,, Do)(*' - *) .
( r 7. 2 8 ) ( 1 7. 2 e ) (17.30) (r / . J.r/
In (17.31) it is implied either that we are evaluating only gaugeinvariant expressions or that we are using the method of the indefinite metric. Slnce it will later be useful to discuss quantities which are not formally gauge-invariant, we shall require from now on that the longitudinal and scalar photons be treated by means of the indefinite metric. The metric operator 4 is prescribed as commuting with ?(o)and 'y(o)'
(*), ,i : o. w\ot(r), "tl: [?(o) For the complete operator
(r7.32)
Ar(x) , we have
l A u @ ) , r t l{:A n @ ) , n \o: ,
/r7 ??\
exactly as for the incoming fie1d. For the complete Dirac field we cannot prescribe the relations correspondingto (I7.32), because the coupled equations of motion do not leave these quantities independent of the degrees of freedom of the incoming electromagnetic field. In order that the quantities present in the theory have the correct reality properties, we must define ,p(x) bV
,t @) : q tp*(x) q y+ . In the theory with coupled fields, tha
( r 7. 3 4 )
the continuity equation for
cr rrrant
a i p @ )- i t 2* . ".p."-l|
o t^t.\ A. ,rr^,'TPV(X)]:0,
/ 1z
? c'l
Sec. 18 Perturbation Calculations in the Heisenberg picture implies for Ar(x) z
aA,:jL: aAwL Atu
(17.36)
0r,
Both (17.35) and 07.36) are operator identities. condition for the incoming field holds in the form
79
If the Lorenrz
+t,p):o,
( r 7. 3 7 )
then it also holds for the complete field. 18. Perturbation Calculations in the Heisenberg Picture For the coupled fields, we cannot exhibit an exact solution ro the equations of motion. Rather, we must attempt to find useful methods of approximation. Because of the small value of the ,e2
1
N , it is straightforward to consider the right * * ( 1 7 . 1 4 ) ( sides of a n d 1 7 . 1 5 )o r ( 1 7 . 2 3 )a n d ( 1 7 . 2 4 ) t o b e " s m a l l , , a n d to attempt a solution in the formof a power seriesin e.We therefore write charge ,
V @ ) : V ( d ( x ) + e t p t t t ( x! ) e z y z \ ( x1) . . .
,
A u @:) A ' f )( r )+ eA t i \@ )+ t ' A e ,@ ) )* " ' ,
08.1)
0s.2)
and, upon introducing these series into the equations of motion, we obtain recursion relations for the different orders of aoproximation:
- + [ to@* x,)y, i toy, \(x,), (x,)\ (x): (jl B . 3 ) y ,(n+t) " / ' y p(,-^) \ " / ) " d.x, ", , 1 \') 2I *Zo
- ' - 't1 i y r t .\t-(- xt ' '.)p, y u r r , - - t ( x ' ) ) d x,' . - to A Y + t\ )' - (t x ) :2+ f D < t(' -x * LOLy (19.4) J '^p
In (18.3)we have symmetrized the right side. Although this is not necessary, it wiII turn out to be convenient. The symmetrization i s c e r t a i n l y a l l o w e d , b e c a u s e b y ( I 7 . 1 2 ) a n d ( 1 7 . 1 3 )t h e o p e r a t o r s Ar(x) and y(x) lor rp(r)] commute for equal times. In the lowest order, we have -i !)(1)(x): I S p ( x - x ' ) y t r Q ) ( x ' ) r t o ) Q t ' ) dt x ' t l t t r t ( x ) :- i ! t t o t ( x ' ) y A t o ) @ 'S) a @ ' - x ) d . x , , Afi(*): D o ( x - t c ' )l l t ) t @ o ) ) , y u r p t o r ( x ' ) l d ,x , + I @"), y,ytot@")l} x ,tz) (x)= 1// So @ ,') T,{!tto)(x'), ftp{ot t x")y*x x Dp(x'- x") d.x'il*" I I S ^ ( r - x ' ) y ,' ,"'S * ( ; (x") (x'), Alo) 6*' tr*" xyot @")} ,' {Atot
08.5) 08.6) (r8.7) I l(18'B) )
G . K i i l l 5 n , Q u a n t u mE l e c t r o d y n a m i c s
80
Sec. lB
, l t ] ) ( x )- + I I D * ( * - x ' ) ( [ r p ( o ) $ ' ) , y u S * ( t ' -x " ) y , p ( o ) ( * " ) l - l ] Or.nl + [ ? ( o ) @ " ) y , S n ( * -" * ' ) , y u , , p t o J ( * 'A) )t )$ I @ " ) d x ' d x " . ) In principle this process can be continued to arbitrary order withThus we can obtain the general equations out any difficulties. for the a-th order approximation to the field operators. The physically observable quantities, for example, the components of the crrrrenf- ca n a I so be constructed in a similar fashion. Because v g r r v r r u ,
we shall later study the current proximations here:
operator,
we give the lowest ap-
(*)), ilP@)-- i t *' (*),v,',p(d
0 B r. o )
(r')] + il]t(") : t I dx' (ptot@),yuSo@- *') y"rtt(o) I - *),ynV(o) -l- [9(0) (r)])A!,0) (*,),J @,)y, Sn@,
(r8.1r)
(r")]}] x (r"),y,rltto) (x'),lEtd dx" lrto)(x),yrSp(x**')y,{y,(o) tfXO:i[ tP\ ' 6 J J[ax' xDp(x'-x")-
O, dx" lt/0)@),y u Sp(x - x') y,, Sp(x' - x") y,og$\(*")l x a r[r[ x {Atl\ (*') ' Al! (*")} -:; (x') y Ato)(x') Sn(*'- x), y, Sp(x - x") x o ,'0,', SEto\ [[ (x")l* XY Alo)(x") tYOt
-,
08.r2)
(")f x " 11,rrd1x')\t, fi (x'- x)yu,V(o) (x"),yst$)(x +i [[a*'ar" [{[v,0, -
-
+ II o'
x") xDo(x' (x")v,,sn(x" x')1,, So(x' x) , yrtp\o)(x)) x dx"lrrrot
x {.4t?(x'),Af)(x")}.
A s i s a l r e a d y a p p a r e n t f r o m ( l B . l 2 ) , t h e h i g h e r a p p r o x i m a t i o n st o the operators rapidly become very complicated. We shall refrain from giving any further explicit examples,l and instead turn to a detailed discussion of the properties of the lowest-order approximations . InEq. (18.6)an advanced singular function appears for the first time. This does not mean that the operators f (r) become tptot(x) as r0->f oo , but rather it is only a consequence of the "reversed" . The same order of writing the difference of coordinates x'-x holds for the advancedfunctions appearing in (18.9), 08.II), and 08.i2). As can be seen from (18.5)through (I8.12), the solutions found in this way contain products of operators ,rptor(x) , ,lt!(x) andsingular functions. This means, for example, that the complete operator rp(z) has a non-zero matrjx element between the vacuum and a state wlth an incoming electron and incoming photon. Even the which allows order rp(r)(z)contains the product Af\(x').rto)(x'), @ysik
2 , I 8 z ( I g s o ) ;2 , 3 7 r ( 1 9 s 0 ) .
Sec. 18 Perturbation Calculations in the Heisenberg Picture
81
this transition: ( 0 1 ? ( r ) lq , k ) = - i e [ S * ( x - x ' ) v , < o l A t o \ ( x ' ) l k ) ( 0 1 ? ( 0 ) ( t ' ) l q ) d x ' - 1 " ' . 0 8 . 1 3 ) Here I q,h) is the state mentioned above; lq) is a state with oniy the electron and l,t) a state with only the photon. The same term also gives a contribution to the matrix element (hlrp(x)lq) z ( k l r p ( x ) l q ) : - i e I S o @ x ' ) y , ( k l A l o t ( x 'l )0 ) ( 0 l V ( 0 ) ( r ' ) l q ) d x ' * . ' . . 1 ( 1 8 . i 4 ) T h e r i g h t s i d e s o f ( 1 8 . 1 3 )a n d ( I 8 . 1 4 ) c o n t a i n i n t e g r a l s w h i c h a r e not actually convergent. We recall that these integrals are to be t a k e n a s t i m i t i n g v a l u e s , a c c o r d i n g t o ( 1 7 . 2 3 )a n d ( I 7 . 2 4 ) , s o t h a t f o r ( 1 8 . 1 3 ,) f o r e x a m p l e,
xl Uyf - m)y et^t
(olylxyiq, k)=-t enm ax, I I rh';H /
eih X --,lfzvu
u- (: q \ : ^ " " ,-' - , " : , , ' ' " '
I
, i Q t ' e - d l x ' +0 1" ' :
l/lR l5) l| \ r v ' r w /
Vv
, e i ( q + h )r r /,\ : - 1' u(q) e (q h)+,nzliy(q* k)- *ly e\^) + fO;
+
I
we have used the followingexpressions: in (18.15) '(^) (o1,llot (*)lP) : ;i? eih', V
( I 8. l 6 a )
( 0 l v ( o ) (, ro)r :
( r 8. l 6 b )
V2
ct)
I c . f . ( s . 2 8 ) ]a n a
){r'o'
,
is not l c . f . ( I 2 . 2 2 ) l , M o r e o v e r , t h e f a c t t h a t ( q+ A ) ' + m 2 : 2 q k zero for all photons (withro$0) has been used. In the theory of free fields, we have seen that the field operators create states with only one particle if they operate on the vacuum. From the calculation above it follows that this is no For example, if we Ionger true for the theory with coupled fields. we obtain, among other states, operate with I on the vacuum, states with two incoming particles (one electron and one photon), according to (18.15). On the other hand, we also obtain the onenarficle sf^tes in this manner. The matrix elementS fOr these ys!
Lrvrv
transitions are different from the corresponding matrjx elements for the free field.
From (lB. B) we have
: (olp(o)(,) (ot,t@)tq) |u>+ + II so('- x')y,X (x")l\ | q) D^(*' - x") d,x'dx"x (0 I {?(o(r'), lrt'@ @"), y,V(o)
- t,e2
f f ^ So(' - x')Y',So(x'- x") Y', x JJ (x")}l q) dx'd.x"{ . . . . (r") {,t!1,(*'), Alo\ x (01?(0)
( I 8. 1 7 )
82
G. fiill5n,
Sec. I8
Quantum Electrodynamics
The expressions appearing in 08.17) can be simplified ably. Beginning with the second term, we have
consider-
( o 1rptor (x'), Ali,Q")} | q) : 1r"; {A!o,t
l
I
frR 1R)
= -I ( o l v ( o ) ( x " ) l z > < z l { A t o ) ( x ' ) , A ltqo)).( ix "")"}" " ' lz>
The second factor of (18.I8) can only give transitions in which the number of photons is changedl Either two photons are produced or the same photon is first produced and then destroyed. The sum over intermediate states in (IB. IB) therefore contains states of either an electron, or an electron and two photons. .For the last class, however, the first factor is zero, and we have
(0 I ?(o)(r") {Al,o,) (*'), Alot(x")} lq) : : (0 I ?(0) (r'') | q) (ql {Al,o,) (*'), A[ot(x")] | q',: : (sl gtto) (x") I q) (o I {Al,| (r'), Alot(x")} loS : : 6,,,"Dn (x,_ *,,) (Ol rpto> 1rS. 1x,,1 In a similar way,
( r 8r.e )
we obtain
(o l {?(o) (x"),y,,,p(o\ (*")l\ | q) : (r'), 7r{o) : - 2 5- ' :
harra
. . ' .[t l[
t-t.- x')y,x
]
. , .[ . S ( r\ .)_1 , . y. r' -' , \D r ( r ' - .x" " )I *L *SK n\ . (. x ' - -x- " )| _D \ t t( l - x " )t Jl x }
x 7 " ( O rl < o l ( x " ) l q ) d * ' d*x.". . .
(18.2I)
I
J
The integrals in (IB.2l) can be done and we shall work them out in Sec. 3I. For the moment, Iet us avoid this and consider (IB.2l) only as an example showing how matrix elements for simple transitions can be developed from complicated expressions like (I8.8). It remainsto establish that the series obtained here satisfy the postulates (17.25), (I7 .26), and that they diagonalize the Hamiltonian. Regarding the diagonalization of the Hamiltonian, it is an immediate result that the conservation 1aw (4.22) for the energymomentum tensor is no longer valid, because the Lagrangian con-
Sec. 18
Perturbation Calculations
in the Heisenberg Picture
83
xu explicitly. They are contained in the tains the coordinates charge. By the equations of motion or by means of considerations similar to those of (4.26), it follows that arpr _ ag a,r-4'
(r8.22)
The right side of (I8.22) is to be understood rather than (4.22), as the derivative with respect to the coordinates with the field operators held fixed. In the present case of a variable charge, we have
orr, _ o9 oe -oe - . ox, oap We consider the time derivative we have
gL:olo
[ a ' r ! oxo }":-;
J
(r8.23)
of the Hamiltonian.
[ 4 s , aoxp? n: -
J
0 8 .2 3 )
From
! . !oxo ' t a'*a oe
08.24)
J
The change in the Hamiltonian during a time interval (ro, ri) can therefore be calculated from the implicit equation u
: H (A(x'),,p (*'))- [ a,"L{ffi,tl)L H (A1x),,t @)) x'
\ff
.6. For example, it is clear that (18.30) does not since this expression is of order of exist in the limit if nrsI, magnitude d.-trLrr if y'o vanishes . A more careful investigation, which we avoid here, shows that the appearance of such terms is connected with the change of the eigenvalue of the energy, and that they are to be understood as series expansions of expressions of the tvpe ,-i6E
lt"-o(u-t)l
08.31)
If we had fixed the initial conditions for a finite time 7, we would have had an expression xo-O (I), rather than ro-O(a-l) . Such an infinite phase in the field operators is of no physical significance and can obviously be avoided by a very careful treatment of the boundary conditions. We do not pursue the matter further because we shall later remove the change in energy from our equations (c.f . Chap. VII) . Apart from these infinite phases, a is present in our solutions only in integrals iike +@
2rl f u _ > 2 n 6 ( 'b- . \ . t rfr,^ore- n a l x o l'-"; p" :" r " : -., p. "t J "ur+
(1g.32)
Sec. 19
S -Matrix
The
8s
or sometimes in inteqrals of the form
I
d x ' o e - * " ) , , - ,-i i,p o @ o - * a :
_@
]-_
: - i p- - | + n a e o ) . Po''""\rtrt.
1xd.+ipo
t( 1 1c9 . 3 3 )
I n t h e l i m i t a + 0 t h e r i g h t s i d e s o f ( I 8 . 3 2 ) a n d ( 1 8 . 3 3 )h a v e s i n g u l a r i t i e s n o s t r vor n r vn Uc r r tLh r ra a rnl d i O n S ri rnr l4y .o .. REpUcq a u ser Ll tq a - l -u ft rU u LnI C U t tl S D u Dr yr q o iul r l e S e expressions must Iater be integrated over p(actually, only spacetime averages of the operators have a physicai meaning), the existence of these expressions is certain. We shall content ourselves with these admittedly incomplete remarks about the adiabatic hypothesis. Actually, we ought to discuss whether or not the series used here are convergent and define a solution, and, if so, whether one is justified in operating termwise with them. We shall later return to these questions and shall even try to carry through the discussion without power series. For the present, we shall not go into this further, but rather study the applications of the theory more closely. 19.
The S-Matrix The field operators which we have considered up to now are nor especially suited to practical applications of the theory. In principle, the electromagnetic field strengths are measurable, as we have indicated in Sec. 10, but this is more a question of the method of interpretation of the theory than of the experiments which are actually done. In actual measurements, the determrnation of interaction cross sections is of great importance. Typically, in these experiments a number of particles of known momenta and energies meet each other. During a rather short time they interact with each other and then they continue again as independent particles with measurable energy-momentum vectors. In general, these new energy-momentum vectors differ from the original ones. Under certain conditions, new particles can be produced in the collision or some of the original ones annihilated. It. is clear that our formulation of quantum electrodynamics with incoming and outgoing particles is quite appropriate for the discussion of such collision problems. In order to relate the incoming and outgoing fields, we have to find-a generalization of the method u s e d i n S e c . 1 1. W e s h a l l d e n o t e r t h e i n c o m i n g f i e l d s , o r i n f i e l d s , b y 1 k r " ) ( x ) a n d , ( e i ' )( x ) a n d t h e o u t g o i n g f i e l d s , o r o u t fields by Af"') (r) and ?c"') (r) . First of all , we know that these quantities obey the same canonical commutation relations. Acl. (Translator's Note) The usual notation in English for the rnfield is /(in) and n61 7(ein). We have retained KAIl5n,s notation only in order to avoid resetting all the equations. Similar remarks apply to ,4(out)- 7(aus).
86
G . K 5 i I I 6 n ,Q u a n t u mE l e c t r o d y n a m i c s
cording to well-known theorem.,l with the following properties: : ttaus)(z)
th"ru must exist a matrjx -S
0e.r)
5-t1p(ei")(z) S ,
0e.2)
At"E-(x): 5-t 7f;i")(r) S ,
0 e. 3 )
SS*:S*S:1. By means of this xo-->+@.
matrix
Sec. 19
we can express
the
Hamiltonian
for
/l'(o)(1(aut,tpl^"")1, as a function
of the Hamiltonian (1(aut, ?("u9) : /1,(0)
for xr-->-
ai
S 1Ii(0)(/(ei'), ,yr("i")) S.
( l e. 4 )
and Introducing the eigenvectors of the operator .i7(o) 12("i"),rp("i")) denoting them simply by ln), we find from (19.4), l7(o)(1(ein), Vkin))ln>: Enln) , 11(0) (1(aus) , ,(aus) ) S-t ln) : E,St lzr).
(le. s) 0e.6)
Tha ctatac oiconcfafae nf fha anarnrr f' n- r- v e r y 'S - r l at \" / a r a t h a r o f o r a large times and consequently the probability that a state ln) makes the transition to a state ln') is given by
unn,:l(ra'lSl")1,
( r e. 7 )
The general problem of finding the outgoing particles, given thosq which are incoming, is therefere solved if the matrjx S is known." Equation (19.3) says nothing about the general structure of the matrjx S except for its unitarity. It is quite clear that S has to be and that even if. e is non-zero, the matrix the unit matrix if. e:0, elements of S can be non-zero only if the states lz) and lz')have the same total energy and total momentum. We therefore expect that S has the general form
(z'lSln):6n,nl
( n ' l R l n ) 6 ( p '- p ) .
( r e .8 )
l. See, for example, P.A. M. Dirac, The Principles of Quantum Mechanics, Third Ed., Oxford,1947 , p. 106. Here it is explicitly assumed that the states lz) form a complete system which is equivalent to the assumption that no bound states exist. See also A. S. Wightman and S. S. Schweber, Phys. Rev. 98,812 (1955), acnaairlltr
v e y v v r s r r l
n
y .
Atc,
v e v .
2. The S-matrix was originally introduced into quantum theory by w. Heisenberg, Z. Physik 120, 513(1943). Thetheory of the S - m a t r j x i n q u a n t u m e l e c t r o d y n a m i c sh a s b e e n d e v e l o p e d , f o r e x a m p l e ,b y F . J . D y s o n , P h y s . R e v. 7 5 , 4 8 6, 7 7 3 6( 1 9 4 9,) a s w e l l a s b y C . N . Y a n ga n d D . F e l d m a n ,P h y s . R e v . 7 9 , 9 7 2 ( 1 9 5 0 ) .
The S -Matrix
Sec.19
87
Here (a'lRla) is a non-singuiar function of the energy-momenrum vectors of the particles considered , and p'and p are the total enerqy-momentum vectors of the states In fact, lrz,) and ln) when we explicitly calculate the S-matrix below, we shall find that these conjectures, implicit in (I9. 8) , are confjrmed. If the expression (19.8) is squared in order to obtain the physically ir teresting transition probabilities (I9.7), the delta function enters quadratically: a completety meaningless result! We have to go back a step in our calculations momentarily and recall that with a (finite) periodic boundary condition, the spatial delta function is replaced by the symbol dpp. and this quantity can be squared without difficulty. If af0 we do not obtain an exact delta function for the energy in (19.8), but rather expressions of the form 1d,
;
{lq
q)
",+W;-p,Y
The expression (I9.9) can be squared without difficulty, but the integral ,.
1
f F(x\azdx
!t!r-", J 1o,a ,qz ,
0e.10)
where F(r) is a regular function, does not exist. erations show that the integral 1
Simple consid-
F(x)as
I r t'"tr f J l a r , r t y d x : 2 -r\0) ",
(19. lr)
does exist. We can thereforewrite symbolically v n n ,: l ( n ' l R l n ) 1 , 6 p n ,6 ( P ' o P o )J 2xt d.
(re. r2)
If a goes.to zero the transition probability becomes very large. In ordinary quantum mechanics, in treating collision problems, the quantity of interest is not the total transition probability but the transition rate. Recalling that the particles have been interacting with each other for a time of the order of. llu, we see from (19,12) that in quantum electrodynamics also, the transition probability per unit time has a finite limit if o( goes to zero. As we shall see in Sec. 20, we can write
( n ' l s l n ) : 6 , , n * ( n ' l R l n ) d o , p6 ( f t -
p o,)
0s.13)
rather than (I9.8), and obtain for these quantities ?won,
at
A , ^ ' o-
:l@'lRln)l'6p'o "o
,n-
po)
.
( r e l. 4 )
From the quantities (19.14), the interaction cross sections, etc., can be obtained in well-known ways. We shall not give general formulas here, because theycannot be written without complicated notation. In later sections we shall often compute particular inter-
G . K i i l l 5 n , Q u a n t u mE l e c t r o d y n a m i c s
88
Sec, 19
action cross sections from the S-matrix. In principle, we can determine the S-matrix from the equations :) S-trtot(r) S : ?0) (r) + I S (* 1rt"*r(z)
x')i e y A(xt) y (x') dx', 0s .$)
(Af*) (x): ) s-re tot(x)S: Af) {4-[ n @- *' )T l',t@'),yu,t'@')fd.x',le . 16) or [S,ytot@)l:
- S/ S(r -
(19.17)
x ' ) i e y A ( x ' ) y ( x ' ) d . x ',
[ S , , + ; o r ( r )S] :I o W - 4 + l r t , @ ' ) , y r r t ' @ ' ) ) d r ' .( l e . 1 8 ) tn (I9.15) through 09.18) we have again used the earlier notation We now develop the matrjx Al|)@) and rp(o)(r)for the in-fields. S in a power series in e , analogous to those for the field operators:
0 e .t s )
S:{f.5tt)-p... I n t h e f i r s t a p p r o x i m a t i o n , f r o m ( I 9 . 1 7 ) a n d ( 1 9 . 1 8,) w e f i n d -i [s(1),?(0)(*)]:
f dx's(x
x ' ) y t r t o t @ ' ) v @ @,' )
09.20)
(41: + [ a"' O (" - *') llt)@) (x'), yrrt(o\(x')). ,4(0) [S(1),
(I9 . 2I)
(r) and rlt\o)(r), it folSince the operator Af)(r) commutes with gJ(o) Iows from (I9 .21) that the first approximation $(r) f e the S -matrix must be of the form S 1 1 ) : _ t I d * , l r t $ ) ( * , ) , y r r l ( o ) ( * , ) ) A(txr ,o)t* s ( r t .
(l9.ZZ)
In (I9. 22J the lstm s(r) is independent of ,lf) (x) . From (19.20) we find that s(1)is also independent of the operators forthe Dirac field and consequentlyis a c-number. This c-number obviously can. not be determined from the commutation relations (I9.17)and (19.I8) S(1): - S(1)* ; From the unitarity of the S-matrix, it follows that that is, the number s(1)must be pure imaginary. Without altering the properties (19.1)through (19.3), we can always multiply the S - m a t r i x b y a f a c t o r e t d ,w h e r e t h e p h a s e d i s a r e a l n u m b e r . W e c a n t h e r e f o r e s e t 5 ( i l i n 1 1 9 . 2 2 )e q u a l t o z e r o b y d e f i n i t i o n ; t h i s only fixes the arbitrary quantity 6. Thus we have as the first approximation to the S-matrix the result
5rr): _ bJ dx, SEtt (*,),yurt'\,1x,)lAtf\(x'). In a similar way given by
gtzt:1
it
+@
_f
dx,
(re.23)
can be shown that the next approximation
(x'),y,,tp
Sec. 20
T i m e - D e p e n d e n tC a n o n i c a l T r a n s f o r m a t i o n
Bg
In (I9 .24) an arbitrary c-number has been set equal to zero, as was done before. The method indicated here leads to rather complicated calculations if the higher orders in the S-matrjx are considered. Despite this, the results are remarkably simple, and it is to be expected that there is a more direct method to calculate this matrix. 20. Treatment of Quantum Electrodynamics by Means of a TimeI)enendent Can6pig6l TranSfOrmation In this section we shall develop another method of solving the differential equations of quantum electrodynamics. This "new" method isthe one which has played the greatest role in the development of the modern theories. It is closely connected with the interaction picture given in Sec. 2. From the fact that the in- and out-fields satisfied the same canonical commutation relations, we concluded in Sec. 19 that there must exist a matrix S with the properties (19.1) through 09.3). Actually, for an arbitrary time xo the field operators satisfy the same commutation relations as the in-fields, and from this we can conclude that there is a timedependent matrix U(ro) which has the following properties:
V @) :
(20.t)
U-, ("0)tp1xYt1, l ,l l.. . lHl) (x;),A fi(x) ...11 -
-(&-r')
,p(x):,t'@@)+ T2 i ! I d r L . . . I d . x f , i *
(20 . 14b)
x [aru1r5,i;, p(o)(r)] ...1]. [...[H(1)(r;), These formulas are not especially suited to practical calculations and several rather complicated transformations are nece_ssarybef o r e t h e s i m p l e f o r m u l a s ( 1 8 . 5 ) t h r o u g h ( I 8 . 1 2 )r e s u l t . I F o r t h e calculation of the field operators, the method of the interaction picture is not especially suitable. The situation is quite different for the S -matrix. Here the series (20.13)essentially contains the result. It is -@
@
S:1 * Zeil" n:l
x[
,?'-1)
2 0f \..1 s ) I a r LI d r ' i . . . j d x , w t H < t t 1 x ' ) . . . H Q(@
_@
_@
_@
F o r w h a t f oal l o w s , i t i s u s e f u l t o t r a n s f o r m t h e e x p r e s s i o n ( 2 0 . 1 5 ) s o m e w h a t .' I f . F ( x r . . . x n ) i s s y m m e t r i c i n a l l v a r i a b l e s , i t i s evident that f
J
brtxn-rbb f -
f
-
O * r a x r . . . d x n F ( x r . . .,x n ) :1 f d, x r . . . f d- . x " F ( x r . . . x *( )2. 0 . 1 6 ) ,t J J J J
&A&aa
The product of the interaction operators in (20.15) is certainly nor symmetric in the time variables, because these do not commute with each other for different times. Despite this, we can transform the reglon of integration according to (20.16) if we require that an operator with a greater tirie always stands to the left of an operator with a smaller tirne. To do this, we introduce a ',time ordering" or simply a " P -symbol " in the fotlowing way:
: P(A(x) a@il) ;:: :t',,:,\ {1\;);"^lT,))
(20.r7)
The generalization ofthe P-symbot to several factors is obvious. The S-matrix (20.15) can now be written as an inteqral from - o to *oo: 91 r :\n f f p ( H | . I ) ( x ; ) . . . a ( , ) ( (r 2 [ ,0) ). )1.8 ) s: r + f -# Jdr;...Jdx? n:I
The expressions (19.23) and (19.24) found earlier for the first apl . S e e , f o r e x a m p l e , J . S c h w i n g e r ,P h y s . R e v . 7 4 , t 4 3 g ( 1 9 4 8 ) 7 5 , 6 5 1( 1 9 4 9 )o; r G . K e i l l 6 nH , elv. phys.Acta22,637 0949). 2 . S e e f o o t n o t ei , p . 9 0 .
gZ
G . K i i l l 5 n , Q u a n t u mE l e c t r o d y n a m i c s
Sec. 20
proximations to the S-matrix are clearly special cases of (20.15). Obviously one can verifyl directly that the formulas (20.14) and ( 2 0 . 1 8 ) s a t i s f y t h e d i f f e r e n t i a l e q u a t i o n s ( t 7 . 1 4 ) , ( 1 7 . 1 5 ) ,a n d t h e r e l a t i o n s ( 1 9 . I 7 ), ( 1 9. I 8 ) . In conclusion, we shall give a proof of the relation (19.Ia) bv means of the methods developed here. We first remark that the s e r i e s ( 2 0 . 1 3 )c o n t a i n s a f a c t o r o f t h e f o r m ( n ' l U ( x o ) l n ): ( n ' l R l n ) d n o ' rt; [,'ot'-n"t";d''
(20'19)
shall therefore supin every order of approximation to U(rr) . ;" pose that all matrix elements of the complete operator [/ have the form (20.19). lf we let ro become very large in this equation, we find ( 2 0. 2 0 ) ( r c 'I S I n ) : ( n ' l R l 6 p p ,6 ( f L - ! o ) , M ' ) * l n ) ) , ") where the matrix R in (20.20) is to be identified with the matrix R in (I9.I3). For finite ro the probability that the system be in state lr?') at time ,o is ,l(n'l(J(xo)ln)l':
tl '(' n - -,-l' R - - l' -z' )' 1-2o^ p p ' J1 ,"r,r, " " d x ' de i ( r i ' - o t t ' t ' - r t : t('2 0 ' 2 r ) l
The time derivative of the quanti;z0.ri probability per unit time:
t, the desired transition
'2 I f" ' - ! l Z I . I Z ) { r p ( *( )t ) , 1 p F()x ' ) }- { V e )@ ) , r p ( (. r ' ) } : { r t t t '()* ) , , p c ) ( * ' ) }
O .( 2 1 . 1 3 )
Now we define the normal product of two or more factors as an operator product where the creation operators always stand to the Ieft of the annihilation operators . We denote the normal product b y : . . . : a n d w e a b b r e v i a l s 4 t o )( r { ' ) ) a s A ( i ) . W e t h e n h a v e t h e normal product of operators Alli (r) , 1. G. C. Wick, Phys. Rev. 80, 268 0950). Theseproducts had already been used by A. Houriet and A. Kind, Helv. Phys. A c t a , 2 2 , 3 l 9 0 9 4 9 ). T h e n a m e " n o r m a l p r o d u c t " w a s i n t r o d u c e d by F. I. Dyson, Phys. Rev. 82, 428(1951).
The P-Symboland the Normalproduct
Sec. 21
YJ
: A ( t ). . .A ( n ) : A @U ) . . .n t +@ ) )I n
+ I /(-) (1) 4t+\(1). . . AetQ- q a*t U+ D . . . ar+)(QI
(2r.r4)
* | tc | (i)A( ) (j)AF)U) . ..4t-)(n)-t i
In (2I.Ia) the normal product is clearly completely svmmetric in all of its variables. It is not necessary to state exactlythe sequence of the 7(+) (or /(-)) operators with each other since, accu n v r ud ri rnrav
fLov (\ 4? rl . 7 t \t t
lLhLaLr er ! n v vnrm r rm r rrrrurLae .
a ur ruar h u r a
statement
is
necessary
for the ?-operators, however, because in (2l.ll) through (21.13) the anticommutators are present. We now give this normal product a well-defined meaning by the conventions
({).. . rt+t : v U)... v @): : e(+) () ... et+t @)+j a. eH (i) eG) @).1 i:r ( 2 1l.s )
* l d . e t - ' 1v4G ) ( i ) r t + ) O . . . r r++. .) .(+e v e O . . . v c , ( n ) . 1 t (2r.rT) : : v O r p ( z ) : r p ( t ) t p ( z ) : y ( r ) y ) ( 2+) :( o l y ( t ) y ( z ) l o,) ( 2 1r.8 ) ,t,U)rp(2): : tp(t)tp(z): = : 9 (1) rp(z): + , (21.te) ,,p(l)rlQ)::tp(r)y(z):*(olrl(r)rp(2)lo>, (zt.zo) ' , t ' U ) r p ( 2 ) : : 1 t , U ) r p((ozj )r :p-(tl ) r p (oz)).f (21.2r) Equations (2I.17) through (2I.2I) can be proved directly from (21.4) through (21.16). As an example of the application of these equations, we shall write the free field current operator as a normal product: I
(x)'y, V@@): + ( 0 | ?(0)(/) y rp(o) (*) lo)-l lqo (t(), yr rp@(x)l : g {: ty (x)lo)y7Sp(x- x') y^(o 1y{o) (x,)| q,) y. x D o @-' x ) .
in the corresponding figure. Obviously the diagram (Fig. 2) gives an intuitive picture of the arrival of an electrofl g, , the emission
Fig. 2.
The graph for the matrix element (22.2).
of a virtual photon, then its absorption, and finally the further propagation of the electron q. One should be cautioned against l. A classification of the various terms in the S -matrix without the use of graphs has been given by E. R. Caianiello, Nuovo Cim. 1 0 , 1 6 3 4 0 9 s 4 ).
r00
G. Kil]6n, Quantum Electrodynamics
Sec. 22
too literal an interpretation of the diagrams, however, since this can easily lead to false results. In general a matrjx element is not represented bya single term a s i n ( 2 2 . 2 ) , b u t r a t h e r t h e r e i s a s u m o f t e r m s , b e c a u s ea l l n o n vanishing normal products must be considered. We can obviously obtain all these terms if we draw a1I the diagrams which can arise by connecting a points with the required number of "external" and "internal" lines. In all practical cases, these diagrams can be drawn immediately. The diagrams which correspond to the n -th approximation (22.1) to the S-matrix can be divided immediately into several groups. Each group contains exactly m! diagramswhich are distinguished from each other only by a different labeting of the variables r' . All these terms are most simply taken into account by dropping the factor a! in the denominator of. (22.I) and actually evaluating only one of the diagrams in each group. A further simplification results if we neglect a1I "disconnected" graphs. A graph is called disconnected if it can be broken into two separate partswhich arp not connected with each other by either electron or photon lines.'
Fig. 3. A disconnected graph.
Fig. 4. A connected graph containing a closed loop.
Thus, for example, the graph shown in Fig. 3 is disconnected, while that ofFig.4 is connected. The basis forthe neglect ofthe disconnected graphs is that their inclusion serves only to multiply all matrtx elements by the same numerical factor. This factor is clearly just the matrix element (0 lsl0), i.e., of the form eid, and therefore without physical significance. The following rules are to be used for transforming a dlagram into an analytical result: l. Each internal photon line corresponds to a factor t 6,,, ?o(ii). 2. Each internal electron line from a pointx'toapoint Nicorrespondsto a factor -* S"(ir). 3. Each external line corresponds to a creation operator if the particle leaves the graph and to an annihilation operator if the I _
(\ rTr qrr ar r nr usL lv ar f o r l sr
p r r vnLive/ )
external lines either! "vacuum bubbles " .
Tho diqcnnnccted
Some authors refer
part is to have no tn
ihaco
narlq
aq
Sec. 22
A Graphical Representation of the S-Matrix l0l particle enters. Note that with our conventions about the ar* rows in the diagrams, a positron is to be considered as running "backwards ". The photons are taken as entering if they are present in the initial state, otherwise they are leaving. 4. Each point rtcorresponds to a factor y,t. 5 . The whole expression is muttiplied by the factor e, (- 7)t+", where a is the number of points and I is the number of closed (electron) loops in the graph. 6. The whole term is multiplied by an overall factor (_1), , where P is determined by the permutation of the factors v@(x\ in the normal product. With the exception of the sign (- {)r in rule 5, these rules require no further discussion since they follow immediately from the considerations of Sec. 2I. The expression ,'a closed loop" of electron lines is quite evident and requires no further elaboratiorL Thus, for example, the graph of Fig. 4 contains just one such ioop. In order to prove the overall sign given in rule 5, we write all the factors :tlt(i)y,,rp(l): which enter the loop next to each other in the P-symbol. This does not affect the sign of the matrix eremenr. We must then evaluate the following expressLon: ,ttUt) y,,,rtt(ir)ii Qr)y,,,,rlt Qr)y (is)yq". . . y,h+ (in).
(22.3)
The contractions can be carried out immediately and give the result * * Sr(i, i,*r) , with the exception of the contraction rp(;r)ri Q,,). The contraction y (r;) rj;(i,1 sives the factor + i SF(i"zl) on account of(21.35). Fromthis the rule for the overall sign in rule 5 follows immediately. In conclusion, we shall discuss one further simplification which can be easily stated in this graphical representation. we can omit ail graphs which contain a closed ioop with an odd number of electron lines. As an example, consider the diagram of Fig. 5.
Fig. 5 . A closed loop having three electron lines.
Fig. 6 . The contribution of this Sraph exactly cancels that of the graph of Fig. 5 .
The assertion is that it may be dropped. for every diagram with a closed loop, which must be added, and in which the is just reversed. To the contribution of
?3486
The basis for this is that there is another diagram direction around the loop Fig. 5 we must therefore
G. Kd116n, Quantum Electrodynamics
IO2
Sec. 22
add the contribution of the graph inFig. 6. For simplicity we shall designate the points which are vertices of the loop by x7,... , x* ' of the diagram with the loop contains a factor The contribution
Sp[7,,sp('lz)y,,Sr(27)...y,,,sr(n1)7'
Q2 '4)
The contribution of the diagram with the reversed direction around the loop differs from the first expression only in the replacement of the spur by (22'5) S p [ 2 , ,S r ( l n ) 7 , * S r ( n , n- 1 ) . . . y , " 5 p ( 2 1 ) ] . In the final result, the sum S p [ 7 , , S p (z1) y , , S o ( 2 7 ) . . . y , , 9 p ( n r ) f *
t\ )- ,- . ^- /\
p S p [ 7 ,S , . ( t n ) y , ^ Sp ' ( n , n - ' t ) . . . y , , s F ( 2 1 ) ] will occur. In order to evaluate this expression, we need a few theorems about the traces of 7-matrices . In what follows, we shall often have to evaluate such traces, so we shall review their ^-nnarJ-ia PI vyvr
c
hara
From the fundamental property of the Trl, * l,Tp:
7-matrices,
2 6 p u,
QZ '7)
we obtain the expression - S p l y u , ) ' , , T h" ' y h ) : S p l y , , y , , l , " " . T ^ f : 2 d , , , , , S PI y , , " ' y , * f n-
I
ttzz.a)
=2Z 6,,,,(-1)' Splv,". . . T,;,1,,*,"' l^f * (-1 )"-l Sply',''' T'^l',1'I i:2 Using the elementarypropertyof a trace, Sp[,aB]: Sp[B /],
(22,9)
and (22 .8) , for n even, we have n
.T^]. Sph", . ..yd :2,6,,"u(- t)' Sp[2,,. .. 7,,-,7,0+,.. Equation(22.I0)
(22'r0)
us to calculate a trace with ?' 7-matrices
"tI.;, 7-matrices are known. In principle, this if all traces with tt-2 result can be used to calculate all traces of an even number of ^fricAc. v/ - m rrrsLrfvvr
For odd r
we write
S p[ ] r " ,... y , , ":f S Ph , , . . . y , , , y ? ) :S PL Z u , .r ., ., y , ^ T s, f
Q2'II)
where 75 is defined bY Tt:
TrTzTsTa,
(22 'I2)
and has the Properties
Tl:r,
(22.13)
Sec.23
The Physical Interpretation of the F-Functions
TsTplTpTs:0,
P:1,2,r,
103
4.
(22.r4)
E r o m ( 2 2 . 1 4 )a n d ( 2 2 . I I ) , i t f o l l o w s d i r e c t l y t h a t Sp hr,,. . . Tol : Sp ly uTn. . . Tat i : e 1)" Sp ly?y,,. . . yul : : ( - 1 ) ' S p [ ) r , ,. . . 7 ^ ] . Therefore, if n is odd, the spur must vanish:
S Ph , , . . . T , , n a :, f o .
\zz. tr) ]
(zz.IG)
Anothersymmetrycondition for traces is often useful: Sp [)r,,. .. y,^]: sp ly"^y,^_,...7,,T,,f.
(22.r7)
Equation (22.I7) is non-trivial only if n is even. For n:2, it is identical with (22.9). Using the expansion (22.I0), the proof for n+2 is readily obtained from the result for n. The general expansion (22.I0) gives the spur of n y_matrlces as a sum of n(ro-z) (n-4)...4.2 t e r m s . B e c a u s et h e r e a r e o n l y four different 7-matrices, related by (22.7), it is to be expected that considerable simplifications are possible in the results for Iarge n. Actually, it can be shownl that such simplifications occur for n>r2. we shall not pursue the matter further because ln our applications there witl not be traces complicated enough to make these simplifications important. F r o m ( 2 2 . 1 7 ) , ( 2 2 . 1 6 ) , a n d f r o m t h e s y m m e t r yp r o p e r t i e s o f t h e functions /n@),
/o@ - /) : /r@' - x) ,
ii /"@ x') : *. Or(*' x),(22.r8)
one can readily show Sp[y"So(12)y,,SoQ3).../,,5p(nt)):
] : ( - 1 ) ' S p [ 7 , ,S o ( 1n ) y , , S p ( n n, - 1 ) . . . 7 * S e ( 2 t n . l Q 2 . I g )
If n is odd, the sum in (22.6) therefore vanishes. We have now proved that all diagrams with closed loops having an odd number of electron lines may be dropped. This result was originally proved by Furry,2 using a completely different method. 23. The Phvsical Jnterpretation of the F-Functions A characteristic feature of the above calculations is the appearance of the functions Dn@- r') and So("*2,) in the matrix elements of the S-matrix. In the original integration of the equations for the field operators, we worked only with the retarded functions. The latter have the obvious interpretation that the value of a field operator at a point r can depend only on quantities at points a' for which the times x[ are smaller than the time xo , IFD
Caianiello and S. Fubini, NuovoCim.9,12ig l2S (1937).
2, W. H. Furry, Phys. Rev.5l,
(igS2).
G . K i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s
I04
Sec' 23
phenomena, This is a natural consequence of the causality of all in complete correspondence with the classical treatment of a field theory by means of integrals over the retarded light cone ' In our expression for the s-matrix this causality has apparently been tost. The integrals over the variables r extend over the entire spaceandtheF-functionswhicharepresentdifferfromzerofor bothsignsofthetimedifference.Despitethiswecanreadily convince ourselves by means of a simple example that the F functions have a "causal." interpretationunder certain conditions. Of course it is not expected that the S-matrix will have as simple a structure as the fietd operators, since the time sequence of two integration points is not fixed from the beginning,' We choose the following term in the S-matrjx' as an example for the analysis of the F-functions:
(r')y^,td (x')| q") x a- ax"(ql rtt\o\ (q,q,| ',il |q,,,q"') : i [[ I O, . r, q"' "'' (x") (q.' (x") y * ) ^'!)o | ) x Dp(x' x") l rtd clearly this integral is obtained in calculating the cross section that two electrons with energy-momentum vectors q" and g"' collide and emerge with new energy-momentum vectors q and q' ' It is a photon straightforward interpretation that the electron q" emits a point t" the g' At momenturn at the point x' and as a result has its consequently and electron other the by the photon is absorbed momentum is changed from q"'to q' , The total probability of this transition contains the sum over all possible combinations of x,' and x,,. serious objections can be made to this naive interpretatiorL ThefunctionDo@)doesnotvanishforspatialseparations;not propeven approximaiely. The emitted photons must therefore be agatedwithspeedsgreaterthanthevelocityoflight.Followlng Fierz,l we shall show that a similar interpretation of the integral ( 2 3 . 1 )i s s t i l l p o s s i b l e i f w e c o n s i d e r n o t e v e n t s a t s l n g l e p o i n t s ' but rather the probability that the photon is absorbed or emitted in a finlte region of space-time R. In order to pursue this idea, we break up the whole four-dimensional space into regions R and rewri-te the integral (23.I) as
( q ' q ' ls " ' l q "' q " ' ) : r / n "a "I " ' '
(23.2)
R"R"
In, p,,:l
)tl
I a*' I dr"(qlrtot@')r^rto)(x')lq")x J
R'
R''
,,, ,,
x D,(x' - x") (q'lrro)@")r^rto)(x") lQ' >' ]
For the following discussion it is not essential that the operator yldlx)be developed in terms of solutions of the Dirac equation field. The only property which with an external, time-independent l.
M . F i e r z , H e l v . P h y s . A c t a 2 3 , 7 3 1( 1 9 5 0 ) .
sec. 23
The Physical Interpretation of the F-Funotions
105
we shall use is the well-defined energy of each state lq). Eram +!r^ ih+^^--l
rru* Lr'elnregrar representations of the functions Do@\,D*(x), and D1(x) ,
D p ( x ) : ? : . " [ 6- 'p- -r ; n ' { p l ^ +' i 'n- "6" (t "kt )z \ \ i (2n)aJ ,r. t
(23.4)
Do@)::-
(23.s)
[^
t d k -e i h , { pA! ., +r -i rnv e v \ (' 'k l ")t d ' ' t(, lA t ' ) .}
QnY1"-
DnQc)::=, \2n)-J
[-
f d h e i h , { p * _ t n e ( h \\6, -(\ h. r z1 ,\ \
(re ^\
\LJ.u,,
|.h.
we find the following relatlons upon taking the positive and negative frequency parts of these three functions:
DF)6):iia,
- }o , n 6 ( n ) l : 1 o f t , ( ,()2, 3 . 7 ) I a n ' , 0 , { n+
n F ( * ) :? t r * r " ' io u , , u , {}o; * i n d ( A , :) } 1 o k , ( , )(. 2 3 . 8 ) io(0 We have, therefore,
t on@):DkI @)+ D? (x).
(23.e)
Introducing the relation (23.9) into (23.3), we find . , tr n'n" : -r t'" ( " ,dx', ..l' dx" (ql V$)(x')yxy)tot 1,1 @,)| q,,) DP @,_ x,,,) \l?, , x . R"
l " ) y ^ y 6 t( x ' , ) x ( q ' l 1 p t i@ 1 q , , , t) -f d,x' I [ , d . x " < q l r p h l p c , ) y ^ y ) @ @ ) 1 q , , ) D- t ) x( x, , ) x x (q'l tptd(x") y^rrot@") lq"'>) '))
I
I l(23.10) |
T h e t l m e d e p e n d e n c eo f t h e f a c t o r < q l l p @ Q t , ) y ^ V h ) @ , ) 1 qi,s, ) g l v e n by e-i"i@l'-qi . A similar expression results for ir," btir", *atri* element. In the tlme integrations in (23.10) the limits of integration are not * oo, but ?,and 7,, , the limlts of the regions R, and R". If we require that the two times I,and T,,are veiy large com_ pared to the times lq'i - qol-, and lq(' - qi l-1, then the time inte_ gratlon is "almost" a delta function. According to (23.7), the first term of. (23.10) gives a contribution only if
ao- ali: q:;'- q;> o.
(23. u)
With these assumptlons we have Io',,p" av-i e2[.dx' d"" : - eJ d,x(q' lptot(r) | o) 7r(0 I ?(0)(r) I s) A^;""(x).
(24.3)
The matrix elementspresentLn (24.3) are given by (o | lpb)(x) | q): + u?)(q)edc" , vv
(24.4)
where the functions w(')(q)are given explicitly in Sec. 12. Since we are now studying the scatterinq of electrons (and not positrons), we must take the lndex z in (24.4) to be I or 2. These two possibilities correspond to the two possible orientations of electron' spin as was discussed lnSec. 13. For a time-independent, external field
A|i*(*): At""(n),
(24.s)
we now obtain
(q - q'). 2n 6(qn- ql,),Qa. 6) (C,lS Iq>: - t att 1n,yr u?t(q)A7|"" wherethe notation (24.2)
At*e) : ! dsx Af;""(a)eiea
has been lntroduced. Accordlng to (20 .22) , the transitlon probabiltty per unlt tlme is then
: (q.- q,)lr. 2n 6(qn- cL) rn,Touv)(q)A?,""" + : + lav't
(24.8)
: - zn$ao (q)yuuv)@)av) (q)y,un(q,)Aii*@- q,) x x Ai"""(q' - C) 6 (So- qi).
To obtainthe last form in (24.8), we have made use of the reality properties of the potentials A?|"'(*). Since AT""@) are real and Ai*'@) is pure imaginarY, we have 'l (avt 1O', yru?t (q),4i'* (q - g'))* -- ux(r)l1l rrrnu{"t (q')x x,4i""(g' - q) - u*t') (q)ua')(q')1i'o(g' - q) : fi (2a.0)
: _uo)@fuuv,r(q,)Afi""(q,_q).
)
To avold resetting the equations, we @") r4ass for the external field. shall follow the German and use the out-field. from 7(aus), This is to be distinguished
Scattering of Electrons and pair production
Sec. 24
I09
From (24.8) we obtain the cross section for the scattering of the state lq) with polarization / into the state lq,)with polarization r' after dividing by the number of incident particles per unit time
and per unit surface area
d,o : -
#
t
or; wn(q')y, u?)(q)no@)y,ot't(q1) x if x A?,"* _ s,r) (q- q,),Ai""" (q,- q)6(qo da. ] , , n . , 0 , W)*
I
tn (24.10) we have required only that the direction of propagation of the outgoing particle lie in a given solid angle and we have s u m m e do v e r a l l e n e r g i e s p o s s i b l e . T h e l a s t f a c t o r i n ( 2 4 . 1 0 ) i s t h e n u m b e ro f s t a t e s p e r e n e r g y i n t e r v a l d i v i d e d b y V l c . t . ( 1 . 4 ) ] . Because of the delta function, the integral can be evaluated imm e d i a t e l ya n d i f w e d e f i n e q , o : A L : E a n d thenwe lq):lq,l:p, find do e 2E 2 - 1 " , , , , , -ai, : - .t671, ao''(o')r,uo,@)n?) (q')x @)r,,wa't X 1i*'(q - q')Ai"""(q'- q). |
,'n'",
If we wished, we could introduce the explicit form for the funslrons uvtQ) and work out the cross section for polarized electrons. In most experlments, however, the incident electrons are unpolarlzed. That is, we must take the average over both directions of polarization for the incident electrons. In general, the polarization of the outgoing electron will not be measured either, and therefore the sum of the cross sectlons for the two polarizations must be calculated. This sum and average can be worked out directly in Eq. (24.II) without havlng to use the expliclt forms of the functrons ot\')(q), which depend on the particular representation employed. Qtr m a as n. . vc rrrv
nf vt
uf
oo
ao
:- --
Pn
uY
.
11, \r!
''L-"-
.
L2),
we
have
1
t,t | ' -t,\, ' S \ - - - r , )' r, - ' , lq')ltuv) (q)u\1) 1n1 r,,wr'')(q')x A Ar"" (q - q') A?,""" x A?,""" (q,* q) :
(2dr T
(24.r2)
: -
TAd, sp ly, (i y q /n)y,(i y q' m)l x x Ai"" (q - q') Al"'"(q'- q) .
The spur in (24.12) can be calculated immediatety by the use of (22.I0) and the relation
sp[I]:a.
(?a 1"\
We obtain
!r; : -
$
(q.-q') A?" (q'-q).(za.ra) La,,{tq'* m2)- q,q;- qLq) A?,""'
Sec. 24
G. Kall5n, Quantum Electrodynamics
It0
Equation (24.I4) is the desired result. The general result (24.14) can be specialized for different Perhaps most important is the case forms of the external field. Here we have of an electrostatic Coulomb field.
A7"'(n): - fj; na,,r.
(24.15)
We obtain the form in f-space from (24.7):
A 1 " \' Y, (t o \ :- i" 6 " p.4. ^ 7 1 .
Q4.16)
e2
In this case p we have
q'and letting
Q'= q'+q'2-
2qq'-
angle,
@ denote the scattering t)
2 p ' ( l - c o s @:)4 p 2 s i n 2 " .
(24.r7)
After substituting in (24.14) and simplifying, there results
H : |^
'""?" * P*1, l'(' ** i7
(24.r8)
Idxrp"sn- 2l
Tr r hr firc,
fL hr reu rr e v r fvo r er ,e
-
ie
r u
tha
ralaiirzicfin
-onefaliZatiOn
known Rutherford scattering cross section.r then it follows from (24.18) that do dQ
I I
Zt' ^
| 16?IlltSrtj*
Of
In fact, if
WelI
the
f'11*'
,
-12
(24.te)
^Al'
I
where *D&_ r
h2
2m
(24.20)
kinetic energy of the electron. Equation is the non-felativistic (24.L9) is the familiar Rutherford cross section in the Heaviside units employed here. A similar problem, whose solution is to be found from the previous formula, is the probability of pair production b:f gUg-ak, external field. Again if the field can be treated in Born approximation, the S-matrix element of interest is given by equations similar to (24.3) and QA.6)z
Q , s ' l Sl 0 ) : - eI d x ' ( q l r t o t ( x , ) l o ) y , ( q , l v {lo0@ ) . 4, )f i " " ( r , )I: : _
_ 7A0 (q)yrut't(_ S;)A";'"(_ q q,) €
-r..t
,
,
I
In (24.2t) lq')is a positronand lq) is an electron. The index r is therefore equal to I or 2 and the index z' is equal to 3 or 4. Furthermore, the external field has been assumed to be time dependent and we have introduced the notation
1. Equation (24.18) was first derived by N. F. Mott., R o y . S o c . L o n d . , A 1 2 6 , 2 s 9( r 9 3 0 ) .
Proc.
Sec. 24
Scattering of Electrons and Pair Production
,4i*'(8): I d* Ai"" 1x1 e;a'.
I11
e4 .zz)
In (24.7) the integration was only over the three spatial coordinates; here we integrate over the time coordinate as well. According to (i9 .7) the probability for the production of a pair lg, g')
)i aau,s'ls I0)1, d8 d,q' -,r,r-!-ar,) (s)y, uv') (- q') x {rnye
(24.23)
x w?')(- Q')y, ,r, (q) Aii,, ? q _ q,) A,i""'(q * q,) dsq dBq,. The total probability
that a pair is produced at ali is therefore
q'- m)y,fx &, .[I ^r*#sp [(tzs m)y,(- i y
- :
x -4i*'(- S - S')A?""(q * q'):
:
#
(24.24)
d.Q Q)Ai*,e) , I r,,,(Q)-4i*'(-
where
r* " (Q ) : t t +: t'::q ' I": s p[(i zq-m)yp(-i yq,-*)y,]6( Q- s- q,) : :q'' J J : -
) - q , ) , 1 m r ) O ( s , ) O ( Qq-' ) x I d , q ' 6 1 q ' z + ?6n(2( Q x Sp [(i 7 (Q- q')- m) yr(i y q' -t nt)y,f ,
@(q): I ft + u{t)t.
(24.25)
(24.25a)
In (24.24),Eq. (I2.I2) has been employed in the summationover r a n d E q . ( 1 2 . 1 3 ) i n t h e s u m m a t i o no v e r / ' . T h e t e n s o r 7 ) " o b v i o u s l y has the transformation properties specified by the indices p and I and depends only on the vector 0. It must therefore be of the following' general form:
Tr" : A(Q\ 6r,+ B (Q2)Q*Q,: 7,,.
(24.26)
Now from
Qr SpLUy Q - q')-n)TuUr r' I m)y,f: +qil(Q-,r' )z+ *\-\ 1zt. zz) - 4(Q,- q!,)lq'' + *'l ) follows
T,Qu:O:Q"lA+BQ'1,
(24.28)
B (Q\ IQ, Q,- 6r,Q'],
(24.2e)
and hence
7,,: with r
Tuu^:1-3s. r e , . l d a '6 ( q ' ' +* ' ) 6 ( ( Q q ' )+' / n ' ) x x@(q')O(Q- q')sp t(ty (Q- q')- *) yuUy q'+ m)y,l.
8 3,2\:
It,n.,or
Sec' 25
G. Kiill6n, Quantum Electrodynamics
lI2
is the The spur in"(24.30) can easily be worked out and the result following integral tor B(Q2) z
B e\=#
o(q'I@Q- s')8lQq'-*'l=\ , *216(Qz-2Qq') Id.q'6(q'21! (za'st)
- q ' ) J- ' # " ' l ) : - .l d q ' b ( q ' ' m + ' )6 ( Q ' - z Q s ) O ( q ' ) O ( Q tf
WecomputetheinvariantfunctionB(Q2)inthatcoordinatesystem is where the spatial components Qs vanish. This assumes that Q immedifollows it time-like, but from the delta function in(24.31) ' ately that the integral must vanish identically for space-Iike Q find we In this waY
'[]l[a't'4v*Iffi* B(-Q,i=1 [,*
']i.I I
x6(zQrll(,1*'-ai):
i
(24.32)
t7 - .I cw?l - 4m') o(Qo) : ,I f, t -lit * z o(QZ + !&"lz"lf
Therefore in an arbitrary coordinate system we have I n2 m 2rz11t .;,' o (q ol- "0-nz).Qa3z) r,,(Q):+2 n l -lr 6l + Q\ \Q Q,-6,, u v,
in By introducing the external current and using the symmetry probability as total space, we can write the
*:la
[oOrf
lg
Q)Iltotlg'1 ,
Q-
Q4'34)
: +"(, - #)[ +So ? + - *'), (24'3s) (Q,) n@) A1:*(Q), ii*(0) : (6p,Q',-QuQ") Q u l i " " ( q :)o '
(24.36) (24.37)
another In a later chapter we shall again find these equations in -*'') connection. Here we remark onlv that the factor O(- t the obviously expresses the condition that the energy given upby system' coordinate every in greater than 2m external field must be 25. Scatteri.ng of Light bv an Elqctron shall work out the cross section for As the next exampGlwe for the scattering of a photonby an the Compton effect, i.e., The first non-vanishing approximation to the S-maelectron. trjx is clearlY
Scattering of Light by an Electron
"'
( o ' . h ' l S I A .a ')
2 J [[ J
1I3
- r')x or'd*"(q'lrro)(x")lo)y,,S"(x" I I
(x')lq)[(oI Ato) xy,,(olrto) @')lk> rs according to (29.6). This is also necessary if the induced current is to depend on the value of the external current only within the retarded light cone. If this were not the case, the theory would clearly be "acausaI". From this property of the function (29.27), it follows that the integral G(p, Fo+ ili : I dxG(x) e-d(pa-Qo+i't,no, exists if 4 is greater than zero. The function is an analytic function of a with no singularities plane. By the well known theorems of analysisr tion, the real and imaginary values, on the real relation
(29.28)
G(p,z) in (29.28) in the upper half for such a funcaxis, satisfy the
+@ -R_ e - ,G \ r (, rov.b l ^\:1
n-
o f J
-€
rmc(P'x)dx x_po
(2e.2e)
Fromthe imaginarypart (29.26),we can thereforecomputethe real part by a simple integration.Using the notation n@)(Pr)introduced I. See, for example, B. A. Hurwitz and R. Courant, Funktionentheorie, Third Ed., Berlin, 1929,p. 335.
Sec. 29
Vacuum Polarization,
Charge Renormalization
137
in (24.35), we find
(2s.30)
ImG(p) : 7ue(p)n@)(pr),
R e G ( / ) : p ' ( - ' ' o ' ,xo-'P- "o t J lxl
t. d r : p i r r r , l o" r - " r '1l xl - P o + x *!P.o] la r = ) J
**
I
*o
- x 2 ), 1 a z \ : p - ' Jo- xf qnu@ r -' ,( -p' z. 1
ltzs.trt
IrQ\(-o\.do :]I6)(p2) , aapz-=
I .
I ,
a
n t i l 6 z 1: p
f
J 0
no)(-"1a"
(2e.32)
p2
a*
Hence we have
G ( p )- G ( 0 ) : f r @ ) ( p -, 7 ) or(0) -tine(f)n(o)(p2). (2e.33) From (24.35) we see directly
that the function II@) (- a) has the value
12 irz
does not converge and the functions n*o(pz) and G(1) actually do not exist. This is clearly aresult of the summation over an infinite number of states in (29.2I). Untit now, we have not paid attention to the convergence of the sum. One might therefore suspect that the theory is not capable of treating the problem of the induced current. If we ignore all this for the moment and just write down a formal expression for the difference II + + II d,x'dx"Kr,(*
and
(r),TpSp(x-x') A @')ll q,q'>+ es(o I ift (x)lI, Q') : :: ax'. rnL(qu* sL)Q | :rp6)
(33.r)
Using the Djrac equation for the free electron, we can write (33.I) as follows:1
(33.3)
or,: t (y,yu- yry,). We see that the electron gets an anomalous magnetic cause of its coupling with the electromagnetic field:
moment be-
(33. 4) ,* A i i ' "( Q )z n6( Q s ) ,
( es. z1
where
- R(0)(0) (q li,,lq') : (q li(f)lq') j - fr rot l 11(0)(0) + R(0)(Q'z) + 1gz1 I,r, - s(0)(0,)j; Ar1 n;12(q\t-V),) 1 Sioi(o)l Q:q'-q. In Eq. (35.3)we have used the fact that the vector Q is Iike, so that the " unbarred " functions R , S, and 11 Using the methods explained in previous sections, we cross section for unpolarized electron scattering in an field:
.,, ( 3 s. 4 )
spacevanish. find the external
1. The second Born approximation in the external field has been calculated by R. C. Newton, Phys. Rev. 97, 1162(1955). See also R. G. Newton, Phys. Rev.9B,1514 (i955); M. Chr5tien, Phys. Rev. 9 8 , I 5 I 5 ( 1 9 5 5 ) ;H . S u u r a , P h y s . R e v . 9 9 , 1 0 2 0 ( 1 9 5 5 .)
Sec.35
#
:
The Infrared Catastrophe
163
-a tu\ ror @r -n0 can be taken in the total cross section. In order to compute (35.11), we have to work out the integral
I : J analn'+p')o(k)r;rV: t l.oiu
: ' f ,4 4 - f !
/8
r
-
oau
,
yna+r,, J (pk- EVFar9 @,n- nll@Jp\
f
zJ oto
:-
(3s.12)
1
kzdlkl
f ,
dQ,
f
^r-.
J'*J
IF;7
itp+9") k-E\?tr1p2_z
For the last transformation in (35.12) we have used the identityl I
lfdu
(3s.13)
t-
a.b
J
lau*b(l
0
11-1sA-integration and gives
in
(35.12) can be done by elementary methods
I
r -c* r -'ru
f du !'t^o2/E J ,4 t^"" tt u
E tno" L '. '>"t @ - J ' " o
I
.-
tVL'-A
I
|,
( 3 s. 1 4 ) (3s.r5)
A:m2lQza(l-a).
B y t h e u s e o f ( 3 5 . 1 4 )a n d ( 3 5 . 1 5 ), w e c a n w r i t e ( 3 5 . 1 1a) s f o l l o w s : do(o) e2 f, 2AE -TTZFltoE r
ddstrabl --7d-:
tr:
-
I
t,
rI
J
l-, -
-l rzl '
( 3 5. l 6 )
"t - t
l
(3s.17)
),
02
mzaaz
r,2=.la.d lA1-6a, *, 0-
i.d e f *++ gr^, =,
-
E tllEj-frii77"EF *\. F lr
I ^^-
-
E_ r
IF_gu-g*"1 T.
I.-A
2P'"o E+p
- 1,.. ,o'
ll'""""'
T h e a - i n t e g r a t i o n i n ( 3 5 . I 7 ) c a n b e done easily, although the integ r a t i o n i n ( 3 5 . 1 8 ) i s q u i t e t e d i o u s . 1 A / oa i r r e n n l v t h e r e s u l t s : Y v v
l.
Y r v v
R. P. Feynman, Phys. Rev. 76, 785 (1949).
The Infrared Catastrophe
Sec.35
2,rtz t, _rtr14\_
r\z lt, ",_,/,^,VTry_l - f
o,=--Zl '(-F V, y r'*@/-t-u\'"-re-') r
. E-
E+b
*IoE B_p
I' ' L 2 m z
lll@. ub Bn - .V r u'g' 1 - ; l L r c *rE
vt*-F
vt
-
e"
' '
l
+, L+(Et ,t\r- ! ,, \ t
e, I
at_l
' : ;kgt._? (#)')-'(- ffi))* - bg++ r"rs##1,
+tq# bgYig!{
@\ ^l P I t + s t n -- '1> ' t ' f;: -r t @\ ^f F{t A stn-1 y:--------=:11 rTU
(3s.20)
'
t
,,, |,,, (3s.22a) r?q ?rh\
d:
(3s,22c)
P:*.,.
("E,
ttA\
T h e f u n c t i o n @ ( x )u s e d i n ( 3 5 . 2 0 ) a n d ( 3 5 . 2 1 ) h a s b e e n d e f i n e d i n ( 3 4. 2 I ) . A d d i n g ( 3 5 . 8 ) a n d ( 3 5 . 1 6 )i n o r d e r t o g e t t h e t o t a l c r o s s s e c t i o n , t h e m a s s p cd r o p s o u t o f t h e r e s u l t b y t h e u s e o f ( 3 4 . 1 8 )a n d ( 3 S . 1 9 ) . '9It i. therefore independent of the phoThe observable quantitv dQ ton mass and the role of the cutoff has-now been assumed by the resolution of the measuring apparatus.r For very small values of 1JE , the probability of emission of many photons has to be- considered also, as Bloch and Nordsieck and Tauch and Rohrlichrhave 1. The first discussion of the infrared problem was given by F. Blochand A. Nordsieck, Phys. Rev. 52, 54 0937). See also W. Pauli and M. Fierz, Nuovo Cim. 15,167 (1938). The treatment used here is due to ]. Schwinger, Phys. Rev. 76,790 (1949). A similar discussion of the higher orders of the problem has been given by J. M. lauch and F. Rohrlich, Helv. Phys. Acta 27, 613 (1954). See also p. 258 of the book by these authors.
166
Sec. 35 d?It re-
G . K i i l 1 5 n , Q u a n t u mE l e c t r o d y n a m i c s By these considerations, it can be shown that
shown.
dQ
mains finite even in the limit AE-->0. In this problem we can regard the motion of the electron as given, so that this is really a problem of the interaction of photonswith a classical current. Just this problem was already discussed in Sec. lland we shall not repeat the details here. At least for the accuracy of measurement presentl_y attainable, such a refinement of the theory is not neces sary. * The complete expression for the total cross section can certainly be obtained from the above equations, but it is rather comnlinrtarl \A/a fharafnge restrict ourselves tO the extreme relativistiC limit. Under these conditions , the cross section simplifies to yrrveLev.
dotot
dt)
doro, -_ --aT ,\t^ -
with (1, f,s d" : ; 'etzl t " g {/ 2; lEsl i " :@ l Jl ' - ; lr[l o
E
rE
-
A\ u)t
(3s.23)
131 17
1
; l + ; - ; s i n 2 ; I"lO , l
(, ^3 s . 2 4 )
and -r -
r
| / n\ -2 / l o ) l - s i n 2 : ) + + - l o g { s i-n- o 2 \i l- l^o' -SO l cto, s z |: - " O ,lll . / : ) l - I' " - ' 2 ) ' t 2 2J--or"-- 2Jl' 2sinzi' 2
3 !S . 2 5 ) \( u
or nf2, the function I For special angles, for example, @:n can be evaluated exactly by means of the formulas given in(34.22) through (34.24). For arbitrary angles there are detailed tablesz of the function @. That a finite cross section results after the renormalization of charge and masswas noted by several authorsJ in 1948. The first careful calculation of the cross sectionwas carried out by J. Schwinger4 and we have used what is essentially his method. A more exact experimental verification of Eqs. (35.23) through (35.25) is still lacking. Certainly many measurements have been made of the scattering of high-energy electrons by nuclei.5 First, the accuracy of these measurements is not high enough so that the results can be said to confirm the theory. Furthermore. composite and for these the spatial extension nuclei are usually involved, lndeed, of the nuclear charge distribution has sizable effects. the deviation from Eq. (35.7) caused bythe nuclear size is much
l . S e e ,h o w e v e r , E . L o m o n ,N u c l e a r P h y s . 1 , 1 0 I( 1 9 5 5 ) ;D . R . Y e n n i e a n d H . S u u r a , P h y s . R e v . 1 0 5 , l 3 7 B ( 1 9 5 7.) 2 . S e e , f o r e x a m p l e ,B . K . M i t c h e l l , P h i I. M a q . 4 0 , 3 5 1 ( 1 9 4 9 ) . , r o g r .T h e o r . P h y s . 3 , 2 9 0 ( 1 9 a 8 ) ; 3 . Z . K o b a a n d S . T o m o n a q aP H . W . L e w i s , P h y s . R e v . 7 3 , I 7 3 ( 1 9 a 8 ) J; . S c h w i n g e r , P h y s . R e v . 7 3 , 4 1 6( 1 9 4 8 ) . 4 . J . S c h w i n g e r ,P h y s . R e v . 7 6 , 7 9 0 0 9 4 9 ) . S e e a l s o L . R . B . E l t o n a n d H . H . R o b e r t s o nP, r o c .P h y s .S o c . L o n d . , A 6 5 ,1 4 5( I 9 5 2 ) . (
Soa
fl vnr r
a vamnla uzrqrrryre,
JL -r rhs o
nnmnrahanqirra vvrlryrv
R e v . M o d . P h y s. 2 8 , 2 I 4 0 9 5 6 ) .
rarriarrz
hrr
v1
P
r\.
T{ofqiedfar
rrvleLsglv'
Sec. 36
The Hyperfine
Structure of the Hydrogen Atom
167
larger than the radiative corrections of (35.23). These measurements are therefore more a determination of the nuclear charge distribution than a verification t\4nrc roconrlrz lhsss -;; of the theorv";;";;";;;"
havebeen
r,trT;t
;;i;;
;il
;;";i:;r""'
by protons "*p"ri." and here the charge distribution is not such a big effect. The experiments have been done so as to determine first the structure of the proton from the angular distribution of the scattered electrons. Then the results can be used for a comparisonZ of the theoretical and experimental values for the total cross section. With an electron energy of about 140 MeV, Tautfest and Panofsky obtained
o""p/o,n"o.:0,988+O,021. Since the theoretically
important
is nf nrrler nf maonitud" ]tog!-, n-'m
/?R
part of the radiative i.e.,
,A\
corrections
about 4% at I40 MeV, the
experimental result (SS.Z0) cannot be viewed as an exact verification of the theory. At least these results are not in contradict i o n w i t h E q s . ( 3 5. 2 3 ) t h r o u s h ( 3 5 . 2 5 ) . 36.
The Hvperfine Structure ofthe HvdrogenAto* As the next application of the results derived in Sec. 35, and as preparation for the discussion of the Lamb shift of Sec. 37, we shall now study the hyperfine structure in the I s state of the hydrogen atom. We are therefore considering a system in which the electromagnetic field is the sum of the external Coulomb fieid of the proton, the field of the magnetic dipole of the proton, and the radiation field. we take the electron in the electrostatic field as the unperturbed system with energy levels given by the Dirac theory of the hydrogen atom. We are concerned with the splitting ofthe Is level underthe influence of the magnetic dipole of the proton, and corrections arising from the coupling of the electron to the electromagnetic radiation field will have to be considered. First we shall consider only the magnetic dipole of the proton as the perturbation and, from the elementary principles of quantum mechanics, we obtain the following expression for the splitting of fha
anarntr
Y7
rl au -v' va fl ., .
6E : - [ d.3x(nl ir@) ln) At r (n) ,
(36.I)
: +# Af;ae(r) Here pr is the magnetic
moment of the
(36.2) proton and
( " 1 i , @ )l n ) i s
1 . R . W . McAllister and R. Hofstadter, Phys. Rev. 102, 851 (1956)E ; . E . Chambers and R. Hofstadter, Phys. Rev. 103, 1454
(1es 6).
2 . G . W . Tautfest and W. K. H. Panofsky, IJOc' (IYb //.
Phys.
Rev. 105 ,
G . K i i l l 6 n , Q u a n t u mE l e c t r o d y n a m i c s
168
Sec. 36
the expectation value of the current operator in the unperturbed For the "unperturbed state" we must state under consideration. Since an exact exconsider the electron and its radiation field. as an pression for this expectation value is clearly not possible, we shall develop it in powers of the additional approximation, radiation field. We cannot use the results of Sec.34 directly befield is strong enough so that it must be cause the electrostatic In principle, we treated exactly in every step of the calculation. can do this by transforming the differential equations for the field
operators, la\
Vz; + m)v@)i ey (Ast'"rt(x) f :
,{coulomu (r))',p(*),
(36. 3) r?A 1\
J Alt'^ht (x) into inte gral equations :
(36' 5) p(x')dx' , v@): rp"(x)- ie J so@,*') y trst'at'r(x') D o ? - x ' ) l r i ? ' ) , v , v ( x ' ) l d x ' .( 3 6 . 6 ) ) ': , 4 f " " \ r (:x )A ' f ' ( r + I In (36.5), yf(x) is the solution of the equation * *)v'@):
Q*
i g v t r c o t t o m o ( r ) r t," @ )
(36.7)
and S^ (x, x') is given by t a \r )* ( x , x ' ) : m ) 5 " 1 x , x ' )- i e y X c o u t o n b (S l, * + So(x, x'):g
,
for
- d(, - *'), (36.8)
xol xt
/? A
o\
This function is a singular function of the same kind as was studied an explicit calculation of this function in Sec. 16. In principle, is possible, since the eigenvalues and eigenfunctions of (36.7) are known. Up to now, this has not been done. Next, if we consider the radiation field in (36.5) as a small quantity, we can iterate these equations in the usual way and thus obtain the first This correction is non-zero correction to the current operator. exactly what results from (31.2) throuqh (31.4) if the singular funcSa(*-*'),and S ( 1()x - x ' ) a r e tions of the free electron, So(r-x'), replaced everywhere by the corresponding functions for the bound S n @ , r ' ) , a n d 5 t t ' ( x , . r ' ). I n p r i n c i p l e , t h e S*(x, x') , electron, expression obtained in this way takes account of the Coulomb field exactly. This procedure leads to
dE:
d E r o*r d E , t r ,
( 36 . 1 0 )
: - + f #x(nllrt"(r),y,rtc(*)lln)Af,s(n), (36.l1) dE(o) 2.1 d r ( l ): -
[
a ' r 3 1 6 i * @ ) l nA) f s ( n ),
/?A ir\
Sec. 36
The Hyperfine Structure of the Hydrogen Atom
169
(x),yuSo@, x')@"(r')l'r ::[ax'76c(x')Sn(x' , *),yr.t"@)l airt l : * Idx' lrt'c --'i- a- a*" lrtc(x'),y^(st1r1z', x)yrSn(x,x") Do(x"- x')I [ /2A
r1\
+Sn(x', x)y,,S(tt(x, x") D^(x'-x")+ So@', x)yoSo@,v") Dtrt(x'-x"))y^tp9(*") (r, r')y,S(r)(*', r)] iltc" (Sp +u:: [[ d.tc' L'P \ T [z,So 4 JJ + Sp [2, 5ttt(x, x')y,St(x', *))) D^("' - ,")lrtt" @"),y,rt" @")f .
@c(x): -t
x')Dn@'-x)*So(r,x')Dtt(x'-x)fy^rp'(*')d'x'.(36.14) Ir^[S(rr(r, We beginby evaluating(36.11). From(36.2)we find d E ( 0 ) :- # l
Fxu,(r)rnun(r)VP.
(36.15)
of the time-independent Here wn(u) is the eigenfunction equation corresponding to the state Iz) : H;o' ,
ei'tr(') ,
(3B.10)
and obtain
H: Hio'+ H f )t H c * n n + J ; d , * l ( . . ) x @ * )( . ) i @ ) 1 , ( 3 8 . 1 1 ) I
Hr;:+I'""les#!e#*ry\Wl r
f
| ^J
t-,\
Ht': + I a'xlv (rr* + *\,p@)), @), Hil:
-
f
;"
: | d|x.ilu@)io@) +
,
(38.11a) ( 3B. 1 1 b )
r
(38.11c) | dtx.{o@)lrp(r),yrve)1.
instead oi trr. 2). Forpf,y.l"uify interesting states, for which the subsidiary condition must be satisfied, the last two terms in (3g.ll) have no significance and can be dropped. In this way the effect of the longitudinal and scalar photons has been replacedl by the Coulomb energy (38.9a).By this procedure,the explicit relativistic l.See Sec.B where it was shown that these degrees of freedom of the field can have no effect for free photons. The method grven there for treating the state vector by means of a limit must actually be used here also in order to eliminate the last two terms in a consistent manner.
G . K t i 1 1 5 n ,Q u a n t u m E l e c t r o d y n a m i c s
I84
covariance has been lost, troublesome.
but for our applications this
S e c . 38 is not
b. The Unperturbed Problem For the unperturbed state of positronium we write the state vector in the form
@ @ ,q ' ) l q ,q ' ) , l z ) : qX ,q
( 3 I. r 2 )
and drop the perturbation Hfr and the last integral in (38.11). The symbol lg, q') stands for an eigenstate of H(0' containlng one elecThe Schroedinger equation tron-positron pair.r
qi)=l(ae.rs) o(q, s')lq,q')+ Zlq, q')(q,q'lHclq,,Q,)@(e,, ^2,@,+ Eq') I
q,q'
:EI
follows
from this.
In r-space
@@,q')lq,q'))
the non-relativistic
approximation
1S
l_
1
A'
L 2m a4
_
r _22
r* fu
-
,2
f...,- _\ : * n2),$B'r4) ;i]'t'@'' n') (E 2m)tp(n''
with ttt(ar, rz) :
I
etttr'+ta'*, @.(q, q') .
q,q'
The binding energy E-2m
takes the values given in (38.1).
c. Treatment of the Transverse Photons by Perturbation Theory For a more complete treatment of the problem, we must also consider state.s with transverse photons in (3 8 .12) . Thus , in the next approximation we write
l z ) : I @ @ ,q ' ) l q ,q ' ) * l a " l n , k ) . q,q'
(38.r6)
n
The second term in (38.16) contains only states which can be produced by the operator H{, from a state with one pair. Thus, except Forfor the photon, the states contain zero, one, or two pairs. mally we write the vector lz) in (38.16)as
lz): edslv) .
( 38 . 1 6a )
Here the vector lq)is aqain of the form (38.12) and the transformation matrix S is determined from HY, in the following way:
H'lv):E19>, H ' - e - i s H e i sx H + i l H , S l - + [ [ H , S ] , S ] a r
s], sl , I x Hs* HY"+ t,tsr,sl + i lHy,,sl- + [tr10, Ho: H$v I H)o',* H"
(JU.r/'
tse'tzut ( 3I . 1 7 b )
l . S t a t e s o f t h i s k i n d m u s t b e clearly differentiated from states with a given number of incoming p a r t i c l e s . C . f . S e c . I l , e s p e c i a i i y E q s . ( 1 1 . 4 7t)h r o u s h ( 1 1 . 5 3 ) .
Sec. 38 Rrr
maan
Pos itr onium c
n{
+ha
185
-n h. .n -i - c e
rtH.,sl:-H{,
( 38 . 1 8 )
all terms which are linear in the creation or annihilation operators of the photons are eliminated in this order. Since aII diagonal elements of the operator l1ff vanish, Eq. (38.18) can be satisfjed by a non-singular matrix S . The commutator on the left side of (38.18) is formally the time derivative of the operator S in an interl^v ' r+' Eq. action picture which is associated with lha nnararnr rrrr 0 L (2.e)1. Equation (3B.IB) is therefore.J.;;;;;v!
S:
- @
J H{ (x'o)d.x".
(38.I8a)
Here HY,@o)is the time-dependent operator represented in this p i c t u r e . w e s u b s t i t u t e ( 3 B . l 8 a ) i n t o ( 3 8 . 1 7 a )a n d t h e n e w H a m iltonian becomes ro
H' : H!,0,+ Hf) + H"- * [ luTt*,), Hfl(xL)ld.x[. -_;
(38.le)
Since the last term in (38.19) contains the commutator of the photon operators (i.e., is a c-number) the photon variables of the problem have been formally eliminated. Using the equation
uf l M u U ) , d , ( x ' ) )-: -(+2 t 1 "t [ "d k- -r , o , ' - "n\ d " t (" \A' "rt )Le " h( tA , €)2f af^, ,l -. \(u3u8 . 2 0 ) l, we then obtain
n{@L)l: LHil@o),
I
: [[ a'*d,x'i,(x)i,(x'\ f(38'2I) ' ' Q n--!1 " t ' . "[- dk rtuu'-od(ft2) " \ ' - l "e(ft) \ ' ' l LIa.," R ' i+Ll k, JJ
lJ
For the transition to a Schroedinger picture, all operators present in fl' have to be taken at the same time. We therefore write the current operator j,(x')in the form
i,@')=-t r(:
at*" dBx",Ly;(x,,,)ynS(x,,,-x,),y15(x,_x,,)yEyt(x,,)J.9g.22) 1.1 ri!:ao
In (38.22)the singular functions of the complete, unperturbed probIem ought to be used. That is. they should include the influence of the term -F1. in the unperturbed Hamiltonian. As in the previous sections, we shall approximate these functions by the singular functions for free particles. Then the time integrai (38.19) can be done and, upon qoing over to the Schroedinger picture, we have
xd(q+ q',- q,- a)far,xlv@'),yrq@'')l
186
G . K l i 1 1 5 n ,Q u a n t u mE l e i t r o d y n a m i c s
Sec. 38
withl V@) -
(38.23a)
[ dsxe-i,t'tr(r) 1,":0.
d. The Fine Structure of Positronium The remainder of the calculation is simple in principle, but somewhat tedious. In the barycentric system the total perturbation enerqv is
d H,+ 6HcI6Hr,+
dll:
dHexchange,
Q , - s l 6 H , t q- ,r,) : - # -r) : (q,- ql6H"lq,, #n:*
(38.24)
d ( q- q , ) ,
(38.2aa)
I (38.24b)
"
l
x lu*Gt (q)uG)(qr)u*G)(q)u( ) (qr)- t7
(q,- ql6H,)q,,-8): -#G:
I
x rF,G)(q)y,ue)(q,)
- (q qr)i(q : qJr u( ) (q.\ x uGtb\ \ 1 L t f 5^, \ r ' t,, t t _ (q
l-a'
q,,1,
''n''
, )" u
-
, , e x c h a n g e< q ,- q t "o n
e2 I r l h , - 8 t )- \ r 4 r -+ t / F ; 6 x f (,r. rn6l x n(-)(q)y* u( ) (q)u( ) @r)yru' n (qr). J
In (38.24), 6H, contains the relativistic corrections to the kinetic energy; 6H",the corresponding terms for the electrostatic energy; rnrl
A Ir
,
fLh1 !av
anmjna
fLaL rr m c
1 '-^rr vrrr
/ 2 ov \v
.
t4 ? \. ul
TO
this
Ofder
We
mUSf
also include exchange effects for the electrons, and this is the origin of the term (38 .24d). In order to obtain this expression, it is simplest to startl with the sum 11, * 6Hr,. Using the approximations
nt*g.ra) :ir-#l''r,
(3B.2sa)
'!.arr (t)- lll'> and the exact relation
(c.f.
Sec. 14)
uc)(-q) : - c uG)(q), l.
(3B.2sb)
'C:TzTr,
G8.26)
If (38.23) and (38.9a) are added, we obtain the result ?2
1Ht ^ u t 3I H e- .z .- -t .r f
I
=X
S f
s3-
x I '" I #
(22)13
.t3.,
d ( s + q , - % - e ' ) L E @ ty) ,
l E @ ' )y, p q ( q ' , ) ) .
v@)l J J \q-qt)' This is just the interaction operator of Sec. 27 f.or the scattering of two electrons from each other. There, however, the whole expresslon was treated as a small perturbation; this is not allowed here.
Sec.38
Positronium
IB7
where I s) is the non-relativistic spin function for the unperturbed problem, we can develop the expression (38.24) in powers of. q2fm2 and (g- q')rlmr. AJter some rather lengthy computations, we ger 6 H : 6 H , + 6 H ,+ 6 H t , + d 4 + d l l e x c h a n s e ,
: -,*l#@(* -ogt1 - #1, (6H,)
G8.27)
Q: e - e,, G8-2za)
(d1/,,):-&+g#y,
(38.27b)
:l,L-s2* w], (d11,)
(38.27c)
(d rl e xch a n se ): ##
v6.t/a)
Here S is the total spin of the positronium and the terms (38.27) have been grouped according to their dependence on the spin. The expectation values of the expression (38.27) can then be cat* culated for the solutions of Eq. (38.14). In this way we obtain the perturbed energy levels of positronium: S:0: ^ rE _m t -177-7u o : -a-arl-t o
+
j
(38.28a)
rill ,
s:{:
6": #t+ - rh]
+* # 6,,, +ffilP
A\j(3B.zlb)
with
I 3t+4 ior l(t+1)(2t+31
a, 1 ' , 1i: 1 -
l\ From this,
ror
1w I
:,-, I(2t -
we find the splitting
1)
; _I !-r-T
|
^ I
(38.28c)
i:t
ror of the states
{3S, and
A E : 1, m,qL: 2,044 . ,tOE Mc/sec.
1156 ,
( 38 . 2 s )
Formulas for the fine structure of positronium have been qiven bv Pirenne,l Landau and Berestetski,2 and by FerreIl.3 Expeiimentai investigations of the energy difference (38.29) have been carried 1 . J . P i r e n n e ,A r c h . S c i . P h y s . N a t . 2 8 , 2 3 3 ( 1 9 4 6 ) ; 2 9 , I 2 I, 2 0 7, 2 6 s ( 1 9 4 7 ) . 2 . L . D . L a n d a ua n d V . B . B e r e s t e t s k i , I . E x p . T h e o r . P h y s . ; . B . B e r e s t e t s k i ,J . E x p . T h e o r . P h y s . U S S R U S S R1 9 , 6 7 3 ( 1 9 4 9 ) V 19, lr3o (1949). 3 . R . F e r r e l l , P h y s . R e v . t 4 - , B 5 B( 1 9 5 1 )t;h e s i s , P r i n c e t o n , 1 9 5 1 .
Sec. 38
G . K d U 5 n , Q u a n t u mE l e c t r o d y n a m i c s
188
out by Deutsch and coworkers.I
They have obtained the value
/E : (2.o))8 + 0.0004)105Mc/sec.
(38.30)
The difference between (38.29) and (38.30) is ^fully explained by Karplus and Kleinz have calculated the next higher term in o( . fLhr roe
cncrorz
el r ur YJ
differcnr:c
account terms of order ma'. ,r
zJtt:
maL 17
a
r- L:
4 3S.
hefuroan
P
:{': n\9
2'
and
4r r qJ o
sf ^fps uLeLvu,
.
takino
into
They find the result 'l
+ 2 l o"g , 2 ) l: 2 . 0 ) } 7 . 1 0 5 M c / s e c . ll
(38.31)
Similar results for the 25 and 2P states have been given by FuIton and Martin.3 e. The Lifetime of Positronium Whether they are bound in positronium or are free particles, an electron and a positron can annihilate each other and radiate two or more photons. This means that positronium is not a stable form, We shall now compute this lifebut that it has a finite lifetime. time. The element of the S-matrjx for a transition from.a state lq, q') with one pair to a state with two photons is givena by the rules
of Sec. 2I and 22 as ( q , q ' I S l k , , k , ) : 6 ( q+ Q '- k t - k , ) t ' ; ) # = u t ' t( q )x , 2V@t@2 ,.liyt\(i /rl._ L
y1q-hr)-*)iyrQ) , iyt(2)(iylq Ar) n\iye{t)l-,,.r, -,\ T\-1). -2ghz _ 1- .ab' t l^ I
(38.32)
The vectors ,(D and e(2)ut" the polarization vectors of the two photons At and ftr. By means of the methods used repeatedly in Chap. V, we obtain the transition probability per unit time zs' for the annihilation of a state le):
l e ) : I @ @ )l q , Q ' ) l q + q , : 0 ,
(38.33)
q
1 . M . D e u t s c h a n d S . B r o w n , P h y s . R e v . 8 5 , 1 0 4 7 ( 1 9 5 2 ); R . Weinstein, M. Deutsch and S. Brown, Phys..Rev. 98, 223 (i955). See also V.W. Hughes, S. Marder and C. S. Wu, Phys. Rev.106,
934 (r9s7). 2 . R . K a r p l u sa n d A . K l e i n , P h y s . R e v . 8 7 , 8 4 8 0 9 5 2 ) . 3 . T . F u l t o n a n d P . C . M a r t i n , P h y s .R e v . 9 5 , 8 I l ( 1 9 5 4 ) . 4. As emphasized before, the states with incoming particles in (38.32) are quite different from the states with "particles at time zero" in (38.12)and (38.16). Only if there is no interaction are these two kinds of states the same. Despite this, we can identify them with each other here because we are concerned with the first non-vanishing approximation in a perturbation calculation. In higher orders this identification is not allowed.
189
( 38 . 3 4 )
U( q , k ) : J L 1 l . t @ ) x X
-h1) -m)i l t y e { t ) l iy ( q -tq", t
y e\z)
' l u ' - ' \ q. r) l-*,-,*,:*
iye@fiy(q*hr)-ml;yeQl,.
- 2 qh z
L"o..t
, p ( r ): * 7
(38.36) o@)',o, VVT tn (38.34) we can go to the non-relativistic limit by the use of ( 3 8 . 2 5 ) a n d ( 3 8 . 2 6 ) ( i . e . , w h e r e l q l < m a n d a rx m ) . If the two photons are polarized parallel to each other, we obtain
U(q,It): s
for e0)le@.
( 38 . 3 7 a )
If the two photons are polarized perpendicular to each other, we have instead
u(s,ts):
i#(,
-
+) ror ei)Leo\.
(38. 37b)
In (38.37b), S again stands for the total spin of the positronrum. Consequently, for triplet states (38.37b) vanishes. Hence these states cannot decay by two photons.l For singlet states,2 ho*e v e r , w e c a n u s e ( 3 8 . 3 4 ) a n d ( 3 8 . 3 7 b )t o o b t a i n y#lg(0)lr: ? ? F i n s-l r e i
ffa,,Z/triplet -
0.
080-1010 d 7 , 6 s e c - 1 ,( 3 8 . 3 g )
(38.3e)
The result (38.39) does not mean that the triplet state is stable, but only that it must decay by at least three photons. The transition probability of this decay must be of the order of ma6 and a detailed calculation using the previous methods 9ives3 the result l. This has been shown only for the order being consi-dered. By the use of conservation of angular momentum, the statement can be shown to be an exact selection rule for a 35-state: Because of momentum conservation, the two photons must be emitted in opposite directions. Either they have total angular momentumzero (mutually perpendicular polarization directions), or they have total angular momentum 2 (mutually parallet polarization directions). Neither case can originate from a 3S-state. See also L. Michel, Nuovo Cim. I0, 3I9 0953). 2. J. A. Wheeler, Ann. N. Y. Acad. Sci. 46, 22I (1946). 3 . A . O r e a n d J . L . P o w e l l, P h y s . R e v . 7 5 , 1 6 9 6 ( 1 9 4 9 ) ;R . Ferrell, thesis, Princeton, 1951.
G. Kiil16n, Quantum Hlectrodynamics
I90
? e r r i p -r c t *
Sec.39
o.7z_'uro, u , , os e c - 1
@,_ g)+6,,r-
(38.40)
The difference between (38.38) and (38.40) has been used experA direct imentally in verifying the existence of positronium.I l:0 has also been made by meas.urement of u)trlpretfor n: I, The resuk, Deutsch.I
,l'ifit", : (0.68+ 0.07). 10?sec{,
( 38 . 4 r )
is in good agreement with (38.40). 39.
A Survev of Radiative Corrections
in Other Processes
a. Compton Scattering Radiative corrections for the Klein-Nishina formula (25.23J have been worked out by several authors. Jost and Corinaldesiz have rroafod thc cnrrcsnnnding problem for particles of spin 0. In the non-relativistic limit, their result isJ L r r v
ao:
v v . r v v y v
t^ / 3 U+ c o s 2 @ d)Q u )1 , t - 9 4
1
lt-i*n
"
[,
-3(i-lcoso*c9s1@)rcos3o l' "obe L LI ( \l -r *^ \ f
c o s O ) l o g = { = . *1 " zlal
a,
l+cos2@
.l,i("'t' 11
where e2
r^:-:-'
4nnt
"
d,
/?q
1n
and
t\
(3e.3)
@<m.
is the resHere a,' isthe frequency of the incident photon and/E olution of the measuring apparatus (as in Sec. 35). As before, the term with AE is present because of the infrared divergence which appears in an integration over virtual photons. Just as in is divergence this fie1d, in an external the scattering of electrons to be understood in terms of properties of the measuring apparatus. For particles with spinl/2 one also obtainsa (39.I) in the nonthe cross section has a rather comrelativistic limit. For ulm, In this case plicated form; we shail forego reproducing it here.a also there is an infrared divergence which is treated in a similar \V^v Ozy^. r ' i.e.,
{\ J?J .9L -l l ) h T form L La - -s t h e r v vn llm g v Lc c f i o n f a c f o r ir rn l l lh s a c the radiative
corrections
vanish
for
at(,/n.
l- + O t uI ' 1 ,.t \2 , |
),
This is not only
l . M . D e u t s c h , P h y s . R e v . 8 2 , 4 5 5 ( 1 9 5 1 )8,3 , 8 6 6 0 9 5 1 ) . 2 . E . C o r i n a l d e s ia n d R . J o s t , H e I v . P h y s . A c t a 2 1 , 1 8 3 0 9 4 8 ) . 3. In this limit the cross section for Compton scattering goes over into the classical Thompsonformula both for particles of spin zero and for the particies of spin I/2 considered previously. At higher energriesthe formula for spin zero differs from the KleinNishina formula. 4 . L . M . B r o w na n d R . P . F e y n m a n ,P h y s . R e v . 8 5 , 2 3 1 0 9 5 2 ). An earlier discussion of the same problem was given by M. R. S c h a f r o t h ,H e l v . P h y s . A c t a . 2 2 , 5 U ( t 9 4 9 ) ; 2 3 , 5 4 2 0 9 5 0 ) .
Sec. 39
Radiative Corrections in Other Processes
tgl
true in lowest order, but can be proved in general.l b. Electron-Electron Scattering The corrections to Eq. (27.20) have been evaluated by Redheadi Here also the complete result is very complicated and we shall restrict ourselves to the non-relativistic limit. If one electron is at rest while the other is incident with velocity u and is scattered through the angle O , then we can write the formula with correc-
tions as follows:
d,o:+nftY!!9 + =cos= + (v
- rr1) d.sin @(t - sin@))* [#t t, - rzacos@ ( l * cos @)) U \
lr Ll:3-tr '' sin20coszO\^' 2
(3e.4)
sin@- cos@))],
(39.aa)
a KI.
In the general case when (39.4a) is not satisfied, the radiative correction terms to this cross section also contain infrared terms which must be treated as in Sec.35. These terms have been dropped from (39.4), since they are multiplied by u2 and this is neglected 1" ^-^
c. The Self-Stress of the Electron Ana+lnr ^-^h'i^* WhiCh Can be treated qimnlrr lrrz fha mctfu6jg J r l l l y l y v y L l l g l l ' g L l
darrolnnad
To do this,
hara
ic
f-ln-e-
SO-Cal.LeO
r r - 1 r - Sr + r ^ ^ ^ r r ^ { S€lf u re > > ur
+L lr -r c' ^ c^ rl c^u^u+r 'i O f l
.
we examine the energy-momentum tensor of our system:
Tr,(x): rjl,)(x)+ Tl?)(*) , r/lt (x): i (lv t*1,T,0,yt(x)l* lrt:@),y, ar y @))\ - lAfitp{x),y,tp@)1 - rctrp (x),yr,,p (x))), J T,!7)@):-+{F,7,1}**d,,4n1n,
a,:+-ieA,(x), -t ti , , _,
aA,@) )Au@) a;____d;.
(39.s) (39. 6) (38.7) /eo
a\
t r ?o
o\
Consider the expectation value of the tensor Tu, for a state with one electron. In the rest frame of this electron, the expectation value of all components of Tu" vanishes, except for the diagonal terms. This follows directly from symmetry considerations. We now define 1 . W . T h i r r i n g , P h i l . M a g . 4 l , 1 1 9 3( 1 9 5 0 ) ;F . E . L o w , P h y s . R e v . 9 6 , 1 4 2 8 ( 1 9 5 4 ) ;M . G e l l - M a n n a n d M . L . G o l d b e r g e r , P h y s . Rev. 96, 1433 0954). 2 . M . L . G . R e d h e a d ,P r o c . R o y . S o c . L o n d . , A 2 2 0 , 2 i 9 ( 1 9 5 3 ) . limit is given byAkAnother treatment of the extreme relativistic hiezer and Polovin, Akad. Nauk USSR90, 55 (1953).
G. K5115n, QuantumElectrodynamics
192
Sec.39
E ( 0 ): * I Q l r n n l q ) l n : o d T x , s(0): I klrul Q)lq:oct"x. Clearly call
E(O) is the total energy of an electron
.S/O) the
been written
Sinco
"splf-stroqq"
for the unrenormalized E(0) :
f\ v?v q. v , ( l
t! Y- 'n. c
theory,
%"*p: mol
narynafia
d^lf-^h^-^y.
rL hr r "t ^v g"Ynr h r
/aa
\ - - . , /
7 ) l' l a v e
we have
(3e.r2)
and dm is the electro-
FfOm
- t
: - t lv $),'p(r)1, T,l',,' rl'): o, --
it follows
(39.11)
at rest and we shall
6 n c,
where rno is the mass of the bare electron
( 3e . 1 0 )
,'rr-
wn
r-,
,
(3s.13a) (3e.13b)
that
3s ( o ) - E(o ): I a " *3 9 ,,1 4 1s)l qo' --im^ f a"* Qll't'(xl'tPtr)Jlt)la:o ,] 1
(3e.r4)
or
m o l 6 m* 2 o ' - ( q l l l t ) u )t ,p ( x ) ) l q ) l q : o3:S ( 0 ) . ( 3 e . r s ) I I f . a i s t h e H a m i l t o n i a n o f t h e u n r e n o r m a l i z e dt h e o r y l S q . 0 Z . t g ) t h r o u s h ( 1 7. 2 2 ) l , t h e n 1
f
-.
hH
I
a
z J d ' x ( q l l v t x l , , t @ l l,s=>Ql sl f t l c ) l no : f t { * , + 6 m ) . ( 3 e . 1 6 ) From (3 f . i5) and (3 f . i6) we concludel
- - tl*,# - u*l:- +hW) s(o)
\JJ.1/,1
Thus it is possible in principle to obtain the self-stress from Jhe self-mass by differentiation. In fact the self-mass is infinite lat least in lowest order perturbation theory; c.f. (32.6)], so that Eq. (39.17) actually has no meaning. We will now show from considerations of invariance that the self-stress (39.17) must vanish (39.17) can then be used to define a identically. In particular, cutoff ,4 in (31.18) as a function of the mass /n0. In order to show this, we consider the electron in a system where it is moving with velocity r; in the x-direction. We denote all quantities in the new coordinate system by a prime, so that the following transformations hold :
-,, (r,,)), (ri): 7)7 { l.
/?q ra\
A . P a i s a n d S . T . E p s t e i n ,R e v . M o d . P h y s . 2 I , 4 4 5 ( 1 9 4 9 ) .
Sec. 39
Radiative Corrections
in Other Processes
i(ri): ,+ ((r,,)-(4,)),
I
-t\ r- l/ t , \ : /t -Px \(,,\ "r -
-"
a
( 3e .r 9 ) (39.20)
dsx,_ll_uzdsx. For the energy E(u) and the momentum \(u) frame, we therefore obtain
I93
in the new coordinate
l l t n \ . '' z r S ( 0 ) )
t"t"'
/tr/')\ -rr S(0)) . \"\"/
,
l | ?q
,11
(3e.22)
J/i_32
Hence, in order that E(u) and {(z) have the transformation properties of a four-vector, it is necessary that S(0) vanish identically. Therefore we must have A l|ml otno \, rnn I
^
(3e.23)
Consequently, the dimensionless quantity 6mlmo cannot depend on ?r,r,o.In a convergent theory this would also be impossible for reasons of dimensionality, since the only dimen'sionless variable which could occur here would be e2. However, in a theory with a cutoff, the mass mo can occur in the combination Al*3 .-(Here a ic t1-.,a a'tnrf I .rur result shows that ,4 must protrruJ uE be L ultu_vlt chosenl portional to mzo. Finally, we remark that a cutoff using the regularization prqcedure of Sec. 30 also givesz the result zero for the self-stress. d.
Scatterinq of Liqht Quanta by an External Field (Delbriick Scatterinq) and Photon-Photon Scattering In Sec. 29 we have seen that an external field can produce virtual pairs and that the vacuum behaves like a polarizable medium. An indirect confirmation of this effect is obtained by the exact ineasurement of the Lamb shift. For a free photonwith energy-momentum vector satisfying h2:0, the effect vanishes, as demonstrated by the charge renormalization of Eq. (29.23). However, ifthe photon passes through an external field, then it can produce virtual pairs which can be scatteredJ by the o.(41.19b) \ r , / I r r r i,t h + /n)ZP (pr))_e: V Zell*l z) (z l-lpt -l p\z):p
The four functions )J*) *u defined by Eqs. ( I.19a) and (4I.I9b). The summations in these equations run over all physical states f o r w h i c h t h e t o t a l e n e r g y - m o m e n t u mv e c t o r i s p ( o r - 2 ) . A g a i n , Z is the volume and we have taken the limit V'->a in (41.19). Frnm fhc
qrzmmctrrr n vrf
cv r hr \ 4aYrv c c
Jl r r r r r r eLJ/
tL h, r cv
tL hr r cv vnr l r, r z
We
Can now
show
that
only two of the functions IJt) atu independent. We start with
If'plz)(z Il;l 0). (4r.zo) lrl,) (p,)+ (iy f t m)rl,t (t'\)"e: - ro,F_fo Now we apply the matrjx
C of Eq. (14.3) to obtain
-V (o Il,l z) (z IloIo) (C')e l lrtr (p,)* (i y f * m)2/,t (p,)7 " " e= p k2C", t:p : c", [x,l-)@,)+ (- iyf +m) z]-t(h,)7"uQ-'),p: : [:J-)(f,) r (iy f -t m)2? (f,))"e.
I
i(41.21) ) I
I n ( 4 1 . 2 1 )w e h a v e u s e d ( 1 4 . 4 ) t h r o u g h ( 1 4 . 6 ) . F r o m t h i s i t f o l l o w s that
,j,) (pr): zt-) (fr)- zo(fr) . Consequently we can write (41.19) as an integral -space: 2
( 41.22) over the
entire
(o|{1"(,),f,(*')}lo>: :
(4r.23) d- p eipv- 4 lz, (p2) + (iv p+m)z,(p\)"Be(p)' ] .rtr J -1
f
N o w w e s h a l l c o n s i d e r t h e e x p r e s s i o n @ ( * - x ' ) ( O l { f ( x ) ,l( L and use the Fourier We write @(x-x'):+(ae1x-x')) tation for e(x-x'),
;(x - x,'\ - l-p .t?v
,J
'S*0". r,,u, 'r,. T
)Jlv).
(4r.24)
Equation for the Constant Iy'
Sec. 42 'I
h l
C
203
^1 1 rA e
tf
anl"
e(x- '')
J
d'Pe;iu-z')t1ilZr(P\:
': : ;;i I aP"'v-"o I #+,2,(t] @oti'): _: -
);
:;4
f
f dbeit6-APf
\2r)"J
(4r.2s)
daz'( '-al alP'
J
and
*
tf
\2n)"
:*
e ( x - x '' ) I d b e i p ( r - a '()i y f * m ) e ( f ) r r @ \ : J
f d 6 r , r r , - ,('i,v b * * l p i ! : t zo t- -- f-'4 - -
t2nJ.J
-
);
f
(4r.26)
I
*o f
;",
tA
dp e l p v-r')Po" O fJffi J J
| -\
2 ,(p,- ( fo* r ) ,) .
The last integral in (41.26) vanishes by reason of symmetry we nave
- i @(x- x')
(At
1A\
206
G. Klill6n, Quantum Electrodynamics
Sec.42
ought to ho1d. Comparison of (42.14) and (42.9) then shows N:1. In a similar way one could conclude from (42.9) and the fact that the functions Et(f') vanish for p2:*mz lsee Eq. (41.28) and the remark following it] that the first term on the right side of (42.10) ought to appear multiplied by a factor l, rather than by I ('r t '- 2- t -1 A / - 4- )t \l ' \'
N2
The answer to both these puzzles is that although does vanish, the convolution integral N-1,^,
"N'
r^r,
( 0 l r t o r ( x ) l q ' ,-
Iso(r-
x')(oll(x')lq)dr'
(Oll(x)lq'2
G2.r5)
is different from zero. In the proof that (Oll(x)lg) vanishes, of the constant ly'. This is we neglected the time derivativel allowed for all finite times because of the adiabatic character of fhc swifchinoT\Teolect of the time derivative is not allowed if integrals from -oo are involved, as in (42.15). Neglecting the time In derivative in such cases can lead to apparent contradictions. Sec. 32, Eq. (32.10) and following, we worked out in detail the approximation to this phenomenon. first perturbation-theoretic There we saw that after the subtraction of the term with tl0) (-*'), vanished because of an the matrix element of l(x) [eq. (:2.i4)] expre s s lon
(q'* m') 6(qz! nf) .
(42.16)
In the convolution (42.I5) there is a factor q2+rnz in the denominator, so that the right side actually has the indeterminate form
n i I*i 6 (q2 * m2 ) q't m'
(42.r7)
This hoids if the adiabatic switching is not taken into account. However, if the switching is carefully carried through, rather than (42,I0), we obtain aterm of the order of magnitude o( [Eq. (32.16)]. A similar term appears IEq. (32.I3)] in place of the denominator qztm2. The behavior of numerator and denominator is well defined in the limit a-+ 0 and gives the unambiguous, non-zero result 'l
I -
N
i n ( 4 2 . 1 5 ) . F o r u n d e r s t a n d i n gt h e e q u a t i o n s a b o v e , i t i s
important to remember that the operator l(r) contains these sin'l nrrlar 9uru
lL- so rr rmr rc D .
S ri lnl nu as u
fha
frrnnfinnqtrr
jfi .\ ff f i 2 \ , /
L-,,^ llqVg
^^Uqglr
'.]^fi-^d ucflll€u
l ay\ 7 u
means of l(x), they contain singular pieces also. This explains If the adiabatic switching is the formal contradiction in (42.10). done carefully, it is evident from the foregoing discussion that the two functions Xr(12) must contain terms of the form
q.'6(p'* m'). These terms give a non-zero contribution l.
See footnote 1. n.
204
(42.r8) in the integral (42.13a)
Sec.43
The Renormalization
of the Charqe
207
w h i c h j u s t e n a b l e s u s t o w r i t e ( 4 2. I 0 ) a s
(ol{,p(x), y(x'))lor:#l
orri!(r'-,) ,@1,6(p, +m")(iy?-m)+ \
l*,.,,
Lzfr(p,)t (iy f t m) zi"r(fz)l#;#]
+ #;#
J
The two new functions ZlrE(fz) are defined so that they are equal to the original functions I,(pz) everVwhere except at the point The singular expressions (42.18) at the point bz:-mz h2:-ruz. hrrzo
haon
ca rvynrlri ua ri tul ryl /
SUD b tIrf aa C C tt ee dd
tf fr O m
ZE, F c ( \br2 \t ,.
sv o _
t.h, .a_t .
.n. o_ f"
Only
does X'"s(- mz) vanish, but also (fra*r1-, Z;,r(pr) can be set equal t:o zero for .pz:-7r2. With these new functions, we obtain from (42.19) not (42.13), but rather
t + ti"s(- m2)+ 2mE'rec, (- m2).
S:
(42.20)
For the following it will usually be simplest to calculate formally with the originai functions J,(p2) and to make the explicit subtraction of the singular parts (42.18) only in the results. 43.
The Renormalization of the Charge We have formally renormalized the mass and the field operator of the electron in Sec. 4l and 42 without the use of perturbation theory. In this section we are going to give a similar treatment for the charge renormalization. Because of the gauge invariance of the theory, the photon has no self-mass and therefore it is not necessary to carry out a renormalization of the energy of the photon. Accordingly, for the matrix elements between the vacuum and a one-photon state lA), there is the same r-dependence for the complete potential Ar(x) as for the in-field A,f,(*) . In this case the most general relativistically covariant relation between the two matrix elements is
'
( o l A , ( ' ) l A )c: l a , , +M z : 3 ; - l ( 0, l , a i ' ) ( x ) l A )(.4 3 . r ) Axpdxyl'
The two constants C and M in(a3.L) are independent of k and. x. A formal proof for (43.1) can be given following the method of proof o f E q . ( 4 1 . 1 5 )i n S e c . 4 1 . U s i n g t h e e q u a t i o n f^, A ' , i ' Q )- . l o ( * - x " ) i p Q " )d x " -
-l
( 4 3. 2 )
D(*- x,,) s1o'nr.{', ) -*r n pAr1r"1ap(*u:,")larx,, \L )oxo l
--JL-6-u\^-^
I c . f . E q . ( 4 I . 7 )] a n d t h e c a n o n i c a l c o m m u t a t i o nr e l a t i o n s , w e f i n d for an arbitrary operator F(.r) , x
: I o Q'* x") F(x),i,@"))dx"-i D (x,,-x) lF(x),A'If'("')l ::+ -* w " t t \ ^ t +l *
. oD(x'- x)
'---
a;l-
oF{x)
" a--d-" Apvl
'sl [t+:
I )
C. fiitt5n, Quantum Electrodynamics
208
In the last two equations /u(.r) stanas equation of motion for A,,(r) :
for the right
Sec. 43 side of the
( 4 3. 4 )
nAu(x):-i,@).
The detailed structure of the current operator is not important for (43.2) and (43.3). Just as in Sec. 4I, the vacuum expectation value of (43.3) gives an expression for the matrix elements of F(,v) between the vacuum and a one-photon state. We have
I f rn,.rr
I
If we take F(r) as 4,,(x) , then (43.1) follows from (43.5) and the values use of invariance properties of the vacuum expectation which appear. The constant C in (43.1) corresponds closely to the constant Al in (41.15) but the constant M has its origin in the particular If the state lA) contains a vector character of the potential. drops out. The contransverse photon, this term automatically stant N of Sec. 42 was not eliminated from the theory by a counte-r term. [The onty counter term in (4I .2) contains the self-mass.J It was isolated by renormalization of the field operator 9(r) and The situation is somewhat it was then considered separately. different for the constant C of (49.1), since now the observable fietd strengths and are therequantities are the electromagnetic The constant C has a fore linear combinations of the potentials. in so far as it connects the units of the significance "physical" also the charge) of the complete fields field strengths (i.e., 4(orlY\ For simnljcitv, we 4 (v\ with fhose of thc in-ficlds shall use the same units for these two kinds of fields and thereThis can only be done by means of a fore we have to take C:1. we shall add an extra term to the curcounter term; accordingly, rent operator: ! ' l t
/ t u \ ^ t
i r@): tt#
\ . " 1
. f, (,),v,,t @l - L J A,"(x)
( 4 3. 6 )
One can substitute this expression with the charge renormalization constant I into (43.4), transfer the term in Z to the left side and divide by 1 - L . This factor can then be understood as a renormalization of the charge. The factor N2 enters the first term in (43.6) because y(x) is now the renormalized Dirac field. In what follows, it will be useful for us to define the charge renormalizaThis is most tion so that the counter term is gauge invariant. as current the renormalized done by defining simply ./-_\
lu\x):
ieN2 r-,
2
, /-.\-- ,ln t t.,l L r p \ x ) , y p y \ x ) )L l u A p \ x )
czAr(x)\
b*"4I
'
(43.7)
Sec. 43
The Renormalization
of the Charqe
209
Ar(r): Af) (r) t J D"(x - x')i,@')dx' , we obtain ?irtx'\ aAp(A_ ae!)Q\ - ^ t ' ' _ "x,) o_@ o*, t ? , u +' J f ?ru 0*),
a,qtt,t ?r,
( 4 3. 8 )
, (43.9)
It follows from this that the difference ot (43.6) and (43 . 7) js of importance only for certain special matrjx elements with scalar and longitudinal photons . Since the new counter term in (43.7) contains the derivatives we must check to see if the canonical commuof the potentials, tation relations are changed by these new terms in the Lagrangian. Our complete Lagrangian is now
9:9,pl9e*
( 4 3. 1 0 )
9w ,
er:-f[rorQ***)v{i)l.tn''tt' - +t- +* v,+ * v@), ('),,t v{")l+ } a* x' l'p @)l,i
- +L W -W) e# - ffi-+ WW,(43.rz) ee: sr:
'*
N2Ar(x) l,t@),yrrp@)1.
( 4 3. i 3 )
The usual rules for quantizationgive {rt@),rp(*'))xo:xi,: nfiA1n - u'), in complete agreement with
(42.2) .
We also obtain
lAr(*),A,(x')1,":,1,:o ,
lY#,
(43.lS)
(43.16)
* n,(*)),"_,r: |r.€,,,6(r t') ,
(43.16a)
tu,:6r,-L6rt6u+, faAp(r)
b.ang'll
l- # ' - # - ) , ": ^ ' :
( 4 3. 1 4 )
2 '
L A , l" T =T \6'n;i + 6'1ar ) ae a' ) ' ( a3' r 7)
For Z :0 , (43 .15) through (43 .17) go over into (17. 8) and (17. 9) . T h e r e l a t i o n s ( 1 7 . 1 0 ) , ( L 7 . I 2 ) a n d ( 1 7 . 1 3 )a r e s t i l l v a l i d i n t h e r e normalized theory. Apart from the replacement of du, by €u, in (43.16), we see that the constant ltrplays an analogous'role for the electromagnetic field to the constant /y' for the electron field. Both appear in a similar way in the commutation relations for the renormalized fields. lf we do not consider the last term in (43.12), we can take the renormalization of charge to mean the replacement of the tield Ar(x) Ay lE=T ,1, (x) and of the charge
2I0
Sec. 43
G. Kiitt6n,'Quantum Electrodynamics
e by 1t1- a . In this way we can understand why the interaction term (43.13) ls formally not affected bV L . The exceptional character of the last term in (43.I2) has its origin in the gauge-invariant In order to obform used in the definition of the current (43.7). tain the expression (43.7) for the current, we need only multiply part of (17.3) bv 1 - L , This form of charge the gauge-invariant was first given by Gupta.l renormalization we are now ready to derive an After these formal preparations, explicit equation for L . To do this, we shall employ the same We therefore calculate method used in Sec. 42 to derive (42.I3). the vacuum expectation value of the commutator of two potentials Ar(*), .,,
(ol LA,(x), A,(x')f| 0) : (0 llAf' (x),,4t0) (r')lI0) + | 0) + (*')l A'j" (ollA,(x)A't'(r), I * | tnu. r, * ( o l l A f ) ( x ) , A , ( x ' ) - A l o ,l(or)' )+l | ( o l l i r @ " ) , * " ' ) x D n ( * ' ) d , x d " , x " D ' p ( x " ) * l! 1,@"')]loS. As before, it is useful to introduce a new notation for the vacuum expectation value in the last term. We therefore write
(P)+ (olli,@),i,@')llrr: ei,(''-x) IrL.,) *f,!d,p | ;fr [ P o< o
with
'nu''n'
a'P,n"o-') nf*)(P),
|
(P): v,.Z(oli, nL',) l,) Qliulo),
(a3.20a)
r l b\(p): v Z : Ir f,)( - p) .
( 43.zob)
p-':p
n\z)--*
From qeneral principles of invariance, functions nL, (p) must have the form
it follows
that the two
nlh)@): A(p,)6,,+ B(p')p,p,: nli) e p): nL;)(p). (43.2r) From the continuity equation for the current, which is also satisfied by the renormalized current, we have
o : ppnfl(il : LA(p,)+ p,B(p\1p,,
(43.22)
and therefore
I (P') nLi (il : (- f'6r, + PrP,) , nQ,): -+ >Qti,lz)(zti,)o ,) -|D-
4 p\z\:f
\1J.LJ)
(43.24)
-,, n (p2) .(43. 2s) o>=;# J Ape;or'' e(il ? f, d,,* f ,,.p,1 1 .
(44.8)
The stated cancellation in (44.4) of the contributions of the scalar and longitudinal photons follows from (44.8). By similar arguments, this cancellation can readily be shown if more than one scalar photon is present. Thus we have shown that the function in (44.4) can be expressed as a finite sum of terms which il(p') are all posttive. A similar proof cannot be given for the functions Do(pz), since t h e o p e r a t o r / ( x ) i r , @ 1 . 1 9 ) i s n o t g a u g e i n v a r i a n t , and therefore a cancellation of longitudinal and scalar photons cannot be ex-
pecreo. Using the positive character of important identitv
0
'
I vAV/ aE
nra-,a Pluvg
lA q \9J.
:--i.nEI them into the equations of motion :::i,rtion of (45 .6) into (45 .4) gives
A - /' l a n d
(Atr,
. ,7 \/
-h rv
-c *r r- lb. aL ^I +- r
(45.4) and (45.5). Sub-
' , ' ,; ', , = m ) t P ( x: ) i e Y A ( r )Q Q ) | 6 m Q ( r )- i I O (x* x') li,@'),i ey A(x)* (r) + 6m,]t(x)lx x A?"*(x')dx' - y, I, !' *' li, @'),tf (x)],4i""(r') .
[ ,n,.,,
I
-"rng (45.6) and (45.7) we can combine the fjrst two terms on the ::;::i side of (45.8) and the four-dimensional integral in the fol- :-,..-ingway: (Terms which are quadratic in ,4|u" (x) may be dropped. )
( ' ) * 6 m Q @ ).' Ai.v)Q - i , i O ( " - * ' ) L i , @ ' ) , ' ; ' e y A *( x( r)) + 6 m , l t ( x ) l A i " " ' ( x ' ) d x , ' :/ / 1 q : i t ^,,-t(x)rp(x) + 6my (x) - ; t *Z
o\
yu A.* @)y (x)'
-: -,'.-eeliminate the time derivatives of second order from the cur-=*' nncrator hv the rrqe of the enrrations nf mofion . fhe .osult :an be written as . v . - - v P u r 9 L v r v j l l r v 4 , L r r g l v
E,
--,t!, i- .\;
tieNz
\"
t)
-_,,
z A, .^, (*v,\1 \ ., \a , - b "u - L6pe! An(x),(4s.10) L P ( r ) ,n * Q ) ) + L
tpt:6pt-
L 6 * a 6 t 1 ',,
(45.11)
_ sing this result and the canonical commutation relations, we can -,..-crkout the commutator in the last term in (45.8) . In this way ','.-e!ind
y 4li,(x'),* (")f ,":,t - - ::3t^ r ^,Q@)6 (u - n') .
(45.r2)
S,:bstitution of (45.12) into (45.8) and application of (4S.9) gives ::-: right side of (45.4). ]n a similar way (45.7) can be verified. Substitution into (45.5) anc a transformation similar to (45.9) gives 1.
For another proof see R. E. Peierls, Proc. Roy. Soc. Lond.
) . ? r 4 ,t 4 3 ( 1 9 5 2 ) .
2tB
G . K l i 1 1 6 n ,Q u a n t u mE l e c t r o d y n a m i c s
Sec.45
zA,,(x)=-1i:! t'rli/tuq@*l qfn@,r,,t't,ll+ I d','lli,{r').o, "i
!!fflot""t' t)+|1 [ o1"- *'yli,c't, + li, --
n,.|::oli,lq'
q')(q" qli'lo)'
( 4 7. r )
The state lq,q') is a state with an in-pair and the matrjx elements of the current operatorwhich enter (47.1) can be taken from the calculation of Sec. 46. From Eqs. (46.8), (46.I2) and (46.13) it foliows that
q'):(oli't'lt,t'>lt- 11Q')+ n @)-irn Q')-zLi *I*r., x
I
' I .x' l1t -. f r t O , l + 1 7 1 o.i\'n_ Lrr(\ ov \) nt t(wo -2),\ zr r N - t ' l 1 _ L 1 I ? ( Q \ ti n e ( Q ) R ( Q \ l - l t n z . z l
-
cll
t i 'n) e s (t ( J , ) ] , rt*(ruts:,)(ql'!rc)10)(01?(0)lq [ S( et O
I
with
( 4 7. 7 a )
Q:q'-q, and
i'y'i!I:*,:":;::::,(Q,) +,{-tj+R(0,) + ir(R(Q,)1-l*, , ,l -
ti )tq? ns(Q1,)1, r* r,t , - S L ) Q l r t ' atl t,)(o1 rtor1S
I
with
(47,8a)
Q:q*q'.
There are the following relations between the new functionsR(Qr), Ft?l , S ( Q r ) , S = ( 0 r ), a n d t h e o l d f u n c t i o n s F , G , H :
n-(?1 + i n e(Q)R(?') : Fuo(q,q'), -
: (q,q') : Goo (- q',- q), *, li(a\ + i n e(Q)S(Qz)f Hoo
for Q:q'-q,
q2:q'2:-m2
and
e ( g ): u ( q , ) : t .
|
I
*,.,,
Equations (47.9) give the most general form which the right side can have on the basis of invariance. From the reality properties of the current operator, it then follows that the four new functions must be real. We now returnto (46.13). Fjrst we note that this equation has a structure similar to Eq. (45.16), i.€., it contains a vacuum expectation value multipiied by @-functions. If we define a func-
G . K i i t l 5 n , Q u a n t u mE l e c t r o d y n a m i c s
226 tion
Fu(P, P') l>v
( 0 1{y 0), i" (')l,l-(4)} 10 ) .:
a
Sec. 47
r r
n dp' eino,)t i p"'4'9, (p,p'\,(47.r0) 1r,,j.J J e
then we can show as in Sec. 44 that this definition (47.10) contains a sum over onlv a finite number of intermediate states. If the re,Tr(f ,1') is finite. normalized operators exist, then the function With the aid of the integral representation
@ ( x ) ) : t2+n t
dr -
t
T -
_J
?e
(47.rr)
r t ' t " z,) "
we then obtain
I
o (x\ (ot{lt (r, i,@)), I(4)}|0): : with
+@ f
dr
, f d I '-t - t € r uP( p ,r
9 , , ( t ' ,b ' \-
, 7 r ,p ' ) : - l _ J x
-@
lTere
r r e r u
.' /f l i s
an
1c\
dx
F u ( p .x ; p ' ) . ( 4 7 . r z a )
.lpo
t€l
wifh
a ynvnv 4qLi 4t iv rv r c
-@
fime-1ikA
a r b' ri Lt rr uar rJ r z
nv vnr :nl rar Ln' 't* t ' . " ) T n a c i m i l a r
lAa
,\ 2 " 'n- ), ',. [l J[ O P , P ' e i P \ 3 ) ) t i P ' t t a t f i u Q , PJ' ) ,
-j-o
=,,
I
Way we
\zAcfnr
Obtain frOm the
fime
com-
secOnd @-funCtion,
6 t ( x \ @ ( i 4 ) ( 0 t { yi0- @ ), )),t(4)}to>:
lI r n r . r r o ,
t = [ [ a b d b , -e i p t s ) )t ,p ' , , ng, ( b , 'b ' 'It.)l " tzn)",J'-r-'r "p\('r/
-"
f
gtn.O'l-
if
I
. g ( h - t " ' r' l
t_it,"p'r
J
"r
@ 7. I Z c )
h ' - n''tr \"
r"gift the functions F , G , H in (47.3) as a In this way we " u r r Hilbert transform of the finite function gr(P, p') typq of generalized in (47.10)plus a similar contribution from the second term in (46.13). Tho real
narf nf fhcqc y s
L
v !
u r r v
r v
evnrpssions v J \ F r
v
oirzes the
functions
R
and
S
in (47.9),while the imaginary part givesthe functions R and S . ISee the similar calculationin Sec. 29, especiatly Eqs. (29.27) throush (29.32). l F r o m ( 4 6 . 2 5 )w e a l s o h a v e
-.' (' (, ,t,lpi ,, ',/?l , l i tl l z + R '(.01 ) l -Lmr-,r, N -ll z ' ,, -' ; + R ( o-) s ( o ) l : o , J or
(47.r4) At this point we must recall that the functions R(p') and S(p') a r e d e f i n e d b y m e a n s o f ( 4 7 . 9 ) , ( 4 7 . 3 ) a n d ( 4 6 . 1 3 )i n t e r m s o f t h e operatorsl(x)of (41.2). These operatorscontain singular expressions of the type (42.18). We must therefore expect corresponding
Sec. 47
Proof that a Renormalization Constant Is lnfinite
227
slngular piecesl of R and S. The calculations ofSec. 46 allow us to isolate these singular pieces in a simple way. The argument that leads from (46.16) to (46.25) was originally carried through for l(x), but it can obviouslybe carried through for just the regular part of l(*). If R."e(p,) and S*c(p2) are the regular parts of the functions R(22) and S(pr) , then we conclude that in addition to (47 .14) we must also have R - " f 3 ) - S - * e ( 0, ) :
'i - \'! r.',' , ' t J " *( - ,w, 21)\|+- , ,r rr n -,r(-*')):1-_'t'(47.15) 1_ 2\-z \
On the other hand, the result (46.26) is independent of the special treatment of the singular terms, and by a transformation similar to that of (33 .2) through (33.6) we can rewrite (47 .8) as
( o j ; i q . s ' > : ( 0 1 ; ; o t t q . o ' ) lIr n Q , ) n , " ' ( Q , r) 5 . ' s 1 9 2 ; I - i.n (II (Q\ - R*c(02)1 S'"s(Q'z))l * l(47.16) + i Q , < o l * l 9 l S , S [' )S ' * ( 0 ' )- i n S ' " s ( Q \ 1 . ) So far we have not madeuse of our assumptionthat the renormalization constants are to be finite and Eq . (47 .16) has been derived using only general assumptions. Now we shail take explicit ac1 - N' the fact that count of and -1=hu.r" been assumed to be I L I- L finite. This means convergence for the integral (43.30) for the function ,F1O; ar well as for the integrals which enter the definition (47.9) according to (47.12) . With this assumption we find
l i m n ( Q ' ) : n @ - ltm -a'** -O'+o
' I
4:*:ff(o)
J 0 \ t+
0
- II(o):o U7.r7)
E)
and irm
Y"\P.f'):
lfo- fi,)- a
:
drdt.' ,. -rrlT** f f ' JJ F="V=;r'e"(P' Po r' : p''l',or') lr"
lim f f l t o p 6 l - r cJ J l . Y y
( 4 7. 1 8 )
dx dyFotp. ); p,, y) (Po p;- ir))ls'- tP6- ie'l)
I. In an earlier work these termswere overlooked. tG. t