On the Transition Matrix and the Green Function in the Quantum

Progress of Theoretical Physics, Vol. 10, No.6, December 1953

On the Transition Matrix and the Green Function in the Quantum Field Theory

Sho TANAKA and Hiroomi UMEZA W A

Institute 0/ Theoretical Physics, Nagoya University

(Received July 26, 1953)

617

The J. Schwinger's formalism of the Green function in the quantum electrodynamics is applied to the transition problem of the state. It is shown that the many body kernel in the Heisenberg representation involves the information about the transition of the state and this is dh'ectly represented by the repeated use of the one body kel'Oel G,@ and the vertex opel'ator rlL defined by J. Schwinger. Further, the renormalization is discussed without use of the usual perturbation theory, although then: remains the difficulty associated with the b-divergence.

§ 1. Introduction

The propagation of the interaction effects between the fields is usually represented by

the Green functions in the present quantum field theory. In the interaction representatioq

these Green functions are the well-known dB, SF. etc. The matrix describing the transi

tion of the state is constructed by the repeated use of these Feynman's functions. As

was shown by Stiickelbergl), introducing the Feynman's functions and the usual Dyson's

S-matrix follows as a logical consequence of the requirements of the c:ausality for the

propagation of the interaction effects. This fact suggests that the Green .function is one

of the fundamental basis of the current quantum field theory.

The importance of these Green function becomes dearer in the Heisenberg representa

tion, because there the state vector is time-independent and the temporal development of

the system is described by the field operators and $0 it can be expected that the Green

function involves the information about the transition of the state. On the other hand,

many authors2) ,3) have expected that the Green function of the 1z-body problem involves

also the information about the stationary state of the nobody problem.

It is the aim of this paper to investigate the detailed property of the Green function

in the Heisenbeg representation. A crucial method to treat the Green function in the

Heisenberg representation ,has been proposed by J. Schwinger.2) In this paper we shall

treat our problem along the same line as he has done and restrict ourselves' only to the

quantum electrodynamics.

In the usual perturbation theory, the Green function of the one body problem has

a special importance. As Dyson4l has shown in the quantum electrodynamics, the S-matrix

element is obtained through the substitution of S;., D;', and r for the electron line, photon

line, and the vertex part in any possible irreducible graphs corresponding to the given

transition process.

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618 s. Tanaka and H. Umezawa

These S~, D~ correspond to the Green functions of the one body problem. In § 4,

we show, without use of the usual perturbation theory, that the mattix element of any

transition is written by an adequate graph which contains the Green function of the one

body problem and J'~ as for the internal line and vertex part, respectively. Further, we

will treat the renormalization problem in our method without use of the perturbation theory.

However, on account of the difficulty associated with the b-divergence, the completion of

the discussion is confined to be left in future.

§ 2. On the transition matrix and the Green function

of the many body problem

We treat the problem in the quantum electrodynamics, whose Lagrange density IS of

the form

L= -1/2 ·Sb{r~(i1~-ieA~) +x} if'+ 1/2· {Sb~+~if') +h.c.

-1/4.F~~.1/2· (a~A~)2 -J~A~, (2·1)

where '1) is a spinor source which anticommutes with if' and Sb, ~ IS given by '1)*r4 and

J~ is a c-number source current and further F~"=a:LA"-a"AiJ..*

From (2.1), the well-known equations of motion in the Heisenberg representation are obtained:

{r~(aiJ.-ieA~) +x}if'='1),

o A~=-J~-j"", j',l- = -ie/2 h~if'+h.c.

(2.2)

(2.3)

(2.4)

Now let us introduce the following notation -for the operators in the Heisenberg representation, A(xJ), B(x2), ••• ,

(2.5)

where 1JY0 is a vacuum state vector defined as the minimum energy state in this representa

tion. Further, we use the interaction representation which coincides with the Heisenberg

representation at a (x) = 0 and denote the minimum energy ,state in this repre.sentation

as (/)0. If the quantities in the Heisenbergrepresentation, are denoted by ...4 (XI) , B(x2) , ... , corresponding to A (XJ ) , B(X2) , ... , respectively, then there are following relations between both quantities:

...4(XJ) =U(a(x1), 0)A(X1) U-\a(x1) , 0), etc. (2.6)

As Gell-Mann and LowE) has shown, 'Fa may be written as

1JY0=.!.U-\±CXl, 0)(/)0/((/)0' U-1(±CXl, 0)(/)0), (2· 7) c

where c is a normalization constant.

* The stat expresses the hermitian conju&ate.

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Oil the Transition Matrix and the Green Function in the Quantum Field Theory 619

Using (2.7), (2.5) is rewritten as

(A(x1), B(X2) , ... ) + = ($0' U( 00, O)P(A (%1)' B(x2) , ... )

xU-I(-ex:., 0)$0)/($0, U(OO. -00)$0)' (2·8)*

where we used the following relation;

U(a, 0)U- 1 (a','0) =U(a, a'). (2.9)

Since the annihilation and creation of a-patticle are described by the positive and

negative frequency parts Q!(x) of a-field operator Q~(x), respectively, we have from

(2.8) the next theorem. Namely, the transition matrix .s~ between the initial state,

where a-particle with the energy-momentum klL and spin state r etc. exists, and the final

state, where a-particle with the energy-momentum k: and spin state r' etc. exists, is

5 r..t(k/, r; ... )(Q (-). a () ) fi=aL< i(k , ..... ) a. Xl , ... , ~" ;rl , ... +,

J.I.' , (2.10)**

where F1~:::5 is the operator which takes out the Fourier's amplitudes refering to "k lL ,

r and the negative frequency part at the world point -Y1 " and "k:, r' and the

posi~ive frequency part at the world point Xl'" a is a normalization constant and ;r1 and

Xl are the coordinates of the particles in the initial and final state, respectively, and so

their time components are -;;- 00 and + 00, respectively. If we bring out the quantities

of Xl in front of tpe one of .:!'·l' we have the following relation:

(Qa.(;rj),"" Qa.(.i)···)+=($o' Q,,(,1;·l)'···' U(oo, -OO)Qa.CX1)···$0)

x 1/($0' U(oo, -00)$0)'

which gives the proof of (2. 10) . Using the the creation and annihilation operators q;, the Fourier transform of Qa. is

expressed in the form

Q,,(x) = lim ~ V-l/~[d"r(k)q: (k)exp i(kr-kot) p-)om k

+ d,.*,.(k) q;: (k) exp -i (kr- kot) ]. (2.11 )

The normalization constant in (2· 10) is expressed by the reciprocal of the product of

* If we denote the interaction Hamiltonian density as H(x) and apply the usual perturbation theory, we have

(2·8)'

The appearance of the denominator in (2·8)' corresponds to the procedure in the usual perturbation which leaves out of consideration the isolated diagram, whose init'al and final states are the vacuum states.

** (2·10) has been applied to the problem of the multiple production of meson; H. Umezawa et aI., Phys. Rev. 85 (1952), 505.

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620 S. Tanaka and H. Umezawa

(d .. t;(k') ...• d .. r(k) ... ).*

Now we define the Green functions for many body problems as follows: electron Green functions

GYj(Zlt zD = al(7)(zD(cfJ(Zl»'

G~(Zl' Z2' z1) = ala~(Z2)(cfJ(Zl)' ¢(zD )+E,

GYj(Zl'X~, z~) = alar;(X~)(cfJ(Xl)' ¢(zD )+E,

GYj(Zl' X2• z~,z~) = al(7)(xD (cfJ(zl),¢(zDcfJ(Z2»+€ etc.

photon Green functions

<MJ.~(~l' ~2) = alij~(~2)(A(J.(~1»' <MJ,~,G(~l' ~2'~3) =a/ij~(~3)(A(J.(~1)' AG(~2»+'

mixed Green functions

K~T,~(Zl' z~, ~1) = a/aJi~l)(cfJ(Zl)' ¢(ZD)H

KYjJ.~(Xl' x~, ~1' ~D = a/aJ~(~D (cfJ(Zl)' ¢(z~), A(J.(~l) )+.

According to the calculation rule given by J. Schwinger, we have

GYj(Zl' zf} =i(cfJ(Zl)' ¢(x~) )+E-t'(cfJ(Zl» (¢(x;),

G~(Zl' X2' z~)=i(cfJ(z]), ¢(X;) , 1'(Z2»+E

-i( cfJ(xt ), ¢(xD) + (1' (Z2) ) E,

KtjJ(Zl' x;, ~])=i(cfJ(ZI)' ¢(.xD, A(~l»+E

-i(A(~]»(1'(Zl) ¢(xD )+E,

(2.12)**

(2.13)

(2 ,14)

(2 ·15)

* (2 ·11) for the case of the field with the arbitrary SpIn has been discussed by Y. Takahashi and H. Umezawa; Prog. Theor. Phys. 9(1953), 14.

** E in the expression ( >+ E means the sign function as to the coordinates appearing in ( >+. For

instance, for the case of (I/>(x), I/>(X2)' if(x/), ¢(xl)+>E, E is given by

E = E (xli x2) E (x/, ;)(1l) E (X1o xl) E (x2, x/) E (x9' xl) E (X1o x'),

where E (x,y) = 1 for Xo >yo,

= -1 for xo<Yo.

In th~ fo110',,:n6 discu~sion W~ shall eliminlte this symbol as far as it does not give rise to mistake.

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On the Transition Matrix and the Green" Function in the Quantum /i"ield Theory 621

Since U(a, a') has non vanishing matrix element only for the transition satisfying

the charge conservation law, we have the following relation for F, in which the numbers

of the¢ and (jj are different each. other ;

(2-·16)

For example, we have

(2.17)

From the invariance of the theory under the charge conjugation, we have the Furry's

theorem, (AfL)J-+o=O. From these relation and (2. 10), (2· 15), we find that the many body Green function

at the limit <'1--,>0, ~--,>O} corresponds to the transition matrix element. Hereafter,we

denote these quantities with (J--'>O, ~--,>O) by those with the super-suffix 0.* For example,

G~(:.rr:r2'i'tx2)' @Ailill2),andKTjo,(x1xtl2} correspond to the Moller scattering of the

two electrons, photon-photon scattering, and Compton-scattering, respectively.

On:e body Green functi()n is connected with the current J;, (x) as follows:

(2.18)

where

GTj(x; x) == lim [GTj(x, x') +GTj(x', x)J/2. X'-»Z

(2·19)

Hereafter, the matrix representation for the coordinates of electron and photon is used and

so the matrices 1, a", ¢, (jj, A",. GTj and @J.~ are 8(x-x'), 8(x-x') a". ¢(x)8(x-x'), (jj(x)8(x.-x'), A,,(x)Q(.i'-X'), GTj(x, x') and @J.~(x, x'), respectively.

From the equations of motion (2·2), (2·3), we have for GTj' @J.~ the equations,

{r fL (a fL -ie(A,,») + M} G~='1 +ier~ (¢·)8 / ij~ «(jj),

{o- P} @J.~=·8,,~ 1.

(2.20)**

(2.21)

In the above expression the mass operator M and polarization operator P are the matrices

whose elements are given by M(r, x'), P (f, f') defined as follows:

J dx"M(x, x")GTj(x", x() = (x-er,,8/ijfL(X»GTJ(x, x'),

J d~IlP (f, f")@J.~(f", fl) = eTr{r,,8/ijv(~I) G.~(f, f)}.

(2·22)

(2.23)

* The limiting process (J ---+0, 1)---+0) should be taken after the variational operation of (2 ·12), (2 ·13) and (2·14).

** The product of the two matrices A and B is defined by'

(X\AB\x')=~ dy(.riA\y)(y\B\x').

FUtther the symbol . in the expression A· or ·A denotes the right or left cooroinateof the matrix A.

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As was shown by J. Schwmger, we have

aiaJ"(~) GY)IJ,Y)+o=eG~I de r~(e) G~@~.~(e ~),

where the vertex operator r~ (~) is. defined as foIlows2) :

From (2.22) and (2.24), we have

and so

M:-x1 +ier",(A",)- e2r", J de·G'ljr~(~')@J,~(~',.) -er[1-'G'Ij a/a}[1-' {ie r~(¢:)aia./';,(¢)G~l} G'Ij'

§ 3. Note On the renormalization theory

(2 ·24)

(2.26)

(2·27)

In this section a short note on the relation of the above theory with Dyson's one

is given. The differential equation (2.21) can be transformed into the integral form;

where

G~(x, x') =5.1'(x-x') -axJ dy 5.1'(x-y) G~(J', x')

-J dydy' 5.1'(x-y) L]°(Y,Y') G~CY', .x'),

~O=MO-x'l, x'=x+ax.

Using (2·27) we have

G~(x, x') =SIl'(x-x') -ax J dy SF(X-y)G~(y, .x')

_e2 J dydy'dy"deSIl'(x-y)G~(J"Y') x r~(e ;y',y")G~(J'''' .x')@J,~(e=',y). (3·2)

If we replace G ~ by S; in (3. 1), this equation cor

responds to the integral equation given by Dyson and

(3.1)

Fig. 1

Fig. 2 (i)

Fig. 2 (ii)

L]0 amounts to the total contribution from the self-energy graph. (3 ·2) agrees with the

final integration of the self-energy graph given by Dyson and corresponds to Fig. 1, in which the electron line, photon line and vertex b correspond to G~, @J. and 1':, respectively.

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On the Transition Matrix and the Green Functio1z in the Quantum Field Theory 623

The fact that r of the point a is not replaced by r~ coorresponds to the Dyson's argu

ment on the b-divergence, because we must take into account only one of the equivalent

graphs (i), (ii) in Fig. 2.

In the quantum electrodynamics we can normalize cp and All- so that in the high energy ·region all possible quantities with dimension of the length are the momenta PII- of particies6)*. In this case the dimensions of G~ and @~ agree with those of SF and DF, respectively, and r~ and r II- are the dimensionless quantities.

Let us separate the infinite constants from G~, @~ and r~ as follows:

G~=Z2G~1'

@~=Zg@~,

(3.3)

where G~l> @~1 and r~l are free from infinity. Substituting (3.3) into (3.2), we have

G~(x, x') =~SF(X-X') -axJ dy SF (x-y)G~l(Y' x') Z2

-e~Z22Z1-1z:~J dydy'dy" derll-SF(X-Y) G~l(J', y') (3.4)

X !'HO(~' ;y', y") G~l(Y'" x')@~,~(~', x).

In the above equation, while the integrand of the third term is finite, its integration may

be divergent. Since this divergence comes from the contribution of the high energy region, the following discussion shows that this integral is at most linearly or logarithmically

divergent. Z1' Z2' and Zg are the function of the upper limit .p ~ co of the integration

concerning the internal momentum and so the dimension of the divergent Z1' Z2' andZg should be zero or negative power of the length. Therefore, the dimension of G~I' @~1

and r~l are [L-n] (n<.3, 1z<2, 11<0), respectively. Since the integral (3.4) is

most strongly divergent in the case of n=3, 2, 0, it is sufficient to treat this case for

the consideration of the highest degree of divergence. Then the integral has a dimension [D2-a-a-a-2J = [L], and can be written as follows:

f dydy'dy"derll-SF(X-Y) G~l(y,y')rs(e ;y',y")G~l(Y'" x')@J,.tce, x)

=Zll[A+B(rll-all-+ x) +C(rll-all-+ x)2+ ... ]

x J dySF(X-y)G~l(Y' x'), (3.5)

where A and B are linearly and logarithmically divergent and C is a finite quantity.

As Fig. 1 is symmetric in association with the two vertices a and b, it is expected that a infinite constant factor ~-1 appears from the vertex a as well as the vertex b after

* In the interaction. of the second kind, the situation is not so simple as in this case, because the coupling constant has the dimension of the length.

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the integration is analogous to Dyson's argument. However, the consistent proof of this

situation is not yet verified in our method. This defect which is due to the, asymmetrical

treatment of the two vertices makes it difficult to compare our method with Dyson's one also in the discussion of the skelton approximation in' the next section. *

Substituting (3· 5) into (3·4), we obtain the following relations asa necessary con

dition for the convergence of the right side of (3.4) :

Z2=1-ie12B,

e1=Zl-lZ2Z 31/2e .

(3.6)

Here eI and x' should be considred as a finite and observable electric charge and mass,'

respectively. Then (3.4) becomes as £ottows:

(3.7)

(3 .6) is in agreement with the condition as to the renormalization constant given by Dyson. Further it is easily found by the same dimensional argument as in the above

(3.5) that the divergent quantities are restricted only to self-energy parts G~. @~ and

vertex part r~. Therefore, the theory is entirely free from divergence, provided that it is shown that

any Feynman diagram of Smatrix is expressed by the irreducible skelton in which internal

lines, vertex, and charge correspond to G~1> @~J' r~l> and t'l' respectively. In this paper the method in which any transition matrix is represented entirely by the G~, @J. and [~ is called the skelton approximation. If we have the formulation of the skelton approximation, any vertex in Fig. 3 is of the form e( G~ @~G~)1/2r~ which can' be reo,

written as follows;

?-lZZl/2(G O ft1 0GO)1/2TlO _ . (Go ft1 0G O)r. ° e.61 2 3 "l)l \:!JJl"l)l 1 (L1- e 1 1}l\:!JJl"1)l ,.1, (3.8)

so that it turns out that there exists no more any infinite quantity in the theory.

Thus it is necessary for completion of our procedure only to investigate the possibility of the above skelton approximation and this is the aim of the next section. As will be shown there, unfortunately, we can not yet find the complete formulation of the skelton approximation and this problem is left to be investigated in future.

§ 4. The skelton approximation

In this section we shall prov.e that the transition matrix element is obtained through

substituting the Green function of the one body problem and r,. into any internal lines and the vertex part of the adequate graph corresponding to its process. However this

graph is not completely equivalent to the skelton and we must often use the uncorrected

* If this deffects are get rid of, one could set up a non-singular theory by using the lagl'angi~n gh'en by G, Takeda7) and applying the variational method in §2;

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On tlte Transition Matri% and the Green Function in tlte Quantum Field Theor), 625

vertexrll-' although it is needless to use SF and DE' According to (2. 10), it turns out that the problem is how to express all the many

body Green function by the one body Green function and the vertex function rll-; i.e.,

the corrected function approximation. In the following we discuss on this problem in some

examples, i.e., the Moller scattering of two electrons and the photon-photon scattermg. (i) Moflir scattering

According to (2. 10), the transition matrix element for the Moller scattering corres

ponds to G~(,:t:"l' <r2, z:, z;). Using (2·12), we have

G~(%l> %2' %{, %D=-J2la~(%~)a~{X2)G'I(XH %D1J'I"O (0)

+ G~(~l X 2 %~ %~),

where

-iG~(xl' %DG~(X2' xD·

This· is represented by the diagrams denoted in Fig. 3.

In this diagram and hereafter it should be noted that

the straight and waved line, the vertex and vertex with

circle correspond to G~, &1J, r 11-' and r~, respectively.

The second term of (4· 1) expresses two independent

electrons scattering without the real interaction. The

effect of the true Moller scattering due to the real

interaction of two .electrons is involved in the first terms

of (4·1). From (2.21), we have

Further using (2.17) and the Furry's theorem, we obtain

Substituting the following. relation

;(.,

(4·1)

(4·2)

Fig. 3

(4·3)

(4.4)

(4.5)

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626 S. Tanaka and H; Umezawa

into (4. 4), we have

(4·6)

where

(2) J G ('%2' %~) =_e2 [G~r(J.G'Il(%2') r(J.(f/) G'IJ'X~)@J(e,.) G'I)

+ G'IlrG'IlC" %DG"l(%2') J r(e)@Jce, . )G'IllIJ,'Il+O' (4·7)

This is represented by the diagrams in Fig. 4,

It can be shown that the second term of

(4, 6) contributes to the skelton higher than

the fourth order. Using (2,22), we have

and this diagrams correspond to e2-skelton.

lJ2 M lJ2 (MG G-1) lJ7} (%~) lJ7} (x 2) lJ1) (X;) lJ~ (x 2) 'I) 'Il

lJ [( lJ G )G-1] x-q(J.-·"I) 'Il . /J1)(x~)/J~(.t'2) lJj~,_

(4.8) Fig, 4

After the tedious calculation, (4, 8) is rewritten in the form;

;Y MI =-e G ~[G-l /J2G'Il ] G-11 . /J1)(%~)a?(X2) J,''1+0 r 'Ilij' 'Il a7}(%~)/J1)(.t'2) 'Il J,"l+O

(4.9)

Therefore, from (4· 6) we have

From (4.3), (2.17), (4.5), we can obtain the relation

(2)

where G is defined by the left side of (4,7) without superscript O. The second term of

( 4 .4) gives to (4. 10) the skelton higher than the e6-approximation, and we write this part

as OCe6). Using (4·10), we can rewrite (4.10) as follows:

(4.12)

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On the Transition Matrix and the Green Function in the Quantum Field Theory 627

The second term of e 4 . 12) corresponds to e4-skelton, and is given by

(4)

G~e ;x2, ,x~)

=-ie4G~rG~ [rG~eX2') f re~")G.,,@JC~"'·) f rC~")GC ,x~)@Jee, :)G"lC:')

This

+rG1)eX2') J rCf") G"lJ rce)G"IJ(,x~)@Jce,. )@Ae',:)G1)(:')

+rG."eX2') J re~")G''le,x~)@./C~'', :)G."C:.) J rce)G",@J(;'" )lJ,.,,~o

-ie4G~rG~rrG.,..(:,) J rc~") G ... e,xD@Jee:",· )G."eX 2') J rce) G.,,@J(e,:)

+rG.,,(:,xD G.,,(x2.) J r(~") G ... (,x~)@Ae',.) J rce) Gy/M ce,:)

+rG.,,(:,x~) G."eX 2') J ree') G.,,@J(e':) J l'(e) G'Ij@JW,') ]IJ,~.o. IS represented by the diagrams in Fig. 5.

A X K X )::( X

Fig. 5

e 4 '13)

As stated in the preceding section, it should be noted here that only some part of all vertices are replaced by r",. For instaric~ in Fig. 4, only one vertex among two is replaced by r",. Thus the contribution which corresponds to Fig. 6 in the usual perturba

tion theory is. not involved in this &agta:m, but in Fig. 5 which belongs to e4-skelton. Of course, if we takemto account infinitely

higher order terms in the present approl{rmation, then the both

vertices will be completely corrected and so by r"" and then the skelton approximation May be obtained. However such a procedure is of a perturbation t:heor~ticif{ concept. This unfavourable situation

of our method is due to the fact that, in the course of obtaining

the higher orderskdton by applying (2.24) to one body Green Fig. 6

function, .only one vertex among two of the self-energy part is replaced by {''''. (ii) Photon-photon scattering

Now we discuss briefly on the photon-photon scattering. This· transition matrix ele-

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ment corresponds to

( 4.14)

where (0)

@J(~lI ~2' ~3' ~4)=-i[@J(~l>'~2)@J(~3' ~4)+@J(~1' ~3)@J(~2' ~4)

+@J(~l' ~4)@J(~2' ~3)J· (4.15)

This is represented by the diagrams in Fig. 7. After the analogous calculation as for a~ .

--. G~I' 'the first term IS trans-a'1)a~ ". formed into the following form:

a2@ I ij(~4) ij(fs) J->O

= -@~ ij(~;~J(~3) L~9@~' (4.16)

J" Fig. 7

(4.17)

Further, we have

a3 G.'1 ij·aJ( f 4) a I( ~3)

(4 ·18)

where the symbol (x---7:J!) means the term obtained by the exchange of X and :JI in

the preceding term. Thus the final result is given by

(4.19)

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On the Transition Matrix and the Green FUllction in the QuantuJn Field Theory 629

where (4)

@~(" '~3' ~4)

=-ie4@~T,.[rG"lS r(e)@J(~/, ~4)G'/jf r(~")@J(~",.)GYjf l'(~"')@J(~'''' ~3)GYj

+rGYjf r(~')@J(~" ~4)GYjf r(~") @J(~'" ~3)G'/jf {'(~"')@i~"',· )G'/j

+rGYjJ rW)@i~', ~3) G'/jJ r(~") @A~",·) GYjf r(~ ")@(~"', ~4) GYj'J.Yj"o

+ (e:4-~)' These correspond to e4-skelton approximation and are represented by diagrams in Fig. 8.

Fig. 8

All other processes be treated in the similar method as to the above two examples.

The authors express their appreciation to Professor S. Sakata for his kind and helpful

interest and his continual encouragement and also thank to Mr. R. Kawabe and Mr. S.

Kamefuchi for their valuable discussions. Their thanks are finally due to Professor M.

Kobayasi for the hospitality to one of them (S.T) at Yukawa Hall.

References

1) E. C. G. Stiickelberg and T. A. Green, Helv. Phys. Acta 24 (1951), 153. 2) J. Schwinger, Proc. Nat. Acad. 37 (1951), 452. 3) E. E. Salpeter and H. A. Bethe,Phys. Rev. 84 (1951), 1232. 4) F. J. Dyson,Ph},s. Rev. 75 (1949), 1736. 5). M. Gell~Mann and F. Low, Ph}, •. Rev. 84 (1951), 350. 6) S. Sakata,H. Umezawa and S. Kamefuchi, Prog. Theor. Ph},s. 7 (1952), 377. 7) G. Takeda, Prog. Theor. Ph},s. 7 (1952), 359.

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On the Transition Matrix and the Green Function in the Quantum

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