PH vsI CAL REVI EN

VOLUME 'PS, NUMBER

FEBRUARY

The Radiation Theories of Tomonaga, Schwinger, and Feynman
F. J. DvsoN
Institute for Advanced Study, Princeton, En@ Jersey (Received October 6, 1948)

A unified development of the subject of quantum electrodynamics is outlined, embodying the main features both of the Tomonaga-Schwinger and of the Feynman radiation theory. The theory is carried to a point further than that reached by these authors, in the discussion of higher order radiative reactions and vacuum polarization phenomena. However, the theory of these higher order processes is a program rather than a definitive theory, since no general proof of the convergence of these eRects is attempted.
The chief results obtained are {a) a demonstration of the equivalence of the Feynman and Schwinger theories, and (b) a considerable simplification of the procedure involved in applying the Schwinger theory to particular problems, the simplification being the greater the more
complicated the problem.

I, INTRODUCTION

and ease of application, while those of Tomonaga-

FeynmScaaonv,reers'iuelsttheooffstuhTbejoemrcetocennaotgfaa,qnu'danSitnucdhmewpienngedeleern,ctt'rodadnyisd--
namics has made two very notable advances. On the one hand, both the foundations an~ the applications of the theory have been simplified by being presented in a completely relativistic way; on the other, the divergence difficulties
have been at least partially overcome. In the
reports so far published, emphasis has naturally been placed on the second of these advances; the magnitude of the first has been somewhat obscured by the fact that the new methods have been applied to problems which were beyond the range of the older theories, so that the simplicity of the methods was hidden by the complexity of the problems. Furthermore, the theory of Feynman differs so profoundly in its formulation from that of Tomonaga and Schwinger, and
so little of it has been published, that its par-
ticular advantages have not hitherto been available to users of the other formulations. The advantages of the Feynman theory are simplicity

Schwinger are generality and theoretical com-
pleteness.
The present paper aims to show how the Schwinger theory can be applied to specific problems in such a way as to incorporate the ideas of Feynman. To ~ake the paper reasonably self-contained it is necessary to outline the foundations of the theory, fo11owing the method of Tomonaga; but this paper is not intended as a substitute for the complete account of the theory shortly to be published by Schwinger. Here the emphasis will be on the application of the theory, and the major theoretical problems of gaugeinvariance and of the divergencies will not be considered in detail. The main results of the paper will be general formulas from which the radiative reactions on the motions of electrons can be calculated, treating the radiation interaction as a small perturbation, to any desired order of approximation. These formulas will be expressed in Schwinger's notation, but are in substance identical with results given previously by Feynman. The contribution of the present paper is thus intended to be twofold: first, to

' Sin-itiro Tomonaga, Prog. Theoret. Phys. 1, 27 {1946);
Koba, Tati, and Tomonaga, Prog. Theoret. Phys. 2, 101 198 (1947);S. Kanesawa and S. Tomonaga, Prog. Theoret.
Phys. 3, 1, 101 (1948); S. Tomonaga, Phys. Rev. 74, 224 (1948).
~ Julian Schwinger, Phys. Rev. V3, 416 (1948); Phys. Rev. 74, 1439 (1948). Several papers, giving a complete exposition of the theory, are in course of publication.
«R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948);
Phys. Rev. 7'4, 939, 1430 (1948); J. A. Wheeler and R. P.
Feynman, Rev. Mod. Phys. 1V, 157 (1945). These articles describe early stages in the development of Feynman's theory, little of which is yet published.

simplify the Schwinger theory for the benefit of
those using it for calculations, and second, to
demonstrate the equivalence of the various theories within their common domain of ap-
plicability. *
* After this paper was written, the author was shown a letter, published in Progress of Theoretical Physics 3, 205 (1948) by Z. Koba and G. Takeda. The letter is dated May 22, 1948, and brieRy describes a method of treatment of radiative problems, similar to the method of this paper.

86

RAD IATIDN THEORIES

II. OUTLINE OF THEORETICAL FOUNDATIONS

Relativistic quantum mechanics is a special

case of non-relativistic quantum mechanics, and

it is convenient to use the usual non-relativistic

terminology in order to make clear the relation

between the mathematical theory and the re-

sults of physical measurements. In quantum

electrodynamics the dynamical variables are the

electromagnetic potentials A„(r) and the spinor
f electron-positron field (r); each component of

each field at each point r of space is a separate

variable. Each dynamical variable is, in the

Schrodinger representation of quantum me-

chanics, a time-independent operator operating

on the state vector C of the system. The nature

of C (wave function or abstract vector) need not

be specified; its essential property is that, given

the C of a system at a particular time, the results

of all measurements made on the system at that

time are statistically determined. The variation

of C with time is given by the Schrodinger

equation

ih[8/Bt]4 =

f H(r)d

4,

where H(r) is the operator representing the total energy-density of the system at the point r. The general solution of (1) is
[ C(i)=exp it/5] —H(r)dr Cp,

with C 0 any constant state vector. Now in a relativistic system, the most general
kind of measurement is not the simultaneous measurement of field quantities at different
points of space. It is also possible to measure independently field quantities at different points of space at different times, provided that the
points of space-time at which the measurements
are made lie outside each other's light cones, so that the measurements do not interfere with each other. Thus the most comprehensive general
type of measurement is a measurement of field quantities at each point r of space at a time t(r),
Results of the application of the method to a calculation of the second-order radiative correction to the KleinNishina formula are stated. All the papers of Professor Tomonaga and his associates which have yet been published were completed before the end of 1946. The isolation of these Japanese workers has undoubtedly constituted a serious loss to theoretical physics.

the locus of the points (r, t(r)) in space-time forming a 3-dimensional surface 0 which is spacelike (i.e., every pair of points on it is separated by a space-like interval). Such a measurement
"will be called "an observation of the system on
0. It is easy to see what the result of the meas-
urement will be. At each point r' the field quantities will be measured for a state of the system with state vector C(t(r')) given by (2). But all observable quantities at r' are operators which commute with the energy-density operator H(r) at every point r different from r', and it is a general principle of quantum mechanics that if
8 is a unitary operator commuting with A, then
for any state C the results of measurements of A are the same in the state C as in the state BC. Therefore, the results of measurement of the field quantities at r' in the state C (t(r')) are the same as if the state of the system were
C(0) =exp —[i/Ii] t(r)H(r)dr Co, (3)

which differs from C(t(r')) only by a unitary
factor commuting with these field quantities. The important fact is that the state vector C(o)
depends only on 0 and not on r'. The conclusion
reached is that observations of a system on r
give results which are completely determined by attributing to the system the state vector
C(o) given by (3).
The Tomonaga-Schwinger form of the Schrod-
inger equation is a differential form of (3).
Suppose the surface 0 to be deformed slightly near the point r into the surface tT', the volume of space-time separating the two surfaces being
V. Then the quotient
[c'(o' ) —c'(o') j/ U

tends to a limit as V~O, which we denote by

BC'/80(r) and call the functional derivative of 4'

with respect to 0 at the point r. From (3) it

follows that

ihc[BC/Bo(r) j=H(r)C,

(4)

and (3) is, in fact, the general solution of (4). The whole meaning of an equation such as (4)
depends on the physical meaning which is at-
" tached to the statement "a system has a constant
state vector Co. In the present context, this
statement means "results of measurements of

F. J. D YSON

" fieM quantities at any given point of space are
independent of time. This statement is plainly non-relativistic, and so (4) is, in spite of appearances, a non-relativistic equation.
The simplest way to introduce a new state
vector + which shall be a relativistic invariant is
to require that the statement "a system has a constant state vector 0"' shall mean "a system
consists of photons, electrons, and positrons,
" traveling freely through space without inter-
action or external disturbance. For this purpose, let
H(r) =Hp(r)+Hi(r),
where II0 is the energy-density of the free electromagnetic and electron fields, and II» is that of their interaction with each other and with any external disturbing forces that may be present.
A system with constant + is, then, one whose H»
is identically zero; by (3) such a system corresponds to a 4 of the form
4(o.) =T(o)4p,
] T(o) = exp [p/h— t(r)Hp(r)dr . (6)

where H, (xp) is the time-dependent form of the
energy-density of interaction of the two fields with each other and with external forces. The left side of (9) represents the degree of departure
of the system from a system of freely traveling particles and is a relativistic invariant; Hi(xp)
is also an invariant, and thus is avoided one of the most unsatisfactory features of the old theories, in which the invariant H» was added to the non-invariant Hp. Equation (9) is the starting point of the Tomonaga-Schwinger theory.
111. INTRODUCTION OF PERTURBATION THEORY
Equation (9) can be solved explicitly. For this purpose it is convenient to introduce a oneparameter family of space-like surfaces filling the whole of space-time, so that one and only one member o(x) of the family passes through any given point x. Let 0.0, a», 0~, ~ ~ be a sequence of surfaces of the fami1y, starting with 0& and proceeding in small steps steadily into the past. By
p pro
Hi(x)dx
o'»

It is therefore consistent to write generally 4(o) = T(o)@(o),

is denoted the integral of Hi(x) over the 4dimensional volume between the surfaces cr» and
(7) o'p ', similarly, by

thus defining the new state vector 4' of any system in terms of the old C. The differential equation satisfied by 4' is obtained from (4), (5), (6), and (7) in the form
phc[B@/Bo(r)] = (T(o)) 'Hi(r) T(o)e. (8)

Now if g(r) is any time-independent field operator, the operator

e(xp) =(T(o)) 'g(r)T(o)

is just the corresponding time-dependent opera-

tor as namics.

'usIutailslya

defined function

in quantum electrodyof the point xa of space-

time whose coordinates are (r, ct(r)), but is the

same for all surfaces 0 passing through this

point, by virtue of the commutation of Hi(r) with Hp(r') for r'Wr Thus (8) m. ay be writ ten

j phc[B%/Bo(xp) ='Hi(xp) 4,

(9)

'See, for example, Gregor Wentzel, Zinfa7hrung in die
Quanteetheorie der W'ellenfelder (Fra»»g Deuticke, Wien, 1943), pp. 18-26.

Hi(x)dx, )t Hi(x)dx

are denoted integrals over the whole volume to the past of o.0 and to the future of 00, respectively. Consider the operator

V=

U(op)

t'
=(

1 —[p/hc7

Hi(x)dx )

(, Xi(1 —[i./hcj

r"
t IZi(x)dx

(10)

O'Q

the product continuing to infinity and the sur-
faces 0.0, cr», . being taken in the limit infinitely
close together. U satisfies the differential equation
phc[B U/Bo(xp) g = Hi(xp) U,
and the general solution of (9) is
+(o) = U( ) Op

with 0'0 any constant vector.

RA D IAT ION TH EOR I ES

Expanding the product (10) in ascending powers of Hj, gives a series
top&0
U = 1+( p/—hc) Hi(xi) dx i+ ( i/—hc)'

p&0
X

Hi(xi)Hi(xp)dxp+ .~ (13)

Further, U is by (10) obviously unitary, and

U-' = U = 1+ (i/hc)

p O'0
Hi(xi)dxi+

(p/hc)'

small perturbation as was done in the last section. Instead, H' alone is treated as a perturbation,
the aim being to eliminate H' but to leave H' in

its original place in the equation of motion of

the system.
Operators S(p) and S(~) are defined by re-
placing Hi by H' in the definitions of U(o) and
U(~). Thus S(o) satisfies the equation

j phc[BS/8~(xp) = H'(x p) S.

(17)

Suppose now a new type of state vector Q(o) to be introduced by the substitution

(pro

)e(sl)

dxi

Hi(xp)Hi(xi)dxp+

. (14)

It is not difficult to verify that U is a function
of op alone and is independent of the family of surfaces of which ro is one member. The use of a
finite number of terms of the series (13) and (14), neglecting the higher terms, is the equivalent in
the new theory of the use of perturbation theory in the older electrodynamics.
The operator U(~), obtained from (10) by
taking ro in the infinite future, is a transformation operator transforming a state of the system in the infinite past (representing, say, converging streams of particles) into the same state in the
infinite future (after the particles have interacted or been scattered into their final outgoing dis-
tribution), This operator has matrix elements
corresponding only to real transitions of the system, i.e., transitions which conserve energy
and momentum. It is identical with the Heisen-
berg 5 matrix. 5

IV. ELIMINATION OF THE RADIATION INTERACTION

In most of the problem of electrodynamics, the
energy-density Hi(xp) divides into two parts—

Hi(xp) =H'(xo)+H'(xp),

(15)

H*'(x.) = —51/cubi. (xo)~p(«).

(16)

the first part being the energy of interaction of the two fieMs with each other, and the second part the energy produced by external forces.
It is usually not permissible to treat H' as a

~ sterner Heisenberg, Zeits. f. Physik 120, 513 (1943), 120, 673 {1943),and Zeits. f. Naturforschung 1, 608 (1946}.

By (9), (15), (17), and (18) the equation of mo-
tion for Q(o) is
j phc[&Q/8n(xo) = (S(o)) 'H'(xo) S(ir) Q. (19)

The elimination of the radiation interaction is

" hereby achieved; only the question, "How is the
new state vector Q(o) to be interpreted?,

remains.

It is clear from (19) that a system with a constant 0 is a system of electrons, positrons, and

photons, moving under the inHuence of their

mutual interactions, but in the absence of ex-

ternal fields. In a system where two or more

particles are actually present, their interactions

alone will, in general, cause real transitions and

scattering processes to occur. For such a system,

it is rather "unphysical" to represent a state

of motion including the effects of the inter-

actions by a constant state vector; hence, for

such a system the new representation has no

simple interpretation. However, the most im-

portant systems are those in which only one

particle is actually present, and its interaction

with the vacuum fields gives rise only to virtual

processes. In this case the particle, including the

effects of all its interactions with the vacuum,

appears to move as a free particle in the absence

of external fields, and it is eminently reasonable

to represent such a state of motion by a constant state vector. Therefore, it may be said that the

operator,

Hr(xo) = (S(o))-'H'(x )S(o),

(20)

on the right of (19) represents the interaction of a physical particle with an external field, in-
cluding radiative corrections. Equation (19) describes the extent to which the motion of a

490

single physical particle deviates, in the external mass, and should have been used instead of

field, from the motion represented by a constant Ho(r) in the definition (6) of T(o). Consequently,

state-vector, i.e., from the motion of an ob- the second bracket should have been used in-

served "free" particle.

stead of Hi(r) in Eq. (8).

If the system whose state vector is constantly

The definition of S(a) has therefore to be

" Q undergoes no real transitions with the passage
of time, then the state vector Q is called "steady. More precisely, Q is steady if, and only if, it satisfies the equation

altered by replacing H'(xo) by'
H'(xo) =H'(xo) +Hs(xo) =H'(xo) —Smc'P(x, )P(x,).

(22)

S(~)n=n.

The value of bm can be adjusted so as to cancel
(21) out the self-energy e&ects in S( ao ) (this is only a

As a general rule, one-particle states are steady formal adjustment since the value is actually

and many-particle states unsteady. There are, infinite), and then Eq. (21) will be valid for

however, two important qualifications to this one-electron states. For the photon self-energy no

rule.

such adjustment is needed since, as proved by

First, the interaction (20) itself will almost Schwinger, the photon self-energy turns out to be

always cause transitions from steady to unsteady identically zero.

states. For example, if the initial state consists

The foregoing discussion of the self-energy

of one electron in the field of a proton, Hp wilI problem is intentionally only a sketch, but it

have matrix elements for transitions of the elec- will be found to be sufficient for practical applica-

tron to a new state with emission of a photon, tions of the theory. A fuller discussion of the

and such transitions are important in practice. theoretical assumptions underlying this treat-

Therefore, although the interpretation of the ment of the problem will be given by Schwinger

theory is simpler for steady states, it is not pos- in his forthcoming papers. Moreover, it must be

sible to exclude unsteady states from con- realized that the theory as a whole cannot be

sideration.

put into a finally satisfactory form so long as

Second, if a one-particle state as hitherto de- divergencies occur in it, however skilfully these

fined is to be steady, the definition of S(0) must divergencies are circumvented; therefore, the
be modified. This is because S(~) includes the present treatment should be regarded as justified

effects of the electromagnetic self-energy of the by its success in applications rather than by its

electron, and this self-energy gives an expecta- theoretical derivation.

tion value to S(~) which is different from unity

The important results of the present paper up

(and indeed infinite) in a one-electron state, so to this point are Eq. (19) and the interpretation

that Eq. (21) cannot be satisfied. The mistake of the state vector Q. The state vector 4' of a

that has been made occurred in trying to repre- system can be interpreted as a wave function

sent the observed electron with its electromag- giving the probability amplitude of finding any

netic self-energy by a wave field with the same particular set of occupation numbers for the characteristic rest-mass as that of the "bare" various possible states of free electrons, positrons,

electron. To correct the mistake, let bm denote and photons. The state vector Q of a system with

the electromagnetic mass of the electron, i.e.,
the difference in rest-mass between an observed
and a "bare" electron. Instead of (5), the divi-

a given 4' on a given surface 0 is, crudely speaking, the 4' which the system would have had in
0 the infinite past if it had arrived at the given on

sion of the energy-density H(r) should have 0 under the influence of the interaction IP(xo)

(r))- taken the form
H(r) = (H, (r)+~mc' 0*(r)Pg
+ (Hi(r)

alone.

The definition of Q being unsymmetrical be-

fimcV*(r)PP(r)

).

—tween
vector

past and future, Q' can be defined

a
by

new type of state reversing the direc-

The first bracket on the right here represents the tion of time in the definition of Q. Thus the Q'
energy-density of the free electromagnetic and of a system with a given 4" on a given 0 is the +

electron fields with the observed electron rest-

~ Here Schwinger's notation P=P*P is used.

RA 0 I ATION TH EO R I Es

which the system would reach in the infinite
future if it continued to move under the inHuence of H'(xo) alone. More simply, Q can be defined
by the equation

Q'(o) =S(~)Q(o).

(23)

Since S( ~ ) is a unitary operator independent of 0, the state vectors 0 and 0' are really only the
same vector in two diferent representations or coordinate systems. Moreover, for any steady state the two are identical by (21).

V. FUNDAMENTAL FORMULAS OF THE SCHViGNGER AND FEYNMAN THEORIES
The Schwinger theory works directly from Eqs. (19) and (20), the aim being to calculate the matrix elements of the "effective external potential energy" H~ between states specified by their state vectors Q. The states considered in
practice always have 0 of some very simple kind, for example, 0 representing systems in which
one or two free-particle states have occupation number one and the remaining free-particle states have occupation number zero. By analogy with (13), S(oo) is given by

O'0

S(o o) = 1+(—i/hc)

p
H'(x~)dx~+

(

i/hc)'—

~

p ~'0

p &(&1)

dxg) H'(xg)H'(x2)dxg+, (24)

satisfactory agreement with experimental re-

sults. In this paper the development of the

Schwinger theory will be carried no further;

in principle the radiative corrections to the

equations of motion of electrons could be calcu-

lated to any desired order of approximation from

formula (25).

In the Feynman theory the basic principle is to preserve symmetry between past and future.

Therefore, the matrix elements of the operator
IIy are evaluated in a "mixed representation;"

the matrix elements are calculated between an

initial state specified by its state vector 0& and a final state specified by its state vector Q~'. The matrix element of Hp between two such
states in the Schwinger representation is

Qm*HpQg = Q2'*S( ~ )HpQg,

(26)

and therefore the operator which replaces IIp in
the mixed representation is

Hp(xp) =S(~)Hp(xp)
=S(~)(S(o)) 'H'(xo)S(o) (27)

Going back to the original product definition of
S(o) analogous to (10), it is clear that S(~)
X(S(o)) ' is simply the operator obtained from
S(a) by interchanging past and future. Thus,

R( ) =S( )(S( )) '=1+( i/hc)—

~ 00
X H'(x&)dx&+( i/hc)'

—dx)

and (S(oo)) ' by a corresponding expression
analogous to (14). Substitution of these series into (20) gives at once

~(*0)

o (*1)

Hp(xo) = Q (~/hc)")I dx() dx2

[, p&(~a-1)

X~

dx„X[HI(x„),

[IP(x,),

]j [H'(») H'(xo)3l. .

(25)

The repeated commutators in this formula are characteristic of the Schwinger theory, and their evaluation gives rise to long and rather dificult analysis. Using the first three terms of the series, Schwinger was able to calculate the second-order radiative corrections to the equations of motion of an electron in an external field, and obtained

X ~ H'(xm)H'(x&)dx2+

(28)

~ &(&x)

The physical meaning of a mixed representation of this type is not at all recondite. In fact, a mixed representation is normally used to describe such a process as bremsstrahlung of an electron in the field of a nucleus when the Born
approximation is not valid; the process of
bremsstrahlung is a radiative transition of the electron from a state described by a Coulomb wave function, with a plane ingoing and a spherical outgoing wave, to a state described by a Coulomb wave function with a spherical ingoing and a plane outgoing wave. The initial and final states here belong to different orthogonal systems
of wave functions, and so the transition matrix
elements are calculated in a mixed representa-
tion. In the Feynman theory the situation is

FJ

analogous, " only the roles of the radiation inter- The restricted integral can then be further

action and the external (or Coulomb) field are divided into (n+1) parts, the j'th part being the

interchanged; the radiation interaction is used integral over those sets of points with the prop-

instead of the Coulomb field to modify the state vectors (wave functions) of the initial and final states, and the external field instead of the

erty that o(xp) lies between o(x; i) and o(x;)

j=1 (with obvious modifications for

and j=n

+1). Therefore,

radiation interaction causes transitions between these state vectors.
In the Feynman theory there is an additional simplification. For if matrix elements are being

g ) ) I„=n!

" tIF(&0)
dx,

p~(&e-I)
dx„

00

Go

calculated betmeen tmo states, either of which is steady (and this includes all cases so far

X, dx, i
o (+o)

i dxiXH'(x, )
~ pr(x2)

considered), the mixed representation reduces to an ordinary representation. This occurs, for

H'(x, i)H'(xp)Hi(x;) IP(x„) (30).

example, in treating a one-particle problem such Now if the series (24) and (28) are substituted as the radiative correction to the equations of into (27), sums of integrals appear which are

motion of an electron in an external field; the precisely of the form (30). Hence finally operator HF(xp), although in general it is not

even Hermitian, can in this case be considered Hp(xp) = Q ( i/hc) "[1—/n!]I.

as an effective external potential energy acting

n=o

on the particle, in the ordinary sense of the words.
This section will be concluded with the deriva-

OQ

OP
=Q (

i/hc—)"[1/n!j I

dxi

..

n=o

OQ

tion of the fundamental formula (31) of the

XP(H'(xp) H'(xi)

IP(x )). (31)

Feynman theory, which is the analog of formula
(25) of the Schwinger theory. If
Fi(xi), , F„(x„)

By this formula the notation Hz(xp) is justified, for this operator now appears as a function of the point xo alone and not of the surface 0.. The

-, are any operators defined, respectively,

points xi,

x„of space-time, then

P(Fi(xi), , F„(x ))

at the (29)

further development of the Feynman theory is mainly concerned with the calculation of matrix elements of (31) between various initial and final states.

mill denote the product of these operators, taken

As a special case of (31) obtained by replacing

in the order, reading from right to left, in which H' by the unit matrix in (27),

the surfaces a(xi), , a(x ) occur in time. In
most applications of this notation F,(x;) will commute with F,(x;) so long as x; and x, are
outside each other's light cones; when this is the
case, it is easy to see that (29) is a function of the
points x~, *, x„only and is independent of the

g(pp) =Q ( i/hc)"—[1/n!$ t dxi

n~o

pc}

I dx
~00

P(Hi(xi) . . . HI(x )) (32

VI. CALCULATION OF MATRIX ELEMENTS

surfaces o(x;). Consider now the integral

In this section the application of the foregoing

theory to a general class of problems will be

dx„P(H'(xp),

explained. The ultimate aim is to obtain a set

H'(xi),

, H'(x, )).

of rules by which the matrix element of the operator (31) between two given states may be

Since the integrand is a symmetrical function of
the points x~, ., x„, the value of the integral is just n! times the integral obtained by re-

written down in a form suitable for numerical evaluation, immediately and automatically. The f'act that such a set of rules exists is the basis of

stricting the integration to sets of points xi,

the Feynman radiation theory; the derivation

x„ for which a.(x,) occurs after a(x;+i) for each i in this section of the same rules from what is

RA D I AT IO N. THEORIES

fundamentally the Tomonaga-Schwinger theory constitutes the proof of equivalence of the two theories.
To avoid excessive complication, the type of

integral of (31); let it be denoted by I' . From

(16), (22), (33), and (37) it is seen that P„ is a

sum of products of (n+1) operators P, (n+1)

P, operators

and not more than e operators A„,

matrix element considered will be restricted in
two ways. First, it will be assumed that the ex-
ternal potential energy is
II'(xo) = —L1/c7j„(xo)A„'(xa), (33)
that is to say, the interaction energy of the electron-positron field with electromagnetic potentials A„'(xo) which are given numerical functions

multiplied by various numerical factors. By Q„
P, may be denoted a typical product of factors
f, and A„, not summed over the indices such
as a and p, so that I' is a sum of terms such as
Q„. Then Q„will be of the form (indices omitted)

'. Q-=0( *o)4(

o)0(

*)4( )
XA

(xi

4(
i).

.

)4(x'.)
A(xi. „),

(39)

of space and time. Second, matrix elements will be considered only for transitions from a state A, in which just one electron and no positron or
8 photon is present, to another state of the
same character. These restrictions are not essential to the theory, and are introduced only for

„ where i o, i&, , i is some permutation of the integers 0, 1, - *, n, and j~, , j„are some, but
not necessarily all, of the integers 1, ~, n in some order. Since none of the operators P and
f commute with each other, it is especially im-
portant to preserve the order of these factors.

convenience, in order to illustrate clearly the

principles involved.

The electron-positron field operator may be

written

4 -(x) = Z 4-(x)~.,

(34)

Each factor of Q„ is a sum of creation and annihilation operators by virtue of (34), (35), and (36), and so Q„ itself is a sum of products of creation and annihilation operators.
Now consider under what conditions a product

where the p (x) are spinor wave functions of
free electrons and positrons, and the a„are

of creation and annihilation operators can give a
non-zero matrix element for the transition A~8.
Clearly, one of the annihilation operators must

annihilation operators of electrons and creation annihilate the electron in state A, one of the

operators of positrons. Similarly, the adjoint creation operators must create the electron in

operator

0-(x) = 2 4-(x)~.,

state 8, and the remaining operators must be
(35) divisible into pairs, the members of each pair

respectively creating and annihilating the same
where a„are annihilation operators of positrons particle. Creation and annihilation operators re-

and creation operators of electrons. The electro- ferring to different particles always commute or

magnetic 6eld operator is

anticommute (the former if at least one is a

A„(x) = Q (A,„(x)b„+A,„'(x)5,), (36) photon operator, the latter if both are electron-
positron operators). Therefore, if the two single

~here b„and 5, are photon annihilation and operators and the various pairs of operators in

creation operators, respectively. The charge- the product all refer to different particles, the

current 4-vector of the electron field is

order of factors in the product can be altered so

j„(x)=f«y(x) p„y(x);

(37)

as to bring together the two single operators and the two members of each pair, without changing

strictly speaking, this expression ought to be the value of the product except for a change of

antisymmetrized to the form'

sign if the permutation made in the order of the

i.(x) = 2~«I|f.(x)A(x) —A(x)k-(x) I (v.)-s. (38)

electron and positron operators is odd. In the case when some of the single operators and pairs

but it will be seen later that this is not necessary of operators refer to the same particle, it is not

in the present theory.

hard to verify that the same change in order of

Consider the product I' occurring in the n'th factors can be made, provided it is remembered

~ See Wolfgang
Eq. (96), p. 224.

Pauli, Rev. Mod. Phys. 13, 203 (1941},

that the division no longer unique,

of the operators into pairs is and the change of order is to

0 YSON

be made f'or each possible division into pairs and

the results added together.

It follows from the above considerations that the matrix element of Q for the transition A —+8
is a sum of contributions, each contribution

arising from a specific way of dividing the factors

of Q„ into two single factors and pairs. A typical

contribution of this kind mill be denoted by 3I.

The two factors of a pair must involve a creation

and an annihilation operator for the same par-
f ticle, and so must be either one P and one or

two A; the two single factors must be one g and
f. one The term M is thus specified by fixing an
integer k, and a permutation ro, r~, , r„of the

j„ integers 0, 1, , n, and a division (si,ti), (s2,t2),
, (si„ti,) of the integers ji, , into pairs;
clearly m=2k has to be an even number; the

term 3I is obtained by choosing for single factors

it(xq) factors
1'i+1,

and f(xri, ), and for associated
(P(x,),P(xr,)) for i =0, 1, , n and (A (xs;),A(xi;)) for i =

pairs of

,
1,

.

k
.

—1, ., h.

In evaluating the term 3f, the order of factors

in Q„ is first to be permuted so as to bring to-

gether the two single factors and the two mem-

bers of each pair, but without altering the order

of factors within each pair; the result of this

process is easily seen to be

Q'=~P(k(»)A'(x o)) P(4(x.) ~t (x"))
)&P(A (xsi),A (xii)) ' P(A (xsg),A (xtg)), (40)

a factor e being inserted which takes the value
f &1 according to whether the permutation of f and factors between (39) and (40) is even or
odd. Then in (40) each product of two associated factors (but not the two single factors) is to be independently replaced by the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle.
Given a bilinear operator such as A„(x)A„(y), the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle is just what is usually called the "vacuum expectation value" of the operator, and has been calculated by Schwinger. This quantity is, in fact (note that Heaviside .units are being used)
(A „(x)A.(y))p ——'1,'tc8„,I D &'&+iD I (x —y),
where D&'& and D are Schwinger's invariant D functions. The definitions of these functions will

not be given here, because it turns out that the vacuum expectation value of P(A„(x),A „(y))
takes an even simpler form. Namely,
(P(A„(x),A.(y))).=',ac~„.D,(x —y), (41)
where Dp is the type of D function introduced by Feynman. Di (x) is an even function of x, with
the integral expansion

Dr(x) = —[i/2''] ~" exp$inx']dn, (42)
~a

where x' denotes the square of the invariant length of the 4-vector x. In a similar way it follows from Schwinger's results that
(P(~t-(x),A(y)))o = 2n(»y) ~n-(x —y), (43)

where
Sps. (x) = —(v, (&/», )+~o)p.~~(x), (44)

Kp is the electron,

reciprocal it(x,y) is

—Co1moprto+n 1waacvceo-rledninggth

of the as &r(x)

is earlier or later than 0(y) in time, and Ap is a

function with the integral expansion

Substituting from (41) and (44) into (40), the matrix element M' takes the form (still omitting the indices of the factors P, P, and A of Q„)

g i&= e (-', it(x;,xr,)Sp(x; —xr;))

(,—. i& Ie
Xrr(-:~ D.
j

,»P(~("),~(".» (46)

The single factors iaaf(xi,.) and P(xri, ) are conveni-

ently left in the form of operators, since the

matrix elements the transition A

of these operators
—+8 depend on the

for effecting wave func-

tions of the electron in the states A and B.

Moreover, the order of the factors iit(xq) and

P(xri, ) is immaterial since they anticommute with each other; hence it is permissible to write

P(P(xi, ),P(xrg)) = s(xg, xri, )g (xg,)P(xri, )

Therefore (46) may be rewritten
g M= ' (-',5 (x;—x.,))g(-,'hcD (x., —x;))

Xg(xi,)P(x.i), (4"/)

with

e' = egit(x;, xr;).

(48)

i

THEORIES

Now the product in (48) is ( —1)&, where P is the

number of occasions in the expression (40) on

which the P of a I' bracket occurs to the left of

the P. Referring back to the definition of c after

E—q1.

(40), it follows that e' according to whether

takes the value the permutation

+1 or
of P

f and factors between (39) and the expression

is even or odd. But (39) can be derived by an even permutation from the expression

4(«)4(x0). 0(&.)4(~.)

(50)

and the permutation of factors between (49)

and (50) is even or odd according to whether the

permutation r p, ~ ~, r „of the integers 0 ~, n is

even
—1

or odd. according

Hence, finally, e' to whether the

in (47) is +1 or
permutation rp,

, r„ is even or odd. It is important that e'

', depends only on the type of matrix element M
considered, and not on the points xp, ' x„;

therefore, it can be taken outside the integrals

in (31).

One result of the foregoing analysis is to justify

the use of (37), instead of the more correct (38),
for the charge-current operator occurring in II'

and H'. For it has been shown that in each
f matrix element such as M the factors and P in

(38) can be freely permuted, so that (38) can be

replaced by (37), except in the case when the two

factors form an associated pair. In the excep-

tional case, M contains as a factor the vacuum

expectation value of the operator j„(x~) at some

point x;; this expectation value is zero according

to the correct formula (38), though it would be

infinite according to (37); thus the matrix ele-

ments in the exceptional case are always zero.

The conclusion is that only those matrix ele-

ments are to be calculated for which the integer
r; differs from i for every i&4, and in these

elements the use of formula (3?) is correct.

To write down the matrix elements of (31) for
the transition A~8, it is only necessary to take

all the products Q„, replace each by the sum of
the corresponding matrix elements 3f given by

(47), reassemble the terms into the form of the
P„ from which they were derived, and finally

substitute back into the series (31).The problem

of calculating the matrix elements of (31) is

thus in principle solved. However, in the follow-

ing section it will be shown how this solution-inprinciple can be reduced to a much simpler and
more practical procedure.

VII. GRAPHICAL REPRESENTATION OF MATRIX ELEMENTS

Let an integer n and a product I' occurring
in (31) be temporarily fixed. The points xo, xi,
, x„may be represented by (n+1) points
drawn on a piece of paper. A type of matrix element M as described in the last section will
then be represented graphically as follows. For
each associated pair of factors (P(x;),P(x.;)) with i&4, draw a line with a direction marked in it
from the point x; to the point x.;. For the single factors g(«), P(xri, ), draw directed lines leading
out from x~ to the edge of the diagram, and in
from the edge of the diagram to . x~A, For each
pair of factors (A (xs;),2 (xi~)), draw an un-
directed line joining the points xa; and x&;. The
complete set of points and lines will be called
the "graph" of M; clearly there is a one-to-one
correspondence between types of matrix ele-
ment and graphs, and the exclusion of matrix
elements with r; =i for i/k corresponds to the
exclusion of graphs with lines joining a point to
" itself. The directed lines in a graph will be called
"electron lines, the undirected lines "photon

lines.
Through each point of a graph pass two electron lines, and therefore the electron lines to-

gether form one open polygon containing the

vertices x~ and x.~, and possibly a number of

" closed polygons as well. The closed polygons will
be called "closed loops, and their number
denoted by /. Now the permutation rp, ~ ~, r„

of the integers 0, , n is clearly composed of

(1+1) separate cyclic permutations. A cyclic

permutation is even or odd according to whether

the number of elements in it is odd or even.

Hence the parity of the permutation r p, ~, r„

is the parity of the number of even-number

cycles contained in it. But the parity of the

number of odd-number cycles in it is obviously

the same as the parity of the total number (n+1)

of elements. The total number of cycles being

c(y1c+le1s),itshe(lp—arnit)y.

of the number of even-number Since it was seen earlier that

the e' of Eq. (47) is determined just by the parity
of the permutation rp, ., r„, the above argu-

F. J. DYSON

ment yieMs the simple formula
( 1)l-a
This formula is one result of the present theory which can be much more easily obtained by intuitive considerations of the sort used by Feynman.
In Feynman's theory the graph corresponding to a particular matrix element is regarded, not merely as an aid to calculation, but as a picture of the physical process which gives rise to that matrix element. For example, an electron line joining x& to x2 represents the possible creation
of an electron at x~ and its annihilation at x2, together with the possible creation of a positron at x2 and its annihilation at x~. This interpretation of a graph is obviously consistent with the
methods, and in Feynman's hands has been
used as the basis for the derivation of most of the results, of the present paper. For reasons of space, these ideas of Feynman will not be discussed in further detail here.
To the product I' correspond a finite number
of graphs, one of which may be denoted by t all possible G can be enumerated without dif-
6 ficulty for moderate values of m. To each
corresponds a contribution C(G) to the matrix element of (31) which is being evaluated.
It may happen that the graph G is discon-
nected, so that it can be divided into subgraphs, each of which is connected, with no line joining a point of one subgraph to a point of another. In such a case it is clear from (47) that C(G) is the product of factors derived from each subgraph separately. The subgraph G~ containing
" the point xo is called the "essential part" of G,
the remainder G2 the "inessential part. There are nom two cases to be considered, according to
whether the points x~ and x.~ lie in G2 or in G~
(they must clearly both lie in the same subgraph). In the first case, the factor C(G2) of C(G) can be seen by a comparison of (31) and (32) to be a contribution to the matrix element
of the operator S(~) for the transition A~8.
Now letting G vary over all possible graphs with the same G~ and different t"2, the sum of the contributions of all such G is a constant C(G&)
multiplied by the total matrix element of S(~)
for the transition A —+B. But for one-particle
states the operator S(~) is by (21) equivalent

to the identity operator and gives, accordingly,
a zero matrix element for the transition A —+B.
0 Consequently, the disconnected for which x~
and xrl, lie in G2 give *zero contribution to the
matrix element of (31), and can be omitted from
further consideration. When xl, and xrj, lie in G~, again the C(G) may be summed over all G consisting of the given G~ and all possible Gm, but
this time the connected graph G~ itself is to be included in the sum. The sum of all the C(G) in this case turns out to be just C(G&) multiplied by the expectation value in the vacuum of the
operator S(~). But the vacuum state, being a
steady state, satisfies (21), and so the expectation value in question is equal to unity. Therefore the sum of the C(G) reduces to the single
term C(G~), and again the disconnected graphs may be omitted from consideration.
The elimination of disconnected graphs is, from a physical point of view, somewhat trivial, since these graphs arise merely from the fact that meaningful physical processes proceed si-
multaneously with totally irrelevant Auctuations
of fields in the vacuum. However, similar arguments will now be used to eliminate a much
more important class of graphs, namely, those
involving self-energy eBects. A "self-energy part"
0 of a graph is defined as follows; it is a set of
one or more vertices not including x0, together
with the lines joining them, which is connected with the remainder of G (or with the edge of the diagram) only by two electron lines or by one or
two photon lines. For definiteness it may be supposed that G has a self-energy part Ji, which
is connected with its surroundings only by one
electron line entering Ii at x~, and another leaving I' at x2, the case of photon lines can be
treated in an entirely analogous way. The points
x~ and x2 may or may not be identical. From G a "reduced graph" Go can be obtained by omitting Ji completely and joining the incoming line
at xj. with the outgoing line at x2 to form a single
electron line in Go, the newly formed line being
). denoted by Given Go and X, there is conversely
a well determined set r of graphs G which are
associated with Go and X in this way; Go itself is
considered also to belong to F. It mill now be
shown that the sum C(F) of the contributions C(G) to the matrix element of (31) from all the graphs G of I' reduces to a single term C'(Go).

RA 0 IATION THEORY ES

Suppose, for example, that the line ) in Go
leads from a point x3 to the edge of the diagram.
Then C(Go) is an integral containing in the integrand the matrix element of

4-(xo)

(52)

for creation of an electron into the state B. Let

the momentum-energy 4-vector of the created

electron be p; the matrix element of (52) is of

the form

Y (xo) =a exp[ —i(P xo)/k]

(53)

with a independent of x3. Now consider the
sum C(I'). lt follows from an analysis of (31) that C(r) is obtained from C(Gp) by replacing
the operator (52) by

P — 00

p 00

( i/kc) "[1/»4!] ' dye

n0

00
dy.

X~+.(x ), II'(y.), ~, ~'(y.)) (54)

(This is, of course, a consequence of the special
character of the graphs of I'.) It is required to
calculate the matrix element of (54) for a transi-
tion from the vacuum state 0 to the state 8, i.e.,
for the emission of an electron into state B. This matrix element will be denoted by Z, ; C(I") in-
volves Z in the same way that C(Go) involves
(53). Now Z, can be evaluated as a sum of
terms of the same general character as (47); it will be of the form

2 Z- = ) &*'(y' —») I'p(y')dy*,

where the important fact is that E; is a function
only of the coordinate differences between y; and xo. By (53), this implies that

Z-= R-p(P) I'p(»),

(55)

with R independent of x3. From considerations of relativistic invariance, R must be of the form
4.R4(P')+(P.v.)p-R=(P')

where p' is the square of the invariant length of
the 4-vector p. But since the matrix element (53) is a solution of the Dirac equation,
P'= —k'«' (Pov.)p. I'p =ik« I'.,

and so (55) reduces to

Z =RgI' (xo),

with Rj, an absolute constant. Therefore the sum C(1') is in this case just C'(Gp), where C'(Go) is obtained from C(Gp)by the replacement

|P(xo)-+R g (x,).

(56)

In the case when the line ) leads into the graph
Go from the edge of the diagram to the point x~, it is clear that C(I') will be similarly obtained
from C(Gp) by the replacement

P(Xo) 4R4'P(Xo).

(57)

There remains the case in which X leads from

one vertex x3 to another x4 of Go. In this case

C(Go) contains in its integrand the function

—',s(x„x4)S»p. (x, —x4),

(58)

which is the vacuum expectation value of the

operator

&(k-(xo),A(x4))

(59)

according to (43). Now in analogy with (54), C(I') is obtained from C(Gp) by replacing (59) by

00
Q

( —i/kc)

"[1/n!]

Qo
dy,

0

00
, dy„

X&(4.( ), 4 (. ), ~'(y ), , ~'(y.)), (60)

and the vacuum expectation value of this opera-

tor will be denoted by

— —'1,7(xo,x4) S'» p, (xo x4).

(61)

By the methods of Section Vl, (61) can be ex-

panded as a series of terms of the same char-

acter as (47); this expansion will not be dis-

cussed in detail here, but it is easy to see that it

leads to an expression of the form (61), with

S»'(x) a certain universal function of the 4-

vector x. It will not be possible to reduce (61) to

a numerical multiple of (58), as Z was in the
previous case reduced to a multiple of F . In-

stead, there may be expected to be a series ex-

pansion of the form

] S»p,

(x)
+

=

(R)So~+paX-4(((xy),+['(4—bti/+~»oh,oo)(+—ao4(o')—pQ'~—o»',)~+po()xo .),

)

(62)

where ' is the Dalembertian operator and the

a, b are numerical coe%cients. In this case C(I')

will be equal to the C'(Go) obtained from C(Go)

by the replacement

S»(xo —x4)-+S»'(xo —x4).

498

F. J. DYSON

6 Applying the same methods to a graph with
a self-energy part connected to its surroundings
by two photon lines, the sum C(F) will be obtained as a single contribution C'(Go) from the

reduced graph Ga, C'(G4) being formed from C(Go) by the replacement

Dg(x4 x4) ~D—i'(x4 x4). — (64)

'I'he function DJ.' is defined by the condition that

i;hc7i„.Di."(x4 x4)—

(65)

is the vacuum expectation value of the operator

responding to that term will coatain the point x;
joined to the rest of the graph only by two electron lines, and this point by itself constitutes a self-energy part of the graph. Therefore, all terms involving Hs are to be omitted from (31) in the calculation of matrix elements. The in-
tuitive argument for omitting these terms is that they were only introduced in order to cancel
out higher order self-energy terms arising from H', which are also to be omitted; the analysis of the foregoing paragraphs is a more precise form of this argument. In physical language, the
argument can be stated still more simply; since bm is an unobservable quantity, it cannot appear in the final description of observable phenomena.

XE(A„(x4), A, (x4), H'(yi), , IP(y„)), (66)
and may be expanded in a series
Dp'(x) = (84+ci '+cm( ')'+ )Dp(x). (67)
Finally, it is not dif6cult to see that for graphs G with se/f-energy parts connected to their surroundings by a single photon line, the sum C(I')
will be identically zero, and so such graphs may be omitted from consideration entirely.
As a result of the foregoing arguments, the contributions C(G) of graphs with self-energy parts can always be replaced by modified contributions C'(G4) from a reduced graph GD. A given G may be reducible in more than one way to give various G0, but if the process of reduction
" is repeated a finite number of times a G4 will be
obtained which is "totally reduced, contains no
self-energy part, and is uniquely determined by G. The contribution C'(Go) of a totally reduced graph to the matrix element of (31) is now to be calculated as a sum of integrals of expressions like (47), but with a replacement (56), (57), (63), or (64) made corresponding to every line in G4. 'Ihis having been done, the matrix element of (31) is correctly calculated by taking into consideration each totally reduced graph once and once only.
The elimination of graphs with self-energy parts is a most important simplification of the theory. For according to (22), Hr contains the subtracted part H8, which will give rise to many additional terms in the expansion of (31). But if any such term is taken, say, containing the factor Hs(x4) in the integrand, every graph cor-

VIII. VACUUM POLARIZATION AND CHARGE RENORMALIZATIO N
The question now arises: What is the physical meaning of the new functions Bp' and Sp', and
of the constant Ri, ? In general terms, the answer is clear. The physical processes represented by the self-energy parts of graphs have been pushed out of the calculations, but these processes do not consist entirely of unobservable interactions
of single particles with their self-fields, and so
" cannot entirely be written ofF as "self-energy
processes. In addition, these processes include the phenomenon of vacuum polarization, i.e., the modification of the Geld surrounding a charged particle by the charges which the particle induces in the vacuum. Therefore, the appearance of Dp', Sp', and R~ in the calculations may be regarded as an explicit representation of the vacuum polarization phenomena which were implicitly contained in the processes now ignored.
In the present theory there are two kinds of vacuum polarization, one induced by the external field and the other by the quantized elec-
" tron and photon fields themselves; these will be
called "external" and "internal, respectively.
s)— It is only the internal polarization which is
represented yet in explicit fashion by the substitutions (56), (57), (63), (64); the external will be included later.
To form a concrete picture of the function D~', it may be observed that the function Dp(y
represents in classical electrodynamics the re-
tarded potential of a point charge at y acting upon a point charge at s, together with the re-

RADIATION TH EOR I ES

tarded potential of the charge at s acting on the

charge at y. Therefore, Dp may be spoken of
" loosely as "the electromagnetic interaction be-
tween two point charges. In this semiclassical

picture, DI' is then the electromagnetic inter-

action between two point charges, including the

efFects of the charge-distribution which each

charge induces in the vacuum.

The complete phenomenon of vacuum po-

larization, as hitherto understood, is included in

the above picture of the function Dp'. There is

nothing left for Sp' to represent. Thus, one of

the important conclusions of the present theory

is that there is a second phenomenon occurring

in nature, included in the term vacuum polariza-

tion as used in this paper, but additional to

vacuum polarization in the usual sense of the

word. The nature of the second phenomenon can

best be explained by an example.

The scattering of one electron by another

may be represented as caused by a potential

energy (the Manlier interaction) acting between
them. If one electron is at y and the other at s,
then, as explained above, the eRect of vacuum

polarization of the usual kind is to replace a
factor Dp in this potential energy by DI'. Now

consider an analogous, but unorthodox, repre-

sentation of the Compton efFect, or the scattering

of an electron by a photon. If the electron is at y and the photon at s, the scattering may be again

represented by a now the operator

Spio(teyn—tiasl)

energy, containing
as a factor; the po-

tential is an exchange potential, because after

the interaction the electron must be considered

to be at z and the photon at y, but this does not detract from its usefulness. By analogy with the

4-vector charge-current density j„which inter-

I acts with the potential DI, a spinor Compton-
eRect density may be defined by the equation

and an adjoint spinor by

These spinors are not directly observable quan-
tities, but the Compton eRect can be adequately
described as an exchange potential, of magnitude
proportional to Si (y —s), acting between the
Compton-eRect density at any point y and the adjoint density at s. The second vacuum polariza-

tion phenomenon is described by a change in the form of this potential from Sp to Sp'. Therefore, the phenomenon may be pictured in physical terms as the inducing, by a given element of Compton-eRect density at a given point, of additional Compton-efFect density in the vacuum around it.
In both sorts of internal vacuum polarization, the functions DI and Sp, in addition to being altered in shape, become multiplied by numerical (and actually divergent) factors Rs and Rm, also the matrix elements of (31) become multiplied by numerical factors such as R&R&*. However, it is believed (this has been verified only for secondorder terms) that all n'th-order matrix elements of (31) will involve these factors only in the form of a multiplier
(eRgR3&) ";
this statement includes the contributions from
the higher terms of the series (62) and (67). Here e is defined as the constant occurring in the fundamental interaction (16) by virtue of (37), Now the only possible experimental determination of e is by means of measurements of the eRects described by various matrix elements of (31), and so the directly measured quantity is not e but eR2R3&. Therefore, in practice the letter e is used to denote this measured quantity, and the multipliers R no longer appear explicitly
in the matrix elements of (31); the change in
" the meaning of the letter e is called "charge
renormalization, and is essential if e is to be identified with the observed electronic charge. As a result of the renormalization, the divergent coe%cients Ri, Rm, and R3 in (56), (57), (62), and (67) are to be replaced by unity, and the higher coefficients a, b, and c by expressions involving only the renormalized charge e.
The external vacuum polarization induced by
the potential A„' is, physically speaking, only a
special case of the first sort of internal polarization; it can be treated in a precisely similar manner. Graphs describing external polarization
" efFects are those with an "external polarization
part, namely, a part including the point xo and connected with the rest of the graph by only a single photon line. Such a graph is to be "reduced" by omitting the polarization part entirely and renaming with the label xo the

F. J. D YSON

point at the further end of the single photon line. A discussion similar to those of Section VII leads to the conclusion that only reduced graphs need
be considered in the calculation of the matrix
element of (31), and that the effect of external
polarization is explicitly represented if in the contributions from these graphs a replacement

A „'(x)~A„"(x)

(68)

is made, After a renormalization of the unit of potential, similar to the renormalization of
charge, the modified potential A„"takes the form
A "(x) = (1+ c, 'yc2( ')'+ )A„'(x), (69)

where the coefficients are the same as in (67).
It is necessary, in order to determine the
functions D~', Sp', and A „",to go back to for-
mulas (60) and (66). The determination of the vacuum expectation values of the operators (60) and (66) is a problem of the same kind as the original problem of the calculation of matrix
elements of (31), and the various terms in the operators (60) and (66) must again be split up,
represented by graphs, and analyzed in detail.
However, since DI' and Sp' are universal func-
tions, this further analysis has only to be carried out once to be applicable to all problems.
It is one of the major triumphs of the
Schwinger theory that it enables an unambiguous interpretation to be given to the phenomenon of vacuum polarization (at least of the first kind), and to the vacuum expectation value of an operator such as (66). In making this interpretation, profound theoretical problems arise, particularly concerned with the gauge invariance of the theory, about which nothing will be said here. For Schwinger's solution of these problems, the reader must refer to his forthcoming papers. Schwinger's argument can be transferred without essential change into the framework of the
present paper. Having overcome the difficulties of principle,
Schwinger proceeded to evaluate the function
Dp' explicitly as far as terms of order n= (e'/
4irAc) (heaviside units). In particular, he found
' for the coeflicient ci in (67) and (69) the value
( n/15s F02) to t—his order. It is hoped to publish
'Schwinger's results agree with those of the earlier, theoretically unsatisfactory treatment of vacuum polariza-
tion. The best account of the earlier work is V. F.Keisskopf,
Kgl. Danske Sels. Math. -Fys. Medd. 14, No. 6 (1936).

in a sequel to the present paper a similar evalua-
tion of the function S~, the analysis involved is
too complicated to be summarized here.
IX. SUMMARY OF RESULTS
In this section the results of the preceding pages will be summarized, so far as they relate to the performance of practical calculations. In effect, this summary will consist of a set of rules for the application of the Feynman radiation theory to a certain class of problems.
Suppose an electron to be moving in an external field with interaction energy given by (33). Then the interaction energy to be used in calculating the motion of the electron, including radiative corrections of all orders, is

XP(H'(xo), H*(xi), , H'(x )) (70)

with H' given by (16), and the P notation as

defined in (29).

To find the effective n'th-order radiative cor-

rection to the potential acting on the electron,

it is necessary to calculate the matrix elements of
J„ for transitions from one one-electron state

to another. These matrix elements can be

written down most conveniently in the form of

an operator E„bilin f, erain P and whose matrix

elements for one-electron transitions are the

same as those to be determined. In fact, the

E„ operator

itself is already the matrix element

to be determined if the P and P contained in it

are regarded as one-electron wave functions;
J„ To write down E„, the integrand I'„ in is

first expressed in terms of its factors f, P, and A,

all suffixes being indicated explicitly, and the
j„. expression (37) used for All possible graphs G

with (n+1) vertices are now drawn as described

in Section VII, omitting disconnected graphs,

graphs with self-energy parts, and graphs with

external vacuum polarization parts as defined in

Section VIII. It will be found that in each graph

there are at each vertex two electron lines and

one photon line, with the exception of xo at

which there are two electron lines only; further,

RA D I ATION TH EOR I ES

E„ such graphs can exist only for even e. is the

sum of a contribution X(G) from each G.
Given G, E(G) is obtained from J„by the

following transformations. First, for each photon

line joining x and y in G, replace two factors
A„(x)A.(y) in P„(regardle ssof their positions) by

', Acb„„-Dp'(x y), —

(71)

X2
FrG. i.
radiative corrections to the motion of an electron

with Do' given by (67) with Ro=1, the function

Do being defined by (42). Second, for each elec-

tron line joining x to y in G, replace two factors

it (x)Po(y) in P„(re gardle ssof positions) by

o S'~o.(x —y)

(72)

with So' given by (62) with Ro —1, the function

in an external field. Let the energy ternal field be
—L1/c]j„(xo)A „'(xp).

of the ex(74)

Then there will be one second-order correction
term
U= [a/157riipo][1/c]j„(xp) 'A„'(xp)

So being defined by (44) and (45). Third, replace
the remaining two factors P(g~(s), fo(w)) in P„
by P~(s)go(oc) in this order. Fourth, replace A„'(xp) by A„"(xp).given by
A„"(x)=A„'(x) —La/15pri~oo] 'A '(x) (73)

arising from the substitution (73) in the zero-

order term polarization

(74). This or Uehling

is

the well-known
term. '

vacuum

The remaining second-order term arises from

the second-order part Jo of (70). Written in ex-

or, more generally, by (69). Fifth, multiply the
whole by ( —1) ', where I is the number of closed
loops in G as defined in Section VII.
The above rules enable X to be written down very rapidly for small values of n. It should be
E„ observed that if is being calculated, and if it
is not desired to include effects of higher order
than the n th'th, en Dp', So', and A„" in (71),
(72), and (73) reduce to the simple functions
J„ DI, S~, and 2„'. Also, the integrand in is.a
symmetrical function of x~, , x„; therefore,
graphs which differ only by a relabeling of the
vertices x~, ~ ., x„give identical contributions to E„and need not be considered separately.
The extension of these rules to cover the calculation of matrix elements of (70) of a more
general character than the one-electron transi-
tions hitherto considered presents no essential
" difficulty. All that is necessary is to consider
graphs with more than two "loose ends, repre-
senting processes in which more than one particle
is involved. This extension is not treated in the
present paper, chieRy because it would lead to
unpleasantly cumbersome formulas.

panded form, J~ is
„ A=ie' dx, J dx,P(4.(xo)(y~).pgp(xo)~~'(xo), tt.(») (V.).oA(»)~. (»),
4.(xo) (~.) rA(xo)~. (»))
Next, all admissable graphs with the three
vertices xo, xy, x2 are to be drawn. It is easy to
see that there are only two such graphs, that G shown in Fig. 1, and the identical graph with x~ and xm interchanged. The full lines are electron lines, the dotted line a photon line. The contribution Z(G) is obtained from 2'o by substituting
according to the rules of Section IX; in this case I=0, and the primes can be omitted from (71),
(72), (73) since only second-order terms are required. The integrand in X(G) can be reassembled into the form of a matrix product,
suppressing the su%xes a, , I. Then, multi-
plying by a factor 2 to allow for the second graph, the complete second-order correction to (74)
arising from J~ becomes

— X. EXAMPLE SECOND-ORDER RAMATIVE
CORRECTIONS

)A„'(xo)— L = —iLe'/Shc]

f dxi

t'
dxoDF(xi

x.

~

As an illustration of the rules of procedure of the previous section, these rules will be used for writing down the terms giving second-order

XP(xi)y So(xo —xi)y„Sp(xo —xo)y f(xo).
9 Robert Serber, Phys. Rev. 48, 49 {i935);E. A. Uehling,
Phys. Rev. 4S, 55 {1935).

D. BOHM

" This is the term which gives rise to the main
part of the Lamb-Retherford line shift, the anomalous magnetic moment of the electron"
" and the anomalous hyper6ne splitting of the
ground state of hydrogen. The above expression L, is formally simpler
than the corresponding expression obtained by Schwinger, but the two are easily seen to be equivalent. In particular, the above expression does not lead to any great reduction in the labor
involved in a numerical calculation of the Lamb shift. Its advantage lies rather in the ease with which it can be written down.
In conclusion, the author mould like to express his thanks to the Commonwealth Fund of New York for financial support, and to Professors Schwinger and Feynman for the stimulating lectures in which they presented their respective theories.
Pates added in proof (To Section II). The argument of Section II is an over-simplification of the method of Tomonaga, ' and is unsound. There is an error in the deriva-
tion of (3); derivatives occurring in H(r} give rise to noncommutativity between H(r) and field quantities at r' when r is a point on a infinitesimally distant from r'. The
' W. E. Lamb and R. C. Retherford, Phys. Rev. 72,
"241 (1947). P. Kusch and H. M. Foley, Phys. Rev. 74, 250 (1948). '~ J. E. Nafe and E. B. Nelson, Phys. Rev. 73, 718 (1948};Aage Bohr, Phys. Rev. 73, 1109 (1948).

argument should be amended as follows. 4 is defined only

for flat surfaces t{r)=t, and for such surfaces (3) and (6)

are correct. 0 is defined for general surfaces by (12) and

(10), and is verified to satisfy (9). For a flat surface, C and
+ are then shown to be related by (7). Finally, since H&

does not involve the derivatives in H, the argument leading

to (3) can be correctly applied to prove that for general 0.
the state-vector +{0) will completely describe results of

observations of the system on 0.,

(To Section I I I), A covariant perturbation theory similar to that of Section III has previously been developed by
E. C G. Stueckelberg, Ann. d. Phys. 21, 367 (1934);

Nature, 153, 143 (1944).

Hz(TgioveSnecbtyion(25V),).buStchiws iHnzge'r='s

"e6ective potential" is not
QHz Q '. Here Q is a "square-

root" of 5( ~ ) obtained by expanding (5( ~ ))& by the

binomial theorem. The physical meaning of this is that
Schwinger specifies states neither by 0 nor by 0', but by an intermediate state-vector 0"=QO=Q '0', whose defi-

nition is symmetrical between past and future. Hp' is also

symmetrical between past and future. For one-particle

states, Hg and Hp' are identical.

Equation {32)can most simply be obtained directly from the product' expansion of S( ~ ).

(To Section VII). Equation (62) is incorrect. The function

Sg' is well-behaved, but its fourier transform has a loga-

rithmic dependence on frequency, which makes an expansion

precisely of the form (62) impossible.
(To Section X). The term I. still contains two divergent

parts. One is an "infra-red catastrophe" removable by

standard methods. The other is an "ultraviolet" diverg-

ence, and has to be interpreted as an additional charge-

renormalization, or, better, cancelled by part of the charge-
renormalizatioti calculated in Section VIII.

PHYSICAL REVIEW

VOLUM E 75, iN UM B ER 3

FEBRUARY 1, 1949

Note on a Theorem of Bloch Concerning Possible Causes of Superconductivity
D. BoHM
Physics Department, Princeton University, Princeton, Peto Jersey {Received September 13, 1948)

Attention is called to a theorem of Bloch, from which it is shown that even when interelectronic interactions are taken into account, the state of lowest electronic free energy corresponds to a zero net current. This result contradicts the hypothesis that superconductivity
is caused by spontaneous currents.

"ANY attempts" have been made to explain

~ ~ superconductivity in terms of spontaneous

currents, which arise because there is a special

group of for which

states of the free

the electron gas as a whole,
energy, Ji=E —'1S, is lower

'%'. ' M.

Heisenberg, Zeits. f. Naturforschg 32, 65 (1948). Born and K. C. Cheng, Nature 161, 1017 (1948).

when a finite current flows than when no current Bows at all. In some of the theories, it is suggested that the current-carrying states in question may have energies which are below that of the state of zero current, while in others, it is suggested that the current-carrying states may have so high a statistical weight that their free energy is