The indicated limit of error in this latter value is the standard deviation derived from seven measurements.
— The fact that the difference 85Bi'" 82Pb'08 is 5 mMU
larger than unity indicates a sharp increase in the slope of the packing fraction curve. This agrees with the
expectation since Bi"' has one proton more than the

magic number 82. The addition of this single proton

adds, in this case, only 3 Mev to the binding energy of

the nucleus. This result is in reasonable agreement with

the diGerence disintegration

of 1.004 mass
data of Harvey.

'units

derived

from

the

~ J. A. Harvey, Phys. Rev. 81, 353 (1951).

PH VSI CAI REVIEW

FEB RUARY 15, T952

Divergence of Perturbation Theory in Quantum Electrodynamics
F. J. Dvsox
Laboratory of XNclear 5tmdk s, Corlell Urliwrs@y, Ithaca, Fnv cwork (Received November 5, 1951)

An argument is presented which leads tentatively to the conclusion that all the power-series expansions currently in use in quantum electrodynamics are divergent after the renormalization of mass and charge. The divergence in no way restricts the accuracy of practical calculations that can be made with the theory, but raises important questions of principle concerning the nature of the physical concepts upon which the
theory is built.

LL existing methods of handling problems in

quantum electrodynamics give results in the form

of power-series in e'. The individual coeKcients in these

series are finite after mass and charge renormalization.

The technique of renormalization can at present be

applied only to the separate coefFicients, and not to the

series as a whole. If the series converges, its sum is a

calculable physical quantity. But if the series diverges,

we have no method of calculating or even of defining

the quantity which is supposed to be represented by

the series.

Several authors have remarked' that the series after

renormalization will be divergent in a trivial way, if the

series represents a scattering amplitude of a free particle,

in circumstances where the particle has a possibility of

being captured into a permanently bound system. In

this situation a perturbation expansion of the scattering

' amplitude will diverge, even in nonrelativistic quantum
mechanics, and in the relativistic theory the series @till
diverge for the same reason. It is to be expected that

such trivial divergences will not impose any funda-

mental limitations on the use of the renormalization

method. In fact,
renormalization

a new method of carrying through
program has been developed,

'

the
a

method which is applicable to problems involving bound

systems and in which divergences of this elementary

nature cannot occur. In the new method the series

cxpRnslon Rx'lscs floIQ R foxIQRl 1Iltegl'Rtlon . of thc

equations of motion over a 6nite interval of time, and

in an elementary nonrelativistic theory such a perturba-

tion expansion would necessarily be convergent. For this

reason it was claimed as probable' that the power series

B.I'erretti, Nuovo cimento 8, 108 (1951);K. Nishijima, Prog.

Th''eFRor...JJP.ohDsytyssa,onnd6,,

37 (1951). A. Pais, Phys. Rev. 82, 840 {1951). Proc. Roy. Soc. (London) A207, 395 (1951).Phys.

Rev. 83, 608, 1207 (1951).
4 Phys. Rev. 83, 608 (1951},Section XII.

arising from the application of the new method in
quantum electrodynamics would always converge. If
the claim had been accompanied by a proof of convergence, then the theoretical framework of quantum elec-
trodynamics could have been considered closed, being
within its limits a complete and consistent theory. The purpose of this note is to present a simple argu-
ment which indicates that the power-series expansions
obtained by integrating the equations of motion in quantum electrodynamics will be divergent after renormalization. The divergence is of a basic character,
di6'erent from the trivial divergences mentioned above,
and is present equally in the results obtained from the new and the older methods of calculation. The argument
here presented is lacking in mathematical rigor and in
physical precision. It is intended only to be suggestive,
to serve as a basis for further, discussions. To me it scclTls convincing enough to Incx'lt publication ln its
present incomplete form", also I am glad to have this
opportunity to withdraw the erroneous argument previously put forward~ to support the claim that the power
series should converge.
The argument for divergence is as follows. According
o FcynIQan) quantum clcctI'odynRIQlcs ls cqulvRlcnt
to a theory of the motion of charges acting on each other by a direct action at a distance, the interaction between two like charges being given by the formula

e'8+(sg2'),

(&)

where e is the electron charge. The action-at-a-distance formulation is precisely equivalent to the usual formulation of the theory, in circumstances where all emitted radiation is ultimately absorbed. We shall suppose that

~ See reference 4. The error in the argument lay in using the concept "the number of times that an interaction operates" in an
intuitive and imprecise way.
6 R. P. Feynman, Phys. Rev. 76, 769 (1949), Eq. (4); Phys. Rev. 440 (1950), Appendix B.

F. J. D YSON

conditions are such as to justify the use of the Feynman formulation of the theory. Then let

F(e) = ao+a2e'+ a4e'+

(2)

be a physical quantity which is calculated as a formal

power series in e' by integrating the equations of motion

of the theory over a finite or an infinite time. Suppose,

if possible, that the series (2) converges for some posi-

tive value of e', this implies that F(e') is an analytic

foufnec,tiFo(n—oef')

e at
will

e=0. Then for sufficiently small values
also be a well-behaved analytic function

with a But
tation.

fcoornFve(r—geen't)
Namely,

power-series expansion.

Fw(e—caen')

also make a physical is the value that

interprewould be

obtained between

for like

Ic' hinargaefsictiistioLu—s ew'8o+r(lsd~2w2)h]eriensttehaed

interaction
of (I). In

the fictitious world, like charges attract each other. The

potential between static charges, in the classical limit of

large distances and large numbers of elementary charges,

will be just the classical Coulomb potential with the

sign reversed. But it is clear that in the fictitious world

the vacuum state as ordinarily defined is not the state of
E lowest energy. By creating a large number of electron-

positron pairs, bringing the electrons together in one

region of space and the positrons in another separate

region, it is easy to construct a "pathological" state in

which the negative potential energy of the Coulomb

forces is much greater than the total rest energy and

kinetic energy of the particles. This can be done without

using particularly small regions or high charge densities,

so that the validity of the classical Coulomb potential

is not in doubt. Suppose that in the fictitious world the

state of a system is known at a certain time to be an

ordinary physical state with only a few particles present.

There is a high potential barrier separating the physical

state from the pathological states of equal energy;

to overcome the barrier it is necessary to supply the

rest-energy for the creation of many particles. Never-

theless, because of the quantum-mechanical tunnel

eGect, there will always be a finite probability that in

any finite time-interval the system will find itself in a

pathological state. Thus every physical state is unstable

against the spontaneous creation of large numbers of

particles. Further, a system once in a pathological state

will not remain steady; there will be a rapid creation of

more and more particles, an explosive disintegration of

the vacuum by spontaneous polarization. In these

circumstances it is impossible that the integration of the

equations of motion of the theory over any finite or

infinite time interval, starting from a given state of the

fictitious world, shouM lead to well-defined analytic

functions. Therefore F( e') cannot be analy—tic and the

the series (2) cannot be convergent.

The divergence of the series in the real world is

associated with virtual processes in which large numbers

of particles are involved. Therefore the divergence

will only become noticeable when terms of very high

order in the expansion (2) are considered. A crude

quantitative estimate indicates that the terms of (2) will decrease to a minimum and then increase again
without limit, the index of the minimum term being roughly of the-order of magnitude 137. This estimate assumes the system to be such that the trivial kind of divergence discussed earlier does not occur. The non-
trivial and unavoidable divergence will not prevent practical calculations being made with the series (2), to an accuracy far beyond anything at present required or
contemplated. Only if similar arguments should be found to be applicable to meson theory, the divergence might impose a severe limitation on the possible accuracy of practical calculations in that field. ~
If the conclusion of the foregoing argument is accepted, then there are.two alternative possibilities for
the future development of quantum electrodynamics.
Alternative A: There may be discovered a new method of carrying through the renormalization program, not making use of power series expansions. In th~s case every physical quantity F(e') will be well-defined and calculable, and the series (2) will be an asymptotic expansion for it in the limit of small e. Since F(e') is not analytic
at e=0, the asymptotic expansion will not be sufhcient
to determine the function uniquely. The additional information necessary to determine F(e') will be obtained
from the existing formalism, using no new physical hypotheses but only some improved mathematical
methods. Alternative B: All the information that can
in principle be obtained from the formalism of quantum
electrodynamics is contained in the coefFicients ao, u2, u4,
of series such as (2). In this case the quantity F(e')
is neither physically well-defined nor mathematically calculable, except in so far as the asymptotic expansion
(2) gives some workable approximation to it. In order to define F(e') precisely, not merely new mathematical
methods but a new physical theory is needed.
I wish to call attention to the attractive features of
alternative B in the present state of physics. If B were true, it would imply that quantum electrodynamics is in its mathematical nature not a closed theory, but only a half-theory giving insufficient information for
the exact prediction of events. experimentally we know
that the world contains one group of phenomena which is accurately in agreement with the results of quantum
electrodynamics, and another group of phenomena
which is not understood at all. We need to develop new
physical ideas to understand the second group, and still we cannot abandon the theory which successfully
accounts for the first. If quantum electrodynamics were a closed theory, this would be a difficult dilemma. But
if the theory itself leaves room for new ideas, no such
dilemma arises. In conclusion, I wish to thank Pro-
fessors Pauli, Bethe, Pais, and Oppenheimer for valuable
discussions of these problems.
C. A. Hurst in a private communication informs me that he has discovered by direct calculation the fact that the S-matrix diverges in the way here described, in the case of a simple scalar meson theory, assuming that certain terms which are not yet calculated do not decisively change the behavior of the series.