367
Letters to tlte Et!i!f11"
respectively. Then, an integral of the following form occurs in the perturbation calculation:
JIF (P(Il, P(2») ap,l) aP!)
(1)
where :F (P(I), PI!)) is a certain function of
and p('l). Owing to the momentum
conservation jaw, (1) can be rewritten as
Pll)
~ G (pi)) a"p:f)
All Improvement 011 the Integrations
appearing in Perturbation Theor,._
H. Umezawa and R. Kawabe.
I'fUIItif:lde oj 7'Iteoret,ical PI"ysics.
NafJOYO- University.
June 2~ 1949
The customary perturbation calculation is
not only ambiguous as to its relativistic
covariance, but often actually gives results
evidently destroying the relativistic covariance.
This can be Seen for instance, from the fact,
indicated by Pais') and others, that a result
with the transformation property of mass
cannot be obtained from perturbation calculations for the self-energy of a moving electron
due to an electromagnetic field. But, since
the system of quantum fiel<l theory is of a
relativistically invariant structpre, it must be
concludeu that the cause of the failure of
relativistic covariancy lay in the process of
the usual perturbation calculations. We intend
to propose a method to remedy tllis defect
of perturbation calculation.
We shall confine the following arguments
to calculati(m!"l in the momentum space. \Ve
consider the case when two particles 1 and
2 are created in the intermediate state, and
denote there momcntum-energy four-vectors
by ,P(1)=(ptIJ, Ep('») and J>:.>}~..,(p(t), E,.<'!»)
(2)
Hitherto, the method has been employed of
taking a sphere as the integration domain
for pil) in (2) and making its radius tend.
to infinity. This method destroys the relativistic covariance in the following two way~.
(i) A sphere is not a relativistically invariant
domain (ii) A momentum-conservatioll relation among particles which do not satisfy
the energy conservation law is not a relativi*tically invariant relation.
la} In order to remove the tlifliculty (ii),
an integration domain must he prescribed
for the integral (1).
(b) In order to rcmove the tlifliculty (I),
the sphere must be replaced by a domain
enclosed by a surface on which the nl'lfllentum-space scalar quantity 1V takes a constant
value. (c) Furthermore, we place the conclition that the above domain hecomes a
momentlim-space !lphere when referred to a
appropriate system of coordmates.
As the four dimensional scalar quantity tI',
we may take the four dimensional scalar
product of the momentum energy lOur-vectors
(P(I), PI!)~.
The actual method or calculation is as
follow; we perform the trans/ormation of
variables 1pO) 1_ w in the intcgral (2), rCwriting it a'! an integration for w. We may
next set the hounds of the integrAtion domain
at t('=a, b (constants) uncler thc condition (a).
The values of a and bare determincll hy
the condition (c). -In l-,'Cnclal, when therc
are 1l particles I, ... ," in the intcrmedi:ttc
state, thc following (,,-1) scalar tjuanllties
must be employed:
:~68
Lrttl'rs to tl", Editor
?I!(k,=,,(P(I),P(2»),
k=2, ... ,11
It can he ascertained, by u0tual calculation
in \"aTiou~ prt.>hlcms, that results haviug" the
corrcct rclati l"istic covariance c..1.n be ohtaiJlcll
hy applyillg thi!; method to perturhation
the. 'ry. Dclaibl rcport:ol (>[ \"J.lioIlS R~slllts
t Ims ~It.t;tlnt:d ·will he g-h'en c1scw here.
1) l'"is; Vcrh, Ac. Amsterdam. V"l; W. 1947.