S.
SOBOTTKA
this
experiment.
Terms
similar
to
those
neglected
in
Eq.
(A1)
have
also
been
neglected
here.
If
we
let
w„('A)A=w(E~,
Ev)dEv,
corresponding
to
an
electron
energy
E1
before
radiation
and
E2
after
radiation,
then
the
probability
that
an
electron
of
initial
energy
Ep
will
have
an
energy
E
after
radiating,
scattering,
and
again
radiating
will
be
~Ep
+Ep
v
(Ep,
E')
dE'
=
dE'
vc(EO,
E&)
Jg,
~~,
Xo
(E~,
Ev)w'(Ev,
E')dRdE,
,
(AS)
where
o
(E~,
Ev)
is
the
theoretical
scattering
cross
section
for
electrons
of
initial
energy
E1
and
final
energy
E2
and
m
and
m'
are
the
probabilities
for
radiation
before
and
after
scattering,
respectively.
For
an
elastic
cross
section,
o(E~,
Ev)
is
a
delta
function
and
the
integrals
of
Fq.
(AS)
can be
evaluated
approximately
to
yield
Eq.
(1)
of
the
text
if
E4
in
that
equation
is
replaced
by
Ep.
Again,
terms
of
the
same
order
as
those
neglected
in
E'q.
(A1)
were
neglected
in
Eq.
(1),
in
addition
to
terms
depending
on
(Eo
—
Es)
but
which
were
considerably
smaller
than
those
that
were
retained.
If
o(Ev,
Ev),
as
a
theoretical
inelastic
cross
section,
is
considered
to
be
a
series
of
many
delta
functions
(elastic
cross
sections),
Eq.
(2)
of
the
text
results,
where
the
summation
has
been
replaced
by
the
integral
sign
of
that
expression.
In
deducing
Eq.
(2),
it
was
assumed
that
the
shape
of
o(
E,
sE)
vas
a
function
of
Ev
with
fixed
E1
does
not
change
with
E1.
This
assumption
gives
adequate
accuracy
for
this
work.
PHYSICAL
REVIEW
VOLUME
118,
NUMB
ER
3
MA
Y
1,
1960
High-Energy
Behavior
in
Quantum
Field
Theory*
STEVEN
WEINBERGER
Department
of
Physics,
Columbia
University,
New
York,
New
York
(Received
May
21,
1959)
An
attack
is
made
on
the
problem
of
determining
the
asymptotic
behavior
at
high
energies
and
momenta
of
the
Green's
functions
of
quantum
field
theory,
using
new
mathematical
methods
from
the
theory
of
real
variables.
We
define
a
class
A„of
functions
of
n real
variables,
whose
asymptotic
behavior
may
be
specified
in
a
certain
manner
by
means
of
certain
"asymptotic
coeKcients.
"
The
Feynman
integrands
of
perturbation
theory
(with
energies
taken
imaginary)
belong
to
such
classes.
We
then prove
that
if
certain
conditions
on
the
asymptotic
coeScients
are
satisfied
then
an
integral
over
k
of
the
variables
converges,
and
belongs
to
the
class
A„z
with
new
asymptotic
coeScients
simply
related
to
the
old
ones.
When
applied
to
perturbation
theory
this
theorem
validates
the
renormalization
procedure
of
Dyson
and
Salam,
proving
that
the
renormal-
ized
integrals
actually
do
always
converge,
and
provides
a
simple
rule
for
calculating
the asymptotic
be-
havior
of
any
Green's
function
to
any
order
of
perturbation
theory.
I.
INTRODUCTION
'
'N
many
respects,
the
central
formal
problem
of
the
~
~
modern
quantum
theory
of
fields
is
the
determina-
tion
of
the
asymptotic
behavior
at
high
energies
and
momenta
of
the
Green's
functions
of
the
theory,
the
vacuum
expectation
values
of
time-ordered
products.

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- Energy, Radiation, asymptotic behavior, J Li